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Deterministic adaptive control based on Laguerre seriesrepresentationCHRISTOS C. ZERVOS a & GUY A. DUMONT ba Department of Electrical Engineering , University of British Columbia , Vancouver, B.C.,Canadab Pulp and Paper Research Institute of Canada and Department of Electrical Engineering ,University of British Columbia , Vancouver, B.C., CanadaPublished online: 18 Jan 2007.

To cite this article: CHRISTOS C. ZERVOS & GUY A. DUMONT (1988) Deterministic adaptive control based on Laguerre seriesrepresentation, International Journal of Control, 48:6, 2333-2359, DOI: 10.1080/00207178808906334

To link to this article: http://dx.doi.org/10.1080/00207178808906334

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INT. J. CONTROL, 1988, VOL. 48, NO. 6, 2333-2359

Deterministic adaptive control based on Laguerre series representation

CHRISTOS C. ZERVOSt and GUY A. DUMONTS

The behaviour of adaptive controllers in the presence of unmodelled dynamics, and the need for reduced a priori information have led us to abandon the usual ARMA transfer function representation for a revresentation bv an orthonormal series. The a p ~ a l of our nc- approach I, that i t eltkinates the need for ~ssumpuons about the ~ lan t order and the tlmr dela! The vlant 1s modelled b\ an orthonormal Lacucrrc network put in state-space f o k A simple predictive c&trol law is propos..&. An explicit deterministic adaptive controller is then designed. Simulations show that it is easy to use, able to handle non-minimum phase plants, and more robust than the conventional model-based approach. Although we chose Laguerre functions, other orthonormal functions may be used. We have already tested some with success.

1. Introduction Over the last two decades numerous adaptive control schemes have been

developed and tested with various degrees of success. The number of applications, though small in relation to the activity in the field, is now sufficient to give credibility to adaptive control techniques. A class of these adaptive control systems introduced as an approximation of the general non-linear stochastic problem, has the capability to provide good control. They are generally known as self-tuning (Kalman 1958, Astrom and Wittenmark 1973, Clarke and Gawthrop 1975). A common characteristic for the schemes so far is that they are model-based. In particular, for the input-output case, ARMAX models are widely used (Astrom 1983).

A thorough treatment of self-tuning schemes is given by Goodwin and Sin (1984). As long as the actual plant can be described by the structure of the model, these schemes behave well. However, when this is not the case, performance degradation with potential destabilization occurs (Rohrs et al. 1985). Current adaptive control schemes deal well with structured uncertainty, but cannot adequately handle unstructured uncertainty. This explains why the behaviour of adaptive controllers in the presence of unmodelled dynamics, has been a topic of concern in the adaptive control community in recent years. A common thread to that work is the use of transfer function models. Our search for robust adaptive control requiring minimal a priori information has led us to the development of unstructured adaptive control. We abandoned the usual ARMAX model for an orthonormal series representation of the plant dynamics. The major advantage of this approach is that any stable plant can be modelled without structural knowledge, i.e. without assumption about the true plant order and time delay.

In 5 2, we present the use of orthonormal functions to model and control plants in L,[O, m). Although we choose Laguerre functions because of their simplicity and their

Received 19 August 1987. Revised 2 November 1987. Department of Electrical Engineering, University of British Columbia, Vancouver, B.C.,

Canada. f Pulp and Paper Research Institute of Canada and Department of Electrical Engineering,

University of British Columbia, Vancouver, B.C., Canada.

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2334 C. C. Zcrvos and C . A . Dirntonr

similarity to transient signals, other orthonormal functions such as Legendre functions may be used. The Laguerre functions exhibit strong features in identifying time delays (common in process control) because of their similarity to Pade approximants. As the model is expressed in a state-space form, state-space control design techniques may be used. We prefer a predictive control law because of its simplicity and ease of use. We present an explicit deterministic adaptive control scheme based on this approach and simulation results in 5 3. We consider robustness in 9: 4 and discuss practical aspects in 5 5. Stochastic and implicit schemes will be presented in a future publication. The common thread and novel aspect for all those schemes is the use of unstructured models based o n orthonormal functions.

2. Modelling and control using a Laguerre network 2.1. Loguerre funcrions

The use of orthonormal functions for obtaining approximations goes back as far as the development of Fourier series. The approximating properties of Fourier series, the sine and cosine terms of which satisfy the orthonormality condition, are well known. The first application of other orthogonal sets t o the transient problem was proposed by Lee (1932). Following that, a renewal of interest and activity in the subject appeared, and several methods of synthesis for special classes of transients and networks were published. For a causal system, the impulse response may exist over the whole positive lime axis. Over this interval, an appropriate and well-known orthogonal basis set is the Laguerre set (Lee 1960). The Laguerre functions, a complete orthonormal set in L,[O, m), have been used often because of their convenient network realization (Lee 1960) and their similarity to transient signals (Young and Huggins 1961, King and Paraskevopoulos 1977, Nurges and Jaaksoo 1981). Wiener (1956) also proposed Laguerres for prediction and for non-linear systems. In continuous time they are described by (Lee 1960)

exp(pr) d i - ' ~ ( 1 ) = A-- [ t i- ' exp ( - ~ p r ) ] ( i - I)! dri- '

where i is the order of the function ( i = 1, ..., N ) and p is the time scale. These functions form an orthonormal set in the time domain [0, m] and the corresponding L:~pl;lce transform for this set is

The orthonormality is preserved in the s-domain and this set is generated by the simple and convenient ladder network of Fig. I . Given a real and continuous function g( t ) in L,[G, b], there exists an integer N and a real number E > 0 such that

where

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Adaptive conrrol based on Laguerrefirncrions

Y(,) Summing Circuit

Figure I . Laguerre ladder network

rT= [r, r2 ... r,,.] and f T = [f, f2 ... j,]. The constant ri is called the Laguerre spectrum gain and for deterministic signals can be computed from (Lee 1960)

ri = g ( r ~ 0 dt ( 5 )

For ergodic stochastic signals the integral in (5) is replaced by the expectation operator giving the cross-correlation at zero lag between the two signals, i.e. r i = ECn(Oh(01.

As the z-transform does not preserve orthogonality, the generation of the discrete- time set from the above continuous one is accomplished by the continuous network compensation method (Jury 1958) for each of the blocks in Fig. 1. I f the plant impulse response is well represented by straight lines between sampling points then a fictitious first-order interpolator is employed to produce a piece-wise straight line approxi- mation of the continuous signal between successive sampling instants. The trans- formed sampled-data control system output coincides with the sampled values of the continuous system output a t the sampling instants.

2.2 Modelling of dynamic sysrems There are several ways to express the Laguerre ladder network of Fig. 1. For our

purpose it is convenient to represent it in a state-space form. This will enable us to derive predictive expressions of plant outputs in a straightforward manner. The outputs li(r), i = I, ..., N, from each block in Fig. 1 are taken to be the states of the Laguerre ladder network. Defining the state vector as

then by discretizing each block it can be readily shown that a discrete-time state space representation of the Laguerre network can be written in the form

where l(r) is the N-dimensional state vector, and u(t) is the system input. A is a lower triangular N x N matrix where the same elements are found respectively across the

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2336 C. C. Zeruos and G. A . Dumonr

diagonal or every subdiagonal. If T is the sampling period and

then

A =

and

The above state-space system is stable ( p > 0), observable and controllable. The output of the process to be modelled is then approximated by the weighted sum of the outputs of the Laguerre filters

The orthonormality of the outputs of the Laguerre network is preserved only when the input to the network is a white noise process. When this last assumption (not satisfied under closed-loop operation) holds then r = c and the above weighted sum in (10) can be taken as the projection of the plant transfer function onto the linear space whose basis is the orthonormal set of Laguerre functions.

The standard calculation of the Laguerre spectrum gains employs (5) with correlation techniques. A more efficient way is to use a least-squares parameter estimation. In the open-loop case the identification can be performed by exciting the system with a white noise sequence. If both the system shown in Fig. 1 and the system whose impulse response is g( t ) are excited by a white noise sequence (or PRBS) and k output sample points are collected, then in the least-squares sense the normal equations can be written in vector-matrix form as

where G is the k-dimensional data vector containing the sampled output values of the system whose impulse response is g(r). The ith column of the k x N matrix M is a k- dimensional vector constructed by the sampled output values of the ith Laguerre filter. The parameters thus obtained are unbiased, i.e. 6 = r (Zervos er al. 1985) even if the output of the plant was corrupted by coloured noise. If the identification is

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Adaptive conrrol based on Laguerre funcrions 2337

performed under closed-loop operation, as is the case when a self-tuning scheme is employed, then the output of the controller is not a white noise sequence. However, input-output can still be collected for an on-line plant identification. Simulations have shown that the least-squares identification produces very good results.

An advantage in using an orthonormal series representation is, that when the model order is increased, the low-order coefficients stay practically constant. Thus, we can change the model order on-line with minimal transient. On the other hand, for an ARMAX model, increasing the model order means all parameters change and thus a significant transient exists. Another advantage is that we can easily represent a time delay since this approach does not require distinction from the dynamics. Laguerre functions show very strong features in this situation because of their similarity to Pad6 approximants. Results from subsequent simulation results indicate both the ad- vantages above.

2.3. Predictive conrrol law The previously derived state-space representation is now used

The convenience of the above state-space representation is that any standard state- space design techniques can be used for state-feedback control. However, simplicity and implementation considerations caused us to develop a predictive control law. In the past decade, several predictive control laws have been proposed, e.g. by Martin- Sanchez (1976), Richalet et al. (1978), Ydstie (1984) and Clarke er al. (1987). Their major advantage is simplicity of use, intuitive appeal, and easy handling of varying time-delay and non-minimum phase behaviour. From (10) we can write for the d-steps ahead output

The recursive use of (7) gives for d future sample points

Using (15) in a continuing recursive substitution and assuming

we obtain the d-steps ahead predictive expression

Substituting (17) into (15) we finally obtain

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2338

where.

C . C. Zrruos and C . A. Dumont

Remark I Let kd be the planl time delay. Then, it is easy to show that d > lid and f i # O are

equivalent. l i d > k,, the right-hand side of the above equation can be equated to the desired

reference trajectory for the plant output. As given by Richalet er a/. (1978), we define a first-order set-point reference trajectory based o n the equations

where 0 < a i I and J ' , ~ is the desired set-point. By recursive substitution y,(t + d) can be written as

,:(r + d) = ady(r) + ( I - ad)y,, (23)

Setting y(r + d) = y,(r + d), and equating the right-hand parts of (18) and (23) we can solve for the required control input u(r) to obtain

Rework 2 The control law (24) can be expressed in velocity form. Equation (24) can be

written as

Using the definition of /I and rearranging

Au(r) = [y, - y(r) - dTAl(r)]P- (26)

where S = (Ad- ' + . . . + I ) , dT = cTSA, Au(I) = ~ ( t ) - u(r - I) and dl([) = I(r) - I(r - I).

R c ~ ~ ~ a r l i 3 An alternate, receding control law as given by Ydstie (1984) can also be derived.

Then, : ~ t each step a control sequence u(r), ..., u(t + d - I) that satisfies y(r + d ) = ):(r + d) and minimizes

is determined, but only u(r) is implemented. Such a control sequence is given by

where, y , = cTAd- l - j b , f o r i = O , ..., d - I

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Adapthe control based on Laguerre functions 2339

Proposition I Let the system be described by (7) and (10). Then the control law, under the

assumption (16), that satisfies y(r + d) = y,(t + d) is the same one as the control law that both satisfies y(r + d) = y,(t + d) and at each step minimizes the cost index

J , = ~ ( 1 ) ~ (29)

Proof Consider y(t + d) - y,(t + d) = O as the constraint equation for the minimization

of the cost function (29) then, the gradient of the lagrangian function with respect to the input u( r ) i s given by

where p i s the lagrangian multiplier and the assumption (16) was used. Solving the above equation for u(r) and substituting in (18) we obtain

Now substituting (31) in (30) we derive for the control law u(t) the same equation as (24). This i s the reason why the equations (24) and (28) give similar simulation results.

0

Theorem I Let the system described by (7) and (10) be controlled by (24) where fl i s non-zero,

and assume y,, constant. Then there i s a prediction horizon d > k,, such that

l im ~ ( t ) = u , - m

where u i s a constant.

Proof

The first part of the proof consists of proving the stability of the closed-loop system. For that we form the closed-loop system equations and we examine the conditions for stability. The closed-loop system can be expressed by

Using (19), one can write

kT = c ~ ( A I - I )

Substituting in (32) and after some simple manipulations one obtains

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2340 C. C. Zeruos and G. A. Dumonr

For stability we examine the matrix

As long as the eigenvalues of the above matrix, for some value of d, are inside the unit disk in the z-plane then the closed-loop system is stable. Now. for sufficiently large d, and while P # O the second term in expression (34) approaches zero because by definition 0 < a < I.

The third term in expression (34) also approaches zero, under the same conditions as above, because the square matrix A is a lower triangular and the eigenvalues of A appear along its main diagonal. It is straightforward then to show that the powers approach zero (Strang 1976) because all the eigenvalues of A are less than one in modulus ((i.,(A)I < 1).

Finally, the first term in expression (34) is always a stable matrix and the closed- loop system is thus stable. Determining the steady-state is then trivial: from (33), i t is obvious that the steady-state is such that y = y,,. 0

Remark 4 It is worthwhile to mention here that in practice. during the simulations, t o

increase d to very large values was hardly ever needed for stability, because as long as d z kd then the number N of the Laguerre filters can be always accordingly adjusted for a proper identification. This is always true for most of the process control loops encountered in practice. Of course, if everything else fails then by increasing d to some large value, stability can always be achieved.

Remark 5 The condition p # 0 is satisfied as long as the prediction horizon d is greater than

the time delay kd of the plant (or d > kd for minimum-phase systems). By definition

This shows that is the sum of the first d sampling points of the impulse response of the plant. Requiring 0 + 0 means looking ahead beyond the time delay of the plant and non-minimum phase behaviour, if present.

Rentark 6 Using (Ad- ' + ... + I ) ( / - A) = ( I -Ad), one can write

When d is sufficiently large then, in the limit, we obtain

p = c T ( ~ - ~ ) - l b

which is the static gain of the transfer function G(z) of the state-space system (A, b, cT)

lim G(z) = c T ( I - A ) - ' b z- 1

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Adapriue control based on Laguerrefunctions 2341

In practice the static gain of the transfer function G(s) can be computed roughly from

where Ti is the identified Laguerre gain,

Remark 7 It is easy to show that the closed-loop characteristic equation is

When d = kd and a =0 , the left-hand side of the above equation is the impulse response of the plant. If it happens to be non-minimum phase, then the regulator is unstable. The best way to shift the regulator poles back inside the unit circle is to increase d.

3. Self-tuning control using Laguerre functions 3.1. Deterministic explicit self-tuner

It is straightforward to design an explicit deterministic adaptive control scheme based on the above formulation. The recursive least-squares (RLS) is used to identify the parameter vector c .

To include immunity against bias in the parameter identification such as those induced by offsets, the least-squares identification scheme uses increments of I((). u(r) and y(t) instead of full values. The control law (24) is then computed at every sampling instant. This adaptive control scheme is globally convergent, as shown by the following theorem.

Theorem 2 Assume that the plant is described by y(t) =c i l ( t ) , then provided that the

projection or least-squares algorithm (36), (37) is used to find. C(t), that dim C = dim c,, and that /J#O (i.e. d 2 kd), then the indirect adaptive control scheme described above is globally convergent in the sense that

(i) {~ ( t ) } , {y(t)) are bounded for all time r

(ii) lim [Y( ' ) - y,,(t)l = 0 I - m

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2342 C. C. Zeruos and C. A. Dumonf

Proof

Using standard arguments (Samson and Fuchs 1981, Goodwin and Sin 1984), the parameter adaptation scheme can be shown to be such .that

(i) (e(r)} i s bounded

(ii) lim lle(f) - e(f - I ) 11 = 0 I - m

(iii) there exist non-negative sequences {r(f)), {$(t)} that converge to zero and such that

Ij(t1 - y(0I < a0 Ill(1) I1 + $(I)

Assuming, for simplicity, that the parameter a in (23) i s zero, one can write the closed- loop system as

From Theorem 1, i t i s easy to show that the free system

I([ + I ) = F(f)l(f)

i s exponentially stable. With

l l v ( N < l l ~ l l l ~ l ~ ' ~ l ~ ~ ~ l + - ~(1 ) l )

i f for simplicity y,, = 0, then using the properties of the adaptation scheme, we can write

Ilv(t) Il < v ( 0 llI(r)11 +&(I)

with

y(t) = l l b l l 5 ( N ~ I - '

&(t)= l lb l l$(~ i I? l~ ' Because both {y(r)) and { S ( t ) } converge to zero, then from Theorem 2.1 in Payne (1987), {I(()} and {v(t)} are bounded and converge to zero. When y,, f 0, {l(t)} and {v(f)} are bounded and i t i s trivial to show that they converge such that

lim v(t) = bp-'y,, z-m

3.2. Simulatiof~ rrsulfs

Some examples showing the potential applicability of the above self-tuning algorithm follow.

Example I Consider the closed-loop system H(q-') of Fig. 2

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Adaptiue confrol bused on Laguerrefuncrions

L...................

Figure 2. System to be controlled in Example I.

with

1 + 0.6048q-' C'(4-I) = K c -0.3697q-1 where K , = 0.3397

Figure 3 shows the step responses of H(q-I ) (i.e. from u to y) with k, = I, with and without noise. Note the oscillatory nature of the response as well as the load disturbance of amplitude 0.2 from time r = 240. For the noisy response, a noise filter with unity C polynomial has been added to corrupt the plant output. The Laguerre adaptive controller is used to control the system H(q- I ) as in Fig. 2, with N = 10 Laguerre filters and p=O.I. Initial parameter estimates are zero. The initial cova- riance matrix is 100 x I, and the forgetting factor is I. Figure 4 shows the behaviour of the adaptive controller with a prediction horizon d = 2 and the driver block filter time constant a = 0.5, both with and without noise. The start-up transient has a very small amplitude. Both the response to the setpoint change and the load disturbance rejection are excellent. As expected from Theorem I there is no steady-state offset. Figure 5 shows runs when the dead time k, in G(q- ' ) is increased from 1 to 3 sampling intervals, both with and without noise. The scheme is exactly the same as in Fig. 4 except for d = 4 and z = 0.7. Again, a load disturbance of amplitude 0.2 was applied from time r = 240. Further simulations have shown the good performance of that scheme for the regulator problem as well.

Example 2 We now consider the non-minimum phase plant described by

This plant was used by Clarke (1984) to demonstrate a pole placement self-tuning controller. Here, we shall use the same sequence of setpoint changes and the same commissioning period as Clarke (1984). Figure 6 shows the output of the above plant tracking a square-wave setpoint when it is under Laguerre self-tuning control, started with zero initial parameter estimates, and set with d = 2 and a = 0.7, i.e. the same conditions as in Fig. 4. The performance is very good, and compares well with that obtained by Clarke with a pole-placement self-tuner based on a model of exactly the

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C . C . Zervos and C. A. Durn0111

Figure

-2.0 4 I 0 50 100 150 200 250 300 350 400

Time in Sornples

(4

2.0

1.5 -

1.0 -

0.5 - -

-2.0 ! C 0 50 100 150 200 250 300 350 400

Time in Samples

( h )

.

Example 1 : output responses of H ( q - ' I . (u ) with and ( b ) without noise, k, = I .

a o o - - 3

0 -0.5 -

-2.0 I C 50 100 150 200 250 300 350 400

Time i? Sooples

- -1.0, ,, A-

- 1 . 5 - / V v ll

-

-

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Adapriue control based on Laguerrejunctions

2.0

1.5

1.0

> 0.5 d

2 0.0 - 2 0 -0.5

- 1.0

-1.5

-2.0 0 50 100 150 200 250 300 350 400

Time in Somples

(4

-2.0 1 I 0 50 100 150 200 250 300 350 400

Time in Somples

( b ) Figure 5. Example I: output responses, (a) with and ( b ) withoui noise, when H ( q - ' ) is under

Laguerre self-tuning control, k, = 3, d = 4.

.

- -

- .

.

r

2.0 '

0 50 100 150 200 250 300 350 400 T ime in Somples

( b ) Figure 4. Example I : output responses, (a) with and (b) without noise, when H ( q e ' ) is under

Laguerre self-tuning control, l id= I , d = 2.

1.5 - 1.0 -

0.5 - - -

v

I

- 1.5 - -2.0

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C. C. Zervos and C . A. Dumonr

-10 1 C 0 50 100 150 200 250 300 350 400

Time in Somples

Figure 6. Example 2: system response under Laguerre self-tuning control, first-order plant.

same structure as the plant. However, as noted by Clarke, his scheme does not behave well in the presence of unmodelled dynamics. Indeed when applied t o the plant

the Clarke pole-placement self-tuner based on a first-order model eventually de- stabilizes the plant. The Laguerre self-tuner was used on this second-order plant with exactly the same design parameters as with the first-order plant. Results presented in Fig. 7 show the excellent bchaviour of this scheme. Note that Figs.4 6 and 7 have all been obtained with the same Laguerre self-tuning scheme and the same initial set-up por:tmeters, although the three plants are all different. This is an indication of the robustness of the Laguerre self-tuner.

0 I' I 0 50 100 150 200 250 300 350 400

Time in Samples

Figure 7. Example 2: system response under Laguerre self-tuning control, second-order plant.

E m ~ i i p l e 3 The present method is limited to stable plants. It is thus interesting to see what

happens when applied to a plant containing an integrator, a common occurrence in

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Adapriue conrrol based on Laguerre functions

process control. A logical way to represent a plant with integrator is

I(t + I ) = Al(t) + bu(l)

Ay(t) = cTI(l)

We now assume

The d-step ahead predictor is then

y(r + d ) = y(c) + Ay(r + I) + ... + Ay(t + d) (47)

y(r + d) = y(e) + dTl(c) + pu(t) (48)

The control law is then (see (26) for definition of d )

Au(c) = [y,(t + d) - y(r) - dTl(t)]P-' (49)

Compare with (24) and (26). Now, consider the plant described by

y(k) = - I.9048y(k - 1) + 0.9048y(k - 2)

+ 0.5[u(k - 1) + 0.0672u(k - 2)] (50)

This plant contains an integrator. Figure 8 shows a simulation run when the plant is under Laguerre self-tuning control and the output is tracking a square-wave setpoint. The Laguerre parameters used were, N = 8, p = 0.25 and d = 3. Good simulation results were also obtained in the range 0.05 < p < 0.5. Note that a load disturbance was introduced from t = 240 to r = 260.

-2.0 0 100 200 300 400 500 600 700 800 900 1000

Time in Samples

Figure 8. Example 3: Laguerre self-tuning control of a system with integrator.

E-~arnple 4 The dynamics of an existing two-link manipulator were simulated using Paul's

equations (Paul 1981) and ACSL (Advanced Continuous Simulation Language) on a DEC VAX-11/750 computer. Each link has a mass of 1 kg and length of 0.5 m. All forces due to gravity, coupling inertia, centripetal acceleration and Coriolis acceler- ation are included in the dynamics of the arm. Two single-input/output Laguerre self-

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2348 C. C. Zervos and C. A. Durnot~t

tuning controllers were implemented, one for each link. Each self-tuner controls the joint angle by the manipulation of the motor armature drive voltage. Several reference trajectories were implemented to test the reliability of the proposed self-tuning algorithm including desired angles of 90" and 270", and circular and linear trajec- tories. Other tests involved increasing the mass at the end point of the second link (equivalent of picking up a payload) part way into the trajectory, and adding armature inductance to test the ability of the controllers to deal with an unexpected pole in the system.

The controllers generally performed quite satisfactorily. The initial values and the parameter settings used for both controllers were: N = 8, p = 2, sampling time T = 0.02 s, d = 4, a = 0.2 for the first 100 points and 0.7 subsequently. The parameter estimates were initialized to zero and the controllers' outputs were limited to +20 volts, the maximum voltage the motors can sustain. The incremental version of the controllers was used. All the initial manipulator angles were set to 0". For the particular experiment regarding the on-line increase of the payload, a t t = 4.0 s during a clockwise circular trajectory centered at (0.5,O.O) of radius 0.1 5, the mass a t the end of the second link has been increased from 0 to 10 kg to simulate the pickup of a load. The arm was still capable of tracking the reference circle (as seen from Fig. 9).

In another experiment we increased the armature inductance in both motors from 100 pH up to 100 mH to see if the unexpected poles could be handled without having to increase N or change p. For values of the inductance L, less than I0 mH the controllers continued to track well (Fig. 10) while a t 100 mH the 2-link manipulator system eventually became unstable. However, by increasing the prediction horizon d

Figure 9. Example 4: manipulator arm tracking a circular trajectory. Payload increase, m, = 10 kg.

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Adapriue corirrol based on Laguerrefunrrions

Figure 10. Example 4: manipulator arm tracking a step trajectory. Response of 2nd joint angle, Inductance increase, d = 4.

to 12 the system was capable again of giving acceptable results (Fig. I I ) with a penalty on the rise time. A side benefit is the reduced overall overshoot.

The results showed that tracking circles, lines and square waves proved to be no problem for the controllers. The only problems were the observed overshoot and some slight torque and controller chatter. Torque chatter was not eliminated although the armature inductor acted like a low pass filter and reduced the amplitude and frequency of the chatter. As far ns the overshoot is concerned the increase of the prediction horizon d contributed toward reducing it.

3.3. Dererminisric implicit selj-runer A deterministic implicit self-tuner can also be derived using the Laguerre

orthonormal set. The controller parameters are then estimated instead of the model ones. The identification would involve the model

where the vector parameter gain k and the input coefficients ji are identified on line. The control law is then

I(([) = [Y. -y(t) - kTl(t)lBi,' (52)

where j,., = 8 , + ... + jd.

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C. C. Zcruos and C. A. Dumonr

Figure I I . Example 4: manipulator arm tracking a step trajectory. Response of 1st joint angle. Inductance increase. d = 12.

4. Robustness Because this method does not rely on a predefined model structure with a fixed

number of poles and zeros, and because it does not separate the delay from the dynamics, we expect it to be more robust than the schemes based on transfer function models. The simple analysis and the examples that follow seem to indicate that this is the case.

Let the true deterministic plant be represented by the state-space equations

Let the Laguerre ladder network model of the above plant be represented by the equations

I,., = Al,+ bit,

p, = eTI,

Where x and I are the state vectors, respectively, not necessarily of the same order. Let the system be under self-tuning control using the (non-linear with respect to

the identified parameter vector) predictive control law derived in 9 2.3, (24), i.e.

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Adaptiue control based on Laguerre functions 2351

where y, is.the d-steps-ahead (d 2 kd) pre-defined reference trajectory as given by

Using the above control law (57) and substituting for it in (53), ( 5 9 , we obtain the following set of equations, respectively

Substituting y, from (54) and y, from (58), the above set ofequations can be written in a form to describe the closed-loop system as

For stability, the A-matrix of the above closed-loop state-space description must have all of its eigenvalues inside the unit disk. If not, the closed-loop system will be unstable. Let us further assume that the output model mismatch between the true plant and the identified (modelled) one can be described by some arbitrary function, say [(t), which has the property that it stays always well bounded for all t, i.e. li(t)l $ Z < m where Z is a positive real number ( Z E 9'). The signal c(t) can be any bounded deterministic or stochastic signal, e.g. measurement noise, sensor drifts, modelling residual. Then we can write

Expressing y, in terms of j, using (62) and (56), the closed-loop state-space description given in (61) can be further written as

Where 0 is an all-zero matrix of the appropriate dimensions. Now let us define the A-matrix in (63) as

We can now present the following theorem concerning the stability of the closed-loop system.

Theorem 3 Let a stable discrete-time system be represented by the set of state-space equations

(53), (54) and let it be sampled every T s and be under predictive control law with

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2352 C. C. Zeruos and G. A . Dumont

d 2 k , as described in 9: 2.3 (i.e. (57)). Assume that the output model mismatch between the true plant and the identified (modelled) one can be expressed by any bounded arbitrary deterministic or stochastic signal [(r) such that l[(r)l< Z < m. Then there is a prediction horizon d such that the closed-loop adaptive system remains always stable.

Proof It is easy to show the validity o f t h e above statement when condition (62) is true by

evaluating the A,, expression given by (64). Under the condition of (62) the closed- loop system can be put in state-space description as shown before by the set of cxpressions in (63). The stability of the overall system is then determined by the upper block-triangular matrix A,, as given by (64). However, because of its special structure the eigenvalues of the matrix A,, are just the eigenvalues of the matrix A, plus the eigenvalues of the matrix

Now since the true plant under study was assumed to be stable then the eigenvalues of the matrix A, are always inside the unit disk. Besides, standard arguments from the proofs of Theorem 1 and Theorem 2 indicate that the matrix A::." is also to be stable and have all its eigenvalues less than unity in modulus (compare (32) with (65)). As a result the closed-loop system remains always stable. 0

An illustrative example follows that makes use of the stability study described above.

Erarnple 5 Let the continuous-time stable plant (Rohrs er 01. 1985) of the form

be sampled very Ts . The input-output data is recursively used at every sampling step (RLS) to derive a discrete model of the plant in terms of a Laguerre orthonormal series as described in 9: 2.2, and a predictive control law is then computed, as described in 9: 2.3, that is applied to the plant on-line thus forming a closed-loop self-tuning system.

First let the sampling time be 0.1 s and the reference input be sin (wr) with w = I rad/s. Underestimating the plant order and assuming that it is of a first order, only one Laguerre gain is estimated during the identification ( N = 1, T=0.1, p = 0.5, d = 2, a = 0.3). Figure 12 shows the plant output y(r) tracking the square-wave reference input y,. Figure 13 shows the Bode plots of the true and identified plant, respectively. The identified Laguerre gain came out to be 1.45. If we evaluate the eigenvalues of the A-matrix from (61) we find out that for the above sampling frequency the system is stable. Increasing the sampling frequency t o T=0 .01 while keeping everything else the same, the system eventually becomes unstable.

Let us further investigate the instability mechanism when the sampling time is T = 0.01 s. According to Theorem 3, if the output model mismatch is bounded (i.e. (62) is true), then there is always a prediction horizon d such that for every sampling time T the closed loop self-tuning system remains stable. In this case if we increased we should expect system stability. Actually when the reference input is I rad/s we d o

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Adapriue conrrol based on Loper re functions

Figure 12. Example 5: output tracking the reference input when N = I. T=0.1. d = 2

I

FREOUENCY tRAO/SECI Figure 13. Example 5: Bode plots for the true (solid lines) and identified (broken lines)

systems, when N = 1 . T = 0.1, d = 2.

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2354 C. C . Zeruos and C. A. Dumont

get stability by increasing d . For values d 2 8 the closed-loop system appears to be stable. Figure 14 shows the system output tracking the reference input when d = 10.

Actually in the single parameter case ( N = I) the Laguerre state-space model given by (55). (56) transforms to simple scalar expressions. The eigenvalues of the A,, matrix then can be evaluated from

%,(A,,) = I . i ( A , ) ~ l . , { A - b[ (1 - a d ) e T + k T ] f i - ' J

Figure 14. Example 5: system output tracking the reference input when N = I , T = 0.01, d = I0 (the two curves are almost identical).

Since the plant under study was assumed to be stable then the eigenvalues or the first term in the right-hand side of (67) are always inside the unit disk. The second term in the right-hand sidc of the above equation is a scalar expression and its value is always lcss than one, as shown below (note that the terms, A = exp(-pT) > 0 and h, E, l a r e now all scalars $0, and la1 < I)

Thus all the eigenvalues of the A,, matrix are inside the unit circle and the closed-loop system is stable.

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Adapriue control based on L~gtrerrvfirnctions

8

Figure 15. Example 5: (a) system output tracking the reference input when N = 2, T = 0.01, d = 2; (b ) an enlargment portion from T = 20 to 23 (the two curves are almost identical).

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2356 C. C . Zeruos and C. A. Dumont

We did obtain stability then by increasing d, when the reference input o was 1 rad/s because the output model mismatch was bounded. However, the system appears to be unstable for any value of d if the reference input is increased to ru = 16.1 rad/s. When condition (62) is not true then the stability of the closed-loop system depends on the stability of the A-matrix in (61). It is easy to see by inspection that for T = 0.01 sampling time and for the particular plant of (66) the A-matrix in (61) has 2 eigenvalues outside of the unit disk, i.e. the closed-loop system is unstable. The reason why the closed loop is unstable is that, for fixed N = 1, the order of identification is inadequate either to represent sufficiently the dynamics of the given plant or make the model mismatch bounded at the specified sampling frequency and at the specified reference input.

In this case, this leads us to 2 possible solutions, either keep the identification error always bounded in case an a priori knowledge of the plant is available, or increase the order of the identification. By increasing the order of the estimatio~l from 1 to 2 the system is always stable for values of sampling times 0.1 and 0.01 and for sinusoid rcference inputs from I to 20 rad/s. Figure 15 shows the output tracking the reference input and Fig. 16 shows the bode plots of the true and the identified plant, respectively ( N = 2, T = 0.01, p = 0.5, d = 2 , 0 = 16.1). By increasing the order of the identified model, stability has been achieved. As the low-order Laguerre gains stay practically constant during an order increase, this can be done on-line without the system going through a transient phase. Flexibility in varying the dimensions of the problem on-line

~ -

without system-upset is an advantage of this new controller.

Figure 16. Example 5: Rode plots for true (solid lines) and identified (broken lines) systems, when N = 2, T=0.01, d = 2.

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Adapriue confro1 based on Loguerrefuncrions 2357

Eso~nple 6 Let a plant be described by the input-output equation

y(k) = 0.9979g(k - I) - 0.0777u(k - 2) + e(k)

- @618e(k - I) - @378e(k - 2) (68)

At the 800th sampling interval, we suddenly switch to the following plant

where e(r) is a white noise sequence N(0,O.I). Figure 17 shows the output of the plant tracking a square wave set-point using the self-tuning scheme mentioned in Example 3. The parameters used were N = 16, p=0.8, d = 10. Despite the sudden transition from one transfer function to another and despite the presence of noise the output is capable of following successfully the setpoint.

-2. - 1 0.00 200. 400. BOO. BOO. 1000. 1200. 1400. 1800,

-5.01 0.00 200. 400. 800. BOO. 1000. 1200. 1400. IBOO

TIME I N SAMPLES

Figure 17. Example 6: system outpul tracking the reference input ( u ) and controller output (h ) . ( N = 16. p = 0.8, d = 10, T = 1-0. A t the 800th point we switch to a diferent plant).

5. Practical aspects and implementation

5.1. Choice of Laguerre filter rime consrunt Although, a s found by simulations, the choice of the parameter p used in the

Laguerre ladder network is not crucial, it does influence the accuracy of the approximation of a given plant a s a truncated Laguerre series. In the simulations, an extensive range of values for the parameter p was found to give acceptable adaptation performance results for a given plant. A method to optimize the parameter p that was tried in practice and found to perform very well was to store an array of plant input-

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2358 C. C. Zeruos and G. A. Dumonr

output data for a period of time and then try a modified constrained Hooke-Jeeves optimization technique (Hooke and Jeeves 1961) on them to obtain an optimum value for the parameter p by min~mizing the residual error obtained from the least- squares identification. The method performed very well in practice with the only drawback being the additional memory requirement and the computational expense involved for the direct search optimization routine to converge. Since a t this time no analytical method is available for the choice of the parameter p, its choice is empirical as explained below.

The ability to control the bandwidth ofadaptive control schemes in order to avoid cxciting unmodelled high frequency dynamics is desirable for robustness. An interest- ing feature of the Laguerre ladder network is that the first block is a first-order low- pass filter with cut-of frequency l /p and the rest of the blocks are all-pass filters. This provides some filtering qualities to the Laguerre self-tuner and allows some control over its bandwidth. Thus the choice of the parameter p can be made to have l/p roughly around the cross-over angular velocity w of the plant.

5.2. Number offilters The number of filters required in the representation relates primarily to the

presence of underdamped modes and the time delay in the plant. The reason is that the orthonormal filters are used to model all dynamics including the delay. For low- order plants with significant delay relative to the dominant time constant, simulations show that 5-10 filters give satisfactory results in many cases. For high-order underdamped plants with substantial delays the number has to increase from 10 to 15 filters. Of course, when the delay is not substantial, fewer filters are required. By monitoring the Laguerre spectrum or the residual error on-line, it is easy to vary the dimension of the problem by increasing or decreasing the order of the identification.

5.3. Choice ofd und a The driver block pole relates to the desired performance and is easy to choose. The

prediction horizon can be automatically altered to make sure that /?is non-zero and that non-minimum phase zeros are not cancelled. The latter can be achieved by checking the roots of (35). A simpler method is to choose d such that B is of the same sign as the estimated process gain, and of significant amplitude. A simple criterion can be

/ ? 2 ~ e ~ ( l - ~ ) - ' b (70)

where ~ ~ 0 . 5 and ( I - A ) - ' b can be precomputed, as it does not depend on the estimates.

6. Conclusions We have presented a novel unstructured adaptive controller based on a Laguerre

ladder network and tested it extensively on simulations. In principle, other ortho- normal functions could be used, although Laguerre functions prove to be a good choice. The result is a robust, simple to use algorithm that requires minimal a priori information. In a future publication we shall present stochastic and implicit versions of this scheme. Currently, we are investigating the theoretical properties of such unstructured schemes. We have already performed successful industrial trials of those schemes, and shall present the results in a future publication.

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Adaptive control hosed on Laguerrefunctions 2359

This research was part ial ly supported by the Natura l Sciences and Engineering Research Counci l o f Canada under grant No . A-5960. We wish t o part icularly thank Mr D. W o n g for performing the simulations for the manipulator a r m example and D r K. Natarajan for his valuable and helpful comments and suggestions while reviewing the manuscript.

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C l i k NJ: Prentice Hall). JURY, I. E.. 1958. Sampled-Darn Control Sysrems (London: Wiley). HOOKE, R., and JEEVFS, T. A,, 1961. J. Ass Compur. Much., 8, 212. KALMAN, R. E., 1958, Trans. Ant. Soc. mech. Engrs, SO. 468. K1h.c. R. E.. and PARASKEVOPOUL~S. P. N.. 1977. Circuil Theorv Aool.. 5. 81. , .. . . LEE. Y. W., 1932, J. Murh. Phys., 11, 83; 1960. Srarisrical Theory ofCommunicurion (New York:

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