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UNIVERSAL REGULATORS FOR DISCRETE-TIME SYSTEMS P A Cook* Introduction The work reported here is an approach to the problem of universal adaptive stabilization for discrete-time systems by means of switching controllers, as previously developed in different ways by MGtensson (1985), Fu & Barmish (1986), Miller & Davison (1989), for continuous- time systems. It is assumed that the plant is linear, time-invariant and can be stabilized by a feedback compensator of known structure, so that the task of the switching algorithm is to search the space of possible controller parameters until stability is achieved. Since the search has to be undertaken on an infinite set, dense in the parameter space, the algorithm is designed so that, in the absence of disturbances, the switching will eventually terminate and the system will then asymptotically approach its equilibrium state. Moreover, by utilizing the internal model principle, the adaptive stabilizer can be embedded in a control scheme to track reference signals with known dynamic properties. Modifications are then considered, so as to allow for the presence of disturbances and the fact that the structure of a stabilizing compensator may not be known. Further, because all systems are in practice subject to bounds on their state variables, the consequences of taking this into account are also examined, in order to show that the switching algorithms still assure the same asymptotic properties for the controlled system. System Description The system to be stabilized, with input U and output y, is described by the controllable and observable state-space representation z(M + 1) = Az(M) + Bu(M), y(M) = Cz(M) where the state-vector z can include the state variables of any internal model required for tracking purposes, besides those of the plant itself. If control is applied by means of a time- varying dynamic compensator with the representation z(M + 1) = F(M)z(kf) + G(M)y(M), u(M) = H(M)z(M) + K(M)y(M) it is convenient to define augmented vectors so that the complete system has the state-space description Z(M + 1) = AqM) + Bii(M), ji(M) = CZ(M), ii(M) = R(M)$(M) with appropriate definition of the matrices &a, c and l?(M). It is assumed that a stabilizing compensator exists, in the sense that there is a matrix I? such that all the eigenvalues of (A + skc) have magnitude less than unity. 'Control Systems Centre, Department of Electrical Engineering and Electronics, UMIST, hlanchester M60 1QD 3/1

[IEE IEE Colloquium on Adaptive Control - London, UK (13 June 1996)] IEE Colloquium on Adaptive Control - Universal regulators for discrete-time systems

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Page 1: [IEE IEE Colloquium on Adaptive Control - London, UK (13 June 1996)] IEE Colloquium on Adaptive Control - Universal regulators for discrete-time systems

UNIVERSAL REGULATORS FOR DISCRETE-TIME SYSTEMS

P A Cook*

Introduction

The work reported here is an approach to the problem of universal adaptive stabilization for discrete-time systems by means of switching controllers, as previously developed in different ways by MGtensson (1985), Fu & Barmish (1986), Miller & Davison (1989), for continuous- time systems. It is assumed that the plant is linear, time-invariant and can be stabilized by a feedback compensator of known structure, so that the task of the switching algorithm is to search the space of possible controller parameters until stability is achieved. Since the search has to be undertaken on an infinite set, dense in the parameter space, the algorithm is designed so that, in the absence of disturbances, the switching will eventually terminate and the system will then asymptotically approach its equilibrium state. Moreover, by utilizing the internal model principle, the adaptive stabilizer can be embedded in a control scheme to track reference signals with known dynamic properties. Modifications are then considered, so as to allow for the presence of disturbances and the fact that the structure of a stabilizing compensator may not be known. Further, because all systems are in practice subject to bounds on their state variables, the consequences of taking this into account are also examined, in order to show that the switching algorithms still assure the same asymptotic properties for the controlled system.

System Description

The system to be stabilized, with input U and output y, is described by the controllable and observable state-space representation

z ( M + 1) = A z ( M ) + Bu(M), y ( M ) = C z ( M )

where the state-vector z can include the state variables of any internal model required for tracking purposes, besides those of the plant itself. If control is applied by means of a time- varying dynamic compensator with the representation

z ( M + 1) = F ( M ) z ( k f ) + G(M)y(M), u ( M ) = H ( M ) z ( M ) + K ( M ) y ( M )

it is convenient to define augmented vectors

so that the complete system has the state-space description

Z(M + 1) = A q M ) + Bi i (M) , j i (M) = C Z ( M ) , i i (M) = R ( M ) $ ( M )

with appropriate definition of the matrices &a, c and l?(M). It is assumed that a stabilizing compensator exists, in the sense that there is a matrix I? such that all the eigenvalues of (A + skc) have magnitude less than unity.

'Control Systems Centre, Department of Electrical Engineering and Electronics, UMIST, hlanchester M60 1QD

3 / 1

Page 2: [IEE IEE Colloquium on Adaptive Control - London, UK (13 June 1996)] IEE Colloquium on Adaptive Control - Universal regulators for discrete-time systems

Switching Algorithm

The function of the switching algorithm is to conduct an on-line search through a sequence of matrices h(i) , dense in the set of possible feedback matrices I?, until it finds one which stabilizes the system. Thus, it sets

where M; are the switching instants. In order to decide when to switch, the algorithm uses a monitoring function defined by

& ( M ) = h(i) [Mi 5 M < Mi,,]

M-1

with 0 < X < 1. The sequence M; is then generated by setting

if this exists, and otherwise Mi+l = 00, where f ( i ) + CO as i + CO. Using this algorithm, it is shown by Lai & Cook (1995a) that, if X is sufficiently close to unity, the switching sequence terminates and 11Z(M)ll + 0 as M -+ CO. Moreover, the same result holds if the search is extended over controllers of increasingly high order, so that the structure does not need to be known in advance. Also, different switching procedures can be used, as in Lai & Cook (1995b), with similar consequences. Further, by choosing the search sequence in a particular way, e.g., so that lIh(i)/l*/f(i) -+ 0 as i -+ ca, it is possible to show that the simpler monitoring function

j=O

can be used instead of T ( M ) .

Tracking

In order to track a reference signal (or reject a disturbance) of known dynamic structure, i.e., satisfying a given recurrence relation, the controller is designed, in accordance with the internal model principle, so as to incorporate a precompensator which duplicates the dynamics required to generate the exogenous signals. By including the state variables of the precompensator in the complete system state vector, the previous analysis can then be extended to show that asymptotic tracking of the reference signal by the plant output is achieved, while the state and input vectors remain bounded provided that the exogenous signals are also bounded (Lai 1995).

Dist urbances

With the algorithm described above, the presence of persistent unknown disturbances would prevent the monitoring function from converging to zero, and hence cause switching to continue indefinitely. In order to avoid this, Lai & Cook (1995a) also consider an alternative procedure which sets

M ; + ~ = min{M : M > ~ i , T ( M ) 2 ~ ~ - ~ i f ( i > ~ ( ~ i ) + g ( i ) )

where f ( i ) --.) CO and g ( i ) + CO as i -+ CO. This algorit!im is shown to ensure termination of the switching sequence, and hence boundedness of the state, if the input disturbance and

2/2

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output measurement noise are bounded, although it does not guarantee asymptotic stability in the disturbance-free case. Nevertheless, the fact that the state remains bounded indicates that the final closed-loop system, after the switching terminates, is in practice almost certain to be asymptotically stable, in the absence of disturbances.

State Bounds

Since the state variables of a real system are usually restricted in magnitude by some a priori bounds, it is natural to ask whether or not the control strategies described here will still work under these conditions. Although it is not clear how such bounds, which in practice arise from nonlinearities, should most appropriately be imposed, the following simple and plausible procedure enables the previous results on asymptotic stability to be generalized (Cook & Lai 1995). The recurrence relation for the augmented state vector is replaced by

Z ( M + 1) = sat[Af(M) + BQ(M)]

Isat[Zlkl I l5kl

where the saturation operator sat[-] has the effect that

for each component ( I C = 1,. . . , n) . If this inequality holds when the state representation is in observable canonical form, then the magnitude of the state vector can be shown to satisfy (in the noise-free case)

for M 2 v, where v is the observability index of the system and c is some constant. Thus, the monitoring function still provides a bound on the state, of the form which is required in order to derive the stability results. Furthermore, because the augmented state vector in general includes the state variables of the compensator, this analysis can also cover cases where saturation applies to the controller instead of, or as well as, the controlled plant.

Simulation Results

To illustrate the operation of the switching algorithms described above, the following simple example is considered here. The state-space equations of the plant are

z l ( M + 1) = sat[zl(M) + 22(M)]

z z (M + 1) = sat[u(M)]

!/(MI = t l (M) + d ( M ) where the saturation operator is given by

if ( 0 1 < 1; sat[v1 = { :in(.), otherwise

and d ( M ) is a noise disturbance, taken to be a normally distributed random sequence with mean 0 and standard deviation 0.1. It thus represemts a cascade of a discrete-time integrator and a one-step delay, subject to saturation on both the state and input variables, and with some Gaussian white noise in the output measurement, Control is applied through the feedback law

213

Page 4: [IEE IEE Colloquium on Adaptive Control - London, UK (13 June 1996)] IEE Colloquium on Adaptive Control - Universal regulators for discrete-time systems

where the gain K ( M ) is switched at instants generated by the switching algorithm using

f ( i ) = g ( i ) = i

and the monitoring function r ( M ) defined with A = 0.95. An arbitrary but convenient way to generate the gain sequence is by using an oscillatory function with slowly increasing amplitude, the one chosen here being

K ( M ) = Asin0.5i EM; 5 M < Mi+l]

which produces

{K(M;)} = {0.4794,1.1900,1.7277,1.8186,1.3382,0.3457,. . .}

starting from i = 1. Since the condition for asymptotic stability of the closed-loop system is

O < . K < l

. it can be seen that the first gain value selected would actually be satisfactory from this point of view but, in the simulation shown below, the switching actually continues, through a sequence of unstable choices, until stability is reached again at i = 6, where the algorithm settles. Also, although the figure shows the results of only a single run, and different runs vary slightly on account of the random disturbance term, the same general features were found on each occasion, with the gain always terminating at the same value. This kind of behaviour appears to be typical of such examples, and is fully compatible with the mathematical analysis, even though the assumed conditions are not strictly satisfied because the disturbance is normally distributed rather than bounded, so that further switching could eventually be triggered by a sufficiently large element in the random sequence.

References

P.A.Cook and W.C.Lai (1995). Universal regulation and tracking for discrete-time systems, Proc. 3rd European Control Conference (Rome, Italy) 1652-1657. M.Fu and B.R.Barmish (1986). Adaptive stabilization of linear systems via switching control, IEEE Trans. Automat. Contr. 31 1097-1103. W.C Lai (1995). Universal Adaptive Stabilization of Discrete Time Systems (Ph.D. Thesis, UMIST, Manchester). W.C.Lai and P.A.Cook (1995a). A discrete-time universal regulator, Int. J. Control 62 17-32. W.C.Lai and P.A.Cook (1995b). A discrete-time universal regulator which provides exponential stability, Control-Theory d Adv. Tech. 10 1843-1850. B.Mttensson (1985). The order of any stabilizing regulator is sufficient a priori information for adaptive stabilization, Syst. d Contr. Lett. 6 87-91. D.E.Miller and E.J.Davison (1989). An adaptive controller which provides Lyapunov stability, IEEE Trans. Automat. Contr. 34 599-609.

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