Singular Systems: Their Origins, General Features, and Non Numerical Solution

Copyright © IFAC System Structure and Control. Nantes. France. 1995

SINGULAR SYSTEMS: THEIR ORIGINS, GENERAL FEATURES, AND NON NUMERICAL SOLUTION XAVIER GRACIA, MIGUEL C. MUNOZ-LECANDA, NARCISO ROMAN-ROY Departament de ,'v[atemci.tica A plicada i Telemci.tica, Universitat Politecnica de Catal1Lnya, Campus Nord edifici CS , 08071 Barcelona, Catalonia . Spain

Abstract. A brief review on singular systems of differential equations is presented. In particular. the problems they pose and their geometric features, including a geometric method for obtaining a non-numerical solution. are analized. The case of linear singular systems is especially studied, and a particular example is presented. Key words. Singular differential equations. linear singular systems, stabilization algorithms.

1

INTRODUCTION

used in several applications.

Roughly speaking, a singular system is a system of differential equations where the maximum order derivatives are not solved . Since any system can be made of first order by adjoining new variables, we can assume that our system is of first order. The general expression of such a system is F(t, x, x) = 0, where t is the independent variable, usually understood as the time, and x is the state vector of the physical system. From now on we consider only autonomous systems, that is, systems where the variable t does not appear explicitly in the equations.

Besides control theory, there are many applications of singular differential equations to other fields like theoretical physics, cicuit theory, chemical engineering, industrial engineering, econometry ...

2

PROBLEMS ARISING WHEN SOLVING SINGULAR SYSTEMS

In these systems we have not a vector field representing the system and there appear some problems when trying to integrate it analytically or numerically. Some of these problems, which are different from the regular case, are the following :

As it will be shown later, even in a local study we are led to work on submanifolds of the initial phase space. So we will suppose from the beginning that the phase space of the system is a differentiable manifold M whose dimension, rn, is the number of free variables used to describe the system. Of course it may be assumed that M is embedded in an suitable euclidean space as a submanifold.

Compatibility In general, the system of differential equations may not be compatible at all the points of the initial phase space M of the system. This means that not any set of values of the variables is admissible as initial conditions for the differential equation. In order to solve the system, the first step of the procedure must be to identify the set of admissible initial conditions. This problem is not solved directly but in an algorithmic way. For instance the differential equation may imply some relations .pa(x) = 0 between the variables x; these relations, called primary constraints in the physical literature , define a subset Ml C M, the primary constraint submanifold. Hence, the first question consists in finding these primary constraints.

'With these conventions , an autonomous differential equation is no more than a submanifold D of T(AI) , the tangent bundle to M, whose elements are the points x and their velocities x with respect to t . This sub manifold is locally defined by a relation F(x, x) = O. A simple example of a singular system comes from a linear control system x = Ax + Bu , where u are the inputs . Suppose that a feedback of the type u = -CX is introduced. Then we obtain the system Ex = Ax, where E = I + BC; this is a singular system if the matrix T is singular . In fact, this type of systems, those of the form Ex = Ax where E is a singular matrix, are often studied and

Nevertheless, the problem does not finish at this point, since another consistency condition is required . In fact, the evolution of the variables is restricted to abride inside the primary constraint 31

submanifold NIl, that is. for every value of the evolution parameter t, the functions x(t) must take values in that set. Hence, the problem is not solved until a set of admissible initial conditions verifying this consistency condition is found. This procedure is repeated until the final constraint submanifold M f is obtained; on this submanifold there exist tangent vector fields representing the system. This is the key idea for constructing algorithms to solve the problem: the set of constraints is obtained by iteration of the tangency condition.

singular systems of differential equations. There are many articles concerning this problem, but, we are not going to deal with it -see for instance [Hairer et al (1989)] [Hairer and Wanner (1991)] for a detailed account. 3

GEOMETRlC ASPECTS OF SINGULAR DIFFERENTIAL EQUATIONS

As we have told above an autonomous first order differential equation is a relation F(x, x) between several functions x in one variable t and their first derivatives x. In geometric terms, a path / : I ~ M is a solution if

Uniqueness Once these problems have been solved, the initial singular system of differential equations is reduced to another one which is compatible and has consistent solution on the points of the final constraint submanifold. Then there are two possibilities:

7(1) c D, where D is a submanifold of T(M) decribing the differential equation. Singular differential equations from this most general point of view have been seldom studied in the literature.

1. If the singularity of the initial system of differential equations arises from an "inadequate" choice of the variables (that is, the initial state space is too large in order to describe the real degrees of freedom of the problem), then in the final constraint submanifold the system has a unique solution.

For instance, in [Menzio and Tulczijew (1978)] this general framework is applied to the hamiltonian dynamics of singular lagrangian systems . More recently the article [Marmo et al (1992)] studies symmetries and constants of motion for these singular equations, and the preprint [Mendella et al (1994)] obtains some conditions of integrability; in this paper "integrability" means the possibility of finding a solution through any point of D .

2. However in some physical problems, the singularity of the system is a consequence of the existence of a certain kind of internal symmetries (the gauge symmetry in theoretical physics terminology). Then the final system of equations may be undetermined; that is , the solution of the system may not be unique. This means that, for every point in the final constraint submanifold which is taken as an initial condition, the evolution of the system is not determined because a multiplicity of integral curves (of different vector fields solution) pass trough it. A reduction procedure can be used to remove this ambiguity: different points of NI] that can be reached through different solutions beginning at the same initial condition must be identified. This quotient is the reduced phase space of the system.

In the same way, but taking as M an euclidean space, [Rabier and Rheinboldt (1994)] studies the existence of solutions of the same problem, and in a certain sense it gives an algorithm to find these solutions under some particular conditions , namely, when D is what the authors call a "completely reducible 7r-submanifold" of T(Rn); the aim of this article is to give a geometric treatment of the authors' previous works, especially [Rabier and Rheinboldt (1991)]-see also [Rheinboldt (1984)] . In the literature the most frequently considered singular differential equations are of a special type: they turn out to combine some restrictions on the base M and some linear transformations (usually singular) of the velocities in T( M). Moreover, the former can be absorbed in the latter, and so the local expression of these equations is

States The physical states of the systems are identified with the points of the reduced phase space . This coincides with the final constraint sub manifold when the vector field solution of the system is unique. Singular control systems Besides these problems, other questions are also of interest when dealing with singular control systems; namely:

A(x)x = b(x),

where A(x) is a matrix, usually singular. We call them linearly singular equations.

reachability, control/ability, stabilizability, observability, realizability and optimality. All these con-

cepts, which are well established for regular systems in control theory, must be reconsidered for singular systems and, as far as we know, they are not completely solved.

The geometric study of these equations has been developed independently in the areas of Theoretical Physics and of Systems Theory. In Theoretical Physics the initial problem [Dirac (1950)] was to obtain a hamiltonian description for singular lagrangian systems; these are lagrangian

Other problems There are also other problems mainly concerning with the numerical solution of 32

For instance, in [Takens (1976)] a "constrained differential equation" is a set of data including a linear restriction on the velocities as well as some constraints on the base manifold obtained from a certain potential function; the existence of discontinuous solutions of this problem is analized . The paper also includes a study of singularities of these constrained equations, and some applications to circuits and singular lagrangians. A related paper is [Shankar Sastry and Desoer (1981)] .

systems for which the hessian matrix of the lagrangian function with respect to the velocities is singular. Some progress was done in the geometrization of this problem -see for instance [Sniatycki (1974)] [Lichnerowicz (1975)]' but the solution did not arrive until [Gotay et at (1978)]. In this article, an algorithm is presented to study the existence of solutions of a "presymplectic system"

ix...;

= a,

where the unknown is a vector field X in a manifold M, w is a presymplectic form and a is a 1form; for instance, one may have M a sub manifold of T*(Q), w the pullback of the canonical symplectic form of Q to }vI and a = dH, where H is a hamiltonian function. :\otice that, since the transformation X 1--+ ixw is not an isomorphism, there is no guarantee of neither existence nor uniqueness of the solutions . In fact, as the question of global existence has negative answer, the authors look for a submanifold of M where the equation is satisfied.

The same geometric framework for constrained equations is presented in [Haggman and Bryant (1984)], with some simplifications. The problem is to find a curve c: I -+ E such that c(I) C I:, T(p)oc=x oc where p: E -+ B is a smooth map, X a vector field along p, and I: is a submanifold of E with the same dimension as M . A particular class of these equations is studied, and the results are applied to circuit theory. Notice that this geometric framework can be understood as a linearly singular equation .

The same ideas are applied to the search of solutions in the lagrangian formalism for the singular case; then the presymplectic equation must be supplemented with a second-order condition [Gotay and Nester (1980)]. In this case a variation of the algorithm allows to include the secondorder condition [Muiioz-Lecanda and Roman-Roy (1992)]. Another way to treat this problem is to work in the manifold .\I T*(Q) xQ T(Q) [Skinner (1983)] [Skinner and Rusk (1983)].

We finally mention the paper [Chua and Oka (1988)], which uses the concept of "generalized vector field", namely, a vector bundle endomorphism A : T( M) -+ T( M) and a vector field v in M. The equation of motion is then

Ao-Y=vo, . Although the solution of this equation is not discussed , the first constraint submanifold is presented . The aim of the paper is to classify the normal forms for generalized vector fields.

=

A more general framework is presented in [Gracia and Pons (1991)] and with more details in [Gracia and Pons (1992)]: a "linearly singular system" IS an expressIOn

4

THE CASE OF LINEAR SINGULAR SYSTEMS

One of the most studied kind of singular systems are the so called linear singular systems

Ao~; =ao"

where the unknown is a path, in M, a is a section of a vector bundle F -+ M , and A is a vector bundle morphism T(JI) -+ F . This problem is also formulated in terms of vector fields. In these articles the algorithm for presymplectic systems is generalized to linearly singular systems -see section 5. This framework includes applications to other problems like singular lagrangian formalism and higher order lagrangians -see for instance [Gracia et at (1991)].

Ex = Ax + Bu, y = Cx + Du , where E, A, B, C and D are constant matrices: E singular . There are different approaches and problems about these systems.

A less general framework, but still including the presymplectic systems : is presented in [MunozLecanda and Roman-Roy (1995)], which correponds to a linearly singular system with F = T*(M) ; besides extending the presymplectic algorithm, it includes a study of the symmetries of the equation .

For instance, in [Campbell (1980)] the system Ax = Bx with A and B constant matrices and A singular is studied. Different problems giving rise to this kind of systems are presented, problems coming from very different areas: optimal control , constrained control, electric circuits and so on. The approach therein is to convert the system into another one with A depending on a parameter c such that A(O) is singular but A(c) is regular for c ::j:. O. In fact these are the so called perturbed problems and they come from boundary layer problems.

On the other hand, some problems in circuit theory led several researchers to consider singular differential equations.

One of the initial problems has been to identify the real state space for a linear singular system . Using operational methods, inconsistent initial conditions for the system Ex = Ax + Bu , y = Cx: 33

where the matrix E is singular, are studied in [Verghese et al. (1981)]. Following the ideas of this work, the notions of observability and controllability are defined for such systems in [Cobb (1984)].

Im A ~ F are vector subbundles, which amounts to saying that A has locally constant rank. The equation A,; . X (x) = 0-( x) for the unknown vector X (x) can be solved only at the points x E M such that the compatibility condition o-(x) E ImA,; (2) holds. This is equivalent to saying that 0-( x) is orthogonal (in the sense of duality) to the kernel of the transposed morphism,

Another problem is studied in [Campbell (1989)]. Consider the regular system x Ax + Bu, y ex + Du, and fix the output y. Do there exist any inputs u giving this fixed output? There it is proved that the solution of this problem is equivalent to solve a singular system on the variables

=

=

(KertA,;,o-(x)) = O. (3) Let lvh = {x E M I o-(x) E ImA,;}. If (sa)r
(x, u) . The methods in these articles are operational. Also wide use is made of matrix pencils. General properties of the set of solutions are not provided and these methods arrive to some inconsistent situations when applied to obtain numerical solutions of some systems .

a = (sa, 0-) . (4) As usual in the theory of constrained systems, one assumes that M1 is a submanifold, the primary

One of the most complete references is [Kaczorek (1992 , 1993)]. In these two volumes a complete study of singular linear systems is performed including the problems of pole-assignement and factor-assignement in closed-loop systems and the precise definitions of reach ability, controllability, observability and constructibility for such systems.

The compatibility condition (2) has lead to consider solutions X which satisfy the equation ofmotion only on M 1 , therefore one is interested only in the values of Xl = XI M1 . Since X (or Xd must be tangent to M 1 , the initial problem becomes the same problem for the subsystem defined by M 1 :

5

primary constraint functions

constraint submanifold.

GEOMETRlC SOLUTION OF SINGULAR EQUATIONS

Here we summarize the geometric algorithm for finding the solutions I of a linearly singular equation A 0 -y = 0- 0 I as presented in [Gracia and Pons (1991)] [Gracia and Pons (1992)]: T( Aif) --,-A~

I

where F1 = FI M1 , and ?Tl , Al and corresponding restrictions to M l .

F

/\/ 1.

are the

The procedure goes on similarly. The compatibility condition for this system yields a subset M 2 , also assumed to be a submanifold. . . In general, let Mo = M, Ai = AIT(Mi) and O-i = o-l Mi , considered as mappings into Fi = FI M , and define recursively ,

M

An alternative description of the dynamics can be given in terms of the equation of motion for a vector field : if N C M is a sub manifold contained in the motion set, and X is a vector field in M tangen t to N , then the integral curves of X contained in N are solutions of the equation of motion if and only if X satisfies 1 A 0 X ::::: 0-. (1)

M i + l := {x E Mi I o-i(X) E ImA i ,;}. (5) This sequence ends in the final constraint submanifold M j . Then o-j(x) E ImAj,; and therefore the eq uation A j 0 X J = 0- J for a vector field X j in Mj has solutions . Moreover, these solutions are not unique when Ker Aj :f 0: their multiplicity is described by the vector fields lying in Ker A j.

N

This algorithm may be presented explicitly in terms of the constraints: the constraints defining M2 are constructed from those defining M l , and so on .

In general this must be considered as an equation both for the submanifolds N C M where the motion can take place and the vector fields X tangent to N .

To begin an algorithm to solve the equation of motion, it is assumed that Ker A ~ T(M) and 1 The

0-1

6

AN EXAMPLE

In order to show the power of geometric methods in the study of singular systems , let us apply

notation::::: means equality at the points of N . N

34

them to a particular linear singular system with constant coefficients. This constancy condition is not relevant to apply the algorithm but we assume it for simplicity.

No arbitrary function can be determined , Ml is the final constraint manifold and the vector field solution of the system depends on two arbitrary functions.

Consider the system Ex = Bx with E and B square constant matrices, E singular. Following section 5, the data in this case are: M = R m (and so T(Al) = R m x R m ), F Rm x Rm , A(x, x) = (x , Ex) and u(x) = (x, Bx) .

=

=

Suppose that the defect of Eis 2, and let Ker lE = span{Ql , Q2}, Ker E = span{Xl ' X2}.

M'2 = {x E MI I
The application of the algorithm begins with the compatibility conditions, which are the linear forms q,(l)(X) = na(Bx), Q' = 1,2; and the primary constraint manifold is

Ml

= {x I
Q'

(3) Both functions
= 1,2}.

Then they define the submanifold Ml where the system is compatible and its general solution IS given by (6).

On the sub manifold Ml the vector fields solution of the system are given by

X=XO+gIXl+g2X2, where gl and g'2 are arbitrary functions.

=

We have a new constraint given by
(6)

The tangency condition yields two equations :

X1(
Observe that both the primary constraints and the general vector field solution can be obtained by application of Gauss elimination method.

( X1(
X2(
9

= _ (

.I~O(
This is a linear system for gl and g'2 . Let D be the matri..x of this system. We have three different cases to study:

Now different situations have to be considered according to the rank of the primary constraints.

(3a) rankeD)

(1) Both constraints
=2.

Then both functions can be determined by solving the linear system. The subset M1 is the final constraint manifold and the solution on M1 is unique.

In this case the final constraint manifold is already Ml R m and the system is undetermined . There are a family of vector fields solution depending on two arbitrary functions , gl and g2. This family is given by equation (6) .

=

(3b) rank ( D)

= 1.

In this case one of the arbitrary functions can be determined in terms of the other one , and in general there appears a new constraint that must vanish to ensure compatibility. If this constraint vanishes on M1 then M1 is the final constraint submanifold and the solution depends on an arbitrary function ; otherwise a new constraint
(2) The set of forms
(3c) rankeD)

= O.

In this case both functions XO(
(2a) One of the numbers X 1(
As a final comment, observe that all the submanifolds obtained are linear submanifolds , and that all the constraints are linear . All the systems that have been solved are linear and, in fact , by iterated application of the Gauss reduction method to

(2b) Both numbers X I (
G. Marmo, G . Mendella, W.M . Thlczijew (1992), "Symmetries and constant of the motion for dynamics in implicit form", Ann. Inst . H. Poincare A 57(2) 147-166. G. Mendella, G. Marmo, W .M. Thlczijew (1994) . "Integrability of implicit differential equations". preprint. M.R. Menzio, W .M. Tulczijew (1978), "Infinitesimal symplectic relations and generalized hamiltonian dynamics" , Ann. Inst . H. Poincare A 28 349-367. M.C. Muiioz-Lecanda, N. Roman-Roy (1992) , "Lagrangian theory for presymplectic systems" , Ann . Inst . H. Poincare A 57 27-45 . M.C. Muiioz-Lecanda, N. Roman-Roy (1995). "Linear singular differential systems : a geometrical approach" , preprint DMAT-UPC . P.J. Rabier, W .C. Rheinboldt (1991) , "A general existence and uniqueness theory for implicit differential-algebraic equations" , Differential Integral Equations 4 563-582. P.J . Rabier , W.C. Rheinboldt (1994), "A geometric treatment of implicit differential-algebraic equations" , J . Differential Equations 109 110146. W.C . Rheinboldt (1984) , "Differential-algebraic systems as differential equations on manifolds" , Math . Comp. 43(168) 473-482 . S. Shankar Sastry, C.A. Desoer (1981) , "Jump behavior of circuits and systems", IEEE Trans . Circuits Syst . CAS-28(12) 1109-1124. R. Skinner (1983) , "First order equations of motion for classical mechanics" , J. Math . Phys. 24 2581-2588 . R. Skinner , R. Rusk (1983), Generalized hamiltonian dynamics I: formulation on T*Q 0 TQ", J. Math. Phys. 24 2589-2594 . J . Sniatycki (1974), "Dirac brackets in geometric dynamics", Ann. Inst . H. Poincare A 20 365372 . F. Takens (1976) , "Constrained ~quations : astudy of implicit differential equations and their discontinuous solutions" , in Structural Stability, the Theory of Catastrophes and Application in the Sciences, P. Hilton ed ., Lectures Notes in Mathematics 525 Springer-Verlag, Berlin . G.C. Verghese, B.C. Levy, T . Kailath (1981) , "A. generalized state-space for singular systems", IEEE Trans . Automat . Contr. AC-26 811-831.

suitable systems the problem has been solved in an algorithmic way. If the defect of the matrix E is different from two then the different possibilities can be reduced to the above studied with minor modifications. 7

REFERENCES

S.L. Campbell (1980) , Singular systems of differential equations, Research Notes in Mathematics, Pitman, San Francisco . S.L. Campbell (1989) "Comments on functional reproducicility of time-variyng input-output systems", IEEE Trans . Automat . Contr. 34 922-924 . L.O . Chua, H. Oka (1988), "Normal forms for constrained nonlinear differential equations. Part I: theory" , IEEE Trans. Circuits Syst . 35 88190l. D. Cobb (1984), "Controllability, observability and dualty in singular systems", IEEE Trans. Automat . Contr. AC-29 1076- 1082. P.A.M. Dirac (1950), "Generalized hamiltonian dynamics", Can ad. J. Math. 2 129-148. M.J . Gotay, J .M . Nester (1980), "Presymplectic lagrangian systems II : the second order equation problem", Ann . Inst . H. Poincare A 32 113. M.J. Gotay, J .M . Nester, G . Hinds (1978) , "Presymplectic manifolds and t he Dirac-Bergmann theory of constraints", J. Math. Phys. 27 23882399 . X . Gracia, J .M . Pons (1991) , "Constrained systems : a unified geometric approach" , Int. J . Theor . Phys. 30 511-516. X . Gracia, J .M . Pons (1992) , "A generalized geometric framework for constrained systems", Diff. Geom. Appl. 2 223-247. X . Gracia, J .M. Pons, N. Roman-Roy (1991), "Higher order lagrangian systems: geometric structures , dynamics and constraints" , J. Math . Phys . 32 2744-2763. B.C. Haggman, P.P. Bryant (1984) , "Solutions of singular constrained differential equations: a generalization of circuits containing capacitoronly loops and inductor-only cutsets", IEEE Trans . Circuits Syst. CAS-31(12) 1015-1025. E . Hairer , C. Lubich , M. Roche (1989) , The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods , Lecture Notes in Mathematics 1409, Springer, Berlin. E. Hairer, G . Wanner (1991), Solving Ordinary Differential Equations Il, Springer, Berlin. T. Kaczorek (1992 and 1993) , Linear control systems, volumes I and I1 , Research Studies Press , Taunton . A. Lichnerowicz (197.5) , "Variete symplectique et dynamique associee a une sous-variete" , C.R. Acad. Sc . Paris 280(A) 523- 527. 36