A semi-infinite horizon LQ self-tuning regulator for ARMAX plants based on RLS

Automatica, Vol.28, No. 2. pp. 401-406. 1992 Printedin Great Britain.

0(~15-1098/92 $5.00+ 0.00 PergamonPressplc 1992 InternationalFederationof AutomaticControl

Brief Paper

A Semi-infinite Horizon LQ Self-tuning Regulator for A R M A X Plants Based on RLS* E. MOSCA~ and J. M. LEMOS$ Key Wards--Adaptive Control; self-tuning regulators; stochastic control robustness.

algorithm having high performance robustness. Indeed, as shown via an ODE analysis (Mosca et al., 1989), and unlike in single-predictor based self-tuners, MUSMAR possible convergence points are close approximations to the minima of the adopted unconditional quadratic cost, even in the presence of mismatching conditions, provided that the control horizon T is chosen large enough. More precisely, T such be chosen such that I~.~1<<1, where AM is the maximum eigenvalue of the closed-loop system. This implies that when I).MI is close to one, T must be large so as to let all the transients decay within the control horizon. This happens for instance in nonminimum-phase plants having zeros close to the unit circle. Since tAMI is a priori unknown, there is no definite criterion for a suitable a priori choice of T. In practice, T is chosen as a compromise between the algorithm computational simplicity, which decreases with T, and the requirement of stabilizing generic unknown plants. In fact, the latter would impose, in principle, T = o% and hence an unacceptable computational load, as well as an irrealizable implementation. In addition, parameter estimates of many-steps ahead predictors tend to be poor. By extending T, one actually overparametrizes the plant. This is however not equivalent to using T = 1 and extending the assumed order of the plant, n. For instance, in the latter case, to stabilize an open-loop unstable nonminimum-phase plant would be impossible, as opposite to what happens when T---~Qo. Further, as shown in Mosca and Zappa (1983), increasing n leads to identifiability problems of the feedback coetficients. On the contrary, this can be avoided if n is kept small and T is enlarged. The above facts motivate the search of the adaptive control algorithms based on a finite number of identifiers and which might yield a tight approximation to the stochastic steady state (T---~oo) LQ control. For the deterministic case Samson and Fuchs (1981) and Ossman and Kamen (1987) proposed a scheme in which a state-space model of the plant is built upon estimates of the one-step ahead predictor. An estimate of the state provided by an adaptive observer is then fedback, the feedback gain being computed via Riccati iterations spread in time. Similar schemes have been developed for stochastic plants (Peterka, 1986). When ARMAX plants are considered, extended least-squares (ELS) or recursive maximum likelihood (RML) identification algorithms must be used. This has the drawback that the inherent simplicity of standard RLS is lost. Here, "simplicity" refers not only to the computational burden, but mainly to the fact that both ELS and RML involve highly nonlinear operations in that their regressor depends not only on the current experimental data, but also on previous estimates. Along this line, it is interesting to establish whether the tuning properties of the classical RLS + GMV (generalized minimum variance) self-tuner can be extended to an RLS + LQ adaptive regulator. Given the above motivations, the problem to be considered in the present paper is the following: Is it possible to suitably modify the MUSMAR algorithm so as to adaptively get the semi-infinite horizon LQ stochastic control for any ARMAX plant by using a small number of predictors

A ~ t r a e t - - S o far no extension of the celebrated self-tuning property of the RLS + GMV adaptive controller is available for the steady-state LQ stochastic control. The above extension is addressed in the paper. Namely, we pose the question: Is it possible to adaptively get the semi-infinite horizon LQ stochastic control for any ARMAX plant by using a finite number of predictors whose parameters are estimated by standard RLS? An adaptive control algorithm solving an underlying control problem that coincides with the semi-infinite LQ stochastic feedback is developed. It embodies a standard RLS separate identification of the parameters of T -> n + 1 predictors of the joint i/o process, n being the order of the ARMAX plant, together with an appropriate control synthesis rule. The proposed algorithm turns out to be a slightly modified version of MUSMAR with the capability of turning itself to the semi-infinite horizon LQ stochastic regulator. 1. Introduction SINCE THE main limitations of standard self-tuning controllers are caused by the adoption of performance critera consisting of a single-step ahead cost functional (Wittenmark and Rao, 1979), a number of long-range adaptive controllers have been developed in the last few years (Grimble, 1984; Peterka, 1984; Clarke et al., 1987; Menga and Mosca, 1980; Lemos and Mosca, 1985). Basically, all such controllers attempt to approximate the stochastic steady-state (s.s.s.) LQ control law. Among these, MUSMAR (Greco et al., 1984) is based on a set of predictive linear-regression models describing, according to the Implicit Modelling Principle (Mosca and Zappa, 1985), the closed-loop joint i/o process, over a control horizon of T steps. The parameters of the above predictive linear-regression models are estimated by standard recursive least squares (RLS). A significant feature of MUSMAR is that all the predictive models are identified directly from experimental data. Thus, no model matching condition is assumed, as is the case in the indirect schemes based on the Certainty Equivalence Control Assumption where the plant predictive behaviour is obtained by extrapolating an identified single-step ahead (ARMAX) model. The redundancy thereby introduced results in an

* Received 4 December 1989; revised 28 January 1991; received in final form 1 August 1991. The original version of this paper was presented at the 3rd IFAC Symposium on Adaptive Control and Signal Processing which was held in Glasgow, Scotland during April 1989. The published proceedings of this IFAC Meeting may be ordered from Pergamon Press plc, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by Associate Editor T. Srderstrrm under the direction of Editor P. C. Parks. f Dipartimento di Sistemi e Informatica, Universit~i di Firenze, Via S. Marta 3, 50139 Firenze, Italy. Author to whom all correspondence should be addressed. ~tINESC, R. Alves Redoi 9, apartado 10105, 1017 Lisboa codex, Portugal. 401

402

Brief Paper

whose parameters are estimated by standard RLS in a separate way? It is shown hereafter how for an n-order A R M A X plant, the semi-infinite horizon LQ feedback can be adaptively obtained by using at least n + 1 i/o predictors, whose parameters are estimated by RLS, together with an appropriate control synthesis rule.

2. Problem formulation Hereafter, the following SISO A R M A X plant A(d)y(k

) = B(d)u(k

) + C(d)e(k

)

nb

tlc

E bi d~ C ( d ) = l +

in

the

y ( t - T + i ) = O i u ( t - T) + ~O's(t- T) + viY(t- T + i )

(8)

u ( t - T + i ) = # i u ( t - T) + dp's(t- T) + v T ( t - T + i).

(9)

This is accomplished with the formulas

- Oi(t - l ) u ( t -- T ) - ~ [ ( t - 1 ) s ( t - t)]

(1)

i~ l

unstable common factors, C is strictly Hurwitz, A, B and C do not share common factors and {e(k)} is a sequence of 2 i.i.d, stochastic variables with zero mean and variance o e. Hereafter, the order of the plant will be denoted by n, n = max {n,,, n h, no}. Associated with the plant, a quadratic cost functional defined over a NT-steps horizon is considered t+NT

k~,='

1

[y2(k+l)+

pu2(k)]}

(2)

with p > 0. In the sequel, it will become clear why in (2) the regulation horizon is denoted by NT. In fact, it will be convenient to increase the regulation horizon by holding T fixed and letting N to become larger. Our goal is to find a convenient procedure in order to select the input u(t) to the plant (1) minimizing the cost (2) for N---~ ~, in such a way that the following requirements are met: R1. The feedback is directly updated according to predictor parameters estimated by standard RLS; R2. The number of operations per single cycle does not grow with N.

[ dPi(t - 1)] + r(t)[u(t -- r + i)

- OAt - 1)u(t - T ) - ~ , ( t - l ) s ( t - r ) l

~ c f l i, such that A and B have no

qJ(t) = [ u ( t - T ) s ' ( t K(t)

z,=Wz, T+Ou,

r +wt

(12)

,~ + ¢p'(t)P(t)cp(t) P(t + 1) = [1 - K(t)cp(t)]P(t)/L

where ~2 ~ diag ~ " ~ ' n

(4)

P""" n'-"--h'--"~ P' T-~'~"~nl''' 1, ~ ) , _ ~v ~ ~ + oP

(5)

and/5 and/~ are, respectively, the matrix of weights and the feedback vector used at time t - 1. 3. Update the augmented vector of feedback gains by F=-(o'eo)

'o'eq J

(6)

with O and qJ replaced by their current estimates, and then apply the control at time t given by

u(t) = F~s(t)

(13)

In practice, a factorized version of these formulas, such as Bierman's U D algorithm is to be used• In the above, 0 i and #i are scalars, ~Pi and ~i are column vectors of dimension 2n, ~ ( . ) and v~'(.) are prediction errors uncorrelated with both u(.) and s(.) and .~ denotes an estimate• Note that since the regressor [u(t - T)s'(t - T)]' is the same for all the models in the RLS algorithms, there is only the need to update one covariance matrix of dimension 2n + 1. This considerably reduces the numerical complexity of the algorithm. The estimates of the matrices 19 and W are then given by:

v/;-

0 T

n+l

~T-

qJ =

O=

ktT

OT j

(3)

where wt denotes a residual vector uncorrelated with S ~ rt t XtZut-lxtlt both z t r and u,_r, zt a [s~'Yt']', t - - l ~ Y t - , + l ) t t - , ) 1 Y,~ t--n i ut--n 1 ~ r [(Yt r + 0 ( t r ) ] , q J i s a 2 T x 2 T m a t r i x a n d O a 2 T x l vector such that the bottom row of qJ is zero, the bottom element of O is 1, and the last 2 ( T - n) columns of ud are zero. Here, x,q2 denotes the column vector [x(tl)x(t ~ 1)'"x(t2)]'. 2. Update the matrix of weights P by the difference Riccati equation P = tIJ~/sW F + Q

(1l)

T)I'

P(t)~o'(t)

OT

3. The adaptive control algorithm The following receding horizon scheme, for any T-> n + 1 will be considered hereafter. MUSMAR-oo algorithm. 1. Given all i/o data up to time t, compute RLS estimates of the parameter matrices tlJ, O in the following set of predictive models:

(10)

1)

=

¢i(t) J

i=l

1 E[JN(t)]=~E{

T-1

Oi(t, ] = [ O-i(t-11)) ] + K ( t ) [ y ( t - T + i ) ~i(t)J LlPi(t-

is considered. In (1), A(d), B(d) and C(d) are polynomials na in the unit delay operator d: A ( d ) = 1 + ~ ai di B ( d ) =

i=l

0 i , ~ o i , i = l . . . . , T and #i,~Pi, i = l . . . . . following set of predictive models:

(7)

where F~ is made-up by the first 2n components of F. 4. Set P = P, F = F, sample the output of the plant and go to step 1 with t replaced by t + 1. []

Remark 1. The estimation of the parameters in (3) is performed by first updating RLS estimates of the parameters

Oi ['~T--n

I

z #l 0-.-0

1

Remark 2. The row vector F has dimension 2T. Given the structure of tlJ, with zeros on the last 2(T - n) columns, the last 2(T - n) entries of F are also zero• 4. Justification o f MUSMAR-oo The aim hereafter is to show that, under suitable assumptions, the LQSR feedback is an equilibrium point of M U S M A R -oo. Some required results on implicit modelling theory are summed up by the following lemma, whose proof is given in Mosca and Zappa (1985): Lemma 1 (Implicit Modelling Principle)• Let the inputs to the A R M A X plant (1) be given by u(k) = Fs(k)

(14)

or equivalently, for suitable polynomials R(d) and S(d), by

R(d)u(k) = -S(d)y(k).

(15)

Let R and S be coprime and such that the closed-loop characteristic polynomial Q = A R + BS be stable and divided by C: C I Q.

(16)

Brief Paper

Fig. 1) must be given by a constant feedback. Since u(t) must be left unconstrained, this implies (see Fig. 1)

If the above conditions are fulfilled for k=t-n

.....

t-l,t+l

.....

403

t+T-I

(17) T -> n + 1.

(23)

then z,+ z has the following representation: z,+ r = qJz, + Ou, + ~,+ r

(18)

~,+r •span {e,+, ,+r }

(19)

where

and ud and O have the properties specified after (3).

[]

Remark 3. As shown in detail in Mosca and Zappa (1985), (15) and (16) amount to assuming that the inputs are given by a stabilizing constant feedback from any Kalman filtered state £,1, of the plant over an interval including, w.r.t, time t, at least n steps in the past as well as T - 1 steps in the future. It is to be remarked that, for the validity of (18) u, must be left unconstrained. The relevance of (18) stems from property (19) which is valid even if, as in (1), C(d):# 1. In fact, (19) allows one to deal with an arbitrary suitably controlled A R M A X plant as if it is of an A R X type. Under such circumstances, W and 0 in (18) can be estimated by standard RLS. Remark 4. The parameters in (18) characterize the dynamics of the closed-loop system. Therefore, they depend on the feedback gain polynomials R and S, as well as on the plant and disturbance dynamics, defined by polynomials A, B and C. It would be burdensome and of little help for the purpose of this paper to give closed-form expressions of W and O in terms of A, B, C, R and S. The point to note, however, is that (18) expresses a prediction of the joint i/o process of a controlled A R M A X plant using a finite set of past i/o data. This should not be confused with the usual technique of developing predictors using Diophantine equations. [] The interest in (18) is that, as soon as W and O are known, (2), which may be written as 1

N

E[JN(t)] = - ~ E { i ~

,lz,+irl'~}

(20)

Clearly, according to the definition of n, (23) already comprises any i/o transport delay that might be present on the plant. The above considerations are summed up in the following iemma. Lemma 2. Let assumptions (14)-(16) be fulfilled for k=t-n

.....

t-l,t+l

.....

t+NT-1.

Then, if T satisfies (23), z,+~r, i = 1, 2 . . . . . N, has the state-space representation (21) irrespective of the plant C(d) innovations polynomial and the value taken on by u r [] Remark 5. Inequality (23) specifies in terms of the plant order n, the minimal dimension of the state z required to carry out in a correct way the minimization procedure under consideration. [] Thus, assuming (23), (21) can be used for all i-> 1. For i -> 2, using (14) in (21), one has Zt+iT = IIJ Fzt+(i--1)T -}- ~t+iT = Zt+iT + Zt+iT

(24)

where kuF is as in (5), £,+ir is the zero input response from the initial state z,+2T, and z,+ir is the response due to ~,+iT from the zero state at time t + 2T. Thus, taking into account that E[~,+ir~;+iT] = 0

(25)

and denoting (20) by ~N(t, F), so as to point out that past and future inputs are given by a constant feedback, one has 1

2

2

~N(t, F) = - ~ E [ l l z , + r l l o + IIz,+zrlle(N) + VN(t, F)] N where VN(t, F ) = I ~ 3

Ile,+~rll~ is

(26)

not affected by u,, and

N--2

can be easily minimized w.r.t, u, when suitable assumptions, to be discussed next, are made on the magnitude of T, and past and future inputs. In fact, if (14)-(17) are assumed, (18) expresses zt+ r in terms of u r Next,

P ( N ) = fl + E (ud~-)i~W~ satisfies the following Lyapunov-

(21)

with A(N)___a(qjFN-~)%qWFN-1. Thus, the first two additive terms within brackets in (26) equal

Zt+iT -~ IIilZt+(i--l)T+ OUt+(i-t)T + ~t+iT

also for all i -> 2 is the inputs u(k) are given by the previous feedback law for k = t - n + ( i - 1)T, . . . . t + i T - 1.

i=l

type equation (Astr/Jm, 1970) P ( N ) = W'FP(N)UJF + f~ - A(N)

2

I t

I t+l

ut

P(N) = - { O ' [ P ( N ) + A(N)IO}-'O'[P(N) + A(N)IW.

lu(t) must be unconstrained

given

t+2T-i \ \ \ \\\ \k \\\\\\\\\\ \x4 k kkk

x \N, k k \ \ \ \ \ N . ~ N ~

I ' " '

2

Consequently arg min &eN(t, F) = # ( N)z, with

inputs oonstrained to be by a ocnstant feedback

ikk~ kk k XkX

__

E[llZ,+ rllQ + IlU2FZ,+rI~P(N)] - E IIZ,+Tlle(N)+ ~(N).

(22)

Since (20) has to be minimized w.r.t, u,, u, must be left unconstrained. Taking i = 2, it is seen that all the inputs between t + ( - n + T) and t - 2T - 1 (the shaded interval on

t+ (-n+T) ~\~\\k\

2

(27)

t+T

f

kl

I

I

I

t+T+l " ' " t+2T-i t+2T t+2T~l . •

left

FIG. 1. Illustration of the minorant imposed on T.

I t+NT

(28)

404

Brief Paper

We now consider the minimization of .ffN(t, F) w.r.t, u, for N---, :~. It is clear that, being W F a stability matrix, one can define &e ___alim &aN(t, F)

(29)

Accumulated lou

divided by tlme

100

9O 80

N ~ 70

Since P(N) + A(N) > P(N) > ff~ >- el, with 0 < c -< min (1, p), F(N) is a continuous function of P ( N ) + A(N). Consequently, since P(N) + A(N)--* P as N---~ o% one has f=a lim F ( N ) = - [ O ' P O ] - ' O ' P W

(30)

00 50 40 30

with P solution of the following Lyapunov equation

P = ItI'FPtll F + if2.

(31)

Theorem 1. Under the same assumptions as in Lemma 2, the input^ at time t minimizing &e®(t,F) in (29) is given by ut=Fz,, with P specified by^(30) and (31). Further, if the procedure used to generate F from F is iterated, the LQSR feedback is as equilibrium point for the resulting iterations. Proof. It remains to show the validity of the last assertion whose proof was given in Greco et al. (1984, Proposition 2). [] By the structure of matrix W, the last T - n elements of f are zero and thus (7) holds. Further, in order to circumvent difficulties associated within possible feedback vectors making temporarily the closed-loop system unstable, and hence the Lyapunov equation (31) meaningless, in MUSMAR-:~ P is updated via the Riccati difference equation

/"

10

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

time

FIG. 2. The accumulated loss divided by time when the plant of example 1 is controlled by standard M U S M A R (T = 3) and MUSMAR-oo. still retaining good performance robustness thanks to its multipredictor implementation.

Example 1. A nonminimum-phase second-order plant is controlled by standard M U S M A R (Greco et al., 1984) and MUSMAR-oo. The plant to be controlled is y(t + 1) - 1.5y(t) + 0.7y(t - 1) = u ( t ) - 1 . 0 1 u ( t - 1) + e ( t + 1)

(4). 5. Algorithmic considerations The matrix W and the vector O can be partitioned in the following blocks:

with input weight p =0.1 in the performance index (2). Here, and in the following examples, {e(t)} is a sequence of independent Gaussian random variables of zero mean and unit variance. Figure 2 compares the accumulated loss divided by time i.e.

with W,[2n x 2n], W, = [ ( T - n ) × 2n], Os[2n], O,[T-n] and the bottom elements of We and O r are zero and 1, respectively, i.e. the predictive model (13) does not impose any constraint on u(t). Let P be partitioned according to

with P~[2n x 2nl, Pr[(T - n) x 2n] and the same for/5. Then, a simple calculation shows that (4) and (6) can be replaced by the equations with smaller dimensions P~ = (W, + O,f~)'/ss(tlJs + O,F,) + Q,

(33)

O'P,W~ + O~QrW , 17 = _ O ' P , O , + O~,Q,O e

(34)

and

with Q~-adiag~,

"~},

Q,=adiag{~~},

and/5~ initialized by/5~ = Q~. The algorithm assumes p 4:0. This in practice constitutes no restriction since p may be made as small as needed.

6. Simulation results The following examples are designed in order to exhibit the features of the M U S M A R - ~ algorithm. Comparisons are made with an indirect LQ adaptive controller (ILQ) based on the same underlying control problem as M U S M A R - ~ , the difference being in the fact that the first identify the usual one-step ahead predictive model and next the LQ control is computed indirectly. Both full and reduced complexity controllers are considered, the aim being to show that MUSMAR-0o is capable of stabilizing plants requiring long control horizons (a feature due to its underlying control law),

t i ~ l y(i)2 + p u ( i - 1) 2

for

standard

MUSMAR

( T = 3 ) and M U S M A R - ~ ( T = 3 ) . In both cases a full complexity controller is used. Since this plant has a nonminimum-phase zero very close to the unit circle, the control horizon T should be very large in order for M U S M A R to behave close to optimal performance. With T = 3 (a value chosen according to the rule T = n + 1), M U S M A R - ~ yields a much smaller cost than standard M U S M A R , exhibiting a loss very close to the optimal one.

Example 2. Since MUSMAR-c~ is based on a state-space model build upon separately estimated predictors, it turns out, according to the implicit modelling principle, not to be affected by a C polynomial different from 1 in the A R M A X plant representation. In this example the following plant with an highly correlated noise is considered: y(t + 1) = u(t) + e(t + 1) - 0.99e(t).

(35)

For p = 1, M U S M A R - ~ converges to F~ =[0.492 0.494]. The optimal feedback is b~, = [0.495 0.495].

Example 3. Standard M U S M A R was shown (Mosca et al., 1989) to be robust against unmodelled dynamics, in the sense that, if T is large enough, M U S M A R converges to the minima of the cost constrained to the chosen regulator regressor. M U S M A R - ~ is expected to inherit the robustness properties of M U S M A R due to the fact that it is based on a set of predictive models that are separately estimated. Consider the open-loop unstable plant

y(t + 1) + 0.9y(t) - 0.5y(t - 1) = u(t) + e(t + 1) - 0.7e(t). (36) Although the plant is of second order, and hence s(t) should be [y(t)y(t - 1)u(t - 1)u(t - 2)], s(t) is instead chosen to be

s,= [y,u, t]'.

(37)

The optimal feedback constrained to the above choice of

Brief Paper fourth-order plant

0.5 0.4

405

' ~ST^RLE

'

I

Y,+4-O. 167yt+3-O.74y,+2-O. 132y,+ I + 0.87y,

0.3

= 0. 132u,+3 + 0.545u,+ 2 + 1.117u~+ i + 0.262u, + e~+4.

0,2

~

o -0.1 -0.2 :

~STAS~

--0.~ --0.4 --0.5 0.2

0.4

0.0

0,8

1

t.2

1.4

1.6

1.5

fl

FIG. 3. The evolution of the feedback calculated by MUSMAR-~ in example 3, superimposed to the level curves of the underlying quadratic cost.

s(t) is, for p = 1 F~*= [1.147 -0.1091. Figure 3 shows the evolution of the feedback when the above plant is controlled by MUSMAR-~ on the space [fir2], superimposed on the quadratic curves of the quadratic cost, constrained to the chosen regulator regressor (37). As is apparent, MUSMARco is able to tune close to the minimum of the underlying cost, despite the presence of unmodelled plant dynamics. Example 4. This example aims at showing the importance of the independent estimation of the predictors in MUSMARo0. A comparison is made with ILO. Consider the nonminimum-phase, open-loop stable, Controller

gains

0.~

-I

-1.5

~ 500

' 1000

' 1500

' 2000

2500

3000

3500

4000

4500

5000

time

FIG. 4. Convergence of the feedback when the plant of example 4 is controlled by MUSMAR-~. A c c u m u l a t e d l o s s divided b y t i m e

14

12

10

6

0

~US~R

500

1000

1500

2000

2500

3000

5500

4000

4500

5000

time

FIG. 5. The accumulated loss divided by time when the plant of example 4 is controlled by ILQ, standard MUSMAR (T = 3) and MUSMAR-~.

Figures 4 and 5 show the results obtained when this plant is coupled to the reduced complexity controller defined by ut=flY,+f2Y, i+f~ u, i+f4 u, 2 and p = 1 0 4. Figure 4 shows the convergence of the first three components of the feedback when MUSMAR-~ is used. Figure 5 shows the accumulated loss divided by time, when ILQ, standard MUSMAR (T = 3) and MUSMAR-~ are used. Although both MUSMAR-~ and ILQ should yield the infinite horizon LQS feedback under model matching conditions, due to the presence of unmodelled dynamics, ILQ presents a large detuning. MUSMAR-~, instead, being based on a multipredictor model, is insensitive to plant unmodelled dynamics.

7. Conclusions An adaptive control algorithm solving an underlying control problem that coincides with the semi-infinite LQ stochastic feedback has been developed. It embodies a standard RLS separate identification of T predictors of the joint i/o process, T>-n + 1, n being the order of the ARMAX plant, together with an appropriate control synthesis rule. The proposed algorithm turns out to be a slightly modified version of MUSMAR with the capability of tuning itself to the semi-infinite horizon LQ stochastic regulator (LQSR). Indeed, it is shown that, in the absence of unmodelled plant dynamics, the LQSR feedback is an equilibrium point of the resulting algorithm, denominated MUSMAR-~. When unmodelled dynamics are present, simulations show that MUSMAR-~ is still capable of tuning itself onto the minima of the underlying cost constrained to the chosen regulator regressor. The idea of performing Riccati iterations spread in time, in order to obtain adaptive control algorithms that tentatively converge to the LQSR feedback, has been used in the past. The work reported here includes two new contributions that are believed to be relevant. First, modelling issues are such that standard RLS are used even when the plant is ARMAX. Second, the redundancy introduced by the separate estimation of a multipredictor set of models, robustify the performance with respect to plant unmodelled dynamics. References Astr6m, K. J. (1970). Introduction to Stochustic Control Theory. Academic Press, New York. Clarke, D. W., C. Mohtadi and P. S. Tufts (1987). Generalized predictive control--Part I. The basic algorithm. Automatica, 23, 137-148. Greco, C., G. Menga, E. Mosca and G. Zappa (1984). Performance improvements of self-tuning controllers by multistep horizons: The MUSMAR approach. Automatica, 20, 681-699. Grimble, M. J. (1984). Implicit and explicit LQG self-tuning controllers. Automatica, 20, 661-669. Lemos, J. M. and E. Mosca (1985). A multipredictor based LQ self-tuning controller. Prec. 7th IFAC Symp. on Identification and System Parameter Estimation. York, UK, pp. 137-142. Menga, G. and E. Mosca (1980). MUSMAR: multivariable adaptive regulators based on multistep cost functionals. In D. G. Lainiotis and N. S. Tsannes (Eds), Advances in Control. Reidel, Dordrecht, pp. 334-341. Mosca, E. and G. Zappa (1983). Overparametrization, positive realness and multistep minimum variance adaptive regulators. In R. S. Bucy and J. M. F. Moura (Eds), Nonlinear Stochastic Problems. Reidel, Dordrecht, pp. 205 -216. Mosca, E. and G. Zappa (1985). ARX modelling of controlled ARMAX plants and its application to robust multipredictor adaptive control. Prec. 24th IEEE Conference on Decision and Control, Ft. Lauderdale, Florida, pp. 856-861; also IEEE Trans. Aut. Control, AC-33 (1988), 371-375.

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Brief Paper

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