input/output stability and passivityyuw/file/sa3.pdf · lasalle theorem example if the origin of...

Input/Output Stability and Passivity

Wen Yu

Departamento de Control AutomáticoCINVESTAV-IPN

A.P. 14-740, Av.IPN 2508, México D.F., 07360, México

(CINVESTAV-IPN) Stability Theory January 27, 2020 1 / 22

LaSalle Theorem - Invariant set theorem

DenitionA set S is an invariant set for a dynmaic system, if every system trajectorywhich starts from a poin in S remain in S for all future time

Theorem(Local Invariant Set Theorem) For some l > 0, the set Ωl is dened by

Ωl : V (x) < l , and V (x) 0, x 2 Ωl

the set R isR Ωl : V (x) = 0

M is the largest invariant set in R, then every solution x 2 Ωl ,x ! M,when t ! ∞


LaSalle Theorem

Theorem

If V (x) 0 and V (x) 0, the set R is the point V (x) = 0 only containthe trajectory x = 0

R Ωl : V (x) = 0, x = 0

then the equilibrium point xe = 0 is asymptotically stable.


LaSalle Theorem

Examplesystem

x + b (x) + c (x) = 0, xc (x) > 0 for x 6= 0b (x) = 0 as x = 0, xb (x) > 0 for x 6= 0

Lyapunov function

V =12x2 +

Z x

0c (y) dy

thenV = x x + c (x) x = xb (x) 0

Because xb (x) > 0 for x 6= 0

xb (x) = 0, i¤ x = 0

Because when x = 0, b (x) = 0, this mplies that

x = c (x) , x = 0

Because x 6= 0, then xc (x) 6= 0, c (x) 6= 0, x 6= 0. If R is dened as

R : x = 0

the largest invariant set M of R containes only one point, x = 0, x = 0and x = 0. So the origion is a locally asymptotically stable point.


LaSalle Theorem

ExampleIf the origin of the closed-loop equation is a stable equilibrium. Dene Ωas

Ω =x (t) =

q, q, ξ

2 R3n : V = 0

For a solution x (t) to belong to Ω for all t 0, it is necessary andsu¢ cient that q = q = 0 for all t 0. Therefore it must also hold thatq = 0 for all t 0. We conclude that from the closed-loop system ifx (t) 2 Ω for all t 0, then

g (q) = gqd= ξ + g

qd,ξ = 0

It implies that ξ = 0 for all t 0. V = 0 if and only if q = q = 0. Weconclude from all this that the origin of the closed-loop system is locallyasymptotically stable.


Barbalat Lemma

LemmaIf f (t) is uniformly continuous (t 0), and the following limit exists

limt!∞

Z t

0jf (τ)j dτ

then limt!∞

f (t) = 0.

Lemma

If f (t) has a nite limit as t ! ∞, f is uniformly continuous, or f isbounded, then

limt!∞

f = 0


Barbalat Lemma

Lemma

If f (t) and f are bounded 2 L∞, and f (t) 2 L2Z ∞

0jf (t)j2 dt < ∞

thenlimt!∞

f (t) = 0


Denitions

For linear time invariant system (LTI), the input-output relations are

time domain: (convolution)

y(t) =Zh(t τ)u(τ)dτ

where h(t) is impulse response

frequency domain

y(s) = H(s)u(s), H(s) = L [h(t)] , Laplace transform

The H can be nonlinear


stability

Lpthe system is said to be Lp stable if

u 2 Lp ) y 2 Lp

function x 2 Lp when

kxkp =Z ∞

0jx (τ)jp dτ

1/p

exists. So we havekykp c kukp , c > 0


stability

Lp

if p = ∞, Lp stability is bounded-input bounded-output (BIBO)stability.

Exponentially weighted L2 norm

kxk2δ =

Z ∞

0eδ(tτ) jx (τ)j2 dτ

1/2

if kxk2δ exist, we say the x 2 L2δ.


Theorems

For LTI, if h 2 L1,

u 2 Lp ) kykp khk1 kukpu 2 L2 ) kyk2 sup jH (jω)j kuk2

There exists a positive denite function V (x , t) and its partial derivativesV , and V (0, t) = 0, following statements


Theorems

For LTI (x = Ax + Buy = Cx

where H(s) = CT (SI A)1 B, if A is asymptotically stable, then

u 2 L∞ ) y 2 L∞, u 2 Lp ) y 2 Lplimt!∞

u = u ) limt!∞

y = H(0)u

u 2 L1 ) y 2 L1 \ L∞, limt!∞

y = 0

u 2 L2 ) y 2 L2 \ L∞, limt!∞

y = 0


Theorems

Small gain theorem: If H1, H2 are bounded, e1, e2 are bounded and

kH1e1k γ1 ke1k+ β1kH2e2k γ2 ke2k+ β2

γ1γ2 < 1

thenke1k (1 γ1γ2)

1 (ku1k+ γ2 ku2k+ β2 + γ2β1)

ke2k (1 γ1γ2)1 (ku2k+ γ1 ku1k+ β1 + γ1β2)

If ku1k and ku2k are bounded then BIBO


Theorems

Bellman-Gronwall Lemma: Allows one to bound a function that satises aintegral inequality by the solution of the corresponding integral equation.If λ (t) , k (t) 0, f satises

f (t) λ (t) +Z t

t0k (s) f (s) ds

then

f (t) λ (t) +Z t

t0λ (s) k (s) e

R ts k (τ)dτds

If, in addition, the function λ (t) is non-decreasing, then

f (t) λ (t) eR ts k (τ)dτ


Passivity-Denitions

Let us consider a SISO nonlinear system given by

x = f (x) + g(x)u,y = h(x)

where x 2 Rn, u 2 R, y 2 R , the vector elds. f and g are assumed tobe in C∞, and h is a CWe say a control u is admissable (u 2 Uad ) if the energy stored insystem is bounded Z ∞

0j y(s)u(s) j ds < ∞

where u 2 U (a known subset of R),for any initial x0, the correspondingoutput

y(t) = h(Φ(t, x0, u))

This assumption is very common for many mechanical andelectromechanical systems and is widely exploited for control purposes.


Passivity-Denitions

DenitionA system is said to be C r -passive if there exists a C r nonnegative functionV : Rn ! R, called storage function, with V (0) = 0, such that, for allu 2 Uad , all initial x0 and all t 0 the following inequality holds:

V (x(t)) V (x0) Z t

0y(s)u(s)ds.

DenitionThe system is said to be C r -passive if there exists a C r nonnegativefunction V : <n ! <, called storage function, with V (0) = 0, such that,for all u 2 Uad , all initial conditions x0 and all t 0 the followinginequality holds

V (z) yu


Passivity-Denitions

DenitionIf

V (x(t)) V (x0) =Z t

0y(s)u(s)ds,

then the system is said to be C r -lossless.

DenitionIf there exists a positive denite function S : Rn ! R such that

V (x(t)) V (x0) =Z t

0y(s)u(s)ds

Z t

0S(s)ds,

then the system is said to be strictly C r -passive.


Passivity-Denitions

DenitionA system is said to be locally feedback equivalent to a C r -passivesystem, or just locally feedback C r -passive, if there exists a feedback law

u = α(x) + β(x)w

(where β(x) 6= 0 in a neighborhood of x = 0) such that the system withthe new input w 2 R is C r -passive. If the closed-loop system is C r -losslessor strictly C r -passive, then the system is said to be locally feedbackC r -lossless or locally feedback strictly C r -passive, respectively.

DenitionA system is said to be locally feedback equivalent to a C r -passive system,or just locally feedback C r -passive, if there exists a feedback law

u = α(z) + β(z)v (1)

where β(z) 6= 0 in a neighborhood of z = 0, and such that the systemwith the new input v is C r - passive.


PassivityDenitions

Consider a class of nonlinear systems described by

x t = f (xt , ut ), yt = h(xt , ut ) (2)

It is assumed that for any x0 = x0 2 <n, the output yt = h(Φ(t, x0, u))of system is such that

R t0 j uTs ys j ds < ∞, for all t 0, i.e,. the energy

stored in system is bounded.


PassivityDenitions

DenitionA system is said to be passive from input ut to output yt , if there exists aC r nonnegative function S (xt ) : <n ! <, called storage function, suchthat, for all ut , all initial conditions x0 and all t 0 the followinginequality holds:

S(xt ) uTt yt εuTt ut δyTt yt ρψ (xt ) , (xt , ut ) 2 <n <m .

where ε, δ and ρ are nonnegative constants, ψ (xt ) is positive semidenitefunction of xt such that ψ (0) = 0. ρψ (xt ) is called state dissipation rate.


PassivityDenitions

Furthermore, the system is said to be

Denition

lossless if ε = δ = ρ = 0 andS(xt ) = uTt yt ;

input strictly passive if ε > 0

output strictly passive if δ > 0

state strictly passive if ρ > 0

strictly passive if there exists a positive denite function

V (xt ) : <n ! < such thatS(xt ) uTt yt V (xt )


PassivityDenitions

DenitionThe system is said to be strictly passive if there exists a positive denite

function V (xt ) : <n ! < such thatS(xt ) uTt yt V (xt )

TheoremIf the storage function S(xt ) is di¤erentiable and the dynamic system is

passive, storage function S(xt ) satisesS(xt ) uTt yt .


input/output stability and passivityyuw/file/sa3.pdf · lasalle theorem example if the origin of...

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