By Torkel Glad; Lennart Ljung
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Extra resources for Control theory : multivariable and nonlinear methods
The main part of this chapter deals with continuous time systems. 6 the corresponding expressions for the discrete time case will be presented. 1 Impulse Response and Weighting Function The output from a linear system is the weighted sum of input values at all times. For a causal system only old values of the input signal are included, and for a time invariant system the contributions are weighted by a factor dependent only on the time distance (not absolute time). 1) The function g() is called the weighting function, and shows how much the value of the input time units ago, now influences the output.
1 (Controllability) The state x* is said to be controllable if there is an input that in finite time gives the state x* from the initial state . The system is said to be controllable if all states are controllable. 2 (Observability) The state is said to be unobservable if, when and , the output is . The system is said to be observable if it lacks ubobservable states. Criteria From a basic control course the criteria for controllability and observability are well known, see for example, Franklin et al.
For a strictly proper transfer function matrix G(s) while the limit is bounded if G is proper. 1: A System With One Output and Two Inputs Consider the system with the transfer function matrix Assume that input number 1 is a unit step, that input number 2 is an impulse, and that the system is at rest for . We then have The Laplace transform of the output is then and the output is We will not use explicit time functions to any great extent in this book, transform tables are therefore not particularly important here.