Step 3: Compute the desired cyclic realisation 4. Thus for a given rational matrix W s there are many pairs of matrices Dl s , Nl s and Dp s , Np s , which give the same matrix W s. Thus the obtained realisation is cyclic. If all the eigenvalues of A and — B have negative real parts, then the unique solution to 6. The Cayley-Hamilton theorem was also extended to non-square matrices, nonsquare block matrices and singular 2D linear systems with non-square matrices Kaczorek, 1995b; 1995c; 1995d. To this end, we pre-multiply the matrix 4. Similarly, if det Lr z 0, then the determinant of the matrix 1.
From 1988 to 1991 he was the director of the Research Centre of Polish Academy of Sciences in Rome. We will show that all these sequences of polynomial solutions have the same sequence of degrees 1. Let T z have the form 5. We say that rational functions w1 s l1 s , w2 s m1 s l2 s m2 s 2. Subtracting from the j-th row i-th column the first row column multiplied by qj1 s by q 1i s , we replace the entry a j1 s a 1i s by the remainder rj1 s r1i s. Analogously, one can define the set of rational matrices whose entries are rational functions of the form 2.
The results of the analysis will be supported by numerical analysis and simulations in the Matlab computational environment. Thus for the matrices 6. Let a least common denominator of the rational matrix 3. Singular and Cyclic Normal Systems 305 5. In order to obtain a realisation with real entries, we additionally transform the complex realisation 4. Dividing the entries a i1 s of the first column and the first row a 1j s by a 11 s , we obtain Polynomial Matrices ai1 s a11 s qi1 s i 2, 3,.
His research interests cover the theory of systems and the automatic control systems theory, specially, singular multidimensional systems, positive multidimensional systems and singular positive 1D and 2D systems. In order to compute the structural decomposition of the matrix 5. Applying elementary operations on columns and carrying out the reduction ª Dl « «I p «¬ 0 2. One can also obtain the solution 5. The elementary divisors of the matrix A are uniquely determined and they determine its essential structural properties. These difference equations allow the realization of the dynamical system by means of delay, multiplication and addition elements in simulation diagrams.
From the assumption that A, B, C and A,B,C are two minimal realisations of the transfer matrix T s , it follows that they satisfy the equality 4. Step 1: Compute the poles s1,s2,…,sp of the matrix Tsp s and their multiplicities m1,m2,…,mp. The book is selfcontained, beginning with important basics such as the CayleyHamilton theorem and definitions and elementary operations of polynomial and rational matrices and moving on to cover such topics as: normal matrices including their realisation rational and algebraic polynomial matrix equations perfect observers for and realisation of linear systems and new results on positive linear discrete and continuoustime systems with delays. The classical Cayley-Hamilton theorem is extended to Drazin inverse matrices and to standard inverse matrices. The following important corollary ensues from Theorem 1. Polynomial Matrices 73 Corollary 1.
Using the solution of Example 2. His research interests cover the theory of systems and the automatic control systems, particularly singular multidimensional systems, positive multidimensional systems and singular positive 1- and 2-dimensional systems. In doing so we generalize the classical Cayley—Hamilton theorem to multivariate polynomial ideals and provide a system theoretic interpretation to the notion of polynomial ideals, leading monomials and Gröbner bases. The text is rounded off with an appendix describing fundamental definitions and theorems relevant to controllability and observability in linear systems. Let s1,s2,…,sp be the poles not necessarily distinct of the rational matrix 3. The proof that if A, C is not observable, then A,B,C is not a minimal realisation, is analogous. From the block-diagonal form 1.
With the basic definitions and operations on rational functions introduced at the beginning, the following issues are subsequently addressed: decomposition into the sum of rational functions, operations on rational matrices, the decomposition of a matrix into the sum of rational matrices, the inverse matrix of a polynomial matrix and its reducibility, the McMillan canonical form of rational matrices, the first factorization of rational matrices and the application of rational matrices in the synthesis of control systems. Then, according to Theorem 4. Let p s be an arbitrary polynomial. Kaczorek, An Extension of the Cayley-Hamilton Theorem for Nonlinear Time-Varying Systems, Int. .
The desired function, according to 2. Taking into account the polynomial matrices in 1. We derive an algorithm that computes the state difference equations for a given set of poles of linear discrete overdetermined autonomous D systems. Consider a square matrix of size mn, given by the formula p w A, B ¦c A ij i Bj , 6. This result is consistent with Theorems 3. If the matrices A and B are similar, i. He is Editor-in-Chief of Bulletin of the Polish Academy of Sciences, Techn.