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On the block decomposition and spectral factors of λ-matrices

Bekhiti Belkacem, Nail Bachir, Dahimene Abdelhakim, Hariche Kamel, Fragulis George F.

Control and Cybernetics

2020

Vol. 49, No. 1

41--76

In this paper we have factorized matrix polynomials into a complete set of spectral factors using a new design algorithm, and with some systematic procedures a complete set of block roots (solvents) have been obtained. The newly developed procedure is just an extension of the (scalar) Horner method to its block form for use in the computation of the block roots of matrix polynomial, the block-Horner method bringing a local iterative nature, faster convergence, nested programmable scheme, needless of any prior knowledge of the matrix polynomial, with the only one inconvenience, which is the strong dependence on the initial guess. In order to avoid this trap, we proposed a combination of two computational procedures, for which the complete program starts with the right block-Q.D. algorithm. It is then followed by a refinement of the right factor by block-Horner’s algorithm. This results in the global nature of the program, which is faster in execution, has well defined initial conditions, and good convergence in much less time.

On the theory of λ-matrices based MIMO control system design

Bekhiti B., Dahimene A., Nail B., Hariche K.

Control and Cybernetics

2015

Vol. 44, no. 4

421--442

In this paper we have described a new design algorithm for the whole set of latent-structure assignments via the approaches of block structure of λ-matrices placement. The procedure that has been developed is based on decoupling of the interactions between control loops in a multivariable plant. The procedure is performed using matrix polynomial solvent reconstruction for the decoupling purposes. However, for the design of the trajectory tracking controller, each input-output pair is treated respectively by designing SISO controllers. A second procedure is the MIMO PID compensator design via the model-matching method. This latter algorithm has been developed in order to avoid the internal or the hidden instability, which may occur in the ﬁrst method, due to the block zeros - block poles cancellation.