Ind. Eng. Chem. Res. 1994,33,889-895
889
A Method for Robust Tuning of Linear Quadratic Optimal Controllers Daniele Semino and Claudio Scali’ Department of Chemical Engineering, University of Pisa Via Diotisalvi 2, 56100 &a, Italy
A methodology for robust tuning of controllers designed according to linear quadratic (LQ)optimal control formulation is presented. It allows a direct approach to the problem when a low-order model and an uncertainty description are available for the system. As a preliminary step, a formal solution for the LQ controller, which depends explicitly on the model, input, and weight transfer functions, is presented. Analytical solutions for this formal expression are possible for common input and system models, including time delays and right-half-plane zeros. The controller tuning is accomplished thereafter by finding the value of the parameter of the weight function that guarantees the required robustness through an iterative numerical procedure. Application to cases of relevant interest in process control are illustrated, and guidelines for the selection of the input weight for the extreme cases of lag-dominant and delay-dominant systems are given.
I
1. Introduction
In the design of control systems for uncertain processes, the compromise to be found is between good performance (highgains to achieve large bandwidth and minimum error) and good robustness (low gains to achieve low sensitivity to process variations). Equivalently, the necessity of reducing regulator gain can be seen as coming from the presence of constraints on control action or from the need of limiting amplification of instrumentation noises. As a consequence, the speed of the closed-loopresponse will be lower and, more in general, the error will be larger than minimum values, which would be possible by fully exploiting the regulator potentiality. In the case of an incorrect or too conservative design, performance of advanced algorithm controllers may result quite comparable with standard structures. For this reason, it is very important to make use of direct procedures for controller design which allow a strict correlation between controller parameters and the desired closed-loop properties. The linear quadratic (LQ) approach is a classicalmethod for control system design (see Kwakernaak and Sivan (1972) for a general presentation; detailed references for specific aspects will be given below). It allows an elegant mathematical formulation to the problem, through the minimization of a quadratic index, which is a suitably weighted function of the output error and of the control action. Qualitatively speaking, by adjusting the input weight closed-looped properties can be affected, but no direct correlation is possible between robustness properties and values of the weight. Some design procedures allow a more direct approach to robustness. For example, adopting the internal model control (IMC) design method (Morari and Zafiriou, 1989), the robust controller is obtained via a two-step procedure. In the first step a nominal controller (ISE optimal) is determined analytically; then a filter, whose structure depends on the type of input, is added to give the global controller. Usually, the filter requires only one tuning parameter, directly correlated with closed-loop properties: speed of response and robustness to process model uncertainty. A complete comparison between the two methods has been presented in Scali et al. (19921,pointing out analogies and differences; also, advantages of the more straightforward IMC design for robustness have been indicated.
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d’
I
I
Figure 1. System with multiplicative uncertainty. Two difficultiesin the way of adoption of LQ controllers in the process industry, where robustness is the main issue, arise from the fact that analytical solutions are not easily available even for simple systems, and nothing equivalent to a tuning rule can be found to assign values of parameters for the LQ controller. The aim of the paper is to face these problems by presenting analytical solutions for the LQ controller for practically significant problems and then to develop a numerical design method to allow robust tuning of the LQ controller. The paper has the following structure: In section 2 the problem of robustness and the LQ approach are briefly reviewed. Section 3 includes analytical solutions for the LQ controllersin cases of common process and disturbance models. In section 4 the method for robust design of LQ controller when an uncertainty description on the process model is available is presented. In section 5 case studies of relevant interest in process control are illustrated and guidelines for the selection of the input weight are given for the extreme cases of lag-dominant and delay-dominant systems. 2. Robustness Issues and the LQ Approach
Assuming that the real plant (process P) can be represented in terms of a nominal plant (model and a multiplicative uncertainty 1, (Figure l), we have for single input-single output (SISO) systems
P(s) = P(s)[l+ l,(s)l
(1)
An upper bound on the magnitude of 1, can be easily found such that ll,(iw)l
t,(W)
(2)
We can therefore define a set II of plants correlated with the model ?! 0 1994 American Chemical Society
890 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
n = (P: I(P-P)P11 s t,(U))
(3)
At each frequency w the real lant P(io) belongs to a disk-shaped region centered at (io) with radius PTm(w). Closed-loop properties can be analyzed in terms of the sensitivity function e and complementary sensitivity function q defined as:
F
+ q = ( 1+ pc1-1 PC e
= (1 PC)-l
qQ
YL
Figure 2. Reference scheme for the LQ problem.
(4)
Y
(5)
For good performance the sensitivity function t should be small to achieve small error e = ( r - y ) = e(r - d). The minimal requirement for all uncertain processes in the set ll is that stability condition is guaranteed; the necessary and sufficient condition for robust stability is
I;(iw)l Ilm-l(w),
vw
(6)
At high-frequensie? when 1, is large, ;has to be small and then e E 1, as ?+e = 1. Therefore implications of the presence of uncertainty on performance are that the control system gives good steady-state behavior (low frequency), while this is not possible with high-frequency inputs. A compromise between good nominal performance (e small) and good robustness (7 small) has to be found. Such a problem could be seen as a robust multiobjective design (RMOD) problem (Dorato, 19911, and appropriate techniques could be attempted. An easier and more straightforward approach is chosen here as explained below. Figure 2 shows the reference scheme for the problem: P is the given process, C the feedback controller to be designed, P d the disturbance model (assumed of deterministic type) and W, a weight function on the control action. The LQ controller minimizes a quadratic index given by the integral of the sum of the square error e = r - y and of the weighted square control action u = W,m,for a particular input d or r; in the time domain: min Jom[e2(t)+ u2(t)1dt c
(7)
The weight function W,allows the relative amount of error and control action to be affected in (7); by increasing the weight, a smaller control action will be required: the resulting effect is a detuning of the controller. This can be seen as a way of accounting for constraints on a manipulated variable or its derivatives, or as a way of making the controller more tolerant to model uncertainty, but this is an indirect way of facing robustness. Qualitatively speaking, once the weight functional form has been chosen, as first step a controller must be found which is optimal according to the defined objective function (7). For a given system, the structure of the controller depends on the type of weight; values of ita parameters change, as a function of the weight parameters, in a way that is not possible to predict and usually must be computed numerically. As second step, robustness must be checked and this requires an iterative procedure (as described in section 4). This happens for every kind of controller having an assumed structure; the numerical procedure related with the first step, instead, is an additional complication of the LQ design method and can be avoided once analytical solutions are available. The conceptual importance of this is evident: the structure of the regulator (poles and zeros, if rational or not) is known from the very first stage of the design and so are the
3. Analytical Solutions to LQ Controller for Common Process Models The approach first presented by Newton et ai. (1957) is followed. The formal derivation of the LQ controller is more directly approached if the feedback scheme is changed into an open-loop structure, as shown in Figure 3; y' = y - d; r' = r - d. The feedback controller C can be obtained once Q is available as
The equivalent problem becomes min I , Q
(9)
I, = c [ e 2 ( t )+ u2(t)3dt
(10)
No specific details are given for the following derivation as it goes through the same mathematics as in Newton et al. (1957), adding a nonzero term WJs).Through Parseval's theorem (9) can be rewritten in the frequency domain as: (11)
By using the Wiener-Hopf factorization, one obtains a formal expression, for which analytical solutions can be obtained if specific conditions are met, as outlined below. This formal expression is
where
D(s) = d(s) d(-s) = D+(s)D-(s) A(s) = P(s) P(-s) + W,(S) W,(-S) = A+(s) A-(s)
(13) (14)
A+@) and D+(s)contain left-half-plane (LHP) zeros and poles of A(s) and D(s) respectively, while the operator { )* denotes that, after a partial fraction of the operand, all terms involving poles in the right half-plane (RHP) are omitted; in this way unstable terms and predictive elements (coming respectively from inversion of RHP zeros and time delays) are canceled and a stable and causal controller Q is obtained. According to whether time delays are present in the process transfer function, the controller C
Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 891 as in (8) may not be rational, requiring therefore an appropriate advanced implementation. Moreover one may argue that, despite Q being stable, such may be not the case for C. However, both internal stability and bounded input-bounded output stability are guaranteed in the equivalent feedback structure as well; therefore the obtained controller can be implemented with confidence. Some remarks about the solution given by (12) are in order at this point: 1. Equation 12 is valid for open-loop systems; for unstable systems, to avoid the pole-zero cancellation between process poles and controller zeros, which would result in an internally unstable system, a more general decomposition has to be used (Youla et al., 1976). 2. Equation 12 can be used for rotational transfer functions with multiplicative delay. 3. A similar expression can also be found in Palmor (1982), for the case of open-loop stable systems, multiplicative delays, and stochastic inputs. Equation 12 is particularly effective for understanding difficulties which arise during a search for analytical solutions. As the analyticity of the solution depends mainly on the spectral factorization in (14), conditions that the process and the weight transfer functions have to satisfy in order to get an analytical solution are easily derived. If P(s)is a proper rational transfer function with a delay and WJs)has a polynomial expression, the polynomial of highest degree to be factored is the numerator of A b ) . If n and m are the degree of P(s) denominator and of the input weight, respectively, the numerator of A(s) is a function of s of degree 2(n m) that can be seen as a function of s2 of degree n + m. If n + m I3, analytical solutions can be found. For first-order processes the order of the weight has to be m I2, for second-order processes it has to be m 5 1, for third-order processes m = 0; no analytical solutions are possible if n > 3. In case one cannot perform an analytical factorization, a number of numerical procedures can be used to address the problem. Spectral factorization of (13) and (14) can be obtained numerically and used in (12)toget the required controller. An alternative solution, which is preferable for high-order systems, can be found via state-space techniques. This latter procedure involves an optimal state feedback controller obtained through the solution of a Riccati equation, as outlined in Scali et al. (1992). In this paper we will direct our attention on first-order transfer functions (with possible delay and zero), which can well represent most higher order industrial processes of interest, including the following as uncertainty: order reduction, nonlinearities, and parameter errors. For robustness purposes W,, should have a magnitude that increases with frequency: the simplest choice is therefore W,, = asrn. The value of the minimum weight order m determines the degree of the derivative of the manipulated variable on which limitations are imposed: m = 0 corresponds to the square value, m = 1 to the square first derivative, and so on. The choice of m depends on the type of disturbance that has to be faced. In particular, if d' in general form is
+
through a first-order dynamics, m 1 1; if it is a ramp through a first-order dynamics, m 1 2. The following cases will be analyzed in detail: n = l , r=1, m = l n=l, r=l, m=2 n = l , r=2, m = 2
The process and disturbance transfer functions are therefore
The disturbance and the control action weight determine the different controller expressions. Case 1. Let the disturbance be d = l/s. If the chosen weight is W, = as,the LQ controller has the following explicit expression:
Q= where
a2 = Ta
'd
+ (a2+ z 2 p + 2KT(U)'/2 + TdK[Td- e4ITd(Td- 2 ) ] TLY+ (a2+ z 2 p + 2K~tr)'/~ +T ~ K
Case 2. Let the disturbance be again d = l/s;the chosen weight is now wu = as2. The LQ controller has the following expression, which depends on the solutions of a biquadratic third-order equation: (76
+ l)(Ufl + 1)
Q = a1s3+ b,s2 + cls + d, where
s6 - -s4 1 T2
the condition m 1 P has to be satisfied in order to get a proper controller. Therefore if the disturbance is a step
+ l)(Ufl + 1) a1s2+ b,s + c,
(TS
+Z2K2
K2
2s2--=o
7
a2T2
(19)
(third order in s2). As such an equation has either six real zeros or two real zeros and four complex ones, two by two conjugate, all the coefficients of previous expressions are real.
892 Ind. Eng. Chem. Res., Vol. 33, No. 4,1994
Case 3. Let the disturbance be d = 1/s2 (rampJ. If the chosen weight is W,,= as2,the LQ controller has the following expression:
Q=
(7s
+ 1)(ag2+ b$ + 1 )
a1s3+ bls2+ cls
+ d,
L
10
(20) 1.
where a,
=TC~ -1
10 2
10
10
1.
10
10
0
Figure 4. (- -) &J-l(u) for P = e+/@+ 1)(equal uncertainty 6 = 15% on the three parameters). (-) Iq(w)l for different values of a: 0.036,0.145,and 0.47.
The parameters Pi are the solution of the same equation as in case 2, and the same considerations guarantee that all coefficients of the expressions are real.
4. Illustration of the Method The procedure to determine the minimum value of the controller parameter a, which guarantees the robust stability condition (6), is illustrated below. Analytical solutions given in the previous section show clearly the nonlinear influence of a on the controller parameters and therefore on the behavior of Iv(w)l. A numerical method to find the minimum value of a to satisfy robust stability is therefore necessary. Details are given for the case of a first-order plus time delay system: P = e+/@+ 1)(uncertainty on all the three parameters), with a step through a lag disturbance d = l/s,Pd = l / ( s + 1 ) and with a first degree input weight W,, = as. A simple bound on multiplicative uncertainty can be found, in the case of parametric uncertainty in the process model, by using the method presented by Laughlin et al. (1987):
with uncertainty on all the parameters: ~ - k ] < h ~(e-$I> T', the disturbance dynamics becomes similar to the one of a different type (with r increased by 1) and the minimum weight order has to be changed consistently. 2. For the case of delay-dominant systems, a second-
Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 895 order weight ( W, = as2) should be adopted, because it gives good performance for all the class of type-1 inputs. 3. For the case of lag-dominant systems, a first-order weight (W,= as)is suggested, but an excessive detuning of the controller takes palce when high uncertainty is present in the process model, for the case of large disturbance lag (Td’of the same order of magnitude as T’). It is worthwhile to underline once more that the choice of FOPTD processes with uncertainty to account for order reduction, nonlinearities, or uncertain parameters includes a great majority of the industrial processes. Moreover, due to the choice of dimensionless variables, the presented cases with different values of T‘, Td’, and parametric uncertainty make the present analysis quite comprehensive. 6. Conclusions
The availability of analytical expressions for the regulator structure, as a function of different process and disturbance models and of different weights on manipulated variables, allows overcoming a typical drawback of the LQ approach to robust design. Analytical results presented for the case of first-order plus time delay process can be used for the robust design of higher order systems. The numerical procedure proposed for a fast computation of values of the weight parameter a,which guarantees robust stability, permits the establishment of a direct correlation between parametric uncertainty and values of a. Therefore a becomes a sort of tuning parameter for the LQ controller as it determines on one side the minimum value of the integral 1 2 and on the other the shape and values of the complementary sensitivity function. From the examined case studies, we can conclude that, for delay-dominant systems, a second-order weight ( W, = as2)gives good performance for the whole class of type-1
inputs, while for lag-dominant systems a first-order weight ( W, = as) is suggested, as no improvement is obtained with higher orders. In this latter case an excessive detuning of the controller may take place when high uncertainty is present in the process model. Acknowledgment This research was partly supported by CNR (Consiglio Nationale delle Ricerche), Grant No. 91.03133.CT03. Literature Cited Dorato, P. A Survey of Robust Multiobjective Design Techniques. In Control of Uncertain Dynumical Systems; Bhattacharyya,Kell, Eds.; CRC Press: Boca Raton, 1991; pp 249-259. Kwakernaak, H.; Sivan, R. Linear Optimal Control Systems; Wiley Interscience: New York, 1972. Laughlin, D. L.; Rivera, D. E.; Morari, M. Smith Predictor Design for Robust Performance. Znt. J. Control 1987, 46 (2), 477-504. Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. Newton, G. C.; Gould, L. A.; Kaiser, J. F. Analytic Design of Linear Feedback Controls; J. Wiley & Sons: New York, 1957; Chapter 6. Palmor,Z. J. Properties of Optimal StochasticControl Systems with Dead-Time. Automatica 1982,18, 107-116. Scali, C.; Semino, D.; Morari, M. A comparison of Internal Model Control and Linear Quadratic OptimalControl for SISO Systems. Znd. Eng. Chem. Res. 1992,31, 1920-1927. Youla, D. C.; Bongiorno, J. d.; Jabr, H. A. Modern Wiener-Hopf Design of Optimal Controllers. Part I: The Single-Input-Output Case. ZEEE Trans. Autom. Control 1976, AC-21 (l),3-13.
Received for review February 2, 1993 Revised manuscript received December I, 1993 Accepted December 16, 1993’
* Abstract published in Advance ACS Abstracts, February 15, 1994.
