8.2 State Feedback and Pole Placement
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1 Pole Placement Design Technique 82 State Feedback and Pole Placement Consider a linear dynamic system in the state space form In some cases one is able to achieve the goal (eg stabilizing the system or improving its transient response) by using the full state feedback, which represents a linear combination of the state variables, that is so that the closed-loop system, given by has the desired specifications The main role of state feedback control is to stabilize a given system so that all closed-loop eigenvalues are placed in the left half of the complex plane The following theorem gives a condition under which is possible to place system poles in the desired locations Theorem 81 Assuming that the pair is controllable, there exists a feedback matrix such that the closed-loop system eigenvalues can be placed in arbitrary locations 1
2 This important theorem will be proved (justified) for singleinput single-output systems For the general treatment of the pole placement problem for multi-input multi-output systems, which is much more complicated, the reader is referred to Chen (1984) If the pair is controllable, the original system can be transformed into the phase variable canonical form, ie it exists a nonsingular transformation such that where s are coefficients of the characteristic polynomial of, that is For single-input single-output systems the state feedback is given by After closing the feedback loop with get from (85), as given by (87), we 2
3 If the desired closed-loop eigenvalues are specified by, then the desired characteristic polynomial will be given by Since the last row in (88) contains coefficients of the characteristic polynomial of the original system after the feedback is applied, it follows from (88) and (89) that the required feedback gains must satisfy The pole placement procedure using the state feedback for a system which is already in phase variable canonical form is demonstrated in the next example Example 81: Consider the following system given in phase variable canonical form It is required to find coefficients such that the closed-loop system has the eigenvalues located at The desired characteristic polynomial is obtained from (89) as 3
4 so that from (810) we have In general, in order to be able to apply this technique to all controllable single-input single-output systems we need to find a nonsingular transformation which transfers the original system into phase variable canonical form This transformation can be obtained by using the linearly independent columns of the system controllability matrix It can be shown (Chen, 1984) that the required transformation is given by where where s are coefficients of the characteristic polynomial of matrix After the feedback gain has been found for phase variable canonical form,, in the original coordinates it is obtained as (similarity transformation) 4
5 & & & Example 82: Consider the following linear system given by The characteristic polynomial of this system is! " Its phase variable canonical form can be obtained either from its transfer function (see Section 312) or by using the nonsingular (similarity) transformation (811) The system transfer function is given by #%$! " Using results from Section 312 we are able to write new matrices in phase variable canonical form representation as & & & The same matrices could have been obtained by using the similarity transformation with # $ # $ with obtained from (811) and (812) as 5
6 ) ( * * Assume that we intend to find the feedback gain for the original system such that its closed-loop eigenvalues are located at, then ' ( ) From equation (810) we get expressions for the feedback gains for the system in phase variable canonical form as ' + + ' * * ' ) ) In the original coordinates the feedback gain is obtained from (813), - Using this gain in order to close the state feedback around the system we get It is easy to check by MATLAB that the eigenvalues of this systems are located at Comment: Exactly the same procedure as the one given in this section can be used for placing the observer poles in the desired locations The observers have been considered in Section 56 Choosing the observer gain such that the closed-loop observer matrix has the desired poles corresponds to the problem 6
7 of choosing the feedback gain such that the closed-loop system matrix has the same poles Thus, for the observer pole placement problem, matrix should be replaced by, replaced by and replaced by In addition, it is known from Chapter 5 that the observability of the pair is equal to the controllability of the pair, and hence the controllability condition stated in Theorem 81 the pair is controllable, which for observer pole placement requires that the pair be controllable is satisfied by assuming that the pair is observable 7
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