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International audience; We further develop a novel control paradigm encoding the core behavior of a reset control system within a continuoustime feedback, described by a differential inclusion. A tuning knob allows adjusting how close the continuous-time feedback resembles the behavior of the underlying reset controller. The Lyapunov conditions that we derive naturally lead to a sum-of-squares formulation allowing for the design of max-of-quadratics Lyapunov certificates certified by polynomial multipliers. A few application examples confirm the practical effectiveness of the proposed solutions.

In this paper we develop a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems in a general configuration. The only assumptions we make about the system are local stability and well-posedness of the algebraic loop. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of norm-bounded disturbances and the nonlinear L 2 gain. The regional analysis is established upon an effective treatment of the algebraic loop and the deadzone function. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a normbounded differential inclusion (NDI), for the description of the original system. The corresponding conditions for stability and performance are derived as Linear Matrix Inequalities (LMIs).

In a companion paper [14], we developed a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems via quadratic Lyapunov functions. The corresponding conditions are expressed in terms of LMIs but can be too conservative in some cases. To obtain less conservative conditions, we use in this paper two types of conjugate Lyapunov functions: the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the LMI conditions for quadratic stability as special cases and hence the reduction of conservativeness is guaranteed. A numerical example demonstrates the effectiveness of this paper’s methods and the great potential of the non-quadratic Lyapunov functions.

In this paper, we report on the experimental application of an anti-windup strategy recently proposed in the literature. The anti-windup law applies to general Euler Lagrange systems but, when specialized to linear plants become similar to a nested saturations approach. We show its effectiveness on a DC motor connected to a load, which acts like an integrating linear plant. The experimental results show highly desirable performance of the compensated closed-loop.

In this paper we propose LMI methods for designing dynamic/static anti-windup compensators to improve regional performance and stability of linear control systems with saturating actuators. Algorithms are developed for minimizing the upper bound on the regional /spl Lscr//sub 2/ gain for exogenous inputs with /spl Lscr//sub 2/ norm bounded by a given value, and for minimizing this upper bound with a guaranteed reachable set or domain of attraction. Based on the structure of the optimization problems, it is shown that for systems whose plants have poles in the closed left-half plane, plant order dynamic anti-windup can achieve semiglobal stability and finite /spl Lscr//sub 2/ gain for exogenous inputs with /spl Lscr//sub 2/ norm hounded by any finite value. The problems are studied in a general setting where the only requirement on the linear control system is well-posedness and internal stability.

Motivated by questions in stability theory for hybrid dynamical systems, we establish some fundamental properties of the set of solutions to such systems. Using the notion of a hybrid time domain and general results on set and graphical convergence, we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and "upper semicontinuous" with respect to initial conditions and system perturbations. The latter means that each solution to the system under perturbations is close to some solution of the unperturbed system on a compact hybrid time domain. The general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets. For example, it is shown that the basin of attraction for a compact attractor is (relatively) open, the attractivity is uniform from compact subsets of the basin of attraction, and asymptotic stability is robust with respect to small perturbations.

This paper proposes a Lyapunov approach to frequency analysis for general systems. The notion of frequency response is extended to general systems through a connection in linear systems. Lyapunov approaches to the characterization of frequency response are established for linear systems, homogeneous systems and nonlinear systems, respectively. In particular, we show that for linear systems, quadratic Lyapunov functions are sufficient for the characterization; for homogeneous systems, homogeneous Lyapunov functions are sufficient; and for general nonlinear systems, locally Lipschitz Lyapunov functions will be used. We also develop a Lyapunov approach for the characterization of the peak of the output. This approach is demonstrated to be effective on linear systems. An LMI based method for performing frequency analysis on linear differential inclusions is developed. Through a numerical example, an interesting phenomenon is observed about the relation between the frequency response and the L/sub 2/ gain of linear differential inclusions.

A novel characterization of bumpless transfer among alternative controllers, both in continuous and discrete time, is introduced. The bumpless transfer problem is to design a compensation scheme guaranteeing an L/sub 2/ (respectively l/sub 2/) bound on the mismatch, after the switching time from one controller to another between the actual plant output and a particular target plant output that corresponds to an ideal response. Minimization of the gain from initial plant state mismatch to L/sub 2/ (respectively, l/sub 2/) plant output mismatch provides improved transients. A solution to the bumpless transfer problem is given, based on the L/sub 2/ anti-windup solution described in Teel and Kapoor (1997) and Grimm et al. (2003), both for plants without input saturation and for plants with input saturation. The solution guarantees anti-windup features in the latter case. The proposed methodology is shown to perform well on a simulation example.

We use a recently developed duality theory for linear differential inclusions (LDIs) to enhance the stability analysis of systems with saturation. Based on the duality theory, the condition of stability for a LDI in terms of one Lyapunov function can be easily derived from that in terms of a Lyapunov function conjugate to the original one in the sense of convex analysis. This paper uses a particular conjugate pair, the convex hull of quadratics and the maximum of quadratics, along with their dual relationship, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a LDI system with an effective approach which enables optimization on the local LDI description. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates that the estimation of the domain of attraction by this paper's methods drastically improve those by the earlier methods.

We introduce a concept of input-to-output practical stability (IOpS) which is a natural generalization of input-to-state stability proposed by Sontag (1989). It allows us to establish two important results. The first one states that the general interconnection of two IOpS systems is again an IOpS system if an appropriate composition of the gain functions of the components is smaller than the identity function. The second one shows an example of gain function assignment by feedback. Applications to the problem of global stabilization via partial-state feedback and output feedback are also considered. >

In this paper, the L/sub 2/ anti-windup approach introduced by A. R. Teel et al. (1997) is applied to the problem of bumpless transfer in (saturated) multi-controller systems, and to the problem of designing highly reliable saturated control systems exploiting hardware redundancy. We illustrate the L/sub 2/ anti-windup technique by means of a simple physical example and illustrate its application to the above mentioned problems using this example and an example taken from the literature.