Computer Aided Design Program for Linear Multivariable Control Systems

COMPUTER AIDED DESIGN PROGRAM FOR LINEAR MUL TIV ARIABLE CONTROL SYSTEMS K. Furuta*, H. Kajiwara* and K. Tsuruoka** *Department of Control Engineering, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo, japan **Engineering Research Association of Nuclear Steel Making, Time-Life Building, Ohtemachl; Chiyoda-ku, Tokyo, japan Abstruct. CAD program package named DPACS-F for a linear multivariable control system analysis and synthesis is presented. The systems to be treated in the CAD can be in the form of state space description, transfer function or inputoutput data, and algorithms based on both state space and frequency domain approaches are provided and anyone of them can be executed by the command line which is given by a simple rule. As an example of a control system designed by CAD, the control system for a position control of a double inverted pendulum on an inclined rail is presented. Keywords_ Computer aided design; Control system synthesis; Computer control

INTRODUCTION

systems connecting arbitrarily is presented, which is called LINK, (iii) the computation of matrices can be achieved during the design in order to check the results or to improve the specification, are given.

It is known that the design of a control system for a linear multivariable system with the high dimension can not be achieved without the help of CAD, since not only complicate computation for the synthesis of a controller but also complicate handling of large s y stem data are required in the course of the design of a control system. When CAD is used, the designer can pay his attention only to the design without being bothered the above problems.

The description of the functions and the structure of the CAD are described in the following. To show the effectiveness of the use of the CAD, a control system is desibned for a position control of a double inverted pendulum on an inclined rail.

The group of Rosenbrock developed CAD based on the frequenc y domain approach, and the several program packages based on the state space approach have been developed, such as by Melsa and Jones(1970), where some of them have presented in the form of CAD and others are sets of programs for the batch processing. These CAD's developed so far are based on either the frequency domain approach or the state space approach. Unbehauen and his colleagues (1975) developed CAD which provides algorithms concerning the state space approach, the frequency domain approach and the system identification. This paper also aims to present a CAD useful in the analysis and synthesis based on state space, frequency domain and input-output data. The CAD is named DPACS-F(Qesign ~~ckage for Control Systems developed bv Furuta Laboratory) -;nd is d~veloped for a minico;;;puterCNOVA with 96KB cpu memory and 10lm disc memo rv), which is modified by ERANS[ for a lar ge computer, and the modified package is named as CONPAK. Special properties of DPACS-F in that (i) as the system data, not only parameters to describe the state space and the input-output relation but also the input-output data can be treated, (ii) the function to make a system from several

267

STRUCTURE OF DPACS-F The structure of DPACS-F concerning the programs and data-files is depicted in Fig.l. DPACS-F treats linear time-invariant systems represented b y one of the following forms. 1) State space description: o x(t)= Ax(t) + Bu(t) y (t)= Cx(t) + Du(t)

(1)

where n-vector x, m-vector u and p-vector y indicate the state, the input and the output res p ectivelv, A ,B,C,D are real matrices of appropriate dimensions, and ox(t) denotes dx(t)/dt in continuous time and x(t+l) in discrete time. 2) Transfer function:

y( a )= G(a )u( a)

(2)

qi (i=l ... n) are scalars, Pi(i=l ... n) and D are p Xm matrices, and 0 denotes s in continuous time and z in discrete time. 3)

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parameters:

K. Furuta, H. Kajiwara and K. Tsuruoka

268

G(a)u(a),

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Markov parameters, Differential or difference equation, Input-output data respectively, (ii) the order n (for @=M, the number of Markov parameters; for @=I, the length of the inputoutput data), (iii) the storing file name, SYSNAME.@i, where i is the pointer to indecate its position in SYSCOM, (iv) the command name and date of the production, and (v) the comments to distinguish it from the other data. Thus it is possible for a designer to handle systems and system-datas by the corresponding names, which makes him not pay attention to the complicate data handling.

(3)

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G(a)

=

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(4)

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+ ... + Qn, + ... + P n ,

Superviser

Qi(i=O ... n) are pXp matrices and Pi(i=l ... n) are pXm matrices, and a denotes d/dt in continuous time and z for discrete time.

The flow chart of the supervisor is depicted in Fig.2. The supervisor has a Command Line Interpreter CLIF as a subprogram, which can interprets the command line of the form:

5) Input-outPllt data: (5) {u(t), y(t)}t=O,l, ... A set of parameters in each representation is called system-data (SYSDATA) in the following.

CMD/SW SYSNAME NEIoISYSNAME/O where CMD denotes the command name corresponding to one of function modules, SIoI specifies a subtask, SYSNAME is the name of the treated system, NEHSYSNAME with the switch "/0" is the name of the system produced as the results of the command. CLIF checks the command line whether SW is appropreate to the command, SYSNAME is already registered, and NEIoISYSNAME is not already resistered. If there is no error, a subprogram STSNF asks a designer which system-data under the system name are used for the command, and then sets all information (tables TSN & OSN) necessary for data-handling in the corresponding function module in a file SYSNAMEF. Afterwards, the function module corresponding to the command is executed. Fig.3 shows a typical flow chart of function modules with loading and storing system-data.

System-data handling The systems resistered are managed using a table SYSLIST stored in a file SYSLISTF which contains the followings for ea ch system: (i) the system name, (ii) the numbers of inputs and outputs, (iii) the sampling interval, and (iv) the number of resistered system-datas with the same input-output relation. A set of system-datas representing a system named SYSNAME is controlled through a table SYSCOM stored in a file SYSNAME.CO which contains the followings for each system-data: (i) the type @ of the representation, where @ denotes one of the symbols S, T, M, D, I corresponding to State space description, Transfer function,

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FUNCTIONS OF DPACS-F

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On system-data management and transformation Storing the s ystem-data of transfer function representation, Polynomial Compiler is provided, which admits a designer to input the data in written formulas. For example, consider the followin g transfer function describin g a system named SYSl with 2-inputs and 2-outputs.

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In Fig.4, the r e lation among functions to transform system-datas is shown. c~m is the function to obtain the minimal realization corresponding to external discriptions of the treated s ystem so that the controllable or observable pair is of the Luenberger's canonical form. Identification programs can be implemented as the f unction C}m/I, and are available from PAI PAC-F (Furuta an d co ll ea gues, 1979 ) . On

Functions of DPACS-F are described with the corresponding command names in TABLE 1.

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system analysis and controller synthesis

There a re two fundamental modules EIG and SVD for the matrix analysis. EIG is provided to obtain eigenvalues and eigenvectors o f a real matrix. For a real matrix A, SVD gives basis matri ces of Ran ge A, (Ra nge A)~, Kernel A, (Kernel A)~ and s ingular values. It is possible to effectively inve stigate the linear independency of vectors by using SVD. Programs of EIG and SVD are made based on EISPACK(Garbow and colleagues,1977).

270

K. Furuta, H. Kajiwara and K. Tsuruoka of DPACS-F. TABLE 1 Discriptions of functions in DPACS-F

Fig.4. System-data transformation S: State space description T: Transfer function M: Markov parameters D: Differential or Difference equation I: Input-output data A fundamental module on the system analysis is LCF to investigate the controllability structure of the pair (A,B) in (1). LCF uses SVD in investigating the linear independency of columns of the controllability matrix. There are two fundamental modules RCT and LOC on the controller synthesis. RCT gives an optimal state feedback gain by solving Riccati equation by Potter's method. LOC gives a state or an output feedback gain to accomplish a specified pole-placement. RCT is supported by EIG and LOC by SVD. CCCO, OPT, PLOC are directly obtained by using LCF, RCT, LOC respectively. In the synthesis of observers, submodules OBS/G, OBS/M, OBS/F and OBS/K are available to obtain Gopinath's state observer, Miller's state observer, functional observer, Kalman filter respectively. In the synthesis of the robust controllers tracking step reference signals, TRACK is used. OBS and TRACK are supported by stabilizing modules RCT and LOC. The above hierarchy structure among function modules results in the speedy and effective design of control systems LINK After the controller synthesis, the responses of the closed-loop system should be simulated in order to evaluate the performance. For this purpose, DPACS-F has a module LINK to connect several systems arbitrarily, and construct the total system. By combining the treated system, the controller and systems representing reference signals and disturbances, the responses of the control system are convieniently simulated as the free response of the total system(Furuta and colleagues,1979). llAT Besides the function modules for the analysis and synthesis of control systems, a module ~lAT is provided for the matrices computations by a designer. Matrices data registered in database can be handled by the names. Using HAT, algorithms not included in DPACS-F may be achieved. tlAT can be executed independent

[1] System-Data 1) SYSIN :to 2) TYPE :to 3) REVISE :to 4) COpy :to 5) DELETE :to 6) RENAME : to 7) LIST :to

Management input system-data type out system-data revise system-data copy system-data delete systems or system-data rename system name list up systems or system-datas

[2] System-Data 8) CMR :to 9) TF :to 10) MRKV :to 11) DIF :to 12) 10 :to

Transformation obtain minimal realization obtain transfer function obtain Markov parameters obtain differential equation obtain input-output data

[3] System Analysis 13) POLE :to investigate stability 14) CCCo :to investigate controllability 15) 16) 17) 18)

NYQST BODE LOCI BAND

:to :to :to :to

draw(inverse)Nyquist diagrams draw Bode diagrams draw root loci draw Gershgorin bands

19) 20) 21) 22)

RDCT COOD DUAL DCMP

:to :to :to :to

obtain reduced order system transform coodinate obtain dual system decompose system

[4] Controller Synthesis 23) PLOC :to solve pole assignment problem 24) OPT : to solve optimal regulator problem 25) TRACK :to solve tracking problem 26) DCPL :to solve decoupling problem 27) OBS :to obtain observer, Kalman filter [5] Simulation 28) LINK :to 29) CLPS :to 30) DIGIT :to 31) StfLT : to

link systems obtain closed loop system digitize continuous system plot free responses

[6] MAT 1) MAT 2) LIST 3) ADD 4) SUB 5) MUL 6) SMUL 7) EIG 8) GEIG 9) }lAP 10) INV 11) LSQ 12) NORH 13) TRACE 14) VEC 15) COOD 16) TRNS 17) PICK 18) EMBED 19) PR}IT 20) NORMZ 21) RSLV 22) LXP 23) IEXP 24) BYE

:to :to :to :to : to :to :to :to :to : to :to :to : to :to : to :to :to :to :to :to :to :to :to :to : to

execute the followings input, revise, copy and type list and delete execute addition execute subtraction execute multiplication execute scalar multiplication obtain eigen-values & vectors solve general eigen-problem obtain basis of range & kernel obtain inverse obtain least square solution calculate norm calculate trace investigate column-independency transform coodinate ob tain transposition pick columns embed columns interchange columns normalize obtain resolvent matrix obtain transition matrix obtain integral of trans.mat. stop llAT

[7] BYE

:to stop DPACS-F

Computer Aided Design Program

~71

POSITION CONTROL OF A DOUBLE INVERTED PENDULUM ON AN INCLUNED RAIL A double inverted pendulum to be controlled is shown in Fig.6. The system consists of a cart moving on an inclined rail, two pendulums hinged to the cart so as to rotate in the plape containing the rail, and a cart-driving means. A block diagram of signal-processing for the system is shown in Fig. 7. A displacement r of the cart, an angle 9 1 between the first pendulum and the vertical line, and an angle 6 2 between the second and the first pendulums are detected by three potentiometers, and then converted to the values of the outputs YI, Y2, YJ by A/D converter respectively. By using the values of Y=[YI, Y2, YJ]', the value of the control input u is calculated and supplied to the power amplifier through D/A converter.

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(2)

~-=:'~

_-l

Block diagram of the control system

Define the state variables x=[q' ,~']', where q=[r/ro, 8 1/ 80 , e 2/ 6 0]' and ra, 6 0 are normalizing factors. A linear mathematical model for the double inverted pendulum system is given by

x=

Ax y= Cx

+ bu + v [13

(7)

O]x

where the constant disturbance v appears because of the inclined rail. The control system to move the double inverted pendulum to a specified position rejecting the disturbance is designed by giving the following sequence of command lines to DPACS-F,as shown in APPENDICES . 1) 2) 3) 4) 5) 6) 7) 8)

SYSIN/S PEND: for the system (7) named PEND DCMP PEND DENDI/O: to remove disturbance v DCMP PENDI PEND2/0: to remove outputs Y2,Y3 TRACK PEND2: to obtain robust controller OBS/F PENDI PENDF/O: for functional observer LINK PEND PENDF PENDT/O: refering to Fig.7 POLE PENDT: for poles of the control system SMLT PENDT: to simulate the responses

Fig.

J.

A double im'ereed p·endultuCl syster-.

-

",- r

Fig.6. Signal transmission diagram

Photo.l. Ascending double inverted pendulum CONCLUTION CAD package DPACS-F is presented, which can treat the system in the state space, transfer function, and input-output data. And it is found effective in the analysis and synthesis of a control system through t he example. REFERENCES Furuta,K., S.Hatakeyama and H.Kominami(1979). Structural identification and software package for linear multivariable systems. 5-th IFAC Symposium on Identification and System Parameter Estimation Furuta,K., T.Nomura and H.Kajiwara (1979). Interactive Simulation ;!e thod for Integrated Li near ~!ultivariable Systems .ISC S . Furuta,K., T.Okutani and H.Sone (1978). Computer control of a double inverted pendulum . Comput.& Elec.Engng,5,pp.67-84. Carbow ,B. S .,J.M. Boyle,J .J.D ongarra and C .B. ~01er(1977). ~atrix EigensYstem Routines - EISPACK Guide L,tension. Springer Verlag. IEE (1973). Computer Aided Control System Design. Conference publication 96. ~elsa,J.L. and S.K.Jones (1970) . Computer programs for computational assistance in the study of linear control theorv. ~cGraw­ Hill. Rosenb r ock ,H. H. (1974). Computer - aided Control System design, Academic Press . Vnbehauen , H., Chr.Schmid, F.Bottiger, B.BaGer B.Gohring (1975). KEDDC:Ein bombiniertes Prozessrechner - Programmsyscem zum Ent"urf und Eirsatz von DDC -Al gorithmen. PD\, report, KFK-PD\' 37.

APPE"DiCES

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