adaptive control and the definition of exponential stability · definition: uniform stability in...

Adaptive Control and the Definition of

Exponential Stability

Travis E. Gibson† and Anuradha M. Annaswamy‡

†

‡

American Control Conference, Chicago ILJuly 1, 2015

Objectives

Prove that the following statement is incorrect

“If the reference model is persistently exciting then the adaptivesystem is globally exponentially stable”

2 / 15

Objectives

Prove that the following statement is incorrect

“If the reference model is persistently exciting then the adaptivesystem is globally exponentially stable”

Prove the following

Adaptive systems can at best be uniformly asymptotically stable inthe large

2 / 15

Objectives

Prove that the following statement is incorrect

“If the reference model is persistently exciting then the adaptivesystem is globally exponentially stable”

Prove the following

Adaptive systems can at best be uniformly asymptotically stable inthe large

Main insights

Indeed if the reference model is PE then after some time the plantwill be PE, but after exactly how much time?

We will show how a PE condition on the reference model implies aweak PE condition on the plant state.

2 / 15

Outline

Definitions Stability Exponential Stability Persistent Excitation (PE) weak Persistent Excitation (PE∗)

Link between PE and Exponential Stability

Link between PE∗ and Uniform AsymptoticStability

Simulation Studies

3 / 15

Uniform Stability in the Large (Global)

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

x0 , x(t0)

Solution s(t;x0, t0)t

x0

s

4 / 15

Uniform Stability in the Large (Global)

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

x0 , x(t0)

Solution s(t;x0, t0)δ

ǫ

tx0

t0

s

Definition: Uniform Stability in the Large (Massera, 1956)

(i) Uniformly Stable: ∀ ǫ > 0 ∃ δ(ǫ) > 0 s.t.‖x0‖ ≤ δ =⇒ ‖s(t;x0, t0)‖ ≤ ǫ.

4 / 15

Uniform Stability in the Large (Global)

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

x0 , x(t0)

Solution s(t;x0, t0)δ, ρ

ǫ

t

η

x0

t0

t0 + Ts

Definition: Uniform Stability in the Large (Massera, 1956)

(i) Uniformly Stable: ∀ ǫ > 0 ∃ δ(ǫ) > 0 s.t.‖x0‖ ≤ δ =⇒ ‖s(t;x0, t0)‖ ≤ ǫ.

(ii) Uniformly Attracting in the Large: For all ρ,η ∃ T (η,ρ)‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ η ∀ t ≥ t0 + T .

4 / 15

Uniform Stability in the Large (Global)

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

x0 , x(t0)

Solution s(t;x0, t0)δ, ρ

ǫ

t

η

x0

t0

t0 + Ts

Definition: Uniform Stability in the Large (Massera, 1956)

(i) Uniformly Stable: ∀ ǫ > 0 ∃ δ(ǫ) > 0 s.t.‖x0‖ ≤ δ =⇒ ‖s(t;x0, t0)‖ ≤ ǫ.

(ii) Uniformly Attracting in the Large: For all ρ,η ∃ T (η,ρ)‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ η ∀ t ≥ t0 + T .

(iii) Uniformly Asymptotically Stable in the Large (UASL)= uniformly stable + uniformly bounded +

uniformly attracting in the large.4 / 15

Exponential Stability

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

x0 , x(t0)

Solution s(t;x0, t0)

ρ

κ‖x0‖

t

x0

t0

κ‖x0‖e−ν(t−t0)

s

Definition: (Malkin, 1935; Kalman and Bertram, 1960)

(i) Exponentially Stable (ES): ∀ ρ > 0 ∃ ν(ρ),κ(ρ) s.t.

‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ κ‖x0‖e−ν(t−t0)

5 / 15

Exponential Stability

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

x0 , x(t0)

Solution s(t;x0, t0)

ρ

κ‖x0‖

t

x0

t0

κ‖x0‖e−ν(t−t0)

s

Definition: (Malkin, 1935; Kalman and Bertram, 1960)

(i) Exponentially Stable (ES): ∀ ρ > 0 ∃ ν(ρ),κ(ρ) s.t.

‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ κ‖x0‖e−ν(t−t0)

(ii) Exponentially Stable in the Large (ESL): ∃ ν,κ s.t.

‖s(t;x0, t0)‖ ≤ κ‖x0‖e−ν(t−t0)

5 / 15

Persistent Excitation

“Exogenous Signal” : ω : [t0,∞) → Rp

Initial Condition : ω0 = ω(t0)

Parameterized Function : y(t, ω) : [t0,∞)× Rp → R

m

6 / 15

Persistent Excitation

“Exogenous Signal” : ω : [t0,∞) → Rp

Initial Condition : ω0 = ω(t0)

Parameterized Function : y(t, ω) : [t0,∞)× Rp → R

m

Definition

(i) Persistently Exciting (PE):∃ T ,α s.t.

∫ t+T

t

y(τ, ω)yT(τ, ω)dτ ≥ αI

for all t ≥ t0 and ω0 ∈ Rp.

6 / 15

Persistent Excitation

“Exogenous Signal” : ω : [t0,∞) → Rp

Initial Condition : ω0 = ω(t0)

Parameterized Function : y(t, ω) : [t0,∞)× Rp → R

m

Definition

(i) Persistently Exciting (PE):∃ T ,α s.t.

∫ t+T

t

y(τ, ω)yT(τ, ω)dτ ≥ αI

for all t ≥ t0 and ω0 ∈ Rp.

(ii) weakly Persistently Exciting (PE∗(ω,Ω)):∃ a compact set Ω ⊂ R

p, T (Ω) > 0,α(Ω) s.t.∫ t+T

t

y(τ, ω)yT(τ, ω)dτ ≥ αI

for all ω0 ∈ Ω and t ≥ t0.

6 / 15

properties of adaptive control

7 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

Control Input u = θT(t)x+ r

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

Adaptive Parameter θ(t)

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

Control Input u = θT(t)x+ r

Error e = x− xm

Parameter Error θ(t) = θ(t)− θ

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

Adaptive Parameter θ(t)

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

Control Input u = θT(t)x+ r

Error e = x− xm

Parameter Error θ(t) = θ(t)− θ

Update Law˙θ(t) = −xeTPB

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

Adaptive Parameter θ(t)

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

Control Input u = θT(t)x+ r

Error e = x− xm

Parameter Error θ(t) = θ(t)− θ

Update Law˙θ(t) = −xeTPB

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

Adaptive Parameter θ(t)

Stability V (e(t), θ(t)) = eT(t)Pe(t) + Trace(

θT(t)θ(t))

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

Control Input u = θT(t)x+ r

Error e = x− xm

Parameter Error θ(t) = θ(t)− θ

Update Law˙θ(t) = −xeTPB

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

Adaptive Parameter θ(t)

Stability V (e(t), θ(t)) = eT(t)Pe(t) + Trace(

θT(t)θ(t))

V ≤ eTQe

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

Control Input u = θT(t)x+ r

Error e = x− xm

Parameter Error θ(t) = θ(t)− θ

Update Law˙θ(t) = −xeTPB

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

Adaptive Parameter θ(t)

Stability V (e(t), θ(t)) = eT(t)Pe(t) + Trace(

θT(t)θ(t))

V ≤ eTQe

‖e‖L∞≤

√

V (e(t0), θ(t0))/Pmin

‖e‖L2≤

√

V (e(t0), θ(t0))/Qmin

8 / 15

Adaptive Control

Plant x = Ax−BθTx+Bu

Reference Model xm = Axm +Br

Control Input u = θT(t)x+ r

Error e = x− xm

Parameter Error θ(t) = θ(t)− θ

Update Law˙θ(t) = −xeTPB

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

Adaptive Parameter θ(t)

Stability V (e(t), θ(t)) = eT(t)Pe(t) + Trace(

θT(t)θ(t))

V ≤ eTQe

‖e‖L∞≤

√

V (e(t0), θ(t0))/Pmin

‖e‖L2≤

√

V (e(t0), θ(t0))/Qmin

The L-norms of e are initial condition dependent!! 8 / 15

Exponential Stability and Adaptive Control

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

9 / 15

Exponential Stability and Adaptive Control

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Theorem: (Morgan and Narendra, 1977)

If x(t) ∈ PE then z(t) = 0 is UASL.

9 / 15

Exponential Stability and Adaptive Control

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Theorem: (Morgan and Narendra, 1977)

If x(t) ∈ PE then z(t) = 0 is UASL.

ESL ES

UASL UAS+ Linear

9 / 15

Exponential Stability and Adaptive Control

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Theorem: (Morgan and Narendra, 1977)

If x(t) ∈ PE then z(t) = 0 is UASL.

ESL ES

UASL UAS+ Linear

Therefore, when x ∈ PE the dynamics z(t) are globallyexponentially stable (Anderson, 1977).

9 / 15

Exponential Stability and Adaptive Control

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Theorem: (Morgan and Narendra, 1977)

If x(t) ∈ PE then z(t) = 0 is UASL.

ESL ES

UASL UAS+ Linear

Therefore, when x ∈ PE the dynamics z(t) are globallyexponentially stable (Anderson, 1977).

The condition of PE for x(t) however does not follow fromxm(t) ∈ PE.

9 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

10 / 15

If xm ∈ PE then x ∈ ?

Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = − (xTh− x

Tmh)(xT

h+ xTmh)

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

‖e‖L∞≤

√

V (z0)Pmin

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

Move xm to the RHS, multiply by −1, and integrateto pT

‖e‖L∞≤

√

V (z0)Pmin

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

Move xm to the RHS, multiply by −1, and integrateto pT

∫ t+pT

t

(xT(τ)h)2dτ ≥

pα−

(

√

V (z0)Pmin

+ 2xmaxm

)

√

pT

∫ t+pT

t

‖e(τ)‖2dτ.

‖e‖L∞≤

√

V (z0)Pmin

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

Move xm to the RHS, multiply by −1, and integrateto pT

∫ t+pT

t

(xT(τ)h)2dτ ≥

pα−

(

√

V (z0)Pmin

+ 2xmaxm

)

√

pT

∫ t+pT

t

‖e(τ)‖2dτ.

‖e‖L∞≤

√

V (z0)Pmin

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

Move xm to the RHS, multiply by −1, and integrateto pT

∫ t+pT

t

(xT(τ)h)2dτ ≥

pα−

(

√

V (z0)Pmin

+ 2xmaxm

)

√

pT

∫ t+pT

t

‖e(τ)‖2dτ.

‖e‖L∞≤

√

V (z0)Pmin

xm ∈ PE

∫ t0+T

t0

xmxTm ≥ αI

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

Move xm to the RHS, multiply by −1, and integrateto pT

∫ t+pT

t

(xT(τ)h)2dτ ≥

pα−

(

√

V (z0)Pmin

+ 2xmaxm

)

√

pT

∫ t+pT

t

‖e(τ)‖2dτ.

‖e‖L∞≤

√

V (z0)Pmin

xm ∈ PE

∫ t0+T

t0

xmxTm ≥ αI

‖e‖L2≤

√

V (z0)Qmin

10 / 15

If xm ∈ PE then x ∈ ?Recall that e = x − xm, then for any fixed unitaryvector h

(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸ ︷︷ ︸

≤‖e‖

(xTh+ x

Tmh)

︸ ︷︷ ︸

=eTh+2xTm

h

(xTmh)2 − (xT

h)2 ≤ ‖e‖

(√

V (z0)

Pmin+ 2xmax

m

)

Move xm to the RHS, multiply by −1, and integrateto pT

∫ t+pT

t

(xT(τ)h)2dτ ≥

pα−

(

√

V (z0)Pmin

+ 2xmaxm

)

√

pT

∫ t+pT

t

‖e(τ)‖2dτ.

Clean the notation∫ t+pT

t

‖x‖2dτ ≥ pα−

(

√

V (z0)Pmin

+ 2xmaxm

)

√

pTV (z0)Qmin

.

‖e‖L∞≤

√

V (z0)Pmin

xm ∈ PE

∫ t0+T

t0

xmxTm ≥ αI

‖e‖L2≤

√

V (z0)Qmin

10 / 15

x ∈ PE∗ x /∈ PE

∫ t+T

t

xm(τ )xTm(τ )dτ ≥ αI

∫ t+pT

t

xT(τ, z)x(τ, z)dτ ≥ pα−

(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

.

11 / 15

x ∈ PE∗ x /∈ PE

∫ t+T

t

xm(τ )xTm(τ )dτ ≥ αI

∫ t+pT

t

xT(τ, z)x(τ, z)dτ ≥ pα−

(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

.

Fixed T, α

11 / 15

x ∈ PE∗ x /∈ PE

∫ t+T

t

xm(τ )xTm(τ )dτ ≥ αI

∫ t+pT

t

xT(τ, z)x(τ, z)dτ ≥ pα−

(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

.

Fixed T, α Free p

11 / 15

x ∈ PE∗ x /∈ PE

∫ t+T

t

xm(τ )xTm(τ )dτ ≥ αI

∫ t+pT

t

xT(τ, z)x(τ, z)dτ ≥ pα−

(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

.

Fixed T, α Free p Initial Condition z0

11 / 15

x ∈ PE∗ x /∈ PE

∫ t+T

t

xm(τ )xTm(τ )dτ ≥ αI

∫ t+pT

t

xT(τ, z)x(τ, z)dτ ≥ pα−

(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

︸ ︷︷ ︸

α′

.

Fixed T, α Free p Initial Condition z0

If the initial condition ‖z(t0)‖ increases (V (z0) increases), then p mustincrease, and thus the time (pT ) must increase to keep α′ constant.

11 / 15

x ∈ PE∗ x /∈ PE

∫ t+T

t

xm(τ )xTm(τ )dτ ≥ αI

∫ t+pT

t

xT(τ, z)x(τ, z)dτ ≥ pα−

(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

︸ ︷︷ ︸

α′

.

Fixed T, α Free p Initial Condition z0

If the initial condition ‖z(t0)‖ increases (V (z0) increases), then p mustincrease, and thus the time (pT ) must increase to keep α′ constant.

Revisit the definitions for PE

(i) Persistently Exciting (PE): ∃ T, α s.t.∫ t+T

t

x(τ, ω)xT(τ, ω)dτ ≥ αI

for all t ≥ t0 and ω0 ∈ Rp.

(ii) weakly Persistently Exciting (PE∗(ω,Ω)): ∃ a compactset Ω ⊂ R

p, T (Ω) > 0, α(Ω) s.t.∫ t+T

t

x(τ, ω)xT(τ, ω)dτ ≥ αI

for all ω0 ∈ Ω and t ≥ t0.11 / 15

Adaptive Control and UASLRevisit the adaptive control problem

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

12 / 15

Adaptive Control and UASLRevisit the adaptive control problem

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Define the following compact set

Ω(ζ) , z : V (z) ≤ ζ

12 / 15

Adaptive Control and UASLRevisit the adaptive control problem

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Define the following compact set

Ω(ζ) , z : V (z) ≤ ζ

Theorem

If xm ∈ PE then x ∈ PE∗(z,Ω(ζ)), for any ζ > 0,and it follows that the dynamics above are UASL.

12 / 15

Adaptive Control and UASLRevisit the adaptive control problem

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Define the following compact set

Ω(ζ) , z : V (z) ≤ ζ

Theorem

If xm ∈ PE then x ∈ PE∗(z,Ω(ζ)), for any ζ > 0,and it follows that the dynamics above are UASL.

Proof.

xm ∈ PE =⇒ x ∈ PE∗(z,Ω(ζ)) from previous slide.

12 / 15

Adaptive Control and UASLRevisit the adaptive control problem

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Define the following compact set

Ω(ζ) , z : V (z) ≤ ζ

Theorem

If xm ∈ PE then x ∈ PE∗(z,Ω(ζ)), for any ζ > 0,and it follows that the dynamics above are UASL.

Proof.

xm ∈ PE =⇒ x ∈ PE∗(z,Ω(ζ)) from previous slide.

PE∗ by definition is a local uniform property

12 / 15

Adaptive Control and UASLRevisit the adaptive control problem

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Define the following compact set

Ω(ζ) , z : V (z) ≤ ζ

Theorem

If xm ∈ PE then x ∈ PE∗(z,Ω(ζ)), for any ζ > 0,and it follows that the dynamics above are UASL.

Proof.

xm ∈ PE =⇒ x ∈ PE∗(z,Ω(ζ)) from previous slide.

PE∗ by definition is a local uniform property

The “Large” part of UASL holds because we can takearbitrarily large Ω

12 / 15

Adaptive Control and UASLRevisit the adaptive control problem

z(t) =

[

A BxT(t)−x(t)BTP 0

]

z(t), z(t) ,

[

e(t)

θ(t)

]

Define the following compact set

Ω(ζ) , z : V (z) ≤ ζ

Theorem

If xm ∈ PE then x ∈ PE∗(z,Ω(ζ)), for any ζ > 0,and it follows that the dynamics above are UASL.

Proof.

xm ∈ PE =⇒ x ∈ PE∗(z,Ω(ζ)) from previous slide.

PE∗ by definition is a local uniform property

The “Large” part of UASL holds because we can takearbitrarily large Ω

Next we prove (by counter example) xm ∈ PE does not imply ESL.12 / 15

Simulation Example

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

z1

z2

z3

z4

z5

z6

z7

z8

z9

θ

e

Plant x = Ax−BθTx+Bu

Reference xm = Axm +Br

Control u = θT(t)x + r

A = −1

B = 1

r = 3

xm(t0) = 3

13 / 15

Simulation Example

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

z1

z2

z3

z4

z5

z6

z7

z8

z9

θ

e

Plant x = Ax−BθTx+Bu

Reference xm = Axm +Br

Control u = θT(t)x + r

A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3

(Jenkins et al., 2013a; 2013b)13 / 15

Simulation Example

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

z1

z2

z3

z4

z5

z6

z7

z8

z9

θ

e

Plant x = Ax−BθTx+Bu

Reference xm = Axm +Br

Control u = θT(t)x + r

A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3

M1 ∪M2 ∪M3 is invariant

(Jenkins et al., 2013a; 2013b)13 / 15

Simulation Example

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

z1

z2

z3

z4

z5

z6

z7

z8

z9

θ

e

Plant x = Ax−BθTx+Bu

Reference xm = Axm +Br

Control u = θT(t)x + r

A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3

M1 ∪M2 ∪M3 is invariant

M3 extends down in anunbounded fashion

(Jenkins et al., 2013a; 2013b)13 / 15

Simulation Example

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

z1

z2

z3

z4

z5

z6

z7

z8

z9

θ

e

Plant x = Ax−BθTx+Bu

Reference xm = Axm +Br

Control u = θT(t)x + r

A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3

M1 ∪M2 ∪M3 is invariant

M3 extends down in anunbounded fashion

maximum rate of change inM3 is bounded

(Jenkins et al., 2013a; 2013b)13 / 15

Simulation Example

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

z1

z2

z3

z4

z5

z6

z7

z8

z9

θ

e

Plant x = Ax−BθTx+Bu

Reference xm = Axm +Br

Control u = θT(t)x + r

A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3

M1 ∪M2 ∪M3 is invariant

M3 extends down in anunbounded fashion

maximum rate of change inM3 is bounded

The fixed rate regardless ofinitial condition implies thatESL is impossible

(Jenkins et al., 2013a; 2013b)13 / 15

Simulation Example Continued

5 5 5 0 0.5 1

0

1

2

z4

θ

e

0 5 10 15-3

-2

-1

0

1

0 5 10 150

1

2

3

4

0 5 10 150

1

2

3

4

0 5 10 15

-10

-8

-6

-4

-2

0

2

e xm

x θ

tt

Jenkins, B. M., T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. (2013a).Asymptotic stability and convergence rates in adaptive systems, IFAC Workshopon Adaptation and Learning in Control and Signal Processing, Caen, France.

Jenkins, B. M, T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. (2013b).Convergence properties of adaptive systems with open-and closed-loop referencemodels, AIAA guidance navigation and control conference.

14 / 15

Simulation Example Continued

5 5 5 0 0.5 1

0

1

2

z4

z5

θ

e

0 5 10 15-3

-2

-1

0

1

0 5 10 150

1

2

3

4

0 5 10 150

1

2

3

4

0 5 10 15

-10

-8

-6

-4

-2

0

2

e xm

x θ

tt

Jenkins, B. M., T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. (2013a).Asymptotic stability and convergence rates in adaptive systems, IFAC Workshopon Adaptation and Learning in Control and Signal Processing, Caen, France.

Jenkins, B. M, T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. (2013b).Convergence properties of adaptive systems with open-and closed-loop referencemodels, AIAA guidance navigation and control conference.

14 / 15

Simulation Example Continued

5 5 5 0 0.5 1

0

1

2

z4

z5

z6

θ

e

0 5 10 15-3

-2

-1

0

1

0 5 10 150

1

2

3

4

0 5 10 150

1

2

3

4

0 5 10 15

-10

-8

-6

-4

-2

0

2

e xm

x θ

tt

Jenkins, B. M., T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. (2013a).Asymptotic stability and convergence rates in adaptive systems, IFAC Workshopon Adaptation and Learning in Control and Signal Processing, Caen, France.

Jenkins, B. M, T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. (2013b).Convergence properties of adaptive systems with open-and closed-loop referencemodels, AIAA guidance navigation and control conference.

14 / 15

Summary PE of the reference model does not imply PE for the state vector

Adaptive control in general can not be guaranteed to be ESL

loria@lss.supelec.fr BibliographyAnderson, B. D. O. 1977. Exponetial stability of linear equations arising in adaptive

identification, IEEE Trans. Automat. Contr. 22, no. 83.

Jenkins, B. M., T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. 2013a.Asymptotic stability and convergence rates in adaptive systems, Ifac workshop onadaptation and learning in control and signal processing, caen, france.

Jenkins, B. M, T. E. Gibson, A. M. Annaswamy, and E. Lavretsky. 2013b.Convergence properties of adaptive systems with open-and closed-loop reference

models, AIAA guidance navigation and control conference.

Kalman, R. E. and J. E. Bertram. 1960. Control systems analysis and design via the

‘second method’ of liapunov, i. continuous-time systems, Journal of BasicEngineering 82, 371–393.

Malkin, I. G. 1935. On stability in the first approximation, Sbornik NauchnykhTrudov Kazanskogo Aviac. Inst. 3.

Massera, J. S. 1956. Contributions to stability theory, Annals of Mathematics 64,no. 1.

Morgan, A. P. and K. S. Narendra. 1977. On the stability of nonautonomous

differential equations x = [A+B(t)]x, with skew symmetric matrix B(t), SIAMJournal on Control and Optimization 15, no. 1, 163–176.

15 / 15

backup slides

16 / 15

Example or recent literature making this mistake

17 / 15

Stability

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

x0 , x(t0)

Solution s(t;x0, t0) δ, ρ

ǫ

t

η

x0

t0

t0 + Ts

Definition: Stability (Massera, 1956)

(i) Stable: ∀ ǫ > 0 ∃ δ(ǫ, x0, t0) > 0 s.t.‖x0‖ ≤ δ =⇒ ‖s(t;x0, t0)‖ ≤ ǫ.

(ii) Attracting: ∃ ρ(t0) > 0 s.t. ∀ η > 0 ∃ an attraction time

T (η, x0, t0) s.t.‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ η ∀ t ≥ t0 + T .

(iii) Asymptotically Stable = stable + attracting.

18 / 15

Uniform Stability

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

x0 , x(t0)

Solution s(t;x0, t0) δ, ρ

ǫ

t

η

x0

t0

t0 + Ts

Definition: Uniform Stability (Massera, 1956)

(iv) Uniformly Stable: δ(ǫ) in (i) is uniform in t0 and x0.

(v) Uniformly Attracting: ρ and T do not depend on t0 or x0

and thus the attracting times take the form T (η,ρ).

(vi) Uniformly Asymptotically Stable, (UAS)= uniformly stable + uniformly attracting.

19 / 15

Uniform Stability in the Large (Global)

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

x0 , x(t0)

Solution s(t;x0, t0) δ, ρ

ǫ

t

η

x0

t0

t0 + Ts

Definition: Uniform Stability in the Large (Massera, 1956)

(vii) Uniformly Attracting in the Large: For all ρ,η ∃ T (η,ρ)‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ η ∀ t ≥ t0 + T .

(viii) Uniformly Asymptotically Stable in the Large (UASL)= uniformly stable +

uniformly bounded +

uniformly attracting in the large.20 / 15

Exponential Asymptotic Stability

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

x0 , x(t0)

Solution s(t;x0, t0)

δ, ρ

ǫ

t

x0

t0

ǫe−ν(t−t0)

s

Definition: (Malkin, 1935; Kalman and Bertram, 1960)

(i) Exponentially Asymptotically Stable (EAS):∀ ǫ > 0 ∃ δ(ǫ),ν(ǫ) s.t.

‖x0‖ ≤ δ =⇒ ‖s(t;x0, t0)‖ ≤ ǫe−ν(t−t0)

(ii) Exponentially Asymptotically Stable in the Large

(EASL): ∀ ρ > 0 ∃ ǫ(ρ),ν(ρ) s.t.‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ ǫe−ν(t−t0)

(iii) Exponentially Stable (ES): ∀ ρ > 0 ∃ ν(ρ),κ(ρ) s.t.‖x0‖ ≤ ρ =⇒ ‖s(t;x0, t0)‖ ≤ κ‖x0‖e

−ν(t−t0)

(iv) Exponentially Stable in the Large (ESL): ∃ ν,κ s.t.‖s(t;x0, t0)‖ ≤ κ‖x0‖e

−ν(t−t0)

21 / 15

Rant about “uniform transients”

22 / 15