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Modern Control Theory -A historical perspective
val, 2019
The purpose of this paper is to present a brief sketch of the evolution of modern control theory. Systems theory witnessed different stages and approaches, which will be very shortly presented. The main idea is that, at present, Control Theory is an interdisciplinary area of research where many mathematical concepts and methods work together to produce an impressive body of important applied mathematics. A general conclusion is that the main advances in Control of Systems would come both from mathematical progress and from technological development. We start with frequency-domain approach and end our historical perspective with structural-digraph approach, passing through time-domain, polynomial-matrix-domain frequential and geometric approaches.
In this section we study state space models of continuous-time linear systems. The corresponding results for discrete-time systems, obtained via duality with the continuous-time models, are given in Section 3.3. The state space model of a continuous-time dynamic system can be derived either from the system model given in the time domain by a differential equation or from its transfer function representation. Both cases will be considered in this section. Four state space forms-the phase variable form (controller form), the observer form, the modal form, and the Jordan form-which are often used in modern control theory and practice, are presented.
Linear systems theory revisited
Automatica, 2008
This paper investigates and clarifies how different definitions of reachability, observability, controllability, reconstructability and minimality that appear in the control literature, may be equivalent or different, depending on the type of linear system. The differences are caused by (1) whether or not the linear system has state dimensions that vary with time (2) bounds on the time axis of the linear system (3) whether or not the initial state is non-zero and (4) whether or not the system is time invariant. Also (5) time-reversibility of systems plays a role. Discrete-time linear strictly proper systems are considered. A recently published result is used to argue that all the results carry over to continuous time. Out of the investigation two types of definitions emerge. One type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions. This paper reveals that time-varying (state) dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and (weak) reachability together with observability at the systems level. Besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time-varying discrete-time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily.
Modelling of Dynamic Systems in State Space
Acta Mechatronica, 2021
This paper deals with the solution of dynamical systems in state space. Complicated differential equations are converted into a simpler form by using state variables in vector matrix. It is used for multi-input and multi-output systems, and the solution is performed using matrix notation. It describes systems with complex internal structure. It allows state models to be manipulated using matrix calculus. Systems described by a state model are characterized by the fact that it is easier to design state control for them.