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System theory of continuous time finite dimensional dynamical systems. The memories of Tsuyoshi Matsuo and R. E. Kalman. (English) Zbl 1436.93003

Studies in Systems, Decision and Control 250. Cham: Springer (ISBN 978-3-030-30479-9/hbk; 978-3-030-30480-5/ebook). x, 221 p. (2020).
The content of this monograph, as his author estimates, may be one of the thing that Tsuyoshi Matsuo and R. E. Kalman aimed for. In this context, it is already known that, based on analysis of state space approach, control problem became an important theme after 1960 to be efficiently used in solving economic problems, industrial technological problems and on the development of digital computers and mathematical programming. It is clear that the modem control design needs the solution of complicated nonlinear matrix equations. The performance obtained by solving matrix equations means that it is often possible to design a control system that works in theory without gaining any engineering intuition about the problem. But, at the same time, a sort of intuition which means the closeness to input, output, and the state had be provided. And the control problems have been solved as algebraically as possible for the first time.
Based on input/output terms, the control problems for a given dynamical system with input and output can be roughly stated as the following three problems:
1. Equilibrium state control. In this case find an input sequence that brings an arbitrary state of the system to the equilibrium state within the size of input values.
2. Fixed value output control. In this case find an input sequence that brings an arbitrary output of the system to fixed value output within the size of input values.
3. Tracking output control. In this case find an input sequence that brings an arbitrary output of the system to a desired trajectory output within the size of input values.
In this monograph, regarding the output sequence to be controlled as the equations to be expressed by terms of input, the control problem is identified as a problem of finding the unique inputs which produce the specified output. If the unique input had not been obtained, the unique input can be obtained by introducing the performance function for inputs to be treated as the square norm, in the sense of energy. This procedure takes a positive attitude toward using computers and mathematical programming. Consequently, a method based on least square norm have been introduced.
For the continuous-time dynamical systems, this monograph also presents that obtaining a dynamical system which describes a given input/output map is equal to determining the rank of the matrix of the input/output map and the coefficients of a linear combination of column vectors in the matrix. It is show that canonical dynamical systems are determined by the reachability (or quasi-reachability) and observability (or distinguishability). This insight of the continuous-time case leads to the ability of discussing fruitful control problems. Clearly, the usual control problems have been mainly discussed in linear systems. On the other hand, there are also some developments for nonlinear systems.
This monograph is of great interest to all which are specialized or will be specialized in system theory. And this because it deals with realization theory and control laws of continuous-time finite-dimensional dynamical systems which include linear and nonlinear input/output relations in the case of input’s space being a set of the function which has the closed interval.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93B15 Realizations from input-output data
93B03 Attainable sets, reachability
93B07 Observability
93C10 Nonlinear systems in control theory
93C05 Linear systems in control theory

Biographic References:

Matsuo, Tsuyoshi; Kalman, Rudolf E.
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