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Autnntatica. Vol. 29. No. 4. pp. 953-968. 1993 0005-1098193 $6.00 + 0.00 Printed in Great Britain. ~ 1993 Pergamon Press Ltd

An Expert System for Multivariable Controller Design*

J. LIESLEHTO,t J. T. TANTI'U~t and H. N. KOIVO

A software package for the design of centralized and decentralized multivariable controllers has been developed. The software package consists of numerical calculation software and a help system implemented using expert system and hypertext techniques.

Key Words--Expert systems; multivariable control systems; PID control; hypertext; computer-aided design.

AIwtr~l- - ln this paper a software package for the design of decentralized and centralized multivariable controllers is described. The software package includes several numerical methods for multivariable controller design as well as an expert system to assist the user. The user interface of the expert system employs hypertext techniques. A continuous- time transfer function matrix is used as a process model. When the process model is known this software package can be used to select a control structure and to tune controllers. The paper first describes the different subtasks of multivariable controller design. Next a short description of the design and analysis methods included in the software package is given. A description of the expert system and a design example finish the paper.

INTRODUCTION

THE TREND IN computer aided control system design has been towards integrated design packages using a command language as a user interface (Rimvall, 1988). One of the first such packages was Matlab (Moler, 1980). Matlab was originally a general purpose linear algebra package, but different Matlab-derived control packages or toolboxes were soon developed. Other control design environments with a command interface include CTRL-C, Matrixx

Received 9 January 1991; revised 6 November 1992; received in final form 25 November 1992. The original version of this paper was presented at the IFAC Symposium on Intelligent Tuning and Adaptive Control (IFAC'91) which was held in Singapore, during January 1991. The Published Proceedings of this IFAC Meeting may be ordered from: Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by Associate Editor K.-E. Arzen under the direction of Editor A. P. Sage. Corresponding author Lieslehto's Telephone 358-31-162657; Fax 358-31-162340.

t Tampere University of Technology, Control Engineering Laboratory, Tampere, Finland.

~t Tampere University of Technology, Information Tech- nology, Tampere, Finland.

Tampere University of Technology, Control Engineering Laboratory, Tampere, Finland.

953

and PC-Matlab. A survey on the use of packages for computer-aided design in control systems in the U.S.A. (Herget, 1988) shows that the aforementioned programs are the most widely used environments for control design. The power of these packages lies not only in their computational capabilities but also in their extendability. For example, half-a-dozen control design packages have been built on top of Matlab, each containing dozens of new com- mands. Because of the extendability, control design packages have developed into integrated tools supporting all subtasks of control design. The increased functionality has caused also increased complexity. Therefore, it is obvious that a considerable amount of knowledge is required from the user before he is able to take full benefit of all the features offered by these complex control design packages.

A control design package can be used efficiently only if the required design knowledge is easily available. One way to achieve this is of course on-line documentation. An on-line documentation system can be made flexible to use by applying modern windowing and hyper- text techniques. However, such a system can only show the design knowledge to a user. The user himself has to apply this knowledge to the problem he is solving. Expert systems have been used to take the integration of numerical calculation software and design knowledge a step further. An expert system is able to apply its knowledge to solve problems. It can help a user to interpret numerical results and to decide how to proceed with a design session.

The use of expert systems in computer-aided control system design has been an active area of research during the 1980s. The Computer-Aided

954 J. LIESLErtTO et al.

Control Engineering (CACE-III) program (James et al. 1985) performs lead-lag precom- pensator design for single-input-single-output systems. When using this program the user does not access numerical analysis programs directly but through an expert system interface. The Intelligent Help System (IHS) expert system (Larsson and Persson, 1987) provides assistance in the system identification problem and supports access to a subset of the IDPAC numerical analysis routines. The expert system compares the actions taken by the user with scripts that represent typical command se- quences and events associated with them. Based on this information it gives advice on how to continue towards the assumed goal. Pang (1991) has developed an expert system for muitivariable controller design. This expert system is designed to be a part of an adaptive controller. It is able to redesign a multivariable controller automatically.

In this paper a software package for the design of centralized and decentralized multivariable controllers is represented. The task of designing controllers for muitivariable systems involves two major design steps. First a designer has to select the control structure and then design and tune the controllers. The software package consists of numerical calculation software and an expert system. The user interface of the expert system employs hypertext techniques. The numerical methods offer several functions for the interaction analysis and control structure selec- tion as well as for the tuning and analysis of multi-input-multi-output (MIMO) PID control- lers. A continuous-time transfer function matrix model of the process is needed as initial data. The expert system monitors the actions taken by the user. Based on the information gathered during the design process it is able to assist the user in problem situations. The software package has been developed on a Macintosh IIfx computer. The numerical methods are coded using Matlab (MathWorks, 1991). The know- ledge base of the expert system is built using Nexpert Object (Neuron Data, 1989) and the user interface is implemented as a HyperCard stack (Apple Computer, 1989). The expert system is historically related to the expert system for interaction analysis (Lieslehto and Koivo, 1987, 1991) and the expert system for tuning PID controllers (Lieslehto et al., 1988).

CONTROLLER DESIGN The design process is divided into four

different subtasks as depicted in Fig. 1. The calculation of interaction matrices and interac-

INTERACTION MATRICES

INTERACTION MEASURES

CONTROLLER TUNING

CONTROLLER ANALYSIS

L.

FIG. 1. Subtasks of controller design.

tion measures are the two subtasks of the control structure selection. After the control structure selection controllers are tuned and analysed. These two steps are often repeated in an iterative manner.

Interaction matrices help the designer to find possible candidates for the control structure of a decentralized controller. These matrices contain information about the interactions between inputs and outputs.

Based on the information from the interaction matrices the designer selects one or more candidates for the control structure. Interaction measures are then used to estimate the performance deterioration of decentralized con- trollers compared to a full centralized controller. If the interaction measures indicate possible problems then the designer has to look at the interaction matrices again and try to find another control structure. Otherwise he is ready to start controller tuning.

There are several methods for the tuning of multivariable PID controllers. These methods usually give some rough tuning matrices and the user can find the final tuning matrices by multiplying the rough tuning matrices with fine tuning parameters. The tuning is typically an iterative process. Based on the results from different analysis methods the values of fine tuning parameters are changed and the control- lers are retuned.

The last step is to analyse the performance and robustness of the designed controller. This is usually done using both simulations and frequency domain methods. If the analysis indicates any problems the designer has to

ES for mutivariable controller design 955

change the fine tuning parameters or try some other tuning methods. Of course, he can also change the control structure.

INTERACTION MATRICES

The control structure should be selected in such way that the inputs and outputs that strongly interact with each other are connected with a feedback. The number of possible control structures rapidly increases as the number of inputs and outputs increases. Several methods proposed in the literature aim to produce an interaction matrix which describes interactions between the inputs and outputs of a MIMO process. These methods help a designer to find possible control structures. The software package includes the classical relative gain array method proposed by Bristol (1966), the dynamic relative gain array method proposed by Tung and Edgar (1981) and the singular value analysis method by Lau et al. (1985). Bristol's relative gain array is probably the most well-known interaction matrix. Each element in a relative gain array is the ratio of two gains representing first the process gain in an isolated loop and, second, the apparent process gain in the same loop when all other control loops are closed. The ratio of these gains defines the relative gain array with elements

#ii = tPijyj, (1)

where ~Po is an element in the steady-state gain matrix G(0) and Yii is an element in the inverse of the steady-state gain matrix G-*(0). Bristol suggested that the proper input-output pair for single loop control is the one having the largest positive #o value.

The relative gain analysis is a steady-state analysis and does not explicitly include dynamic effects. Several extensions including also the dynamic effects have been proposed. Out of them the toolbox includes the dynamic RGA proposed by Tung and Edgar. The elements of the dynamic RGA are calculated as follows:

ec, j = g,i( s ) ]/ ji (2)

where g,j(s) is an element in the process transfer function matrix G(s) and '/i~ is an element in the inverse of the steady-state gain matrix G-~(0). Both the RGA and the dynamic RGA can be calculated only if the number of inputs is equal to the number of outputs.

The third method in the toolbox is proposed by Lau, Alvarez and Jensen. This method is based on singular value analysis and it is applicable also for process models with unequal numbers of inputs and outputs. Interactions are analysed using the singular value spectral

representation of a m x n open loop matrix transfer function G(s) whose rank is q. This representation is

q

G(s) = ~ o,(s)W~(s), (3) i=1

where o~ are the singular values of G and W~ are the dyadic expansion matrices defined in terms of singular vectors as follows:

Wi(s) = zi(s)v~(s). (4)

Through this expansion the system transfer function is expressed as a linear combination of the q nodal contributions. Each of the nodal terms consists of a scaling factor oi and a rotational transformation W~. Lau, AIvarez and Jensen suggested that the maximum entries in the rotational matrices define the input-output pairings for single loop control.

A new interaction matrix is also introduced. This method is based on the scaling of input and output variables. Although large gain between an input and an output indicates strong interaction, the process gain matrix cannot be directly used for interaction analysis, because it depends on the scaling of input and output variables. Anyway, by rescaling the input and output variables we can find a scaled gain matrix the elements of which are directly comparable with each other. In the scaled gain matrix the average absolute value of the elements of each row and column is one. So, values larger than one indicate strong interaction and values smaller than one indicate weak interaction.

INTERACTION MEASURES

Using interaction matrices we are able to find possible candidates for the control structure. Interaction measures are used to find out whether these control structures can be used or not. Interaction measures give the user informa- tion about the stability of the decentralized control and the loss in performance caused by these control structures. The interaction meas- ures included in the software package are the spectral radius of the Jacobi iteration matrix (Mijares et al., 1986) and the structured singular value interaction measure (Grosdidier and Morari, 1986).

The interaction measure proposed by Mijares and others is based on the idea of diagonal dominance. They suggest that a matrix is diagonally dominant when the off-diagonal terms make only a small contribution to the inverse of the matrix. The inverse of a matrix G can be calculated using the Jacobi method of iteration. The full matrix G and the iteration matrix A are

956 J. LIESLEHTO et al.

FIG. 2. Interaction as additive uncertainty.

related as follows:

G = (~(1 - A), (5)

where G is the block diagonal matrix which has as its elements the block diagonal elements of G. The inverse of G is now

G -1 = (1 - A)-I(~ -t , (6)

G- '=( I + A + A2 + A3 + . . . )G - ' . (7)

The matrix series converges if the spectral radius of A is less than one. In such a case the matrix G is block diagonally dominant. Mijares and others suggest that the control structure should be selected in such way that the corresponding Jacobi iteration matrix has the smallest spectral radius of all possible control structures.

Grosdidier and Morari (1986) have studied a system depicted in Fig. 2 where interactions are represented as additive uncertainty. A controller

C = diag {Ct, C2 . . . . . C,}, (8)

is to be designed for the system

G = diag {GI,, G22 . . . . , G , ,} , (9)

such that the block diagonal closed-loop system with the transfer matrix

!:I = 0C(1 + GC) -I, (10)

is stable. The interaction measure expresses the constraints imposed on the choice of the closed-loop transfer matrix

/ t = diag {H,,/q2 . . . . . /q,}

for the block diagonal system, which guarantee that the full closed-loop system

H = GC( I + GC) -~ (11)

is stable. The SSV interaction measure gives the following constraint for the stability of the closed-loop system H:

#(/~,(jco)) O, (12)

where /~ denotes the structural singular value and LH is the relative error matrix

L,, = (G - G)O- ' . (13)

It is assumed that G(s) and t~(s) have the same RHP poles and that /...