CASAM — an interactive package for computer aided system analysis and modelling

- AN IRTERACTIVE PACKAGE FOR COWUTER AIDED SYSTEM ARALYSIS AIVDYODELLING

CASAY

A.

VARGA*

This paper prcaents an interactive package - CASAM, for the Abstract. oomputer aided analysra and rodelling of multivariable ayatama daacribed either by state equations or by transfer-matrices. CASAY ia baaed on several powerful, portable Fortran subroutine packages (BIMASC, BIMAS, EISPACK, LINPACK, ODEPACKIwhich implement the latrst advances in numerical algorithms. All functions are performed by the means of a command language. The CASAM pack8 e is implemented on the Romanian family of minicomouters INDEPENDEa T and CORAL. as well a8 on the DEC PDP-li systems. * aided

deaign;

Numerical

INTRODUCTION In the last few years, major developments have been achieved in elaborating efficient and reliable algorithms for most computational problems modelling and simulation of linear and nonlinear syain the analysis, terns. The existence of high performance linear algebra packages LINPACK differential equations [2] and EISPACK [5], [ 0] as well as the ordinary solvers package ODEPACK [b] contributed decisively to reliable computer implementat ion of these algorithms. Recently, two powerful packages of BIRAS [13] and BIMASC [lb] have been deportable Fortran subroutines, veloped for the computer aided system analysis and design. These packages implement the latest advances in numerical elgorithms using the highest quality a&able numerical software. CASAM is an interactive Computer Aided system Analysis and bJodelling paakage which is mainly based on BIYAS and BIMASC packages. Most of modelling and analysis functions are provided for lineer continuous or discrete models in standard state-space form. However several programs csn deal with linear models in generalized state-spece form.Simulation programs are provided for both linear and nonlinear systems. Graphical facilities are included for plotting simulation results. All functiona are performed organization

by the means of a command language. A flexible data allows an easy communication among various CASAMprograms. LINEAR SYSTEMS MODELLING

The purpose of modelling primarily consists of determining thoao forma of systems models which are appropriate for the analysis of aystema properties or which can be used in the simulation. Some modal transformationa we provided in order to obtain better conditioned state-space parameterization of models or reduced order approximate models. The uae of #Central Institute for Bd.Hiciurin, No.8-10, 120

Mane ement and Informetica 7131! Bucharest, Romania.

B’ = Q-‘B,

A’ = Q-'AQ,

where Q is a given to reduce

or computed

by orthogonal

berg or ordered

real

C’ - CQ, D’ = D traneformation

tranaformatidne,

Schur forma.

matrix.

(6) TSO can be used

the etste-metrix

TSI uaea non-orthogonal

A to Hessen-

trensf orme-

tiono to reduce I, to condensed forms (balanced, Heasenberg, Schur, blockdiagonal 1 ). TCF uses orthogonal similarity transformation6 to compute standard contro=bility-obsarvability forms of the system [ll]. TBAL and w perform system balencing transformations [7]. TRED can alao be

[I

used to compute reduced

order

approximete

For GSSM,two progrema m and m trensformetiona of the form

models.

perform

orthogonel

similarity

E’ = QTEZ, A' = QTAZ, 8’ = QTB, C’ = CZ, D’ = D (7) where Q end Z are orthogonel trensformation matrices. GTSOreduces the peir (E,A) to trienguler-Heasenberg or triengular-ordered reel Schur forms. GTCF computes controllability or observability cenonical forma

bl*

SYSTEM ANALYSIS The CASAM packege provides several programs for the anelyaia of properties of linear state-space systems. The programs Syriaand OsyA can be used for the enalysia of stability, controllebility-stebilizebility end observebility-detectability of SSSb! and GSSM, respectively [ll] ,[9]. Both programs also computes the system poles. MZE end GAZE ere programs for computing the multiveriable systems zeros using the algorivhms in

bl Sgl l

The simulation

of systems servea as a powerful tool for anelysia.SDS ie a simulation progrem for discrete lineer systems, while -SCS and SSCS ere progrems for simulating continuous linear systems. SSCS is eppropriete for systems described by “stiff” differential equetiona or for high accuracy requirements. Ossc can be used for the simulation of linear systems described by GSW. SCS uws the Runge-Kutta-Fehlberg method [a], while SSCS end GSSC are baaed on implicit Adama end beekwerd difference methods 6 . All these programs exploit the lineerity of the system, by reducing Nonlinear equations

[I

the number of operations aystema described x(t) y(t)

or the implioit

required

by the explicit

= f(x(t), = g(x(t),t) ayrtem of differential

to eveluate

derivatives

system of differential

[-I

l

u(t),t) equations

= f(x(t),u(t),t) A(r(t),t)x(t) y(t) = g(x(t),t) ten be simulated ueiag the prqgamr s and oslps, rrspectively. This programe are baaed on the LSCDP ld LSODI packegee which belong to the DDEFACEcollection of ordinary differential equation solvers [6]. Simuletion reaulta can be plotted using the progrem PLUI!. 121

CASAYmodelling

programs assumes the availability

of the linrarisod

models of the physical systems which are studied. These nodelr are de& termined either from model building ueing the basic physical principle6 or through system identification. The CASAMmodelling described

program3 deal with linear

by standard Ax(t) y(t)

state-space

= Ax(t) = Cx(t)

model3

time-invariant

+ au(t) + Du(t)

or by generalized state-space models Eb(t) = Ax(t) + Butt)

eystens

(SSSM) of the form (1)

(GSSM) of the form

(2) y(t) = Cx(t) + Du(t) where x, u end y are the state, input and output vectors, respectively, A, B, C, D, E are constant coefficient matrices, and where x is the differential operator d/dt for continuous systems or the forward shift operator Xx(t) = x(t+l) for discrete ayatems. An alternative eyetem description accepted by some CASAMprograms is the input-output description

given

by

Y(h 1 = G(X) U(h) where Y and U are the transform In continuous case, the Laplace

(3) output and input vectora, respectively. - transform is used, while in discrete

case the Z - transform is used. The matrix matrix and generally is a rational matrix. ee list

below the modelling

facilities

G(A) ie the system transfer

implemented

in the CASAMprograms.

a. Discretization TMCDperforms the discretization of continuous input-output models using TSCD performs the discretization of the matrix exponential method SSSM using the matrix exponential method with Pad6 approximations combined with a very efficient block-diagonalization procedure 1

bl* -

b. Transformation3 between inout-outnut These two system descriptions are related G(x) = C(AI-A)-lB for

and state-3DaCe by the relation3

+ D

cl03 ) [I .

descrintions (4)

SSZM, and

+ D G(A) = C(%A)-‘B (5) Por GSSM. The programs m and m construct state-space realizations Par proper and non-proper transfer aetrices, respectively. These programs determine generally non-minimal order state-space models and therefore they are usually followed by m and ayNR, respectively, which extract minimal order model13 from non-minimal ones, using orthogonal similarity transformations The program3 TSY and OTsy computer [ 11 the transfer matrix corresponding to a SSSY or GSSN, respeatively [151,

lIC4

Cl61 * c. Similarity

Several 122

transformations CASAMprograms performs‘

similarity

trsnsfcrmetionr

of the farm

CASAY OPERATION AND IYPLEXENTATION 6x1 functions of CASAM are activated by the means of a simple commend language. Each program, when it is activated, fetches its input data of most pr ogrems from the diek and puts its rectulta on disk. The outputs can be used 8s inputs to other programs of the package. A flexible data organizetion allows an easy ~~~icatio~ emmong various CASAM programs, CASAM can solve problem8 with 35-40 state variables, all computations being done in double precision. The progrems are written in Fortran,

excepting a few routines written in MACRO-11 language. CASAM is implemented on the Romanian family of minicomputers I~EPE~ENT and CORAL, and on a DEC PDF-11/34

1 II

and G.E. Stewart: An al orithm for computing reducing Bavely, C.A., subspaces by block diagonalieation, a4 AM J. Numer. Anal. l&1979), 359-367 LINPACK Dongarra I J.J., J.R.Bunch, C.B.Moler, and S.W.%ewart: User's Guade, SIAM, Phyladelphia 1979. Emami-Naeini, A., end P. Van Dooren: Computation of zeros of linear mult ivariable systems. Aut omet ica. 16 (19821, 415-430. Forsythe, G.E., M.A. Malcolm, and Cz, Moler: Computer methoda for mathematical computations. Prentice-Hall, Englewood CliPfs,1977. Garbow, B,S., J.M,Boyle, J.J.Dongarra, and C.B.Moler: Metrlx eigensystem routines - EISPACK guide extension, Lect. Motes in Comp. jcie., ~01.51, springer Verlag, Berlin 1977. Hindmarsh, A.C.: Lsrge ordinary differantial equation systems and software. IEEE Control System4 Magazine. 2 (1982) 4, 24-30, Moore, B.C,: Principal component analysis-in linear systems: controllability, observability end model reduction. IEEE Trens. Autom. Control. AC-26 (1981) 17-32. Smith, B.T.,.M.Boyle, J.J.Dongarra, B,S.Gerbow, Y.Ikebe, V.C. Klema, and C.B. Moler: Matrix elgensystem routines - EISPACK guide, Lect. Notes in Comp. &ie., ~01.6, Springer Verlag, Berlin, 1976. Van Dooren, P.: The generalized eigenstructure problem in linear system theory. IEEE Trans. Autom. Control. AC-26 (1981), 111-129. Van Loan, C.F.: Computing integrals involvingmatrix exponentiala. IEEE Trans. Autom. Control. AC-2 (1978), 395-404. Verge, A.: Numerically steble ---+ a gorithm for standard controllability form determination. Electronics Letters. u 119811, 74-75. Verga, A., and V.Sima: A numerically stable algorithm for transfer function matrix evaluation. Tnt. J.Control. 22 (19811, 1123-133. Varga, A. end V.;ima: BIMAS - A basic mathematical package for cornputer aided analysis and design. Preprints of the IFAC 9th JYorld Congress, Budapest, 2-6 June, 1984. Varga, A., and A.Ravidovi~iu: BIMASC - A package of Fortren subprograms for analysis, design and simulation of control systems. Preprints of 3rd LFAC Symp.on CAD in Control and Engineering Systens, Copenhagen, July 31 - August 2, 1985. Varga, A., V.Sima, end C.V.Verga: On numerical eimulation of linesr continuous control systems. Preprints of SI~TION'83 Symp., Prague, June, 1983. Varga, A.: Transfer functions eveluation of generalized state-space models. Report ICI, TR-05.83, 1983. l

2 [.I

[I3 [I4 5 Cl c 61 [I7 [I8 [79

E21 17 [I 14 Cl 15 [I 16 [I

system.