An analysis and design method for linear systems subject to actuator saturation and disturbance

We present a method for estimating the domain of attraction of the origin for a system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. This condition is shown to be less conservative than the existing conditions which are based on the circle criterion or the vertex analysis. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. This condition is then extended to determine the invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements. The effectiveness of the developed methods is illustrated with examples.     

Analysis and design for singular discrete linear systems subject to actuator saturation

A method to estimate the domain of attraction for a singular discrete linear system under a saturated linear feedback is established. Simple conditions are derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. These conditions are expressed in terms of linear matrix inequalities. The largest contractively invariant ellipsoid can also be determined by solving an optimization problem with linear matrix inequality constraints. This result is extended to the design of feedback gain that results in the largest contractively invariant ellipsoid, which is also a linear matrix inequality optimization problem. A numerical example demonstrates the applicability and effectiveness of the presented method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Exact characterization of invariant ellipsoids for single inputlinear systems subject to actuator saturation

We present a necessary and sufficient condition for an ellipsoid to be an invariant set of a linear system under a saturated linear feedback. The condition is given in terms of linear matrix inequalities (LMIs) and can be easily used for optimization-based analysis and design     

Set Invariance Conditions for Singular Linear Systems Subject to Actuator Saturation

In this note, we establish a set of conditions under which an ellipsoid is contractively invariant with respect to a singular linear system under a saturated linear feedback. These conditions can be expressed in terms of linear matrix inequalities, and the largest contractively invariant ellipsoid can be determined by solving an optimization problem with LMI constraints. With the feedback gain viewed as an additional variable, this optimization problem can be readily adapted for the design of feedback gain that results in the largest contractively invariant ellipsoid. Moreover, in the degenerate case where the singular linear system reduces to a regular system, our set invariance conditions reduce to the existing set invariance conditions for normal linear systems.     

On IQC approach to the analysis and design of linear systems subject to actuator saturation

This paper establishes IQC-based (Integral Quadratic Constraints) conditions under which an ellipsoid is contractively invariant for a single input linear system under a saturated linear feedback law. Based on these set invariance conditions, the determination of the largest such ellipsoid, for use as an estimate of the domain of attraction, can be formulated and solved as an LMI optimization problem. Such an LMI problem can also be readily adapted for the design of the feedback gain that achieves the largest contractively invariant ellipsoid. While the advantages of the proposed IQC approach remain to be explored, it is shown in this paper that the largest contractively invariant ellipsoid determined by this approach is the same as the one determined by the existing approach based on expressing the saturated linear feedback as a linear differential inclusion (LDI), which is known to lead to non-conservative result in determining the largest contractively invariant ellipsoid for single input systems.     

An analysis and design method for discrete-time linear systems under nested saturation

This paper considers a discrete-time linear system under nested saturation. Nested saturation arises, for example, in systems with actuators subject to both magnitude and rate saturation. A condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant. Moreover, this condition is shown to be equivalent to linear matrix inequalities (LMIs) in the actual and auxiliary feedback gains. As a result, the estimation of the domain of attraction for a given set of feedback gains is then formulated as an optimization problem with LMI constraints. By viewing the feedback gains as extra free parameters, the optimization problem can be used for controller design.     

An analysis and design method for linear systems under nested saturation

This paper considers a linear system under nested saturation. Nested saturation arises, for example, when the actuator is subject to magnitude and rate saturation simultaneously. A condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant. Moreover, this condition is shown to be equivalent to linear matrix inequalities (LMIs) in the actual and auxiliary feedback gains. As a result, the estimation of the domain of attraction for a given set of feedback gains can be formulated as an optimization problem with LMI constraints. By viewing the feedback gains as extra free parameters, the optimization problem can be used for controller design.     

Controlled invariance of ellipsoids: linear vs. nonlinear feedback

Several equivalent conditions or statements for set invariance were obtained for systems with one saturating actuator in a recent paper. In particular, it was shown that the existence of a nonlinear feedback that makes an ellipsoid invariant is equivalent to the existence of a feedback linear inside the ellipsoid that makes it invariant. In this paper, we will show that this equivalence property holds conditionally for systems with multiple saturating actuators. We will provide a criterion to check if the largest ellipsoid made invariant by nonlinear feedback can also be made invariant by a feedback linear inside the ellipsoid. Numerical examples reveal that this criterion is usually satisfied. The equivalence of other set invariance conditions will also be investigated.     

The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method.      

Composite quadratic Lyapunov functions for constrained control systems

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.