Adaptive internal model control schemes

In order to implement the IMC-based controllers of the last section, the plant must be known a priori so that the `internal model' can be designed and the IMC parameterQ…s†calculated. When the plant itself is unknown, the IMC- based controllers cannot be implemented. In this case, the natural approach to follow is to retain the same controller structure as in Figure 1.1, with the internal model being adapted on-line based on some kind of parameter estimation mechanism, and the IMC parameterQ…s†being updated pointwise using one of the above control laws. This is the standard certainty equivalence approach of adaptive control and results in what are called adaptive internal model control schemes. Although such adaptive IMC schemes have been empirically studied inthe literature, e.g. [2, 3], our objective here is to develop adaptive IMC schemes with provable guarantees of stability and robustness.

To this end, we assume that the stable plant to be controlled is described by P…s† ˆZ0…s†

R0…s†‰1‡m…s†Š; >0 …1:7†

whereR0…s†is a monic Hurwitz polynomial of degreen;Z0…s†is a polynomial of degree l with l < n; Z0…s†

R0…s†represents the modelled part of the plant; and m…s†is a stable multiplicative uncertainty such that Z0…s†

R0…s†m…s† is strictly proper. We next present the design of the robust adaptive law which is carried out using a standard approach from the robust adaptive control literature [9].

1.3.1 Design of the robust adaptive law We start with the plant equation

yˆZ0…s†

R0…s†‰1‡m…s†Š‰uŠ; > 0 …1:8†

where u, y are the plant input and output signals. This equation can be rewritten as

R0…s†‰yŠ ˆZ0…s†‰uŠ ‡m…s†Z0…s†‰uŠ

Filtering both sides by 1…s†, where …s† is an arbitrary, monic, Hurwitz polynomial of degreen, we obtain

yˆ…s† R0…s†

…s† ‰yŠ ‡Z0…s†

…s†‰uŠ ‡m…s†Z0…s†

…s† ‰uŠ …1:9†

The above equation can be rewritten as

yˆT‡ …1:10†

where ˆ ‰1T; 2TŠT; 1, 2 are vectors containing the coecients of

‰…s† R0…s†Š and Z0…s† respectively; ˆ ‰T1; T2ŠT; 1ˆan 1…s†

…s† ‰yŠ, 2ˆal…s†

…s†‰uŠ;

an 1…s† ˆsn 1;sn 2;. . .;1T al…s† ˆsl;sl 1;. . .;1T and

ˆD m…s†Z0…s†

…s† ‰uŠ …1:11†

Equation (1.10) is exactly in the form of the linear parametric model with modelling error for which a large class of robust adaptive laws can be developed. In particular, using the gradient method with normalization and parameter projection, we obtain the following robust adaptive law [9]

_ˆPr‰"Š; …0† 2 C …1:12†

"ˆy y^

m2 …1:13†

^

yˆT …1:14†

m2ˆ1‡n2s; n2s ˆms …1:15†

m_sˆ 0ms‡u2‡y2; ms…0† ˆ0 …1:16†

where >0 is an adaptive gain;Cis a known compact convex set containing ;Pr‰Šis the standard projection operator which guarantees that the param- eter estimate…t†does not exit the setCand0>0 is a constant chosen so that m…s†, 1

…s† are analytic in Re‰sŠ 0

2. This choice of o, of course, necessitates some a priori knowledge about the stability margin of the

unmodelled dynamics, an assumption which has by now become fairly standard in the robust adaptive control literature [9]. The robust adaptive IMC schemes are obtained by replacing the internal model in Figure 1.1 by that obtained from equation (1.14), and the IMC parameters Q…s† by time- varying operators which implement the certainty equivalence versions of the controller structures considered in the last section. The design of these certainty equivalence controllers is discussed next.

1.3.2 Certainty equivalence control laws

We ®rst outline the steps involved in designing a general certainty equivalence adaptive IMC scheme. Thereafter, additional simpli®cations or complexities that result from the use of a particular control law will be discussed.

. Step 1: First use the parameter estimate …t† obtained from the robust adaptive law (1.12)±(1.16) to generate estimates of the numerator and denominator polynomials for the modelled part of the plant6

Z^0…s;t† ˆT2…t†al…s†

R^0…s;t† ˆ…s† T1…t†an 1…s†

. Step 2: Using the frozen time plant P…s;^ t† ˆZ^0…s;t†

R^0…s;t†, calculate the appro- priateQ…s;^ t†using the results developed in Section 1.2.

. Step 3: ExpressQ…s;^ t† as Q…s;^ t† ˆQ^n…s;t†

Q^d…s;t† where Q^n…s;t† and Q^d…s;t† are time-varying polynomials withQ^d…s;t†being monic.

. Step 4:Choose1…s†to be an arbitrary monic Hurwitz polynomial of degree equal to that of Q^d…s;t†, and let this degree be denoted by nd. . Step 5:The certainty equivalence control law is given by

uˆqTd…t†and 1…s†

1…s† ‰uŠ ‡qTn…t†and…s†

1…s†‰r "m2Š …1:17†

where qd…t† is the vector of coecients of 1…s† Q^d…s;t†; qn…t† is the vector of coecients of Q^n…s;t†; and…s† ˆ ‰snd; snd 1; . . .;1ŠT and and 1…s† ˆ ‰snd 1; snd 2; . . .;1ŠT.

The robust adaptive IMC scheme resulting from combining the control law (1.17) with the robust adaptive law (1.12)±(1.16) is schematically depicted in

6In the rest of this chapter, the `hats' denote the time varying polynomials/frozen time `transfer functions' that result from replacing the time-invariant coecients of a

`hat-free' polynomial/transfer function by their corresponding time-varying values obtained from adaptation and/or certainty equivalence control.

Figure 1.2. We now proceed to discuss the simpli®cations or additional complexities that result from the use of each of the controller structures presented in Section 1.2.

1.3.2.1 Partial adaptive pole placement

In this case, the design of the IMC parameter does not depend on the estimated plant. Indeed, Q…s†is a ®xed stable transfer function and not a time-varying operator so that we essentially recover the scheme presented in [4].

Consequently, this scheme admits a simpler stability analysis as in [4] although the general analysis procedure to be presented in the next section is also applicable.

1.3.2.2 Model reference adaptive control

In this case from (1.1), we see that the Q…s;^ t† in Step 2 of the certainty equivalence design becomes

Q…s;^ t† ˆWm…s†P…s;^ t† 1

…1:18†

Our stability analysis to be presented in the next section is based on results in the area of slowly time-varying systems. In order for these results to be applicable, it is required that the operator Q…s;^ t† be pointwise stable and also that the degree ofQ^d…s;t†in Step 3 of the certainty equivalence design not change with time. These two requirements can be satis®ed as follows:

. The pointwise stability of Q…s;^ t† can be guaranteed by ensuring that the frozen time estimated plant is minimum phase, i.e.Z^0…s;t†is Hurwitz stable for every ®xedt. To guarantee such a property forZ^0…s;t†, the projection set C in (1.12)±(1.16) is chosen so that 82 C, the corresponding Z0…s† ˆT2al…s† is Hurwitz stable. By restricting C to be a subset of a Cartesian product of closed intervals, results from Kharitonov Theory [10]

can be used to ensure that C satis®es such a requirement. Also, when the

r ‡

qTn…t†

qTd…t†

u

T P…s†

Regressor generating block

yˆT‡

y^ˆT

"m2 and…s†

L1…s†

and 1…s†

L1…s†

‡

‡

‡

Figure 1.2 Robust adaptive IMC scheme

projection set C cannot be speci®ed as a single convex set, results from hysteresis switching using a ®nite number of convex sets [11] can be used.

. The degree ofQ^d…s;t†can also be rendered time invariant by ensuring that the leading coecient ofZ^0…s;t† is not allowed to pass through zero. This feature can be built into the adaptive law by assuming some knowledge about the sign and a lower bound on the absolute value of the leading coecient ofZ0…s†. Projection techniques, appropriately utilizing this knowl- edge, are by now standard in the adaptive control literature [12].

We will therefore assume that for IMC-based model reference adaptive control, the set C has been suitably chosen to guarantee that the estimate …t† obtained from (1.12)±(1.16) actually satis®es both of the properties mentioned above.

1.3.2.3 AdaptiveH2 optimal control

In this case,Q…s;^ t†is obtained by substitutingP^M1…s;t†,B^P1…s;t†into the right- hand side of (1.3) whereP^M…s;t†is the minimum phase portion ofP…s;^ t†and B^P…s;t†is the Blaschke product containing the open right-half plane zeros of Z^0…s;t†. ThusQ…s;^ t†is given by

Q…s;^ t† ˆP^M1…s;t†RM1…s†‰B^P1…s;t†RM…s†ŠF…s† …1:19†

where ‰Š denotes that after a partial fraction expansion, the terms corre- sponding to the poles ofB^P1…s;t†are removed, andF…s†is an IMC ®lter used to force Q…s;^ t†to be proper. As will be seen in the next section, speci®cally Lemma 4.1, the degree ofQ^d…s;t†in Step 3 of the certainty equivalence design can be kept constantusing a single ®xed F…s†provided the leading coecient of Z^0…s;t†is not allowed to pass through zero. Additionally Z^0…s;t†should not have any zeros on the imaginary axis. A parameter projection modi®cation, as in the case of model reference adaptive control, can be incorporated into the adaptive law (1.12)±(1.16) to guarantee both of these properties.

1.3.2.4 AdaptiveH1 optimal control

In this case,Q…s;^ t†is obtained by substitutingP…s;^ t†into the right-hand side of (1.5), i.e.

Q…s;^ t† ˆ 1 W…b^1† W…s†

" #

P^ 1…s;t†F…s† …1:20†

whereb^1is the open right half plane zero ofZ^0…s;t†andF…s†is the IMC ®lter.

Since (1.20) assumes the presence of only one open right half plane zero, the estimated polynomial Z^0…s;t†must have only one open right half plane zero and none on the imaginary axis. Additionally the leading coecient ofZ^0…s;t†

should not be allowed to pass through zero so that the degree ofQ^d…s;t†in Step

3 of the certainty equivalence design can be kept ®xed using a single ®xedF…s†.

Once again, both of these properties can be guaranteed by the adaptive law by appropriately choosing the setC.

Remark 3.1 The actual construction of the sets C for adaptive model reference, adaptive H2 and adaptive H1 optimal control may not be straightforward especially for higher order plants. However, this is a well- known problem that arises in any certainty equivalence control scheme based on the estimated plant and is really not a drawback associated with the IMC design methodology. Although from time to time a lot of possible solutions to this problem have been proposed in the adaptive literature, it would be fair to say that, by and large, no satisfactory solution is currently available.