5.4 Lyapunov Feedback Stabilization
It is possible to stabilize a system by state feedback using a certain Lyapunov equation solution. The following method, due to Armstrong (1975), stabilizes a continuous-time system with a damping factor .
1. Choose the damping factor such that the eigenvalues of have positive real parts.
2. Solve the Lyapunov equation for
3. Obtain the stabilizing feedback matrix .
For the discrete-time system, the method has the same format. The Lyapunov equation to be solved in this case is and the state feedback matrix is given by , where is chosen such that . The feedback matrix is such that (see Datta (2003) for details). The method in the discrete-time case was developed by Armstrong and Rublein (1976).
The aforementioned algorithms are accessible via the function StateFeedbackGains using the method LyapunovShift. Additionally, Advanced Numerical Methods provides the method PartialLyapunovShift that reassigns only those eigenvalues that lie outside the specified region, which can be given only as DampingFactorRegion or SettlingTimeRegion[, ]. This typically results in feedback matrices with a smaller norm. However, the algorithm in this case requires computing the ordered Schur decomposition before performing the Lyapunov shift and is computationally more expensive.
Methods for Lyapunov feedback stabilization.
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This is the typical state-space system arising in the magnetic tape control problem.
The partial Lyapunov feedback stabilization method is applied to the same state-space system.
The system has been stabilized with the damping factor .