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EECE 574 - Adaptive Control Recursive Identification in Closed-Loop and Adaptive Control

Guy Dumont

Department of Electrical and Computer Engineering University of British Columbia

January 2010

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 1 / 43

Tracking Time-Varying parameters

Tracking Time-Varying Parameters

All previous methods use the least-squares criterion

V(t) = 1 t

t

∑ i=1

[y(i)− xT(i)θ̂ ]2

and thus identify the average behaviour of the process.

For standard RLS, the estimation gain eventually converges to zero and adaptation stops.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 2 / 43

Tracking Time-Varying parameters Forgetting Factor

Forgetting Factor

When the parameters are time varying, it is desirable to base the identification on the most recent data rather than on the old one, not representative of the process anymore. This can be achieved by exponential discounting of old data, using the criterion

V(t) = 1 t

t

∑ i=1

λ t−i[y(i)− xT(i)θ̂ ]2

where 0 < λ ≤ is called the forgetting factor.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 3 / 43

Tracking Time-Varying parameters Forgetting Factor

Forgetting Factor

The new criterion can also be written

V(t) = λV(t−1)+ [y(t)− xT(t)θ̂ ]2

Then, it can be shown (Goodwin and Payne, 1977) that the RLS scheme becomes

RLS with Forgetting

θ̂(t +1) = θ̂(t)+K(t +1)[y(t +1)− xT(t +1)θ̂(t)] K(t +1) = P(t)x(t +1)/[λ + xT(t +1)P(t)x(t +1)]

P(t +1) = {

P(t)− P(t)x(t +1)x T(t +1)P(t)

[λ + xT(t +1)P(t)x(t +1)]

} 1 λ

In choosing λ , one has to compromise between fast tracking and long term quality of the estimates. The use of the forgetting may give rise to problems.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 4 / 43

Tracking Time-Varying parameters Forgetting Factor

Forgetting Factor

The smaller λ is, the faster the algorithm can track, but the more the estimates will vary, even the true parameters are time-invariant.

A small λ may also cause blowup of the covariance matrix P, since in the absence of excitation, covariance matrix update equation essentially becomes

P(t +1) = 1 λ

P(t)

in which case P grows exponentially, leading to wild fluctuations in the parameter estimates.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 5 / 43

Tracking Time-Varying parameters Forgetting Factor

Variable Forgetting Factor

One way around this is to vary the forgetting factor according to the prediction error ε as in

λ (t) = 1− kε2(t)

Then, in case of low excitation ε will be small and λ will be close to 1. In case of large prediction errors, λ will decrease.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 6 / 43

Tracking Time-Varying parameters EFRA

Exponential Forgetting and Resetting Algorithm

The following scheme1 is recommended:

EFRA Algorithm

ε(t +1) = y(t +1)− xT(t +1)θ̂(t)

θ̂(t +1) = θ̂ T(t)+ αP(t)x(t +1)

λ + xT(t +1)P(t)x(k +1) ε(t)

P(t +1) = 1 λ

[ P(t)− P(t)x(t +1)x

T(t +1)P(t) λ + x(t +1)TP(t)x(t +1)

] +β I− γP(t)2

where I is the identity matrix, and α , β and γ are constants.

1M.E. Salgado, G.C. Goodwin, and R.H. Middleton, “Exponential Forgetting and Resetting”, International Journal of Control, vol. 47, no. 2, pp. 477–485, 1988.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 7 / 43

Tracking Time-Varying parameters EFRA

Exponential Forgetting and Resetting Algorithm

With the EFRA, the covariance matrix is bounded on both sides:

σminI ≤ P(t)≤ σmaxI ∀t

where

σmin ≈ β

α−η σmax ≈

η γ

+ β η

with

η = 1−λ

λ With α = 0.5, β = γ = 0.005 and λ = 0.95, σmin = 0.01 and σmax = 10.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 8 / 43

Identification in Closed Loop The Identifiability Problem

Identification in Closed Loop

The Identifiability Problem Let the system be described by

y(t)+a · y(t−1) = b ·u(t−1)+ e(t)

with u(t) = g · y(t)

Let â and b̂ be closed-loop estimates of a and b.

Then,the closed-loop system can be written as:

y(t)+(â− b̂ ·g)y(t−1) = e(t)

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 9 / 43

Identification in Closed Loop The Identifiability Problem

Identification in Closed Loop

Hence any estimates â and b̂ that satisfy

â− b̂ ·g = a−b ·g

will give the same value for the identification criterion.

All estimates such that

â = a+ k ·g b̂ = b+ k

will give a good description of the process.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 10 / 43

Identification in Closed Loop The Identifiability Problem

Identification in Closed Loop

If the identification is performed using two feedback gains g1 and g2 or if the parameter a is fixed, then the system becomes identifiable, because we have as many equations as unknowns.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 11 / 43

Identification in Closed Loop Definitions

Definitions

Let the discrete plant be described by

y(t) = GD(q−1)u(t)+GN(q−1)e(t)

where GD and GN are linear rational transfer functions in the backward shift operatorq−1 (i.e. q−1y(t) = y(t−1) that can be parameterized by a vector θ , {e(t)}= N(0,σ). Let S denote this true system. Let us assume that the identification is performed with the feedback controller R such that u(t) = R · y(t). The problem is then to find θ̂ , an estimate of θ such that the modelM(θ̂) given by

y(t) = ĜD(q−1)u(t)+ ĜN(q−1)e(t)

where ĜD(q−1) and ĜN(q−1) are parameterized by θ̂ , describes the system S. Assume that an identification method denoted I is used to obtain the estimate θ̂ .

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 12 / 43

Identification in Closed Loop Definitions

Definitions

A loose and intuitive definition of identifiability is that M(θ̂) describes S as the number of measurements N tends to infinity. Let us define

DT(S,M) = {θ̂ |ĜD(q−1)≡ GD(q−1) and ĜN(q−1) = GN(q−1)∀q}

This is the set of desired estimates which corresponds to models M(θ̂) with the same plant and noise transfer functions as the actual system S(θ). Note that this set does not depend on the regulator R nor on the identification method I. Note also that the orders of the model transfer functions can be greater than those of the system, in which case there exist some pole-zero cancellations.

The actual estimates θ̂ depend on the number of measurements, the system and its model, the regulator and the identification method and can be written as θ̂(N;S,M, I,R).

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 13 / 43

Identification in Closed Loop Definitions

Definitions

Definition 1: The system S is said to be system identifiable under M, I,R, i.e.SI(M, I,R) if

θ̂(N;S,M, I,R)−→ DT(S,M)w.p.1 as N→ ∞. Definition 2: The system S is said to be strongly system identifiable under I and R, i.e. SSI(I,R) if it is SI(M, I,R) for all M s.t. DT(S,M) 6= φ where φ denotes the empty set. Definition 3: The system S is said to be parameter identifiable under M, I and R, i.e.PI(M, I,R) if it is SI(M, I,R) and DT(S,M) consists of only one element.

According to those definitions, the system of Example 1 is neither SI nor PI for the class of

models (2). Moreover, since for the model (2) DT(S,M) 6= φ , the system is not SSI either. However, when two regulators are used, the system becomes PI. It is also important to note

that when a system is SSI(I,R), the fact that the identification is performed under closed-loop is irrelevant and the identification can then be performed as if the system were operating in

open-loop.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 14 / 43

Identification in Closed Loop Definitions

Example

Consider the system

A(q−1)y(t) = B(q−1)q−ku(t)+C(q−1)e(t)

with the feedback F(q−1)u(t) = G(q−1)y(t)

The closed-loop system can then be described as

(AF−q−kBG)y(t) = CFe(t)

It is then obvious that any estimates Â and B̂ such that

(ÂF−q−kB̂G) = (AF−q−kBG)

will describe the above system. Hence, if L(q−1) is an arbitrary polynomial, any Âand B̂ s.t.{

Â = A+LG B̂ = B+qkLF

(1)

are possible estimates. This means that the true order of the system cannot be established from this type of closed-loop experiment and thus, has to be known à-priori.

Guy Dumont (UBC EECE) EECE 574 - Adaptive Control January 2010 15 / 43

Identification in Closed Loop Identifiability Conditions

Identifiability Conditions

Let the system S be described by the following ARMAX process:

A(q−1)y(t)