ME675c5_IOStability_t0

7/29/2019 ME675c5_IOStability_t0

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I/O Stability Lecture Notes by B.Yao

1

Input-Output (I/O) Stabil ity

-Stability of a System

Outline:

Introduction

White Box and Black Box Approaches Input-Output Description

Formalization of the Input-Output View

Signals and Signal Spaces

The Notions of Gain and Phase The Notion of Passivity The BIBO Stability Concept

I/O Stability Theorems

The Small Gain Theorem The Passivity Theorem Positive Real Functions and Kalman-Yakubovich Lemma

For LTI System


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1.Introduction

Systems can be modeled from two points of view: the internal (or state-space)

approach and the external (or input-output) approach. The state-space approach takesthe white box view of a system, and is based on a detailed description of the inner

structure of the system. In the input-output approach, a system is considered to be a

black box that transforms inputs to outputs. As a consequence, the system is modeled

as an operator. The operator can be represented by either a verbal description of the

input-output relationship, or a table look-up, or an abstract mathematical mapping that

maps an input signal (or a function) in the input signal space (or a functional space) to

an output signal (or a function) in the output signal space.

2. Formalization of I/O View2.1 Signals and Signal Spaces

2.1.1Single-Input Single-Output (SISO) Systems

In SISO systems, the input and output signals can be described by real valued timefunctions, e.g., an input signal is denoted by a time-function u inR, i.e.,

( ),u u t t R

Note that to specify an input signal u, it is necessary to give its value at any time, not

values at a point or an interval.


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The input and output signal spaces can be described by normed spaces; certain norms

have to be defined to measure the size of a signal. For example,

The super norm orL-norm

0

sup ( )t

u u t

(I.1)

Application examples of using the super norm are the problem of finding

maximum absolute tracking error and the problem of dealing with control

saturation.

2L -norm

1

22

2 0( )u u t dt

(I.2)

If we are interested in the energy of a signal, 2L -norm is a suitable norm tochoose.

pL -norm

1

0( )

p p

pu u t dt , 1 p (I.3)


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Corresponding to different norms, different norm spaces are defined:

L-space

the set of all bounded functions, i.e., the set of all functions with afiniteLnorm.

: for some 0L x x M M (I.4)

pL -space

the set of all functions with a finite pL -norm., i.e.,

1

0: or ( ) for some 0

p p

p pL x x M x t dt M M

(I.5)

In general, these normed spaces may be too restrictive and may not include certain

physically meaningful signals. For example, the 2L -space does not include the

common sinusoid signals (why?). To be practically meaningful, the following

extended space is defined to avoid this restrictiveness:


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Definition [Extended Space]

IfXis a normed linear subspace ofY, then the extended space eX is the set

: for any fixed 0e TX x Y x X T , (I.6)

where Tx represents the truncation ofx at Tdefined as

( ) 0( )

0T

x t t Tx t

t T

(I.7)

The extended 1L space is denoted as 1eL , the extended 2L space as 2eL , .

Example:

2 2( ) sin but( ) but

e

e

u t t u L u Ly t t y L y L


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2.1.2Multi-Input-Multi-Output (MIMO) Systems

Multiple inputs and multiple outputs can be represented by vectors of time functions.

For example, m inputs can be represented by a m-dimension of time functions u, i.e.,

: 0 mu R with

1

( )

( )

( )

m

m

u t

u t R

u t

(I.8)

Norms corresponding to (I.1)-(I.3) are defined as

The super norm

0

sup ( )t

u u t

(I.9)

where, for avoiding confusion, ( )u t is used to represent the norm of ( )u t inmR space1.

1The super-norm of ( )u t in mR is ( ) max ( )ii

u t u t

.


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The 2L -norm

22 20 0

( ) ( ) ( )Tu u t dt u t u t dt (I.10)

The pL -norm

1

0( )

p p

pu u t dt

, 1 p (I.11)

where ( )u t represents the norm of ( )u t in mR space2.

The corresponding norm spaces are represented bym

L , 2m

L andmpL respectively. The

extended spaces arem

eL , 2m

eL andmpeL respectively.

2 Thep-norm of ( )u t in mR is

1

1

( ) ( )m pp

ipi

u t u t

.


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2.1.3 Useful Math Facts

The following lemmas state some useful facts about pL -spaces.

Lemma I.1 [Page 144 of REF1] (Holders Inequality)

If , 1,p q , and1 1

1p q

, then, pf L , qg L imply that 1fg L , and

1 p qfg f g (I.12)i.e.,

1 1

0 0 0( ) ( ) ( ) ( )

p qp qf t g t dt f t dt g t dt

(I.13)

Note:

When 2p q , the Holders inequality becomes the Cauchy-Schwartz inequality,

i.e.,

1 1

2 22 2

0 0 0( ) ( ) ( ) ( )f t g t dt f t dt g t dt

(I.14)


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An extension of the Lemma I.1 to peL space can be stated as

Lemma I.2

If , 1,p q , and1 1 1p q

, then, pef L , qeg L imply that 1efg L , and

1

T Tp qT

fg f g , 0T (I.15)

or

1 1

0 0 0( ) ( ) ( ) ( )

T T Tp qp qf t g t dt f t dt g t dt 0T (I.16)

Lemma I.3 [Page 144 of REF1] (Minkowski Inequality)

For 1,p , , pf g L imply that pf g L , and

p p pf g f g (I.17)

I/O St bilit L t N t b B Y


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Note:

(a) A Banach space is a normed space which is complete in the metric defined by itsnorm; this means that every Cauchy sequence is required to converge. It can be

shown that pL -space is a Banach space. [Pages 76-82 ofREF1]

(b) Power Signals: [REF5]

The average power of a signaluover a time span Tis2

0

1( )

Tu t dt

T (I.18)

The signal uwill be called a power signal if the limit of (I.18) exists asT , andthen the square root of the average power will be denoted bypow( )u :

12

2

0

1pow( ) lim ( )

T

Tu u t dt

T

(I.19)

I/O Stability Lecture Notes by B Yao


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(c) The sizes of common pL spaces are graphically illustrated below [REF5]

Note:

1 2L L L

Example: (verify by yourself)

1( )

1f t

t

2( )f t L L , but 1( )f t L , 1( ) ef t L

0 if 0

1( ) 0 1

0 1

t

f t tt

t

1( )f t L , but ( )f t L , 2( )f t L

2L

1L

pow

L

I/O Stability Lecture Notes by B Yao


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2.2 Gain of A System

2.2.1Definitions

From an I/O point of view, a system is simply an operator Ssuch that maps a inputsignal u into an output signaly. Thus, it can be represented by

: ie oeS X X with y Su (I.20)

where the extended norm space ieX represents the input signal space (i.e., ieu X ) and

oeX is the output signal space. The gain of a linear system can thus be introduced

through the induced norm of a linear operator defined below:

Definition [Gain of A Linear System]

The gain ( )S of a linear system S is defined as

, 0( ) supeu X u

Su

S u (I.21)

where u is the input signal to the system. In other words, the gain ( )S is the

smallest value such that

( )Su S u , eu X (I.22)



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For a linear system described by a matrix m nijA a R , the gain ( )S is theinduced norm of the matrix given by

maxi ijj a (row sum) forsuper norm,

1maxj iji a (column sum) for1-norm, and

2T

M MA A A (maximum singular value) for2-norm.

Nonlinear systems may have static offsets (i.e., 0Su when 0u ). To accommodatethis particular phenomenon, a bias term is added in the above definition of gains:

Definition[Page 147 of REF1] [Gain of A Nonlinear System]

The gain ( )S of a nonlinear system Sis defined to be the smallest value such that

( )Su S u , eu X (I.23)

for some non-negative constant.



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Definition[Page 197 of REF7] [Finite Gain Input-Output Stability]

A nonlinear system S is called finite-gainLp input-output stable if the gain ( )S

defined in (I.23) is bounded (or finite), in which theLp

norm is used for input and

output signals.

Note:

Whenp=, the above finite gainL stability results in bounded-input bounded-

output(BIBO) stability. The converse is in general not true. For example, the staticmappingy=u2 is BIBO stable but does not have a finite gain.

The above definitions are valid for all systems including acausal systems. Physical

dynamic systems are always causal, which is captured by the following definition

which further simplifies the analysis:

Definition [Page 146 of REF1] [Causal Nonlinear Operators]

A mapping : n mpe peS L L is said to be causal if for every input u and every T>0, itfollows that

, 0, nT peT TSu Su T u L

In other words, the truncation of the output to u(.) up to time Tis the same as theoutput to uT(.) up to time T.



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2.2.2Linear Time Invariant Systems

An LTI system with a TF matrix G(s) can be described in input/output forms by

( ) ( ) ( )Y s G s U s in s-domain

0( ) ( * )( ) ( ) ( )

ty t g u t g t u d in time-domain (I.24)

where 1( ) ( )g t L G s is the matrix of unit impulse responses. The unit impulseresponse matrix of a strictly proper stable ( )G s has the following properties:

(a) ( ) , 0tmg t g e t for some constants 0mg and 0

(b) ( ) (0 ) ( ) , 0tmdg t g t g e t for some constants 0mdg , where

( )g t represents the induced norm of the matrix ( )

m m

g t R

.



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Lemma I.4 For 1,p (note that p includes 1 and ), if 1g L and pu L ,then

1

*p p p

y g u g u (I.28)

Proof:

1,p , let q be1 1

1p q

. Then,

0

1 1

0 0

( ) * ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

t

t tp q

y t g u t g t u d

g t u d g t u g t d

Applying Holders inequality (Lemma I.2):11

0 0

11

1 0

( ) ( ) ( ) ( )

( ) ( )

q qt tppq

t ppq

y t g t u d g t d

g g t u d

Thus, 0,T



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1

1 0 0

1 0 0

1 0

1 0

1 0 0

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

pp T t pp p

qT p

p T t pq

p T T pq

p T T pq

p T pq

g

y g g t u d dt

g g t u d dt

g g t u dt d

g g t dt u d

g g t dt u d

1 10( )

p q T p p pq

pg u d g u

So,

1 1

p p p

p p p py g u y g u

which proves the lemma for 1,p . With the same change of order of theintegration as above, the case for 1p can be proved easily. Forp we can take

u out of the integral and the result follows. Q.E.D.



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THEOREM I.1 [Page 59 of REF2]

Let ( ) m mG s R be a strictly proper rational transfer function matrix of s.

(a) The system is stable (i.e. G(s) is analytic in Re[ ] 0s or has all poles in LHP)iff 1( )g t L .

(b)Suppose that the system is asymptotically stable (or exponentially stable), then,

(i) 1 1m m mu L y L L , 1my L , y is absolutely continuous and

( ) 0y t as t ;

(ii) 2 2m m m

u L y L L , 2m

y L , y is continuous, and ( ) 0y t as

t ;(iii)

m mu L y L ,my L , y is uniformly continuous;

(iv)mu L and if ( )u t u as t , then, in addition to (iii),

( ) (0)y t G u as t , and the convergence is exponential;

(v) For (1, )p , ,m mp pu L y y L .



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Sketch of Proof:

As G(s) is strictly proper rational TF, 1( ) and ( )g t g t L . Thus, from Lemma I.4,

[1, ],p m m

p pu L y L ,

and

m

py L .

As 1m

y L , ( ) ( )y t y . Since 1m

y L , ( ) 0y . Also, the fact that 1m

u L andmg L leads to

my L . Thus, (i) is true as the absolutely continuous of ( )y t can beproved by definition.

The fact that 2,m

y y L means that 1T

y y L .Noting2 2

02 ( ) (0)

t Ty ydt y t y ,

1Ty y L leads to the results that

2( ) my t L and ( )y t is continuous and converges

as t . Since 2m

y L , it must be ( ) 0 asy t t , which proves (ii).

THEOREM I.2

If G(s) is strictly proper and stable, then, the LTI system is finite gain pL input-

output stable for [1, ]p with1

( )S g . #

Proof:Follows directly from Theorem I.1 (a) and Lemma I.4. Q.E.D.



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We introduce two norms forG(s):

-Norm

( ) : sup ( )G s G j

(I.25)

2-Norm1

22

2 1( ) ( )2G s G j d

(I.26)

Note that ifG(s) is stable, then by Parsevals theorem,

1 1

2 22 2

2 20

1( ) ( ) ( )

2G s G j d g t dt g

THEOREM I.3

Assume that G(s) is stable and strictly proper. Then, its typical input and output

relationship can be summarized by the following two tables.



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Table I.1: Output Norms and Pow For Two Typical Inputs

( ) ( )u t t ( ) sin( )u t t

2y

2( )G s

y

g

( )G j

pow( )y 0 1

( )2

G j

Table I.2: System Gains ( )S

2u u

pow( )u

2y ( )G s

y

2

( )G s 1

g

pow( )y 0 ( )G s

( )G s



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2.3 The Notion of Phase

The phase angle between two vectors 1v and 2v innR space is

1 21 2

1 2 1 22 2 2 2

cosT v vv v

v v v v (I.29)

Similarly, we can define an inner product on the signal space 2mL as

1 2 1 20( ) ( )Tx x x t x t dt

1 2 2, mx x L (I.30)

It is obvious that the induced norm by the inner product is the 2L -norm defined early.

The phase ( )u of the system Sfor a given input u may thus be defined as

2 2 2 2

cos ( )y u Su u

uu y u Su

(I.31)

Note that unlike the gain ( )S , the above definition of phase depends on the input

signal.



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2.4 Passivity

The notion of passivity is introduced as an abstract formulation of the idea of energy

dissipation. It is observed that a physical system consisting of passive elements only

(e.g. resistors, capacitors, and inductors in electrical systems, mass, spring, and

damper in mechanical systems) can only store and dissipate energy. As a result, the

output of the system to any input will follow (or conform to) the input since the

system does not have active energy to fight against the input. This phenomenon ismathematically captured by the requirement that the inner product of the input signal u

and the output signaly is non-negative, i.e.,

0y u 2eu L (I.32)

which is equivalent to saying that the phase ( )u in (I.31) is between ,2 2

, i.e.,

( )2 2

u 2eu L (I.33)

The notion of passivity can thus be formally defined as follows


D fi iti [P 154 f REF1] [P i it S t ]


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Definition [Page 154 of REF1] [Passivity Systems]

A system with input u and outputy ispassive if there exists a constant 0 independent of the control input u(t), t>0, such that

y u 2eu L (I.34)

In addition, the system is input strictly passive (ISP) if there exists 0 independent ofu such that

2

2y u u 2eu L (I.35)

andoutput strictly passive (OSP) if there exists 0 independent ofu such that2

2y u y 2eu L (I.36)

Note:

In the above definition, a bias term is added to the inequality (I.32) to

represent the effect of non-zero initial conditions (or non-zero initial energy).

Note:

The passivity condition (I.34) is equivalent to

T Ty u 0T 2eu L or


T


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0( ) ( )

Ty t u t dt 0T 2eu L

Similarly, (I.35) and (I.36) are equivalent to2

2T T Ty u u 0T 2eu L and

2

2T T Ty u y 0T 2eu L

respectively since the physical system is causal.

Example I.1:A Mass System is Passive but neither ISP nor OSP

Consider a mechanical system consisting of a pure mass m shown below

The input to the system is the applied force Fand the output is the velocity v of the

mass. Then, 2eu L

2 2 2 2

0 0

1 1 1 1( ) ( ) ( ) ( ) ( ) (0) (0)

02 2 2 2

T T

T T

Ty u v t F t dt v t mv t dt mv mv T mv mv

which indicates that the system is passive. To see that the system is neither ISP nor

OSP, choose

1

ms

u=F y=v


( ) ( ) sinu t F t t


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( ) ( ) sinu t F t t

0

1 1( ) ( ) (0) ( ) (0) (1 cos )

ty t v t v F d v t

m m

From (I.40), 0T , T Ty u is bounded. Since2 2

2 2T Tu F and

2 2

2 2T Ty v asT , there does not exist an 0 such that (I.35) or(I.36) is satisfied. Thus, the system is neither ISP nor OSP. The physical explanation

is that an inertia element only stores energy but does not dissipate energy.

Example I.2:A Damper is Passive, ISP and OSP

A damper is described by:

( ) ( ) ( ) ( )y t F t bv t bu t Thus

2 22

2 20 0

1( ) ( ) ( )

T T

T T T T y u y t u t dt b u t dt b u y

b

which indicates that the system is ISP and OSP. The physical explanation is that a

damper dissipates energy.

Lemma I.5 [Passivity & Lyapunov Formulation]

Consider a system with input u and output y. Suppose that there exists a positivesemi-definite function ( , ) 0V x t such that

u=v(t) y=F(t)

b



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( ) ( ) ( )TV y t u t g t (I.41)

Then, the system is passive if 0g . In addition, the system is

(i) ISP if2

1 2g u , 2eu L , for some 0

(ii) OSP if2

1 2g y , u for some 0 .

Note:

(I.41) is an abstraction of the energy-conservation equation of the form

Stored Energy (kinetic potential)

External Power Input Internal Power Generation

d

dt

Thus, the system with the form (I.41) is normally referred to as a system in a powerform.


Proof:


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Proof:

From (I.41):

0 0

0

( ) ( ) (0) ( )

(0) ( )

T T

T T

T

y u V g t dt V T V g t dt

V g t dt

Thus, if ( ) 0g t , the system is passive. If2

1 2

g u , then the system is ISP.

If2

1 2g y , then the system is OSP. Q.E.D.



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2.5 Input-Output Stability

From an I/O point of view, a system is stable if a bounded input will result in a

bounded output, which is described by the following formal definition:

Definition [Page 197 of REF7] [Input-OutputL Stability]

A system :m qe eS L L is I/OL stable if there exists a class Kfunction , defined

on [0, ) , and a non-negative constant such that

T LT LSu u 0T meu L (I.44)

whereL

represents one of the norms defined for a function or a signal.

It is finite-gainL stable if there exist non-negative constants and independent

ofu such that

T LT LSu u 0T meu L (I.45)



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3.I/O Stability Theorems

Having defined the notions of BIBO stability and passivity, some useful criteria can

be developed to judge the I/O stability and passivity of a complex system based on the

I/O stability and passivity of its subsystems. Specially, consider the following

feedback system Swhich is an inter-connection of two subsystems 1Sand 2S :

Fig. I.1 Feedback Connection of 1Sand 2S

We would like to judge the I/O stability ofSbased on the I/O stability of 1S and 2S .

- 1Seu

22

S + n2u


THEOREM I.4 [The Small Gain Theorem (SGT)]


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THEOREM I.4 [The Small Gain Theorem (SGT)]

Consider the system shown in Fig. I.1. Suppose that 1S and 2S are finite-gain pL

I/O stable, i.e., there exist1

,2

0 and constants1

and2

such that 0T ,

1 1 1T Tp pT py S e e pee L

2 2 2 2 2 2T Tp pT py S u u 2 peu L (I.46)

if

1 2 1 (I.47)

then the closed-loop system is finite-gain pL stable with the gain from the input u

to the output y less than 1 1 21 . In fact,

1 1 2 1 2 11 2

1

1T T Tp p p

y u n

(I.48)

Note:

(a) The small gain theorem (SGT) is useful when we analyze the robustness of acontrol design to unmodeled dynamics; the stable nominal closed-loop system can

be considered as the subsystem1

S and the unmodeled dynamics can be

represented by the subsystem 2S .



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(b) The SGT is in general quite conservative since the condition (I.47) is quitestringent. For example, suppose that 1Sand 2S are stable LTI systems with TF

1( )G s and 2 ( )G s respectively. Then, their 2L gains are 1 1( )G s and2 2 ( )G s respectively. In order to satisfy the condition (I.43), it is necessary

that the Nyquist plot of the open-loop TF 1 2( ) ( )G s G s lies within the unit circle.

This is quite conservative compare to the Nyquist Stability Criterion which only

requires that the Nyquist plot does not encircle the 1,0 point.

Proof:

Note that2e u y 2T T Te u y

2u y n 2T T Tu y n (I.49)Substituting (I.49) into (I.46), we have

1 2 1

1 1 2 2 2 1

1 1 2 1 2 1

T T T

T T p

T T T

y u y

u u

u y n

(I.50)

which leads to (I.48) if (I.47) is satisfied. Q.E.D.


3.2. Passivity Theorems


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y

3.2.1 General Passivity Theorems

Lemma I.6 [Parallel Connection]Consider the parallel connection of the subsystem 1S and 2S shown below

System S

Fig. I.2 Parallel Connection of 1S and 2S

Assume that 1Sand 2S are passive. Then, the resulting system S is passive. In

addition,

(i) if any one of 1S and 2S is ISP, then S is ISP.

(ii) if both1S

and2S

are OSP, then S is OSP. Proof:

1S

2S

u y

2y

1y

+


Noting that


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g

1 2T T T T T T y u y u y u the proof of the lemma is straightforward. Q.E.D.

Lemma I.7 [Feedback Connection]

Consider the feedback connection of the subsystem1

S and2

S in Fig. I.1 with n=0.

If 1S and 2S are passive, then S is passive. In addition,

(i) if 1S is OSP or 2S is ISP, then S is OSP.

(ii) if 1S is ISP and 2S is OSP, then S is ISP

Note:

If the two subsystems 1S and 2S are in the power form (I.41) with the associated

energy function and the dissipative function being 1 1,V g and 2 2,V g respectively, then Sin either the parallel connection in Fig. I.2 or the feedback

connection Fig. I.1 can be described in the power form with 1 2V V V and

1 2g g g .


Proof:


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Noting that

2 2T T T T T T T T T y u y e y y e y y (I.51)the proof of the passivity ofSand OSP ofS in (i) are obvious. The following is to

prove (ii).

Assume that 1S is ISP and 2S is OSP. Then, there exist 1 0 , 2 0 , 1 , and

2 such that2

1 12T T Ty e e

2

2 2 2 22T T Ty y y (I.52)

Let 1 2min , , and 1 2 0 . From (I.51),

22 2

2 22 2 2 2

2 2

2 2 2

2

2 2

T T T T T T

T T T

y u e y e y

e y u

This shows that Sis ISP. Q.E.D.


THEOREM I.5 [Passivity Theorem]


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If a system S is OSP, then, the system S is I/O finite-gain 2L stable.

Proof:

Since Sis OSP, there exists 0 and 0 such that

22T T T T T

y u Su u y 0T (I.53)

Noting2 2T T T T

y u y u , we have

2

2 2 20T T Ty y u (I.54)

which leads to

2

2 2 22

4

2

T T T

T

u u uy

u (I.55)

which proves the theorem. Q.E.D.


3 2 2 Passivity of LTI Systems


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3.2.2 Passivity of LTI Systems

An important practical feature of the passivity formulation is that it is easy to

characterize passive LTI systems in terms of the positive realness of its TF. Thisallows linear blocks to be straightforwardly incorporated or added in a nonlinear

control problem formulated in terms of passive maps

Definition [Positive Real (PR) and Strictly Positive Real (SPR)]

A rational transfer function G(s) with real coefficients is positive real (PR) if

Re ( ) 0G s Re( ) 0s

and is strictly positive real (SPR) if ( )G s is positive real for some positive.

THEOREM I.6 [Page 127 of REF3] [Conditions For PR]

A rational TF G(s) with real coefficients is PR iff

(i) G(s) has no poles in the right half plane (RHP).(ii) If G(s) has poles on thej-axis, they are simple poles with positive residues.

(iii) For all for which s j is not a pole of G(s), one has Re ( ) 0G j


THEOREM I.7 [Page 127 of REF3] [Condition for SPR]


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Let( )

( )

( )

N sG s

D s

be a rational TF with real coefficients andn be the relative

degree of G(s) (i.e., deg ( ) deg ( )n D s N s ) with 1n . Then G(s) is SPR

iff

(i) G(s) has no poles in the closed RHP (i.e., all poles are in LHP only).

(ii) Re ( ) 0G j , , .

(iii) (a) When 1n , 2lim Re ( ) 0G j .

(b) When 1n ,( )

lim 0G j

j

.


Example:


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(1)1

( )G s

s

, ors is PR but not SPR.

(2)1

( )G ss

, ors , 0 is SPR.

(3)3

( )( 1)( 2)

sG s

s s

is PR but not SPR since it violates (iii) (a) of

Theorem I.7. In fact, you can verify that

2

2 2

6Re ( ) 0

2 9G j

,

but 0 ,

2 2

2 22 2 2

3 2 3

Re ( ) 02 3 3 2

G j

,

for large . Thus 0 , ( )G s will not be PR, andG(s) is not SPR.


Corollary I.1

1


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(i) G(s) is PR (SPR) iff 1( )G s

is PR (SPR).

(ii) If G(s) is PR, then, the Nyquist plot of ( )G j lies entirely in the closed RHP.

Equivalently, the phase shift of the system in response to sinusoidal inputs is

always within 90o .

(iii) If G(s) is PR, the zeros and poles of G(s) lie in closed LHP. If G(s) is SPR, thezeros and poles of G(s) lie in LHP (i.e., G(s) is asymptotic stable and strictly

minimum phase).

(iv) If 1n , then G(s) is not PR. In other words, if G(s) is PR, then 1n .(v) (Parallel Connection) If 1( )G s and 2 ( )G s are PR (SPR), so is

1 2

( ) ( ) ( )G s G s G s .

(vi) (Feedback Connection) If 1( )G s and 2( )G s are PR (SPR), so is1

1 2

( )( )

1 ( ) ( )

G sG s

G s G s

.


Example:


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2

1( )

sG s

s as b

is not PR.

2

1( )

sG s

s s b

is not PR.

2

1( )G s

s as b

is not PR.

2

1

( )s

G s s as b

is SPR for 1, 0a b , as

2

22 2 2

( 1)Re ( ) 0,

b aG j

b a

.


THEOREM I.8 [PR and Passivity]


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Assume that ( )G s is stable (i.e., all poles in LHP). Then,

(i) The system G(s) is passive iff G(s) is PR.

(ii) The system G(s) is ISP iff there exists 0 such that

Re ( ) 0G j , (I.57)

(iii) The system G(s) is OSP iff there exists 0 such that

2

Re ( ) ( )G j G j , (I.58)

Proof:

Since G(s) is stable, it follows from Parsevals theorem that


0

1( ) ( ) ( ) ( )

2y u y t u t dt Y j U j d


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0

2 2

0

( ) ( ) ( ) ( )2

1

( ) ( ) ( )2

1 1( ) ( ) Re ( ) ( )

2

y y j j

G j U j U j d

G j U j d G j U j d

(I.59)

where YandU are Laplace transforms ofy andu, respectively.

(i) Assume that G(s) is PR, then, Re ( ) 0,G j . From (I.59), it is clear that

0y u and the system is passive.

Assume that G(s) is passive, then 0 independent ofu, s.t.

,y u u (TI.8.1)

Suppose G(s) is not PR, i.e., 0 such that 0Re ( ) 0G j .By continuity, 1 2 s.t. 0 1 2, and Re ( ) 0G j c for some

0c and 1 2, . Let a control input 1( ) ( )u t L U j

be


1 2 2 1, ,( )

kU j


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( )0 else

U j

Then, for this input, the output satisfies (I.59):

2 2

1 1

22

22 1

1Re ( ) Re ( )

( )

ky u G j k d G j d

k c

Thus, ifkis chosen such that

2

2 1( )k

c

then y u and a contradiction is obtained. This proves that G(s) is PR.

(ii)Notice that G(s) is ISP iff 0 and 0 , s.t.2 2

2 0

1( ) ,y u u U j d u

From (I.59), the above is equivalent to

20

1Re ( ) ( ) ,G j U j d u

Following the same argument as in the proof (i), the above is true iff Re ( )G j


which proves the theorem.


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(iii)The same proof as in (ii) by noting

2 2 2 2

2 0 0

1 1( ) ( ) ( )y Y j d G j U j d

#


Example:

2 2

21 1s


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2

2 22

2 22

2 2

2 22

22

22

1 1( ) , ( ) 1 2

( 1) 1

1 1( ) 1 4

1

1

1Re ( ) 0

1

sG s G j j

s

G j

G j

G(s) is PR but not SPR (why?)since there are a pair of zeros on jaxis at 1,2z j . Obviously, for 1 ,

Re ( ) 0G j and thus by Theorem I.8, G(s) is not ISP.

Since

2

Re ( ) ( )G j G j , from (iii) of Theorem I.8, G(s) is OSP.

The following example shows that in general there is no direct connection between

SPR and ISP (or OSP).


Example:


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1( ) 2G s s is SPR, and it is also ISP, but not OSP.

2

1( )

s 2G s

is SPR, and it is OSP, but not ISP.

2

3 21( )1

sG ss

is PR, but not SPR, and it is OSP but not ISP.

However, if G(s) is proper, then G(s) being SPR implies OSP. The proof is omitted.


The following theorems make the connections between PR concepts and Lyapunov

stability formulation in state space.


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y p

THEOREM I.9 [PR and Lyapunov -- Kalman-Yakubovich (KY) Lemma]

[Page 240 of REF7]

Let1( ) ( )G s C sI A B D be a p p transfer matrix where (A,B) is

controllable and (A,C) is observable. Then, G(s) is positive real iff there exist

matrices 0TP P , , andL Wsuch thatT T

A P PA L L (I.60)T T

B P C W L (I.61)T T

W W D D (I.62)

THEOREM I.10 [SPR & Lyapunov--Kalman-Yakubovich-Popov (KYP) Lemma]

[Page 240 of REF7]

Let1( ) ( )G s C sI A B D be a p p transfer matrix where (A,B) is

controllable and (A,C) is observable. Then, G(s) is strictly positive real iff there

exist matrices 0TP P , ,L andW, and a positive constant such thatT T

A P PA L L P (I.63)T T

B P C W L (I.64)T T

W W D D (I.65)


In many applications of SPR concepts to adaptive systems, the TF G(s) involves stable

pole-zero cancellations, which implies that the system associated with (A,B,C) is


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p p y

uncontrollable or unobservable, and the above KY or KYP lemma cannot be applied

since they require the controllability of the pair(A,B). In these situations, the following

MKY lemma should be used instead.

THOREM I.11 [Page 129 of REF3] [Meyer-Kalman-Yakubovich (MKY) Lemma]

Given a stable matrix A, vectors B, C and a scalar 0d . If1( ) ( )G s d C sI A B

is SPR, then, for any given s.p.d. matrix L>0, there exists a scalar 0 , a vector qand an s.p.d. matrix P such that

T TA P PA qq L

and

2T TB P C d q (I.65)

Note:

The above PR and SPR concepts for a TF G(s) can be extended to MIMO systems

with a TF matrix G(s). For details, see [REF7].


References


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[REF1] Shankar Sastry (1999),Nonlinear Systems: Analysis, Stability, and Control,

Springer-Verlag, New York, Inc.

[REF2] C. A. Desoer and M. Vidyasagar (1975), Feedback Systems: Input-Output

Properties, Academic Press, New York, Inc.

[REF3] Ioannou, P.A., and Sun, Jing (1996), Robust Adaptive Control, Prentice-Hall.

[REF4] Slotine, J.J.E. and Li, Weiping (1991),Applied Nonlinear Control, Prentice-

Hall.

[REF5] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum (1992), Feedback Control

Theory, Macmillan Publishing Co.

[REF6] AstromK. J. and Wittenmark B. (1995) Adaptive Control, Second edition,

Addison-Wesley.

[REF7] Khalil, H. K. (2002), Nonlinear Systems, Third edition, Prentice-Hall.

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