Pole assignment at infinity and polynomial matrix equations

This paper presents a one-to-one correspondence between a pseudo-observable form and the matrix fraction description (MFD) of a MIMO system so that by given one form, the other may be derived accordingly. In the development, we consider only a left coprime MFD and pseudo-observable form. The right coprime MFD and pseudo-controllable form can be derived in a dual sense. >

In §21, we discovered (c.f. Theorem 21.9) that a real m-input, p -output time-invariant continuous- or discrete-time input-output linear system with impulse response t → H(t) or k → H(k) was realizable if and only if its transfer function s → G (s) or z → G (z) agreed on its region of convergence ROC(H) with a real proper (p × m) matrix rational function of s or of z. The constant realizations for such a system was shown in Facts 21.10 to lie in one-to-one correspondence with matrix quadruples (A, B, C, D) which satisfied $$\begin{aligned} & & & G\left( s \right) = C{\left( {s{I_n} - A} \right)^{ - 1}}B + D,\quad s \in ROC\left( H \right), \\ or \\ G\left( z \right) = C{\left( {z{I_n} - A} \right)^{ - 1}}B + D,\quad z \in ROC\left( H \right). \\ \end{aligned} $$

On the equivalence between MFD models and pseudo-observable forms of MIMO systems

Controllability Analysis of Linear Time-Invariant Descriptor Systems∗∗This work was supported by CSIR, New Delhi (Grant No. 25(0195)11/EMRII).

Based on the matrix fraction description (MFD) consideration and polynomial matrix algebra, interesting results for the multivariable discrete linear quadratic regulator and optimal estimator are derived. The solutions are obtained by the spectral factorization of polynomial matrices in concise forms. The difficulties of solving the discrete algebraic Riccati equation can thus be avoided. In the derivation, some connections between the state space approach and the frequency domain approach are further investigated.

The problem of block decoupling of singular systems via state feedback and regular input transformation is solved by following an approach based on the matrix fraction description (MFD) of the system. The solvability condition is obtained from the 'numerator' matrix of a coprime and column reduced MFD of the given system. The characterization of the fixed poles is given and it is shown that all the non-fixed poles can be arbitrarily assigned.

This article deals with the problem of dissipative control synthesis for a class of descriptor systems with uncertainties in the derivative matrix. Attention is focused on the design of a proportional plus derivative (PD) state feedback, which guarantees that the closed-loop system is robustly stable and strict dissipative. Firstly, a sufficient condition for the closed-loop system is robustly stable and strict dissipative is presented by using a simple idea of changing the problem to the corresponding problem of an augmented uncertain system. Then, a PD controller is constructed by solving LMIs. Finally, a numerical example is given to demonstrate that the proposed method is effective.

In this paper an algorithm is presented for obtaining a relatively prime matrix fraction description (MFD) for a time-invariant linear system in state space form not necessarily of minimal dimension. The algorithm first performs a sequence of simple coordinate transformations on the state vector of the system. The observability (controllability) indices of the system are also determined at this time. The relatively prime factors of an MFD are then obtained via a recursive procedure involving the submatrices of the state and the input matrix of the transformed state space system. A numerical example is given to illustrate the use of the algorithm.

Feedback design for regularizing descriptor systems

A procedure is given for finding the partial-fraction expansion of strictly proper rational transfer functions that are commonly found in systems analysis. Eight formulae for the expansions are given to include the cases: (a) scalar transfer functions, (b) matrix transfer functions, (c) left and right matrix fraction descriptions, (d) transfer function matrices, and (e) block expansion of left and right matrix fraction descriptions. These formulae are derived from system theory using the observability canonical form and do not require the use of derivatives, the particular nature of the Jordan form, or interpolating polynomials; rather, it is required that the Vandermonde matrix be formed from the roots, latent roots, eigenvalues or solvents of the denominator. The main features of the procedure include that all the residues are found at the same time by simple matrix manipulation and can be used for the repeated and/or non-repeated case. The structured form of the formulae is very simple and fil...

Implementation of column-pseudoproper right matrix fraction descriptions with cascaded or parallel active RC networks

A rational function whose denominator polynomial has no zeros inside the unit circle, is called a generalized polynomial, and has properties similar to a polynomial. The ring of generalized polynomials is denoted by R∑(σ). A generalized polynomial matrix has the ∑-Smith form similar to the Smith form of a polynomial matrix. Linear neutral time-delay systems can be represented as systems over the field of rational functions whose argument is σ, where σ is a fixed delay operator. R∑(σ)-controllability which is similar to R[σ]-controllability for a time-delay system of the retarded type, is defined. Theorem 1 gives a necessary and sufficient condition for R∑(σ)-controllability. This condition is represented by the ∑-Smith form for the controllability matrix.∑-pole assignability which is the extension of pole assignability for a retarded system and guarantees asymptotic stability of the closed loop system by phiscally realizable state feedback, is defined. Theorem 2 states that R∑(σ)-controllability is equivalent to ∑-pole assignability. Theorem 3 gives a sufficient condition for decoupling by physically realizable state feedback. Finally, it is shown that under the condition of Theorem 3 and R∑(σ)-controllability, there exists a differential-difference compensator of the neutral type to achieve decoupling and arbitrary ∑-pole assignment.

Let [Ctilde](s)[Dtilde](s)−1 be a right coprime matrix fraction description of the (r × m) strictly proper transfer function matrix T(s) of a linear multivariable system Σ = (A, B, C). In this paper we establish a direct relation between (i) the algebraic concept of a right divisor C R(s) of the ‘numerator’ matrix [Ctilde](s), i.e. a (not necessarily square) polynomial matrix C L(s) that satisfies: [Ctilde](s) = C L(s)C R(s) for some polynomial matrix C L(s), and (ii) the geometric concept of an (A, B)-invariant subspace which is contained in the kernel of C. A special class of right divisors C R(s) of C(s) which correspond to (A, B)-invariant subspaces in ker C is characterized by a number of properties and a simple formula is presented which expresses the above correspondence. By generalizing the concept of a greatest right divisor (GRD) of a polynomial matrix to include also possibly non-square polynomial matrices, it is shown that GRDs of the numerator matrix C(s) correspond to the maximal (A...

In this paper, a theory of polynomial and rational matrix interpolation is introduced and applied to problems in systems and control. The polynomial matrix interpolation theory is first outlined and then applied to solving matrix equations; it is also used in pole assignment and other control problems. Rational matrix interpolation is also discussed and it is used to solve rational matrix equations including the model matching problem.

Aircraft Flutter Modal Parameter Identification Using a Numerically Robust Least-squares Estimator in Frequency Domain