Post on 17-Apr-2020
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 ! ∞
(CINVESTAV-IPN) Stability Theory January 27, 2020 2 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 3 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 4 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 5 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 6 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 7 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 8 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 9 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 10 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 11 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 12 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 13 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 14 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 15 / 22
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
(CINVESTAV-IPN) Stability Theory January 27, 2020 16 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 17 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 18 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 19 / 22
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.
(CINVESTAV-IPN) Stability Theory January 27, 2020 20 / 22
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 )
(CINVESTAV-IPN) Stability Theory January 27, 2020 21 / 22
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 .
(CINVESTAV-IPN) Stability Theory January 27, 2020 22 / 22