Linear Control Systems pp 73-205 | Cite as

# Continuous linear systems

## Abstract

This Chapter contains solved problems about continuous-time linear control systems. It begins with the background material on linear differential equations and matrices (Sections 3.1, 3.2, and 3.3). It continues with a discussion of the advantages of the state-space representation of linear systems over their input-output representation (Sections 3.4 and 3.5). In Sections 3.6 and 3.7 we investigate three fundamental properties of systems: stability, state controllability, and state observability. In Section 3.8 we examine the canonical forms of linear systems and their properties. Section 3.9 shows that by using the state feedback we can arbitrarily place the poles of the system. The condition for this so-called modal controllability is, quite amazingly, the state controllability and observability. Next, in Section 3.10, we describe how the feedback gain should be picked so that the quadratic optimality is achieved. In Section 3.11 we explain the design of the state observers. In Section 3.12 we investigate how to pick the observer gain so that the effects of noise are minimized in a mean-square sense. The result is the Kalman-Bucy filter. Finally, in Section 3.13, we describe the reduced-order observers.

## Keywords

Impulse Response State Feedback Inverted Pendulum Signal Flow Graph Canonical Realization## Preview

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