Guaranteed Time Control using Linear Matrix Inequalities
For control engineers designing safety-critical systems, this provides a method to compute guaranteed convergence times with formal bounds, though the approach is incremental over existing Lyapunov-based techniques.
This paper presents a synthesis approach to guarantee a minimum upper-bound on the time to reach a target set under uncertainties and constraints, using a harmonic transformation of the Lyapunov function and piecewise quadratic representations solved via linear matrix inequalities. The method is demonstrated on three examples, showing effectiveness in reducing worst-case convergence time.
This paper presents a synthesis approach aiming to guarantee a minimum upper-bound for the time taken to reach a target set of non-zero measure that encompasses the origin, while taking into account uncertainties and input and state constraints. This approach is based on a harmonic transformation of the Lyapunov function and a novel piecewise quadratic representation of this transformed Lyapunov function over a simplicial partition of the state space. The problem is solved in a policy iteration fashion, whereas the evaluation and improvement steps are formulated as linear matrix inequalities employing the structural relaxation approach. Though initially formulated for uncertain polytopic systems, extensions to piecewise and nonlinear systems are discussed. Three examples illustrate the effectiveness of the proposed approach in different scenarios.