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NASA Contractor Report 4713 



Robust Stabilization of Uncertain Systems 
Based on Energy Dissipation Concepts 


Sandeep Gupta 


Contract NASI -19341 
Prepared for Langley Research Center 


March 1996 



NASA Contractor Report 4713 



Robust Stabilization of Uncertain Systems 
Based on Energy Dissipation Concepts 


Sandeep Gupta 

ViGYAN, Inc. • Hampton, Virginia 


National Aeronautics and Space Administration 
Langley Research Center • Hampton, Virginia 23681-0001 


Prepared for Langley Research Center 
under Contract NAS1-1 9341 


March 1996 



Printed copies available from the following: 


NASA Center for AeroSpace Information 
800 Elkridge Landing Road 
Linthicum Heights, MD 21090-2934 
(301) 621-0390 


National Technical Information Service (NTIS) 
5285 Port Royal Road 
Springfield, VA 22161-2171 
(703) 487-4650 



Abstract 


Robust stability conditions obtained through generalization of the notion of energy 
dissipation in physical systems are discussed in this report. Linear time-invariant 
(LTI) systems which dissipate energy corresponding to quadratic power functions are 
characterized in the time-domain and the frequency-domain, in terms of linear matrix 
inequalities (LMIs) and algebraic Riccati equations (AREs). A novel characterization 
of strictly dissipative LTI systems is introduced in this report. Sufficient conditions 
in terms of dissipativity and strict dissipativity are presented for (1) stability of 
the feedback interconnection of dissipative LTI systems, (2) stability of dissipative 
LTI systems with memoryless feedback nonlinearities, and (3) quadratic stability of 
uncertain linear systems. It is demonstrated that the framework of dissipative LTI 
systems investigated in this report unifies and extends small gain, passivity and sector 
conditions for stability. Techniques for selecting power functions for characterization 
of uncertain plants and robust controller synthesis based on these stability results are 
introduced. A spring-mass-damper example is used to illustrate the application of 
these methods for robust controller synthesis. 


Keywords: State space characterization of dissipative LTI systems, stability of inter- 
connected dissipative systems, Linear Matrix Inequalitiess (LMIs) for robust stability. 



Acknowledgements 


Several people provided assistance in development of the results presented in this 
work. Foremost of all is Dr. Suresh M. Joshi, who provided essential nurturance 
and guidance, apart from extensive technical discussions to clarify numerous con- 
cepts (and misconceptions). I have learned immensely about clear thinking and lucid 
expression by his example. Dr. Peiman G. Maghami taught me the nuts-and-bolts 
aspects of exploring new ideas, for example, the practice of verifying intuitive notions 
numerically (in Matlab), while attempting their analytical proofs. Dr. Ernest S. 
Armstrong sparked my interest in linear multivariable control theory while teaching 
a course by that name. Dr. Daniel P. Giesy, Sean P. Kenny and George A. Tan 
were very generous with their assistance, particularly in getting the computers to 
work right! Finally, I gratefully acknowledge the support provided by the Guidance 
and Control Branch at NASA Langley Research Center, Hampton, VA 23681, under 
Contract No. NASl-19341. 


IV 



Contents 


Abstract 

Acknowledgements 

1 Introduction 1 

2 Dissipative LTI Systems 6 

3 Time-Domain Characterization 11 

4 Algebraic Riccati Equation Characterization 20 

5 Frequency-Domain Properties 25 

6 Strictly Dissipative Systems 35 

7 Stability of Feedback Interconnection 41 

8 Stability with Feedback Nonlinearities 48 

9 Quadratic Stability 32 

10 Selection of Quadratic Power Functions 57 

11 Spring-Mass-Damper Example 62 

12 Summary ^ 

A Signals and Systems ^7 

References 


v 



List of Figures 

7.1 Negative Feedback Interconnection of Dissipative LTI Systems 41 

8.1 LTI System, E, with Negative Feedback Nonlinearities 48 

9.1 Uncertainty Configuration for Quadratic Stability. 52 

11.1 Three Spring-Mass-Damper System 62 

11.2 Singular Value Plot of the Spring-Mass-Damper System 72 

11.3 Bode Plot of the SISO Model 72 

11.4 Nyquist Plot of the SISO Model 73 

11.5 Nyquist Plot of the SISO Model, with Smallest Circle 73 


vi 



List of Tables 


11.1 Parameters of Three Spring-Mass-Damper System 63 

11.2 Natural Frequencies and Damping Ratios for Vibration Modes. ... 63 

11.3 State Space Realization for Spring-Mass-Damper Model 64 


vii 




Chapter 1 


Introduction 


Intuitively, a system which dissipates energy would eventually lose all of its initial 
energy, and would approach a configuration corresponding to zero energy if energy is 
not added to the system. Feedback interconnection of dissipative systems would be 
stable if the interconnection is such that individual subsystems dissipate their energy. 
Using mathematical abstractions of the notions of physical power and energy, the 
concept of energy dissipation has been employed to develop sufficient conditions for 
stability with dissipative systems. Numerous stability results in the literature such as 
small gain conditions, passivity conditions, and sector conditions for stability follow 
naturally as special cases from this framework of dissipative systems. 

Energy dissipation in a passive mechanical system is reviewed, first, as an intro- 
duction to the characterization of dissipative systems. Consider small oscillations 
of a passive mechanical system, such as a spring-mass-damper system[l], about its 
equilibrium configuration. Let q = q n } T be a vector of n generalized coor- 

dinates characterizing the kinematic configuration of this system. These generalized 
coordinates are selected such that q = 0 is an equilibrium configuration of the sys- 
tem. Small oscillations of the system about this equilibrium are being investigated. 
Let / = {/i,...,/ n } T be the corresponding vector of generalized forces. For small 
oscillations, kinetic energy of the system is T = \q T Mq , where M = M T > 0 is a 
symmetric, positive definite mass-inertia matrix of the system, and q corresponds to 
generalized velocities. Potential energy of the system is expressed as V = \q T Kq, 
where K = K T > 0 is a symmetric, positive definite, stiffness matrix of the sys- 


1 



tem. Energy dissipation occurs within the system due to forces proportional to the 
generalized velocities, which resist the motion of this system [1]. These forces are 
represented by Rayleigh’s dissipation function, 7 Z = q T Dq , where D — D T > 0 
is a symmetric, positive semidefinite matrix, characterizing damping in the system. 
Lagrangian equations of motion for natural dynamic systems, including Rayleigh’s 
dissipation function[l, 2], are given as 



where L is the Lagrangian, L = T — V. This leads to the equations of motion for small 
oscillations of the mechanical system about its equilibrium configuration as 

Mq + Dq + Kq = f 

Natural outputs of this system are generalized velocities, y = q, such that the dot 
product (or inner product) of input forces and natural outputs, y ■ f (equivalently, 
y T f ) gives the power input to the system. 


The total energy of this system, E, is the sum of its kinetic energy and its potential 
energy, E = T + V = | (q T Mq + q T Kqj . Note that the total energy is a quadratic, 
positive definite function of the state of the system. The rate of change of total energy 
of the system is 

^ | ( q T Mq + q T Mq + q T Kq + q T Kqj 

= q T {Mq + Kq) 

Substituting for (Mq + Kq) from the equations of motion of the system, and using 
y = results in the power balance equation, 

f = 

= y T f-<L T Dq 


This power balance equation mathematically states that the rate of change of total 
energy of the system is equal to the power input into the system minus the rate of 
energy dissipation in the system. Since the Rayleigh dissipation function is nonneg- 
ative for passive systems (7Z > 0), the power balance equation leads to the following 
inequality, which is known as the dissipation inequality, 



( 1 . 1 ) 


2 



This inequality states that for a passive mechanical system, rate of change of total 
energy is less than or equal to power input to the system. 

Integrating the power balance equation over an arbitrary interval of time [fo> ti] 
leads to the energy balance equation, 

f y T fdt — f 'E.dt + E(ti) — E(to) 

J to «/to 

The energy balance equation expresses the fact that the total energy input to the 
system over an arbitrary interval of time, given by the time integral of input power 
over that interval, is equal to the energy dissipated by the system in that time period 
plus the net change in energy of the system. The energy dissipated by a passive 
system over any time interval is nonnegative since its integrand, Rayleigh’s dissipation 
function, is nonnegative. Thus, the passive mechanical system satisfies the following 
integral form of the dissipation inequality, 

f y T fdt > E(ti) — E(t 0 ) (1-2) 

Jt 0 

over an arbitrary time interval and all admissible inputs /. Admissible inputs 

axe the inputs for which the equations of motion have a well-posed solution. This 
restriction is placed on the inputs for technical reasons, and is satisfied by most 
inputs encountered in physical systems. Note that the integral form of the dissipation 
inequality may be obtained directly by integrating Eq. 1.1) over an arbitrary time 
interval. It states that the total energy input to the system over an arbitrary time 
interval is greater than or equal to the net change in energy of the system, the 
difference being the energy dissipated by the system. 

The conditions imposed by energy dissipation in a passive mechanical system are 
given in differential form by Eq. (1), and in integral form by Eq. (2). In these inequal- 
ities, the inner product of the input forces and output velocities, y T f , represents the 
physical power input to the system, p ; and the total energy function, E, represents 
physical energy of the system, since it is the sum of the kinetic and potential energies. 
Consider a generalization of the expression of power to a quadratic function of the 
input and the output; and that of the energy function to an arbitrary, positive defi- 
nite, quadratic function of system states. Linear time-invariant (LTI) systems which 
satisfy the energy dissipation inequality with respect to these generalized power and 
energy functions are studied in this report. Note that the quadratic power functions 


3 



and quadratic energy functions may not have any physical interpretation correspond- 
ing to the notions of power and energy in mechanics. However, employing quadratic 
power functions and energy functions leads to the characterization of a large class of 
dissipative LTI systems, which includes many types of systems investigated in the lit- 
erature, such as gain bounded systems, passive systems, and sector bounded systems. 
Furthermore, parallels from the physical notions of power and energy are exploited 
to develop stability results for interconnection of dissipative systems. This results in 
a framework unifying and extending a number of stability results in the literature. 

A very general description of dissipative dynamic systems is presented in Refs. 
[3, 4, 5]. Consider a dynamic system in state space form, x = g(x, /, t), y = 
h(x , /, t), where x denotes the system state, / represents input to the system, y is the 
system output, and the nonlinear functions, g and h, describe the system dynamics. 
This system is said to be dissipative, according to Ref. [3], if there exists an absolutely 
integrable function of the input and the output, the power function, p(y, /), (referred 
to the supply rate in Ref. [3]), and a function of the system state, the energy 
function, E(x), (referred to as the storage function in Ref. [3]), such that 

/ P(y,f)dt > E(x(h)) - E(x(t 0 )) (1.3) 

J to 

for all admissible inputs, /(<), and arbitrary time intervals [f 0 ,<i], y(t) being the 
response of the dynamic system. The references provide concise results for character- 
ization and stability of general dissipative dynamic systems. 

This report restricts attention to linear time-invariant (LTI) systems, which are 
dissipative with respect to quadratic power functions. This allows the development 
of specific expressions characterizing dissipative LTI systems, and computational al- 
gorithms for determining dissipativity with respect to quadratic power functions[6]. 
Time-domain and frequency-domain conditions characterizing dissipative LTI sys- 
tems are developed in terms of linear matrix inequalities (LMIs) and algebraic Riccati 
equations (AREs). These state space characterizations of dissipative LTI systems are 
shown to be generalizations of the corresponding characterizations of gain bounded 
systems, sector bounded systems, as well as positive real systems. Novel concepts 
of input gain-matrix bounded LTI systems, and output gain-matrix bounded LTI 
systems are introduced as generalizations of gain bounded LTI systems. These sys- 
tems are also shown to be dissipative with respect to certain quadratic power func- 
tions. Strictly dissipative LTI systems are proposed as a further restricted class of 


4 



dissipative LTI systems, and these systems are characterized in time-domain and 
frequency-domain. Sufficient conditions are presented for (1) stability of the feedback 
interconnection of dissipative LTI systems, (2) stability of dissipative LTI systems 
with memoryless feedback nonlinearities, and (3) quadratic stability of uncertain lin- 
ear systems. Results for the three stability problems mentioned above obtained using 
small gain, passivity and sector criteria, are shown to be special cases of the results 
for dissipative LTI systems. New stability results for feedback interconnection of LTI 
systems, in terms of input/output gain-matrix bounded LTI systems, also follow as 
special cases of the stability results for dissipative LTI systems. Finally, numerical 
techniques for tight characterization of plant uncertainty employing convex program- 
ming techniques for linear matrix inequalities, are presented; and, an approach for 
robust dissipative controller synthesis is discussed. A numerical example of a spring- 
mass-damper system is used to demonstrate the application of the results developed 
in this report for robust controller synthesis. 


5 



Chapter 2 


Dissipative LTI Systems 


Consider a linear, time-invariant system, £, given by 


x(t) = Ax(t) + 

y(t) = Cx(t) + Df(t), (2.1) 


where y(t) is the p x 1 output vector, /(<) is an m x 1 input vector, z(i) is an n x 1 state 
vector and the system matrices ( A , B , C, D ) describe the dynamics of the system. The 
pxm transfer function matrix for this system is G(s) = C(sl — A)~* B + D. A general 
quadratic power function of the input and the output is expressed as 


p{y,f) 


f T 


Q N 
N T R 


y 

f 


( 2 . 2 ) 


where Q = Q T is a real symmetric p x p matrix, R = R T is a real symmetric m x m 
matrix, and N is a real pxm matrix. The LTI system, E, is defined to be dissipative 
with respect to a quadratic power function, p(y, /), as follows (Ref. [3]). 


Definition 2.1 A stable LTI system, £ : x = Ax + Bf , y = Cx + Df, where 
(A, B , C, D) is a minimal system realization, is dissipative with respect to the quadratic 
power function, 


p(yj) 


y T f T 


Q N 
N T R 



6 



if there exists a positive definite, quadratic energy function, E(x ) = x T Px, with P = 
P T > 0, which satisfies the dissipation inequality 


[ T P (y , f)di > E(x(T)) - E(x( 0)), (2.3) 

Jo 

for all T 6 [0, oo) and for all f € CSf e . 


The extended space of square integrable functions, is defined in the Appendix. 
Essentially this space characterizes a set of well-behaved input functions, such that 
well-posed solutions to the dynamic equations exist, when the input belongs to this 
space. This space includes almost all signals encountered in practical applications. 
The condition in Eq. (2.3) is the integral form of the dissipation inequality presented 
in the introduction. However, note that for the definition above, to = 0 and t\ = T 
can be used without loss of generality since linear time-invariant systems axe being 
considered. This inequality ensures that for a dissipative LTI system, the time integral 
of power input over any interval, that is, the total energy input to the system, is 
greater than or equal to the net change in total energy of the system. The difference 
between the total energy input to the system and the net change in energy of the 
system is the energy dissipated by the system. This is why systems which satisfy 
the inequality above are called dissipative systems. Note again that the quadratic 
power function, p(y,f ), and the quadratic energy function, E(x), may not have any 
physical interpretation, but are mathematical abstractions with properties similar to 
of those of physical power and energy. The dissipativity condition can be expressed 
in differential form as 

j^E(x) < p(y,f) (2.4) 

by differentiating the expression in Eq. (2.3) with respect to time. In differential form, 
the dissipation inequality states that the rate of change of the stored energy is less 
than or equal to the input power, the difference being the rate of energy dissipation. 


Observe that if the coefficient matrix for the quadratic power function in Eq. (2.2) 
is positive semidefinite, that is, if 


Q N 
N t R 


> 0 


it follows from linear regulator theory that the dissipation inequality in Eq. (2.3) is 
satisfied by all stable systems [7, 8]. Further, if the matrix is negative definite, the 



dissipation inequality is not satisfied by any nontrivial system. Thus, dissipative LTI 
systems are characterized by quadratic power functions whose coefficient matrix is 
indefinite, that is, it is neither positive semidefinite, nor negative semidefinite. The 
set of LTI systems which satisfy the dissipation inequality when the coefficient matrix 
is indefinite is a restricted subset of stable LTI systems which is said to be dissipative 
with respect to the quadratic power function. The analysis of dissipative LTI sys- 
tems with quadratic power functions may be considered as an extension of the linear 
regulator theory[7, 8]. Conditions characterizing dissipativity of LTI systems, with 
respect to quadratic power functions, appear in terms of linear matrix inequalities 
and algebraic Riccati equations, parallel to those of linear regulator theory. 

Many systems considered in the literature can be treated as special cases of dissi- 
pative LTI systems as defined above. In particular, gain bounded systems, positive 
real systems, and sector bounded systems are dissipative with respect to specified 
quadratic power functions, as shown below. 

Linear time-invariant systems whose 77oo norm bounded by unity, (also referred to 
as bounded real systems in the literature [9, 10]) satisfy 

J o y T (t)y{t)dt < f T (t)f(t)dt 

for all T € [0, oo) and / € C,™ t - Rewriting this condition as 

[ T {f T (t)m - y T (t)y(t)}dt > 0 (2.5) 

JO 

it is seen that bounded real systems are dissipative LTI systems with a quadratic 
power function p(y,f ) = f T (t)f(t) - y T (t)y(t), that is, the general quadratic power 
function in Eq. (2.2), with R = 7, Q = -7, and N = 0. 

Similarly, general norm bounded systems with || G(s) < 7 satisfy 

/ y T (t)y(i)dt < 7 2 / f T (t)f(t)di 

Jo JO 

for all T G [0, co) and / € C™ e . Thus, these systems are dissipative with respect to 
?(y?/) = If 2 f T (t)f (t) — y T that is, the general quadratic power function in Eq. 
(2.2) with R = 7 2 7, Q = -7, and N = 0. 

Passive systems are characterized by the input-output property [9, 11] 

F y T {t)f(t)dt> 0 ( 2 . 6 ) 


8 



for all T E [0,oo) and / 6 C™ e . This condition corresponds to dissipativity with 
respect to a quadratic power function = y T (t)f(t) + f T (t)y(t) or the general 

quadratic power function in Eq. (2.2) with R — 0, Q = 0, and N = I. In fact, 
as noted in the first section, dissipative systems are obtained as a generalization of 
passive systems. 

Sector bounded systems also are special cases of dissipative LTI systems. For 
example, consider an LTI system inside sector [ a,b ] with b > 0 > a. By definition[12, 
13], the input and the output of this system satisfy 

/ {y(0 - a/(0) T {y (0 “ W)} dt < 0 

Jo 

for all T € [0, oo) and / 6 C™ e . Rewriting this condition as 

^— a bf T (t)f(t) J r(a-\-b)y T (t)f(t)-y T (t)y(t)^dt >0 (2.7) 

shows that the sector bounded LTI system is dissipative with respect to the quadratic 
power function p(y,/) = {— abf T (t)f(t) + (a + b)y T (t)f(t) — t/ r (t)j/(f) j or the gen- 
eral quadratic power function in Eq. (2.2) with R = —abl, Q = —I, and N — al, 
where a = (a + b)j 2. 

These three examples show the generality of the class of LTI systems that are 
dissipative with respect to quadratic power functions. These classes of systems are 
obtained as special cases of dissipative LTI systems, simply by substituting specific 
values for the matrices Q , N and R. Moreover, note the special cases are obtained by 
substituting scalar matrices in the coefficient matrix for quadratic power functions. 
On the other hand, the following extensions of the notion of gain bounded LTI systems 
provide examples of cases where Q and R are full matrices. 

An LTI systems, E, with transfer function, G(s) = C(sl — A)~ 1 B + D and fioo 
gain bounded by 7, that is ||G(s)|| 00 < 7, satisfies ||y|| 2 < II7/II2, or equivalently, 
So y T ydt < Jq 7 2 f T fdt. As an extension to characterization of systems by a positive 
scalar, its norm, consider MIMO systems being characterized by symmetric, 
positive definite matrices rather than positive scalar gains. A stable LTI system is 
defined to be input gain-matrix bounded with respect to a symmetric, positive definite 
matrix, T t - = Tf > 0, if ||j/|| 2 < ||ri/|| 2 , or equivalently, if / 0 T y T ydt < / 0 T f T T?fdt , 
for all / € C™ e . Input gain-matrix bounded systems are easily seen to be dissipative 
with respect to the quadratic power functions with Q = —I,R = T^, and JV = 0. 


9 



To see input gain-matrix bounded systems as an extension of gain bounded systems, 
note that an LTI system whose gain is bounded by 7 is input gain-matrix bounded 
with respect to the scalar matrix, T, = 7 1. General positive definite values for T, can 
be employed for a tighter characterization of LTI systems, as discussed later in this 
report. 

Similarly, noting that a gain bounded LTI system also satisfies satisfies ||^y||2 < 
H/U2, a stable LTI system is defined as output gain-matrix bounded with respect to a 
symmetric positive definite matrix, r o = Tj > 0, if ||r~ 1 j/||2 < II/II2, or equivalently, 
if Jo y T ^o 2 v dt < jJ f T fdt for all / € C™ e . Output gain-matrix bounded LTI systems 
are seen to be dissipative with respect to quadratic power functions with R = I ,Q = 
— r~ 2 , and N = 0 . Note that an LTI system whose gain is bounded by 7 is output 
gain-matrix bounded with respect to the scalar gain matrix T 0 = 7 1. 

Combining these two ideas, an LTI system is input-output gain-matrices bounded 
with respect to symmetric positive definite matrices Ti = Tf > 0 and T 0 = Tj > 0, 
if lir^Sflk < ||r,-/|| 2 , or equivalently, / 0 r y T T~ 2 ydt < /J f T T 2 fdt for all / € ££• 
This system is seen to be dissipative with respect to a quadratic power function with 
R = r?, Q — Tj 2 , and N = 0. It should be noted that the gain matrices T, and 
T 0 are not independent of each other in the sense that if a system is input-output 
gain-matrices bounded with respect to F, and r o , then it is also bounded with respect 
to ar,- and ^r o , where a is any positive scalar. Again, to see that these systems form 
a generalization of gain bounded systems, note that a system whose gain is bounded 
by 7 is input-output gain-matrices bounded with respect to scalar matrices T, = 7 : / 
and r 0 = 7 0 I with 7 = 7 ,7 0 . 

Frequency-domain characterization of gain-matrix bounded LTI systems in a later 
section will further clarify the notion that constraints imposed by the definitions 
above are generalizations of gain bounded systems. Moreover, stability results for 
gain-matrix bounded systems will be obtained from the general stability result for 
dissipative LTI systems in later sections. 


10 



Chapter 3 


Time-Domain Characterization 


Time-domain characterizations of dissipative LTI systems axe developed in this sec- 
tion. The first characterization is presented in terms of a constrained solution of a 
system of three matrix equations. This characterization is referred to as the dissi- 
pativity lemma, since it is a generalization of the Kalman- Yakubovich Lemma (or 
positive realness lemma) for positive real systems, and the bounded realness lemma 
for gain bounded systems[9, 10]. The conditions of the dissipativity lemma can be 
equivalently expressed in terms of a linear matrix inequality (LMI). The LMI charac- 
terization of dissipative LTI systems is very important in application of these results 
for tight characterization of uncertain plants in terms of dissipativity, as described 
in later sections. LMI characterizations of gain bounded, positive real, and sector 
bounded systems [6] follow directly from the LMI characterization of dissipative LTI 
systems by substituting their respective power functions. 

State space characterization of LTI systems which are dissipative with respect to 
quadratic power functions is presented in the following Theorem. 


Theorem 3.1 Consider a stable LTI system, E : x = Ax-\- Bf, y = Cx-\-Df, where 
( A,B,C,D ) is a minimal realization of the system. The following statements are 
equivalent. 


11 



1. The LT I system, £, is dissipative with respect to a quadratic power function, 


p(yJ ) 



Q N 
N T R 


y 

f 


2. (Dissipativity Lemma) There exists a symmetric, positive definite matrix, P = 
P T > 0, and matrices L and W which satisfy the following system of three 
matrix equations, 

PA + A t P = C t QC - L t L 

PB = C t (QD + N)-L t W (3.1) 

R + N t D A D t N A D t QD = W T W 


3. (LMI Characterization) There exists a symmetric, positive definite matrix, P = 
P T > 0, which satisfies the following linear matrix inequality 


C T QC -PA- A t P C t (QD A N) - PB 

(i QD + N) T C - B T P R + N T D + D T N + D T QD J 


> 0 


(3.2) 


Proof: (1) ■$= (2) : Assuming that there exists a symmetric, positive definite matrix, 
P = P T > 0, which satisfies the conditions of the dissipativity lemma, Eq. (3.1), 
it is shown that E(x) = x T Px is a quadratic energy function which satisfies the 
dissipativity inequality in differential form, Eq. (2.4). Consider 

= i T Px + x T Pi 

= x t (A t P + PA)x + f T B t Px + x T PBf 
Using the first two relations in Eq. (3.1) gives 

4-E(x) = -x t L t Lx + x t C t QCx + f{QD + NfCx + 
at 

x t C t (QD + N)f - f T W T Lx - x T L T Wf 
Adding and subtracting f T W T W f for “completing the square” leads to 

j t E(x ) = -x T L T Lx-x T L T Wf-f T W T Lx-f T W T Wf 

+x T C T QCx A x T C T QDf + f T D T QCx 

x T C T N f A f T N T Cx + f T W T Wf 


12 



Using the last relation in Eq. (3.1) to substitute for f T W T W f gives 

^-E(x) = —x t L t (Lx + W f) — f T W T (Lx + Wf) 

+x t C t Q{Cx + Df ) + f T D T Q(Cx + Df ) 

(x T C T + f T D T )Nf + f T N T (Cx + Df) + fRf 

Thus, using y = Cx + Df, it follows that 

j t E(x) =-(Lx + W}) t (Lx + Wf) + p(y,f) (3.3) 

Since (Lx -f W f) T (Lx + W f) > 0, for any input, / <E the differential form of the 
dissipativity inequality, -^E(x) < p(y,f ), follows. Integrating this inequality over an 
arbitrary time interval, [0,T], leads to the integral for of the dissipation inequality in 
Eq. (2.3). 


(2) <= (3) : Let M be a Cholesky factor of the matrix in the linear matrix inequality 
condition, Eq. (3.2), that is, 

C T QC — PA - A T P C T (QD + N) — PB } =m t m>0 
(QD + Nfc - B T P R + N r D + D T N + D T QD J 


Partitioning M as M = [ L W J conformally to the partitions on left-hand side, 
leads to 


' C T QC -PA- A T P 

C T (QD + N) - PB 


' L T L 

L T W ‘ 

(QD + N) T C - B T P 

R + N t D + D T N + D T QD 


W T L 

W T W 


(3.4) 


Conditions of the dissipativity lemma in Eq. (3.1) follow by equating the submatrices 
in Eq. (3.4). 


(2) => (3) : This follows by reversing the steps above. Forming the matrix in the LMI 
condition, Eq. (3.2), using the conditions of the dissipativity lemma, Eq. (3.1), leads 
to 


' C T QC -PA- A T P 

C T (QD + N) - PB 


' L T L 

l t w ' 

(QD + N) T C - B T P 

R + N T D + D T N + D T QD 


W t L 

W T W 


The right-hand side of Eq. (3.5) may be written as M T M, with M = [ L W ]. Thus, 
the right-hand side of Eq. (3.5) is nonnegative definite, which is the LMI condition 
of (3). 


13 



(1) => (3) : The dissipation inequality in differential form, Eq. (2.4), ensures that 
there exists a quadratic energy function E(x ) = x T Px , with P = P T > 0, which 
satisfies p(y,f ) — -^E(x) > 0. Algebraic manipulations of this conditions, shown 
below, lead to the LMI condition in (3). 


First, expand the quadratic power function in terms of the state and the input as 
follows. 

P(yj) = y T Qy + y T Nf + f T N T y + f T Rf 

= (Cx + Df) T Q(Cx + Df) + ( Cx + DffNf + f T N T (Cx + Df ) + f T Rf 
= x t C t QCx 4- x T C T (QD + N)f + f(QD + N) T Cx 
+ f T (D T QD + D t N + N t D + R)f 


P 


C T QC C T (QD + N ) 

J l (QD + N) T C R + N T D + D T N + D T QD J [ / J 


(3.6) 


Further, express the derivative of the energy function, j- t E{x), as a quadratic in terms 
of the state and the output as follows. 


—E(x) = x T Px + x T Px 
at 


= x 1 (A 1 P + PA)x + f T B T Px + x T PBf 


P ] 


pa + a t p pb 

B T P 0 



x 




(3.7) 


Substituting from Eq. (3.7) and Eq. (3.6) into the differential form of the dissipation 
inequality gives 


y p 


C T QC -PA- A r P C T (QD + N) - PB 
[ ( QD + N) T C - B T P R + N t D + D T N + D T QD J [ / J 


> 0 


Since the dissipation inequality must be valid for all / 6 £^e, and controllability of 
the system realization implies that the inequality must be satisfied for all x, it follows 
that 

' C T QC -PA- A T P C t (QD + N) — PB 
(QD + N) T C - B T P R + N t D + D T N + D T QD 

This is the LMI characterization of dissipative LTI systems in (3). □ 


> 0 


Theorem 3.1 presents constraints on the system matrices of a minimal realization of 
an LTI system for the system to be dissipative with respect to a given quadratic power 


14 



function. The generality of the results in Theorem 3.1 for dissipative LTI systems 
is emphasized by noting that the corresponding results for gain bounded systems, 
positive real systems, and sector bounded systems, follow simply by substituting 
their respective power functions in the general results. 

The bounded realness lemma for characterizing unity gain bounded systems, or 
bounded real systems[9, 10] and the LMI form of this condition[14, 15] follow by 
substituting the power function for bounded real systems, that is, setting R = /, Q = 

—I, and N = 0, in the results of Theorem 3.1. 

Corollary 3.1 Consider a stable LTI system, S, with a minimal realization, (A, B , C , D). 
The following statements are equivalent. 


1. The LTI system, S, is bounded real. 

2. ( Bounded Realness Lemma) There exists a symmetric, positive definite matrix, 
P = P T > 0, and matrices L and W satisfying 

PA + A t P = -C T C - L t L 

PB = -C t D - L t W (3.8) 

I - d t d = W T W 


3. There exists a symmetric, positive definite matrix, P = P T > 0, which satisfies 
the following linear matrix inequality 


' PA + A T P + C T C C T D + PB 
D t C + B T P D t D - I 


< 0 


(3.9) 


The Kalman- Yakubovich lemma, also known as the positive realness lemma, for 
characterizing positive real LTI systems[10, 16], and the LMI characterization of pos- 
itive real systems[17] follow directly from the results of Theorem 3.1 by substituting 
the power function for positive systems, that is, by setting N = /, Q = 0 and R = 0. 


Corollary 3.2 Consider a stable LTI system, S, with a minimal realization ( A , B , C , D). 
The following statements are equivalent. 


1. The LTI system, S, is positive real. 


15 



2. (Positive Realness Lemma) There exists a symmetric, positive definite matrix, 
P = P T > 0, and matrices L and W satisfying 


PA + A t P = -L t L 

PB = C T — L t W (3.10) 

D + D T = W T W 


3. There exists a symmetric, positive definite matrix, P = P T > 0, which satisfies 
the following linear matrix inequality 


PA + A t P PB - C T 
B T P — C ~(D + D T ) 


< 0 


(3.11) 


State space characterization of LTI systems inside sector [a, b] is presented in terms 
of sector boundedness lemma and the corresponding linear matrix inequality in Ref 
[6, 18]. These characterizations follow directly from the results of Theorem 3.1 by 
setting R — —abl, N = al and Q = —I where a = (a + b)j 2. 


Corollary 3.3 Consider a stable LTI system, Y^,with a minimal realization (A, B , C , D). 
The following statements are equivalent. 


1. The LTI system, E, is inside sector [a, 6]. 

2. (Sector Boundedness Lemma.) There exists a symmetric, positive definite ma- 
trix, P = P T > 0, and matrices L and W satisfying 

PA + A t P = -C T C - L r L 

PB = C T (aI -D)- L T W (3.12) 

—abl + a(D + D T ) - D T D = W T W 

where a = (a + b)j 2. 

3. There exists a symmetric, positive definite matrix, P = P T > 0, which satisfies 
the following linear matrix inequality 


PA + A t P + C T C PB - C T (al - D ) 
B t P - (a/ - D) T C abl - a(D + D T ) + D T D 


< 0 


(3.13) 


16 



State space characterization of other sector bounded LTI systems in terms of LMIs 
can be obtained in a similar fashion by substituting respective power functions. 

Next three corollaries present the LMI characterization of gain-matrix bounded 
LTI systems. Again, these results follow simply by substituting their respective power 
functions into the result of Theorem 3.1. For input gain-matrix bounded systems, 
the following result is established by setting Q = —I,N = 0, and R = T 1 2 . 


Corollary 3.4 Consider a stable LTI system, E, with a minimal realization, ( A , B, C , D) 
The following statements are equivalent. 


1. The LTI system, E, is an input gain-matrix bounded system, with respect to a 
symmetric, positive definite matrix, T,-. 

2. There exists a symmetric, positive definite matrix, P = P T > 0, and matrices 
L and W satisfying 

PA + A t P = -C T C - L t L 

PB = —C t D — L t W (3.14) 

r? - d t d = w T w 


3. There exists a symmetric, positive definite matrix, P = P T > 0, which satisfies 
the following linear matrix inequality 


PA + A T P + C t C C t D + PB 
D T C + B T P D T D - T? 


< 0 


(3.15) 


A similar result follows for output gain-matrix bounded LTI systems with respect 
to a symmetric positive definite matrix, r o , by setting Q = —T~ 2 ,N = 0, and R = I. 


Corollary 3.5 Consider a stable LTI system, E, with a minimal realization, ( A , B , C , D ) 
The following statements are equivalent. 


1. The LTI system, E, is output gain-matrix bounded with respect to a symmetric, 

positive definite matrix, r o . 


17 



2. (Bounded Realness Lemma) There exists a symmetric , positive definite matrix, 
P = P T > 0, and matrices L and W satisfying 

pa + a t p = -c T v~ 2 c - L t L 

PB = -C t T; 2 D - L t W (3.16) 

/ - D t T~ 2 D = W T W 


3. There exists a symmetric, positive definite matrix, P = P T > Q, which satisfies 
the following linear matrix inequality 


PA + A T P + C T V~ 2 C 
D T I~ 2 C + B T P 


C t T~ 2 D + PB 
D t T~ 2 D - 1 


< o 


(3.17) 


Finally, the result for input-output gain-matrices bounded LTI systems with re- 
spect to a symmetric positive definite matrices, T,-, and r o , follows by setting Q - 
-T~ 2 ,N = 0, and R = T 2 . 

Corollary 3.6 Consider a stable LTI system, E, with a minimal realization, (A, B, C, D ). 
The following statements are equivalent. 

1. The LTI system, E, is input-output gain-matrices bounded with respect to a 
symmetric, positive definite matrices, I\, and r o . 

2. There exists a symmetric, positive definite matrix, P = P T > 0, and matrices 
L and W satisfying 


pa + a t p = -c T r; 2 c - l t l 

PB = -C T r; 2 D - L t W (3.18) 

r? - d t t; 2 d = w T w 


3. There exists a symmetric, positive definite matrix, P = P T > 0, which satisfies 
the following linear matrix inequality 


' PA + A t P + C T T~ 2 C C t Yz 2 D + PB 

d t t~ 2 c + b t p d t t~ 2 d - r? 


< 0 


18 


(3.19) 



In concluding this section, it is noted that the LMI characterization of dissipative 
LTI systems in Theorem 3.1 is a very useful result in applications. Dissipativity of 
an LTI system with respect to a given power function may be posed as a feasibility 
problem with the LMI in Eq. (3.2). Moreover, tight characterization of plants can be 
developed as of optimization of linear objective functions with LMI constraints using 
the result of Theorem 3.1. Efficient convex programming algorithms are available [19] 
for the solution of such LMI problems. These techniques will be discussed in greater 
detail in a later section. The point emphasized here is that the LMI characterization 
of dissipative LTI systems in Theorem 3.1 is essential for enbaling such analysis. 


19 



Chapter 4 


Algebraic Riccati Equation 
Characterization 


The linear matrix inequality characterizing dissipative LTI systems is equivalent to 
quadratic matrix inequalities (QMIs), or algebraic Riccati inequalities (ARIs) under 
an additional constraint. Extremal solutions of the ARIs can be determined from 
solutions of the corresponding algebraic Riccati equations (AREs)[20, 21, 22]. This 
leads to the characterization of dissipative LTI systems in terms of algebraic Riccati 
equations. Again, the ARE characterizations of gain bounded systems, positive real 
systems, and sector bounded systems follows directly by substituting their respective 
quadratic power functions[6]. 

The first result presents characterization of certain dissipative LTI systems with 
algebraic Riccati inequalities. 


Theorem 4.1 Consider a stable LTI system, £ : x = Ax + Bf, y = Cx + Df, where 
(A, B,C,D) is a minimal realization of the system, and a quadratic power function, 


p{y,f ) 


f T 


Q N 
N T R 


y 

f 


Assume that the matrix R = (R - f N T D + D T N + D T QD ) is positive definite. Then, 
S is dissipative with respect to this quadratic power function if and only if there exists 
a symmetric, positive definite matrix, P = P T > 0, which satisfies the algebraic 


20 



Riccati inequality Q — NR 1 N T > 0, where Q = C T QC — PA — A T P and N = 
C T (QD + N)~ PB. 


Proof: The result follows by using the Schur complements identity to show that 
the algebraic Riccati inequality above is equivalent to the linear matrix inequality 
in Theorem 3.1, under positive definiteness of R. Note that rearranging terms, Q — 
NR ~ x N t > 0 can be written as 


C T (Q - ( QD + ^R-'iQD + Nf) C -PA- A T P - PBR~ 1 B T P > 0 (4.1) 

where A = A — BR~ 1 (QD + N) T C. In this form it is clear that this condition is a 
quadratic matrix inequality in P. 

Since R > 0, and Q — NR~ 1 N T > 0, 


Q-NR-'N T 0 
0 R 


> 0 


Post-multiplying by the nonsingular matrix T = 
by T t gives 


I 0 

R~ A N T I 


, and pre-multiplying 


1 NR- 1 

0 I 


Q - NR~ 1 N t 0 
0 R 


I 0 

R-'N 7 ' I 


Q N 
N T R 


> 0 


Thus, with R > 0, a symmetric, positive definite matrix, P = P T > 0, satisfies the 
algebraic Riccati inequality if and only if it satisfies the LMI. Thus, the result follows 
from Theorem 3.1. □ 


Feasibility of the algebraic Riccati inequality can be determined from the solutions 
of the corresponding algebraic Riccati equation, using comparative theorems for these 
solutions, and results for extremal solutions of algebraic Riccati equations[21, 22]. 
Furthermore, conditions for the existence of a symmetric, positive definite solution 
to algebraic Riccati equations have been studied extensively, in terms of conditions 
on eigenvalues of the corresponding Hamiltonian matrix[23]. These conditions can 
be used to establish dissipativity of LTI systems, as summarized in the following 
Theorem. 


21 



Theorem 4.2 Consider a stable LTI system, E, with a minimal system realization 
(A, B,C, D), and a quadratic power function, 

y 

f . 


p{y,f) = 


f T ] 


Q N 
N t R 


Assume that the matrix R = (R + N T D + D r N + D T QD) is positive definite. Then 
the system is dissipative with respect to this quadratic power function, if and only 
if there exists a symmetric, positive definite solution, P = P T > 0 to the algebraic 
Riccati equation 


A T P + PA + PBR~ 1 B t P - C t [Q - ( QD + N)R~\QD + N) T }C = 0, 
where A = A — BR~ l (QD + N) T C. Equivalently, the Hamiltonian matrix 


(4.2) 


H = 


A BR~ l B T 

C T [Q - {DQ + N)R~ l ( DQ + N) T ]C -A T 


does not have eigenvalues on the imaginary axis. 


Algebraic Riccati equation and Hamiltonian matrix characterization of the special 
cases follow by substituting their power functions in the results of Theorem 4.1 and 
Theorem 4.2. These results are well known for gain bounded systems and positive 
real systems[22, 24]. The following corollaries demonstrate that these results follow 
directly from the general result for dissipative systems by substituting the power 
functions for bounded real and positive real systems. 


Corollary 4.1 Consider a stable LTI system, E, with a minimal realization ( A , B , C, D). 
Assume the matrix R = (I — D T D) is symmetric and positive definite. Then, the sys- 
tem is bounded real if and only if there exists a symmetric, positive definite matrix, 

P = P T > 0, which satisfies the algebraic Riccati inequality 

A t P + PA + ( C t D + PB)R~ X ( C t D + PB) t + C T C <0 (4.3) 

Equivalently, E is bounded real if and only if there exists a symmetric positive definite 
solution to the algebraic Riccati equation 

A T P + PA + PBR ~ l B t P + C T (i + DR- 1 D T )C = 0 (4.4) 


22 



where A = A + BR l D T C , or if the Hamiltonian matrix 


A + BR~ l D T C 

BR-'B T 

-C T (/ + dr~ x d t )c 

—(A + BR~ l D T C) T 


does not have eigenvalues on the imaginary axis. 


Corollary 4.2 Consider a stable LT I system, E, with a minimal realization (A, B , C, D ). 
Assume the matrix R = (D + D T ) is symmetric and positive definite. Then, the sys- 
tem is positive real if and only if there exists a symmetric, positive definite matrix, 

P = P T > 0, which satisfies the algebraic Riccati inequality 

A t P + PA + ( PB - C^R- 1 ( PB - C T ) T < 0 (4.5) 

Equivalently , £ is positive real if and only if there exists a symmetric } positive definite 
solution to the algebraic Riccati equation 

A t P + PA + PBR- 1 B t P + C t R~ 1 C = 0, (4.6) 


where A = A — BR~ X C, or if the Hamiltonian matrix 

\A-BR~ X C BR~ l B T 
H ~[ -C T R~ X C -{A-BR~ 1 C) t 

does not have eigenvalues on the imaginary axis. 


Algebraic Riccati equation characterization of sector bounded and gain-matrix 
bounded LTI systems follows from Theorem 4.2 by substituting their respective 
quadratic power functions. 

Algebraic Riccati equation and Hamiltonian matrix characterization of dissipative 
LTI systems allows the use of well established numerical techniques to determine 
whether a given LTI system is dissipative with respect to specified quadratic power 
functions. This provides an alternative to the LMI technique presented in the previ- 
ous section for establishing dissipativity of an LTI system with respect to a quadratic 
power function. Both approaches have their merits. Though efficient convex program- 
ming techniques are available for determining the feasibility of LMIs, the algebraic 
Riccati equation approach is computationally faster for high order systems. However, 


23 



the algebraic Riccati equation approach can be used only when R is well- conditioned 
and can be inverted reliably. In general, it has been observed that numerical accuracy 
of the LMI approach is superior when the problem data is not well conditioned. Fur- 
ther, LMIs are very useful in determining power functions such that all plants from 
a given uncertainty set of plants are dissipative with respect to that power function. 
Thus, it is concluded that both LMI and ARE characterizations of dissipative LTI 
systems are useful in characterizing dissipative LTI systems. 


24 



Chapter 5 


Frequency-Domain Properties 


Frequency- domain properties of dissipative LTI systems are examined in this sec- 
tion. A frequency-domain inequality that is satisfied by all dissipative LTI systems 
is presented first. Then, it is shown that this inequality establishes dissipativity with 
respect to quadratic power functions that have negative semidefinite coefficient ma- 
trix Q. In other words, the frequency-domain conditions axe necessary and sufficient 
for power functions with Q = Q T < 0, and are necessary for all dissipative LTI sys- 
tems. Note the class of dissipative systems with Q = Q T < 0 includes all the special 
cases being considered and numerous generalizations. Thus, frequency- domain char- 
acterization of gain bounded[9, 10], positive real, and sector bounded systems[18, 12] 
follows from the results of this section by substituting their respective power func- 
tions. Further, frequency-domain conditions for gain-matrix bounded systems are 
also presented. 

The first Theorem presents the frequency-domain inequality satisfied by all dissi- 
pative LTI systems. 


Theorem 5.1 If a stable LTI system, E, with transfer function matrix, G(s), is 
dissipative with respect to a quadratic power function 


p{y,f) 


y T f T 


Q N 
N T R 


y 

f 


25 



then 



for all u>. 

Proof: Let (A, B, C, D ) be a minimal realization of E. Since E is dissipative with 
respect to p(y,f), there exists a symmetric, positive definite matrix, P = P T > 0, 
and matrices L and W which satisfy the following conditions of the dissipativity 
lemma, 

PA + A t P = C t QC - L t L 
PB = C t (QD + N) - L T W 
R + N t D + D t N + D t QD = W T W 

Using these relations, the result follows from algebraic manipulations outlined below. 
Since G(ju) = C{ju>I - A)- l B + D, and G*{ju) = B T (-jul - A T )~ 1 C T + D T , 

4>(ju) = [B T {-juI - A t )~ x C t + D t ] Q [C(ju>I - A)~ l B + D] 

+ [. B T (-juI - A t )~ 1 C t + D t ] N + N T [C(jul - A)~'B + D]+R 

Expanding and collecting the terms 

= B T {-juI - A T )~ l C T QC(jul - A)~ l B 

+B t (-jl>I - A T )~ 1 C T (QD + N) + ( QD + N) T C(juI - A)~ l B 
+D t QD + D t N + N t D + R 

Substituting for the relations of the dissipativity lemma, Eq. (3.1), 

<KM = B T (—ju>I - A T )~ 1 (PA + A T P + L T L) {jul - A)~ l B 
+B T (-juI - A t )~\PB + L t W ) 

+{B t P + W T L)(ju:I - A)~ l B + W T W 

Rearranging terms, 

<f>(ju) = B T {-juI - A T )~ l L T L(juI - A)~ l B + B T (-juI - A T )~ l L T W 
+W T L(juI - A)~ l B + W T W 
+B T (-juI - A T )~ X PB + B T P(juI - A)~ l B 
+B T (-ju;I - A T )~\PA + A T P){juI - A)~ l B 


26 



Note that 


B T (-ju>I - A t )~\-PA - A T P)(ju:I - A)~ l B 
= B t (-ju;I - A 7 )- 1 \{-juI - A t )P + P(jul - A)] (jul - A)~ 1 B 
= B T P(juI - A)~'B + B T (-juI - A T )~ l PB 


Thus, 

4,(,u) = ( L(juI-A)-'B+Wy(UjwI-A)-'B + W ) 

where V(juj) = L{jul - A)~ X B + W. Since V*{ju)V{juj) > 0 for all a?, the result 
follows. □ 

The next Theorem establishes sufficiency of the frequency-domain inequality for 
dissipativity of LTI systems with respect to quadratic power functions with negative 
semidefinite coefficient matrix, Q. 


Theorem 5.2 Consider a stable LTI system,, E, with transfer function matrix , G(s ), 
and a quadratic power function, 


p{y,f)={y T f T 


Q N] \y ‘ 

n t r J [ / . 


with Q = Q T < 0. If the LTI system, E, satisfies the frequency-domain condition 


4>{ju) = [ G*(ju) I 


' Q 

N ‘ 

' G{ju) ‘ 

. NT 

R 

I 


> 0 


for all u>, then E is dissipative with respect to the quadratic power function, p(y,f)- 


Proof: The details of this proof are involved, but the key idea is that the matrix 
relations of the dissipativity lemma follow by comparing minimal realizations on either 
side of a spectral factorization identity for <j> [25, 26]. The proof is a generalization 
of the frequency-domain proofs for bounded realness lemma and positive realness 
lemma. 

First, recall some notation and results reviewed in the Appendix. Paraconjugate 
of a rational transfer function matrix, M(s), is A/~(s) = M T (-s). If ( A,B,C,D ) is 


27 



a realization of M(s), then (- A T , -C T ,B T , D T ) is a realization of M~(s). A matrix 
<j>(s) is said to be parahermitian if <f>~(s) — 4>(s). Finally, the spectral factorization 
theorem states that a parahermitian matrix, 4>(s), which is positive semidefinite on 
the imaginary axis can be factored as <j>(s) = M~(s)M(s), where the spectral factor, 
Af(s), is proper, stable, and its transmission zeros are in the closed left-half plane. 


Note that 


<K*) = [ G~(s) I ] 


Q N 
N T R 


G(s) 

I 


is parahermitian. Further, 4>(s) is assumed to be positive, semidefinite on the imagi- 
nary axis. Thus, the spectral factorization theorem ensures that there exists a stable, 
proper, spectral factor, M(s), of 4>(s), that is, 


<Ks) = [ G~(s) 



' Q 

N 1 ‘ 

_n t 

R J 


G(s) 

I 


M~(s)M(s) 


(5.2) 


The matrix relations of the dissipath ty lemma are obtained by comparing minimal 
realizations on either side of the spectral factorization identity, Eq. (5.2). Recall from 
Theorem 3.1 that satisfying the conditions of the dissipativity lemma implies that the 
LTI system is dissipative with respect to the corresponding quadratic power function. 


Let (AijBi, Ci, Di) be a minimal realization of M($). Then, M~(s) has a realiza- 
tion (— Af, -Cf , Bf , Dj). Therefore, the product of these transfer function matrices, 
M~($)M(s) has a realization 


ft Ai 0 
Vk-CfCi -A* 


Bi 

—CjDi 



Since the state space realization of M(s) is minimal, (A\,C\) is observable, and there 
exists a symmetric positive definite matrix, Xi = Xj > 0, such that 


A^Xi + XiAi + Ci C\ = 0 


(5.3) 


Applying the state transformation 
as 


/ 0 
Xi I 


gives another realization for M~($)M(s) 


( 


Ar 0 

0 -Afj 


Bi 

-XiBi - C\Di J 


,{B?X 1 + D'[C 1 B? ] ,DjDi) (5.4) 


28 



Note that this realization is minimal since all modes of M(s) are stable, and all modes 
of M~(s) are unstable. 

Next, a minimal realization of the left hand side of the spectral factorization iden- 
tity in Eq. (5.2) is obtained. Let (A,B, C, D) be a minimal realization of G(s). Then, 
a realization of QG{s ) is (A, B , QC , QD ), that of G~(s) is (— A T , —C T , B T ,D T ), and 
a realization of G~ (s)QG(s) is 


(\ A 0 
{[ -C T QC -A T 


B 

-C T QD 


D T QC B t ],D t QD 


Since A is stable and Q = Q T < 0, there exists a symmetric, positive semidefinite 
matrix, X = X T > 0, such that 


A x X + XA = C t QC 


(5.5) 


Applying the state transformation 
as 


I 0 
-X I 


gives another realization for G~ (s)QG(s) 


A 0 
0 -A T 


B 

XB - C T QD 


, [ -B T X + D T QC B T I ,D t QD 
L ) 

Further, a minimal realization of G~(s)N -f N T G(s ) is 


(\ A 0 

U o -^ T 


B 

-C T N 


N T C B t },N t D + D t N 


Thus, a realization of <f>(s) = G~ (s)QG(s) + G~ (s)N + N T G(s) + R is (A r , B r , C r , D r ), 
where 


A r 

B T 

C T 

D r 


A 0 

0 -a t 

B 

XB - C T (QD + N) 

[ -B t X + {QD + N) t C B t ] 
R + N t D + D t N + D t QD 


(5.6) 


Note again that this is a minimal realization, since modes of A are stable, and modes 
of — A T are unstable. 


29 



Now compaxe the minima] realizations on either side of the spectral factorization 
identity, Eq. (5.2), given by Eq. (5.4) and Eq. (5.6). It follows that there exists a 
nonsingular state transformation matrix, T, such that 

A 1 = T~ X AT 
Bi = T~ X B 

B\X x +D{Cx = ( -B t X + ( QD + NfC) T 

D[ Dx = R+ N t D + D t N + D t QD (5.7) 

Detailed arguments for guaranteeing the existence of such a transformation matrix, 
T, follow from the same arguments as those presented in Ref. [27], 

Finally, manipulations with the matrix relations in Eq. (5.7) leads to the relations 
of the dissipativity lemma. Setting W = D\ gives 

R + N t D + D t N + D t QD = W T W 

Substituting A\ = T~ l AT in Eq. (5.3), premultiplying by T~ T and postmultiplying 
by T _1 , results in 

A t {T~ t X 1 T- ') + (T^XxT-^A = -T- t C^CxT~ 1 (5.8) 

Adding Eq. (5.5) and Eq. (5.8), and setting P = X -\-T~ T XiT* 1 > 0 and L = CiT -1 , 
results in 

A t P + PA = C t QC - L t L 

Further, using the second and third relations in Eq. (5.7) leads to 

B t (T~ t XiT~ 1 ) + DjCxT' 1 = (QD + N) T C - B 1 X 

Rearranging the terms, and substituting for P = P T > 0, L , and W gives the remain- 
ing relation of the dissipativity lemma, 

PB = C t (QD + N)~ L t W 

Thus, it follows from the hypothesis of the theorem that there exists a positive definite 
matrix, P = P T > 0, and matrices L and W such that 

PA + A t P = C t QC - L t L 
PB = C t (QD + N)~ L t W 
R + N t D + D t N + D t QD = W T W 


30 



It follows from Theorem 3.1 that the system, E, is dissipative with respect to the 
quadratic power function, p(y, /). □ 

Theorem 5.1 and Theorem 5.2 establish that the frequency-domain inequality in 
Eq. (5.1) is necessary and sufficient to characterize dissipative LTI systems with 
respect to a quadratic power function for which Q = Q T < 0. This condition is 
satisfied for the special classes of dissipative LTI systems being considered. Therefore, 
necessary and sufficient frequency- domain conditions characterizing gain bounded, 
positive real, and sector bounded systems follow by substituting their power functions 
into the results of Theorems 5.1 and 5.2, and are summarized in the corollaries below. 
Note that often these equivalent frequency- domain characterizations are used to define 
these special classes of dissipative systems. 

Corollary 5.1 Consider a stable LTI system, E, with a minimal realization (A, B , C, D), 
and transfer function matrix, G{s). The LTI system is bounded real if and only if its 
frequency response, G(jcv), satisfies 

= 1- G-(ju)G{ju>) > 0 (5.9) 

for all u. Equivalently, there exists a symmetric, positive definite matrix, P = P T > 0, 
and matrices L and W which satisfy 

PA + A t P = -C T C - L T L 
PB = -C t D - L t W 
I - D t D = W T W 

if and only if the frequency- domain inequality in Eq. (5.9) is satisfied for all u. 

Corollary 5.2 Consider a stable LTI system, E, with a minimal realization (A, B , C, D), 
and transfer function matrix, G(s). The LTI system is positive real if and only if its 
frequency response, G(ju > ), satisfies 

<f>(M = G*(ju>) + G{ju) > 0 (5.10) 

for alluj. Equivalently, there exists a symmetric, positive definite matrix, P = P T > 0, 
and matrices L and W which satisfy 

PA + A t P = -L t L 
PB = C t - L t W 
D t + D = W T W 


31 



if and only if the frequency-domain inequality in Eq. (5.10) is satisfied for all u>. 

Corollary 5.3 Consider a stable LTI system, S, with a minimal realization ( A , B . C,D ), 
and transfer function matrix, G(s). The LTI system is inside sector [a, 6], for a < 

0 < b, if and only if its frequency response, G(ju>), satisfies 

= —abl + a(G*(juj) + G(ju>)) — G*(jui)G(ju) > 0 (5-11) 

for all u>, where a = (a + b)/ 2. Equivalently, there exists a symmetric, positive definite 
matrix, P = P T > 0, and matrices L and W which satisfy 

PA + A t P = -C T C - L t L 
PB = C T (aI — D) - L t W 
-abl + a(D T AD)- D T D = W T W 

if and only if the frequency-domain inequality in Eq. (5.11) is satisfied for all u. 

These results also provide the necessary and sufficient frequency- domain condi- 
tions for gain-matrix bounded LTI systems. As with gain bounded systems, these 
equivalent frequency-domain conditions may also be considered as definitions of gain- 
matrix bounded systems. Also, these conditions further clarify the interpretation of 
gain-matrix bounded systems as extensions of gain bounded systems. The frequency- 
domain conditions for these systems follow directly by substituting their power func- 
tions into the general results for dissipative LTI systems. 

First, consider input gain-matrix bounded LTI system, with respect to the input 
gain-matrix, T,. The corollary below follows by setting Q = —I,N = 0, and R = Th 

Corollary 5.4 Consider a stable LTI system, £, with a minimal realization ( A , B , C, D ), 
and transfer function matrix, G(s). The LTI system is input gain-matrix bounded with 
respect to a symmetric positive definite matrix, I",, if and only if its frequency response, 
G(ju;), satisfies 

<t>{ju) = T, 2 - G'{ju)G{ju) > 0 (5.12) 

for all u. Equivalently, there exists a symmetric, positive definite matrix, P = P T > 0, 
and matrices L and W which satisfy 

PA + A t P = -C T C - L t L 
PB = -C t D - L t W 

r 2 - d t d = w T w 


32 



if and only if the frequency-domain inequality in Eq. ( 5.12 j is satisfied for all u. 


Similarly, frequency-domain conditions for output gain-matrix bounded systems 
follow by setting Q = — N = 0, and R— I. 

Corollary 5.5 Consider a stable LTI system, E, with a minimal realization (A, B , C , D), 
and transfer function matrix, G(s). The LTI system is output gain-matrix bounded 
with respect to a symmetric positive definite matrix, T 0 , if and only if its frequency 
response, G(ju), satisfies 

<t>(M = I - G* (ju)T~ 2 G{ju) > 0 (5.13) 

for all u. Equivalently, there exists a symmetric, positive definite matrix, P = P T > 0, 
and matrices L and W which satisfy 

pa + a t p = -c T r; 2 c - l t l 

PB = -C t T~ 2 D - L t W 
I - D t T~ 2 D = W T W 

if and only if the frequency-domain inequality in Eq. (5.13) is satisfied for all u. 

Results for input-output gain-matrices bounded LTI systems with respect to sym- 
metric positive definite matrices, I\ and r o , follow by setting Q = — r~ 2 ,iV = 0, and 

r = r 2 . 

Corollary 5.6 Consider a stable LTI system, E, with a minimal realization (A, B,C,D ), 
and transfer function matrix, G(s). The LTI system is input-output gain-matrices 
bounded with respect to symmetric positive definite matrices, I\- and T 0 , if and only 
if its frequency response, G(ju), satisfies 

*0'w) = r? - G-(jw)T?G(ju) > 0 (5.14) 

for all u. Equivalently, there exists a symmetric, positive definite matrix, P = P T > 0, 
and matrices L and W which satisfy 

PA + A t P = -C t T: 2 C - L T L 
PB = —C r T~ 2 D - L t W 

r 2 - d t t- 2 d = w T w 

if and only if the frequency- domain inequality in Eq. (5.1 f) is satisfied for all uj. 


33 



Again, to see that gain-matrix bounded systems are extensions of gain bounded 
systems, consider an LTI system whose norm is bounded by 7. It is straightfor- 
ward to check from the frequency- domain conditions that this system is input gain- 
matrix bounded with respect to the matrix T, = 7 1; it is output gain-matrix bounded 
with respect to the matrix T 0 =7/; and it is input-output gain-matrix bounded with 
respect to the matrices T, = 7 ,/ and T 0 = 7 a I, where 7, and 7 0 are scalars such 
that 7 = 7,7o- With the scalar structure of the symmetric, positive definite gain- 
matrices above, gain-matrix bounded systems are reduced to the usual gain bounded 
systems. However, when these matrices are not restricted to a scalar structure, but 
are allowed to be general symmetric, positive definite matrices, gain-matrix bounded 
systems characterize a much larger class LTI systems. Stability results for gain-matrix 
bounded systems axe also derived from those of general dissipative systems and are 
presented in later sections. 

The frequency-domain inequality characterization provides insight into the frequency- 
domain behavior of the dissipative LTI systems, and this interpretation is very useful 
in selecting quadratic power functions for uncertain systems. Frequency-domain con- 
ditions for positive real systems have been used in the literature for graphically deter- 
mining positive realness of a system. However, with the efficient numerical methods 
for LMIs and AREs available now, frequency- domain conditions are not used for de- 
termining dissipativity of LTI systems. However, these conditions are very useful in 
exhibiting the frequency-domain characteristics of dissipative LTI systems, and are 
often used to define the special cases of dissipative systems. 


34 



Chapter 6 


Strictly Dissipative Systems 


A key notion for the stability of interconnected dissipative systems is that of strictly 
dissipative systems. As will be seen in the next section, dissipativity of LTI sys- 
tems establishes Lyapunov stability of interconnected dissipative systems only. A 
more restrictive notion than dissipativity is needed to establish asymptotic stabil- 
ity of the interconnected dissipative systems. However, the literature on dissipative 
systems [3, 4, 5] lacks a consistent approach to the notion of strictly dissipative sys- 
tems. In this section, a novel characterization of strictly LTI dissipative systems is 
presented, which has been motivated by the requirements of stability of intercon- 
nected dissipative systems. State space characterization of strictly dissipative LTI 
systems, in terms of further constraints beyond the dissipativity lemma, are also 
presented. Frequency-domain implication of the additional constraints is that the 
frequency- domain inequality (FDI) must be satisfied in a strict sense. It is shown 
that this definition of strict dissipativity is consistent with strict bounded realness 
and strict positive realness. 

Consider the energy balance equation for LTI systems which are dissipative with 
respect to quadratic power functions to explore further restrictions required for strictly 
dissipative systems, 

/(*))*= l T Kdt + E(T)-E(0) (6.1) 

Jo Jo 

where / 0 T 7 Zdt represents the energy dissipated by the system. The definition for 
dissipativity requires that the dissipated energy, / 0 r 7 Zdt, be greater than or equal to 


35 



zero, so that the dissipation inequality in Eq. (2.3), namely, 

f T p(y(t),f(t))dt>E(T)-E( 0) 

JO 

is satisfied. However, this definition allows dissipative systems to have motion along 
which no energy is dissipated. That is, it is possible to have / 0 T 7 Zdt = 0 along certain 
nontrivial state trajectories for dissipative LTI systems. For these state trajectories, 
the energy balance equation reads / 0 T p(y(t), f(t))dt = E(T) — E( 0), or energy input 
is equal to the net change in energy of the system, and no energy being dissipated. 

Strictly dissipative LTI systems are defined below as systems which dissipate en- 
ergy along almost all state trajectories of the system. A finite number of trajectories 
along which no energy dissipation occurs are exponentially stable trajectories, with 
exponentially decreasing input. An exponentially decreasing input, f(t), is an input 
of the form f(t ) = p(t)e st , where p(t) is a polynomial vector in t of the same di- 
mension as /, and Re{s} < 0. It is shown later in this section that these exceptional 
trajectories correspond to stable transmission zeros of a real rational transfer func- 
tion matrix. In other words, energy must be dissipated by a strictly dissipative LTI 
system for almost all motion of the system, and energy dissipation may be equal to 
zero only for a finite number of exponentially stable trajectories of the system. The 
proposed definition of strictly dissipative LTI systems based on these consideration 
is as follows. 


Definition 6.1 A stable LTI system, E : x - Ax + Bf, y = Cx + Df , where 
( A , B, C, D) is a minimal realization, is strictly dissipative with respect to a quadratic 
power function, 


p{y,f)=[y T f T 


' Q 

N ‘ 

y 

. NT 

R 

J . 


if there exists a positive definite, quadratic energy function, E(x) = x T Px, with P = 
P T > 0, which satisfies the strict dissipation inequality 


[ T p(y , f)dt > E(x(T)) - E(x( 0)) (6.2) 

J 0 

for all T € (0, oo) and for all nonzero 1 e £“«. except for a finite number of exponen- 
tially decreasing inputs. 


State space characterization of strictly dissipative LTI systems is developed next. 
Note that a strictly dissipative LTI system is obviously dissipative with respect to 


36 



its quadratic power function; however, it satisfies additional constraints beyond those 
required by dissipativity. Thus, a strictly dissipative system must satisfy the dissipa- 
tivity lemma, and some additional constraints, presented in the next Theorem. 


Theorem 6.1 A stable LT 7 system, £ : x = Ax+Bf, y = Cx+Df , where (A, i?,C, D) 
is a minimal realization of the system , is strictly dissipative with respect to a quadratic 
power function, 


p(y,f ) 


y T f T 


Q N 
N T R 


y 

f 


(6.3) 


if and only if there exists a symmetric, positive definite matrix, P — > 0, and 

matrices L and W satisfying the dissipativity lemma, 


PA + A t P = C t QC - L t L 
PB = C t {QD + N) - L T W 
R + N t D + D t N + D t QD = W T W 


the matrices ( A , L ) are observable, and all transmission zeros of the transfer function , 
X>(s) = L(sl — A)~ l B + W, are in the open left-half plane. 


Proof: Since a strictly dissipative system also satisfies the conditions for dissipativity, 
there exists a symmetric, positive definite matrix, P = P T > 0, and matrices L and 
W, which satisfy the dissipativity lemma. Thus, from the proof of Theorem 3.1 it 
follows by intergrating Eq. (3.3), over the interval [0, T], that 

[ T p(y(t), f(t))dt = l T d T (t)d(t)dt + E(T) - E(0) 

Jo Jo 

where d = Lx + W f. Thus, the rate of energy dissipation function, 7 Z(t) in Eq. (6.1), 
is given as 7 Z(t) = cF(t)d(t ). 

First assume that the LTI system is strictly dissipative. If (A, L) is not observable, 
it is possible to have nonzero state trajectories in the unobservable subspace of ( A , L) 
along which d = Lx + W f is identically zero. Therefore, exponentially increasing 
input can be generated, which excites unobservable states of the system only, such 
that 7 £(f) = <P(t)d(t) = 0. The system does not dissipate energy along these unstable 
trajectories with exponentially increasing input, which implies that the system is not 
strictly dissipative, a contradiction. Thus, (T, L) must be observable. Since (A, B ) 


37 



is controllable by the minimality assumption for E, it follows that ( A , B, L, W) is 
a minimal realization for T>(s) = L(sl - A)~ l B + W. Furthermore, d = Lx + Wf 
is identically zero when the system input is along the direction corresponding to 
a transmission zero of V(s). If V{s) has a transmission zero in the closed right-half 
plane, then there exists an input / corresponding to this transmission zero, that is not 
exponentially decreasing, for which d = Lx + W f = 0 and 7 Z(t) = <F(t)d(t) = 0. This 
again contradicts the hypothesis for a strictly dissipative LTI system, therefore, all 
transmission zeros of V{s) must be stable. Thus, for strictly dissipative LTI systems, 
there must exist a symmetric, positive definite matrix, P = P T > 0, and matrices 
L and W, which satisfy the dissipativity lemma, the matrices ( A , L) are observable, 
and all the transmission zeros of T>( s ) must be stable. 

Conversely, assume that there exists a symmetric, positive definite matrix P = 
P T > 0, and matrices L and W, which satisfy the dissipativity lemma, (A, L) is 
observable, and all the transmission zeros of 'D(s) are in the open left-half plane. 
Then, d = Lx + W f = 0 only for input / € £!£, corresponding to stable transmission 
zeros of T>(s), that is, only for a finite number of exponentially decreasing input. For 
all other / € ££, the energy dissipated, / 0 r <F(t)d{t)dt > 0, that is, the system is 
strictly dissipative with respect to p(y , /). □ 

Note from the proof of Theorem 6.1 that a finite number of system inputs for 
which a strictly dissipative LTI system does not dissipate energy correspond to stable 
transmission zeros of T>(s) = L(sJ - A)~ l B+W. For all other inputs / e ££, a strictly 
dissipative LTI system must dissipate energy, that is, / 0 T P.{t)dt = / 0 T <F(t)d{t)dt > 0. 

Further, it should be noted that the differential form of the dissipation inequality 
in Eq. (2.4), namely, f t E{x) < p{y,f), is not strenghtened to j t E{x) < p{y,f), for 
strictly dissipative LTI systems. It is possible for d = Lx + Wf to be equal to zero at 
certain time instants, even for strictly dissipative LTI systems. However, d(t) must 
not be identically equal to zero over a finite time interval. In terms of the matrices 
L and W, satisfying the dissipativity lemma, this implies that L and W do not have 
to be full column rank for strict dissipativity. The necessary and sufficient conditions 
on L and W for strict dissipativity are that (A,L) is observable, and transmission 
zeros of T>(s) = L(sl - A)~ l B + W are stable. 

For the frequency-domain conditions, recall from Theorem 5.1 that a dissipative 


38 



LTI system satisfies 




I 


Q N 
N T R 


G(ju) 

1 


V'{ju)V{ju). 


for all u, where V(s) = L(sl — A)~ l B + W. Since V(s) is minimal and does not 
have transmission zeros on the imaginary axis (its transmission zeros are stable), it 
follows that V*(ju>)V{ju>) > 0 for all u. Thus, if an LTI system with transfer function 
G(s) is strictly dissipative with respect to a quadratic power function, p(y,f), then 
its frequency response, G(ju>), satisfies 


<f>{ju) - [ G‘{ju) I 


Q N 
N T R 


G{ju) 

I 


> 0 


(6.4) 


for all u. 


The converse is true for the case where Q = Q T < 0 is symmetric, negative 
semidefinite. In this case, if the frequency response, G(juJ ), of an LTI system satisfies 


4>(juj) --- [ G m (ju>) I 


Q N 
N T R 


G{ju) 

I 


for all u>, then from Theorem 5.2 it follows that there exist matrices P = P T > 0, L 
and W, which satisfy the dissipativity lemma, so that 


4>{ju) 


G'{ju) I 


Q 

N ' 

■ <?(,•«) ■ 

N T 

R 

i 


= Tr<ju>yD<ju>) > o 


for allu, where V(s) = L(sI-A)~ l B+W. Spectral factorization, <f>(ju) = V*(ju)V(ju) 
can always be performed such that T>(s) is minimum phase and stable with ( A , B, L , W) 
being a minimal realization. This implies that ( A , L ) is observable and V(s) does not 
have transmission zeros in the closed right-half plane. Thus, the strict frequency- 
domain inequality in Eq. (6.4) ensures strict dissipativity of the LTI system when 

Q = Q T < 0. 


For Q = Q T < 0, the strict frequency-domain inequality of Eq. (6.4) is a neces- 
sary and sufficient condition for strict dissipativity of the LTI system. This observa- 
tion shows that the definition of strict dissipativity is consistent with strict bounded 
realness and strict positive realness. Substituting the power function for bounded 
realness shows that a system with transfer function, G(s ), is strictly bounded real 


39 



if its frequency response satisfies (f>(juj) = I — G*(ju)G(ju>) > 0, for all w, that is, 
if ||(?(.s)|| 0O < 1. This is the definition for strictly bounded real LTI systems in Ref. 
[9, 10]. Similarly, using the power function for positive real systems in the definition 
of strictly dissipative systems shows that an LTI system is strictly positive real if and 
only if = G“(ju>) + G(ju>) > 0, for all u>. Again this definition is consistent 

with the definition of strictly positive real systems in Ref. [28]. Finally, substitut- 
ing the power function for systems inside sector [a, b] leads to the frequency-domain 
characterization of LTI systems to be strictly inside sector [a, 6] as 

4>(j“ = ~abl + a (G*(ju>) + G(joj)) - GT(ju)G(ju) > 0 
for all w, where a = (a + 6)/2 as in Refs. [12, 18]. 

Frequency-domain criteria for LTI systems which are strictly gain-matrix bounded 
follow by substituting their respective power functions. A stable LTI system is strictly 
input gain-matrix bounded with respect to a symmetric positive definite matrix, 
r, = rf > 0, if and only if <f>(ju) = T? - G’ (ju)G(ju:) > 0 for all u. A stable LTI 
system is strictly output gain matrix bounded with respect to a symmetric positive 
definite matrix, r o = Tf > 0, if and only if its frequency response satisfies <j>{jui) = 
I —G*(ju})T~ 2 G(juj) > 0 for all u. Finally, a stable LTI system is strictly input-output 
gain-matrix bounded, with respect to matrices, T, = Tf > 0, and r o = Tj > 0, if 
and only if its frequency response satisfies 4>{ju) = T? — G m (ju>)T~ 2 G(juj) > 0 for all 
u>. These frequency-domain conditions are necessary and sufficient, and can be used 
to define these strictly gain matrix bounded systems. 


40 



Chapter 7 


Stability of Feedback 
Int er connect ion 


A central result of this report, namely, sufficient conditions for closed-loop stability of 
the standard feedback interconnection of dissipative LTI systems, is presented in this 
section. This stability result is shown to be a very general result, in that small gain 
conditions, passivity conditions, and a number of sector conditions for stability[12, 29] 
follow as special cases of this result. Further, new stability conditions are obtained for 
gain-matrix bounded LTI systems by substituting their quadratic power functions, 
demonstrating the generality of the result for dissipative systems. 



Figure 7.1: Negative Feedback Interconnection of Dissipative LTI Systems. 

Consider two stable linear, time-invariant systems, Si and E 2 , in the standard 
negative feedback interconnection, as shown in Fig. 7.1. Assume that (A,-, Bi, C,-, D,) 
are minimal realizations of these systems, such that their dynamics are given by the 


41 





state space equations, 


X{ — A{X i T Bifi 

y, = CiXi + Difi, *' = 1,2 (7.1) 

The standard negative feedback interconnection imposes the conditions f 2 = yi, and 
/i = — 3 / 2 - Further, assume that these two systems are dissipative with respect to 
quadratic power functions respectively, 




Vi 


fl] 


Qi N> 

N? Ri 



z = l,2 


(7.2) 


The following Theorem gives sufficient conditions on the power functions under which 
the feedback interconnection is stable. 


Theorem 7.1 If there exist positive scalars, Qi > 0 and a 2 > 0, such that 

aiM3/ii/i) + Q 2j>2(-/i,yi) <0 (7.3) 

for all yi and fi, then the standard feedback interconnection of dissipative LT I systems, 
Ei and E 2 , is Lyapunov stable. Furthermore, if either of these systems is strictly 
dissipative, then the closed-loop system is asymptotically stable. 


Proof: A weighted sum of the energy functions of these two dissipative systems is 
used as a Lyapunov function to establish the stability results. 

Since the LTI systems, E j, for i = 1,2 are dissipative with respect to quadratic 
power functions, p,(y,, /,), by Theorem 3.1, there exist symmetric, positive definite 
matrices, Pi = Pj > 0, and matrices L,, W , for both systems (that is, i = 1, 2), which 
satisfy the conditions of the dissipativity lemma, 

Pi A, + Aj Pi = CjQiCi - Lj Li 
PiB t = Cj{QiDi + Ni) - LjWi 
Ri + NjDi + Dj Ni + Dj QiDi = WfW { 

Consider a Lyapunov function, V{x l ,x 2 ) = aiEi(x) + a 2 E 2 (x), where £,(x f ) = 
xf PiXi, correspond to the quadratic energy functions for Ei and E 2 . Since c*i > 


42 



0,a2 > 0, and the energy functions, £,(x,), are positive definite functions, the Lya- 
punov function, V (xi , X 2 ), is a positive definite function of the states of the closed-loop 
system, Xi,X 2 - The derivative of this Lyapunov function along system trajectories is 

jV(x u x 2 ) = ct x ^E\{x\) + a 2 ^E 2 (x 2 ) 

From the proof of Theorem 3.1, it follows that 

= “(te + WJ t ) T (L tXi + W^ + piiyiJi) 

at 

for i = 1,2. Thus, 

■j-V(xi, x 2 ) = —ai(LiX X + W 1 /i) r (Lix 1 + W x f x ) + aipi(yi,/i) 

at 

—a 2 (L 2 x 2 + W 2 f 2 ) T (L 2 x 2 + W 2 f 2 ) + cx 2 p 2 (y 2 , f 2 ) 

Since a,- > 0, and (Z.x,- + W'</j) T (X,x i + W;/;) > 0, for i = 1,2, it follows that 

^V(xx,x 2 ) < Q!Pi(yi,/i) + a 2 p 2 (y 2 ,f 2 ) 

Using the conditions for the standard feedback interconnection, that is, f 2 — y\ and 
2/2 = — / 1 , gives 

— V(xi,x 2 ) < Qjpi(yi,/i) + ot 2 p 2 (- 

Since, by hypothesis, aipi (yi, / 1 ) + ot 2 p 2 (— / 1 , t/i ) < 0, the derivative of the Lyapunov 
function along closed-loop system trajectories is ^U(xi,X 2 ) < 0. Thus, by Lyapunov’s 
Second Theorem [27], the closed- loop system is Lyapunov stable. 

For asymptotic stability, without loss of generality, assume that S 2 is strictly 
dissipative with respect to the quadratic power function, P 2 {y 2 ifi)- It is shown that 
the trivial solution, that is, xi = 0 and x 2 = 0, is the only possible trajectory when 
^V(x-i,x 2 ) = 0, so that asymptotic stability of the closed-loop system is established 
using LaSalle’s Theorem [30, 27]. Since each of the terms in the expression for the 
derivative are nonpositive, ^U(xi, X 2 ) = 0, implies that d 2 = L 2 x 2 + W 2 f 2 = 0. Since 
strict dissipativity of S 2 implies that the system is observable and transmission zeros 
are stable, either f 2 = 0 or f 2 decreases exponentially with time. Stability of E 2 
implies that X 2 and y 2 are also exponentially decreasing functions of time. Thus, 
input and output to the first system f\ and y\ are exponentially decreasing, from 
the requirements of the feedback interconnection. Stability and minimality of the 


43 



realization of Ei implies that Xi is decreasing exponentially. Now, since x x and x 2 are 
exponentially decreasing with time, ■^V(x 1 ,x 2 ) < 0, which is a contradiction. Thus, 
the equilibrium configuration at the origin is the only possible system trajectory with 
f i V(x 1 ,x 2 ) = 0. Hence, the result. If Ei is strictly dissipative, interchanging the 
indices of the systems, the argument above again establishes stability of the closed- 
loop system. □ 

Theorem 7.1 is a very powerful result on stability of feedback interconnection of 
linear time-invariant systems. Next, a series of corollaries are presented, which show 
that a number of stability results in the literature follow directly by substituting 
specific power functions and values of scalars to satisfy the sufficient condition in Eq. 
(7.3). The feedback interconnection of LTI systems, whose stability is characterized in 
the corollaries, is the standard negative feedback interconnection, shown in Fig. 7.1. 
The Small Gain Theorem for stability of the feedback interconnection of LTI systems 
follows by using a, = 1, and power functions for bounded real system, p,(y,, /,) = 
fffi - yfyi, for i = 1,2 in Theorem 7.1. 


Corollary 7.1 (Small Gain Theorem) The feedback interconnection of two bounded 
real LTI systems (that is, systems satisfying HG.^Hoo < 1, for i = 1,2) is Lyapunov 
stable. The closed-loop system is asymptotically stable if either of the systems is 
strictly bounded real (that is, ||G,-(s)||oo < 1 , for i = 1 or i = 2.) 


A more general result for small gain conditions, which is essentially a scaled version 
of the result above, states that the standard feedback interconnection is stable if the 
gains |jGi(s)i|oo < 7* for i = 1,2, satisfy 7172 < 1. This result follows by using power 
functions for gain bounded systems, p,(t/,, /;) = 7 iff fi — yfyi , and scalars Qi = 1 
and a 2 = 7?. 

Corollary 7.2 (Passivity Theorem) The feedback interconnection of two positive 
real systems is Lyapunov stable. If either system is strictly positive real, then the 
closed-loop system is stable. 


These passivity conditions for stability for the feedback interconnection of passive 
LTI systems follows by using a,- = 1, and power functions for positive real system, 
Pi(yufi) = fjyi + yf ft, for * = 1,2 in Theorem 7.1. 


44 



The sector stability results, presented in the next series of corollaries, represent 
a refinement of the results in the literature[12, 13, 31] since the definition of LTI 
systems satisfying the sector conditions in a strict sense is weaker than that assumed 
in the literature. 

Corollary 7.3 The feedback interconnection of an LTI system, Ei, inside sector 
[a, 6], where a < 0 < b, with another LTI system, E 2 , which is inside sector [— j], 

is Lyapunov stable. If either system satisfies its sector constraint in a strict sense, 
the closed-loop is stable. 


This result follows using c*i = 1, a 2 = -ab, and the power functions corresponding 
to the sector conditions, that is, Pi(yi,fi) = — (2/1 — a/i) r (yi — fy/i) an d Plinth) = 
—(2/2 + f 2 /b) T {y 2 + /2/a) in Theorem 7.1. 

Corollary 7.4 The feedback interconnection of an LTI system, Ei, inside sector 
[a, 6], where 0 < a < 6, with another LTI system, E 2 , which is outside sector [— — 5 ], 
is Lyapunov stable. If either system satisfies its sector constraint in a strict sense, 
the closed-loop is stable. 


Using Qi = 1, Q ! 2 = ab, and the power functions corresponding to the sector condi- 
tions, that is, pi (2/1 , /1) = —{y\—af\) T (yi — byi) and p 2 (j/ 2 , / 2 ) = ( y2+f2/a) T {y2+f2/b ) 
in Theorem 7.1 leads to this result. 

Corollary 7.5 The feedback interconnection of an LTI system, Ej, inside sector 
[0, b], where 0 < b < 00, with another LTI system, E 2 , which satisfies {^-Vljb) 
is positive real, is Lyapunov stable. If either system satisfies its constraints in a strict 
sense, the closed-loop is stable. 


Using ot\ = 1, o 2 = b, and the power functions corresponding to the sector conditions, 
that is, pi(j/i,/i) = — y/(yi — by\) and p 2 (y 2 ,/2) = fI{V 2 + f 2 /b) in Theorem 7.1 leads 
to this result. 

Other combinations of sector conditions for stability of the feedback interconnec- 
tions can be obtained in a similar fashion. Note that all the results presented thus 


45 



far follow by substituting scalar matrices for the coefficient matrices of the quadratic 
power functions, namely, matrices Qi , TV,-, jR,-, for i = 1, 2. These results are essentially 
extensions of the single-input single-output (SISO) stability results to multi-input, 
multi-output (MIMO) systems. However, assigning general nonscalar values to the 
matrices Qi,Ni,Ri, leads to stability which are truly MIMO stability results in that 
they provide a larger number of parameters to characterize stability characteristics 
as opposed to system gain (a scalar) or sector conditions (two scalars). For obtain- 
ing general results for MIMO systems, consider a matrix oriented expression of the 
stability condition in Eq. (7.3). Since the equation must be true for all j/i,/i, the 
condition in Eq. (7.3) is equivalent to 


&1Q1 + o 2 -ft 2 — Q.2N2 
ot\N^ — a 2 iV 2 ol\R\ + G 2 Q 2 


< 0 


(7.4) 


This form of the sufficient condition for stability immediately leads to the following 
corollary, which is a very significant result itself. 


Corollary 7.6 The feedback interconnection of an LTI system, Ei, which is dissipa- 
tive with respect to the quadratic power function, 


P\{V\,h) = Qi [ Vi fi] 


' Q 

N ' 

3/1 

_n t 

R 

.A . 


with another LTI system, £ 2 , which is dissipative with respect to the quadratic power 
function 


P2(V2,f2) = Ol 2 



\-R N T 1 

3/2 

1 

.h. 


for any 01,02 > 0, is Lyapunov stable. If either system is strictly dissipative with 
respect to its quadratic power function, then the closed-loop is stable. 


Proof: Using the power functions for these dissipative systems in Eq. (7.4) leads to 


0i0 2 


Q-Q N-N 

n t -n t r-r 


= 0 < 0 


Thus, Eq. (7.4) is satisfied, and the result follows from Theorem 7.1. □ 


The next three corollaries present new stability results for gain-matrix bounded 
LTI systems. Since gain-matrix bounded LTI systems form an extension of gain 


46 



bounded systems, the following results may be viewed as extension of the small gain 
conditions. 


Corollary 7.7 The feedback interconnection of an input gain-matrix bounded LTI 
system with respect to F, with another LTI system which is output gain-matrix bounded 
with respect to T -1 , is Lyapunov stable. If either LTI system is strictly bounded with 
respect to its gain-matrix, then the closed-loop is stable. 

Proof: Note that the power functions for these systems are Q\ = —R 2 = —I, ft 1 = 
N 2 = 0, and R\ = — Q 2 = T 2 . The result follows substituting these in Eq. (7.4), with 
Oj = a 2 = 1 and Theorem 7.1. □ 

Corollary 7.8 The feedback interconnection of an output gain-matrix bounded LTI 
system with respect to T, with another LTI system which is input gain-matrix bounded 
with respect to T _1 , is Lyapunov stable. If either LTI system is strictly bounded with 
respect to its gain-matrix, then the closed-loop is stable. 


Proof: The power functions for these systems are Q 2 = —R\ = —I, Ni = N 2 — 0, an d 
R 2 = —Qi = r~ 2 . The result follows substituting these in Eq. (7.4), with Qi = a 2 = 1 
and Theorem 7.1. □ 

Corollary 7.9 The feedback interconnection of an input-output gain-matrices bounded 
LTI system with respect to F, and F 0 , respectively, with another LTI system which is 
input-output gain-matrices bounded with respect to T” 1 and r~ 1 , respectively, is Lya- 
punov stable. If either LTI system is strictly bounded with respect to its gain matrices, 
then the closed-loop is stable. 

Proof: Note that the power functions for these systems axe Q 2 — —R\ — — T 2 , 
Ni — = 0, and R 2 = —Qi = T" 2 . The result follows substituting these in Eq. 

(7.4) with Oj = 0-2 — 1 and Theorem 7.1. □ 

Numerous other stability results for MIMO systems, like the ones presented above, 
can be obtained simply by substituting different power functions in the result of 
Theorem 7.1. This demonstrates that the sufficient conditions for stability of a closed 
loop system presented in Theorem 7.1 is a very powerful and comprehensive stability 
result. 


47 



Chapter 8 


Stability with Feedback 
N onlinearit ies 


Application of the framework of dissipative LTI systems is studied in this section for 
stability of linear time-invariant systems, with memoryless (perhaps time-varying) 
nonlinearities in negative feedback. LTI systems with feedback nonlinearities are re- 
ferred to as Lure systems in the literature[ll, 30]. Many systems which axe primarily 
characterized as linear time-invariant systems except for a few nonlinear components 
such as saturating actuators, can be represented as Lure systems. Therefore, tradi- 
tionally, there has been significant interest in the stability of such systems. 



Figure 8.1: LTI System, S, with Negative Feedback Nonlinearities. 

The stability results are available primarily for sector bounded nonlinearities, with 
positive real constraints on a transformed linear system [11, 30]. Stability of Lure 
systems is established in this section for a large class of nonlinearities, with dissipa- 


48 





tivity conditions on the LTI system. The results for sector bounded nonlinearities, 
norm bounded nonlinearities and passive nonlinearities follow as special cases of these 
results. 

Consider an LTI system, E, with a minimal state space realization, (A, B, C, D), 
such that its dynamics are described by x = Ax + Bf , y = Cx+Df , where x is an n x 1 
vector, and /, y are mxl vectors. The feedback nonlinearity is represented by ty(y, t), 
which is memoryless, that is, no dynamics are associated with the feedback loop, as 
shown in Figure 8.1. Also, the nonlinearity ^(y, t ) has m outputs with m inputs, y(t). 
Since negative feedback is assumed, / = — ty(y,t). Thus, the time-varying, nonlinear 
closed-loop system is given by 


x = Ax — Bty(y,t) 

y = Cx-D-*{y,t) ( 8 . 1 ) 

Note that the measurement equation for the system above is a nonlinear equation, 
which is implicit in y(t ), output of the linear system. The nonlinearity, 'F(yA), is 
assumed to satisfy smoothness conditions such that the closed-loop system in Eq. 
(8.1) is well-posed, that is, its solution exists and is unique. Specifically, it is assumed 
that the nonlinearity, \t(y,t) is locally Lipschitz in y and uniformly Lipschitz in t , so 
that the Lure system in Eq. (8.1) is well-posed [11, 30]. Furthermore, it is assumed 
that ^(0, t) = 0, for all t, so that the origin is an equilibrium of the closed-loop 
system. 


Theorem 8.1 If the nonlinearity ty(y,t) satisfies 

T Q $ + <S T Ny + y T N T $ + y T Ry > 0 (8.2) 


for all t and for all y € K 71 , and the LTI system, E, is dissipative with respect to the 
quadratic power function, 


p(yj) 


T tT 

y f 


' -R N t ‘ 

y 

N -Q 

J . 


then the origin is a Lyapunov stable equilibrium of the Lure system in Eq. (8.1). If E 
is strictly dissipative with respect to p(j/, /), then the origin is a globally asymptotically 
stable equilibrium. 


49 



Proof. Since the LTI system, S, is dissipative with respect to the quadratic power 
function, p(y,f), there exists a symmetric, positive definite matrix, P = P T > 0, 
such that 

PA + A t P = -C r RC - L t L 
PB = C t (N t - RD) - L t W 
-Q + ND + D t N t - D t RD = W T W 

Consider the energy function of the dissipative LTI system, £, as a Lyapunov function, 
V(x) = x T Px, which is a positive definite function, since P = P T > 0. Proceeding 
in parallel to the proof of Theorem 3. 1 , the time derivative of the Lyapunov function 
along the trajectories of the system is 

^V(x) = -(Lx + W f) T (Lx + Wf) +p(y,f ) 

Since (Lx + W f) T (Lx + W /) > 0, and using the feedback relationship, / = — 
shows that 

jV(x) = - + <H T Ny + y T N T V + y T Ry) 

From the hypothesis, it follows that f t V(x) < 0, so that the origin is a Lyapunov 
stable equilibrium. 

If £ is strictly dissipative, then global asymptotic stability follows by showing 
that the trivial trajectory at the origin is the only possible system trajectory when 
dt^’( x ) = 0, by the Invariance Theorems in Ref. [27]. Since ^V(x) = 0 implies that 
d ~ Lx + Wu = 0, and 'D(s) = L(sl — A)~ X B -f W has only stable transmission 
zeros, either input, /, and state, x, are identically zero, or they are exponentially 
decreasing. However, since P = P^ > 0, exponentially decreasing states contradict 
the condition, ^ V(x ) = 0, the trivial state, x(t) = 0, is the only possible trajectory. 
Hence the result. □ 

Note that Theorem 8.1 requires that the condition in Eq. (8.2) is satisfied globally, 
and this leads to global asymptotic stability of the Lure system shown in Figure 
8.1. However, if the condition in Eq. (8.2) is satisfied in some neighborhood of the 
origin, but not globally, then the arguments above establish asymptotic stability of 
the equilibrium at the origin. 

The following three corollaries present the results for the special cases, which follow 
simply by substituting their respective power functions. 


50 



Corollary 8.1 If the nonlinearity ty(y,t) satisfies y T y — < 0 for all t and for 

all y € .ft” 1 , and the LTI system, E, is bounded real, then the origin is a Lyapunov 
stable equilibrium of the Lure system in Eq. (8.1). IfEis strictly bounded real, then 
the origin is a globally asymptotically stable equilibrium. 


Corollary 8.2 If the nonlinearity ^(yfi) satisfies y Tl ^ > 0 for all t and for all 
y € ft 1 ", and the LTI system, E, is positive real, then the origin is a Lyapunov stable 
equilibrium of the Lure system in Eq. (8.1). If E is strictly positive real, then the 
origin is a globally asymptotically stable equilibrium. 


Corollary 8.3 If the nonlinearity '$(y,t) satisfies — ay ) T ($ — by) < 0, a < 0 < b, 
for all t and for all y € ft m , and the LTI system, E, is inside sector [— £], then the 

origin is a Lyapunov stable equilibrium of the Lure system in Eq. (8.1). IfH satisfies 
the sector condition in a strict sense, then the origin is a globally asymptotically stable 
equilibrium. 


Finally, the following result follows by using the power function of a system outside 
a sector. This result is usually expressed as the absolute stability result [27, 30]. 


Corollary 8.4 If the nonlinearity ty(y,t) satisfies — ay) T (ty — by) < 0, 0 < a < b, 

for all t and for all y € R m , and the LTI system, E, is outside sector [— £, — £], then the 
origin is a Lyapunov stable equilibrium of the Lure system in Eq. (8.1). IfT, satisfies 
the sector condition in a strict sense, then the origin is a globally asymptotically stable 
equilibrium. 


For a SISO system, the result above corresponds to the Circle Criterion [30], since a 
SISO LTI system being outside sector [—£,—£] implies that the frequency response 
of the system lies outside the circle intersecting the real axis at — ^ and — £ in the 
frequency plane. 

Finally, the framework of dissipative LTI systems extends the results for stability 
of Lure systems, beyond unification of the SISO results, to MIMO stability results, 
which may be obtained by substituting their specific power functions in the stability 
result of Theorem 8.1. 


51 



Chapter 9 


Quadratic Stability 


Quadratic stability deals with stability of uncertain linear systems for all time- varying 
parameter variations within a predefined uncertainty set. 



Figure 9.1: Uncertainty Configuration for Quadratic Stability. 

The time-varying linear systems, whose stability is being investigated, are given 
as 

x(t) = {A + AA(<)) x(f) (9.1) 

where x(t) is an n x 1 state space vector, A is the constant, known part of the system 
matrix, and AA(t) is the unknown, time-varying part of the system matrix. The 
unknown time-varying component of the system matrix is assumed to be within an 
uncertainty set, that is, AA(t) € A, where A is a known, predefined uncertainty set. 
The uncertain linear system in Eq. (9.1) is said to be quadratically stable if there exists 
a parameter-independent quadratic Lyapunov function which guarantees stability for 


52 





all time- varying parametric variations within the uncertainty set. Quadratic stability 
is contrasted with a weaker concept of robust stability, where the uncertain parameters 
in AA are constant, but unknown. Robust stability may be established by ensuring 
that the system eigenvalues are in the open, left-half plane for all uncertain values of 
the parameters within the uncertainty set. A discussion of the distinction between 
quadratic stability and robust stability can be found in Ref. [32]. 

The uncertain component of the system matrix, AA(t), may depend on a smaller 
set of parameters, represented by an mxm matrix, F(t ), such that AA(<) = — BF(t)C , 
where B is a n x m matrix, and C is a m x n matrix. The matrices B , C describe 
the distribution of the uncertain parameters in the matrix, F(t), onto the uncertain 
component, A A(t). The uncertain parameter matrix, F(t ), belongs to a parametric 
uncertainty set, A p, so that F(t) E A f implies that A A(t) E A. Thus, the uncertain 
linear system can be represented as shown in Fig. 9.1, as a known linear system with 
state space realization, (A,B,C, 0), with the matrix of uncertain parameters, F(t), 
in negative feedback. The minus sign in the description of system matrix uncertainty 
had been selected to be consistent with the negative feedback paradigm. A general 
approach to representing uncertain linear systems as a known LTI system, with the 
parametric uncertainty in a negative feedback matrix, is referred to as 71 pulling out 
the A's” in the literature[24, 22]. In general, the direct feed through matrix (the D 
matrix) of the linear system may not be zero, as in the simpler description, above. 
In this general case, the uncertain component of the system matrix is expressed as 
A A{t) = - BF(t ) (I + DF{t )) _1 C. 

Quadratic stability of uncertain linear systems has primarily been studied for norm 
bounded parametric uncertainty, that is, for systems where the parametric uncertainty 
set A p is defined in terms of norm bounds. The next result develops the extension 
of quadratic stability results to problems where the known linear, time-invariant part 
of the uncertain system is dissipative. 


Theorem 9.1 Consider an uncertain system, E : x(t) = (A + AA(t)) x(t), where 
AA(t) = — BF(t) (I + T>F(t)) _1 C, and F(t) is a matrix of uncertain, time-varying 
parameters , which satisfy 

Q - NF(t) - F T (t)N T + F T (t)RF(t) < 0 (9.2) 

If (A, B,C, D) is a minimal realization of an LTI system, which is dissipative with 


53 



respect to the quadratic power function 


p{y,f ) =[y T f T ] 


Q N 
N T R 


y 

f 


then the origin is a Lyapunov stable equilibrium of the uncertain system, E. If the 
LTI system is strictly dissipative, then the uncertain system is quadratically stable. 


Proof: Since the LTI system is dissipative with respect to the quadratic power func- 
tion, p{y,f), there exists a symmetric, positive definite matrix, P — P T > 0, and 
matrices L and W which satisfy the dissipativity lemma, 

PA A A t P = C t QC - L t L 
PB = C t (QD + N)~ L t W 

R + N t D + D t N + D t QD = W T W (9.3) 

Consider the energy function of this dissipative LTI system as a Lyapunov function 
for the uncertain linear system, V(x) = x T Px, where P = P T > 0 is a symmet- 
ric, positive definite matrix which satisfies the dissipativity lemma above. The time 
derivative of this Lyapunov function along the trajectories of the uncertain linear 
system is 

jV{x) = x t (PA A A t P)x - x T PBF(t)(I + DF(t))~ l Cx 

—x T C T (I + DF(t))~ T F T (t)B T Px (9.4) 

Let H(t ) = (7 + DF{t))~ l , and drop the argument t from H(t) and F(t) to simplify 
the following expressions. Using the first two relations in Eq. (9.3) gives 

fv{x) = x t (C t QC - L T L)x - x T (C t (QD + N) - L T W) FHx 

—x T H T F T (( QD + N) t C - W t L ) x 
= —x T L T Lx + x t L t WFHx -f x T H T F T W T Lx + x T C T QCx 

—x T C T (QD + N)FHx - x t H t F t (QD + N) T Cx (9.5) 

Adding and subtracting x T H T F T W T WFHx to Eq. (9.5) for completing the square, 
and using the last relationship in Eq. (9.3), leads to 

^U(x) = -x t L l x + x t L t WFHx + x t H t F t W t Lx - x t H t F f W t WFHx 

at 

+x t C t QCx - x t C t (QD A N)FHx - x t H t F t (QD + N) T Cx 
+x t H t F t (R A N t D A D t N A DQD)FHx 


54 



Collecting terms and simplifying 

^-V(x) = -x t (L - WFH) t (L - WFH)x + x T (C - DFH) T Q(C - DFH)x 

at 

-x T {C - DFHfNFHx - x T H T F T N T (C - DFH)x 
+x t H t F t RFHx (9.6) 

Note the identity, H = (C - DFH), which may be verified readily from the definition 
of H. Using this identity in Eq. (9.6) gives 

—V(x) = —x t (L — WFH) t (L — WFH)x 
dt 

+x t H t (Q - NF - F t N t + F t RF ) Hx (9.7) 

Since x T (L - WFH) T (L — WFH)x > 0, and by the hypothesis, the second term 
in Eq. (9.7) is nonpositive, it follows that f t V(x ) < 0. Thus, the uncertain, linear, 
time- varying system is Lyapunov stable about the origin. 

If the LTI system G(s) = C(sl - A)~ l B + D is strictly dissipative with respect to 
the quadratic power function, p(y,f), then 'D(s) = L(sl — A)~*B + W is a minimal 
realization of a stable, minimum phase transfer function. Denote / = —FHx. so that 
y' = (L — WFH)x — Lx + Wf. Thus, from Eq. (9.7) it follows that ^U(x) = 0 
implies y' = 0. Since T>(s) is minimum phase, y' = 0 implies that either x = 0, or 
that / = —FHx and x are exponentially decreasing functions of time. The later 
possibility leads to a contradiction, since it implies j^V(x) < 0. Hence, x = 0 is the 
only possible trajectory when £ t V(x) = 0. Therefore, the uncertain system, E, is 
asymptotically stable for all F(t) € by the Invariance Theorems in Ref. [27], that 
is, the uncertain linear system, E, is quadratically stable. □ 

The stability result of Theorem 9.1 represents a general conditions for quadratic 
stability of a large class of uncertain linear systems. Particular results when paramet- 
ric variations belong to uncertainty sets characterized as in Eq. (9.2) follow simply 
by substituting the corresponding power functions in Theorem 9.1. Specifically, the 
case for norm bounded parametric uncertainty is presented in the following corollary, 
since this case has been studied extensively in the literature. 

Corollary 9.1 Consider an uncertain system, E : x(t) = (A + AA(t)) x(Z), where 
A A(t) = -BF(t)(I + DF(t)) -1 C, and F(t) is a matrix of uncertain, time-varying 


55 



parameters , is norm bounded ; that is it satisfies 


I - F T (t)F(t) > 0 (9.8) 

for all t. If (A,B,C,D) is a minimal realization of an LTI system, which is bounded 
real, then the origin is a Lyapunov stable equilibrium of the uncertain system, £. If 
the LTI system is strictly bounded real, then the uncertain system is quadratically 
stable. 


This corollary follows from Theorem 9.1 by substituting Q = R = /, and 
N = 0. Using the Riccati equation characterization for strictly bounded real systems, 
the corollary above states the quadratic stability result in Ref. [32], Thus, corollary 
9.1 presents the Riccati equation conditions for quadratic stability with norm bounded 
uncertainty [32], in terms of bounded real systems, which has a more general LMI 
characterization. 

Similar results can be obtained for quadratic stability when the known LTI system 
is positive real, by substituting N - I and Q = R = 0, and for sector bounded LTI 
systems, by substituting Q = abl, N = al , and R = /, where a = (a + b)/2. These 
results are not stated as separate corollaries to avoid repetition. Thus, it is seen that 
the framework of dissipative LTI systems presents a general framework for quadratic 
stability of a large class of uncertain linear systems. 


56 



Chapter 10 


Selection of Quadratic Power 
Functions 

Characterization of stable LTI systems in terms of dissipativity with respect to cer- 
tain quadratic power functions is necessary for the application of the stability results 
developed in the three previous sections. The selection of quadratic power functions, 
such that a given LTI system is dissipative with respect to the power function, is 
addressed in this section. This problem is posed as optimization of linear objec- 
tive functions with LMI constraints, or positive semidefinite programming problems, 
which can be solved using efficient convex programming techniques [19, 33, 34]. 

The LMI characterization of dissipative LTI systems, given by Theorem 3.1, forms 
the foundation for the techniques presented in this section. Let a minimal realization 
of a given stable LTI system be ( A , B, C, D ), and the coefficient matrices for a given 
quadratic power function be Q = Q T , R = i? T , and N. First, consider the problem 
of determining whether this LTI system is dissipative with respect to the quadratic 
power function. An LMI approach to this problem, based on the result of Theorem 
3.1 is discussed here. This feasibility problem is posed as maximization of a linear 
objective function, J = t, with respect to a scalar variable, <, and a symmetric matrix, 
P = P T , such that (1) P - tl > 0 and 

[ C T QC -PA- A T P C T (QD + N) — PB 1 
( ) (QD + N) t C-B t P r + n t d + d t n + d t qd J- 

If the parameter t attains a positive value, then the first condition implies that the 


57 



symmetric matrix P = P T is positive definite, and the second condition implies that 
it satisfies the dissipativity LMI, so that the LTI system is dissipative. Otherwise, the 
system is not dissipative with respect to the given quadratic power function. Note that 
the problem of maximization of a linear objective function subject to LMI constraints 
is a convex programming problem. Thus, a global maximum for the parameter t exists, 
and may be computed using efficient numerical technqiues [14, 34], 

Often, it is desirable to obtain the coefficient matrices, Q — Q T ,R — R T , and 
N of the power function, such that a given LTI system is dissipative with respect 
to that power function. Note that the matrix inequality of the dissipativity lemma 
is linear with respect to the matrix P = P T as well as the coefficient matrices, 
Q = Q T ,R = R T , and N. Therefore, the problem of selecting quadratic power func- 
tions can be posed as an LMI problem, with the coefficient matrices also being con- 
sidered as optimization variables. However, since every stable LTI is dissipative with 
respect to numerous quadratic power functions, it is desirable to determine power 
functions which somehow provide a tight characterization of the LTI system under 
consideration. The motivation for tight characterization of plants is to obtain a larger 
class of controllers that stabilize the closed-loop system, using the stability result of 
Theorem 7.1; or, to enlarge the uncertainty sets described in Theorems 8.1 and 9.1. 
A number of approaches to such selection of power functions are discussed in the rest 
of this section. 

Consider the power function corresponding to the / H (Xl norm of a stable LTI system, 
E, whose transfer function is G(s) = C(sl- A)~ l B+D , and (A, B, C, D) is a minimal 
realization. From Section 2, it follows that if a parameter, 7, is larger than the 
norm of E, that is, if HG^sjHoo < 7, then the system is dissipative with respect to 
a quadratic power function with coefficient matrices, Q = — /, R = 7 2 7, and N = 0. 
Thus, to select a tight power function of this form, the parameter 7 must be minimized 
until it attains its minimum value, which is However, since 7 2 appears in 

the dissipativity LMI, rather than 7, the objective function J = 7 2 leads to a positive 
semidefinite programming problem. Thus, selection of quadratic power functions 
with the structure of system gain is accomplished by minimiz ing a linear objective 
function, J = 7 2 , with respect to a symmetric, positive definite matrix, P = P T > 0, 
and positive scalar, 7 2 under the LMI constraint for bounded realness, that is, 


PA + A t P + C t C C t D + PB 
D T C + B T P D T D - 7 2 / 


< 0 


58 



By definition, the optimum value is J = | |G r (^s) 1 1^. Thus, this approach also pro- 
vides a convex programming approach to computing the Ti^ norm of a stable LTI 
system. It is noted here that this approach to computing Hoc norm of a stable LTI 
system is very robust and efficient, since it involves positive semidefinite program- 
ming for the minimization of a linear objective subject to LMI constraints. This is in 
contrast to computing the Hoo norm using a bisection technique along with checking 
imaginary axis eigenvalues of a Hamiltonian matrix, which becomes an ill-conditioned 
problem as the parameter 7 gets closer to the norm. For large dimension systems, 
this positive semidefinite programming approach is computationally intensive, but 
provides accurate results. 


Next, an LMI approach is presented to obtain a tight characterization of an LTI 
system in terms of input gain-matrix boundedness. The approach is to minimize the 
size of the input gain-matrix, as measured by its Frobenius norm, with respect to 
which the system is input gain-matrix bounded. Note that for the special case of 
gain bounded systems, with scalar input gain-matrices, the Frobenius norm of the 
scalar input gain-matrix is proportional to the gain of the system. Thus, for this 
special case, minimizing the Frobenius norm of the input gain-matrix is equivalent to 
minimizing the gain of the system. For the general case, the approach for obtaining a 
tight characterization of an LTI system in terms of input gain-matrix boundedness is 
to deter mi ne the minimum Frobenius norm, input gain-matrix. Recall that coefficient 
matrices for the quadratic power functions of input gain-matrix bounded systems are 
Q = — N = 0, and R = F - . The square of the Frobenius norm of T t is the trace 
of the symmetric positive definite matrix, R = R T > 0. The problem is reduced to 
minimizing the trace of a positive definite matrix, R = RF > 0, (with Q — I and 
N = 0 remaining fixed), while satisfying the dissipativity LMI. A minimum Frobenius 
norm input gain- matrix, I\, is given by the square root of R. Thus, the LMI problem 
is to minimize a linear objective function, J — Trace R. with respect to positive 
definite matrices, P = P ^ > 0 and R = RF > 0, under the LMI constraint, 


' PA + A T P + C T C C t D + PB 
D T C A B T P D T D - R 


<0 


An input gain-matrix bounding the given LTI system with minimum Frobenius norm 
is given by T; = R R 2 . 


A similar approach can be used to determine an output gain-matrix bounding an 
LTI system, r o = Y T 0 > 0, such that Frobenius norm of its inverse is maximized. From 


59 



section 2, the coefficient matrices for the quadratic functions of output gain-matrix 
bounded systems are R = /, N = 0, and Q = — T~ 2 . Let V = T~ 2 , so that the trace 
of this symmetric positive definite matrix, Trace V, is the square of the Frobenius 
norm of T” 1 . The LMI problem becomes to maximize a linear objective function, 
J = Trace V, with respect to symmetric, positive definite matrices, P = P T > 0, and 
V = V T > 0, such that 


PA + A t P + C T VC C t VD + PB 
D T VC + B T P D T VD - I 


< 0 


An output gain matrix, bounding the LTI system, is given by r o = V -1 / 2 . 


An LMI approach to determine parameters, a, b, such that a < 0 < b and a 
given LTI system lies inside sector [a, 6] is presented next. The characterization of 
tightness of sector boundedness is motivated by the frequency-domain interpretation 
of a sector bounded single-input single-output (SISO) LTI system in the frequency 
plane. A SISO LTI system inside sector [a, 6] has its frequency response within a 
circle in the frequency plane, which intersects the real axis at a and 6, with the center 
of this circle being on the real axis. The center of this circle is at (a + b )/ 2 and the 
radius of this circle is equal to ( b — a)/2. When a = —7 and b = 7, this corresponds 
to the Hoo gain of the system being bounded by 7, which implies that its frequency 
response lies within a circle centered at the origin, having radius 7. Minimizing 7 2 , 
therefore, corresponds to minimizing the square of the radius of a circle centered at 
the origin (which is proportional to the area of the circle), such that the frequency 
response lies within the circle. Sector boundedness allows the center of this circle to 
lie anywhere on the real axis of the frequency plane. Thus, a tight characterization of 
a SISO LTI system in terms of sector boundedness is to determine the smallest circle 
centered on the real axis, such that the frequency response of the LTI system lies 
within the circle. The smallness of the circle is measured in terms of the area of the 
circle, which is proportional to square of the radius of this circle, that is, (6 — a) 2 / 4. 

With motivation from the frequency-domain interpretation above, the optimization 
problem is to select the parameters a and b such that (6 — a) 2 / 4 is minimized, with 
the LTI system being inside sector [a, 6]. Note that ( b — a) 2 /4 = a 2 + <5j, where 
a = (a + b)j 2, and = —ab > 0. Furthermore, to obtain a linear objective, set 
J = 61 + 62, with 62 > ct 2 . These manipulations are performed to formulate the 
optimization problem of selecting the parameters o, b as the following LMI problem: 
To minimize the linear objective function, J = + S 2 , with respect to a symmetric 


60 



positive definite matrix, P = P T > 0, and positive scalars 61, a, under the following 
LMI constraints: 


PA + A T P + C T C PB - C T (aI - D) 
B t P -fa/ - D) T C *- T 


taT \ i taT ta 


< 0 


and 


62 a 

a 1 


> 0 


The optimal values of a and b are obtained as a — a — \/o 2 + <*>i and b = a + y/a 2 + 61, 
from the optimal values for a and 61 . 


Another tight characterization of LTI systems can be obtained in terms of sym- 
metric positive definite matrices, R = R T > 0, and general nonzero matrix, N, with 
Q = —I, as described by the following LMI problem. Minimize a linear objective 
function, J = Trace R -f 8\, with respect to symmetric positive definite matrices, 
R = R t > 0, P = P T > 0, and a general matrix, N, which satisfy the LMI con- 


straints, 


' PA A A t P + C t C C t (D-N) + PB 

(D -N) T C + B T P D T D- N T D - D T N - R 


< 0 


and 


8J 

N 


N T 

I 


> 0 


A similar LMI optimization could be performed with R = /, arbitrary nonzero matrix, 
JV, and Q = Q T < 0. 


Once power functions providing a tight characterization of uncertain plants to be 
controlled have been obtained, controllers are synthesized to enhance system perfor- 
mance while ensuring that they satisfy the dissipativity criteria for robust stability. 
Synthesis of robustly stabilizing dissipative controllers which enhance overall perfor- 
mance of the closed-loop system is an open research area, which will be explored in 
the future. 


61 



Chapter 11 


Spring-Mass-Damper Example 


This section demonstrates the application of the results developed in this report for 
robust controller synthesis using a spring-mass-damper system. First, characteriza- 
tion of this LTI system in terms of various power functions is presented. The later 
part of this section discusses an approach for synthesis of linear, quadratic Gaussian 
(LQG) controllers which satisfy stability criteria for dissipative systems. 


XI 



* 1 

x2 

yi 



K2 


k3 

ml 

— VWWVV- 

m2 

— vwww- 

fl 

dZ 


d3 



Figure 11.1: Three Spring- Mass- Damper System. 

The system used for the numerical example consists of three masses interconnected 
by springs and dampers as shown in Figure 11.1. Values used for the masses, spring 
constants, and damping coefficients are shown in Table 11.1. Input forces are applied 
at masses 1 and 3, and velocities of masses 2 and 3 are measured. Equations of 
motion for this system are developed using a Lagrangian approach, as described in 


62 



i 

m % 

di 

k 

1 

2.0 

0.5 

5.0 

2 

1.0 

0.2 

2.0 

3 

2.0 

0.15 

2.0 

4 


0.45 

5.0 


Table 11.1: Parameters of Three Spring- Mass-Damper System. 


l 

Open-loop Eigenvalue 

u>, (rad/sec) 

Pi 

1 

-0.0812 ± 1.3145j 

1.3170 

0.0616 

2 

-0.1625 ± 1.8643j 

1.8713 

0.0868 

3 

-0.2563 ± 2.3867j 

2.4005 

0.1068 


Table 11.2: Natural Frequencies and Damping Ratios for Vibration Modes. 


the Introduction. Natural frequencies and damping ratios for three modes of vibration 
of this system are given in Table 11.2. Six states for a minimal realization of these 
dynamics as a state space model are positions of the three masses followed by their 
velocities. This state space realization is given in Table 11.3. Thus, the model used is a 
multi-input, multi-output (MIMO) system, with noncollocated sensors and actuators. 
The singular value plot for this system is shown in Figure 11.2. 

Before proceeding with the examples for the MIMO system, some examples are 
presented for a single-input, single-output (SISO) system, corresponding to input 
force applied at mass 1 with velocity of mass 3 being measured. This is done because 
the results for SISO systems can be visualized, in terms of the frequency response 
of the system, and circles in the frequency plane exhibiting the frequency domain 
conditions for dissipative LTI systems. 

Bode magnitude plot for this system are shown in Figure 11.3, and its Nyquist 
plot is shown in Figure 11.4. Hoc norm of the system is computed to be 1.516. Using 
a gain bound of 7 = 1.75, it follows that this system is dissipative with respect 
to a quadratic power function, Q = — 1 ,R = 3.0625, and N = 0. Using convex 
programming techniques and software from Refs. [33, 34] to solve the feasibility 


63 





0.00 

0.00 

0.00 

1.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

1.00 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

1.00 

-3.50 

1.00 

0.00 

-0.35 

0.10 

0.00 

2.00 

-4.00 

2.00 

0.20 

-0.35 

0.15 

0.00 

1.00 

-3.50 

0.00 

0.075 

-0.30 


0.0 

0.0 











0.0 

0.0 











0.0 

0.0 

C = 

' 0.0 

0.0 

0.0 

0.0 

1.0 

0.0 ' 

D = 

' 0.0 

0.0 

0.5 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

TO 

0.0 

0.0 

0.0 

0.0 











0.0 

0.5 












Table 11.3: State Space Realization for Spring- Mass-Damper Model. 


LMI of the previous section, a symmetric, positive definite matrix which satisfies the 
dissipativity LMI, Eq. (3.2), with the values above for Q,N, and R, is computed to 
be 



57.180 

-37.370 

22.555 

-1.901 

-6.451 

-0.099 

-37.370 

60.795 

-49.074 

3.984 

2.387 

-3.491 

22.555 

-49.074 

83.735 

0.353 

1.775 

2.463 

-1.901 

3.984 

0.353 

13.335 

-4.157 

4.279 

-6.451 

2.387 

1.775 

-4.157 

13.752 

-8.123 

-0.099 

-3.491 

2.463 

4.279 

-8.123 

21.873 


All eigenvalues of this matrix P are positive, with the minimum eigenvalue being 
7.024. All eigenvalues of the symmetric matrix on the left-hand side of the dissipativity 
LMI in Eq. (3.2) are nonpositive for this symmetric matrix, P, thus showing that the 
LMI is satisfied. Note that since R used in section 4 is nonsingular, the same matrix, 
P, also satisfies the algebraic Riccati inequality of Eq. (4.1). The dashed circle in 
Figure 11.3 is a circle of radius 1.75 centered at the origin. The frequency response 
of this system is seen to lie within the dashed circle, as expected from the frequency 
domain conditions of section 5. 


64 



Next, a minimum 7 satisfying the gain boundedness LMI is computed for the SISO 
system, using the LMI approach presented in the previous section. Using convex 
programming software [33, 34] for the positive semidefinite program of the previous 
section, the minimum value of 7 is computed to be 7 = 1.516, which is the Woo norm 
of the system. A symmetric positive definite matrix satisfying the gain boundedness 
LMI for this value of 7 is computed as 


7.804 

-0.319 

1.590 

0.153 

0.022 

-0.175 

-0.319 

5.296 

-0.399 

-0.169 

-0.184 

-0.638 

1.590 

-0.399 

8.429 

0.404 

0.650 

0.458 

0.153 

-0.169 

0.404 

2.869 

0.853 

0.863 

0.022 

-0.184 

0.650 

0.853 

1.759 

0.901 

-0.175 

-0.638 

0.458 

0.863 

0.901 

3.016 


All eigenvalues of the matrix, P, are positive, with the minimum eigenvalue being 
1.1570, and all eigenvalues of the symmetric matrix on the left-hand side of the gain 
boundedness LMI are nonpositive. The dotted circle in Figure 11.3 is centered at 
the origin with a radius of 1.516, and it can be seen that this is the smallest circle 
centered at the origin such that the frequency response of the system remains within 
the circle. 

Finally, using the LMI approach of the previous section, tight parameters a and b 
are determined corresponding to the smallest circle that is centered on the real axis, 
and contains the frequency response of the system. The optimization results in a = 
—0.488 and b = 1.525 as the tightest parameters such that the LTI system lies inside 
sector [a, b]. Therefore, the system is dissipative with respect to a quadratic power 
function with Q = -1,7? = 0.744, and N = 0.519. A symmetric positive definite 
matrix satisfying the sector boundedness lemma for a = —0.488 and b = 1.525, that 
is, the dissipativity LMI with values of Q,N , and R as above, is 


8.001 

-3.884 

4.739 

-0.092 

-0.324 

0.413 

-3.884 

7.736 

-4.066 

0.320 

-0.027 

-0.593 

4.739 

-4.066 

8.417 

-0.334 

0.498 

0.293 

-0.092 

0.320 

-0.334 

2.168 

-0.015 

1.346 

-0.324 

-0.027 

0.498 

-0.015 

1.903 

-0.088 

0.413 

-0.593 

0.293 

1.346 

-0.088 

2.296 


The minimum eigenvalue of the matrix P is 0.819, and it can be verified that all 
eigenvalues of the matrix on the right-hand side of the sector boundedness LMI are 


65 



nonpositive. Figure 11.5 shows the Nyquist plot again. The dashed circle is centered 
on the real axis, and intersects the real axis at a = —0.488 and b = 1.525. It is 
seen that this is the smallest circle centered on the real axis such that the frequency 
response of the LTI system remains within the circle. For comparison, the dotted 
circle, of radius 7 = 1.516 centered at the origin, corresponding to the Hoc norm of 
the system, is also shown in Figure 11.5. It is obvious from Figure 11.5 that sector 
boundedness provides a tighter characterization of this system as opposed to the 
Hoo norm characterization. Thus, this example demonstrates that the use of more 
general quadratic power functions, than those for gain-boundedness, can lead to a 
tighter characterization of the plant for robust stabilization. 

The two-input, two-output model of the spring- mass-damper system is be used for 
the following numerical examples. First, the Hoc norm of the system is calculated 
using the LMI approach presented in the previous section, that is, by minimizing the 
gain bound, 7, under the constraint that it satisfies the gain boundedness lemma. The 
minimum value is computed to be 7 = 2.4788, which indeed is the H x norm of the 
system. Thus, one tight characterization of the LTI system is that it is dissipative 
with respect to a quadratic power function with coefficients Q — —I,N = 0, and 
R = 6.14477, where the identity matrix I is of order 2. A symmetric positive definite 
matrix satisfying the dissipativity LMI for this power function is 



10.666 

0.145 

0.919 

0.651 

0.595 

-0.029 

0.145 

7.289 

-1.839 

-0.727 

-0.058 

0.232 

0.919 

-1.839 

14.737 

-0.077 

-0.304 

0.009 

0.651 

-0.727 

-0.077 

3.711 

1.295 

0.956 

0.595 

-0.058 

-0.304 

1.295 

2.348 

0.893 

-0.029 

0.232 

0.009 

0.956 

0.893 

4.646 


The minimum eigenvalue of this matrix, P, is 1.518, and the LMI for gain boundedness 
is satisfied. 

Next, tight sector bounds for this MIMO system are computed using the LMI 
approach as above. The rationale for this optimization is that the process minimizes 
the sum of areas of the circle in frequency plane for each channel. In practice, the same 
approach is used as for selecting tight sector bounds for SISO systems. The solution 
to the LMI problem gives the optimal parameters as a = —0.582 and b = 2.662. 
Unfortunately, there is no simple way to visualize this result, and its verification 
follows simply by noting that the sector boundedness lemma with parameters a = 


66 



—0.582 and b = 2.662 is satisfied by the following symmetric positive definite matrix, 



10.784 

-4.256 

4.984 

0.087 

0.091 

0.589 

-4.256 

9.044 

-4.631 

0.083 

-0.041 

-0.275 

4.984 

-4.631 

11.340 

-0.584 

0.064 

0.112 

0.087 

0.083 

-0.584 

3.164 

0.075 

1.458 

0.091 

-0.041 

0.064 

0.075 

2.276 

0.022 

0.589 

-0.275 

0.112 

1.458 

0.022 

3.170 


It can be verified that this matrix is positive definite, and its smallest eigenvalue is 
1.680. Note that there the system is not positive real, so no results can be computed 
for the positive realness lemma. 


Next the computation of a minimum Frobenius norm input gain matrix is per- 
formed, such that the LTI system is input matrix gain bounded, as discussed in 
section 2. This computation is implemented as a linear objective with LMI con- 
straints, presented in the previous section. The optimal value of a input matrix gain 


bound is T, = 


1.771 0.697 
0.697 1.820 


Equivalently, the LTI system is dissipative with re- 


spect to a quadratic power function with coefficient matrices Q = — /, N = 0, and 


3.624 2.504 
2.504 3.799 


The dissipativity LMI is satisfied for these coefficients of the 


quadratic power function by the symmetric, positive definite matrix, 


11.256 

-2.987 

6.073 

-2.987 

9.975 

-3.333 

6.073 

-3.333 

12.132 

0.397 

-0.734 

0.313 

0.581 

0.002 

-0.471 

-0.112 

0.413 

-0.194 


0.397 

0.581 

-0.112 

-0.734 

0.002 

0.413 

0.313 

-0.471 

-0.194 

3.699 

0.655 

2.187 

0.655 

2.893 

0.625 

2.187 

0.625 

3.900 


The smallest eigenvalue of this matrix is 1.529, demonstrating that it is positive 
definite; and it can be verified that the eigenvalues of the right-hand side of the LMI 
are nonpositive. 


Similarly, following the approach presented in the previous section, an output gain 

2.379 1.535 “ 

1.535 2.479 


matrix bound is computed as T 0 = 


. The LTI system is dissipative 


with respect to a quadratic power function with coefficients Q 


0.678 -0.595 

-0.595 0.640 


67 



N — 0, and R = /, satisfying the dissipativity LMI with 



5.073 

-3.005 

1.940 

0.262 

0.344 

-1.108 

-3.005 

4.592 

-3.407 

-0.726 

0.002 

0.575 

1.940 

-3.407 

6.119 

1.181 

-0.313 

-0.121 

0.262 

-0.726 

1.181 

1.265 

-0.353 

0.337 

0.344 

0.002 

-0.313 

-0.353 

0.973 

-0.396 

-1.108 

0.575 

-0.121 

0.337 

-0.396 

1.543 


The minimum eigenvalue of P is 0.680. 

Many other quadratic power functions can be obtained such that the given system 
is dissipative with respect to that power function. The quadratic power functions 
computed in this section for the same system demonstrate that a system is dissipative 
with respect to many quadratic power functions. 

Once the plant has been characterized in terms of dissipativity with respect to 
a quadratic power function, synthesis of a controller that is dissipative with respect 
to another power function which satisfies the sufficient condition for stability is per- 
formed for stability robustness. Various approaches for such robust controller syn- 
thesis are possible. Hoo control theory provides a framework for synthesis of robust 
controllers for gain bounded systems, and synthesis of positive real controllers is dis- 
cussed in Refs [15, 35]. Extension of these techniques to general dissipative systems 
is being pursued currently. An approach to design MIMO controllers employing op- 
timal linear regulators and state estimators such that the overall controller satisfies 
the stability criteria is discussed in the next. 

Full state feedback is assumed for the design of the feedback gain matrix which 
optimizes a quadratic performance index using the linear regulator theory. Linear 
state estimators are designed such that the overall controller satisfies dissipativity 
requirements for robust stability. The approach for synthesis of state estimators is 
that of design of optimal Kalman filters, except that the noise covariance matrices 
are design parameters rather than a description of actucal process and measurement 
noise statistics. This approach is an extension to dissipative systems of the approach 
described in Ref. [36] for positive real systems. 

Linear regulator theory provides the optimal state feedback for minimizing a 
quadratic objective function as follows. For an LTI system, x = Ax + Bf , with full 


68 



state feedback control law, / = — C c x, the linear regulator problem is to determine 
the feedback gain, C c , such that a quadratic objective function 

roo 

J= / x T Q r x + fRrfdt 

Jo 

is minimized, where Q r = Qj >0, and R r = R% > 0, are the weighting matrices for 
state deviations and control effort. The optimal gain is C c = R~ l B T P c , where P c is 
the stabilizing solution of the Riccati equation, 

a t p c + p c A - p c br; 1 b t p c + Q r = 0 

For output feedback controllers, since the system state is not measured, state esti- 
mators are required to provide an estimate of the state. If the covariance matrix of 
Gaussian process noise in the system is Vj = Vj > 0, and the covariance matrix of 
Gaussian measurement noise is Wj = Wj > 0, then the optimal Kalman filter for 
state estimates is given by 

'x = {A - P f C T Wf l C)x + Bf + P f C T Wj l y 

where Pj is the stabilizing solution of the Riccati equation, 

APf + P S A T - P f C T W]- l CP } + V f = 0 

so that x is an optimal estimate of the state. Combining the estimator with the 
state feedback linear regulator results in a controller with a realization ( A c , B c . C c , 0), 
where 

A c = A - br; 1 b t p c - PjC T WJ l C 
B c = P f C T Wj l 

Cc = r; 1 b t p c 

Using the separation theorem, LQG theory establishes that this controller minimizes 
the following objective function, 



where £ {•} denotes the estimated value of the argument. Optimality of this objective 
function holds when the matrices V f = Vj > 0 and Wj = Wj > 0, represent noise 
covariance matrices for process noise and measurement noise, respectively. However, 


69 



in the current design approach, these matrices are treated as design parameters for 
synthesis of state estimators such that the overall controller satisfies desirable dissi- 
pativity criteria for robust stability. Therefore, the approach is to select weighting 
matrices Q r = Qj > 0 and Rr = Rj > 0 for the quadratic objective function, and 
then design a state estimator, using Vj = Vj > 0 and Wj — Wj > 0, as design 
parameters, such that the resulting controller satisfies the dissipativity constraints. 

Three controller designs are presented for the spring-mass-damper system to illus- 
trate the application of the stability results of dissipative systems to robust control 
synthesis. For all these controllers, the weighting matrices for the linear regulator 
objective function are chosen as Q r = 100 * C T C and R = I. Design parameters for 
the state estimator were chosen as V f = p\C T C and Wj = p^I. Scalar parameters 
Pi,P 2 were designed for the controller to satisfy desired robust stability conditions. 
The performance measure, V, used for a comparison of these controllers is the Tin 
norm of the closed-loop system, or equivalently, the root mean square (RMS) value 
of the output, with zero mean, unit intensity white noise applied at the input. 

First, a controller is designed while ensuring that its norm satisfies the small 
gain condition for stability in feedback around the spring-mass-damper system. With 
Pi = 1 and p 2 = 15, a controller is obtained which satisfies the small gain stability 
condition, that is, its 'H 0 0 norm is less than 1/2.4788. This may be verified by solving 
the dissipativity with a positive definite matrix, P. The performance value, V, for 
this controller is 0.809. 


Next, a controller is designed such that it is output bounded with respect to the 
output matrix gain, T D = F” 1 , which guarantees stability for all plants which are 

input matrix gain bounded with respect to T, = ^ 0.697 

[ 0.697 1.820 _ 

the spring-mass-damper system being considered. It can be verified that such a 
controller is obtained with p\ = 1.0, and P 2 = 13. The performance of this controller 
is V = 0.800. 


which includes 


Finally, a controller is designed which is inside sector [—0.3757, 1.718]. This would 
guarantee stability of the system with the spring-mass-damper in the feedforward 
loop, using the sector stability condition, since the plant lies inside sector [—0.528, 2.662] 
A sector-bounded system, as desired, is obtained with p\ = 1.2, p 2 = 15. The perfor- 
mance of this controller is V = 0.798. 


70 



Note that there is not much difference in performance of these controllers. This 
is because the aim in designing these controllers was to present a strategy for syn- 
thesizing robust, dissipative controllers, rather than minimizing the two-norm of the 
closed-loop system. Synthesis techniques which for robust dissipative controllers that 
optimize closed-loop performance will be addressed in the future. 


71 




Figure 11.2: Singular Value Plot of the Spring-Mass-Damper System. 



Figure 11.3: Bode Plot of the SISO Model. 


72 






Figure 11.4: Nyquist Plot of the SISO Model. 



Figure 11.5: Nyquist Plot of the SISO Model, with Smallest Circle. 


73 






Chapter 12 


Summary 


A detailed investigation of linear time-invariant (LTI) systems which are dissipative 
with respect to quadratic power functions has been presented in this report. In this 
framework, robust stability results have been developed for a large class of systems, 
employing mathematical abstractions of the notions of physical power and energy. 
Gain bounded systems, positive real systems, and sector bounded LTI systems are 
shown to be dissipative with respect to certain quadratic power functions. Novel con- 
cepts of gain-matrices bounded LTI systems have been introduced, and are shown to 
be a class of dissipative LTI systems. It is demonstrated that dissipative LTI systems 
represent a large class of LTI systems. Stability results presented for dissipative LTI 
systems have been developed, unifying and extending a number of stability results 
available in the literature. Specifically, small gain, positivity, and sector conditions for 
stability are shown to be special cases of the stability results for dissipative LTI sys- 
tems; and new stability results for input/output gain-matrices bounded LTI systems 
have been presented. 

State space characterization of dissipative LTI systems has been presented in terms 
of the dissipativity lemma, which provided a generalization of the bounded realness 
lemma and the Kalman- Yakubovitch lemma or the positive realness lemma. The state 
space characterization is equivalently expressed as a linear matrix inequality (LMI) 
in terms of a minimal state space realization of the LTI system. For certain cases, the 
LMI characterization has been shown to be equivalent to a quadratic matrix inequality 
(QMI), which led to an algebraic Riccati equation (ARE) characterization of dissi- 


74 



pative LTI systems. Frequency domain characterization of dissipative LTI systems 
was explored. Necessary conditions for dissipative LTI systems have been presented 
in terms of frequency domain inequalities (FDIs), and these conditions were shown 
to be sufficient as well for a large class of dissipative LTI systems. Strictly dissipa- 
tive LTI systems, which are essential in the development of robust stability results 
for dissipative systems, are defined as a further restricted class of dissipative LTI 
systems. Time-domain and frequency- domain characterizations of strictly dissipative 
LTI systems have also been developed in this report. State space characterizations, 
and time-domain as well as frequency-domain properties of bounded real, positive 
real and sector bounded systems have been shown to follow directly from the results 
of dissipative LTI systems. 

The framework of dissipative LTI systems has been employed to develop general 
robust stability results. In particular, three stability results involving dissipative LTI 
systems have been presented in this report. Sufficient conditions were presented for 
(1) stability of the feedback interconnection of dissipative LTI systems, (2) stability 
of dissipative LTI systems with memoryless feedback nonlinearities, and (3) quadratic 
stability of uncertain linear systems. The Lyapunov function approach has been used 
to establish these results, with the energy functions of the dissipative LTI systems 
being the Lyapunov functions. Stability conditions for these problems, derived from 
small gain, positivity and sector criteria, were shown to be special cases of the results 
for dissipative LTI systems. New stability results for feedback interconnection of LTI 
systems, in terms of input/output gain-matrix bounded LTI systems were also shown 
to follow as special cases of the stability results for dissipative LTI systems. Thus, 
stability results for dissipative LTI system have been shown to be general results, 
which unify and extend a number of stability results from the literature. 

Numerical techniques for tight characterization of given LTI systems, in terms of 
dissipativity with respect to quadratic power functions, have also been presented in 
the report. This approach utilized recently developed positive semidefinite program- 
ming techniques to solve linear matrix inequalities. A number of formulations have 
been presented for selection of power functions with prescribed structure. Robust con- 
troller synthesis techniques, based on the stability results for dissipative LTI systems, 
have been discussed. In particular, an approach for dissipative controller synthesis, 
employing optimal linear regulators and state estimators, has been presented. The 
state estimators were designed such that the overall compensator is dissipative with 
respect to required power functions for robust stability. A numerical example of a 


75 



spring-mass-damper system has been employed for a demonstration of the application 
of results presented in this report. 

Future work would involve further investigation into approaches for tight char- 
acterization of uncertain plants, with parametric uncertainty and structured uncer- 
tainties. Also, robust controller synthesis techniques which enhance overall system 
performance need to be developed further. Finally, characterization of nonlinear sys- 
tems which are dissipative with respect to specified power functions, and development 
of specific stability results for these systems could be pursued in the future. 


76 



Appendix A 

Signals and Systems 


This appendix summarizes some results from signals and systems theory used in this 
report. First, function spaces for the input and output signals are described, and 
then the representations of linear time-invariant (LTI) systems in the state space 
form and the frequency-domain, along with certain operations on these systems, are 
presented. Properties of bounded real, positive real, and sector bounded LTI systems 
are reviewed, and bilinear transformations between these systems are presented. The 
last part of this appendix discusses the spectral factorization theorem, which is used 
in the frequency- domain characterization of dissipative LTI systems. 

Extended spaces of square-integrable functions form the mathematical framework 
for the input and the output signals of continuous-time systems. The space of 
(Lebesgue) square-integrable functions, that is, real-valued functions / : 3J+ — ► 
which satisfy / 0 °° f T (t)f(t)dt < oo, will be denoted as An inner product on 
this space is defined as (y,/) = / 0 °° l / T (t)f(t) dt, for all y, / C ™ . With this 
inner product, is a Hilbert space, and the natural norm, induced by the in- 
ner product, is expressed as || / ||= [(/, Z)] 1 ^ 2 - The Fourier transforms of sig- 
nals in £™ also form a Hilbert space. This space, denoted by is a space 

of complex functions, / : C — * C m . which are analytic in the closed, right-half 
plane and satisfy dm < oo. The inner product in this space is 

(yj)c? + (j&) = if -oo fT(J u )fU u ) du i and the induced norm in this space is || / || 
= [(/, Z)] 1 ^ 2 . The subspace of real, rational, proper functions in is denoted 

as . Matrices with real, rational, proper elements which are analytic in 


77 



the closed, right-half plane, form an inner product space denoted by 
The extended Parseval’s theorem states that for any f,y G £™, the inner product 
{yj) = Io° y T (t)f(t)dt = J-oo y m (ju)f(ju)du; = ( y,f ), where yj G £™+0'&) are 
Fourier transforms of y,f respectively, and vice versa. Thus, the Fourier transform 
provides a an isometric isomorphism between C™ and £™ + (j$t). 

In response to “well-behaved” input functions, the output of unstable dynamic 
systems may increase without bound as time increases; specifically, for the inputs in 
£™, the outputs may not be £™. In fact, one definition of the stability of a dynamic 
system (bounded input, bounded output stability) is that the outputs be in £™ f° r 
all inputs from C Therefore, for the study of the stability of dynamic systems, a 
notion of extended spaces, which contain both the “well-behaved” signals and the 
“exploding” signals, is needed. 


The truncation operator or truncation projection is needed to define extended 
spaces. Given any signal, / : 3£ + — ► 3? m , the truncated signal, denoted as / r (t), for 
T G [0, oo), is defined as 



for t <T 
for t > T 


This is a mathematical statement of the intuitive concept of truncating a signal at 
time T. The extended space, corresponding to the space £™, denoted by ££, is defined 
as follows 

^e = {/i/:^ + ^» m ,/r€£^ V Tg[0,oo)}. 

Note that is only a linear space; that is, it is not an inner product space or 
a normed space. For all / G £™, it follows that f T G for all T G [0, oo). 

Thus, C 2 is a subspace within ££. Also, given any function, / : 9? + — ► 0i m , if 

fr € for all T G [0,oo), then || fr || is a nondecreasing function of T. In this 

case, if limr-,.^ || f T || exists, then / G C™, and limr^ || f T || = || / || . Thus, 

this extended space contains both “well-behaved” functions as well as “exploding” 
functions. For example, / x = e ot for a < 0 is in both spaces, C™ and ££; however, 
f 2 = t at for a > 0 belongs to but does not belong to £™. The extended space 
of square integrable functions, ££, is the universal set for inputs and outputs of 
continuous time LTI systems examined in this work. 


Next some properties of linear, time-invariant (LTI) systems are reviewed. A state 


78 



space realization of a linear, time-invariant system, E, is given by 

x = Ax + Bf 
y = Cx + Df 

where y(t) is the p x 1 output vector, f(t) is an m x 1 input vector, x(t) is an n x 1 state 
vector and the system matrices (A, B , C, D ) describe the dynamics of the LTI system. 
The p x m transfer function matrix for this system is G{s) = C(sl — A)~ l B + D. 
The impulse response matrix for the system, E, is given by G(t ) = Ce At B + D8(t). 
This system is stable if and only if the eigenvalues of A are in the open left-half 
plane. If A is a stable matrix, then G(s) € 7L£^+ m (jX). A state space realization, 
(A, B,C, D), is a minimal realization if and only if (A, B) is controllable and (A,C) 
is observable. If z = Tx, where T is a nonsingular, state transformation matrix, the 
transformed state space realization of G(s) is given as (X 1-1 AT,T 1 B,CT,D). Two 
minimal state space realizations of a transfer function matrix are related by a state 
space transformation. Paraconjugate transpose of a transfer function matrix, G(s), 
is given by G~(s) = G T (— s) = B T (—sI — A T )~ 1 C T A D T . Thus, if (A, B , C, D ) is a 
state space realization of G(s), then (—A T ,—C T ,B T ,D T ) and (—A T ,C T ,—B T ,D T ) 
are state space realizations of G~(s). 


Next, consider state space realizations for parallel, series and feedback interconnec- 
tions of two LTI systems, in terms of their individual realizations. Let (A l5 Bi,C\,D\) 
be a state space realization of Gi(s) and (A 2 , B 2 , C 2 , D 2 ) be a state space realization 
of G 2 (s). A state space realization of the parallel connection of Gi(s) and G 2 (.s), that 
is, a state space realization of Gi(s) + G 2 (s) is 


(\ Al ° 

\ , o a 2 


B a 
B 2 



C 2 J , D\ A D 2 


) 


A state space realization of the series connection of Gi(s) and G 2 (s), that is, 


is 


' Ai BiC 2 ' 


BiD 2 

0 A-2 


B 2 

for the series interconn' 

0 

CN 

1 


b 2 

[ BiC 2 A! j 


BiD 2 


Cl DiC 2 },DiD 2 ^J 
ion is 

DiC 2 Ci],DiD 2 ^J 


Finally, the closed-loop transfer function of the negative feedback interconnection 
of LTI systems G\(s) and G 2 (s) is T(s) = Gi(s) [/ + G ? 2 (>s)G ! i(s)] . A state space 


79 



representation for this interconnection in terms of the states of C?i(s) and G 2 (s) is 
given by (Ad, B ch C d , D d ), where 


A d 


B d 


C d 
D d 


Ai - BiD 2 (I + D\D 2 )~ x C\ -B^I + d 2 d 1 )- 1 c 2 
B 2 (I + D 1 D 2 )~ 1 A 2 -B 2 D 1 (I + D 2 D 1 )~ l C 2 

B 1 (I + D 2 D 1 )- 1 
B 2 D\(I + D 2 D\)~ 1 

(/ + DtD 2 )-'Ci -(/ 4- D 1 D 2 )~ 1 D 1 C 2 ] 

D 1 (I + D 2 D 1 )- 1 


Properties of bounded real, positive real and sector bounded systems are discussed 
next. Bounded real systems are systems with finite H <*> norm [9, 10]. Consider the 
systems with unity gain, that is, || C?(s) ||oo< 1. Recall that norm of a system 
is the induced operator norm with the norm for the input, /, and the output, 
V , [30]. Therefore, the condition for norm of a system being bounded by unity 
imlpies that 

f y T (t)y(t)dt< l°° y T (t)y(t)dt< f° /f (t)f T (t)dt = f f T (t)f{t)dt 

JO JO Jo Jo 

for all T € [0, oo) and / € £!£, with y(t) being the system response to the truncated 
input, /t( 0- Thus bounded real systems satisfy 

J* {f mm- y T (t)y(t)}dt> o, 

for all T € [0, oo) and / G C™ e . In the frequency-domain, an LTI system with trans- 
fer function, (?(s), is bounded real if / — ^ 0 for all u>. For single- 

input, single-output systems, the bounded realness condition can be visualized in the 
frequency-domain as the frequency response being within a unit circle centered at the 
origin in the frequency plane. A system is strictly bounded real if it satisfies the con- 
ditions for bounded realness in a strict sense. Small gain conditions for stability state 
that the feedback interconnection of a bounded real systems and a strictly bounded 
real system is stable. 

Passive systems are characterized by the input-output property / 0 T y T (t)f(t)dt > 0, 
for all T G [0, oo) and / G C™ e [11]. Equivalently, passive systems satisfy 

£{« T mm+f T mt)}dt>o, 


80 



for all T £ [0, oo) and / £ C™ e . In the frequency-domain, an LTI system is passive, or 
equivalently, the transfer function is positive real, if G’(ju> ) + G(ju>) > 0, for all u;. 
Strictly positive real systems satisfy these conditions in a strict sense. Passivity con- 
ditions for stability state that the feedback interconnection of a positive real system 
and a strictly positive real systems is stable. 

A number of sector boundedness conditions for LTI systems axe described in the 
literature. An LTI system inside sector [a, 6], with oo > b > a, satisfies (( y — af ), ( y — 
bf))j < 0, for all T £ [0, oo) and / £ C™e [13, 29]. A memoryless system is inside 
sector [a, 6] if its graph lies within a conical region in the input-output space defined 
by this inequality. If the memoryless system is time-varying, then the shape of the 
graph of a time- varying nonlinearity changes shape with time, however, it must stay 
within this conical region for all time, if the nonlinearity is sector bounded. For an LTI 
system, the sector boundedness condition may be rewritten as / 0 T {y(t)—af(t)) T (y(t) — 
bf(t))dt < 0, or, equivalently, 

£ {-abf T (t)f(t) + (a + b)y T (t)f(t) - y T (t)y(t)}dt > 0, 

for all T £ [0, oo) and / £ C™ e . In the frequency-domain, a transfer function, G(s), is 
inside sector [a, b] if herm{[G(ju;) - a/]*[G(ja>) - bl]} < 0, for all u>, where herm(-) 
stands for Hermitian part of the argument, that is, herm(Af) = 0.5(M* + M). The 
frequency plane provides a simple visualization for sector bounded SISO systems. 
The frequency response of a SISO system inside sector [a, 6] lies within a circle in 
the frequency plane, which is centered on the real axis and intersects the real axis 
at a and b. LTI systems that satisfy these conditions in a strict sense are strictly 
inside sector [a,b\. For b > 0 > a, a sector stability result states that the feedback 
interconnection of an LTI system inside sector [a, 6] with another LTI system which 
is strictly inside sector [—£, — £] is stable. 

Bilinear transformations between positive real systems, bounded real systems and 
sector bounded systems are reviewed next [9, 10, 12]. Let S be a bounded real 
system, so that y = Sf satisfies (y,y) — {/, /) < 0. Noting that {/, /) — (j/,y) = 
((/-y) 5 (f + y)) > 0? it follows that Z = (I - 5)(/ + 5) -1 is positive real. Conversely, 
if Z is positive real, then S = (I - Z)(I + Z)~ x is bounded real. Further, if a system, 
T , is inside sector [a, 6], then the system, S = r -1 (T — al), where r = (b — a)/2 and 
a = (a + b)/2, is bounded real [37]. Conversely, if S is bounded real, then T = rS + al 
is inside sector [a, 6]. This fact can be derived from the following manipulations. Let 


81 


y — Tf and y' = 5/; then, y' = r l (y - a/), and 


(: v '» y') - (/» /) 


^ {(y - «/,y - <*/> - r 2 (/,/)} 

“J {(y,y) - 2a(y,/) + (a 2 - r 2 )(/, /)} 
■^{(y,y) ~(a + b)(yj) + ab(fj)} 

^{( V-af,y~bf ')} 


This shows that (y',y') - (/,/} < 0 if and only if (y - af,y - bf) < 0, hence the 
result. 

Finally, the spectral factorization theorem is discussed, since it is used in the devel- 
opment of frequency-domain characterization of dissipative LTI systems. A transfer 
function matrix, $(s), is called a parahermitian matrix if it satisfies $~(s) = $(s). 
The spectral factorization theorem essentially states that a parahermitian transfer 
function matrix, which is positive semidefinite on the imaginary axis, can be fac- 
torized with stable factors. This may be thought of as an extension of the concept 
of Cholesky decomposition of positive semidefinite matrices. The theorem states 
that a given parahermitian transfer function matrix $~(s) = $( 5 ), which satisfies 
^ 0, for all uj € 3?, can be factorized as $(s) = M~(s)M(s), where M(s ) is a 
stable transfer function matrix with transmission zeros in the closed left-half plane. 

This appendix has presented the notation and some results which have been used 
in this work. 


82 



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86 




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4. TITLE AND SUBTITLE 

Robust Stabilization of Uncertain Systems Based on Energy Dissipation 
Concepts 

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C NASI -19341 
WU 233-10-14-12 

6. AUTHORS) 

Sandeep Gupta 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

Vigyan, Inc. 

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Hampton, VA 23666 

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13. ABSTRACT (Maximum 200 word*) 

Robust stability conditions obtained through generalization of the notion of energy dissipation in physical 
systems are discussed in this report. Linear time-invariant (LTI) systems which dissipate energy corresponding 
to quadratic power functions are characterized in the time-domain and the frequency-domain, in terms of linear 
matrix inequalities (LMIs) and algebraic Riccati equations (AREs). A novel characterization of strictly dissipative 
LTI systems is introduced in this report. Sufficient conditions in terms of dissipativity and strict dissipativity are 
presented for (1 ) stability of the feedback interconnection of dissipative LTI systems, (2) stability of dissipative 
LTI systems with memoryless feedback nonlinearities, and (3) quadratic stability of uncertain linear systems. It 
is demonstrated that the framework of dissipative LTI systems investigated in this report unifies and extends 
small gain, passivity; and sector conditions for stability. Techniques for selecting power functions for 
characterization of uncertain plants and robust controller synthesis based on these stability results are 
introduced. A spring-mass-damper example is used to illustrate the application of these methods for robust 
controller synthesis. 

14. SUBJECT TERMS 

State space characterization of dissipative LTI systems 
Stability of interconnected dissipative systems 
Linear Matrix Inequalities (LMI's) for robust stability 


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