Regarding self-tuning controllers for nonminimum phase plants

0005-1098/87 $3.00+ 0.00 Pergamon Journals Ltd. © 1987 International Federation of Automatic Control

Automatica, Vol. 23, No. 3, pp. 405 408, 1987 Printed in Great Britain.

Technical Communique

Regarding Self-tuning Controllers for Nonminimum Phase Plants* G. I. VOSS,~" H. J. CHIZECK~§ and P. G. KATONA~ Key Words--Adaptive control; self-tuning control; extended horizon controllers; nonminimum phase systems; mean arterial pressure control.

Nonminimum-phase discrete-time plants can be divided into two classes: Type I plants, where the initial response to a unit step input is in the same direction as the final value; and Type H plants, where the initial response is opposite in direction from the steady-state step response. Self-tuning controllers, which depend only on the initial response of the plant for the sign of the control gain, do not in general provide satisfactory control of Type II nonminimum-phase plants. Recently, several selftuning controllers were evaluated in simulation studies using a nonminimum-phase plant with a known I/O transport delay and zero noise (Clarke, 1984). The evaluation was restricted to nonminimum-phase plants where the initial response to a unit step input is in the same direction as the final value (Type I). This correspondence describes an adaptive control algorithm, the Control Advance Moving Average Controller (CAMAC), which is suitable for both Type I and Type II nonminimumphase plants. The CAMAC as well as other recent algorithms of Ydstie (1984) are natural extensions of the minimum variance self-tuning controller. Like the generalized minimum variance self-tuning controller, they are designed to minimize a given cost function at a specific future time advance. This control advance, however, is not restricted to the I/O transport delay. Since the output at this future time advance may be a function of future inputs, the solution which minimizes the cost function is not unique. The added constraints which specify the control law will significantly influence the closed loop system. The Extended Horizon Controller (EHC) of Ydstie (1984) implements a series of T inputs, where T is the integer number of control advance steps, based on a constraint which minimizes the sum of the squared inputs. Since the output equals the setpoint only at n T ( n = 1,2,3,...), the resulting closed loop system may exhibit cyclic behavior, especially for large T. The Receding Horizon Controller (RHC) of Ydstie (1984) implements only the first input, which is determined by the EHC, and recalculates the control at each interval. Although the RHC reduces the cyclic behavior of the EHC it may produce an offset error in control to a nonzero setpoint. CAMAC, the EHC and RHC are related to the numerical Dynamic Matrix Central algorithm of Cutler and Ramaker (1980). It also computes inputs for the present and a number of future times (different for each time) in order to compensate for process dead times. Another related approach is given by Wittenmark and Astrrm (1984, pp. 603-604), where over-estimation of the dead time (i.e. increased prediction horizon) using the ]tstr6m and Wittenmark (1973) self-tuning regulator is suggested for the control of nonminimum-phase plants. The CAMAC algorithm described here utilizes a constraint which minimizes the change in future inputs. As demonstrated in simulation, this choice of constraint avoids the cyclic behavior of the EHC and the offset error of the RHC.

Abstract--This correspondence is in response to the recent excellent survey of Clarke (1984, Autoraatica, 20, 501-517), which points out that minimum variance self-tuning controllers fail in the control of nonminimum-phase plants. It is also in response to recent algorithms of Ydstie (1984, Preprints 9th IFAC World Congress, Budapest) which have application in the control of such systems. A new control algorithm, designed to control nonminimum-phase plants, is described here. This ~Control Advance Moving Average Controller" (CAMAC) determines the input that minimizes the variance between the output and the setpoint at a time advance equal to or greater than the input-output (I/O) transport delay, based upon an assumption made on future inputs. The CAMAC algorithm is tuned by determining the appropriate control time advance. This approach is similar to that of Ydstie (1984, Preprints 9th IFAC World Congress, Budapest), but uses a different assumption on future inputs which yields substantially different controller performance. Its performance (in simulation) compares very favorably with that of self-tuning control algorithms reviewed in Clarke (1984, Automatica 20, 501-517) and those described in Ydstie (1984, Preprints 9th IFAC World Congress, Budapest) in the control of nonminimum-phase plants. Since it is not restricted to controlling the plant at the assumed I/O transport delay, it is also suitable for plants with unknown or time-varying dead times. The CAMAC algorithm has been successfully implemented in the control of mean arterial pressure in anesthetized animals. 1. Introduction self-tuning controllers, considerable concern has existed regarding the closed loop stability of the controlled plant. Minimum variance self-tuning controllers and model reference adaptive controllers, which attempt to cancel the poles and zeroes of the plant, produce instability when applied to nonminimum-phase plants. To provide stable control of nonminimum-phase plants, the minimum variance self-tuning controller must be modified such that the control gain has the correct sign and that the bandwidth of the controller is sufficiently reduced. Several techniques, including pole-placement (Wellstead et al., 1979), generalized minimum variance controller (Clarke and Gawthrop, 1975), and a factorization method (Gawthrop, 1982) have been used to modify the minimum variance self-tuning controller so as to control nonminimum-phase plants. SINCE THE INTRODUCTION o f

* Received 19 February 1985; revised 18 June 1985; revised 15 September 1986. This paper was recommended for publication by past Editor A. H. Lewis. t Formerly of Biomedical Engineering of Case Western Reserve University, Cleveland, OH 44106, U.S.A.; now at Lilly Research Laboratories. ~:Departments of Biomedical Engineering and Systems Engineering, Case Western Reserve University, Cleveland, OH 44106, U.S.A. § Author to whom all correspondence should be addressed.

2. Plant Consider a diserete-time single input-single output plant which is described by the following equation: ~.(q- ~)y(t) = B(q- ~)u(t - 1)q-d + ~(t)

405

(1)

406

Technical Communique

where q 1 is the backward shift operator, d is the system dead time, y is the output, u is the input, and z(t) is a general disturbance. Often z(t) is expressed as C(q-1)e(t), where e(t) is Gaussian white noise. Here A(q-1), B(q -~) and C(q -1) are polynomials of order (m × 1), (n x 1) and (p × 1), respectively such that:

J*=

~ i=

[u(t+i)-u(t+i-

1)] 2.

(10)

1

Since J * is minimized by setting future inputs equal to the current input, u(t), then (9) can be written as

A(q -1) = 1 + a l q -1 + 0 2 6 / - 2 + '.- + a.q-"

B(q -1) = bo + blq -1 + b2q -2 + "'" + b , , - l q " - 1

C(q -1) = 1 + clq -1 + c2q -2 + "'" + cpq -p.

In addition, the inputs are chosen to simultaneously minimize the following:

J = HY* - Fy'(t) - G"u'(t)q -1 - g'u'(t)ll 2

(11)

(2) where g' = go + gl + "" + gk. J in (11) is minimized by:

Let T(q-1) be an arbitrary observer satisfying the following

identity: T(q - a) = E(q - 1)A(q- I) + q-(~+d) F( q - 1)

(3)

where E ( q - 1) and F ( q - 1) can be determined from T ( q - 1), A ( q - 1) and an arbitrarily defined positive integer k. Then (1) can be written as: Ty(t + k + d) = Fy(t) + Gu(t)q ~ + Ez(t + k)

where G(q -1) = E ( q - l ) B ( q - l ) . Letting y'(t) = y(t)/T, (4) can be written as:

u'(t) = u(t)/T

y(t + k + d) = Fy'(t) + Gu'(t)q k + Ez(t + k)/T.

(4) and

(5)

Note that y(t + k + d) can be estimated using (5) for any k, based upon present and past outputs and inputs (and, if k > 0, future inputs). Also note that the F, G and E polynomials are functions of this k. 3. Controller

The objective of any controller is to minimize a given cost function subject to certain constraints. Although the control objective can be generalized to include the weighted cost of control and cost of change in control (Clarke and Gawthrop, 1975), for simplicity only the minimization of the variance between the output and a constant (or slowly time-varying) setpoint y*, while maintaining stability is considered here. It is also assumed that T = C and e(t) = 0 for convenience. The cost function is given by: J = Ily* - ~ t + k + d)[[2.

(6)

Replacing y (t + k + d) in (6) with the estimate of y(t + k + d) given by (5), where e(t) is set to zero and T = C gives: J = Ily* - Fy'(t) - Gu'(t)qkH 2.

u(t) = (y* - Fy'(t) - G"u'(t)q-1)/g,.

(12)

The control law (12) is determined at each interval, as in the RHC algorithm, but with this different constraint that all future inputs equal the current input. 4. Selection o f control advance k

The selection of the control advance, k, is a critical step. Note that when this "control advance" k = 0, the CAMAC control law is identical to that for the minimum variance self-tuning controller. Under conditions when this controller provides stable control, so will CAMAC. By increasing k, the number of poles in the closed loop equation for u(t) is increased and the poles are shifted. For all plants with only minimum phase zeroes and for nonminimum-phase plants that are open loop stable, by making k sufficiently large the poles of u(t) can be shifted such that they lie inside the unit circle. For systems that are not open loop stable, this is not always possible. Details arc given in Voss (1986). Increasing k tends to decrease oscillations in u(t), but it also results in an increase in the deviation between y(t) and y*. CAMAC chooses the control advance that is the minimum k such that the poles of the closed loop equation for u(t) are stable and (recursively) estimated future inputs do not oscillate beyond specified bounds. Details of this choice of k and imposition of oscillatory and other input bounds are contained in Voss (1986). 5. Simulation results

To compare the C A M A C with other self-tuning controllers with nonminimum-phase plants, the authors performed simulation studies that paralleled those of Clarke (1984) (i.e. using the same examples). The plants are identified using a recursive least-squares estimator. The input to the plant is bounded such that - 100 ~< u ~< 100. Closed loop control begins at time t = 25, after an initial period of system identification. During control the setpoint switches between 20 and 50 at every twentyfifth sample for 250 samples. Noise was not added. The Type I nonminimum-phase plant used by Clarke (1984) to evaluate the various self-tuning controllers is given by:

(7) y(t) = u(t - 1) + 2 u ( t - 2) + 0.7y(t - 1).

Introducing the identity G(q- 1) = G'(q- 1) + G"(q- l)q-(k + I)

(8)

where G'(q -1) = g o + g l q -1 + "'" + g ~ q - k G"(q -1) = gk+l + "'" + gk+=q-"

the present and future inputs can be separated from past inputs as given by J = [lY* - (Fy'(t) + G"u'(t)q- 1 + G'u'(t)q-1 + G,u,(t)q~)ll2.

(9)

Using the CAMAC (Fig. 1 top), the control time delay, k, was initially selected to be I and was increased to 2, 3 and 4 at t -- 100, 150 and 200, respectively to demonstrate the effect of increasing k on damping of control. Increasing k from 0 (the minimum variance self-tuning controller) to I changed the poles of the closed loop equation for u(t) from - 2 (unstable) to 0.204 + 0.685i and 0 . 2 0 4 - 0.685i. Further increases in k increased the number of poles and shifted the poles closer to zero. For comparison, the EHC (Fig. 1 middle) and RHC (Fig. 1 bottom) of Ydstie (1984) were also used to control the same plant. For both trials the control horizon, T(where T = k + d), was initially set at 2 (at t = 25) and increased to 3, 4 and 5 at t = 100, 150 and 200, respectively. Note that the CAMAC (Fig. 1) avoids the oscillatory behavior of the EHC as well as the offset error of the RHC. This illustrates a significant influence of the input constraint on performance. Unlike the pole-placement algorithn~ the CAMAC (for k > 0)

Technical Communique

407

'"'a Y"

75-

y

CAMAC

....

o

-25-1

125"

s'o

r

,

r------,=__J

16o

i u

260

Time

0- + - c - + _ 0 _ { _ o _ Extended Horizon

75-

0 -50 "J

50

100

150 Time

200

250

300

FIG. 3. GMV control of a Type II nonminimum-phase plant using an overestimation of dead time: (a) open loop identification interval; followed by control with (b) iambda ='64, (c) 16, (d) 4, (e) 1 and (f) 0.25.

25" 0 -25.J 125-

5'o

16o

260

Time

to evaluate the generalized minimum variance (GMV) controller. The cost function being minimized is given by:

a I" 0--+-c--+ Receding

J = (y* - y(t + 1))2 +

2(u(t) -

u(t -

1))2.

aly(t

-

Horizon

75-

The plant was modelled by: y(t) =

o

-25 J

5'0

16o

Time

15o

26o

2go

FIG. 1. Comparison of CAMAC (top), EHC (middle) and RHC (bottom) control of a Type I nonminimum-phasc plant. In the top figure, (a) denotes an open loop identification interval; followed by control with (b) k = 1, (c) k = 2, (d) k = 3 and (e) k = 4. For the EHC and RHC, (b) T = 2, (c) T = 3, (d) T = 4 and (e) T = 5. Note increasing oscillatory behavior of input and output for the EHC and the increasing offset error for the RHC. = 751

1) +

blu(t -

2) +

1).

Since bo is negative the sign of the control is negative. Hence the controller would supply a negative input to increase the output. Thus the GMV was unstable for any positive value of 2. The GMV controller, like other control algorithms which rely on the initial plant response for the sign of control gain, does not satisfactorily control Type II nonminimum-phase plants. Under such conditions the control needs to be reinitiated with an increased dead time in the estimation. By using another model in estimating the system parameters: y(t)

.

bou(t -

= bou(t -

2) +

blu(t -

3) +

aly(t

-

1)

the GMV controller provides stable control since the estimated bo with the new model is positive. Figure 3 displays the control of the GMV controller for 2 = 64, 16, 4, 1 and 0.25 at t = 25, 100, 150, 200 and 250 respectively. For any choice of 2 the performance of the GMV was inferior to that of the CAMAC. 6. E x p e r i m e n t a l

o

5'0

i;o

26o

2;0

-50 J Time F]o. 2. CAMAC control of a Type [I nonminimum-phase plant: (a) open loop identification interval; followed by control with (b) k = 2, (c) k = 3, (d) k = 4 and (e) k = 5 fixed. also effected stable control of Clarke's Type I nonminimumphase plants with unmodelled poles or zeroes. Increasing k dampened the oscillatory effects of high frequency unmodelled dynamics on both the input and the output. Of course, performance was degraded by the poor plant identification. To evaluate the performance of the CAMAC with Type II nonminimumphase plants the following plant was used: y(t) = -- lu(t -- 1) + 2u(t -- 2) + 0.7y(t -- 1). The control time delay, k, was initially selected to be 2 at t = 25 and was increased to 3, 4 and 5 at t = 100, 150 and 200, respectively (Fig. 2). For comparison, the same plant was used

performance

The CAMAC has been implemented to control mean arterial pressure in anesthetized dogs using sodium nitroprusside (Vuss, 1986). It achieved good control despite the presence of intentionally induced large disturbances in this nonminimum-phase system possessing unknown and varying input-output delays (Fig. 4). By extending the B parameter matrix in the estimator both the minimum and maximum possible dead times were covered in the system identification. Thus the CAMAC satisfactorily handled the unknown I/O transport delay like other nonminimum-phase plants. 7. C o n c l u s i o n s

The CAMAC, for a sufficiently large k, was shown to provide stable control of nonminimum-plants even with unmodelled poles or zeroes. Since the CAMAC is not restricted to controlling the plant at the assumed I/O transport delay, it can control plants with unknown or time-varying I/O transport delay. In these simulation studies of the control of Type I nonminimumphase plants, the CAMAC performed as well or better than other self-tuning algorithms reported in Clarke (1984) and Ydstie (1984). For some example systems under special conditions, however, the discretization of the tuning parameter (k) in the CAMAC may lead to a slightly more conservative control than

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I00"

Animal Experiment

Acknowledgements--Supported by the National Science Foundation under grant ECS84-00765.

a= 80E E {3-

<

Note added in proof--ln Ydstie (1985) an assumption equivalent

60-

to (10) is presented as one of several variants of the EHC and RHC. 40-

Identification Control ,[ 06- ~I/ . Halothane = 1 % - - ~ Halothane = ~

150/0

0.4-

E o_ 0.2-

0

o

4b (minutes)

FIG. 4. CAMAC control of mean arterial pressure (MAP = upper tracing) in an anesthetized dog with sodium nitroprusside (SNP = lower tracing) input. The sample/control interval is 20s. An initial period of open loop identification (t = 0-8 rain) is followed by control of MAP from 74 to 50 mmHg (t = 8-180min). To test the controller, the inspired halothane concentration is changed at t = 12.5 min from 1% to 0.5%. This necessitates an increased control level to maintain the same pressure. is necessary. The CAMAC also provided adequate control of Type II nonminimum-phase plants, unlike controllers which depend solely on the initial plant response for the sign of the control gain and so provide poor control.

References Astrom, K. J. and B. Wittenmark (1973). On self-tuning regulators. Automatica, 9, 185-199. Clarke, D. W. (1984). Self-tuning control of nonminimum-phase systems. Automat~ca, 20, 501-517. Clarke, D. W. and P. J. Gawthrop (1975). Self-tuning controller. Proc. lEE, 122, 922-934. Cutler, C. R. and B. L. Ramaker (1980). Dynamic matrix controi--a computer control algorithm. Proc. JACC, Vol. 1. Gawthrop, P. J. (1982). A continuous-time approach to discretetime self-tuning control. Opt. Control Applic. Meth., 3, 399414. Voss, G. I. (1986). A self-tuning controller for drug delivery. Ph.D. Thesis, Dept. Biomed. Engng, Case Western Reserve University, Cleveland, Ohio. Wellstead, P. E., J. M. Edmunds, D. Prager and P. Zanker (1979). Self-tuning pole/zero assignment regulators. Int. J. Control, 30, 1-26. Wittenmark, B. and K. J. Astr6m (1984). Practical issues in the implementation of self-tuning control. Automatica, 20, 595605. Ydstie, B. E. (1984). Extended horizon adaptive control. Preprints 9th IFAC World Congress, Budapest (paper 14.4/E-4). Ydstie, B. E., L. S. Kershenbaum and R. W. H. Sargent (1985). Theory and application of an extended horizon self-tuning controller. AIChE Jl, 31, 1771.