Input-Output (I/O) Stability

I/O Stability ME689 Lecture Notes  by B.Yao

-Stability of a System

Outline:

¾ Introduction

• White Boxes and Black Boxes • Input-Output Description

¾ Formalization of the Input-Output View

• Signals and Signal Spaces • The Notions of Gain and Phase • The Notion of Passivity • The BIBO Stability Concept

¾ I/O Stability Theorems

• The Small Gain Theorem • The Passivity Theorem • Positive Real Functions and Kalman-Yakubovich Lemma For LTI System

1 I/O Stability ME689 Lecture Notes  by B.Yao

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 takes the 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 View

2.1 Signals and Signal Spaces

2.1.1 Single-Input Single-Output (SISO) Systems

In SISO systems, the input and output signals can be described by real valued time functions, e.g., an input signal is denoted by a time-function u in R, i.e., u =Œ{}ut(), 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.

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 or L• -norm u suput ( ) (I.1) • = t≥0 Application examples of using the super norm are the problem of finding maximum absolute tracking error and the problem of control saturation.

• L2 -norm 1 • 2 u = Ê utdt2 () ˆ (I.2) 2 Ë¯Ú0

If we are interested in the energy of a signal, L2 -norm is a suitable norm to choose.

• Lp -norm 1 • pp u = Ê ut() dtˆ , 1 £<•p (I.3) p Ë¯Ú0

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 a finite L• norm. LxxM=£: for some M > 0 (I.4) • {}•

2 I/O Stability ME689 Lecture Notes  by B.Yao

• Lp -space

the set of all functions with a finite Lp -norm., i.e., Ï¸1 ÔÔ• p p LxxM=£: or Ê utdtM ( )ˆ £ for some M > 0 (I.5) p Ì˝p ËÚ0 ¯ Ó˛ÔÔ

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

meaningful signals. For example, the L2 -space does not include the common sinusoid signals (why?). To be practically meaningful, the following extended space is defined to avoid this restrictiveness:

Definition [ Extended Space]

If X is a normed linear subspace of Y, then the extended space X e is the set

XxYxXeT=Œ{}: Œ for any fixed T ≥ 0 , (I.6)

where xT represents the truncation of x at T defined as Ïx(tt ) 0 ££T xtT ()= Ì (I.7) Ó0 tT>

The extended L1 space is denoted as L1e , the extended L2 space as L2e , " .

Example: u(t )=ﬁŒœ sinw t uL but uL 22e y(tt )=ﬁŒœ yL••e but yL

2.1.2 Multi-Input Multi-Output (MIMO) Systems

Multi inputs and multi 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., u :0[ •Æ] Rm with

È˘ut1() u()tRÍ˙# m (I.8) =ŒÍ˙ Î˚Í˙utm ()

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

• The super norm

u suput ( ) (I.9) • = t≥0 where u()t represents the norm of ut()in Rm space1.

1 ∞ m = The superior -norm of u(t) in R is u(t) ∞ max ui (t) . i

3 I/O Stability ME689 Lecture Notes  by B.Yao

• The L2 -norm

••2 u ==ut() dt uT () tutd () t (I.10) 22ÚÚ00

• The Lp -norm 1 • pp u = Ê ut() dtˆ , 1 £<•p (I.11) p Ë¯Ú0 where u()t represents the norm of ut()in Rm space2.

m m m The corresponding norm spaces are represented by L• , L2 and Lp respectively. The extended m m m spaces are L•e , L2e and Lpe respectively.

2.1.3 Useful Facts

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

Lemma I.1 [Ref.1] (Holder’s Inequality) 11 If pq,1,Œ•, and +=1, then, fLŒ , g ŒL imply that fg ŒL , and [ ] pq p q 1

fg1 £ fpq g (I.12) i.e., 11 •••pqpq f() t g () t dt£ ÊÊ f () t dtˆˆ g () t dt (I.13) ÚÚÚ000ËË¯¯ Note: When pq==2 , the Holder’s inequality becomes the Cauchy-Schwartz inequality, i.e., 11 •••2222 f() t g () t dt£ ÊÊ f () t dtˆˆ g () t dt (I.14) ÚÚÚ000ËË¯¯

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

Lemma I.2 11 If pq,1,Œ•, and +=1, then, fLŒ , g ŒL imply that fgLŒ , and [ ] pq pe qe 1e fg£ f g , "≥T 0 (I.15) ( )T 1 TTpq or 11 TTTpqpq ftgtdt() ()£ ÊÊ ft () dtˆˆ gt () dt "≥T 0 (I.16) ÚÚÚ000ËË¯¯

1 m  p  p 2 The p-norm of u(t) in R m is u(t) =  u (t)  . p ∑ i  i=1 

4 I/O Stability ME689 Lecture Notes  by B.Yao

Lemma I.3 [Ref.1] (Minkowski Inequality)

For p Œ•[1, ] , fg, Œ Lp imply that fgL+Œp , and

fg+£ppp f + g (I.17)

Note:

(a) A Banach space is a normed space which is complete in the metric defined by its norm; this

means that every Cauchy sequence is required to converge. It can be shown that Lp -space is a Banach space.

(b) Power Signals: [Ref.5] The average power of a signal u over a time span T is 1 T u2 ()t dt (I.18) T Ú0 The signal u will be called a power signal if the limit of (I.18) exists as T Æ•, and then the square root of the average power will be denoted by pow(u ) : 1 2 Êˆ1 T 2 pow(u )= Á˜ limutdt ( ) (I.19) Ë¯TÆ• T Ú0

pow L2

L∞

Note:

L12«ÃLL•

Example: (verify by yourself)

1 ft()= ﬁ f ()tLLŒ«2 , but f ()t œL1 , f ()tLŒ 1e 1+ t •

Ï0 if t £ 0 Ô Ô 1 ft( )=<<Ì 0 t 1 ﬁ f ()t ŒL1 , but f ()tLœ • , f ()tLœ 2 Ô t ÔÓ0 t > 1

5 I/O Stability ME689 Lecture Notes  by B.Yao

2.2 Gain of A System

2.2.1 Definitions

From an I/O point of view, a system is simply an operator S such that maps a input signal u into an output signal y. Thus, it can be represented by

S : XXieÆ oe with y = Su (I.20)

where the extended norm space X ie represents the input signal space (i.e., u Œ X ie ) and X oe 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 [Ref.2] [Gain of A Linear System] The gain g ()S of a linear system S is defined as Su g ()S = sup (I.21)

uXŒ e u uπ0 where u is the input signal to the system. In other words, the gain g ()S is the smallest value such that

SuS£ g ()u , "Œu X e (I.22)

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

Definition [Gain of A Nonlinear System] The gain g ()S of a nonlinear system S is defined to be the smallest value such that

Su£+g () S u b , "Œu X e (I.23) for some non-negative constant b .

Definition [ Finite Gain Input-Output Stability] A nonlinear system S is called finite-gain Lp input-output stable if the gain g ()S defined in

(I.23) is bounded (or finite), in which the Lp norm is used for input and output signals.

Note:

When p=∞, the above finite gain Lp stability, i.e., L∞ stability, results in bounded-input bounded-output (BIBO) stability. The converse is in general not true. For example, the static mapping y=u2 is BIBO stable but does not have a finite gain.

2.2.2 Linear Time Invariant Systems

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

Y ()sGsU= () ()s in s-domain t y()tgutgtud==- ( * )() (ttt ) ( ) in time-domain (I.24) Ú0 where g()tLGs= -1 {} () is the unit impulse response.

6 I/O Stability ME689 Lecture Notes  by B.Yao

Theorem I.1 [Ref.1] Let G(s) be a strictly proper rational function of s. (a) The system is stable (i.e., G(s) is analytic in Re[s ]≥ 0 or has all poles in LHP) if and only if

gLŒ 1 . (b) Suppose that the system is asymptotically stable, then, (i) g decays exponentially, i.e., g()tMe£ -at for some M ,0a > .

(ii) u ŒﬁŒ«LyLL11• , yL Œ 1 , y is continuous except on a set of measure zero, and yt( ) Æ 0 as t Æ•.

(iii) u ŒﬁŒ«LyLL22• , y ŒL2 , y is continuous except on a set of measure zero, and yt( ) Æ 0 as t Æ•.

(iv) For p Œ•[1, ] , u ŒﬁLyyLpp, Œ and y is continuos except on a set of measure zero. ∆ We introduce two norms for G(s):

• -Norm G(sGj):sup( ) (I.25) • = w w 2-Norm 1 Êˆ1 • 2 2 Gs()= Á˜ G ( jww ) d (I.26) 2 Ë¯2p Ú-• Note that if G(s) is stable, then by Parseval’s theorem, 1 1 Êˆ1 ••222 Ê ˆ 2 G()sGjdgtdtg===Á˜ (ww ) () (I.27) 22Ë¯2p ÚÚ-• Ë 0 ¯

Theorem I.2 Assume that G(s) is stable and strictly proper. Then, its typical input and output relationship can be summarized by the following two tables.

Table I.1: Output Norms and pow For Two Typical Inputs u()t = d ()t u(t)sin(= wt) • y 2 G()s 2 y g G()j • • ª w pow( y) 0 1 G()jw 2

Table I.2: System Gains g ()S u u pow(u) 2 • y G()s • • 2 • y G()s g • • 2 1 pow( y) 0 G()s G()s £ • •

7 I/O Stability ME689 Lecture Notes  by B.Yao

Lemma I.4 [Ref.1] For p Œ•[1, ] , if gLŒ 1 and u ŒLp , then

y p =£gu* p g1 up (I.28)

Proof: By causality, gt( -=t ) 0 , ">t t . Thus, "Œp [1, •) , t • ytgutgtud()==-=- *() (ttt )() gtud ( ttt )() ÚÚ00 ••11 £-g (tudttt ) ( ) =- gtugtd ( t )pq ( t ) ( - t ) t ÚÚ00

ŒŒLLpq 11 where +=1. Note that the first term is in L and the second term is in L . Hence, by pq p q Holder’s inequality 1 1 ••p p È q ˘ q yt()£-È˘ gt (ttt ) u ( ) d gt ( - t ) q d t Î˚Í˙ÚÚ00Í ˙ Î ˚ 1 1 • p p £-ggtudq È˘ (ttt ) ( ) 1 Î˚Í˙Ú0 Thus, 1 Ï¸p p 1 ÔÔ••È˘p p yt()£- gq Ì˝ gt (ttt ) u ( ) d dt p 1 ÚÚ00Î˚Í˙ Ó˛ÔÔ Using Tonelli’s theorem to change the order of integration, we obtain 1 1 •• p p yt()£- gq È˘ gt (ttt ) dtu ( ) d p 1 ÚÚ00Í˙ { Î˚} 1 111• p p ££=g qqpgu (tt ) d g g u g u 11{}Ú0 111pp which proves the lemma for p 1, . For p , we can take u out of the integral and Œ•[ ) =• • the result follows. Q.E.D.

Theorem I.3 If G(s) is strictly proper and stable, then, the LTI system is finite gain Lp input-output

stable for p Œ•[1, ] with g ()s £ g 1 . #

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

8 I/O Stability ME689 Lecture Notes  by B.Yao

2.3 The Notion of Phase ϕ n The phase angle between two vectors v1 and v2 in R space is vvT vv cosj ==12 12 (I.29) v vvv 1222 12 22

m Similarly, we can define an inner product on the signal space L2 as • x xxtxt= T () ()dt "Œx , xLm (I.30) 12Ú0 1 2 12 2

It is obvious that the induced norm by the inner product is the L2 -norm defined early. The phase j ()u of the system S for a given input u can thus be defined as y uSuu cosj (u ) == (I.31) u yuSu 22 2 2 Note that unlike the gain g ()S , the phase j depends on the input signal.

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 is mathematically captured by the requirement that the inner product of the input signal u and the output signal y is non-negative, i.e.,

yu ≥ 0 "ŒuL2e (I.32) È pp˘ which is equivalent to saying that the phasej ()u in (I.31) is between Í- , ˙ , i.e., Î 22˚ pp -£j()u £ "ŒuL (I.33) 22 2e The notion of passivity can thus be formally defined as follows

Definition [Passivity Systems]

A system with input u and output y is passive iff there exists a constant b ≥ 0 independent of the control input u(t), t>0, such that

yu ≥-b "ŒuL2e (I.34) In addition, the system is input strictly passive (ISP) if there exists e > 0 independent of u such that 2 yu≥-eb u2 "ŒuL2e (I.35) and output strictly passive (OSP) if there exists e > 0 independent of u such that 2 yu≥-eb y2 "ŒuL2e (I.36)

Note that in the above definition, a bias term -b is added to the inequality (I.32) to represent the effect of non-zero initial conditions (or non-zero initial energy).

9 I/O Stability ME689 Lecture Notes  by B.Yao

Note: The passivity condition (I.34) is equivalent to

yuTT≥-b "≥T 0 "ŒuL2e or T ytutdt() () ≥-b "≥T 0 "ŒuL (I.37) Ú0 2e Similarly, (I.35) and (I.36) are equivalent to yu u2 T 0 uL (I.38) TT≥-eb T2 "≥ "Œ2e and yu y2 T 0 uL (I.39) TT≥-eb T2 "≥ "Œ2e 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

u=F 1 y=v ms

The input to the system is the applied force F and the output is the velocity v of the mass. Then,

"ŒuL2e TT y u==◊ vtFtdt() () vt () mvtdt () TT ÚÚ00 (I.40) 11T 1 1 ==mv22 mv ( T ) - mv 2 (0) ≥- mv 2 (0) 220 2 2 which indicates that the system is passive. To see that the system is neither ISP nor OSP, choose 11t u()tFt== () sint ﬁ y()tvtv==+ () (0) Fd (tt ) =+- v (0) (1cos)t m Ú0 m From (I.40), T 0 , y u is bounded. Since uF22and yv22as "≥ T T TT22=Æ•TT22=Æ• T Æ•, there does not exist an e > 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 only stores energy but does not dissipate energy.

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

A damper is described by:

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

Thus TT221 y u==== ytutdt() () b u2 () tdt bu y TTÚÚ00 T22b T which indicates that the system is ISP and OSP. The physical explanation is that a damper dissipates energy.

10 I/O Stability ME689 Lecture Notes  by B.Yao

Lemma I.5 [Passivity & Lyapunov Formulation] Consider a system with input u and output y. Suppose there exists a positive semi-definite function Vxt(,)≥ 0 such that V =-ytutgT () () ()t (I.41) 2 Then, the system is passive if g ≥ 0 . In addition, the system is ISP if g 12≥ e u , "Œu L2e , for 2 some e > 0 , and OSP if g 12≥ e y , "u for some e > 0 .

Note: (I.41) is an abstraction of the energy-conservation equation of the form d Stored Energy (kinetic+= potential) External Power Input + Internal Power Generation (I.42) dt [][][] Thus, the system with the form (I.41) is normally referred to as a system in a power form.

Proof: From (I.41): TT y uVgtdtVTVgtd=+ () =-+ ( ) (0) () t TT ÚÚ00( ) T ≥-Vgtdt (0) + ( ) N Ú0 b 2 2 Thus, if gt()≥ 0, the system is passive. If g 12≥ e u , then the system is ISP. If g 12≥ e y , then the system is OSP. Q.E.D.

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 [Input-Output L Stability] mq A system S : LeeÆ L is I/O L stable if there exists a class K function a , defined on [0, •] , and a non-negative constant b such that Su£+ab u "≥T 0 "Œu Lm (I.44) ( )T L ( T L ) e

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

It is finite-gain L stable if there exist non-negative constants g and b independent of u such that Su£+g u b "≥T 0 "Œu Lm (I.45) ( )T L T L e

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 S which is an inter-connection of

two subsystems S1 and S2 .

11 I/O Stability ME689 Lecture Notes  by B.Yao

u e y + S 1

y 2 u2 + n S2

Fig. I.1 Feedback Connection of S1 and S2

We would like to judge the I/O stability of S based on the I/O stability of S1 and S2 .

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

Consider the system shown in Fig. I.1. Suppose that S1 and S2 are finite-gain Lp I/O stable,

i.e., there exist g 1 , g 2 ≥ 0 and constants b1 and b2 such that "≥T 0 , ySee=£+g b "Œe L TTpT( 111) p p pe ySuu=£+g b "Œu L (I.46) 222222TTpT( ) p p 2 pe If

g 12g < 1 (I.47)

then the closed-loop system is finite-gain Lp stable with the gain from the input u to the output

y less than g 112(1-gg) . In fact, 1 yun (I.48) TTTppp£+++g 112121gg g b b 1-gg12( )

Note: (a) The small gain theorem (SGT) is useful when we analyze the robustness of a control design to unmodeled dynamics; the stable nominal closed-loop system can be considered as the subsystem

S1 and the unmodeled dynamics can be represented by the subsystem S2 . (b) The SGT is in general quite conservative since the condition (I.47) is quite stringent. For

example, suppose that S1 and S2 are stable LTI systems with TF Gs1()and Gs2 () respectively. Then, their L gains are G ()s and G ()s respectively. In order to satisfy the 2 g 11= • g 22= •

condition (I.43), it is necessary that the Nyquist plot of the open-loop TF G12()sG ()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 that

e =-u y2 ﬁ eTT£+uy2T

u2 =+y n ﬁ u2T £+ynTT (I.49) Substituting (I.49) into (I.46), we have

12 I/O Stability ME689 Lecture Notes  by B.Yao

yuy TT£++gb121( T ) £+gggbbuu ++ (I.50) 112221TT()p

£+g 112uynTTTgg() +++ gb 121 b which leads to (I.48) if (I.47) is satisfied. Q.E.D.

3.2. Passivity Theorems

3.2.1 General Passivity Theorems

Lemma I.6 [ Parallel Connection]

Consider the parallel connection of the subsystem S1 and S2 shown below

S 1 y1

u y +

y2 S2

System S

Fig. I.2 Parallel Connection of S1 and S2

Assume that S1 and S2 are passive. Then, the resulting system S is passive. In addition,

(i) if any one of S1 and S2 is ISP, then S is ISP.

(ii) if any one of S1 and S2 is OSP, then S is OSP.

Proof: Noting that

yT uyuyuTTTTT=+12 the proof of the lemma is straightforward. Q.E.D.

Lemma I.7 [Feedback Connection]

Consider the feedback connection of the subsystem S1 and S2 in Fig. I.1 with n=0. If S1 and S2 are passive, then S is passive. In addition,

(i) if S1 is OSP or S2 is ISP, then S is OSP.

(ii) if S1 is ISP and S2 is OSP, then S is ISP

Proof: Noting that

yT uyeyyeyyTTTTTTTT=+=+22 (I.51) the proof of the passivity of S and OSP of S in (i) are obvious. The following is to prove (ii).

Assume that S1 is ISP and S2 is OSP. Then, there existe1 > 0 ,e2 > 0 , b1 , and b2 such that 2 ye e TT≥-eb11 T2

13 I/O Stability ME689 Lecture Notes  by B.Yao

2 yy y (I.52) 2222TT≥-eb T2

Let eee= min{}12 , , and bbb=+≥120 . From (I.51), 22e 2 yuey≥+-≥+eb ey - b TTTT( 2222) () TT 22 2 ee2 2 ≥+ey -=-bb u 22()TT2 22 T This shows that S is ISP. Q.E.D.

Note: If the two subsystems S1 and S2 are in the power form (I.41) with the associated energy function

and the dissipative function being (V11, g ) and(V22, g ) respectively, then S in either the parallel connection in Fig. I.2 or the feedback connection Fig. I.1 can be described in the power form

with V =+VV12 and g =+gg12.

Theorem I.5 [Passivity Theorem]

If a system S is OSP, then, the system S is I/O L2 stable.

Proof: Since S is OSP, there exists e > 0 and b ≥ 0 such that yu Suu y2 T 0 (I.53) TT=≥-()T Teb T2 "≥ Noting y uyu, we have TT£ T22 T yyu2 0 (I.54) ebTTT222--£ which leads to uu2 4 TT22++eb yT £ "u (I.55) 2 2e This shows that y is bounded if u is bounded, and proves the theorem. Q.E.D. T 2 T 2

3.2.2 Passivity In 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. This allows linear blocks to be straightforwardly incorporated or added in a nonlinear control problem formulated in terms of passive maps

Definition [Ref 1] [Positive Real (PR) and Strictly Positive Real (SPR)] A rational transfer function G(s) with real coefficients is positive real (PR) if Re(Gs ( )) ≥ 0 ">Re(s ) 0 and is strictly positive real (SPR) if Gs()- e is positive real for some positive e .

Theorem I.6 [Ref 1] [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).

14 I/O Stability ME689 Lecture Notes  by B.Yao

(ii) If G(s) has poles on the jw -axis, they are simple poles with positive residues. (iii) For all w for which s = jw is not a pole of G(s), one has Re(Gj (w )) ≥ 0 (I.56)

Theorem I.7 [Ref 1] [Condition For SPR] N()s Let Gs()= be a rational TF with real coefficients and n* be the relative degree of G(s) D()s (i.e., n* =-degDs ( ) deg N (s ) ) with n* £ 1. 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{}Gj (w )> 0 , "Œ-••w [ , ] . (iii) (a) When n* = 1, limww2 Re{}Gj ( )> 0 . w Æ• Gj()w (b) When n* =-1, lim> 0 . w Æ• jw Example: 1 (1) Gs()= , or s is PR but not SPR. s 1 (2) Gs()= , or s + l , l > 0 is SPR. s + l s + 3 (3) Gs()= is PR but not SPR since it violates (iii) (a) of Theorem I.7. (1)(2)ss++ In fact, you can verify that 6 Re{}Gj (w )=>2 0 , "w ()29-+ww22 but ">e 0 , (323--+-eeeew) 22 ReGj (we-= )( ) < 0 , for large w . {} 2 2 ()23-+-ee22 w + w 2() 32 - e Thus ">e 0 , Gs()- e will not be PR, and G(s) is not SPR.

Corollary I.1 [Ref 1] 1 (i) G(s) is PR (SPR) iff is PR (SPR). G()s (ii) If G(s) is PR, then, the Nyquist plot of G()jw 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, the zeros and poles of G(s) lie in LHP (i.e., G(s) is asymptotic stable and strictly minimum phase).

(iv) If n* > 1 , then G(s) is not PR. In other words, if G(s) is PR, then n* £ 1.

(v) (Parallel Connection) If Gs1() and Gs2 ()are PR (SPR), so is Gs()=+ G12 () s G () s.

15 I/O Stability ME689 Lecture Notes  by B.Yao

Gs1() (vi) (Feedback Connection) ) If Gs1() and Gs2 ()are PR (SPR), so is Gs()= . 1()()+ GsGs12

Example: s -1 Gs()= is not PR. s2 ++as b s +1 Gs()= is not PR. s2 -+s b 1 Gs()= is not PR. s2 ++as b s +1 Gs()= is SPR. s2 ++as b

Theorem I.8 [Ref 2] [PR and Passivity] Assume thatGs()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 existse > 0 such that Re(Gj (we )) ≥> 0 , "w (I.57) (iii) The system G(s) is OSP iff there existse > 0 such that 2 Re()(Gjwew) ≥ Gj (), "w (I.58)

Proof: Since G(s) is stable, it follows from Parseval’s theorem that ••1 y uytutdtYjUjd==() () (www )* ( ) ÚÚ0 2p -• 1 • =-Gj (ww ) Uj ( ) U ( j ww ) d (I.59) 2p Ú-• 1 • 2 = Re{}Gj (www ) Uj ( ) d p Ú0 where Y and U are Laplace transforms of y and u, respectively. It is thus straightforward to prove the theorem. The details can be worked out if you would like to know the complete proof. Q.E.D.

The following example shows that in general there is no direct connection between SPR and ISP (or OSP).

Example:

G1()ss=+ 2 is SPR, and it is also ISP, but not OSP. 1 Gs2 ()= is SPR, and it is OSP, but not ISP. s2+ s2 +1 Gs3()= 2 is PR, but not SPR, and it is OSP but not ISP. ()s +1

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

16 I/O Stability ME689 Lecture Notes  by B.Yao

The following theorems make the connections between PR concepts and Lyapunov stability formulation in state space.

Theorem I.9 [Ref2] [SPR and Lyapunov Formulation —Kalman-Yakubovich (KY) Lemma] Let the LTI system x =+Ax Bu (I.60) yCx= be controllable and observable. Then, the TF G()sCsIA=- ( )-1 B is SPR iff there exit symmetric positive definite (s.p.d.) matrices P and Q such that

AT PPA+=-Q (I.61) and BT P = C (I.62)

Theorem I.10 [Ref 1] [PR & Lyapunov —Kalman-Yakubovich-Popov (KYP) Lemma] Given a square matrix A with all eigenvalues in the closed LHP, a vector B such that (A,B) is controllable, a vector C and scalar d ≥ 0 . The TF defined by G()sdCsIA=+ ( - )-1 B (I.63) is PR iff there exist an s.p.d. matrix P and a vector q such that

ATTPPAqq+=- and BTTPC-=±( 2 dq) (I.64)

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 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.

Theorem I.11 [Ref 1] [Meyer-Kalman-Yakubovich (MKY) Lemma] Given a stable matrix A, vectors B, C and a scalar d ≥ 0 . If G()sdCsIA=+ ( - )-1 B is SPR, then, for any given s.p.d. matrix L>0, there exists a scalarg > 0 , a vector q and an s.p.d. matrix P such that ATTPPAqq+=--g L and BTTPC-=±( 2 dq) (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 [Ref 1].

17 I/O Stability ME689 Lecture Notes  by B.Yao

References [Ref1] Ioannou P. A. and Sun Jing (1996), Robust Adaptive Control, Prentice-Hall. [Ref2] Astrom K. J. and Wittenmark B. (1995) Adaptive Control, Second edition, Addison-Wesley. [Ref3] Slotine, J.J.E. and Li, Weiping (1991), Applied Nonlinear Control, Prentice-Hall. [Ref4] Khalil, H. K. (1996), Nonlinear Systems, Prentice-Hall.