On The Estimation And Enlargement Of Domain Of Attraction For Linear Systems Subject To Actuator Saturation

Over the past decades, systems with saturation nonlinearities have been extensively studied in the research community, due in part to their ubiquity in engineering and in part to the theoretical challenges they pose in control theory. Enormous amount of research has been devoted to the systematic analysis and synthesis of such systems, including global stabilization, semi-global stabilization and regional stability and stabilization. Among the results on regional stability and stabilization, however, conservativeness in some aspects, such as the treatments of saturation functions and the choice of Lyapunov functions, still exists. To reduce this conservativeness, further investigations for regional stability of saturated systems are in order.This dissertation investigates linear systems subject to actuator saturation for the further enlargement of the estimates of the domain of attraction of such systems. We have proposed several new approaches, such as multiple auxiliary matrices, partitioning of the convex hull representing saturated linear feedback, partitioning of the virtual input space, the generalized piecewise quadratic Lyapunov function and design of the switching anti-windup compensators, to estimate and enlarge the domain of attraction of saturated systems. These new approaches also apply to estimate or reduce the nonlinear L2 gain of saturated systems. The main contribution of this dissertation is stated as follows.1. Improved convex hull representation of saturated linear feedback via multiple auxiliary feedback matrices and its applications. We introduce a separate auxiliary feedback matrix for each vertex of the traditional convex hulls respectively representing single-layer and nested saturated linear feedbacks, and construct the improved convex hull representations for both of these saturated linear feedbacks. As another application of the idea of introducing multiple auxiliary feedback matrices, we assign a group of independent auxiliary feedback matrices for a composite quadratic Lyapunov function, and establish a set of less conservative sufficient conditions under which a level set of composite quadratic Lyapunov function is contractively invariant and can be as an estimate of the domain of attraction. Based on these conditions, we formulate an optimization problem for maximizing such a level set. The results obtained from this optimization problem show that the resulting maximized estimate of the domain of attraction is significantly larger than the existing estimates.2. Determination of the maximal contractively invariant ellipsoid for saturated linear systems with multiple inputs. We propose an algebraic computation approach for the determination of the maximal contractively invariant ellipsoid. We partition the state space into several regions according to the saturation statuses of each input,and then compute the maximal possible contractively invariant ellipsoids on all these regions and their intersections. The minimal one among these maximal possible invariant sets is the maximal contractively invariant ellipsoid. This approach involves to solve a set of polynomial equations or to compute the eigenvalues of certain specified matrices. Moreover, we propose an LMI-based criterion to determine if the optimal ellipsoid obtained by the optimization problem via the improved convex hull representation of saturated linear feedback is the maximal contractively invariant ellipsoid.Several numerical examples demonstrate the effectiveness of our results.3. Design of the switching anti-windup compensator via partitioning of the convex hull representing saturated linear feedback. We propose to divide the convex hull into several convex sub-hulls, each of which is defined as the convex hull of a subset of the auxiliary feedbacks. Whenever the value of the saturated linear feedback falls into a sub-hull, it can be expressed as a linear combination of a subset of the auxiliary linear feedbacks that define the sub-hull and thus a less conservative result can be expected.A separate anti-windup gain is designed for each sub-hull by using a common quadratic Lyapunov function and implemented when the value of the saturated linear feedback falls into this sub-hull. Simulation results indicate that such a saturation-based switching anti-windup design has the ability to significantly enlarge the domain of attraction of the closed-loop system.4. Estimation of the domain of attraction and the nonlinear L2 gain for a saturated system with an algebraic loop. We construct a virtual input space from the algebraic loop contained in systems, and divide this virtual input space into several regions.This partitioning enables some special properties of the virtual inputs to emerge in different regions of the virtual input space. They are combined with an existing piecewise quadratic Lyapunov function of an augmented state vector composing the system states and the saturation function to arrive at a set of less conservative stability and performance conditions. On the other hand, considering the relationship between the system states and the saturation function, we propose a generalized piecewise quadratic Lyapunov function, which results from adding a term that characterizes the regional sector condition of saturation function to the piecewise quadratic Lypunov function.The matrix associated with the generalized piecewise quadratic Lyapunov function is not required to be positive definite, and thus another set of less conservative stability and performance conditions are established. Simulation results indicate that each of the proposed approaches have the ability to obtain a larger estimate of the domain of attraction and a significantly tighter estimate of the nonlinear L2 gain than the existing methods.