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CHAPTER 6 Stability Concepts for Input-Output Systems The Lyapunov stability concept is a property of the system’s equilibria. It is not a system property in the sense that we can say a nonlinear system is Lyapunov stable or not. For a variety of reasons, this notion of stability does directly not capture what many engineering applications are actually interested in. In particular, if we think of a system as as mapping G : L in ! L out from an input space of signals to an output space of signals, then what many engineering applications want to assure is that in some sense the “input” is rejected at the ”output”. This consideration leads to a number of different input-output stability concepts that include L p stability, input-to-state stability (ISS), and passivity or dissipativity. The objective of this chapter is to review these stability concepts, establish their fundamental properties and inter-relationships. 1. Vanishing Perturbations The most natural way to start a discussion regarding input-output stability is to consider the input as an additive perturbation of a homogeneous system. Rather than think of an external input, we first consider the case when this perturbation is itself a state dependent function that perturbs the original system. There are two cases to consider; vanishing and non-vanishing perturbations. This section examines the vanishing perturbation case. Consider the dynamical system ˙ x = f (t, x)+ g(t, x) Suppose x =0 is exponentially stable for the unperturbed system (i.e. g(t, x)=0) and further suppose that g(t, 0) = 0 for all t 0. We refer to such systems as having a vanishing perturbation since the perturbation vanishes at the equilibrium point x =0. Since the unperturbed system is exponentially stable, we can use the converse theorem to infer the existence of a C 1 function, V : R n ! R, that serves as a Lyapunov function for the unperturbed system. For this V there exist non-negative constants, c 1 , c 2 , c 3 , and c 4 such that c 1 |x| 2 V (t, x) c 2 |x| 2 ˙ V (x) -c 3 |x| 2 @ V @ x c 4 |x| 145
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Page 1: Stability Concepts for Input-Output Systemslemmon/courses/ee580/lectures/chapter6.pdf · CHAPTER 6 Stability Concepts for Input-Output Systems The Lyapunov stability concept is a

CHAPTER 6

Stability Concepts for Input-Output Systems

The Lyapunov stability concept is a property of the system’s equilibria. It is not a system property in thesense that we can say a nonlinear system is Lyapunov stable or not. For a variety of reasons, this notion ofstability does directly not capture what many engineering applications are actually interested in. In particular,if we think of a system as as mapping G : L

in

! Lout

from an input space of signals to an output space ofsignals, then what many engineering applications want to assure is that in some sense the “input” is rejectedat the ”output”. This consideration leads to a number of different input-output stability concepts that includeLp stability, input-to-state stability (ISS), and passivity or dissipativity. The objective of this chapter is toreview these stability concepts, establish their fundamental properties and inter-relationships.

1. Vanishing Perturbations

The most natural way to start a discussion regarding input-output stability is to consider the input as anadditive perturbation of a homogeneous system. Rather than think of an external input, we first considerthe case when this perturbation is itself a state dependent function that perturbs the original system. Thereare two cases to consider; vanishing and non-vanishing perturbations. This section examines the vanishingperturbation case.

Consider the dynamical system

x = f(t, x) + g(t, x)

Suppose x = 0 is exponentially stable for the unperturbed system (i.e. g(t, x) = 0) and further suppose thatg(t, 0) = 0 for all t � 0. We refer to such systems as having a vanishing perturbation since the perturbationvanishes at the equilibrium point x = 0.

Since the unperturbed system is exponentially stable, we can use the converse theorem to infer the existenceof a C1 function, V : Rn ! R, that serves as a Lyapunov function for the unperturbed system. For this V

there exist non-negative constants, c1

, c2

, c3

, and c4

such that

c1

|x|2 V (t, x) c2

|x|2˙V (x) �c

3

|x|2�

@V

@x

c4

|x|

145

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146 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

We further suppose that the perturbation satisfies the bound

|g(t, x)| �|x|

for all time t > 0 where � > 0.

We use V as a candidate Lyapunov function for the perturbed system. The directional derivative of V withrespect to the perturbed system is

˙V =

@V

@t+

@V

@xf(t, x) +

@V

@xg(t, x)

Using the above bounds on V , we infer

˙V �c3

|x|2 +

@V

@x

|g(t, x)|

�c3

|x|2 + c4

�|x|2

If we can ensure � < c3c4

then

˙V (t, x) �(c3

� �c4

)|x|2 < 0

which implies that the additively perturbed system’s equilibrium is also exponentially stable if the “gain” ofthe perturbation is small enough.

As an example consider the system

x = Ax + g(t, x)

where |g(t, x)| �|x| and we assume A is Hurwitz. Since A is Hurwitz there is a unique positive definitematrix P = PT > 0 that satisfies the Lyapunov equations

PA + ATP + Q = 0

for any matrix Q = QT > 0.

We let V (x) = xTPx serve as a candidate Lyapunov function for the perturbed system. We already knowthat

�(P)|x|2 V (x) = xTPx �(P)|x|2

so we select c1

= �(P) and c2

= �(P). The directional derivative of V with respect to the perturbed systemis

˙V =

@V

@xAx +

@V

@xg(t, x)

= xT(PA + ATP)x + xTPg(t, x) + gT (t, x)Px

= �xTQx + xTPg + gTPx

��(Q)|x|2 + 2�(P)�|x|2

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2. NON-VANISHING PERTURBATIONS 147

The first term in the above inequality is negative definite, but the second term is positive definite. Select0 < ✓ < 1 and take part of the negative definite first term and share it with the positive definite term to obtain

˙V �(1 � ✓)�(Q)|x|2 � ✓�(Q) + 2�(P)�|x|2

The first term in the above inequality is still negative definite and note that the last two terms combined arenegative definite if

✓�(Q) � 2�(P)� > 0

This will be true if � satisfies

� <✓�(Q)

2�(P)

(86)

So we can then say that with this restriction on � that

˙V �(1 � ✓)�(Q)

�(P)

V (x)

So by the comparison principle we can conclude that

V (t) V (t0

) exp

�(1 � ✓)�(Q)

�(P)

(t � t0

)

(87)

which establishes that the perturbed system is still exponentially stable provided we can ensure the perturba-tion has a linear bound, �, that satisfies equation (86). Another nice thing about this analysis is that we canbound the rate of decay on V in equation (87). This in turn can be used to bound the rate of decay of x(t), sowe have a rate of “convergence” for the perturbed system.

2. Non-vanishing Perturbations

This section now considers an additively perturbed system in which the perturbation does not vanish at theorigin of the original system. In this case, it is impossible to assure that the unpertubed system’s equilibrium isLyapunov stable. All we can guarantee is that the state remains within a bounded neighborhood of the originalsystem’s equilibrium. This notion of so-called uniform ultimate boundedness or UUB will be important whenwe later begin to study the input-output stability concept known as input-to-state stability.

Consider additively perturbed systems

x(t) = f(t, x) + g(t, x)

where x = 0 is the equilibrium point of the unperturbed system x = f(t, x), but where the additive perturba-tion no longer vanishes at the equilibrium. In other words g(t, 0) 6= 0. In this case the origin is no longer anequilibrium point of the perturbed system. Since there is no “equilibrium” we need to develop an extensionof the Lyapunov stability concepts that describes what it means for the system to be well behaved, i.e. stable.This extension is sometimes called uniform ultimate boundedness or UUB.

Solutions of x = f(t, x) are said to be UUB if there exists c > 0 such that for all a < c there exists a realconstant, b, and T > 0 such that if |x(t

0

)| < a then |x(t)| b for all t � t0

+ T . Fig. 1 provides a graphical

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148 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

interpretation of this concept. Essentially, what this figure shows is that any trajectory that originates ina ball Nc(0) about the equilibrium of the unperturbed system will enter and remain within a closed ballNb(0) within a finite time T . Since this finite time, T , and target neighborhood are independent of the initialtime, the property holds in a uniform manner. In other words, all trajectories are ultimately convergent toa uniformly bounded set about the origin. The size of the target neighborhood Nb(0) is called the ultimatebound, b. The left hand plot shows what such trajectories and neighborhoods look like in a two-dimensionalphase space. The right hand plot shows the time history of various trajectories starting in Nc(0) and reachingthe set Nb(0) by time T .

N (0)c

N (0)b

T

b

c

-c

-b

0

time (t)

FIGURE 1. Uniform Ultimate Boundedness (UUB)

We want to be able to characterize how the size of the ultimate bound, b, varies as a function of the strengthof the non-vanishing perturbation. The following UUB theorem provides this characterization of the ultimatebound.

THEOREM 68. (UUB theorem) Consider the system x = f(t, x) where f : [0, 1) ⇥ D ! Rn is piecewisecontinuous in t and locally Lipschitz in x. Let V : [0, 1) ⇥ D ! R be C1 such that for all t,

↵(|x|) V (t, x) ↵(|x|)˙V �↵(|x|), for all |x| � µ > 0

where µ > 0 and the functions ↵, ↵, and ↵ are class K. Then there exists a class KL function � and a finitetime T > 0 such that solutions for the system x = f(t, x) satisfy

|x(t)| �(|x(t0

)|, t � t0

), for all t0

t < T

x(t) 2 �

x 2 Rn: |x| ↵�1

(↵(µ))

, for all t > T

Proof: This theorem’s proof closely follows the techniques we used in the previous chapter to establish theLyapunov stability of time-varying systems. We already know from the theorem’s assumptions that ˙V isnegative definite for x lying outside of the closed ball Bµ(0) = {x : |x| µ}. For notational conveniencewe let ⌘ = ↵(µ) for the µ in the theorem. We also define the set

⌦t,⌘ = {x : V (t, x) < ⌘}Note that Bµ(0) ⇢ ⌦t,⌘ so that ˙V is also negative definite outside of ⌦t,⌘ .

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2. NON-VANISHING PERTURBATIONS 149

For any state x outside of ⌦t,⌘ , one can see that

˙V �↵(|x|) �↵(↵�1

(V ))

Since ↵ � ↵ is class K, we know from theorem 62 in chapter 5 that there exists a class KL function, �, suchthat

V (t, x(t)) �(V (t0

, x(t0

)), t � t0

)

for all x(t) outside of ⌦t,⌘ . Since � is class KL this means that V (t, x(t)) is decreasing until it eventuallyenter the set ⌦t,⌘ . Let t

0

+ T be the time when this occurs.

From the assumed bounds on V (t, x) and ˙V , however, we can deduce that for t0

t < t0

+ T that

↵(|x|) V (t, x) �(↵(|x(t0

)|), t � t0

)

Applying ↵�1 to both sides of this inequality yields,

|x(t)| ↵�1

(�(↵(|x(t0

)|), t � t0

))

for all t0

t < t0

+ T . From theorem 61 in chapter 5 we can readily deduce that �(r, s) = ↵�1

(�(↵(r), s))

is class KL.

Finally, we know that once x(t) enters ⌦t,⌘ that it remains within that set because ˙V < 0 for all x outside theset. For t > t

0

+ T , we know V (t, x(t)) < ⌘ and again using our assumed bounds on V we obtain

|x| ↵�1

(V (t, x)) ↵�1

(⌘) ↵�1

(↵(µ))

which is the ultimate bound and completes the proof. }

Example: Consider the system

x1

= x2

x2

= �(1 + x2

1

)x1

� x2

+ M cos!t

When M = 0, the origin is asymptotically stable and the Lyapunov function for the unperturbed system is

V (x) = xT

"

3/2 1/2

1/2 1

#

x +

1

2

x4

1

= xTPx +

1

2

x4

1

When M > 0, we apply the UUB theorem 68 using the preceding V (x) as a candidate certificate function.The directional derivative of V is

˙V = �x2

1

� x4

1

� x2

2

+ (x1

+ 2x2

)M cos!t

= �|x|2 � x4

1

+

h

x1

x2

i

"

1

2

#

M cos!t

�|x|2 � x4

1

+ M |x|�

"

1

2

#

�|x|2 � x4

1

+ Mp

5|x|

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150 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

The first two terms above are negative definite. The last term is indefinite. We use the same trick of introduc-ing a constant ✓ 2 (0, 1) to shift part of the first two terms to cover the last term. This leads to

˙V �(1 � ✓)|x|2 � x4

1

� ✓|x|2 + Mp

5|x|The above term implies that

˙V �(1 � ✓)|x|2 � x4

1

< 0, for all |x| > Mp

5

So the inequality in the UUB theorem 68 is satisfied for µ = Mp

5/✓.

The value of the ultimate bound is computed as follows. Note that

V (x) � xTPx � �(P)|x|2

We also can see that

V (x) xTPx +

1

2

|x|4

�(P)|x|2 +

1

2

|x|4

These last two inequalities suggest we should select ↵(r) = �(P)r2 and ↵(r) = �(P)r2

+

1

2

r4. The ultimatebound is b = ↵�1

(⌘) where

⌘ = max

|x|µ↵(|x|) ↵(µ)

So we can conclude that the ultimate bound is

b = ↵�1

(↵(µ))

= ↵�1

�(P)µ2

+

1

2

µ4

=

s

�(P)µ2

+ µ4/2

�(P)

The notion of uniform-ultimate boundedness provides the “bridge” between Lyapunov stability and someinput-output stability concepts. To build that bridge we need to shift from viewing the perturbation g(t, x) asbeing functionally dependent on the state, to an input signal w in some appropriately defined Lp signal spacethat is generated exogeneously to the original unperturbed system. Probably the easiest way of doing this isthrough the stability concept known as input-to-state stability or ISS, which is the next section’s topic.

3. Input-to-State Stability

We now turn to study the “stability” of a dynamical system with an external input w : R ! Rm. In particular,consider the system equations

x(t) = f(x(t), w(t))

where x : R ! Rn and w : R ! Rm with f(x, w) being locally Lipshcitz in x and w and f(0, 0) = 0. Weassume that w is a bounded piecewise continuous function of time.

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3. INPUT-TO-STATE STABILITY 151

The system x = f(x, u) is said to be input-to-state stable or ISS if there exists a class KL function � anda class K1 function � (sometimes called the gain) such that for any initial state x(0) = x

0

2 Rn thecorresponding state trajectory, x : R�0

! Rn, for any w 2 L1 satisfies the inequality

|x(t)| �(|x0

|, t) + �(kwkL1)(88)

for all t � 0.

Since � > 0 and � > 0 it can be readily shown that

max(�, �) � + � 2 max(�, �)

So an alternative and equivalent characterization of ISS is that the state trajectory satisfies the inequality

|x(t)| max (�(|x0

|, t), �(kwkL1))(89)

for all t � 0. In the following discussion we will use both definitions of ISS in equations (88) and (89)interchangeably.

Fig. 2 provides a graphical view of what the condition in equation (89) means. In particular, it says that thereare two terms; one that bounds the initial transient decay of the system by a class KL function and an ultimatebound that is a function of the amplitude of the disturbance, w. To be ISS requires that any trajectory be liebelow the maximum of these two bounds.

|x(t)|

time (t)

β(|x |,t)

γ(||w|| )

0

ℒ∞

FIGURE 2. Input-to-State Stability (ISS)

Note that there is a strong similarity between what is shown in Fig. 2 and what is shown for UUB in Fig. 1. Inparticular, it is important to identify the way in which ISS and UUB differ. Probably the most important dif-ference is that ISS is a global property of the system whereas UUB is only a local property of the unperturbedsystem’s equilibrium. Nonetheless the similarity between both concepts suggests that we can use the methodemployed in certifying whether a system is UUB to also certify whether a system is ISS. In particular, wewill establish that the existence of an ISS-certificate function (also known as an ISS-Lyapunov function) issufficient for a system to be ISS.

A C1 function V : Rn ! R will be called an ISS-certificate function for the system

x(t) = f(x, w)

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152 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

if there exist class K1 functions ↵, ↵, and ↵ and a class K function ⇢ : R ! R such that

↵(|x|) V (x) ↵(|x|)

and for |x| > ⇢(kwkL1) that

˙V =

@V

@xf(x, w) �↵(|x|)(90)

Note that if V is an ISS-certificate for x = f(x, w), then it is a Lyapunov function for the unforced systemx = f(x, 0). What we will now prove is that the existence of an ISS-certificate is sufficient for the system tobe ISS. The proof will show that the existence of an ISS-certificate establishes that the system is UUB fromwhich the input-to-state stability of the system is readily established.

THEOREM 69. (ISS-certificate) If V : Rn ! R is a C1 function that is also an ISS-certificate for x =

f(x, w), then the system is ISS.

Proof: The UUB theorem 68 implies that there exists KL functions � and T � 0 such that

|x(t)| �(|x0

|, t)

for 0 t < T and the ultimate bound is

|x(t)| ↵�1

(↵(r))

for T > t where r = ⇢(kwkL1). Let

�(kwkL1) = ↵�1

(↵(⇢(kwkL1)))

Note that � is clearly class K and so we can see that

|x(t)| max(�(|x0

|, t), �(kwkL1)) �(|x0

|, t) + �(kwkL1)

which establishes the system is ISS. }

An alternative test to see where the system has an ISS-certificate is given in the following theorem. In thistheorem the condition on ˙V is replaced by a dissipative inequality that is somewhat easier to verify than thecondition in equation (90).

THEOREM 70. (ISS certificate - dissipative inequality) Consider the system x(t) = f(x(t), w(t)) where f

is Lipschitz and w 2 L1. A C1 function V : Rn ! R is an ISS-certificate for this system if and only if thereexist K1 functions ↵, ↵, and ↵ and a class K function � such that

↵(|x|) V (x) ↵(|x|)

and

˙V =

@V

@xf(x, w) �↵(|x|) + �(|w|)(91)

for all x 2 Rn and all w 2 Rm.

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3. INPUT-TO-STATE STABILITY 153

Proof: Suppose V satisfies the condition in equation (91) and define

⇢(r) = ↵�1

(k�(r))

for some k � 1. Then the requirement that |x| � ⇢(|w|) for all w implies

˙V =

@V

@xf(x, w) �↵(|x|) + �(|w|)

�k � 1

k↵(|x|)

which means V satisfies our earlier definition in equation (90) for an ISS-certificate.

Conversely, let us assume V is an ISS-certificate then for |x| � ⇢(|w|) we know

@V

@xf(x, w) �↵(|x|)

So let us define the function � : R ! R as

�(r) = max

|u|=r,|x|⇢(r)

@V

@xf(x, w) + ↵(⇢(|w|))

This is the maximum value that ˙V +↵ can achieve within the bounded set |x| ⇢(|w| = r). Since this is thelargest value for ˙V + ↵ for inputs with magnitude equation to r, we can deduce that

@V

@xf(x, w) �↵(|x|) + �(|w|) �↵(|x|) + max(0,�(r))

The last term max(0,�(r)) can be overbounded by a class K function which suffices to show that V satisfiesequation (91). }

The condition in equation (91) is sometimes called the ISS dissipative inequality. For convenience the pair(↵, �) that characterize the dissipative inequality (91) in theorem 70 is called an ISS-pair. We now turn toseveral examples illustrating the use of ISS certificates in verifying whether a system is ISS or not.

Example: Consider the linear system

x = Ax + Bu

and assume that A has eigenvalues with negative real parts (i.e. Hurwitz). Let P = PT > 0 denote theunique solution to the Lyapunov equation PA + ATP + I = 0. Note that V (x) = xTPx Note that

�(P)|x|2 V (x) �(P)|x|2

So we can define K1 functions ↵(r) = �(P)r2 and ↵(r) = �(P)r2. Also note that

˙V =

@V

@x(Ax + Bu) �|x|2 + 2|x| kPk kBk |u|

Pick 0 < ✓ < 1 and share some of the negative definite first term of the preceding inequality with theindefinite second term. This yields,

@V

@x(Ax + Bu) �(1 � ✓)|x|2 � ✓|x|2 + (1 � ✓)c|x||u|

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154 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

where c =

2

1�✓kPk kBk. For |x| > cr we readily see that the first and third terms together are negativedefinite. So for such x we can guarantee

@V

@x(Ax + Bu) �✓|x|2

which verifies condition (90) for ISS stability with the gain �(r) =

�(P)

�(P)

cr.

Example: Now consider the single-input single-output (SISO) system

x = �axk+ bxp�(w)

with a > 0 and � : R ! R being a C1 function with �(0) = 0. We assume that k is an odd integer and p

is an integer such that p < k. Since a > 0 and k is odd, we can readily establishing that the equilibrium atx = 0 for x = �axk is globally asymptotically stable. Choose a candidate ISS-certificate of the form,

V (x) =

1

2

x2

The directional derivative of V is

˙V =

@V

@xf(x, w) = �axk+1

+ bxp+1�(w)

Let ✓ : R ! R be a class K function such that

|�(w)| ✓(|w|)

then for all w 2 R

˙V �a|x|k+1

+ |b| |x|p+1✓(|w|)= |x|p+1

(�a|x|q + |b|✓(|w|))

where q = k � p.

Consider the class K1 function

↵(r) = ✏rk+1

with ✏ > 0. If we choose x so that

�✏|x|q � �|x|q + |b|✓(|w|)

for a > ✏, then ˙V becomes

˙V |x|p+1

(�a|x|q + |b|✓(|w|)) �✏|x|k+1

Since k +1 is even, this means ˙V �↵(|x|) where ↵ is class K1. Assume a > ✏, we then get ˙V < �↵(|x|)when

|x| >

✓ |b|✓(|w|)a � ✏

1/q

= ⇢(|w|)

which implies the system is ISS.

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3. INPUT-TO-STATE STABILITY 155

Note that it was critical that p < k in the above example. If this inequality were not to hold then the systemmay fail to be ISS as demonstrated in the following example,

x = �x + xw

For this system if w(t) = 2 for all t � 0, then the system simply becomes x = x whose origin is clearlyunstable.

Example: Now consider the 2-dimensional system where n = 2 and m = 1

z = �z3

+ zy

y = az2 � y + w

where a is a real parameter. We select a candidate ISS-certificate

V (z, y) =

1

2

(z2

+ y2

)

Taking the directional derivative yields,

˙V =

@V

@xf(x, w) = �z4

+ (1 + a)z2y � y2

+ yw

We know that

z2y 1

2

z4

+

1

2

y2

So for any � > 0 we can show

yw �

2

y2

+

1

2�w2

This allows us to bound ˙V as

˙V �z4

+ (1 + a)

1

2

z4

+

1

2

y2

� y2

+

2

y2

+

1

2�w2

=

�1 +

|1 + a|2

z4

+

�1 +

|1 + a|2

+

2

y2

+

1

2�w2

If |a| < 1, then the first term is negative. The second term will be negative for � > 0 such that

�1 +

|1 + a|2

+

2

< 0

So this means we can find a and � such that there exist positive constants d1

and d2

for which

˙V �(d1

z4

+ d2

y2

) +

1

2�w2

�↵(|x|) + �(|w|) �↵(|x|) + �(kwkL1)

where ↵ and � are both class K1. So ˙V satisfies the ISS dissipative inequality and so this system is ISS.

Example: Consider a plant of the form x(t) = f(x, u) where f(0, 0) = 0 and where u is a control input thatsatisfies u = k(x+ e) for some Lipschitz function k. Let us assume that k has been chosen so that the closed

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156 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

loop system is ISS with respect to the input e. From theorem 70 this means there is an ISS-cerificate V suchthat

@V

@xf(x, k(x + e)) �↵(|x|) + �(|e|)

where ↵ is class K1 and � is class K.

Now let us consider a sampled-data version of this system. In particular, let us consider a sequence ofsampling instants {⌧i}1

i=0

where ⌧i < ⌧i+1

and for any T > 0 there exists N such that ⌧i > T for all i > N

(i.e. forward progress). Let us assume that instead of using x in the feedback law, we use the state sampledat time instant ⌧i. In other words, our sampled data system satisfies the following differential equation

x(t) = f(x(t), k(x(t))

for all t where the sampled state x is a piecewise constant signal that takes values

x(t) = x(⌧i)

for all t 2 [⌧i, ⌧i+1

) with i = 0, 1, . . . , 1.

The question is whether the origin of this sampled data system is asymptotically stable or not. In particular,let us define the sampling error as

e(t) = x(t) � x(t)

With this change of variables, one readily sees that

x(t) = f(x(t), k(x(t))) = f(x(t), k(x(t) + e(t)))

We already know that k was chosen so that this system was ISS with respect to e, so we can deduce that forall x and e that

@V

@xf(x, k(x)) �↵(|x|) + �(|e|)

Now select 0 < ✓ < 1 and divide the first term with the second to obtain@V

@xf(x, k(x + e)) �(1 � ✓)↵(|x|) � ✓↵(|x|) + �(|e|)

If we can guarantee that the error satisfies that the inequality

|e| ��1

(✓(↵(|x|)))(92)

for all time, then ˙V < 0 which is sufficient to assure the global asymptotic stability of the origin of thissampled data system.

So how can we ensure that the inequality in equation (92) is always satisfied? We simply monitor the samplingerror e = x � x. If that gap exceeds the state dependent threshold ��1

(✓↵(|x|)) then the state is sampled,thereby setting the error to zero. This condition is sufficient to ensure the asymptotic stability of the sampleddata system. This approach to sampled data control is sometimes referred to as Lebesgue-sampling or state-based event-triggering [Tab07]. In general, the intersample times, ⌧i+1

� ⌧i, will not be regular and so the

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4. Lp

STABILITY 157

sampling of the state is highly sporadic. However, the average intersampling time is usually much longerthan what would be seen if one had decided to employ the Nyquist sampling condition [WL09, WL11].

4. Lp Stability

The input-to-state stability concept is still tightly tied to the state space realization of the dynamical system.There is, however, another way of thinking about systems that was alluded to in chapter 2. In that chapter wealso defined a dynamical system as an operator G : W ! Y that mapped input signals from a normed linearspace W to output signals in a normed linear space Y . We refer to this as an input-output system.

W YG

w y

FIGURE 3. I/O System

It is customary to represent such systems by the block diagram shown in Fig. 3.In this figure the system is represented by the block marked with the operatorlabel, G. The inputs and outputs of the system are marked by directed arrowsthat point into and out of the system block, respectively. For the system inFig. 3, the input signal w : R ! Rp is a continuous-time real vector valuedsignal that is a member of a complete normed linear space, W . In a similar waythe output signal y : R ! Rq is also a continuous-time real valued vector thatis again in a normed linear space, Y .

The input-output description of the system as an operator is extremely unconstrained. This stands in directcontrast to the topological description of a dynamical system or even the ODE representations of a dynamicalsystem. The I/O description places very few constraints on the nature of the system and may not even havethe notion of “state” that we have grown accustomed to using. In fact, this description is too general to be ofdirect use without imposing some additional restrictions.

The main restriction we need is one that recognizes that “time” only moves in the forward direction. From ourstandpoint this means that our system, G, must be a map that ensures some type of forward causality betweenthe system’s inputs and outputs. We introduce this notion of causality by first defining the truncation of atime-domain signal w : R ! Rp with respect to time instant T as the signal wT : R ! Rp that takes values

wT (t) =

(

w(t) t < T

0 t � T

If we then let G : W ! Y be a map between signal spaces W and Y , then we say the system G is causal ifand only if

(G[w])T = (G[wT ])T

for all T � 0 and any w 2 W . Essentially what this definition does is state that the output of the systemprior to time T using the untruncated input w is the same as that of the system using the truncated signal wT .Intuitively, this means that “future” inputs after time T have no impact on the system’s response prior to timeT , thereby capturing our notion of what it means for the system’s output to be causally related to its input.

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158 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

Throughout these lectures whenever we refer to systems as operators between input and output spaces wewill always assume the system map is causal.

With regard to the causal input-output system shown in Fig. 3, we can then define a notion of “stability” thatis frequently used in engineering design. In this usage, the input signal, w, is considered to be a disturbanceand what we want to do is to determine how much of that disturbance appears at the system output. A“stable” system G is then seen as one for which a suitably bounded input signal always produces a suitablybounded output signal. This is the notion of input-output stability that is developed in this section. We needto formalize this notion to make it clear what it means for the input and output to be bounded.

Recall that the system G : W ! Y is a map between two normed linear signal spaces, so now we get specificabout the signal spaces of interest to us. We let W be the set of all time-domain signals w : R ! Rp. Welet Y be the set of all time-domain signals y : R ! Rq . In particular, let Lp(W ) denote the Banach spaceconsisting of all functions w 2 W such that the signal’s Lp norm,

kwkLp

=

Z 1

�1|w(t)|pdt

1/p

is finite. We define the set Lpe(W ) as the set of all functions such that wT 2 Lp(W ) for all T . We callLpe(W ) the extension of Lp(W ) since it should be apparent that Lp(W ) ⇢ Lpe(W ). With regard to oursystem, G, we now think of it as a map between Lpe(W ) and Lpe(Y ). We say the system G : Lpe(W ) !Lpe(Y ) is Lp-stable if for any w 2 Lp(W ) then the output G[w] 2 Lp(Y ). In other words, for any input, w,that is bounded with respect to the Lp(W ) norm, then its associated output G[w] is also bounded with respectto the Lp(Y ) norm.

Note that we can provide an equivalent definition of Lp stability in terms of comparison functions. In partic-ular, it can be shown that a system G : Lpe(W ) ! Lpe(Y ) is Lp-stable if and only if there exist a class Kfunction, ↵, and a non-negative constant � such that

k(G[w])T kLp

↵(kwT kLp

) + �

for any w 2 Lpe(W ) and any T � 0.

The map G : Lpe(W ) ! Lpe(Y ) has a finite Lp gain if there exist constants � and � such that for all T � 0

and any w 2 Lpe(W )

k(G[w])T kLp

�kwT kLp

+ �

The system’s gain is defined as

�(G) = inf

� : k(G[w])T kLp

�kwT kLp

+ �, for any w 2 Lpe(W )

The following theorem provides a well-known way to estimate a system’s L2

gain when the system is “affine”with respect to the input. This theorem characterizes the L

2

gain with respect to the satisfaction of aninequality H(x) 0 known as the Hamilton-Jacobi inequality or HJI.

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4. Lp

STABILITY 159

THEOREM 71. Consider the time-invariant system

x = f(x) + g(x)w

y = h(x)

with f(0) = 0 and h(0) = 0. Let � be a positive constant and suppose there exists a C1 positive semi-definitefunction V : Rn ! R, such that

H(x) ⌘ @V

@xf(x) +

1

2�2

@V

@xg(x)gT (x)

@V

@x

◆T

+

1

2

hT(x)h(x) 0(93)

Then the system is finite gain L2

stable with a gain less than or equal to �.

Proof: We use a completing the square argument on ˙V . This means

˙V =

@V

@x+

@V

@xgw

= �1

2

�2

u � 1

�2

gT✓

@V

@x

◆T�

2

+

@V

@xf +

1

2�2

@V

@xg(x)gT (x)

@V

@x

◆T

+

1

2

�2|w|2

The theorem assumes that H(x) 0, which means that

@V

@x+

1

2�2

@V

@xggT

@V

@x

◆T

< �1

2

hT(x)h(x)

Inserting this into our expression for ˙V above and using the fact that y = h(x) yields,

˙V =

@V

@xf(x) +

@V

@xg(x)w

1

2

�2|w|2 � 1

2

|y|2 � 1

2

u � 1

�2

gT✓

@V

@x

◆T�

2

1

2

�2|w|2 � 1

2

|y|2

Now note that this allows us to infer that

V (x(⌧)) � V (x0

) 1

2

�2

Z ⌧

0

|w(t)|2dt � 1

2

Z ⌧

0

|y(t)|2dt

Since V (x(⌧)) � 0 this implies thatZ ⌧

0

|y(t)|2dt �2

Z ⌧

0

|w(t)|2dt + 2V (x0

)

Taking the square root of both sides and using the fact thatp

a2

+ b2 a + b for a � 0 and b � 0 allows usto conclude that

kyT kL2 �kwT kL2 +

p

2V (x0

)

which means the system is L2

stable with a gain less than � and a bias ofp

2V (x0

). }

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160 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

Example: Consider the LTI system

x = Ax + Bw

y = Cx

and let V (x) = xTPx. In this case the Hamilton-Jacobi inequality becomes

H(x) = xT(PA + ATP)x +

1

2�2

xTPBBTPx +

1

2

xTCTCx 0

and we can conclude that the L2

gain is less than � if the algebraic Riccati equation (ARE)

0 = PA + ATP +

1

2�2

PBBTP + CTC

has a positive definite solution. The use of the ARE in determining a linear system’s L2

-gain is an importanttechnique used in control system synthesis.

Example: Consider the system

x1

= x2

x2

= �ax3

1

� kx2

� w

y = x2

where a > 0 and k > 0. Consider the function

W (x) = ax4

1

4

+

x2

2

2

Note that W is a Lypaunov function for the unforced system where w = 0. Now define the function V (x) =

↵W (x) where ↵ is a constant to be determined. We will check to see if V satisfies the HJI. In particular wesee

@V

@x= ↵ax3

1

x2

+ ↵x2

(�ax3

1

� kx2

) = �↵x2

2

1

2�2

@V

@xggT

@V

@x

◆T

=

1

2�2

h

↵ax3

1

↵x2

i

"

0

1

#

h

0 1

i

"

↵ax3

1

↵x2

#

=

1

2�2

↵2x2

2

1

2

hTh =

1

2

x2

2

Taking these terms and assembling the HJI yields

H(x) =

�↵k +

↵2

2�2

+

1

2

x2

2

0

For H(x) to be negative semi-definite we need

�↵k +

↵2

2�2

+

1

2

0

which we can rearrange to get a bound on the L2

gain, �

�2 � ↵2

2↵k � 1

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4. Lp

STABILITY 161

We need to select ↵ to make the right hand side of the above inequality as small as possible. In particularnote that

@

@↵

↵2

2↵k � 1

=

(2↵k � 1)2↵� ↵2

2k

(2↵k � 1)

2

Setting this equal to zero implies

2↵2k � 2↵ = 0 ) k =

1

Inserting this into our bound for � shows that �2 � 1

k , which represents the tightest bound we can get for theL

2

gain using this particular certificate function.

Note that one might think of k as a controller gain. In other words, the system in this example may beinterpreted as

x1

= x2

x2

= �ax3

1

+ u � w

y = x2

where u is a control input that we select to ensure the closed-loop system is ISS with respect to the disturbancew. In particular, this example shows that we an select the control to be u = �ky (an output feedback controllaw) where we showed how to select the control gain k to minimize our upper bound on the induced L

2

gain of the closed-loop system. This is a common objective in regulation problems that seek to minimize theimpact that a disturbance has at the system’s output. In particular, this example demonstrates that the HJIequation can be used to obtain an ”optimal” control gain in the sense of minimizing the closed-loop system’ssensitivity to the applied disturbance.

Example: We now consider a system of the form

x = f(x) + G(x)u + K(x)w

y = h(x)

where u is a control input and w is the external disturbance. The functions f , G, K, and h are sufficientlysmooth with f(0) = 0 and h(0) = 0. We assume there exists a � > 0 and a smooth positive semi-definitefunction V : Rn ! R that satisfies

@V

@xf(x) +

1

2

@V

@x

1

�2

K(x)KT(x) � G(x)GT

(x)

�✓

@V

@x

◆T

+

1

2

hT(x)h(x) 0

for all x 2 Rn. Let us assume that the control input satisfies

u = �GT(x)

@V

@x

◆T

We will use the HJI to prove that the closed-loop map from w to

"

y

u

#

is finite gain L2

stable.

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162 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

In particular, the closed loop system equations are

x = f(x) � G(x)GT(x)

@V

@x

◆T

+ K(x)w

The closed loop map from w to

"

y

u

#

is given by

x = fc(x) + Gc(x)w

=

"

f � GGT

@V

@x

◆T#

+ K(x)w

yc =

"

y

u

#

= hc(x) =

"

h(x)

�GT(x)

@V@x

�T

#

For this closed loop system the HJI is

Hc(x) =

@V

@xfc +

1

2�2

@V

@x

GcGTc

@V

@x

◆T

+

1

2

hTc hc

=

@V

@x

"

f � GGT

@V

@x

◆T#

+

1

2�2

@V

@x

KKT

@V

@x

◆T

+

1

2

hTh +

1

2

@V

@x

GGT

@V

@x

◆T

=

@V

@xf +

1

2

@V

@x

1

�2

KKT � GGT

�✓

@V

@x

◆T

+

1

2

hTh

By the theorem’s assumption this means Hc(x) 0 and so we can conclude that the given V satisfies theHJI and so the closed loop system is finite gain L

2

stable with a gain that is less than �.

It is possible to relate Lp stability back to the Lyapunov stability concept of chapter 5. To do this, however, weneed to have a concrete representation for the input-output system, G : Lpe ! Lpe, where the notion of stateand equilibrium are clearly identified. This is usually done by considering a system whose state equations areof the form,

x(t) = f(t, x, w), x(0) = x0

y(t) = h(t, x, w)

(94)

where x is the system state, y is the system’s output, and w is the exogneous input. The usual assumptionsare made on f : R ⇥ Rn ⇥ Rp ! Rn to ensure the existence of unique solutions. We assume that x = 0 isan equilibrium point for the unforced system, x = f(t, x, 0). The output function, h : R ⇥ Rn ⇥ Rp ! Rq ,is usually assumed to be piecewise continuous with respect to t and Lipschitz with respect to u and x. Withthe realization of G given in equation (94) we can then provide conditions under which exponential stabilityof the unforced system’s equilibrium implies that the input-output system in equation (94) is finite gain Lp

stable.

THEOREM 72. (Relation between Lp stability and Lyapunov stability) Consider the input-output systemgiven in equation (94) where the origin, x = 0, is an exponentially stable equilibrium of x = f(t, x, 0).

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4. Lp

STABILITY 163

Assume there exist positive constants L, r, ru, ⌘1

, and ⌘2

such that

|f(t, x, u) � f(t, x, 0)| L|u||h(t, x, u)| ⌘

1

|x| + ⌘2

|u|(95)

for all x such that |x| < r and u where |u| < ru. If there exists a C1 function V : R ⇥ Rn ! R andnon-negative constants c

1

, c2

, c3

, and c4

such that

c1

|x|2 V (t, x) c2

|x|2@V@t +

@V@x f(t, x, 0) �c

3

|x|2�

@V@x

� c4

|x|(96)

then for all |x| < r. Then for all initial conditions x0

2 Rn such that |x0

| < rq

c1c2

and all w 2 Lpe(W ) with

kwkL1 sufficiently small there exist positive constants � and � such that the outputs, y, solving the systemequation (94) also satisfy

kykLp

�kwkLp

+ �

In other words, the system is finite gain Lp stable.

Proof: Consider ˙V along trajectories of the forced system

˙V =

@V

@t+

@V

@xf(t, x, 0) +

@V

@x[f(t, x, w) � f(t, x, 0)]

Using the assumed bounds in equation (96) and the Lipschitz constant for f yields,

˙V �c3

|x|2 + c4

L|x||w|

Take W (t) =

p

V (t, x(t)) and note that

˙W =

˙V

2

pV

We can then see that

˙W �1

2

c3

c2

W +

c4

L

2

pc1

|w(t)|

By the comparison theorem we can therefore conclude that

W (t) e� c32c2

tW (0) +

c4

L

2

pc1

Z t

0

e�(t�s)c32c2 |w(s)|ds

which implies that

|x(t)| c2

c1

|x0

|e� c32c2

t+

c4

L

2c1

Z t

0

e�(t�s)c32c2 |w(s)|ds

and so we can conclude that if

|x0

| r

2

r

c1

c2

sup

s|w(s)| c

1

c3

r

c2

c4

L

1

2

then |x(t)| r for all time.

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164 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

This preceding deduction means that the bound on h in equation (95) holds for all time and so

|y(t)| k1

e�at+ k

2

Z t

0

e�a(t�s)|w(s)|ds + k3

|w(t)|

where k1

=

q

c1c2

|x0

|⌘1

, k2

=

c4L⌘1

2c1, k

3

= ⌘2

, and a =

c32c2

. Assign to each of the terms in the aboveequation a signal, y

1

, y2

, y3

, respectively. If w 2 Lpe with kwkL1 sufficiently small, then for any T > 0

ky2T kL

p

k2

akwT kL

p

ky3T kL

p

k3

kwT kLp

ky1T kL

p

k1

where ⇢ =

(

1 if p = 1(1/ap)

1/p otherwise. So that since

kyT kLp

ky1T kL

p

+ ky2T kL

p

+ ky3T kL

p

then we can use the above bounds on the Lp norm of y1

, y2

, and y3

, to conclude that

kyT kLp

k1

⇢+

k2

akwT kL

p

+ k3

kwT kLp

=

k2

a+ k

3

kwT kLp

+ k1

⇢ = �kwT kLp

+ �

which identify the finite gain and bias for this system. This system therefore is finite gain Lp stable. }

Example: Consider the single-input single-output (SISO) system

x1

= x2

x2

= �x1

� x2

� a tanh(x1

) + w

y = x1

where a > 0. We can view this as a perturbation of a linear system

x =

"

0 1

�1 �1

#

x = Ax

The A matrix is Hurwitz with eigenvalues �1,2 = � 1

2

±p

3

2

j. Since A is Hurwitz there exists a matrixP = PT > 0 that satisfies the Lyapunov equation

PA + ATP + I = 0

Solving for P we see P =

"

1.5 .5

.5 1

#

. We then use V (x) = xTPx as the certificate function in theorem

72 and check to see in what sense it satisfies the inequalities (96). In this case we see that

˙V = �x2

1

� x2

2

� ax1

tanh(x1

) � 2ax2

tanh(x1

)

�|x|2 + 2a|x1

||x2

| �(1 � a)|x|2

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4. Lp

STABILITY 165

This is negative definite for all a < 1. We can then easily check to see that theorem 72 holds with

�(P)|x|2 V (x) �(P)|x|2

|f(t, x, u) � f(t, x, 0)| |u||y| = |x

1

| |x|

so we can use the theorem to conclude that this system is finite gain Lp stable.

We can also establish a relationship between input-to-state stability (ISS) and Lp stability through the fol-lowing theorem. In this case, it is necessary to limit the gain of the output function h.

THEOREM 73. (ISS versus Lp stability) Consider the input-output system realized by equations (94). Letf : [0, 1) ⇥ Rn ⇥ Rm ! Rn be piecewise continuous in t and locally Lipschitz in x and u. Let h :

[0, 1)⇥Rn⇥Rm ! Rm be piecewise continuous in t and continuous in x and u. Suppose the x = f(t, x, w)

is locally ISS and assume there exist class K functions ↵1

, ↵2

and a non-negative constant ⌘3

such that

|h(t, x, w)| ↵1

(|x|) + ↵2

(|w|) + ⌘3

(97)

Then there is a constant k1

> 0 such that for all initial conditions with |x0

| < k1

, the system in equation (94)is L1 stable.

Proof: Since the system x(t) = f(t, x, w) is ISS, we know there exist class KL function � and class Kfunction � such that

|x(t)| �(|x0

|, t) + �(kwkL1)

for all w 2 L1. From the assumption on h, we can then assert that

|y(t)| ↵1

(�(|x0

|, t) + �(kwkL1) + ↵2

(|w(t)|) + ⌘3

↵1

(2�(|x0

|, t)) + ↵1

(2�(kwkL1)) + ↵2

(|w(t)|) + ⌘3

where we used the property of class K functions that

↵1

(a + b) ↵1

(2a) + ↵1

(2b)

We can therefore deduce that

kyT kL1 �0

(kwtkL1) + �0

where �0

= ↵1

� 2� + ↵2

and �0

= ↵1

(2�(|x0

|, 0)) + ⌘3

and the proof is complete. }

Example: Consider the SISO system

x1

= �x3

1

+ g(t)x2

x2

= �g(t)x1

= x3

2

+ w

y = x1

+ x2

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166 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

where g(t) is continuous for all t � 0. Take V =

1

2

(x2

1

+ x2

2

). Then

˙V = �x4

1

� x4

2

+ x2

w

Using the fact that

x4

1

+ x4

2

� 1

2

|x|4

and selecting ✓ 2 (0, 1), we repartition ˙V as

˙V �1 � ✓

2

|x|4 � ✓

2

|x|4 + |x| |w|

�1 � ✓

2

|x|4, for all |x| �⇣

2|w|✓

1/3

So from theorem 69 we know that this system is ISS. The function h = x1

+x2

clearly satisfies the condition(97) and so we can conclude that this system is also L1 stable.

5. Passive and Dissipative Systems

A very common idea in nonlinear control is that stable systems are those that dissipate energy. Lyapunovfunctions may be seen as a generalization of this energy idea for “closed” systems that have no externalinputs. We’d like to extend this “energy” approach to “open” or input-output systems. In this framework, theinput signal is seen as injecting energy into the system and the output is seen as drawing energy out of thesystem. The difference between what flows into and what flows out of the system must be “stored” within thesystem and for a system to be “stable” in this framework we want that stored energy to always be less than orequal to the difference of what was injected or delivered to the system. This notion of “stability” is referredto as passivity and dissipativity, depending on precisely how it is defined. The objective of this section isto introduce this “energy” based view of input-output systems, to present the fundamental properties of thisenergy-based input-output stability concept and to relate it back to previously defined stability concepts. Wewill start by introducing the passivity stability concept and then move on to generalize this concept to definethe dissipativity concept.

Consider the electrical device shown on the left side of Fig. 4 where u : R ! R is an applied voltage andy : R ! R is the current injected into the device. The voltage-current relationship is given by u = Zy

where Z is the impedance function. We view the device as an input-output system whose input is u (theapplied voltage) and whose output is y (the injected current). Since we are interested in “energy”, we knowthe instantaneous power, p : R ! R, injected into the device is given by p(t) = u(t)y(t). If p(t) � 0 thenenergy is being absorbed or delivered to the device and if p(t) < 0, then the device is acting as a source inthat it is delivering energy (power) to the outside world. When the device is absorbing energy (i.e. p(t) � 0),then we say it is passive.

Let us now consider a specific electrical device that is shown on the right hand side of Fig. 4. This shows thedevice to be an RLC circuit. The current through the inductor is i

2

and the voltage across the capacitor is vc.Since the inductor and capacitor are the only energy storage devices in the circuit, we can treat i

2

and vc as

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5. PASSIVE AND DISSIPATIVE SYSTEMS 167

1

2

3

FIGURE 4. Electrical Circuit

state variables that we denote as x1

and x2

, respectively. With these conventions the state equations for ourinput-output system now become

Lx1

= u � h2

(x1

) � x2

Cx2

= x1

� h3

(x2

)

y = h1

(u) + x1

where h1

: R ! R is a nonlinear admittance function representing the current through nonlinear resistor R1

,h

2

: R ! R is the nonlinear impedance function representing the voltage across resistor Rw and h3

: R ! Ris the nonlinear admittance function representing the current across resistor R

3

.

The “energy” stored within this electrical circuit is given by

V (x) =

1

2

Lx2

1

+

1

2

Cx2

2

namely, the energy stored in both the inductor and the capacitor. According to the notion of passivity heuris-tically introduced above, this circuit is passive if the total energy injected into the system is greater than whatwhat is stored in the system. In other words,

Z t

0

u(s)y(s)ds � V (x(t)) � V (x(0))(98)

energy injected into system over [0, t) � change in energy stored over [0, t)

The function V : R2 ! R that we introduced above is called a storage function since it represents how muchenergy is stored in the system.

If we take the derivative of the integral equation (98), then we obtain

u(t)y(t) � ˙V (x(t))

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168 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

So we can conclude that the system is passive if the derivative of the storage function is less than the injectedinstantaneous power p(t) = u(t)y(t) for all t. For this particular system this means that

˙V = Lx1

x1

+ Cx2

x2

= x1

(u � h2

(x1

) � x2

) + x2

(x1

� h3

(x2

))

= x1

(u � h2

(x2

)) � x2

h3

(x2

)

= (x1

+ h1

(u))u � (uh1

(u) + x1

h2

(x1

) + x2

h3

(x2

))

= uy � positive definite term

This last inequality, of course, implies that ˙V uy and so we can conclude that this circuit is indeed“passive”.

Our problem is now to generalize the preceding discussion to any input-output system that has a state-basedrepresentation of the form

x = f(x(t), u(t))

y = h(x(t), u(t))(99)

where f : Rn ⇥ Rm ! Rn is Lipschitz in both x and u and h : Rn ⇥ Rm ! Rm is also Lipschitz. Notethat we are assuming this system has the same number of inputs as outputs. We will say that this system ispassive if there exists a C1 positive semidefinite function (aka the storage function), V : Rn ! R, such that

uT(t)y(t) � ˙V (t) ⌘ @V

@xf(x(t), u(t))

for all time t.

It will be convenient to introduce the following “strict” forms of passivity. In particular one says the systemis

• memoryless if uT y =

˙V

• strictly input passive if uT y � ˙V + uT�(u) where uT�(u) > 0 for all u 6= 0

• strictly output passive if uT y � ˙V + yT ⇢(y) where yT ⇢(y) > 0 for all y 6= 0

• strictly passive if uT y � ˙V + (x) for some positive definite .

These strict notions of passivity will be useful later in relating stability to passivity.

Note that there are some limitations to the preceding passivity concept. In the first place, it assumed avery specific form of “power” that made sense for circuits, but may not apply to other dynamical systems.Secondly the notion of power assumes the system has as many inputs as outputs. This also may not generallybe true of the systems of interest to us. It will therefore become convenient to generalize the passivity conceptin a manner that removes these limitation. That generalization is the notion of dissipativity [Wil72].

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5. PASSIVE AND DISSIPATIVE SYSTEMS 169

Consider an input-output system G : Lpe ! Lpe with the state space realization

x(t) = f(x(t), u(t)), x(t0

) = x0

, t � t0

y(t) = h(x(t), u(t))

where x(t) 2 Rn is the state, y(t) 2 Rp is the output and u(t) 2 Rm is the input. The usual Lipschitzassumptions on f and h are enforced to ensure that the state trajectories are unique. The system is dissipativewith respect to the supply rate function, r : Rm ⇥ Rp ! R if

0 Z t

t0

r(u(s), y(s))ds(100)

is satisfied for all t � t0

and all u 2 Lpe.

Let us now assume there exists a continuous non-negative definite function V : Rn ! R with V (0) = 0 suchthat

V (x(t)) V (x(t0

)) +

Z t

t0

r(u(s), y(s))ds, t � t0

(101)

where x and y satisfy the input-output system equations. This inequality is called the dissipative inequalityand if it holds then clearly the system will satisfy the condition in equation 100. Since V is smooth, we candifferentiate it with respect to time to establish that the system is dissipative provided

˙V (t) r(u(t), y(t))(102)

for all time t. Note that this is precisely the same relationship we used to describe a passive system. In thatcase, however, the supply rate function was r(u, y) = uT y. By introducing the supply rate function, r, we nolonger require y and u to be of the same dimension.

A second choice for the supply rate is

r(u, y) =

1

2

�2kuk2

U � kyk2

Y

where � � 0, k · kU is the norm for the input signal space and k · kY is the norm for the output signal space.The system, G, is dissipative with respect to this supply rate if and only if there exists a positive semi-definitefunction V : Rn ! R such that for all t � 0, x

0

, and input u,

1

2

Z t

0

�2|u(s)|2 � |y(s)|2� ds � V (x(t)) � V (x0

) � �V (x0

)

and soZ t

0

|y(s)|2ds �2

Z t

0

|u(s)|2ds + 2V (x0

)

which is the same as

kyT k2

L2 �2kuT k2

L2+ 2V (x

0

)

for any T � 0. If we take advantage of the fact thatp

a2

+ b2 a + b when a, b � 0 then the precedinginequality implies

kyT kL2 �kuT kL2 +

p

2V (x0

)

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170 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

So we can conclude that the systems that are dissipative with respect to this supply rate are also finite gainL

2

-stable.

We need to have a way to certify whether a system G is dissipative with respect to a given supply rate r. Thefollowing theorem provides one answer to this question.

THEOREM 74. Consider the input-output system in equation (99). The system is dissipative with respect tosupply rate r(u, y), if and only if

Va(x) = sup

u(·), T�0

n

� R T0

r(u(s)y(s))ds for x(0) = xo

(103)

is finite for all x 2 Rn. Furthermore, if Va(x) is finite for all x then Va is a storage function and all otherpossible storage functions V satisfy Va(x) V (x) for all x 2 Rn.

Proof: Suppose Va is finite. Clearly Va � 0, so compare Va(t0) with Va(x(t1

)) � R t1t0

r(u(s), y(s))ds

for a given u; [t0

, t1

] ! Rm and resulting state x(t1

). Since Va is given as the supremum over all u(·), itimmediately follows that

Va(x(t0

) � Va(x(t1

)) �Z t1

t0

r(u(s), y(s))ds

and so Va is a storage function, proving that G is dissipative with respect to the supply rate r.

Conversely, let G be dissipative with respect to supply rate r. Then there exists a C1 function V � 0 suchthat for all u,

V (x(0)) +

Z T

0

r(u(s), y(s))ds � V (x(T )) � 0

which shows that

V (x(0)) � sup

(

�Z T

0

r(u(s), y(s))ds

)

= Va(x(0))

therefore proving that Va is finite as well as equation (103). }

The quantity Va(x0

) can be interpreted as the maximum “energy” that can be extracted from the systemG starting at initial condition x

0

. The function Va, therefore, is called the available storage. The precedingtheorem states that G is dissipative with respect to r if and only if this available energy is finite for every initialcondition. It is actually possible to reduce the set of initial states for which we need to check the availablestorage to a single point if the system is reachable. Consider the system x = f(x, u) where u : R ! Rm

is a control input. Let x(t; x0

, u) denote the state trajectory for this system under a given control input, u,from initial time x(0) = x

0

. We say this controlled system is reachable from initial state x⇤ 2 Rn if for anyx 2 Rn there exists a control input u and a finite time T such that x(T ; x⇤, u) = x. In other words, therealways exists a control input that can drive the system state from x⇤ to x in finite time T . The followingtheorem asserts that for systems that are reachable from x⇤, it is only necessary to check the available storageat x⇤ to certify that the system is dissipative with respect to the given storage function.

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5. PASSIVE AND DISSIPATIVE SYSTEMS 171

THEOREM 75. (Average Storage and Reachability) Assume that the input output system, G, in equation(94) is reachable from x⇤ 2 Rn. Let Va : Rn ! R denote the available storage in equation (103) withrespect to supply rate, r : Rm ! Rn ! R. Then G is dissipative if and only if Va(x⇤

) is finite.

Proof: If G is dissipative, then by theorem 74 the available storage, Va, is finite for all x 2 Rn, including x⇤.Conversely, suppose Va(x⇤

) is finite but that there exists an x 2 Rn for which Va(x) = 1. In other wordsthe system is not dissipative with respect to r. Since by reachability we can steer x⇤ to x in finite time, thiswould imply using time-invariance that Va(x⇤

) = 1, which contradicts our assumption so there can be no x

for which Va(x) = 1 and so the system G must be dissipative with respect to r. }

Let us now examine the relationship between dissipative/passive systems and our earlier stability concepts.Throughout the remainder of this section, we consider storage functions, V , that are C1. Returning to thedifferential dissipative inequality in equation (102) we see the system is dissipative if

˙V =

@V

@xf(x, u) r(u, y) = r(u, h(x, u))(104)

for all x and u. The similarity of the differential dissipative inequality to the Lyapunov stability-certificatecondition that ˙V 0 suggests that we should be able to use the storage function to relate dissipativity to anequilibrium’s Lyapunov stability.

THEOREM 76. (Dissipativity and Stability) Let V : Rn ! R�0

be a positive semi-definite C1 storagefunction for the system G in equations (99) and that ˙V (t) r(u(t), y(t)). Assume that the supply rater : Rm ⇥ Rp ! R satisfies

r(0, y) 0

for all y 2 Rp. Suppose that x⇤ 2 Rn is a strict local minimum for V . Then x⇤ is a stable equilibriumof the unforced system x = f(x, 0) with Lyapunov function W (x) = V (x) � V (x⇤

) � 0 for x sufficientlyclose to x⇤. Furthermore suppose that {x⇤} is the largest invariant set of x = f(x, 0) in the set {x 2 Rn

:

r(0, h(x, 0)) = 0} for all t. Then x⇤ is an asymptotically stable equilibrium which is globally asymptoticallystable if V � 0 is proper (i.e. radially unbounded).

Proof: By equation (104) and the assumption that r(0, y) 0, we can deduce

˙V =

@V

@xf(x, 0) r(0, h(x, 0)) 0

and so V (x(t)) is non-increasing along solutions of the unforced system x = f(x, 0). This means, therefore,that f(x⇤, 0) = 0 and the claim follows from the invariance principle in theorem 54 of chapter 5. }

Let us consider a particularly important class of systems that are affine in the input u without a direct feedthrough term. So G is an input output system realized by the following state-space equations

x = f(x) + g(x)u

y = h(x)

(105)

where g(x) is an n ⇥ m matrix of functions.

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172 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

We now examine conditions for this affine system to be dissipative with respect to a number of passivitysupply rate functions. In particular, If this system is dissipative with respect to the passivity supply rater(u, y) = uT y, then equation (104) implies

@V

@x[f(x) + g(x)u] uTh(x)

which can be shown to be equivalent to the so-called Hill-Moylan conditions [HM76, HM77]

@V@x f(x) 0 and @V

@x g(x) = hT(x).(106)

For the strictly output passive supply rate r(u, y) = uT y � ✏|y|2 with ✏ > 0 we get the condition

@V

@x[f(x) + g(x)u] uTh(x) � ✏hT

(x)h(x)

which is equivalent to

@V@x f(x) �✏hT

(x)h(x) and @V@x g(x) = hT

(x)(107)

On the other hand, for the strict input passivity supply rate r(u, y) = uT y ��|u|2 with � > 0 the dissipativityinequality leads to the condition

@V

@x[f(x) + g(x)u] uTh(x) � �uTu

which could never happen. Finally for the L2

supply rate r(u, y) =

1

2

�2|u|2 � 1

2

|y|2 with � � 0, thedissipativity inequality becomes

@V

@x[f(x) + g(x)u] 1

2

�2|u|2 � 1

2

|h(x)|2(108)

The main condition for Lyapunov stability in theorem 76 was that the storage function, V , have a strict (local)minimum at the equilibrium x⇤. In fact this property is implied when the system is zero-state observable. Inparticular, we say the system Ga in equation (105) is zero-state observable if u(t) = 0 and y(t) = 0 for allt � 0 implies that the state trajectory x(t) = 0 for all t � 0.

THEOREM 77. (Storage Rate for Zero-State Observable Affine Systems) Let V � 0 be a solution to thedissipative inequality in equation (106), (107), or (108) for the affine system Ga in equation (105). If Ga iszero-state observable, then V (x) > 0 for all x 6= 0.

Proof: Let us first consider the L2

supply rate inequality (108). So let V � 0 be a solution to the dissipativeinequality in equation (108). Substituting u = 0 into this inequality and integrating yields

V (x(T )) � V (x(0)) �1

2

Z T

0

|y(t)|2dt

Since V (x(T )) � 0, this means

V (x(0)) � 1

2

Z T

0

|y(t)|2dt

Now let V (x(0)) = 0, then y(t) = 0 for all t � 0 and so by the zero-observability property we knowx(0) = 0. This therefore establishes htat V (x) > 0 for all x 6= 0.

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5. PASSIVE AND DISSIPATIVE SYSTEMS 173

A similar argument holds if V � 0 is a solution to the strictly output passive supply rate inequality in equation(107).

So let us turn to the case where V � 0 is a solution to the passivity supply rate inequality (106). Take thefeedback law u = �y and then the integral form of the dissipative inequality becomes,

V (x(T )) � V (x(0)) �Z T

0

|y(t)|2dt

and so V (x(0)) � R T0

|y(t)|2dt, where h(t) = h(x(t)) is generated by x = f(x) � g(x)h(x). Now withV (x(0)) = 0 then y(t) = h(x(t)) = 0 for all t � 0 with x(t) generated by x = f(x) and so by zero stateobservability x(0) = 0. }

To actually establish the asymptotic stability of the equilibrium when the affine system is strictly outputpassive and has a finite L

2

gain, we require a weaker notion of observability. We say the system is zero-statedetectable if u(t) = 0 and y(t) = 0 for all t � 0 implies limt!1 x(t) = 0. For affine systems with zero-state detectability, the following theorem asserts that strict output passivity or a finite L

2

gain are sufficientto ensure the asymptotic stability of the equilibrium.

THEOREM 78. (Passivity and Asymptotic Stability) Let V � 0 be a solution to the dissipative inequalitywith either the strictly output passive supply rate (107) or the L

2

supply rate (108) with V (0) = 0 andV (x) > 0 for x 6= 0. If the affine plant Ga in equation (105) is zero-state detectable then x = 0 is anasymptotically stable equilibrium of x = f(x). Moreover if V is radially unbounded then the origin isglobally asymptotically stable.

Proof: From theorem 76, we know that x = 0 is a stable equilibrium of x = f(x). Taking u = 0 in eitherthe strict output passive or L

2

dissipative inequalities yields,

˙V (x) =

@V

@x �✏|h(x)|2

with ✏ >). Asymptotic stability follows from the Invariance Principle in Theorem 54 of chapter 5 since˙V (x) = 0 implies h(x) = 0. }

We now turn to examine the relationship between dissipativity and input-to-state stability. Recall from theo-rem 70 if we consider the system

x = f(x, u)

that this plant is ISS if there exists a C1 function V : Rn ! R, class K1 functions ↵, ↵, ↵, and class Kfunction � such that

↵(|x|) V (x) ↵(|x|)

and

˙V =

@V

@xf(x, u) �↵(|x|) + �(|u|)(109)

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174 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

We can think of this as an input-output system in which the system’s output y = x is the state. This means,then that the above condition on ˙V may be seen as the dissipative inequality with respect to the supply ratefunction r(u, y) = �(|u|) � ↵(|y|) and that the ISS-certificate is essentially a supply rate function for thesystem. In this regard, we can see that if the system is ISS, it is also dissipative with respect to the ISS pair,(�,↵), where y = x.

6. Stability of Feedback Interconnections

x2

u

x1

FIGURE 5. Feedback Interconnection ofISS Systems

One of the most useful aspects of an input-output system isthat they can be interconnected. The output of one system canbe used as the input to another system to form a larger sys-tem with more desirable properties. This section reviews re-sults that determine when a feedback interconnection satisfiesan input-output stability concept such as ISS, Lp stability, orpassivity. These results are important for they provide the foun-dation upon which robust control systems can be developed.

ISS Small Gain Theorem: We first investigate input-to-state stability (ISS) of a feedback interconnection oftwo ISS systems [JTP94]. Fig. 5 shows the interconnection of two ISS systems

x1

= f1

(x1

, x2

), x1

(0) = x10

x2

= f2

(x1

, x2

, u), x2

(0) = x20

(110)

where x2

is the input to the first system with vector field f1

and (x1

, u) are the inputs to the second systemwith vector field f

2

. The initial conditions for the first and second system are x10

and x20

, respectively.

THEOREM 79. (ISS Small Gain Theorem) Consider the interconnected system in equation (110) wheref1

(0, 0) = 0 and f2

(0, 0, 0) = 0. Assume that the first system is ISS with respect to input x2

so that whenx

2

2 L1, there exist class KL function �1

and class K function �1

such that

|x1

(t)| max {�1

(|x10

|, t), �1

(kx2

kL1)}Assume that the second system is ISS with respect to inputs x

1

and u so that for any x1

, u 2 L1 there existclass KL function �

2

and class K functions �2

and �u such that

|x2

(t)| max {�2

(|x20

|, t), �2

(kx1

kL1 , �u(kukL1)}If for all r > 0, we can verify that �

1

(�2

(r)) < r, then the interconnected system is ISS with respect to inputu.

Proof: Consider initial conditions x10

2 Rn1 and x20

2 Rn2 and let u 2 L1. Assume that the upper andlower systems are ISS so there exist class K1 functions �

01

, �1

, �02

, �2

, and �u such that for all t,

|x1

(t)| max {�01

(|x10

|), �1

(kx2

kL1}|x

2

(t)| max {�02

(|x20

|), �2

(kx1

kL1), �u(kukL1)}

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6. STABILITY OF FEEDBACK INTERCONNECTIONS 175

We first show that these state trajectories must remain bounded. To do this choose R such that

R > max {�01

(|x10

|), �1

� �02

(|x20

|), �1

� �u(kukL1)}R > max {�

02

(|x20

|), �2

� �01

(|x10

|), �u(kukL1)}

If the state trajectories are not bounded then for this R there exists a time T such that |x1

(T )| > R or|x

2

(T )| > R. So define the truncated signals for i = 1, 2

xiT (t) =

(

xi(t) if t 2 [0, T ]

0 if t > T

Let x1

denote the response of the upper system to the truncated input x2T . Since the truncated signal is

bounded on [0, 1), we have

|x1

(t)| max {�01

(|x10

|, �1

(kx2T kL1)}

for all t � 0. By causality, we know that x1

(t) = x1

(t) for all t 2 [0, T ]. We can therefore deduce that

|x1T |L1 = max

t2[0,T ]

|x1

(t)| max {�01

(|x10

|), �1

(kx2T kL1}(111)

In a similar way, let x2

denote the response of the lower system to inputs x1T and u. The boundedness of

these signals over [0, 1) allows us to conclude that

kx2T kL1 = max

t2[0,T ]

|x2

(t)| max {�02

(|x20

|), �2

(kx1T kL1), �u(kukL1)}(112)

Inserting the inequality (112) for kx2T kL1 into inequality (111) yields,

kx1T kL1 max {�

01

(|x10

|), �1

� �02

(|x20

|), �1

� �u(kukL1)}

and in a similar way we also show that

kx2T kL1 max {�

02

(|x20

|), �2

� �01

(|x10

|), �u(kukL1)}

Note that the right hand side of the preceding two inequalities are less than the R introduced above. So wecan conclude that |x

2

(T )| < R and |x1

(T )| < R, which contradicts the assumption that the state trajectoriesare unbounded.

Having established that the state trajectories are bounded and well-defined for all t � 0, we now move toestablish the ISS property. In particular, the assumptions that the upper and lower systems are ISS means that

kx1

kL1 max {�01

(|x10

|), �1

(kx2

kL1)}kx

2

kL1 max {�02

(|x20

|), �2

(kx1

kL1), �u(kukL1)}

Combining both inequalities and using the fact that we assume �1

(�2

(r)) < r, we obtain

kx1

kL1 max {�01

(|x10

|), �1

� �02

(|x20

|), �1

� �u(kukL1}kx

2

kL1 max {�02

(|x20

|), �2

� �01

(|x10

|), �u(kukL1)}

since the composition of K1 functions is again K1, we can use the above inequalities to conclude that theinterconnected system is ISS. }

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176 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

Example: Let us consider the following system

x1

= �x3

1

+ x1

x2

x2

= ax2

1

� x2

+ u

where a is a real parameter. We can regard this as the feedback interconnection of two scalar systems. Forthe upper system

x1

= f1

(x1

, x2

) = �x3

1

+ x1

x2

we view x1

as the state and x2

as the input. Consider the ISS-certificate

V (x1

) =

1

2

x2

1

and its directional derivative with respect to the upper system is

˙V =

@V

@x2

f1

(x1

, x2

) �|x1

|4 + |x1

|2|x2

|

So choose 0 < ✓ < 1 and redistribute the negative definite term to obtain

˙V �(1 � ✓)|x1

|4 � ✓|x1

|4 + |x1

|2|x2

|

For x1

such that

(1 � ✓)|x1

|2 � |x2

|

we can see that

˙V �↵(|x1

(t)|)

where ↵(r) = ✓r4, which is clearly a class K1 function. So ˙V �↵(|x|) when |x1

| � ⇢(|x2

|) =

q

|x2|1�✓

which shows that f1

is ISS with respect to x2

.

For the lower system

x2

= f2

(x1

, x2

, u) = ax2

1

� x2

+ u

we let x2

be the state and (x1

, u) be the inputs. The candidate ISS-certificate will be V (x2

) =

1

2

x2

2

whosedirectional derivative with respect to f

2

is

˙V =

@V

@x2

f2

(x1

, x2

, u) |x2

|(|a||x1

|2 � |x2

| + |u|)

Again select 0 < ✓ < 1 and redistribute the negative definite term to rewrite ˙V as

˙V = �(1 � ✓)|x2

|2 � ✓|x2

|2 + (a|x1

|2 + |u|)|x2

|

So if x2

satisfies

(1 � ✓)|x2

| � |a||x1

|2 + |u|

then we can conclude ˙V �↵(|x2

|) where ↵(r) = ✓r2 is also K1. In particular this means that if

|x2

| � max{�2

(|x1

|), �u(|u|)}

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6. STABILITY OF FEEDBACK INTERCONNECTIONS 177

where �2

(r) =

2|a|r21�✓ and �u(r) =

2r1�✓ , then ˙V �↵(|x

2

|). This is sufficient to show that the lower systemis ISS with respect to x

1

and u.

So we now have the gains for the two ISS systems. The gain of the upper system is �1

(r) =

q

r1�✓ . The gain

for the lower system is �2

=

2|a|r21�✓ . We now check the small gain condition in theorem 79 to obtain

�2

(�1

(r)) =

2|a|1 � ✓

r

1 � ✓=

2|a|r(1 � ✓)2

which if |a| < 1

2

shows that �2

(�1

(r)) < r and so for this range of a the small gain condition is satisfied andwe can conclude the full system is ISS.

Lp Small Gain Theorem: We now turn to a similar small gain result that holds for the feedback intercon-nection of a pair of Lp stable systems. The system under consideration is shown in Fig. 6 where there are twosystems G

1

: Lpe ! Lpe and G2

: Lpe ! Lpe. The Lp small gain theorem is similar to the ISS small gaintheorem in that the Lp stability of the interconnected system is guaranteed if the product of the Lp gains ofthe subsystems is less than one. The technique used to prove this theorem, however, is much different fromthat used to prove the ISS small gain theorem.

G

G

1

2

Σ

Σ

+

+

+

_

w1

w2y2

y1u1

u2

FIGURE 6. Feedback Interconnection of Lp Stable Systems

THEOREM 80. (Lp Small Gain Theorem) Consider the interconnection shown in Fig. 6 of two systemsG

1

: Lpe ! Lpe and G2

: Lpe ! Lpe where both subsystems are finite gain Lp stable. This means,therefore, that there exist non-negative constants �

1

, �1

, �2

, and �2

such that

ky1T kL

p

�1

ku1T kL

p

+ �1

ky1T kL

p

�2

ku2T kL

p

+ �2

for any T > 0. Then the interconnected system is finite gain Lp-stable if �1

�2

< 1.

Proof: Consider the operator S2

: Lpe ! Lpe defined by the equation

S2

[u2T ] = w

2T + (G1

[w1T + (G

2

[u2T ])T ])T

Consider two Lp signals u2T and u

2T and examine the Lp norm of the difference between S2

[u2T ] and

S2

[u2T ]. This consideration yields,

kS2

[u2T ] � S

2

[u2T ]kL

p

= kG1

[w1T + (G

2

[u2T ])T ] � G

1

[w1T + (G

2

[u2T ])T ]kL

p

�1

kG2

[u2T ] � G

2

[u2T ]kL

p

+ �1

�1

�2

ku2T � u

2T kLp

+ �1

+ �2

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178 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

By assumption �1

�2

< 1, so that S2

is a contraction mapping and we can infer the existence of a unique Lpe

function u2

such that u2

= S2

[u2

]. A similar argument establishes the existence of a unique u1

2 Lpe thatsatisfies

u1

= S1

[u1

] = w1

+ G2

[w2

+ G1

[u1

]]

Since the loop in Fig. 6 is well-posed (i.e. the internal signals, u1

and u2

, exist in Lpe), we can now look atthe Lp norm of the internal signals. Since G

1

and G2

are finite gain Lp stable, we can see for u1

that

ku1T kL

p

kw1T kL

p

+ k(G2

[u2T ])T kL

p

kw1T kL

p

+ �2

ku2T kL

p

+ �2

kw1T kL

p

+ �2

(kw2T kL

p

+ �1

ku1T kL

p

+ �1

) + �2

= �1

�2

kw1T kL

p

+ (ku1T kL

p

+ �2

kw2T kL

p

+ �2

+ �2

�1

)

solving for ku1T kL

p

in the above inequality yields,

ku1T kL

p

1

1 � �1

�2

�kw1tkL

p

+ �2

kw2T kL

p

+ �2

+ �2

�1

and a similar argument shows that

ku2T kL

p

1

1 � �1

�2

�kw2T kL

p

+ �1

kw1T kL

p

+ �1

+ �2

�2

which means the Lp norm of both signals is finite and so the interconnected system is finite gain Lp stable.}

Example: Consider the system

x = f(t, x, v + d1

(t))

✏z = Az + B(u + d2

(t))

v = Cz

where f is smooth, A is Hurwitz, �CA�1B = I , ✏ is a small (perturbation) parameter, and d1

, d2

areexternal Lp disturbances. We assume the input u is a control signal u = k(t, x) where k is chosen toattenuate the impact that disturbances d

1

and d2

have on the state. We therefore want to select k(t, x) so thatthe closed-loop input-output map from (d

1

, d2

) to x is less than �; a constant that we will use to specify thecontrolled system’s performance level.

In the above system, we think of x as the system state. This means that the function f characterizes theplant dynamics. The other dynamical state, z, represents the dynamics associated with the plant’s actuator.In general we assume that the actuator’s dynamics are much faster than those of the plant. This means,therefore, that ✏ is small.

To simplify the controller design problem, we assume that the control engineer neglects the dynamics of theactuator. So he/she designs a controller under the assumption that ✏ = 0 and lets v = �CA�1B(u + d

2

) =

Page 35: Stability Concepts for Input-Output Systemslemmon/courses/ee580/lectures/chapter6.pdf · CHAPTER 6 Stability Concepts for Input-Output Systems The Lyapunov stability concept is a

6. STABILITY OF FEEDBACK INTERCONNECTIONS 179

u + d2

to obtain the reduced order model for the controlled plant,

x = f(t, x, u + d)

where d = d1

+ d2

. Assuming the state variables are available for measurement, the control engineer usesthis reduced order model of the plant to design a state feedback control law, u = k(t, x) that meets the designobjective. Let us assume we have such a control, then because of the performance requirement, this controlledsystem is Lp stable where

kxkLp

�kdkLp

+ �

where � < �. This relationship holds under the assumption that we can neglect the actuator dynamics. Thequestion is whether this is really true. In other words, if we include the actuator dynamics (i.e. ✏ 6= 0) willthe system still be Lp stable and if so will its gain � still be less than the performance requirement �? We canuse the Lp small-gain theorem 80 to answer this question.

When the controller is applied to the actual system, the closed-loop equations take the form

x = f(t, x, Cz + d1

(t))

✏z = Az + B(k(t, x) + d2

(t))

Let us assume that d2

is differentiable with ˙d2

2 Lp. We introduce the following change of variables,

⌘ = z + A�1B(k(t, x) + d2

(t))

so that the closed-loop equations become

x = f(t, x, k(t, x) + d(t) + C⌘)

✏⌘ = A⌘ + ✏A�1B(

˙k +

˙d2

(t))

where

˙k =

@k

@t+

@k

@xf(t, x, k(t, x) + d(t) + C⌘)

The above closed-loop system can be represented as shown in Fig. 6 where G1

is the system

x = f(t, x, k(t, x) + u1

)

y1

=

˙k =

@k

@t+

@k

@xf(t, x, k(t, x) + u

1

)

and the lower system G2

has the state space realization

⌘ =

1

✏A⌘ + A�1Bu

2

y2

= �C⌘

with w1

= d1

+ d2

= d and w2

=

˙d2

. In this representation, the system G1

is the nominal reduced-orderclosed-loop system and system G

2

represents the actuator dynamics that were initially neglected when wedesigned the controller, k.

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180 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

Let us assume the feedback control law, k(t, x), satisfies the following regularity property�

@k

@t+

@k

@xf(t, x, k(t, x) + u

1

)

c1

|x| + c2

|u1

|

for all t, x, and u1

where c1

and c2

are non-negative constants. Combining this property with the fact thatkxkL

p

�kdkLp

+ � yields

ky1

kLp

�1

ku1

kLp

+ �1

where �1

= c1

� + c2

and � � 1 = c1

�. Since G2

is an LTI system where A is Hurwitz, we can easily showthat G

2

is finite-gain Lp stable for any p 2 [1, 1) and so

ky2

kLp

�2

ku2

kLp

+ �2

⌘ ✏�fku2

kLp

+ �2

where

�f =

2�2

(Q)kA�1BkkCk�(Q)

�2

= ⇢kCkk⌘(0)ks

�(Q)

�(Q)

⇢ =

8

<

:

1 if p = 1⇣

2✏�(Q)

p

1/pif p 2 [0, 1)

and Q satisfies the Lyapunov equation QA + ATQ + I = 0. So assuming the feedback connection is well-posed, we conclude from the Lp small-gain theorem that the input-output map from w to u is finite gain Lp

stable and that

ku1

kLp

1

1 � ✏�1

�f

�kw1

kLp

+ ✏�fkw2

kLp

+ ✏�f�1

+ �2

Using the fact that kxkLp

�ku1

kLp

+ � we can bound the state x as

kxkLp

1 � ✏�1

�f

kdkLp

+ ✏�fk ˙d2

kLp

+ ✏�f�1

+ �2

+ �

which implies the internal stability of the interconnected system for sufficiently small ✏. In particular, thesmall gain condition requires

✏ <1

�1

�f

Recall that �1

is the I/O gain of the controlled nominal system from u1

to y1

and �f is the gain of the actuatorsystem.

Passivity Theorem for Feedback Interconnections: We now turn to study the passivity of feedback inter-connects. This result differs significantly from the prior small-gain results in that it asserts that any feedbackinterconnect of passive systems is again passive. In particular, we see that passivity is preserved under feed-back interconnections. This is useful in studying large complex networked dynamical systems [MH78] thatconsist solely of feedback interactions. It suggests that if you have a large-scale passive system, then thefeedback interconnection of the large-scale system with another passive system does not destroy the passivity

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6. STABILITY OF FEEDBACK INTERCONNECTIONS 181

of the larger system. This result, therefore, provides a modular way to build passive networked systems ofarbitrarily large scale.

G

G

1

2

Σ

Σ

+

+

+

_

u1

u 2y2

y1e 1

e 2

FIGURE 7. Feedback Interconnection of Passive Systems

THEOREM 81. (Passivity of Feedback Interaconnects) Consider the feedback connection shown in Fig. 7of two systems G

1

and G2

with state equations,

xi = fi(xi, ei)

yi = hi(xi, ei)

for i = 1, 2 where e1

= u1

� y2

and e2

= u2

+ y1

. If systems G1

and G2

are passive, then the feedbackconnection is also passive.

Proof: Let V1

and V2

be storage functions for G1

and G2

, respectively. Since these systems are passive, weknow

˙Vi eTi yi

Let V = V1

+ V2

and from our feedback connections we see that

eT1

y1

+ eT2

y2

= (u1

� y2

)

T y2

+ (u2

+ y1

)

T y2

= uT1

y1

+ uT2

y2

This implies that

uT y = uT1

y1

+ uT2

y2

� ˙V1

+

˙V2

=

˙V

and so the feedback system is also passive. }

One of the reasons we are interested in the feedback connection being passive is that it can be used todetermine if the connection is L

2

-stable. The following theorem shows how we can use theorem 81 toestablish the L

2

stability of two interconnected passive systems.

THEOREM 82. (L2

-stability of Passive Feedback Systems) Consider the feedback connection in Fig. 7where G

1

and G2

are two passive systems with storage functions V1

and V2

, respectively such that

eTi yi � ˙Vi + ✏ieTi ei + �iy

Ti yi

for i = 1, 2. Then the feedback system is finite gain L2

stable if ✏1

+ �2

> 0 and ✏2

+ �1

> 0.

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182 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

Proof: Let V = V1

+ V2

and note that we can rewrite ˙V as

˙V =

˙V1

+

˙V2

�yT

"

(✏2

+ �1

)I 0

0 (✏1

+ �2

)I

#

y � uT

"

✏1

I 0

0 ✏2

I

#

+ uT

"

I 2✏1

I

�2✏2

I I

#

Let a = min{✏2

+ �1

, ✏1

+ �2

}, b =

"

I 2✏1

I

�2✏1

I I

#

� 0, and c =

"

✏1

I 0

0 ✏2

I

#

� 0, then

˙V �a|y|2 + b|u||y| + c|u|2

= � 1

2a(b|u| � a|y|)2 +

b2

2a|u|2 � a

2

|y|2 + c|u|2

k2

2a|u|2 � a

2

|y|2

where k2

= b2

+ 2ac. Integrating over [0, T ] and using the vector hat V (x) � 0, we obtain

kyT kL2 k

akuT kL2 +

p

2V (x(0)/a

which establishes that the feedback interconnection is finite gain L2

stable with a gain less than ka and a bias

ofp

2V (x(0))/a. }

Example: Consider the feedback connection

G1

:

(

x = f(x) + g(x)e1

y1

= h(x)

G2

: y2

= ke2

with k > 0. Suppose a positive definite V : Rn ! R exists such that

@V

@xf(x) 0

and

@V

@xg(x) = hT

(x)

This means that the directional derivative of V is

˙V =

@V

@xf(x) +

@V

@xg(x)e

1

yT1

e1

which means that G1

is passive. The second system is memoryless and so it too is passive. Now note that

eT2

y2

= keT2

e2

= �keT2

e2

+

1 � �

kyT2

y2

So the conditions in theorem 82 are satisfied with ✏1

= �1

= 0, ✏2

= �k and �2

=

1��k . So by the preceding

theorem, this means the entire interconnected system is finite gain L2

stable.

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6. STABILITY OF FEEDBACK INTERCONNECTIONS 183

Example: Now consider a specific case of the prior example in which the upper system is

x1

= x2

x2

= �ax3

1

� �(x2

) + e1

y1

= x2

with the second system being y2

= ke2

. We assume � 2 [�↵, 1) where a, ↵, and k are positive. For theupper system’s storage function we select

V1

=

a

4

x4

1

+

1

2

x2

2

The directional derivative of V1

is

˙V1

= ax3

1

x2

� ax3

1

x2

� x2

�(x2

) + x2

e1

= ↵x2

2

+ x2

e1

= ↵y2

1

+ y1

e1

where ✏1

= 0 and �1

= �↵. For the second system we see that

e2

y2

= ke2

2

= �ke2

2

+

1 � �

ky2

2

where 0 < � < 1. This means that we take ✏2

= �k and �2

=

1��k . If k > ↵ we can choose � so that

�k > ↵. This would imply that ✏1

+ �2

> 0 and ✏2

+ �1

> 0. So again we can use our earlier theorem toconclude that the feedback connection is finite-gain L

2

stable.

We now consider the feedback interconnection of two passive systems when the external input u = 0. In thiscase we want to know whether the resulting interconnection is asymptotically stable. The following theoremprovides such conditions.

THEOREM 83. Consider the feedback connection of two time-invariant dynamical systems where u = 0. Theorigin is asymptotically stable if

• both feedback components are strictly passive or• both feedback components are output strictly passive and zero state observable.

Proof: Let V1

and V2

be the storage functions for G1

and G2

, respectively. Let V (x) = V1

(x1

) + V2

(x2

) bea candidate Lyapunov function for the closed loop system. In the first case,

˙V uT y � 1

(x1

) � 2

(x2

) = � 1

(x1

) � 2

(x2

)

with u = 0 where 1

and 2

are positive definite functions. This is sufficient to establish that the origin isasymptotically stable. In the second case,

˙V �yT1

⇢1

(y1

) � yT2

⇢2

(y2

)

where yTi ⇢i(yi) > 0 for i = 1, 2 and all yi 6= 0. Here ˙V is only negative semi-definite and ˙V = 0 implies

y = 0. To use the Invariance principle, we need to show that y(t) = 0 for all t implies x(t) = 0. Note thaty2

(t) = 0 implies e1

(t) = 0 and the zero-state observability of G1

implies that if y1

(t) = 0, then x1

(t) = 0.

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184 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

A similar argument applies for G2

and so the origin must be asymptotically stable by the Invariance principle.}

The proof uses the idea that the sum of the storage functions for the feedback components can be used as acandidate Lyapunov function for the feedback connection. The preceding analysis is restrictive in the sensethat for ˙V =

˙V1

+

˙V2

< 0, we require both ˙V1

0 and ˙V2

0. This is not necessary. One term, ˙V1

, forinstance could be positive as long as ˙V

2

is sufficiently negative that the sum of both is negative. This idea isexploited in the following examples.

Example: Consider the feedback connection

G1

:

8

>

>

<

>

>

:

x1

= x2

x2

= �ax3

1

� kx2

+ e1

y1

= x2

G2

:

8

>

>

<

>

>

:

x3

= x4

x4

= �bx3

� x3

4

+ e2

y2

= x4

where a,b, and k are positive constants. Let V1

=

a4

x4

1

+

1

2

x2

2

as the storage function for H1

. we obtain

˙V1

= ax3

1

x2

� ax3

1

x2

� kx2

2

+ x2

e1

= �ky2

1

+ y1

e1

So H1

is output strictly passive. With e1

= 0, we have

y1

(t) ⌘ 0 , x2

(t) ⌘ 0 ) x1

(t) ⌘ 0

which shows that H1

is zero-state observable. Using V2

=

b2

x2

3

+

1

2

x2

4

as the storage function for H2

, weobtain

˙V2

= bx3

x4

� bx3

x4

� x4

4

+ x4

e2

= �y4

2

+ y2

e2

So H2

is strictly output passive and with e)2, we have

y2

(t) ⌘ 0 , x4

(t) ⌘ 0 ) x3

(t) ⌘ 0

which shows H2

is zero-state observable. Thus by the second case in the above theorem, and the fact that V1

and V2

are radially unbounded, we conclude that the origin is globally asymptotically stable.

Example Now consider the feedback connection of the previous example, but change the output of G1

toy1

= x2

+ e1

. From the expression

˙V1

= �kx2

2

+ x2

e1

= �k(y1

� e1

)

2 � e2

1

+ y1

e1

we conclude G1

is passive, but we cannot conclude strict passivity or output strict passivity. Therefore wecannot apply the preceding theorem. Using

V = V1

+ V2

=

1

4

ax4

1

+

1

2

x2

2

+

1

2

bx2

3

+

1

2

x2

4

as a candidate Lyapunov function for the closed loop system we obtain

˙V = �kx2

2

+ x2

e1

� x4

4

+ x4

e2

= �kx2

2

� x2

x4

� x4

4

+ x4

(x2

� x4

)

= �kx2

2

� x4

4

� x2

4

0

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7. STABILITY OF CASCADED INTERCONNECTIONS 185

Moreover, ˙V = 0 implies that x2

= x4

= 0 and

x2

(t) ⌘ 0 ) ax3

1

(t) � x4

(t) ⌘ 0 ) x1

(t) ⌘ 0

x4

(t) ⌘ 0 ) �bx3

(t) + x2

(t) ⌘ 0 ) x3

(t) ⌘ 0

So by the invariance principle and the fact that V is radially unbounded we conclude the origin is globallyasymptotically stable.

7. Stability of Cascaded Interconnections

Based on the definitions for the various input-output stability concepts, it should be apparent that the parallelcomposition of two stable systems will preserve that stability. At first glance, one might also think that thecascade (series) connection of any two stable systems will also preserve stability, but this is not always trueas can be demonstrated in the following example.

Example: Consider the cascaded system where the driving system G1

in Fig. 2 has the state space realization

˙⇠1

= ⇠2

˙⇠2

= ��2⇠1

� 2�⇠2

+ u

y1

= ⇠2

(113)

and the driven system, G2

, has the state space realization

⌘ = � 1

2

(1 � y1

)⌘3

y = ⌘(114)

Note that the driving system is a linear system and where � > 0. The question is whether the origin of thiscascaded system is asymptotically stable.

u yy1

G1 G2

FIGURE 8. Cascade Connection of Two Input-Output Systems

The above system is a cascaded system in which the driving system when u = 0 is a linear system of theform

"

˙⇠1

˙⇠2

#

=

"

0 1

��2 �2�

#"

⇠1

⇠2

#

The state transition matrix for this linear system is

�(t) =

"

(1 + �t)e��t te��t

��2te��t(1 � �t)e��t

#

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186 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

Note that the (2, 1) element has a �2 term so that for large enough �, this term may have an extremely largepeak. In particular if we let the initial condition for G

1

be ⇠1

(0) = 1 and ⇠2

(0) = 0, then ⇠2

(t) = ��2te��t

whose plot is shown on the left side of Fig. 9 for � = 10. Note that we see a large negative excursion in ⇠2

before it returns to 0. This phenomena is referred to as peaking [SK91].

time, t0 1 2 3 4 5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.2 0.4 0.6 0.80

5

10

15

20

25

30

35

40

45

time, t

FIGURE 9. Peaking in the driving system triggers a finite escape in the driven system- (left)The state of the driving system, ⇠, over 5 seconds - (right) The state of the driven systemshowing a finite escape at 0.755 sec.

While ⇠2

is negative, we see that ⌘(t) will be increasing. In fact if we insert our closed form expression for⇠2

(t) into the differential equation for G2

, we get

⌘ = �1

2

(1 � �2te��t)⌘3

This ODE is separable and we can therefore integrate it to obtain the following

⌘2

(t) =

⌘2

0

1 + ⌘2

0

(t + (1 + �t)e��t � 1)

Note that the denominator may go to zero for a finite t. This would mean that ⌘(t) becomes unbounded at afinite time. The specific parameters chosen for Fig. 9 exhibit this finite escape time at t ⇡ 0.755. The righthand pane of the figure shows this finite escape.

What this example shows is that even though both cascaded systems are asymptotically stable when thereis no input, the cascade combination of these “stable” systems is unstable. Note that both of these systemsare passive, and so clearly, the cascade combination of the two passive systems may not necessarily lead to astable system. The question is whether the other stability concepts we’ve introduced (ISS and Lp-stability)also suffer from the same problem. It is relatively easy to show that cascades of ISS or Lp stable systems willpreserve the underlying stability concept. This is one important way in which these other stability conceptsdiffer from passivity.

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7. STABILITY OF CASCADED INTERCONNECTIONS 187

Finally, it is important to say something about why such cascaded systems are of interest to us. In particu-lar, one important way of synthesizing controllers for nonlinear systems is through a feedback linearizationprocess that was introduced in chapter 1. This linearization automatically generates a cascade of linear sys-tems. Clearly, for this case the stability or stabilizability of such cascades will be an important theme in thedevelopment of nonlinear control systems.

The fact the the cascade of two Lp-stable system will again be Lp is rather easy to establish. This result isformalized in the following theorem.

THEOREM 84. Consider the cascade connection of a driving system G1

: Lpe ! Lpe and a driven systemG

2

: Lpe ! Lp. If G1

and G2

are both finite gain Lp stable then the cascaded system G2

G1

is also finitegain Lp stable.

Proof: Since G1

and G2

are both finite gain Lp stable there exist positive constants �1

, �2

, �1

, and �2

suchthat for all T > 0

ky1T kL

p

�1

ku1T kL

p

+ �1

ky2T kL

p

�2

ku2T kL

p

+ �2

Since the output of system G1

is driving the input to system G2

, we can rewrite the second inequality as

ky2T kL

p

�2

�1

ku1T kL

p

+ �1

+ �2

= �2

�1

ku1T kL

p

+ (�2

�1

+ �2

)

which shows that the cascaded system is finite-gain Lp stable with a gain of �2

�1

and a bias of �2

+ �1

+ �2

.}

A similar result an be established with the cascade in Fig. 2 consists of two input-to-state stable (ISS) systems.Since we are dealing with ISS systems, we will take y

1

= h1

(x1

, u) = x1

and y2

= h2

(x2

, y1

) = x2

inFig. 2. Recall that a system x = f(x, u) is ISS is there exists a C1 function V : Rn ! R, class K1 functions,↵, ↵, ↵ and a class K function � such that

↵(|x|) V (x) ↵(|x|)(115)@V

@xf(x, u) �↵(|x|) + �(|u|)(116)

for any x 2 Rn and u 2 Rm. Recall also that we refer to the pair (↵,�) as an ISS pair for the systemx = f(x, u). If we are to establish that the cascaded system is ISS, we need to show how to find an ISS pairfor the cascade from the ISS pairs of the driving and driven subsystems. The following theorem provides atechnical lemma that will help us do this.

THEOREM 85. (ISS Pairs) Assume (↵,�) is an ISS pair for x = f(x, u).

• Let � be a class K function such that limr!1�(r)�(r) is finite (i.e. � is o(�) as r ! 1). Then there

exists a class K1 function ↵ such that (↵, �) is an ISS pair for the system.

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188 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

• Let ↵ be a class K1 function such that limr!0

↵(r)↵(r) is finite (i.e. ↵ is o(↵) as r ! 0). Then there

exists a class K function � such that (↵, �) is an ISS-pair for the system.

Proof: We consider a candidate ISS-certificate W (x) of the form

W (x) = ⇢(V (x))

where V is an ISS-certificate for the system x = f(x, u) that satisfies the inequalities (115-116) and where ⇢is a class K1 function defined by an integral of the form

⇢(s) =

Z s

0

q(t)dt

in which q : R�0

! R�0

is a smooth nondecreasing function such that q(s) > 0 for s > 0 (we refer to suchfunctions as being of class SN ).

Let us examine the directional derivative of W with respect to f(x, u) to get

@W

@xf(x, u) = q(V (x))

@V

@xf(x, u) q(V (x)) [�↵(|x|) + �(|u|)](117)

Set ✓(s) = ↵(↵�1

(2�(s))) and note that the right hand side of inequality (117) can be rewritten in terms of✓ to get

@W

@xf(x, u) �1

2

q(↵(|x|))↵(|x|) + q(✓(|u|))�(|u|)(118)

The theorem’s first assertion will be established if we can show that it is possible to find q and ↵ such that

q(↵(s))↵(s) � 2↵(s)(119)

q(✓(r))�(r) �(r)(120)

We can, without loss of generality, suppose that � is class K1 (since if it is not we can only majorize it by aclass K1 function). This would mean that ✓ is also class K1 and we can then define two functions � and ˜�

from the relations

�(r) = �(✓�1

(r)), ˜�(r) = �(✓�1

(r))

We can readily verify that both � and ˜� are also class K1. So we can use this fact to see that there exists aclass SN function q (smooth and non-decreasing) such that

q(r)�(r) ˜�(r)

for all r 2 R�0

. Therefore

q(✓(r))�(r) �(r)(121)

which establishes inequality (120).

To finish the first part, we need to verify that inequality (119) holds. In particular, define ↵ by the relation

↵(s) =

1

2

q(↵(s))↵(s)(122)

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7. STABILITY OF CASCADED INTERCONNECTIONS 189

This is class K1 because ↵ and q are both class SN . Taking inequalities (121-122) together verifies thatinequality (119) holds and this therefore establishes the first assertion of the theorem.

To prove the second part, we need to find q and � such that inequalities (119-120) hold. So let us definefunctions � and ˜� by the relations

�(s) =

1

2

↵(✓�1

(s)), ˜�(s) = ↵(✓�1

(s))

These functions are such that lims!0

˜�(s)�(s) is finite. Using this property, we can show that there exists a class

SN function q such that

˜�(s) q(s)�(s)

for all s 2 R�0

. This means, therefore, that

�1

2

q(✓(s))↵(s) �↵(s)

which establishes inequality (119). We can establish inequality (120) by simply selecting

�(r) = q(✓(r))�(r)

and then proving that � is class K1, thereby establishing the theorem’s second assertion and completing theproof. }

Theorem 85 can now be readily used to establish that the cascade of two ISS systems is again ISS. This resultis formalized in the following theorem.

THEOREM 86. (Cascade of ISS Systems) Consider the cascaded

x = f(x, z)

z = g(z, u)

where f(0, 0) = 0, g(0, 0) = 0 with f and g being locally Lipschitz. Suppose that the upper (driven) systemis ISS with respect to input z. Suppose that the lower (driving) system is ISS with respect to input u. Then thecascaded system is ISS with respect to input u.

Proof: By hypothesis, there exist an ISS pair (↵,�) for the driven system and an ISS pair (�, ⇣) for thedriving system. Define a function ˜beta as

˜�(s) =

(

�(s) for small s

�(s) for large s

Then by the second assertion in the ISS-pair theorem 85 there exists ˜⇣ such that (

˜�, ˜⇣) is an ISS pair for thedriving system. Also define a function � as

�(s) =

1

2

˜�(s)

Then by the first assertion in the ISS-pair theorem 85, there exists ↵ such that (↵, 1

2

˜�) is an ISS pair for thedriven system.

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190 6. STABILITY CONCEPTS FOR INPUT-OUTPUT SYSTEMS

This means, therefore, there exist positive definite and proper functions V and U such that@V

@xf(x, z) �↵(|x|) +

1

2

˜�(|z|)@U

@zg(z, u) �˜�(|z|) +

˜⇣(|u|)Consider the sum of these two functions, W (x, z) = V (x) + U(z) and note that it satisfies

@W

@xf(x, z) +

@W

@zg(z, u) �↵(|x|) � 1

2

˜�(|z|) +

˜⇣(|u|)and so W is an ISS certificate for the cascaded system and the proof is complete. }

Let us return to our peaking system and re-examine the driving and driven system with regard to Lp-stability,ISS stability, and passivity. The simulation result in Fig. 9 show that the driven system in equation (114) isnot Lp and is not ISS. This is true because the input to the driven system is ⇠ and the simulation establishesthe existence of an input ⇠ with finite Lp norm such that the driven system’s output becomes unbounded. Soclearly we cannot apply theorems 84 or 86. In fact, this example shows that ensuring the two systems are ISSor Lp stable are critical conditions that when violated can lead to unstable cascades.

If we examine the driving and driven system, with regard, to passivity, we see that the driving system inequation (113) is a linear system whose A matrix is Hurwitz. This is sufficient to establish that the drivinglinear system is strictly passive. If we examine the driven system, let us consider the storage function

V (⌘) =

1

2

⌘2

and if we compute its directional derivative with respect to the driven system, we obtain

˙V = ⌘(�1

2

(1 � u)⌘3

) = �1

2

(1 � u)⌘4

= �1

2

⌘4

+

1

2

u⌘4

If we define the supply rate function r(u, ⌘) =

1

2

u⌘4, then clearly

r(u, ⌘) ˙V + (⌘)

where (r) =

1

2

r4 is positive definite. We can therefore conclude that the driven system is also strictlydissipative. In this regard, both systems are strictly dissipative and so both are asymptotically stable whenthe input is zero. However, our simulation results show that the cascade is not asymptotically stable and soestablishing the dissipative nature of each subsystem is not sufficient to assure the stable or dissipative natureof the whole.

8. Concluding Remarks

This chapter reviewed some of the most popular input-output stability concepts that are used in the design ofnonlinear control systems. The basic concepts reviewed here were input-to-state stability (ISS), Lp stability,and passivity/dissipativity. In all cases we showed how these concepts were related back to the traditional no-tion of Lyapunov stability which only applies to undriven systems. So, if we take an input-output system andtake the input as a control signal based on the system’s output, it becomes possible to establish the Lyapunovstability of the interconnected “plant” and “controller” provided we can assure these two subsystems possess

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8. CONCLUDING REMARKS 191

some form of input-output stability. For the Lp and ISS stability concepts, we needed to place a small-gaincondition on the interconnection. For strictly dissipative systems, we do not need to place any restrictionslike the small gain theorem. In later chapters, we will therefore take advantage of these stability concepts insynthesizing controllers that solve the regulation problem for a nonlinear dynamical system.

The material in this chapter was taken from a variety of sources. The material on perturbed and UUB systemsand Lp stability was taken from [Kha96]. The ISS material was based on the work in [Isi99] and the passivitysection was based on [Van00]. It is important to note that these are not the original references in which theseconcepts were introduced. Dissipativity and ISS stability concepts are relatively recent and it is important tonote some of the original sources for this work. The concept of dissipativity was introduced by J. Willemsin the context of so-called behavioural control [Wil72]. A great deal of work was done to characterize therelationship between passivity and Lyapunov stability [HM76, HM77, BIW91]. The notion of Input-to-Statestability originates with E. Sontag [SW95, SW96, ASW00]. The peaking phenomenon that was used in thediscussion of cascaded systems will be found in [SK91]. The one example using ISS to study event-triggeredcontrol originated in [Tab07] and was generalized to networked control systems in [WL11].