Saturation Control of LTI Systems

Abstract

In this thesis, we consider the stabilization and disturbance attenuation problems for linear systems subject to actuator saturations using gain-scheduling output feedback control. By utilizing convex hull expression of saturating linear feedback law, we parameterize the proposed nonlinear output feedback law in the form of a quasi-LPV system. Conditions that ensure the system stabilizability (stabilization) within a Lyapunov level set as well as achieve disturbance attenuation capabilities (performance) are then established in terms of the coefficient matrices of the controller in the quasi-LPV form. Determination of the controller coefficient matrices is formulated and solved as linear matrix inequalities (LMIs) problem with a variable to be optimized. For stabilization problem the optimization variable is the largest stability region could be obtained; while for performance problem it refers to the smallest L2 gain or L2 to L-infinity gain for magnitude and energy bounded disturbances. The proposed method applies to general linear systems including strictly unstable ones and is presented in both the continuous-time and discrete-time settings, whenever it is possible.

The proposed saturation control approach is further generalized from two aspects: The first one is about the extension of the Lyapunov function from constant to parameter-dependent. Both of the stabilization and disturbance attenuation problems are reformulated within the framework of discrete-time systems. Synthesis conditions can be casted into LMI optimization problems by introducing an extra intermediate matrix. This extra degree of freedom is used to synthesize the parameter-dependent Lyapunov functions. Numerical examples showed that the approach based on parameter-dependent Lyapunov functions indeed provide less conservative results in terms of larger domain of attraction and better disturbance attenuation characteristics than constant Lyapunov functions. The second generalization is about the form of the saturation functions. We generalize the proposed design approach, which is discussed in the context of standard saturation inputs, to a class of saturation-like inputs described by multiple bends piecewise-linear functions. The form of the saturation-like input is represented by fixing a set of tunable parameters including input slopes and bend points, to certain values. Its control synthesis condition can also be solved as LMI optimization problems. The modified saturation control approach is applicable to a wider class of saturation functions, even the class of continuously nonlinear inputs that could be approximated by piecewise concave/convex linear functions.