Control theory : multivariable and nonlinear methods by Torkel Glad; Lennart Ljung

By Torkel Glad; Lennart Ljung

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It has a special focus on multivariable systems. Other than that, the material is of general character; corresponding descriptions of signals and systems and their properties can be found in circuit theory, signal processing and basic control theory. 25 Chapter 2 Representation of Linear Systems In the general classification of systems in Chapter 1, we defined linear, time invariant and causal systems. Systems with these three properties constitute, by far, the most common models of control objects.

Those with a finite 2-norm, which give well defined outputs y (which need not have finite 2-norm). The gain may very well be infinite. Remark. For this definition of gain to be meaningful, the zero levels of the signals should be defined so that corresponds to . 5 Gain and Signal Size 17 and where equality holds for at least one x, say x*. This gives The gain is therefore less than or equal to K. On the other hand, we can , which gives equality in the above expression. use the input The gain of the system is therefore equal to K.

20) The interpretation is completely analogous to the continuous time case. 3). Everything concerning continuous systems also applies to discrete time systems if the Laplace table is exchanged for the -transform table. 23) if the left hand side exists. 6 Discrete Time Systems 39 Transfer Operators The transfer operator is considerably more straightforward in discrete time than in continuous time. 26) We then have The transfer operator has, formally, been obtained by exchanging z for q in the definition of the transfer function matrix.

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