Study of Inverse Systems Using Matrix Methods

Inverse systems have many important applications in areas such as control theory, filtering and coding theory. The computation of the so-called zero initial state system inverses for linear time-invariant state-space systems is essentially equivalent to de- terming generalized inverses of the associated transfer-function matrices. For square and invertible systems, the computation of inverses can be performed by explicit formulas either in the standard state-space or in a descriptor system formulation. For non-square systems, explicit formulas can be employed only in the full-rank case to determine left or right inverses, provided that the system feedthrough matrix has full rank too. However, these direct formulas do not allow to arbitrarily choose the spurious poles which appear in the computed left or right inverses. In the more general case of systems with transfer-function matrices of arbitrary ranks, no explicit formulas can be used. In this paper we address the numerically reliable computation of generalized inverses of rational matrices by using orthogonal matrix pencil reduction techniques. We put particular emphasis on the computation of two classes of inverses: the weak generalized inverse, also known as the (1,2)-inverse, and the Moore-Penrose pseudoin- verse. The (1,2)-inverses can be computed using a numerically reliable approach based on the reduction of the system matrix pencil to a particular Kronecker-like form. The computation of the Moore-Penrose pseudoinverse is performed by employing full-rankfactorizations resulting from appropriate column/row compressions with all-pass factors. By combining the underlying computational techniques, other types of inverses of rational matrices can be computed as well. We present some numerical examples to illustrate our methods, and report on recently developed MATLAB software devised by the author to compute some generalizedinverses.

Consider a p x m rational matrixwith realcoefficients. Throughout the paper we assume that rankover rationals (i.e.,has rank r for almost all values of). The zeros ofare those values of(finite or infinite) where loses its maximal rank r. The poles ofare those valuesof(finite or infinite) where the elements ofbecome infinite. We callproper if it has only finite zeros (i.e.,is finite). In a system theoretical setting,can be interpreted as the transfer-functionmatrix either of a continuous-time system, if is thecomplex variable s appearing in the Laplace transform,or of a discrete-time system, ifis the complexvariableappearing in the Z-transform. Accordingly, we callstable if all its poles lie in the appropriatestability domain (i.e., the open left half complex planefor a continuous-time system or the interior of the unitcircle for a discrete-time system). Depending on the type of the system, the conjugate ofis the matrixdefined asfor a continuous-time system orfor a discrete-time system. We say thatis all-pass ifA stable all-pass matrix is called aninner matrix. Letbe an mxp real rational matrix. Consider thefollowing Moore-Penrose relations (see, e.g., (BenIsrael and Greville, 1976)):

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The well-known Moore-Penrose pseudoinverseis unique and satisfies all the four Moore-Penrose relations. Depending on the interpretation offorthe same rational matrixwe have two different pseudoinversesand for a continuous-time and a discrete-time system, respectively. In solving practical problems, the uniqueness of the Moore-Penrose pseudoinverse is rather a disadvantage, since no flexibility is provided, for example, in assigning the poles of the corresponding inverse system to desired locations. Therefore, often other types of generalized inverses are preferred. The weak generalized inverse satisfies only the first two Moore-Penrose conditions and is therefore called a (1,2)-inverse. Weak generalized inverses are useful in solving rational matrix equations or in computing inner-outer factorizations (Varga, 1998). In what follows we will use the notationfor all types of inverses satisfying one or more Moore-Penrose conditions. Of particular importance for applications are (Ben Israel and Greville, 1976; Campbell and Meyer, 1991): the (1)-inverse to solve systems of equations with rational matrices, the (1,3)-inverse to compute least-squares solutions of rational matrix equations, the (1,4)-inverse, also known as the minimum norm inverse, to compute minimum norm solutions of matrix equations. With this nomenclature, the Moore-Penrose pseudoinverse is an (1,2,3,4)-inverse. The left and right inverses, frequently used in the control literature, are particular (1,2)-inverses of full column rank or full row rank matrices, respectively. Note. Our approach to determine generalized inverses relies on the standard system inversion concepts widely used in the control literature. Therefore, to define the generalized inverses using the Moore- Penrose conditions, we will not assume that has constant rank for all(as is done elsewhere (Sontag, 1980)). It follows that the generalized inverses computed by our methods satisfy pointwise conditions 1-4 (or part of them) for almost all values ofexcept a finite set of points, which includes certainly the poles and zeros of

The numerical computation of Moore-Penrosepseudoinverses was only recently addressed in thecontrol literature (Oara and Varga, 2000). Anumerically reliable approach to computepseudoinverses of rational matrices can be devisedalong the lines of the recently developed generalmethods to compress rational matrices to fullrow/column-rank matrices by using left/rightmultiplication with inner factors. This computation isthe main part of the recently developed algorithms tocompute inner-outer factorizations of rational matrices(Oara and Varga, 1999; 2000). By using "orthogonal"compression techniques with inner factors, thecomputation of the Moore-Penrose pseudoinverse of a givencan be performed as follows:

wherehas full row-rank andis square inner

whereis invertible andis square inner Note. We have now the overall "orthogonal" decomposition ofas

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This computational approach employing the recent compression techniques developed in (Oara and Varga, 1999; 2000) is applicable to an arbitrary rank rational matrix regardless of whether it is polynomial, proper or improper. Furthermore, can have stable or unstable poles and zeros, and even poles and zeros on the imaginary axis or on the unit circle. The main computations in this approach are the row compressions performed in Steps 1 and 2. The methods proposed in (Oara and Varga, 1999; 2000) to perform this compression are based on reducing the system pencilto a particular Kronecker- like form which isolates the left singular structure ofThen the compression is achieved by solving for the stabilizing solution a standard algebraic Riccati equation of orderwhereis the sum of the left minimal indices ofNote thatis usually much smaller than the order n of a minimal descriptor realization of

In this section we show how we can compute (1,2,3)- and (1,2,4)-inverses by combining the underlying techniques to determine (1,2)- and (1,2,3,4)-inverses. We also discuss shortly a general approach to determining inverses using row/column compression techniques. For notational convenience, in this section we will denote by G simply a rational matrix In the previous section we discussed compression techniques of rational matrices by premultiplying them with all-pass factors. The following procedure can be used to compute a (1,2,3)-inverse of a given G by combining row compression with inner factors and (1,2)-inverse computation:

where R has full row-rank and U is square inner.

It is easy to check thatis a (1,2,3)-inverse. Indeed, and the first three Moore-Penrose relations aresatisfied. To compute a (1,2,4)-inverse, the above procedure can be applied to the transposed matrix An alternative approach to calculation of generalizedinverses can be devised using exclusively row/columncompression techniques. With a row compressionfollowed by a column compression, we can determine afull rank factorization of G in the form with L, T and R being invertible rational matrices. Thisdecomposition of G allows us to determine severalinverses with a specific choice of the transformationmatrices L and T. In general, a (1,2)-inverse of G cansimply be computed as If we use an inner matrix L for the row compressionand a general matrix T for column compression, then the corresponding inverseis a (1,2,3)-inverse, whilefor general L and inner T thecorresponding inverse is a(1,2,4)-inverse. If L and T are inner, then the inverseis the Moore-Penrose pseudoinverse. Compression techniques with inner matrices wererecently developed in (Oara and Varga, 1999; 2000).For compression with general matrices, methodsproposed in (Oara, 2000) can be employed. Thesemethods rely also on pencil algorithms using theequivalent descriptor representation of the rationalmatrix G. A particular advantage of this approach is theflexibility in choosing the poles and zeros of thecompressing factor.

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All presented computational methods to determine a generalized inverse of a rational matrix G(_) have overall computational complexity O(n3), where n is the order of the underlying descriptor representation G. The main advantage of using pencil techniques to solve computational problems involving rational matrices is that the whole arsenal of well-developed linear algebra techniques to manipulate matrix pencils can be employed to devise numerically reliable algorithms for these rather complex problems. In contrast, rational matrix manipulations using polynomial techniques are generally considered to be numerically less robust than methods based on state-space representations. It is to be expected that all presented techniques can be extended to more general systems, as, for instance, periodic time-varying and even general time-varying systems. For the computation of weak generalized inverses with arbitrarily assigned spurious poles and for the computation of Moore-Penrose pseudoinverses, Matlab functions are available in a recently developed Descriptor Systems toolbox (Varga, 2000). The main functions rely on a collection of mex-functions providing easy-to-use gateways to highly optimized Fortran codes from the SLICOT library (Benner et al., 1999) developed within the NICONET project.1 The computational layer of basic mex-functions provides efficient and numerically robust computational tools to perform reductions to Kronecker-like forms, reordering of generalized real Schur forms, balancing of descriptor systems, minimal realization of rational matrices, and other computations.

We have proposed numerically reliable methods to compute two classes of generalized inverses of rational matrices: the (1,2)- or weak inverses and the Moore-Penrose pseudoinverses. The proposed methods are completely general, being applicable to rational matrices of arbitrary ranks regardless of whether they are polynomial, proper or improper. Particular emphasis has been put on reliably computing (1,2)-inverses, because of their relevance to many practical applications. The proposed approach provides flexibility to cope with various conditions on the spurious poles of the computed (1,2)-inverses. For instance, a stable (1,2)-inverse can easily be computed whenever exists. The computation of other types of inverses has been also addressed, by combining "orthogonal" compression techniques using inner factors with left/right inverse computation. For the proposed methods, robust numerical software has been implemented. Interesting open computational problems in the context of determining various types of inverses are posed bythe exploitation of intrinsic parametric freedom. Twoaspects could be relevant for control applications: (1)determining particular inverses with special properties(e.g., a minimal Mc-Millan degree or stable withminimal Mc-Millan degree), and (2) generating thewhole class of inverses by exploiting theparametrizations of inverses (see, e.g., (Campbell andMeyer, 1991)). In this paper we partially addressedonly the first aspect. However, a detailed elaboration of algorithms to compute least order inverses is stillnecessary.

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