Variable-structure systems and system zerosO.M.E. El-Ghezawi, B.Sc, M.Eng., Ph.D., S.A. Billings, B.Eng.,Ph.D., M.lnst.M.C., A.F.I.M.A.,C.Eng., M.I.E.E.,

and A.S.I. Zinober, M.Sc. (Eng.), Ph.D., A.F.I.M.A., C.Eng., M.I.E.E.

Indexing terms: Multivariable control systems, Poles and zeros

Abstract: The analysis of variable-structure systems in the sliding mode yields the concept of equivalentcontrol, which leads naturally to a new method for determining the zeros and zero directions of squarelinear multivariable systems. The analysis presented is conceptually easy and computationally attractive.

1 Introduction

Numerous methods have been proposed for calculating systemzeros. The concepts underlying such calculations range fromthe use of high-gain output feedback [1,2] and generalisedeigenvalue QZ algorithms [3,4] to geometric formulations [5,6] and the use of generalised inverses [7]. Other algorithmsobtain the zeros as the poles of the system minimal-orderright or left inverse [8] or by pole-zero cancellation techniques[9].

In the present study the relationship between variable-structure systems (VSS) and the zeros of the square lineartime-invariant multivariable system S(A,B, C)

x = Ax+Bu

y = Cx

xSRn

y<ERm

ueRr

m<n

(1)

(2)

is investigated. It is assumed throughout that B and C havefull rank and \CB\ =£ 0. (The case where CB is singular can beconsidered with suitable modifications and will be reportedelsewhere.)

A new method of computing the zeros of S(A, B, C) isderived by considering the theory of variable-structure systems.It is shown that the system zeros are the eigenvalues of areduced-order matrix which arises naturally in variable-structuresystems design. The algorithm is computationally simple andyields insight into the operation of variable-structure systemsin the sliding mode. A singular-perturbation analysis [2] ofhigh-gain linear multivariable systems has revealed the relation-ships between fast and slow modes, multivariable transmissionzeros, cheap control problems and VSS systems in the slidingmode.

This paper begins with a brief overview of variable-structuresystems design. The application of this theory to the com-putation of system zeros and zero directions is then investigated,and the physical interpretation of the method is discussed.Additional properties of the proposed algorithm are investigatedby decomposing the state space, and worked examples areincluded to illustrate the validity of the method.

2 Variable-structure systems in the sliding mode

Variable-structure systems [10, 11] are characterised by adiscontinuous control action which changes structure uponreaching a set of switching surfaces. The control has the form

U; =st(x) > 0

st(x) < 0(3)

Paper 2289D, received 22nd March 1982Dr. El-Ghezawi and Dr. Billings are with the Department of ControlEngineering and Dr. Zinober is with the Department of Applied &Computational Mathematics, University of Sheffield, Sheffield S102TN, England

where ut is the ith component of u and st(x) is the ith of them switching hyperplanes which satisfy

s(x) = Cx = 0 s(x)ER' (4)

The above system with discontinuous control is termed avariable-structure system (VSS) since the effect of theswitching hyperplanes is to alter the feedback structure of thesystem.

Sliding motion occurs if, at a point on a switching surfacest(x) = 0, the directions of motion along the state trajectorieson either side of the surface are not away from the switchingsurface. The state then slides and remains for some finite timeon the surface s{(x) = 0 [10, 11]. The conditions for slidingmotion to occur on the rth hyperplane may be stated innumerous ways. We need

0 and lim s: > 0s; -> 0 '

(5)

or equivalently

(6)

in the neighbourhood of st(x) = 0. In the sliding mode the sys-tem satisfies the equations

Si(x) = 0 and Si(x) = 0 (7)

and the system has invariance properties, yielding motionwhich is independent of certain system parameters and dis-turbances. Thus variable-structure systems are usefully em-ployed in systems with uncertain and time-varying parameters.

Consider the behaviour of the system dynamics duringsliding when the sliding mode exists on all the hyperplanes,assuming that the nonunique control u has been suitablychosen. During sliding, eqn. 4 and its derivative

i = Cx = 0 (8)

hold, and the equations governing the system dynamics maybe obtained by substituting an equivalent control ueq for theoriginal control w. From eqns. 1 and 8 the linear control is

'eq = -(CB)~l CAx (9)

and substituting in eqn. 1 yields the equations governing thesystem dynamics in the sliding mode:

x = [I-B(CB)~lC\ Ax = Aeqx (10)

Note that during sliding m state variables can be expressed interms of the remaining (n —m) state variables from eqn. 4. Thisallows a reduction in the order of the system matrix (seeSection 4).

3 System zeros

The use of variable-structure systems theory in calculating thesystem zeros is motivated by the observation that variable-

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structure systems in the sliding mode are very closely relatedto the output zeroing problem.

3.1 Output zeroing problemMacFarlane and Karcanias [12] state that a necessary andsufficient condition for an input

«(0 = *exp(z0 l (0 (11)

to yield rectilinear motion in the state space of the form

x(t) = x0 exp(zf)l(0 (12)

such that

y(t) = 0 forr>0 (13)

is that

hi-A -B

C 0= 0 = P{z) (14)

where z is a system zero, x0 is the related state zero direction,g is the input zero direction and 1(?) denotes the Heaviside unitstep function. At the complex frequency s=z, the state(x0) and input (g) zero directions satisfy

GN{P(z)}

where TV {P{z)} is the null space or kernel of P{z).

(15)

3.2 Zero calculation using VSS theoryConsidering the switching functions s to be the system outputsy, VSS and the output zeroing problem reduce to the selectionof a control u and a state vector JC such that the output y{t) =0 for t > 0. Calculation of the system zeros using VSS theoryexploits the fact that, i f>(0 = 0 for t > 0, then y{t) = 0 fort > 0. The algorithm which consists of determining the eigen-values of the matrix Aeq which arises when the feedbackcontrol yieldingj;(0 = 0 is applied to S(A, B, C) is summarisedas follows:

(a) Calculate u using y = Cx = 0. This yields

C(Ax+Bu) = 0

and

u = -{CB)'xCAx (16)

which is the equivalent control ueq of VSS theory.

(b) Substitute in eqn. 1 to yield

x = [I-B(CB)~lC]Ax = AeQx (17)

(c) Determine the eigenvalues and eigenvectors of Aeq.(d) Any eigenvector xl

0 satisfying Cxl0 - 0 is a state zero

direction and the corresponding eigenvalue is a system zero.(e) The corresponding input zero directions are given bySi = = -(CBylCAx\> (18)

In practice we need consider only the (n — m) eigenvaluesX,- G a(Aeq) — {0}m in steps (c) and (d). This becomes clear inSection 4 (see eqn. 33), since it is evident that only the eigen-vectors associated with (n—m) eigenvalues X,- of Aeq lie inthe null space of C and satisfy Cxl

0 = 0. The (n - m) eigenvaluesX,- are not necessarily nonzero-valued.

Consider an example to illustrate the procedure.

Example 1

x =

0 1

0 0

- 6 - 1 1

0

1

-6

x +

0

0

1

y = [20 9 1]JC

The eigenvalues of

leg = [I-B(CBylC]A =

1

0

- 2 0

0

1

- 9

are 0 , - 4 , - 5 with corresponding eigenvectors

l l ' 1 [ 1

ex = 0 e2 = — 4 e3 = — 5

OJ L 16 |_ 25

Since Cex =£ 0, Ce2 = 0 and Ce3 = 0, the zeros of the systemare —4 and — 5 with state zero directions

1

- 4

16

and

1

- 5

25

The respective (scalar) input zero directions are, from eqn.18,

g

and

= [6 - 9 - 3 ]

g= [6 - 9 - 3 ]

- 4

16

" 1

- 5

25

= — 6

= - 2 4

Note that the zero directions are obtained without resortingto the higher-dimensional (n + m) system matrix P(s).

3.3 Physical interpretationThe (VSS) equivalent control is such that it assigns some ofthe closed.-loop eigenvalues and the corresponding eigenvectorsto coincide with the system zeros and state zero directions,thus driving the system to be unobservable. Consider theobservability matrix

c\(r A \ = \C CA CA2 CAn~11 TJ (19)

Since

CAeq = C[I-B(CBYlC}A = 0 (20)

it follows that

CAleq = 0 / = 1 , 2 , 3 , . . . , / i - 1

and

rank [0(C,Aeq)] = rank (C) = m<n

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The action of the equivalent control ueq is therefore to drivethe system to be unobservable.

4 Decomposition of the state spaceThe calculation of the system zeros using the proposed methodrelies on pole-zero cancellation through appropriate statefeedback using the equivalent control ueQ. The zeros of thesystem are given by the eigenvalues of a certain matrix as-sociated with the (n — m)-dimensional unobservable subspace.To find this matrix an unobservability decomposition isemployed [13].

Introduce the similarity transformation

x = Tx

where

mT=\C-P\}n-m

(21)

(22)

C is the output matrix which spans the observable subspace,and P is a matrix chosen to ensure that T is nonsingular.Therefore

x = TAeqT~xx

y = CT1 x

Define the partitioned inverse of T as

T~l = [V: W ]}n

(23)

(24)

(25)

and recall that a generalised inverse of a (k x /)-matrix S is amatrix S8 satisfying [14]

sses = s (26)

Using these definitions and exploiting the identity 7T * = /„ ,it can be readily shown that

PW = / „ - ,

PV = 0

CW = 0

(27)

(28)

(29)

(30)

Therefore V and W can be taken to be generalised inverses ofC and P such that the conditions given in eqns. 29 and 30 aresatisfied, i.e. PC8 = 0 and CP8 = 0. From eqns. 17 and 23,

x = leq [C8 P8] X =q*

qi

and

y = C[C8\Pe]x

From eqn. 20, CAeq = 0 and therefore

0m 01x =

l* i

y = [im:0]x

(31)

(32)

(33)

(34)

This is the standard unobservability decomposition, and thezeros are therefore given by the (n — rri) eigenvalues ofPAeqP

8.The calculation of the inverse of T yields V and W. How-

ever, if the computation of the matrix inverse is to be avoided,the matrix P should be chosen such that eqns. 29 and 30 are

satisfied. Furthermore, since P8 E.N{C}, a possible choice forP8 is M. P is then equal to M8, subject to the conditionM8M = Im and the zeros of the system are therefore equal tothe eigenvalues of M8AeQ M, where M is a basis matrix for thenull space of C.

Since the state eigenvectors xl0 (or state zero directions) lie

in the null space of C, they can be expressed as

x'o = Mat 0 i = \,2,...,n-m (35)

Thus

AeqXlQ = AeqMcti = ZiMttj

where z{ is the zero associated with XQ and

M8AeqM(xi - ZiM8M<Xi = zixi (36)

Therefore a ,- is an eigenvector of M8AeqM corresponding to thezeroz,-.

To calculate the state zero directions xl0, the eigenvectors

Oj of the (n — m)th-order matrix M8AeqM should be deter-mined and substituted in eqn. 35. The input zero directionsg( are given by replacingx0 byMo,- in eqn. 18 to yield

t = -(CB)-1CAMai (37)

The algorithm presented above resembles that of the NAMmethod of MacFarlane and Kouvaritakis [5]. In both methodsthe zeros are determined as the eigenvalues of an (n — m)-dimensional matrix. It can be easily shown that the matrixNe =M8[I-B(CB)~1C] qualifies for the matrix N in theNAM algorithm. This is because

NeB = 0

and

NeM = In.m

where the matrix M is the same in both methods. However,the approach proposed in this paper offers the advantage ofcalculating the state and input zero directions without resortingto the null space of the (n + m)-dimensional system matrixP(s). Our technique involves only certain matrix multipli-cations in the calculation of the zero directions.

Example 2Consider the system [5]

A =

- 1 1 3

0 - 1 - 1

0 - 1 - 3

0 3 - 1

2

- 1

- 1

B =

1

1

1

2

0

2

1

2

c =1 - 1 3 0

0 - 1 - 3 2

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From eqn. 10, qtA =

AeQ = -

- 2 - 6 6 4

1 12 42 - 2 9

1 6 12 - 1 1

2 15 39 - 3 1

Also, we can choose M as a basis for TV {C}:

P8 = M =

From eqn. 36, the zeros are given by the eigenvalues of

- 0 . 5 - 1 . 5

0.5 - 2 . 5

2

2

0

1

- 6

0

2

3

M8AegM =

Therefore the system has two zeros at — 1 and — 2. Theeigenvectors a,- of M8AeQM are

and

The state zero directions are given by xl0 = Mcc,- (eqn. 35) and

hence

xh = [0 6 2 6]T

and

xl = [ - 4 2 2 4 ] T

The input zero directions are (eqn. 37)

0

- 2 1 " 1 0and £2 =

5 Classification of the zeros

Since CB is nonsingular, the system S(A, B, C) has the maxi-mum number of zeros, i.e.n—m, and the zeros are the systemzeros [12]. Since the system is square, all the input, outputand input-output decoupling zeros are invariant zeros, andtherefore the invariant zeros are the system zeros. The set oftransmission zeros ZT is the set of invariant zeros Zj minusthe set of decoupling zeros ZD. Thus

zT = z 7 — zD

An output decoupling zero satisfies [12]

and

t = 0

(38)

(39)

(40)

where pt is a nonzero eigenvector of A associated with the zerozt. Eqn. 14 implies that gt = 0, i.e. the input zero direction iszero.

An input decoupling zero satisfies [12]

and

QtB = 0

(41)

(42)

where qt is a nonzero left eigenvector of A associated with thezero z,-. If eqns. 39—42 are satisfied, we have an input-outputdecoupling zero. Alternatively, the output decoupling zeroscan be determined by noting that for gt = 0 to hold we needfrom eqn. 18 gt = — (CB)~1CAxl

0 = 0, and, since CB is non-singular, CAX1Q = 0. The transmission zeros are determinedby establishing which zeros do not satisfy eqns. 39 and 40 oreqns. 41 and 42.

ExampleConsider

A =

n

3the system with

0

0

- 6

0

0

C = [10 7

From eqn. 17

0

0

(D

1

0

- 1 1 -

1]

1

0

- 1 0 -

0

1

6

0

1

7

with eigenvalues {0, — 2, — 5}. The zeros are {0, — 2, — 5} —{0} = {— 2, —5}, and the corresponding state zero directionsare

* 6 =

1

- 2

4

and2 -

1

- 5

25

Since — 5 is not an eigenvalue of A, the zero z2 = — 5 is atransmission zero. Zj = — 2 is an eigenvalue of A. SinceCAXQ = 0, the input zero direction corresponding to zl = — 2is zero. So — 2 is an output decoupling zero. The outputdecoupling zero Zj = — 2 is not an output decoupling zerosince the left eigenvector qx = [3 4 1] does not satisfy

. 42).

6 Computational aspects

Two techniques for determining zeros have been described. InSection 3.2 the set of eigenvalues of the nth-order matrixAeq, i.e. o(Aeq), needs to be determined. The n—m zeros aregiven by o(Aeq) — {0}m. In Section 4 the zeros are obtained asthe eigenvalues of the (n — w)th-order matrix MeAeqM>

where M is a basis for the null space of C. The choice ofmethod depends upon the values of m and n and the ease ofthe calculation of the basis M. The method of Section 3.2 isto be favoured for n > m.

We have assumed throughout the system order to be n andthat rank B, rank C and rank (CB) are given. Otherwise, theymay be determined using suitable algorithms [15]. Ill con-ditioning of (CB), Aeq and kernel C may cause difficulties. A

IEEPROC, Vol. 130, Pt. D, No. 1, JANUARY 1983

highly accurate method for determining the zeros such as theQZ-type algorithms for the solution of nonsymmetric gener-alised eigenvalue problems [3, 4] should then be used.

7 Conclusions

It has been shown that the study of a reduced-order systemwhich arises in the design of variable-structure systems leadsto a new method of calculating multivariable system zeros.The resulting algorithm is computationally fast and simpleto implement. It has particular advantages when determiningthe state and input zero directions since the calculation ofthe (n + m)-dimensional null space of the system matrixP(s) [12] is avoided.

8 Acknowledgment

The authors wish to acknowledge the guidance of Dr. D.H.Owens in the preparation of this paper.

9 References

1 DAVISON, E.J., and WANG, S.H.: 'Properties and calculation oftransmission zeros of linear multivariable systems', Automatica,1974, 10, pp. 643-658

2 YOUNG, K.D., KOKOTOVIC, P.V., and UTKIN, V.I.: 'A singularperturbation analysis of high-gain feedback systems', IEEE Trans.,1977, AC-22, pp. 931-937

3 LAUB, A.J., and MOORE, B.C.: 'Calculation of transmission zerosusing QZ techniques', Automatica, 1978, 14, pp. 557—566

4 PORTER, B.: 'Computation of the zeros of linear multivariablesystems',Int. J. Syst. Set, 1979, 10, pp. 1427-1432

5 KOUVARITAKIS, B., and MacFARLANE, A.G.J.: 'Geometricapproach to the analysis and synthesis of system zeros. Parts 1 and2', Int. J. Control, 1976, 23, pp. 149-182

6 OWENS, D.H.: 'Invariant zeros of multivariable systems: A geometricanalysis', ibid., 1977, 26, pp. 537-548

7 LOVASS-NAGY, V., and POWERS, D.L.: 'Determination of zerosand zero directions of linear time invariant systems by matrixgeneralized inverses', ibid., 1980, 31, pp. 1161-1170

8 PATEL, R.V.: 'Multivariable zeros and theix properties', inFALLSIDE, F. (Ed.) 'Control system design by pole-zero assign-ment' (Academic Press, 1977), pp. 145-165

9 WOLOVICH, W.A.: 'Multivariable system zeros', in FALLSIDE, F.(Ed): 'Control system design by pole-zero assignment', (AcademicPress, 1977), pp. 226-236

10 UTKIN, V.I.: 'Variable structure systems with sliding mode: Asurvey', IEEE Trans., 1977, AC-22, pp. 212-222

11 UTKIN, V.I.: 'Sliding modes and their application in variablestructure systems' (MIR, Moscow) (English translation, 1978)

12 MacFARLANE, A.G.J., and KARCANIAS, N.: 'Poles and zeros oflinear multivariable systems: A survey of the algebraic, geometricand complex variable theory', Int. J. Control, 1976, 24, pp. 33-74

13 KUDVA, P., and GOURISHANKAR, V.: 'On the stability problemof multivariable model following systems', ibid., 1976, 24, pp. 801 —805

14 GRAYBILL, F.A.: 'Introduction to matrices with applications instatistics' (Wadsworth Publishing Company, 1969), chap. 6

15 DAVISON, E.J., GESING, W., and WANG, S.H.: 'An algorithm fordetermining the minimal realization of a linear time-invariantsystem and determining if a system is stabilizable-detectable', IEEETYans., 1978, AC-23, pp. 1048-1054

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