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By Torkel Glad; Lennart Ljung

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This definition of stability of solutions to differential equations is also called Lyapunov-stability. 4 and Chapter 12 we shall return to stability of linear and nonlinear systems, respectively. 6) • defined signal norms • shown the small gain theorem: that a closed loop system is stable if the product of the gains in the loop is less than 1. system gain and stability Literature Mathematical foundations. Control theory builds on a mathematical foundation and we shall use concepts and results from a variety of mathematical subfields.

6 Discrete Time Systems 37 Assume now that we realize the system from input 1 to the SISO output in an observer canonical from. Assume then that we realize the SISO system from input 2 in the same way. The realizations will have the same A and C matrices. The actual output is the sum of the parts originating from the two input signals, and we can simply directly add the effects as follows: The example, with obvious modifications, can also be used to realize a transfer function with one input signal and several output signals in controller canonical form.

The algebra for the relationship between state space representations and other forms of representation for a linear system is therefore the same for time continuous and discrete time models. 5 thus apply also for discrete time, by letting q replace p. 7 Comments The Main Points of the Chapter We have shown how linear systems can be represented in different forms. One observation is that the multivariable case in this respect is a straightforward generalization of the SISO case: it is “just” a question of interpreting certain variables such as matrices and vectors.

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