Design, parametrization, and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules

Abstract

Control design belongs to the most important and difficult tasks of control engineering and has therefore been treated by many prominent researchers and in many textbooks, the systems being generally described by their transfer matrices or by Rosenbrock equations and more recently also as behaviors. Our approach to controller design uses, in addition to the ideas of our predecessors on coprime factorizations of transfer matrices and on the parametrization of stabilizing compensators, a new mathematical technique which enables simpler design and also new theorems in spite of the many outstanding results of the literature: (1) We use an injective cogenerator signal module F over the polynomial algebra D=F[s] (F an infinite field), a saturated multiplicatively closed set T of stable polynomials and its quotient ring DT of stable rational functions. This enables the simultaneous treatment of continuous and discrete systems and of all notions of stability, called T-stability. We investigate stabilizing control design by output feedback of input/output (IO) behaviors and study the full feedback IO behavior, especially its autonomous part and not only its transfer matrix. (2) The new technique is characterized by the permanent application of the injective cogenerator quotient signal module DTFT and of quotient behaviors BT of DF-behaviors B. (3) For the control tasks of tracking, disturbance rejection, model matching, and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed loop behaviors, parametrize all such compensators as IO behaviors and not only their transfer matrices and give new algorithms for their construction. Moreover we solve the problem of pole placement or spectral assignability for the complete feedback behavior. The properness of the full feedback behavior ensures the absence of impulsive solutions in the continuous case, and that of the compensator enables its realization by Kalman state space equations or elementary building blocks. We note that every behavior admits an IO decomposition with proper transfer matrix, but that most of these decompositions do not have this property, and therefore we do not assume the properness of the plant. (4) The new technique can also be applied to more general control interconnections according to Willems, in particular to two-parameter feedback compensators and to the recent tracking framework of Fiaz/Takaba/Trentelman. In contrast to these authors, however, we pay special attention to the properness of all constructed transfer matrices which requires more subtle algorithms.

Про iнварiантний пiдхiд до розв’язностi проблеми синтезу зворотного зв’язку для лiнiйної системи керування

“…Let Θ 0 ∈ Ω be some fixed vector. With the help of the methods, which were presented in [2,3], and [9], we find the feedback matrix K = K(Θ 0 ). Now already it is possible to estimate a radius of sphere S := { Θ − Θ 0 ≤ r} such that ∀Θ ∈ S the matrix A(Θ)+B(Θ)K(Θ 0 )C(Θ) is stable.…”

“…An important problem, which can arise up at the solution of feedback design problem, is a robustness problem of the got matrix K [2,3,9]. It is clear that for research of the robust properties of this matrix methods using transformations of coordinates become practically useless.…”

Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection

“…4.6]. It enables the transfer of Vidyasagar's LTI stabilization and control theory [16] to periodic behaviors and the application of [3]. The algebra A is canonically a subalgebra of the matrix algebra B := Z N ×N .…”

“…In Section 5 we also discuss the characteristic variety or set of poles of an autonomous behavior and define and characterize the stability of autonomous and of input/output systems. By means of the algorithms from [3] all results of this paper are constructive, but have not yet been implemented in the periodic case. History: A completely different approach to stabilization and control of discrete periodic systems given by state space equations is exposed by Bittanti and Colaneri [2, pp.…”

“…the bibliographies of [16] and [14] for important contributors to the latter field) and its application to stabilization problems is due to Quadrat, cf. [14], and was also applied in [3]. In this approach a commutative domain S of stable operators, often a Banach algebra, with its quotient field K is given.…”