Weehong Tan

Stability Region Analysis Using Polynomial and Composite Polynomial Lyapunov Functions and Sum-of-Squares Programming

We propose using (bilinear) sum-of-squares programming for obtaining inner bounds of RoAs for dynamical systems with polynomial vector fields. We search for polynomial as well as composite Lyapunov functions comprised of pointwise maximums of polynomial functions. Results for several examples from the literature are presented using the proposed methods and the PENBMI solver.

Some controls applications of sum of squares programming

We consider nonlinear systems with polynomial vector fields and present two algorithms based on sum of squares programming, that may answer system theoretic questions. The first algorithm provides a bound for the reachable set of a system driven by a unit-energy disturbance, while the second synthesizes a polynomial state feedback controller to enlarge the provable region of attraction. We also outline a variant of the second algorithm for handling systems with input saturation. Both algorithms are demonstrated using two-state nonlinear systems.

Control Applications of Sum of Squares Programming

We consider nonlinear systems with polynomial vector fields and pose two classes of system theoretic problems that may be solved by sum of squares programming. The first is disturbance analysis using three different norms to bound the reachable set. The second is the synthesis of a polynomial state feedback controller to enlarge the provable region of attraction. We also outline a variant of the state feedback synthesis for handling systems with input saturation. Both classes of problems are demonstrated using two-state nonlinear systems.

Quasi-LPV modeling and LPV control of a generic missile

In this paper, we will apply linear parameter-varying (LPV) control theory to the design of gain-scheduled missile autopilot. A nonlinear coupled 3-axis generic missile model is used as our design example. We will show that this model can be reduced to quasi-LPV form so that LPV design methodology can be applied. Simulation of the gain-scheduled controller with the nonlinear missile model yields excellent results and serves to validate both the quasi-LPV modeling procedure and the LPV control theory in practice.

Stability region analysis using sum of squares programming

We propose using (bilinear) sum of squares programming for enlarging a provable region of attraction of polynomial systems using polynomial Lyapunov functions. In order to keep the number of decision variables low, we propose the use of composite Lyapunov functions. We also present a method of finding outer bounds of attractive invariant sets for polynomial systems. Excellent results for some examples were obtained using the proposed methods and the PENBMI solver

Local gain analysis of nonlinear systems

We analyze the induced Lscr₂ rarr Lscr_infinand Lscr₂ rarr Lscr₂ gain for nonlinear systems using sum-of-squares optimization techniques. Two examples are presented, illustrating the ideas and highlighting the advantage of direct nonlinear analysis over simplified analysis of the linearized systems

Simulation-aided reachability and local gain analysis for nonlinear dynamical systems

We analyze reachability properties and local input/output gains of systems with polynomial vector fields. Upper bounds for the reachable set and nonlinear system gains are characterized by polynomial Lyapunov (storage) functions satisfying certain bilinear constraints. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates are assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute upper bounds for the reachable set and nonlinear system gains and to initialize further coordinate-wise affine optimization. We illustrate the method on several examples from the literature.

Modeling and control design of an AGV

The modeling and control design of an automated ground vehicle (AGV) for transporting cargo containers is presented. As the AGV dynamics has uncertainties, H/sub /spl infin// loop-shaping synthesis is used for lateral dynamics control design. Design considerations and implementation are also presented.