Simplified parameter adaptive control

OPTIMAL CONTROL APPLICATIONS & METHODS, VOL. 8, 37-48 (1987)

SIMPLIFIED PARAMETER ADAPTIVE CONTROL

K . WARWICK Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, U. K.

K . Z. KARAM Department of Electrical and Electronic Engineering, University of Newcastle Upon Tyne, Newcastle Upon Tyne,

NEI 7RU. U.K.

AND

M. T. THAM Department of Chemical and Process Engineering, University of Newcastle Upon Tyne, Newcastle Upon Tyne,

NEI 7RU, U.K.

SUMMARY

A simple parameter adaptive controller design methodology is introduced in which steady-state servo tracking properties provide the major control objective. This is achieved without cancellation of process zeros and hence the underlying design can be applied to non-minimum phase systems. As with other self-tuning algorithms, the design (user specified) polynomials of the proposed algorithm define the performance capabilities of the resulting controller. However, with the appropriate definition of these polynomials, the synthesis technique can be shown to admit different adaptive control strategies, e.g. self- tuning PID and self-tuning pole-placement controllers. The algorithm can therefore be thought of as an embodiment of other self-tuning design techniques. The performances of some of the resulting controllers are illustrated using simulation examples and the on-line application to an experimental apparatus. KEY WORDS Adaptive control Servo control Self-tuning PID Pole placement

1. INTRODUCTION

Many self-tuning/adaptive control schemes have been proposed for the regulation and servo control of both stochastic and deterministic systems whose parameters are either unknown and/or time-varying. - They can be classified as either implicit or explicit algorithms. In the implicit approach, the process model is reformulated in terms of controller parameters. The latter are estimated on-line and used directly to generate the control signal which will satisfy an objective function. In the case of the explicit approach, the parameters of a model of the process are determined on-line. These are then used to design a controller. The final calculation of the appropriate control signal is therefore an indirect procedure, as a result of this intermediate design stage.

At first sight, it would seem that higher overheads, in terms of computational requirements, would be incurred in the implementation of an explicit self-tuning control scheme. Take for example the implicit generalized minimum variance (GMV) self-tuning controller of Clarke and Gawthrop. The computational burden during each iteration is lower than when the synthesis is formulated explicitly. However, with the GMV algorithm, the number of parameters to be

01 43-2087/87/0 10037- 12$06.00 0 1987 by John Wiley & Sons, Ltd.

Received October 1985 Revised July 1986

38 K. WARWICK, K. Z. KARAM AND M. T. THAM

estimated increases with the magnitude of the time-delay and hence the rate of adaptation is decreased. For a process with a large time delay to sampling interval ratio, this could lead to degradation in control behaviour. Although the sampling interval can be increased to reduce this ratio, the remedy can again result in unacceptable performance if the process has relatively small time constants. In addition, non-minimum phase behaviour and/or variable time delays can easily lead to closed-loop instability when applying the GMV controller. Robustness of performance can be achieved when a self-tuning controller based on pole-assignment ' is used. Although the latter is an explicit algorithm, robust control is achieved at the expense of optimality of performance, as well as significantly higher overheads due to the need to solve a set of simultaneous equations.

Adaptive control algorithms which require high computational effort are limited to areas of applications where the process being controlled has dominant dynamics the order of minutes. When rapid adaptive control is required, as in the control of robot manipulators, the number of calculations per sample iteration becomes an important factor in determining its applic- ability.

In this paper, a methodology for the synthesis of simple adaptive controllers is proposed. The objective is an algorithm where computational requirements are kept to a minimum, while bearing in mind the need to incorporate robustness properties. The algorithm is applicable to systems disturbed by random effects and set-point inputs and can cope with processes which exhibit variable time delays and non-minimum phase behaviour.

The basic structure of the algorithm is first introduced, and it will be shown how simple modifications can be made in order to impart different performance capabilities, namely an adaptive controller with integrating properties, an adaptive pole-placement controller where the solution of a set of simultaneous equations is not necessary and a self-tuning PID controller. The performances of the various forms of the algorithm are illustrated by several examples.

THE SYSTEM AND CONTROLLER

The system is considered to be described by Lhe equation

where u ( t ) and y ( t ) are the process input and output, respectively, at the time instant c. The signal ( e( t ) , t = 0, 1,2, . . . ) is a sequence of zero-mean random inputs of finite variance and is assumed to be uncorrelated with the input and output signals. z-' is the backward shift operator and is defined by

z-'y(t) = y ( t - i) The integer k is the time delay of the system expressed as an integer multiple of the sampling time. Further, k 2 1 such that bo is non-zero in the polynomial definitions:

It is further assumed that the polynomial A (z-') has roots which lie strictly within the unit cir- cle; and B(1), i.e. the sum of the coefficients in the B(z- ' ) polynomial, is assumed to be non- zero.

A set-point (desired ouput) value, o( i ) , is introduced to the system by means of a scalar feed-

SIMPLIFIED PARAMETER ADAPTIVE CONTROL 39

forward term, s, to form an auxiliary error function r ( t ) , where

r ( t ) = su(t ) - y ( t ) (3) The control input, u( t ) , is related to the auxiliary output via a general rational transfer function, namely

G(z-')r(t) F(z- )

u ( t ) = - (4)

where F(z - ' ) is monic, i.e. with unity leading coefficient in the same form as A ( 2 - ' ) as defined in equation (2).

The characteristic equation of the closed-loop system can then be easily shown to be

A ( z - ' )F(z - ' ) + z-~B(z-')G(z-') = 0 ( 5 ) which reflects a general control law.6 Depending on how the polynomials F(z-' ) and G(z - ' ) are defined, it then follows that different closed-loop performances can be achieved. For example, the characteristic polynomial, i.e. the LHS of equation (9, may be required to satisfy

(6)

Here, the zeros of the user-specified polynomial, T(z- ' ) , are the desired poles of the closed- loop system. The solution of F(z-') and G(z-') using equation (6), and the calculation of the control signal using equation (4) results in a pole-placement control scheme.

This paper is, however, not immediately concerned with pole-placement algorithms. Rather, the emphasis is on versatile algorithms which require relatively low computational effort. A method is described in the following section to illustrate the synthesis of simple self-tuning controllers. Later, two straightforward extensions/special cases are considered to demonstrate the generality of the technique.

A(z- ' )F(z- ' ) + z-~B(z-')G(z-') = T(z-')C(z-')

SIMPLE SELF-TUNING ALGORITHMS

The first, and perhaps the simplest, form of the controller is obtained when the polynomials G ( z - ' ) and F(z- ' ) are defined as

G(2-l) = A(z - ' )

and (7) F(z- ' ) = 1 - z-kB(z-1)

From equation (4), the control signal can therefore be calculated as

u ( t ) = A ( z - ' ) r ( t ) + B ( z - ' ) u ( t - k ) (8) Note that the coefficients of the system polynomials A (2 - ' ) and B(z- ' ) make up the parameters of the resulting controller. On substitution of this input signal into the system equation (A), the closed-loop relationship is obtained as

It is clear that for zero offset at the steady state, the value of the scalar feedforward term, s, must be chosen as

s = l/B(l)

40 K . WARWICK, K. Z. KARAM AND M. T. THAM

Set-point tracking is therefore achieved with an error which can be expressed as

Since, by definition, e ( t ) is a random zero-mean sequence, the tracking error will, therefore, also be a zero-mean sequence.

However, e ( t ) may not be a zero-mean sequence, as in situations where a d.c. bias is present in the disturbance term. In this case, e ( t ) can be redefined as

(1 1)

Here, e ( t ) is considered to be made up of two components: e ' ( t ) , which is a white sequence with zero mean, and a time independent constant, d. In this case, the expected value of tracking error is no longer zero, i.e. an offset results which is given by

e ( t ) = e ' ( t ) + d

where E ( . ) is the expectation operator. Further, the constant, d , can be regarded as an unknown deterministic load disturbance. This implies that if such a load disturbance manifests, then the algorithm described will perform poorly unless suitable measures are taken, e.g. supplementary estimation of the disturbance and incorporating it in the control calculation. The problem can, however, be easily resolved by considering the following modifications to the algorithm. Instead of using the definitions given in equation (7), G(z - ' ) and F(z- ' ) are redefined as

G(z- ' ) = A ( z - ' )

and (13) F(z-1) = B ( l ) - z - k B ( z - l )

Substitution of equation (13 ) into equation (4 ) enables the calculation of the control signal as

Further substitution of equation (14) into equation ( 1 ) will yield the following closed-loop relationship:

In this case, steady-state set-point tracking is obtained by setting s = 1 . Comparison of equation (8) and equation (15) will reveal that the computation requirement is in fact the same. The latter algorithm has the same deterministic servo characteristics as the first algorithm, whose control signal was defined by equation (8). However, the modification negates the effects of any bias, d , since in the steady state

[B(1) - z -kB(z- ' ) ] = 0 The use of this technique imparts integrating properties to the control law and is discussed elsewhere in more detail.'$' It is also noted that the use of either equation, (8) or (15), does not lead to cancellation of the zeros of the B polynomial. Therefore, in the control of non- minimum phase systems, unstable poles will not be introduced into the closed loop expression.


As a result, both forms of the controller can be applied to the servo control of non-minimum phase processes.

IMPLEMENTATION

The model of the system given by equation (1) can be rewritten in vector form as

u(t) = eTx(t) where

eT(0 = (a1(t), . * * , an(t); bo(t), * . . , b n ( 9

XT(f)=(-y( t - l ) , . . . , - y ( f - n a ) ; u ( t - k ) , . . . , U ( f - k - n b ) )

(17)

(18)

and

The parameter vector, OT(t), can be estimated on-line at each sample/control instance using an appropriate parameter estimation algorithm. In self-tuning control, the recursive least squares procedure* is commonly used, with the estimation error obtained at each recursion from

E ( t ) = ~ ( t ) - eT(t - i ) ~ ( t ) (19) The estimates of the system parameters, i.e. the coefficients of the polynomials A ( z - ' ) and B(z-' ) , are subsequently used to calculate the control signal from equation (8).

For the sake of simplicity, it is assumed that C(z - ' ) = 1, in the following discussion. Noise coloration effects can, nevertheless,, be taken into account by using an extended least squares parameter estimation algorithm. This would, however, increase the number of parameters to estimate. Not only will this increase computational overheads, but the larger number of parameters will also result in slower adaptation. Moreover, it is often found in practice that a white noise model, and hence the use of an ordinary least squares estimator, is sufficient. It has also been assumed that the time delay, k, is known exactly. With the proposed controller, this assumption can be relaxed. Since the parameters of the model describing the process are being estimated, as long as the maximum value of the delay is available, the order of the B(z- ' ) polynomial can be extended to account for the uncertainty in time delay value.'

ILLUSTRATIVE EXAMPLE

The performance of the above controller, using the definitions given by equation (13), was investigated when applied to a system whose open-loop characteristics are given by

(1 - 1 .3z-'+ 0.4z-2)y(t) = (z-' + 1.5~-~)~(t) + (1 - 0*65z-' + 0- lz-2)e(t) (20) The time delay (k) in this case is 1. The variance of the zero mean disturbance term, e( t ) , was 0.1 and the set-point input, u(t ) , was varied between the values ? 50. Set-point changes were effected after every 75 recursions.

In this example, it was assumed that C(z-') = 1. Therefore, only four parameters have to be estimated: a l ( t ) , az(t), bo(t) and b l ( f ) . The control signal was calculated at each recursion according to

where

42 K . WARWICK, K. Z. KARAM AND M. T. THAM

Figure 1 . System output signal y ( t )

Figure 2. System control input signal u ( t )

The performance of the controller over a period of 600 recursions is shown in Figures 1 and 2. Figure 1 shows the plot of the output signal, y ( t ) , and Figure 2 shows the plot of the cor- responding control signal, u(t) . It can be observed that after a short tuning-in period, the algorithm successfully controlled the output signal to set-point demands.

ILLUSTRATIVE EXAMPLE 2

This example demonstrates the robust nature of the proposed simple self-tuning controller (SSTC) by application to a non-minimum phase process with a variable time delay. The

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44 K. WARWICK, K . Z . KARAM AND M. T. THAM

simulated process is given by

(1 - 1.4z-'+ 0 * 4 9 ~ - ~ ) y ( t ) = ~ - ~ ( 0 - 5 + 1 - 4 z - ' ) u ( t ) + e ( t ) where e ( t ) was a white noise sequence with zero mean and variance of 0.1. The set-point input was a square wave of magnitude k 50. A step change in set-point demand occurred every 100 iterations. The time delay was, in the first instance, set to unity. This was subsequently changed on the 951st iteration to be equal to 2. The degree of the estimated B polynomial was set to 2 to account for the variation in time delay values.

Figure 3 shows the output of the process under the control of the SSTC. The set-point change at iteration 1000 resulted in an initial sharp overshoot. However, as the controller retuned to the new process conditions, the overshoots on subsequent set-point demands were significantly reduced in magnitude.

The GMV algorithm of Clarke and Gawthrop4 was also applied to the same process with the time delay estimated as 1 and the following weightings were specified: output weighting P = 1, input weighting Q = 0.5 and set-point weighting R = 1 -024.

Initially, the control was good but the system became unstable upon the change in process delay to 2 at the 951st iteration. Although overall stable control could be achieved when the estimated time delay was set to 2, as shown in Figure 4, the performance was poor. Large overshoots occurred at each set-point change and unacceptable oscillations were observed when the process delay was changed at the 951st iteration. The performance of the SSTC is clearly superior to that shown by the GMV for this example.

VARIATIONS OF THE BASIC ALGORITHM

The structure of the SSTC algorithm is fairly versatile and simple modifications can be made to alter its performance capabilities. It has been shown earlier that an integrating controller can be synthesized from the basic structure by simply redefining the polynomials G ( z - ' ) and F(z- ' ) . In this section, two additional forms are discussed.

Selftuning PID controller

here is described by Although many various forms of discrete PID controllers exist, 9B10 the algorithm of interest

u ( t ) = [Kp + Ki/(l - 2 - l ) + Kd(1 - z - ' ) ] [ u(t) - y ( t ) ] (23) Kp, Ki and Kd are the parameters associated with the proportional, integral and derivative elements, respectively. This expression can, however, be rewritten as

(1 - z - ' )u( t ) = K ( z - ' ) [ u ( t ) - y(t)l (24) where

K(z- ' ) = kl + k22-I + k3Zp2 k l=Kp+Ki+Kd

k2 = - (Kp + 2Kd) and


Recalling equation (14) and rearrangement leads to

[B(1) - z-%(z-')l u ( t ) = A ( z - ' ) [ su ( t ) - y( t ) l (25)

In order for equation (25) to take the same form as equation (23), the following conditions must hold:

(a) s = 1 (b) K ( z - ' ) = A ( 2 - I ) (c) k = 1 (d) B(1) = B ( z - ' ) = bo

Therefore, by restricting the model of the process to be a second-order process with no zeros, and with only unit delay (sampling delay), the SSTC can be forced into having the PID structure of equation (23). Although the restrictions (a) to (d) do limit the capabilities of the resulting controller, it should be noted that they are quite common in the design of self-tuning PID algorithms. 9 + 1 0 To calculate the control action at each time step, the following expression is used:

u ( t ) = u ( t - 1 ) + ( 1 + alz-' + azz -2 ) [u ( t ) - y(r)l/bo (26)

Simple pole-placement controller

explicit, T(z- ' ) is defined as The user-defined polynomial T(z - ' ) was introduced previously in equation (6). To be more

T ( z - ' ) = 1 + t1z-1 + t2z-2 + . . . + t,tZ-n' (27) where nt is the degree of T(z - ' ) . The user specifies values for the coefficients ti. Since the zeros of T(z- ' ) are the poles of the closed-loop system, the user is effectively specifying the closed- loop performance of the system. In order to achieve this property, G ( z - ' ) and F ( z - ' ) are defined as

F(z - ' ) = T ( z - ' ) - 2-kB(z-1) (28) G ( t - ' ) = A ( 2 - l )

Use of equation (28) will lead to the closed-loop expression

z - k~ ( z - ' )su ( t ) + ~ ( z - ' ) c ( z - ' ) e ( t ) Y ( t ) = T(z- ' ) A (2- ' )T(z - 1

For zero error is set-point tracking at the steady state, the scalar s is now given by

s = T ( l ) / B ( l )

From equation (29), it can be seen that the resultant controller ensures that the relationship between set-point input and the system output is characterized by the polynomial T(z - ' ) . The pole-placing control signal is then calculated from

[ T ( z - ' ) - ~ - ~ B ( z - ' ) l ~ ( t ) = A(z - ' ) [T( l )U( t ) - B ( l ) y ( t ) ] / B ( l ) (30)

Unlike the method proposed in Reference 5 , this algorithm does not require the solution of a set of simultaneous equations and can therefore be considered a simpler approach to pole- placement self-tuning control.

46 K . WARWICK, K. Z. KARAM A N D M. T. THAM

*

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Tank 1 Tank 2

ILLUSTRATIVE EXAMPLE 3

As a final example, the pole-placement form of the SSTC (SPPSTC) was applied to an experimental coupled tank system. A schematic diagram of the process is given in Figure 5. The manipulated variable is a 0-12V d.c. signal which drives a variable speed pump. This pump draws water from a reservoir and releases it into Tank 1. Tank 1 is connected to Tank 2 by an orifice of fixed diameter. There is an outflow from Tank 2 which allows the water back to the reservoir. The controlled variable is the level of water in Tank 2 and this signal is in the form of an amplified voltage from a differential pressure cell.

The SPPSTC was applied to the above system with the pole-polynomial arbitrarily chosen to be

T(Z- ') = 1 - 0.52-

The system was assumed to be adequately described by a second-order model:

A ( z - ' ) y ( t ) = B(z - ' )u ( t - k)+ e ( t )

deg(A (z- ')) = 2; deg(B(z- I)) = 1 and k = 1

For the sake of future comparison, an open-loop identification test was performed to determine the coefficients of the model. With a sampling interval of 1, the resulting discrete model was found to be

(1 - 1 ~ 2 9 ~ - ' + 0 ~ 3 l ~ - ~ ) y ( t ) = ( 0 - 5 4 + 0 - 2 6 z - ~ ) u ( t - 1)+ e ( t )

In application of the SPPSTC however, the process was assumed to be unknown apart from the time delay. Thus, the estimates of A (z- ') and B(z- ') were identified on line using a least- squares-based parameter-estimation algorithm. The estimates were then used in place of A (z- I ) and B(z- ') in equation (33) to calculate the control signal. Identification and control were carried out at 1 s intervals. Since there is a possibility of time-varying dynamics, the scalar set-point weighting variable was also calculated on-line as

s = OqB(1)

The level of water in Tank 2 was required to follow set-point demands which varied between 15 cm and 20 cm every 100 s.

The estimated coefficients of both the A(z - ' ) and B(z- ') polynomials converged to approximately the values obtained from the open-loop identification tests. Convergence of the

Outlet Valve

SIMPLIFIED PARAMETER ADAPTIVE CONTROL

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47

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a, 3 i m ' 0.0 M, a U a,

-0 .5

time ( s e c s ) 500

Figure 6 . Parameter estimates for B(2-I )

Figure 7 . Tank 2 water level under simplified pole placement control

coefficients of the B(z-' ) is shown in Figure 6 , and Figure 7 shows the controlled output and the set-point sequence. From both Figures, it can be seen that good set-point tracking was achieved with no offset once the parameters have tuned in. It was noted that although the output was corrupted by a significant level of noise, the simplifying assumption of no coloration effects did not seem to degrade the performance of the controller.

CONCLUSIONS

A method for designing simple self-tuning controllers has been outlined. The scheme is seen to belong to the class of self-tuning algorithms in which the control objective is more concerned with steady-state properties.

In realizing the zero error set-point tracking objective, it was not necessary to perform cancellation of process zeros. Thus, the algorithm can be used in the control of non-minimum phase systems. It was also shown how different self-tuning strategies can be arrived at by suitable definitions for the two user-defined polynomials. In particular, the pole-placement

48 K. WARWICK, K. Z. KARAM AND M. T. THAM

form of the algorithm can be implemented without the need to solve a set of simultaneous equations, and hence considerable savings in computational overheads can be achieved. The proposed technique thus provides a general approach to the design of simple self-tuning algorithms.

The parameters of the various controllers which may result are also the parameters of the model of the controlled process. This has the advantage that the estimated parameters are directly related to the process and are therefore more meaningful. Another result of the flexibili- ty of design is that if the various definitions of user-specified polynomials can be incorporated into a single control system package, then a single device can perform the role of an auto-tuner for three term controllers as well as a stand-alone adaptive controller. The former can relieve the tedium of tuning numerous PID loops, and the latter can be applied to tackle more difficult control problems.

A limitation of the algorithm is quite apparent. Because of the appearance of the A ( z - ) polynomial in the closed-loop expressions for the forms of the algorithm considered in the paper, it is required that the controlled process be open-loop stable. It was also assumed that B(1) # 0. However, both the stability requirement and the assumption are generally true for many processes.

REFERENCES

1. Astrom, K. J . and B. Wittenmark, On self-tuning regulators, Automatica, 9, 185-199 (1973). 2. Makto, D. and R. Schumann, Self-tuning deadbeat controllers, Int. J . Control, 40, (2), 393-402 (1984). 3 . Warwick, K., Self-tuning regulators - a state space approach, Int. J . Control, 33, (9, 839-858 (1981). 4. Clarke, D. W. and P. J . Gawthrop, Self-tuning controller, Proc. IEEE, 122, 929-934 (1975). 5. Wellstead, P., D. Prager and P. Zanker, Pole assignment self-tuning regulator, Proc. IEE, 126, (8), 781-787

6 . Clarke, D. W., Model following and pole placement self-tuners, Optim. control appl. methods, 3 , 323-335

7. Tuffs, P. S. and D. W. Clark, Self-tuning control of offset: a unified approach, Report 1539/84, Oxford Universi-

8. Isermann, R., Parameter adaptive control algorithms - a tutorial, Automatica, 18, 513-528 (1982). 9. Warwick, K . , Further developments in self-tuning control, in Harris, C. J . and S. A. Billings (eds), Selftuning

10. Gawthrop, P. J., Self-tuning PI and PID controllers, Proc. IEEE Conf. on applications of Adaptive and

(1 979).

(1982).

ty Engineering Labs.

and Adaptive Control, 2nd revised edition, Peter Peregrinus, 1985, Chapter 15 .

Multivariable Control, Hull, U.K., 1982.

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Simplified parameter adaptive control