Genetic Tuning Of Digital Pid

ear devices. Apart from large scale numerical methods, this is the only method known at present for carrier-injected ridge waveguides in which the two-dimensional mixing of the fields beneath the ribs can be modelled with some accuracy. Acknowledgment: S. V. Burke acknowledges with thanks the financial support from BT Laboratories. P. C. Kendall is grateful to the Wolfson Foundation for its support.

14th March 1992 S . V. Burke; P. C. Kendall, P. N. Rohson and G. J. Rees (Dept. of Electronic and Electrical Engineering, Uniuersity of Sheffield,Mappin St., SheffieldSI 3JD, United Kingdom) M. J. Adam (BT Laboratories, Martlesham Heath, Ipswich IP5 7RE, United Kingdom)

* Present address: Dept. of Physics and Astronomy, University of Wales College of Cardiff, PO Box 913, Cardiff CFl 3TH. United Kingdom

However, in both cases, such controllers need to be tuned carefully to produce acceptable closed-loop behaviour. This tuning process can frequently present difficulties, even if realtime expert systems are used to automate the tuning process [4]. Indeed, such real-time expert systems for the tuning of digital P I D controllers are frequently unattractive in practice because of problems such as knowledge elicitation. Therefore, in this Letter, the techniques of genetic algorithms [S, 61 are proposed as an alternative approach to the tuning of such controllers. These techniques are very simple to apply and yet promise to provide a means of readily incorporating in digital P I D controllers practically meaningful performance characteristics. Genetic tuning methodology: The digital PID controllers proposed by Porter [l] are governed on the discrete-time set {0, T, 2 T , , .. ,k T , . . .} by control-law equations of the form [23 U, = T K , e ,

+ T K , z , + K,(e, - e,-])

(1)

References BENSON. T. M.,KENDALL, P.e., STERN, M. s., and QULNNEY, U.A.: 'New results for rib waveguide propagation constants', IEE Proc. J, 1989,136, pp. 97-102 ROBSON, P. N., and KENDALL, P. c. (Eds.):'Rib waveguide theory by the spectral index method' (Research Studies Press and Wiley, 1990) MACE, U. A. H., ADA-, M. J., SINGH, I., FISHER, U. A., HENNING, 1. O., and D U N r A N , w. J.: 'Twin-ridge laser amplifier crosspoint switch', Electron. Lett., 1989, 25, pp. 987-988 PAIS, I., and HARDY, A.: 'A coupled-mode analysis of twin-stripe index-guided lasers', IEEE J. Quantum Eleciron., 1969, QE-25, pp. 1609-1 616 SNYDER. A. w., and LOVE, J. D.: 'Optical waveguide theory' (Chapman and Hall, London, New York, 1983) SCHIFF, L. I.: 'Quantum mechanics' (McGraw-Hill, 1968),p. 248 BURKE, s. v., KENDALL, P. e., RITCHIE, s., ROBERTSON, M. J., and R O E S " P. N.: 'Analysis of rib waveguide coupler filters', IEE Proc. J . 1992,139, pp. 59-65

GENETIC TUNING CONTROLLERS

OF DIGITAL PID

B. Porter and A. H. Jones Indexing terms: Adaptive control, Digital control The techniques of genetic algorithms are proposed as an alternative means of tuning digital PID controllers. This use of genetic algorithms is particularly attractive because the same basic approach can always be readily used, even in the case of digital PID controllers for complex multivariable plants with highly interactive dynamics. Introduction: In recent years, the following guidelines have been explicitly adopted by Porter [l] for the design of digital controllers for multivariable plants : (a) use only plant input and output data (i.e. regard state a s a

mathematical abstraction) (b) model plants using only such data (e.g. step-response or impulse-response matrices) (c) develop effectivecontrol laws that use only such data

(d, develop effective procedures

that identify such data in real

time ( e ) develop effective algorithms that tune controllers in real

time. In this way, a continuum of design methodologies has been developed for both nonadaptive [2] and adaptive [3] digital controllers. ELECTRONICS LETTERS 23rd April 1992

Vol. 28

where T E R + is the sampling period, e, = U - y, error vector, U E R" is the set-point vector, and

zk = z t -

I

+ Te,-

E

R"

E

R" is the (2)

is the integral of the error vector. The P I D controller matrices K , e R m x m ,K , E R " ~ " , and K , E R " ~ " are given by the design equations 121

K, = H-'(T)n

(34

K , = G-'(O)E

(36)

K,

(34

and =

H-'(T)A

where H(T)E R" " and G(0)E R" K m are, respectively, the open-loop step-response matrix and the steady-state transfer function matrix. In addition, in eqn. 3, II E R""", Z ER""", and A E R"" are positive diagonal tuning matrices whose elements are chosen by the designer to achieve satisfactory closed-loop control. In implementing such nonadaptive digital P I D controllers in the case of fixed-parameter plants [Z], it is assumed that the matrices H(T) and G(0) are known from off-line step-response tests. However, for variable-parameter plants, the matrices H(T) and G(0) can still be obtained from on-line tests conducted in real time. The resulting adaptive digital PID controllers [3] accordingly incorporate identifiers that recursively estimate the required matrices H(T) and G(0). However, in both the nonadaptive [2] and adaptive [3] cases, the selection of a set {n,Z, A} of tuning matrices is frequently a difficult task that cannot be simplified even by the use of real-time expert systems [4] for this purpose. These considerations motivate the investigation of the possibility of using genetic algorithms to tune both nonadaptive [2] and adaptive [3] digital P I D controllers. T o use genetic algorithms in this way, it is only necessary to encode the elements of the positive diagonal tuning matrices in each set {n,E, A} involved in the design equations (eqn. 3) in accordance with a system of concatenated, multiparameter, mapped, fixed-point coding [6]. Thus, each set (n, X, A) of tuning matrices is represented by a string of binary digits. Then, following a random initial choice, entire generations of such strings can be readily processed in accordance with the basic genetic operations of selection, crossover, and mutation [6]..In particular, the selection process ensures that the successive generations of digital P I D controllers produced by genetic algorithms exhibit progressively improving behaviour in respect of some fitness measure such as minimum rise time or minimum integral square error. This use of genetic algorithms for tuning digital P I D controllers [2, 31 is particularly attractive because the same basic approach can always be readily used, even in the case of controllers for complex multivariable plants with highly interactive dynamics. Indeed, the efforts of control engineers can thus be concentrated on the

No. 9

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843

formulation of appropriate scalar fitness measures to reflect the needs of engineering practice. Illustrative example: These general considerations can be con-

veniently illustrated by designing a digital P I D controller for the single-input/single-output plant with transfer function 1 c(sj =

(s

References ‘Issues in the design of intelligent control systems’, I E E E Control Systems Magazine, 1989, 9, pp. 97-99 PORTER, E., and JONES, A. H . : ‘Design of tunable digital set-point tracking PID controllers for linear multivanable plants using stepresponse matrices’. Proc. 25th IEEE Conf. on Decision and PORTER, E.:

Control, Athens, 1986, pp. 1502-1507 and PORTER, B.: ‘Design of adaptive digtal set-point tracking PID controllers for multivariable plants’, I E E E Trans.. 1987, AC-32. pp. 459-462 PORTER, 8.. JONES, A. H., and MTKEOWN, c. n.: ‘Real-time expert tuners for PI controllers’, Proc. I E E , 1987, 134, pp. 260-263 HOLLAND, I. H.:‘Adaptation in natural and artificial systems’ (The University of Michigan Press, Ann Arbor, 1975) GOLDBERG, D. E.: ‘Genetic algorithms in search, optimization and machine learning’(Addison Wesley, Reading, 1989) POLAK, E., and MAYNE,D. Q: ‘An algonthm for optimization problems with functional inequality constraints’, I E E E Trans., 1976, AC-21, pp 184-193 GESING, w., and OAVISON, E. J.: ‘An exact penalty function algorithm for solving general constrained parameter optimization problems’, Aufomafica,1979, IS, pp. 175 -188

JONES, A. H.,

(4)

+ 3Xs2 + 2s + 2)

It is of considerable interest to design such a controller for this plant using genetic algorithms since Polak and Mayne [7] and Gesing and Davison [8] previously optimised a P I D controller for this plant using rather complicated nongenetic optimisation algorithms. In the present case, it is evident from eqn. 4 that the steady-state transfer function (5)

G(0) = 0.1667

and that the step-response function

H ( T ) = 0.1665 x

(6)

when T = 0001 s. It is required to tune a digital P I D controller governed by equations of the form of eqns. 1-3 such that the integral square error (ISE) is minimised. 0 155

LOW-TEMPERATURE MBE-GROWN In,.,,Ga,.,,AI,.,,As/lnP OPTICAL WAVEGUIDES

I

H. Kiinzel, N. Grote, P. Albrecht, J. Bottcher a n d C . Bornholdt Indexing terms Optical waveguide, Integrated optics 0 131

5

10 generotlon

15

20

Fig. 1 Generation average integral square error

The results of solving this problem by means of a genetic algorithm with a population size n = 16, a crossover probability p , = 0.95, and a mutation probability p , = 0.01 are shown in Figs. 1 and 2 over 20 generations. In Fig. 1 the generation average ISE is plotted against generation number, and in Fig. 2 the best-of-generation ISE is plotted. These figures clearly indicate that the ISE of the best genetically tuned digital P I D controller steadily approaches the value 0.1251 previously obtained [7, 81 by nongeneric optimisation algorithms. In addition, the tuning parameters obtained by the genetic algo{II, Z, rithm steadily approach the values A) = (5.203 x 1667 x lo4, 1.665 x corresponding to the controller settings previously obtained [7, 81. O

’35r 129

0 127

Fig. 2 Best-ofigenerafion integral square error

Conclusion: In this Letter, the techniques of genetic algorithms [ 5 , 61 have been proposed as an alternative means of tuning digital P I D controllers. This genetic approach is much simpler than that of the rather complicated nongenetic optimisation algorithms previously needed [7, 81 even in the case of simple plants. 13th March 1992

B. Porter and A. H.Jones (Centre for lnstrumentation and Automation, Uniumsity of Salford, Salford M5 4 W T , United Kingdom)

a44

Inrroducrion: Optical waveguides represent basic elements of

111-V photonic integrated circuits where they are employed for on-chip optical signal routing as well as for passive and electro-optic waveguide devices. For practical applications, low optical propagation losses of less than I-ZdB/cm have to be achieved. In addition, high electrical resistivity of the waveguide layers will be advantageous in different ways: Obviously, free carrier absorption is virtually eliminated in such waveguides. When interconnecting electrically controlled optoelectronic or electro-optic devices high resistivity waveguide layers help minimise electrical crosstalk. Another application deals with vertically integrated waveguide-coupled pin-FET receivers as described in Reference 1, where the integrated waveguide adds an extra ‘buffer’ layer to the FET located on top. Here, a reasonably high resistivity is required for suppressing excess FET leakage current flow over this ‘buffer’. On InP, optical waveguiding layers have been prefer= 1-1.3pmj whereas waveguides ably made of InGaAsP (i.g o n InGaAlAs have not been as widely investigated. Optical losses hitherto reported were greater than 2 dB/cm [2-51, and electrical properties were hardly considered. Furthermore, there is a lack of reliable data on the refractive index. In this work we have investigated In, ,,Ga, 18A10.03A~ (Ag = I.O6pm)/InP optical waveguides grown by MBE. In particular, the impact of the substrate temperature on the optical and electrical properties was studied. An estimate of the refractive index at i= I-55pm will be given.

‘33b 7 7

w O 0 131

Eo

The MBE growth of In, ,,Ga, ,aAl, 3 0 A ~(jlB= 1-06pm) layers in the temperature range of 400-450°C was demonstrated to give high-quality optical waveguides which not only exhibit low propagation losses as low as 0.5dB/cm at i.= 1.55pm but concomitantly high resistivity of > 104Clcm. The refractive index of In, ,,Ga, 18A1Uo,As was estimated to be 3-207 0.03 at 1 = I-55pm.

M B E growth: The epitaxial In, 52Ga,.,8AlozoAs layers were deposited by solid-source MBE onto ‘epi-ready’ InP 2” substrates mounted In-free. Except for I/V-characterisation purposes semi-insulating substrate material was used. As the parameter of primary interest, the growth temperature was ELECTRONICS LETTERS 23rdApril 1992 Vol 28 No. 9

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