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�Adaptive Control

Part 9: Adaptive Control with Multiple Models and Switching

I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 1

Outline

1. Introduction

2. Multimodel Adaptive Control with Switching

3. Stability of the Adaptive System

4. Stability of the Switching System

5. Stability with Minimum Dwell Time

6. Application to the Flexible Transmission System

7. Effects of the Design Parameters


Introduction

Consider controller design for a system with very large uncertainty

Robust Control : A fixed controller does not necessarily exist thatstabilizes the system, or if it exists, it does not givegood performances.

Adaptive Control : The classical adaptive control givesunacceptable transient adaptation for large and fastparameter variation.

Solution: Multimodel adaptive control

◮ Classical multimodel adaptive control

◮ Robust multimodel adaptive control

◮ Multimodel adaptive control with switching

◮ Multimodel adaptive control with switching and tuning


Classical multimodel adaptive control


Robust multimodel adaptive control


Robust multimodel adaptive control

◮ Very similar to the classical multimodel adaptive control.◮ The controllers are robust with respect to unmodeled

dynamics and uses output feedback instead ofstate-feedback.

◮ The control input is the weighted sum of the outputs ofthe controllers (no switching).

u(t) =

N∑

i=1

Pi(t)ui(t)

where Pi(t) is the posterior probability of i -th estimator.◮ The posterior probabilities are computed as:

Pk(t + 1) =

[

βke−

12r ′k(t+1)S−1

krk (t+1)

∑N

i=1 βie−

12r ′i(t+1)S−1

iri (t+1)Pi(t)

]

Pk(t)

◮ The stability of this scheme is not guaranteed.I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 6

Multimodel adaptive control with switching


Multimodel adaptive control with switching and tuning

P

P1

P2

Pn

Plant

C (θ)

C1

C2

Cn

-

-

-

-

-

6

6

6

6

-

-

-

-

+

+

+

+

-

-

-

-

yn

y2

y1

y0

εn

ε2

ε1

ε0

y-

-

)-

-

3

w

u0

u1

u2

un

u

-

-

-

-

-

-

-

After a parameter variation (a large estimation error)◮ First the controller corresponding to the closest model (fixed

model) is chosen (switching).◮ Then the adaptive model is initialized with the parameter of

this model and will be adapted (tuning).I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 8

Structure of Multimodel Adaptive Control with Switching

Plant: LTI-SISO (for analysis) with parametric uncertaintyand unmodelled dynamics:

⋃

θ∈Θ

P(θ)

where P(θ) = P0(θ)[1 + W2∆] with ‖∆‖∞ < 1.Other type of uncertainty can also be considered.

Multi-Estimator: Kalman filters, fixed models, adaptive models. IfΘ is a finite set of n models, these models can beused as estimators (output-error estimator). If Θ isinfinite but compact, a finite set of n models withand adaptive model can be used.

Multi-Controller: We suppose that for each P(θ) there exists C (θ)in the multi-controlller set that stabilizes P(θ) andsatisfies the desired performances (the controllers arerobust with respect to unmodelled dynamics).


Structure of Multimodel Adaptive Control with Switching

Monitoring Signal: is a function of the estimation error to indicatethe best estimator at each time.

Ji (t) = αε2i (t) + β

t∑

j=0

e−λ(t−j)ε2i (j)

with λ > 0 a forgetting factor, α ≥ 0 and β > 0weightings for instantaneous and past errors.

Switching Logic: Based on the monitoring signal, a switchingsignal σ(t) is computed that indicates which controlinput should be applied to the real plant. To avoidchattering, a minimum dwell-time between twoconsecutive switchings or a hysteresis is considered.

The dwell-time and hysteresis play an important role on thestability of the switching system.


Switching LogicStart

σ(t) = arg mini

Ji (t)

Jσ ≤ (1 + h)Ji

?

?

�

-

No

Yes

Start

σ(t) = arg mini

Ji (t)

Wait Td

?

?

�

HysteresisDwell-time

◮ A large value for Td may deteriorate the performance and asmall value can lead to instability.

◮ With hysteresis, large errors are rapidly detected and a bettercontroller is chosen. However, the algorithm does not switchto a better controller in the set if the performanceimprovement is not significant.


Stability of Adaptive Control with Switching

Trivial case:

◮ No unmodelled dynamics and no noise,

◮ the set of models is finite,

◮ parameters of one of the estimators matches those of theplant model,

◮ plant is detectable.

Main steps toward stability:

1. One of the estimation errors (say εk) goes to zero.

2. εσ(t) = y(t) − yσ(t) goes to zero as well.

3. After a finite time τ switching stops (σ(τ) = k t ≥ τ).

4. If εk goes to zero, θk will be equal to θ and the controllerCk stabilizes the plant P(θ):

(Certainty equivalence stabilization theorem)


Stability of Adaptive Control with Switching

Assumptions: Presence of unmodelled dynamics and noise.Existence of some“good” estimators in the multi-estimator block.The plant P is detectable.

1. εk for some k is small.

2. εσ is small (because of a“good”monitoring signal).

3. All closed-loop signals and states are bounded if:The injected system is stable.

Multi-Controller Multi-Estimator

Plant

+

-

+

+

εσ

y

yu

yσ

uσ

Injected System

- -

-

? -

�?

6 6


Stability of Switching Systems

Each controller stabilizes the corresponding model in themulti-estimator for frozen σ.Question: Is the injected system stable for a time-varyingswitching signal σ(t)?

Is fσ(x) stable? No


Stability of Switching SystemsConsider a set of stable systems:

x = A1x , x = A2x , . . . x = Anx

then x = Aσx is stable if :

◮ A1 to An have a common Lyapunov matrix (quadratic

stability).

◮ If the minimum time between two switchings is greaterthan Td (minimum dwell time).

◮ If the number of switching in the interval (t, T ) does notgrow faster than linearly with T (average dwell time).

Nσ(t, T ) ≤ N0 +T − t

Td

∀T ≥ t ≥ 0

N0 = 1 implies that σ cannot switch twice on any intervalshorter than Td .


Stability of Switching Systems

Common Lyapunov Matrix : The existence of a commonLyapunov matrix for A1, . . . ,An guarantees the stability of Aσ.This can be verified by a set of Linear Matrix Inequalities(LMI):

Continuous-time

AT1 P + PA1 < 0

AT2 P + PA2 < 0

...

ATn P + PAn < 0

Discrete-time

AT1 PA1 − P < 0

AT2 PA2 − P < 0

...

ATn PAn − P < 0

I The stability is guaranteed for arbitrary fast switching.

I The stability condition is too conservative.


Stability of Switching Systems by Minimum Dwell TimeTheorem : Assume that for some Td > 0 there exists a set ofpositive definite matrix {P1,P2, . . . ,Pn} such that:

ATi Pi + PiAi < 0 ∀i = 1, . . . ,N

andeAT

i Td PjeAiTd − Pi < 0 ∀i 6= j = 1, . . . ,N

Then any switching signal σ(t) ∈ {1, 2, . . . ,N} withtk+1 − tk ≥ Td makes the equilibrium solution x = 0 of

x(t) = Aσ(t)x(t) x(0) = x0

globally asymptotically stable.

◮ The first group of LMIs are always feasible because A1 to An

are stable.

◮ The second LMIs are always feasible if Td is large enough.

◮ Td can be minimized using LMIs.


Stability of Switching Systems by Minimum Dwell Time

Proof : Consider the Lyapunov function V (x(t)) = xT (t)Pσx(t).We should show that for any tk+1 = tk + Tk with Tk ≥ Td > 0 wehave V (x(t + k)) < V (x(t)). Assume that σ(t) = i fort ∈ [tk tk+1) and σ(tk+1) = j . We have :

V (x(t + 1)) = xT (tk+1)Pjx(tk+1)

= xT (tk)eATi

Tk PjeAiTk x(tk)

< xT (tk)eATi

(Tk−Td )PieAi (Tk−Td )x(tk)

< xT (tk)Pix(tk)

< V (x(tk))

◮ This can be proved for discrete-time systems as well.

◮ If A1 to An are quadratically stable, i.e.P = P1 = P2 = . . . = Pn then the LMIs are feasible for anyTd > 0.


Stability of Switching Systems by Minimum Dwell Time

Comments : There are some conservatism in this approach.

1. Minimum of Td is an upper bound on the minimum dwelltime.

2. Vi(x(tq)) < Vi(x(tp)) where σ(tp) = σ(tq) = i is sufficient forthe stability.


Application to the Flexible Transmission System

Φm

motor axis

DCmotor

Positionsensor

Controller

axisposition

+- Φref

PCDAC

ADC

load

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−50

−40

−30

−20

−10

0

10

20

30

f/fe

dB

Unloaded model

Half loaded model

Full loaded model


Multiple Model with Switching and Tuning

P

P1

P2

P3

Plant

P

Supervisor????

-

?

-

-

-

-

6

6

6

6

-

-

-

-

-

-

-

-

- 6

-A.A.P.

?7-

- y

y

εCL

u

u

r(t)

ε3

ε2

ε1

ε0

y3

y2

y1

y0

P ∈ {P , P1, P2, P3}

+

-

- +

- +

- +

- +

�

PolePlacementController

PolePlacementController


Design ProcedureMulti estimator Design:

◮ Number of estimators: more estimator → more accuracy butmore complexity.

◮ Type of estimators: output error, AMAX predictor, Kalmanfilter, etc.

◮ Fixed or adaptive: All fixed needs too many models for adesired accuracy. With all adaptive estimators, a persistentlyexcitation signal is needed. A trade-off is a few number offixed model such that at least one of them stabilizes the plantmodel and an adaptive model to improve the accuracy.

◮ Adaptive model: It should be initialized with the parameters ofthe closest fixed model. It can be a classical RLS or a CLOEadaptive model. For regulation problem adaptive model is notproposed.

For flexible transmission system we chose 3 output error fixedestimators in 0 % 50% and 100% load and one adaptive model

with CLOE.I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 22

Design ProcedureRLS versus CLOE : Disturbance and noise affect the parametersof RLS estimators (drift problem) but not the parameters of CLOEestimators. Therefore, in the absence of excitation signal, theclosed-loop system may become unstable.

T1S

G

R

-- -6

+

-- ?

�

y(t)u(t)r(t)

w(t)

T1S

G

R

- -6

+

-

�

y(t)u(t) -

?

6-εCL(t)

-

-

AAP

?

�

+

-

++

3

3

3

>

θ

3

3

Pole Placement �


RLS versus CLOE

Experimental results

Classical adaptive control CLOE adaptive control


Design Procedure

Multi-controller design : Robust pole placement

Desired closed-loop poles: Complex and simple poles are chosen.

I A pair of complex poles with the same frequencyas the first resonance mode and a dampingfactor of 0.8.

I A pair of complex poles with the same frequencyas the second resonance mode and a dampingfactor of 0.2.

I 6 auxiliary poles at 0.2.

Fixed terms in the controller: A fixed pole at 1 (integrator) and azero at -1 for reducing the input sensitivity functionat high frequencies.

Remark : Adaptive model is initialized with the parameters of thelast switched estimator and the desired closed-loop poles arechosen based on this fixed model.


Design procedure

Monitoring signal and switching logic:

Ji (t) = αε2i (t) + β

t∑j=0

e−λ(t−j)ε2i (j) α ≥ 0, β > 0, λ > 0

I α� β: More weights on instantaneous errors → fast reaction.This leads to fast parameter adaptation but poor performancew.r.t. disturbance.

I α = λ = 0: Monitoring signal is the two-norm of the error foreach estimator. The reaction to parameter variation is slowbut leads to good performance w.r.t. disturbance.

I The minimum dwell time should be chosen to assure thestability. If the theoretical minimum is too large, a hysteresiscycle with an average dwell time is preferred.


Experimental results on the flexible transmission system

Multimodel adaptive control versus robust control

Load changes from 0 to 100 % in four steps (9,19,29 and 39s)



Multiple model with CLOE

Load changes from 100% to 0% in two stages (19 and 29s)α = 1, β = 1, λ = 0.1



Same case with classical adaptive control (simulation)

Load changes from 100% to 0% in two stages (19 and 29s)Unstable in real time experiment



Real system does not belong to the fixed models set

Load changes from 75% to 25% in one stages (19s)


Effects of design parametersDesign parameters: Number of fixed and adaptive models, choiceof adaptation algorithm, forgetting factor λ, dwell time.Test conditions: Spaced parameter variation and frequentparameter variation are simulated using following signals:

Tc � T represents spaced parameter variation and2T < Tc < 100T frequent parameter variation.

Performance criterion: Jc(t) =

(1

Tf

∫ Tf

0[r(t)− y(t)]2

)1/2


Effects of design parameters

Number of fixed and adaptive models

2 fixed models

7 fixed models

2 fixed + 1 adaptive models

0.8 1 1.2 1.4 1.6 1.8 2

x 10−3

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

parameter changing rate (fc)

pe

rfo

rma

nce

in

de

x (

Jc)

2 fixed models

7 fixed models

2 fixed+1 adaptive models

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

parameter changing rate (fc)

pe

rfo

rma

nce

in

de

x (

Jc)

Spaced parameter variation Frequent parameter variation

I One adaptive model can reduce the number of fixed models.

I Adaptive models have more effects for spaced parametervariation.



Choice of forgetting factor

0 0.5 10.008

0.009

0.01

0.011

0.012

0.013

lambda

Jc

(a) spaced variations

0 0.5 10.015

0.02

0.025

0.03

0.035

0.04

0.045

lambda

Jc

(b) frequent variations

0 0.5 10.02

0.03

0.04

0.05

0.06

lambda

Jc

(d) noisy environment

0 0.5 10.01

0.012

0.014

0.016

0.018

lambda

Jc

(c) output disturbance

I Spaced variation → λ small. Frequent variation → λ large.I Disturbance and noisy at the output → λ small.I In this example λ = 0.3− 0.4 is a good trade off.



Choice of dwell time

frequent

spaced

5 10 15 20 25 300.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Dwell time Td in sampling period

Pe

rfo

rma

nc in

de

x (

Jc)

Performance criterion versus dwell time for spaced and frequentparameter variation

The smallest dwell time that assures the stability should bechosen



Choice of adaptation algorithm

0 20 40 60 80 100 120−2

−1

0

1

2

Time (s)

Reference and plant output ( RLS algorithm )

0 20 40 60 80 100 1200

1

2

3

Time (s)

switching diagram

0 20 40 60 80 100 1200

1

2

3

Time (s)

switching diagram

0 20 40 60 80 100 120−2

−1

0

1

2

Time (s)

Reference and plant output ( CLOE algorithm )

RLS adaptation algorithm CLOE adaptation algorithm

I Three fixed models (no-load, half-load, full-load) inmulti-estimator.

I Fixed plant model (25% load).



Choice of adaptation algorithm

RLS algorithm

CLOE algorithm

0.01 0.02 0.03 0.04 0.05 0.06 0.070.016

0.0165

0.017

0.0175

0.018

0.0185

0.019

0.0195

Noise variance

Pe

rfo

rma

nc in

de

x (

Jc)

Performance criterion versus noise variance for RLS and CLOECLOE gives to switching control a better performance and

switching control assure the stability of adaptive control withCLOE


References:1. D. Liberzon. Switching in Systems and Control. Birkhauser, 2003.

2. B.D.O Anderson et al. Multiple model adaptive control. part 1: finitecontroller coverings. Int. J. of Robust and Nonlinear Control 2000;10(11–12):909–929.

3. J. Hespanha et al. Multiple model adaptive control part 2: switching. Int.J. of Robust and Nonlinear Control 2001; 11(5):479–496.

4. K. S. Narendra, J. Balakrishnan. Adaptive control using multiple models.IEEE-TAC, 1997; 42(2):171–187.

5. S. Fekri et al. Issues, progress and new results in robust adaptive control.Int. J. Adaptive Control and Signal Processing, 2006; 20:519–579

6. J. C. Geromel, P. Colaneri. Stability and Stabilization of Continuous-TimeSwitched Linear Systems. SIAM J. Control Optimal, 2006,45(5):1915–1930.

7. A. Karimi, I.D. Landau. Robust Adaptive Control of a FlexibleTransmission System Using Multiple Models. IEEE-TCST, 2000,8(2):321-331.

8. A. Karimi, I. D. Landau, N. Motee. Effects of the design paramters ofmultimodel adaptive control on the performance of a flexible transmissionsystem. Int. J. of Adaptive Control and Signal Processing,2001,15(3):335–352.


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