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# lec25 - 25 Further Robustness of the LQR The most common...

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25 Further Robustness of the LQR The most common robustness measures attributed to the LQR are a one-half gain reduction in any input channel, an infinite gain amplification in any input channel, or a phase error of plus or minus sixty degrees in any input channel. While these are general properties (Continued on next page)

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134 25 FURTHER ROBUSTNESS OF THE LQR that have a clear graphical implication on the Bode or Nyquist plot, other useful robustness conditions can be developed. These include robustness to uncertainty in the real coeﬃcients of the model (e.g., coeﬃcients in the A matrix), and certain nonlinearities, including control switching and saturation. We will use the Lyapunov stability and the LMI formulation of matrix problems in this section to expand these ideas. Saturation nonlinearities in particular are ubiquitous in control system application; we find them in ”railed” conditions of amplifiers, rate and position limits in control surface actuators, and in well-meaning but ad hoc software limits. As shown below, moderate robustness in saturation follows from the basic analysis, but much stronger results can be obtained with new tools. When the LQR is used to define the loop shape in the loop transfer recovery method (as opposed to the Kalman filter in the more common case), these robustness measures hold. 25.1 Tools 25.1.1 Lyapunov’s Second Method The idea of Lyapunov’s Second Method is the following: if a positive definite function of the state γx can be found, with V ( γx ) = 0 only when γx = γ 0, and if dV ( γx ) /dt < 0 for all time, then the system is stable. A useful analogy for the Lyapunov function V ( γx ) is energy, dissipated in a passive mechanical system by damping, or in a passive electrical system through resistance. 25.1.2 Matrix Inequality Definition Inequalities in the case of matrices are in the sense of positive and negative (semi) definite- ness. Positive definite A means γx T Aγx > 0 for all γx ; positive semidefinite A means γx T Aγx 0 for all γx . With A and B square and of the same dimension, A < B means γx T Aγx < γx T Bγx , for all γx. (300) Also, we say for the case of a scalar ρ , A < ρ means A ρI < 0 . (301) 25.1.3 Franklin Inequality A theorem we can use to assist in the Lyapunov analysis is the following, attributed to Franklin (1969). 1 A T B + B T A ρA T A + B T B , for all real ρ > 0 . (302) ρ The scalar ρ is a free parameter we can specify. It is assumed that the matrices A and B are of consistent dimensions.
25.1 Tools 135 25.1.4 Schur Complement Consider the symmetric block matrix defined as follows: A B M = B T . (303) D The Schur complements of blocks A and D are defined, respectively, as D B T A 1 B A = T D = A BD 1 B , (304) and an important property is that M > 0 ⇒∀ A > 0 and D > 0 and A > 0 and D > 0 . (305) The Schur complements thus capture the sign of M , but in a smaller dimensioned matrix equation. The fact that both A and D have to be positive definite, independent of the Schur γ ] T complements, is obvious by considering γx that involve only A or D , e.g., γx = [1 1 × n A γ 0 1 × n D ,

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lec25 - 25 Further Robustness of the LQR The most common...

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