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 the following

Adaptive systems can at best be uniformly asymptotically stable inthe large

2 / 15

Objectives



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


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


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

x0 , x(t0)

Solution s(t;x0, t0)δ, ρ

ǫ

t

η

x0

t0

t0 + Ts



(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


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

x0 , x(t0)

Solution s(t;x0, t0)δ, ρ

ǫ

t

η

x0

t0

t0 + Ts



(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


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


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

x0 , x(t0)


ρ

κ‖x0‖

t

x0

t0

κ‖x0‖e−ν(t−t0)

s


(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





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





m

Definition

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

∫ t+T

t



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

p, T (Ω) > 0,α(Ω) s.t.∫ t+T

t


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



-

∑

∑

θT

r

xm

x

e

Reference Model

Plant

Unknown Parameter θ

8 / 15

Adaptive Control



Control Input u = θT(t)x+ r

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant


Adaptive Parameter θ(t)

8 / 15

Adaptive Control




Error e = x− xm

Parameter Error θ(t) = θ(t)− θ

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant



8 / 15

Adaptive Control




Error e = x− xm


Update Law˙θ(t) = −xeTPB

-

∑

∑

θT

r

xm

x

e

Reference Model

Plant



8 / 15

Adaptive Control




Error e = x− xm



-

∑

∑

θT

r

xm

x

e

Reference Model

Plant



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

θT(t)θ(t))

8 / 15

Adaptive Control




Error e = x− xm



-

∑

∑

θT

r

xm

x

e

Reference Model

Plant




θT(t)θ(t))

V ≤ eTQe

8 / 15

Adaptive Control




Error e = x− xm



-

∑

∑

θT

r

xm

x

e

Reference Model

Plant




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




Error e = x− xm



-

∑

∑

θT

r

xm

x

e

Reference Model

Plant




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


z(t) =

[


]

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


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]



ESL ES

UASL UAS+ Linear

9 / 15


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]



ESL ES

UASL UAS+ Linear

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

9 / 15


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]



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


(xTmh)2 − (xT

h)2 = −(xTh− x

Tmh)

︸︷︷︸

≤‖e‖

(xTh+ x

Tmh)

︸︷︷︸

=eTh+2xTm

h

10 / 15


(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


(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


(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


(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

)


∫ 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


(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

)


∫ 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


(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

)


∫ 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


(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

)


∫ 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


∫ t+T

t


∫ t+pT

t


(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

.

Fixed T, α

11 / 15


∫ t+T

t


∫ t+pT

t


(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

.

Fixed T, α Free p

11 / 15


∫ t+T

t


∫ t+pT

t


(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

.

Fixed T, α Free p Initial Condition z0

11 / 15


∫ t+T

t


∫ t+pT

t


(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

︸︷︷︸

α′

.


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


∫ t+T

t


∫ t+pT

t


(√V (z0)Pmin

+ 2xmaxm

)√

pTV (z0)Qmin

︸︷︷︸

α′

.


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


(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) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]

12 / 15


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]

Define the following compact set

Ω(ζ) , z : V (z) ≤ ζ

12 / 15


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]


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


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]


Ω(ζ) , z : V (z) ≤ ζ

Theorem


Proof.

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

12 / 15


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]


Ω(ζ) , z : V (z) ≤ ζ

Theorem


Proof.


PE∗ by definition is a local uniform property

12 / 15


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]


Ω(ζ) , z : V (z) ≤ ζ

Theorem


Proof.



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

12 / 15


z(t) =

[


]

z(t), z(t) ,

[

e(t)

θ(t)

]


Ω(ζ) , z : V (z) ≤ ζ

Theorem


Proof.



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


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




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




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




A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3


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




A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3



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




A = −1

B = 1

r = 3

xm(t0) = 3

θ

e

−xm(t0)

M1

M2

M3



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


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



14 / 15


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



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

[email protected] 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)


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


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

x0 , x(t0)


ǫ

t

η

x0

t0

t0 + Ts


(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)


δ, ρ

ǫ

t

x0

t0

ǫe−ν(t−t0)

s


(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

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

Documents