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Additional info for Control theory : multivariable and nonlinear methods
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4 Let A be an matrix and B an matrix. 11) has full rank. ) 48 Chapter 3 Properties of Linear Systems Equivalently formulated: Let C be a matrix. 12) has full rank. The proof is given in Appendix 3A. 4 can be extended to show that if a system is not controllable, the controllable modes (eigenvalues) can be modified, but the uncontrollable cannot. 4) it is particularly interesting to know if it is controllable, and thereby can be modified. The following concepts therefore are useful. , it has all the eigenvalues in the stability region).
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.
24) We may interpret |A| as the “gain” of the mapping. Based on this interpretation of the operator norm, we define the concept of gain also for general systems. , 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*.