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REFERENCESR. Inigo and J Morton, Simulation of the dynamics of an industrialrobot, IEEE Trans. Educ., vol. 34, no. 1, pp. 89-99, Feb. 1991.H. Pota, A prototype flexible robot arm-An interd isciplinary under-graduate project, IEEE Trans. Educ., vol. 35, no. 1, pp. 83-90, Feb.1992.W. Richard, An educational image processing/machine vision system,IEEE Trans. Educ., vol. 34, no. 1, pp. 129-133, Feb. 1991.C. Vibet, Elements to teach robot control design, IEEE Trans. Educ .,vol. 36, no. 1, pp. 45-51, Feb. 1993.N. Vira and E. Tunstel, Use of symbolic computation in roboticseducation,ZEEE Trans. Educ., vol. 35, no. 1, pp. 18-31, Feb. 1992.

MIMO Systems--Transfer Function to State-SpaceHemanshu Roy Pota

Abstruct-In this short note we present a very simple method, usefulfor classroom teaching, to get a minimal state-space representation withthe exception of systems where there is a pole-zero cancellation) froma transfer function matrix. The method is direct and does not involvethe intermediate step of obtaining nonminimal realization-which furtherrequires system reduction routines-and hence is suitable as a compactself-contained topic which has been hitherto neglected in undergraduatelinear control theory curriculum.

I. INTRODUCTIONThe undergraduate linear control theory curriculum can be d ividedinto two parts. The introductory part of the curriculum deals withclassical frequency domain material and the advanced part i s devotedto state-space theory. Single-input single-output (SISO) transfer func-tions are central to the teaching of the classical frequency domaintheory. Considerable part of the curriculum is devoted to derivingstate-space canonical forms from the SISO transfer functions [ 2 ] ,

[4] 5 ] The idea of minimal state order and the associated notionsof controllability and observability are discussed as a prelude tothe state-space theory. With the state-space theory, the fact that thesystem under study is SISO or multi-input multi-output (MIMO) isless relevant. Hence, the knowledge of the state-space theory enablesthe students to attack MIM O control problems, provided a state-spacerepresentation is available; this is the case in some practical situations,while in other situations transfer function matrices arise naturally.Many books on linear systems theory [4], [ 5 ]discuss the derivationof a state-space representation from a transfer function matrix.The discussion normally starts with obtaining either block observeror block controller state-space (nonminimal) realization, which isfollowed by algorithms to obtain minimal realization (both con-trollable and observable) from these nonminimal realizations. Thesealgorithms [5] are suitable only for a digital computer implementationand are more of an exercise in linear algebra than in controllerdesign. The consequence is that this material has to be left out ofclassroom teaching (even in standard text books such as [ l ] and[2], this material is omitted) and the students get little confidence

Manuscript received December 31 1992; revised June 9, 1993The author is with the Department of Electrical Engineering, UniversityCollege, University of New South Wales, Australian Defence Force Academy,Canberra ACT 2600, Australia.Publisher Item Identifier S 0018-9359(96)02136-X.

is attacking problems w here transfer function matrices arise naturallyfrom the physics of the problem [6]. Most of the practical controlsystems are multivariable systems, and the students should be taughtthe connection between a transfer function matrix and its state-spacerealization in a simple way. Students, when taught this connectionusing methods given in [4] and [ 5 ] , ose the physical feel behindthe definition of system state and presume that controllability andobservability are more important issues than the system representationitself.In this short note, a simple procedure is given that can be used toget a minimal state-space realization (with the exception of systemswhere there is a pole-zero cancellation) directly from a transferfunction matrix by hand calculation, without going through theintermediate nonminimal realization stage. The theory behind thisrealization is not new [3] and the main motivation for writing thisshort note arises from the fact that the author could not find asingle book where such a simple method is given. This work aimsat providing material in a suitable form to enable the inclusion ofthe realization of state-space from a transfer function matrix in anundergraduate curriculum, giving students a tool to have a first goat multivariable control design for problems such as those that arisein [6].

11. TRANSFER UN CTI ONO STATE-SPACELet a general multivariable system 1 inputs and p outputs) transferfunction matrix be given as follows:

Let us consider two examples before giving a general procedure toderive a state-space representation of the above MIMO system 1).A . Examples

The first example is of a multi-input single-output system and thesecond example is of a two-input two-output system. In the generalmethod, to be presented later in this paper, every multivariable systemI inputs and p outputs) is first reduced to p multi-input single-output systems. Each of the inputs to these p multi-input single-outputsystems is an output of a single-input single-output system. Theprocedure to ge t a state-space representation for a multi-input single-output system used in the first example can be used to get the finalstate-space representation. The secon d example is chosen so that thistwo-step procedure is made clear to the reader.

Example 1: The transfer function matrix for this example systemis1Y s )= [ s a ) . b ) s

The first step in writing state-space equations for a MIS0 system isto pull out all the common factors in the denominator polynomials.Notice that the factor s a ) s common to both the columns. Pullingthis term out, the above transfer function matrix can be written as

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As a first step in getting the state-space equations, the system left aftertaking the common factor s a s considered. Since there is nocommon term left between the two transfer functions, each of themcan be considered as a separate SISO system and the state variablesdefined as follows:Zll(S) ELM Z,,(S) A z+ds+e)s + b )211 bz11 = u l t )Lc1 = 211 2 2 = ai, 2 3 = 22121 = -bzl u l ( t )

321 2 1 d ezzl = u ~ t )kz = -dxZ ex3 u ~ ( t )5 3 = x2.

A .

Note that sZ,, s) = i z , t ) lower case is used for the variablerepresentation in the ti me -d om m and the upper case for the Laplacedomain). With the above definition of state variables and the inter-mediate output variables Z l l s ) and Z ~ l s ) ,he system equationscan be rewritten as follows:

The above system 8) can be seen as two separate two-input one-output systems with the following state-space representation:1 by1 = 211 Z21

2 3 = Yl23 = -b23 2 2

2 cy2 = Z l l Z212 4 = yz5 4 = C Z ~ ~1 2 .

Again collecting all the differential equations involving the statevariables, a compact matrix representation can be written as follows:

The state-space representation of the above system can be written byusing the standard observer canonical form of the SISO systems [ ](note from above that zl1 = x1 and 221 = x3)y a y = xi xz cz3 .

Let 2 4 6 t ) , then we can write54 = 21 2 2 ex3 ax4 .

Putting all the above equations together in a compact m atrix notation,we have- b 0 00 -d - e0 1 0

1 l c=

Example 2: The transfer function m atrix for the second examplesystem is given below.

To get a state-space representation of 7),we first consider one inputat a time, i.e., one column of the transfer function matrix Here, letus consider both the columns side by side and definea U Z S )

dZZld = uz(t)Zll(S) 5 -s+a) 221(s) = s+d)51 = --a21 u 1 t ) iz = -ax2 W t ),211 a511 = u i t )2 = Zll 2 2 = zz1A A

There is an order in how the various intermediate variables, Z s ) ,are defined. First, all of the common factors in the denominators ofcolumn 1 are pulled out and 2 s ) s defined as the intermediateoutput due to that common denominator; 221 s ) s similarly defined.With the above definitions of Z ~ i ( s )nd 221 s), we can write thesystem transfer function matrix (7) as

The state order four, which is as minimal as can be verified fromthe result in [3].B. General Procedure

In this section a procedure to derive a state-space representationof a general MIMO system 1) is given. Each term of the matrix,,s a ratio of rational polynomials such that the order ofd, , s) is greater than or equal to the order of nthat a transfer function matrix is available with all its constituentpolynomials given in pole-zero form. The procedure to get a minimalstate-space realization of system ( l) , directly frofunction matrix, can be stated as:

1) Set i = 1.2) Select the ish column of the transfer function matrix.3 ) Find the largest commo n factor in the denominator polynomialsof the ith column vector. Let this commo n factor be D z l s ) ,then define Z,, (s) m.4) Pull out the common factor D zl s ) rom the ith column andthen find a factor that is common to most of the remain-ing denominator polynomials, call it DZ2 s ) , nd then define5 ) Pull out D12 s ) from the column vector in Step 4) and thenrepeat Step 4) until there is no comm on factor between any ofthe two denominator polynomials of the column vector.6) If i < I, set i = i 1 and go to Step 2).7) After all the common terms of each column have been dulyconsidered, we can rewrite system (1) as

z22 s)z loD , z a ) .

wherea) W z 3 s )input variables for the modified system) ISequal to the last Z,, s) that was pulled oelement of the transfer function matrixIn Example 2, W ~ l ( s ) Zll(s).

h Z 3 s )re the terms of the denominator of the (i,g)thelement of the transfer function matrix that remain afterb)

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having pulled out all the common terms in Steps 3)-5).In Example 2, h l l s ) = 3 b . representation can be written by hand calculation-an ideal situationfor classroom teaching.8) Now the above system (11) can be seen as p independentsingle-input multi-output systems i = 1, . . p )

Note that the order of this multi-input single-output system [3]is the least commo n denominator of all the terms in (12)above. The obvious, mechanical procedure to wnte the state-space representation of the system (12) is to write the right-handside of (12) over its lowest com mon denominator an d then writean observer canonical [5] form of state-space representation.Although considerable savings in efforts to write the equationscan be achieved if common factors in h t3 s ) re first spottedand pulled out as was done in Example 1.

The procedure given above will result in a minimal realizationprovided there is no system pole-zero [7] cancellation; it shouldbe noted that it is important to consider the cases where thereis a pole-zero cancellation, but for a majority of cases [6] onecan proceed without considering it and adequate check can beperformed in due course. This method can be easily implemented ona computer, and in most cases where the denominator polynomials ofthe transfer function matrix has large common factors, a state-space

111. CONCLUSIONA direct method to write down state-space equations

(of minimal order) for a given transfer function matrix is presentedin this paper. The method is simple enough for hand implementationand does not divert the attention from the sim ple physical meaning ofthe system s tate to issues relating to controllability and observability.This method is ideal for classroom teaching and is just as useful inanalyzing many practical systems [ 6 ] .REFERENCES

[ I ] G. F. Franklin, J D. Powell, and A. Emami-Naeini, Feedback Controlof Dynamic Systems.[2 ] J J DAzzo and C. H. Houpis, Linear Control Systems Analysis andDesign.[3] H. H. Rosenbrock, Efficient computation of the least order for a giventransfer function matrix, Electron. Lett., vol. 3, pp. 413-414, 1967.[4] M. R. Chidambara and S . Ganapathy, An Znfroduction to Control ofDynamic Systems. Faridabad, India: Sehgal Educational Consultants,1979.[5] T. Kailath, Linear Systems. Englewood C liffs, NJ: Prentice-Hall, 1980.[6] H. R. Pota and T. E. Alberts, Multivariable transfer functions fora slewing piezoelectric laminate beam, ASME J. Dynamic Systems,Measurement, and Control, vol. 117, no. 3, pp. 352-359, Sept. 1995.[7] P. K. Sinha, Multivariable Control. New York Marcel Dekker, 1984.

Sydn ey, Australia: Addison-Wesley, 1986.Sydney, Australia: McGraw-Hill, 1981.