## Abstract

In this chapter, we consider state space representations of linear systems of ordinary differential equations with several inputs and outputs (MIMO systems). As in the SISO case, a linear system arises when a nonlinear system is linearized at an equilibrium state. All control theory methods developed in this book refer to linear systems and the nonlinearity of the original system has to be taken into account indirectly. After having introduced the linearization procedure, we give solution formulas for linear MIMO systems, characterize the stability and calculate the frequency response. Then, by deriving a state space representation for a general SISO transfer function, we are led in a natural way to four very important properties of linear systems, namely controllability, observability, stabilizability and detectability. Several characterizations of these properties are given and the Kaiman decomposition of the state space is presented. They are used in the formulation and the proofs of the theorems describing *H* _{2} and *H* _{∞} optimal controllers.

## Keywords

Transfer Matrix MIMO System Linear Dynamical System Lyapunov Equation State Space Representation## Preview

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## Notes and References

- The material presented in this chapter is standard and can be found in a similar form in many books where state space methods for control systems are analyzed. Basic references are Chen [23], Kailath [52], Wonham [105], and also Kwaker-naak and Sivan [58]. The basic concepts of controllability and observability were introduced by Kaiman [53]; see also Kaiman [54]. An introduction to stochastic processes which goes far beyond the material presented in Sect 5.8 can be found in Papoulis [74].Google Scholar