The worst-case norm of uncertain linear systems under inputs with magnitude bound and rate limit

Abstract

For many years, researchers in control systems have widely employed the worst-case norms (WCNs) as performance indices measuring how worst control systems can be affected by specific disturbances, which in turn indicates how well a selected controller can reject undesired effects. In this dissertation, we consider the WCN of uncertain linear time-invariant systems when the disturbance inputs are modeled to have bounded magnitude and limited rate. The first part of this work is devoted to analysis and computation of the WCN in the absence of uncertainty. The WCN computation is formulated as an optimal control problem, whose solution yields characterization of the worst-case input. A novel algorithm called Successive Pang Interval Search (SPIS) is developed to construct the worst-case input and compute the WCN. The second part of this work involves analysis and computation of the WCN in the presence of uncertainty. The WCN analysis and computation of linear systems without uncertainty has now become the solid groundwork for the case of uncertain linear systems. The WCN computation in this case is formulated via discretization approach, which leads on an NP-hard convex maximization problem. Novel upper and lower bounds of the WCN are introduced and can be obtained by solving sparse linear programming problems. To compute the exact WCN, we develop an algorithm called Hierrarchical Branch-and-bound (HBB) algorithm. This algorithm employs a standard branch-and-bound (BB) technique with a procedure called reduction of Ambiguity Magnitude Threshold (RAMT). We validate the algorithm and compare numerical results with that obtained by an exhaustive search and a standard BB algorithm. HBB algorithm yields correct results with excellent computational speed and outperforms existing standard algorithms; hence, it is deemed to provide a viable means to attain the WCN computation of high dimensional problems. In addition, we demonstrate a practical use of HBB algorithm by applying to multi-objective PID tuning for an active suspension system subjected to variable load mass and road disturbances. The PID design in achieved with reasonable time and the simulation results clearly show that all design specifications are satisfied.