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By Torkel Glad; Lennart Ljung

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All angles are measured in radians and the time unit is seconds. 15) This expression gives the relationship between state representation and the transfer operator (or transfer function matrix: exchange p for s). 16) 36 Chapter 2 Representation of Linear Systems The step from state space form to transfer operator is straightforward matrix algebra. It can be more difficult to move in the other direction. 18) Using this starting point it is also easy to represent a transfer function matrix with one output and several inputs in observer canonical form.

A simple one is input-output stability for general systems. With this is meant that an input with bounded norm must lead to an output with bounded norm. 2 (Input-Output Stability) A system is input—output stable if it has finite gain. The Small Gain Theorem A very simple, but useful result on stability of general closed loop systems is given by the following theorem. 7. 7: A closed loop system. ) The loop gives the following relations Take the norm of the signals and use the triangle inequality and the definition of gain: which gives The gain from r1 and r2 to e1 is consequently finite.

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. , the output that is obtained when the input is an impulse δ(t) (Dirac’s delta function).

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