Piecewise Linear Lyapunov Functions Research Articles

Counterexample-guided computation of polyhedral Lyapunov functions for piecewise linear systems

This paper presents a counterexample-guided iterative algorithm to compute convex, piecewise linear (polyhedral) Lyapunov functions for continuous-time piecewise linear systems. Polyhedral Lyapunov functions provide an alternative to commonly used polynomial Lyapunov functions. Our approach first characterizes intrinsic properties of a polyhedral Lyapunov function including its “eccentricity” and “robustness” to perturbations. We then derive an algorithm that either computes a polyhedral Lyapunov function proving that the system is asymptotically stable, or concludes that no polyhedral Lyapunov function exists whose eccentricity and robustness parameters satisfy some user-provided limits. Significantly, our approach places no a-priori bound on the number of linear pieces that make up the desired polyhedral Lyapunov function. The algorithm alternates between a learning step and a verification step, always maintaining a finite set of witness states. The learning step solves a linear program to compute a candidate Lyapunov function compatible with a finite set of witness states. In the verification step, our approach verifies whether the candidate Lyapunov function is a valid Lyapunov function for the system. If verification fails, we obtain a new witness. We prove a theoretical bound on the maximum number of iterations needed by our algorithm. We demonstrate the applicability of the algorithm on numerical examples.

Analysis of sliding-mode control systems with relative degree altering perturbations

We consider sliding-mode control systems subject to unmatched perturbations. Classical first-order sliding-mode techniques are capable to compensate unmatched perturbations if differentiations of the output of sufficiently high order are included in the sliding variable. For such perturbations it is commonly assumed that they do not affect the relative degree of the system. In this contribution we consider perturbations that alter the relative degree of the process and study their impact on the closed-loop control system with a classical first-order sliding-mode design. In particular we consider systems with full (nominal) relative degree subject to a perturbation reducing the relative degree by one and analyse the resulting closed-loop system. It turns out that the sliding-manifold is not of reduced dimension and the uniqueness of the solution may be lost. Also attractivity of the sliding-manifold and global stability of the origin may be lost whereas the disturbance rejection properties of the sliding-mode control are not impaired. We present a necessary and sufficient condition for the existence of unique solutions for the closed-loop system. The second-order case is studied in great detail and allows to parametrically specify the conditions obtained before. We derive a necessary condition for the global asymptotic stability of the closed-loop system. Further we present a constructive condition for the global asymptotic stability of the closed-loop system using a piece-wise linear Lyapunov function. Each of the prominent results is illustrated by a numerical example.

Stochastic stability of positive Markov jump linear systems with fixed dwell time

This paper studies the stochastic stability of positive Markov jump linear systems with a fixed dwell time. By constructing an auxiliary system that originated from the initial system with state jumps, sufficient and necessary conditions of stochastic stability for positive Markov jump linear systems are obtained with both exactly known and partially known transition rates. The main idea in the latter case is applying a convex combination to convert bilinear programming into linear programming problems. On this basis, multiple piecewise linear co-positive Lyapunov functions are provided to achieve less conservative results. Then state feedback controller is designed to stabilize the positive Markov jump linear systems by solving linear programming problems. Numerical examples are presented to illustrate the viability of our conclusions.

Mode-dependent impulsive control of positive switched systems: Stability and L1-gain analysis

Abstract This paper studies the stability and L1-gain problems for impulsive positive switched systems (IPSS) consisting of both unstable and stable subsystems, and proposes a mode-dependent impulsive control (MDIC) method for the first time. By means of a piecewise linear co-positive Lyapunov function (PLCLF) and using average dwell time (ADT) approach, some sufficient criteria for global exponential stability (GES) and standard L1-gain are established in terms of linear programming (LP), which can be verified conveniently by Matlab. A special impulsive controller is designed to guarantee the positivity and stability of the addressed system. Several numerical examples with simulations are provided to demonstrate the effectiveness of the obtained results.

Observer-Based Control for Cyber-Physical Systems With Periodic DoS Attacks via a Cyclic Switching Strategy

This article studies the observer-based output feedback control problem for a class of cyber-physical systems with periodic denial-of-service (DoS) attacks, where the attacks coexist both in the measurement and control channels in the network scenario. The periodic DoS attacks are characterized by a cyclic dwell-time switching strategy, such that the resulting augmented system can be converted into a class of discrete-time cyclic dwell-time switched systems including a stable subsystem and an unstable subsystem. By means of a cyclic piecewise linear Lyapunov function approach, the exponential stability and $l_2$ -gain analysis, and observer-based controller design are carried out for the augmented discrete-time cyclic switched system. Then, the desired observer and controller gains in piecewise linear form are determined simultaneously so as to ensure that the resulting closed-loop system is exponentially stable with a prescribed $\mathcal {H}_{\infty }$ performance index. Finally, a practical application of unmanned ground vehicles under periodic DoS attacks is provided to verify the effectiveness of the developed control approach.

Problem of Stabilizing a Switching System Using a Piecewise-Linear Control System

The problem of stabilizing a mathematical hybrid system with switchings between the operating modes is solved. Each of these modes is associated with nonlinear differential equations that have control parameters. The switching instances (conditions) are control components. A stabilizer must be designed in positional form that allows the trajectory of the entire nonlinear system to reach the target set in the phase space for a (prescribed) finite time. To solve the problem, k]an apparatus of continuous piecewise-linear Lyapunov functions is used along with the corresponding piecewise-linear control functions. A theorem concerning the sufficient conditions for the stabilizability of a hybrid system in the considered class of controls is proved. An algorithm for constructing the Lyapunov functions and the stabilizer is given.

Модифицированный алгоритм определения полных областей устойчивости нестационарных нелинейных систем

The paper proposes a numerical algorithm for constructing Lyapunov spline functions for investigating the absolute stability of nonlinear nonstationary systems. In the case of asymptotic stability of the system, the implementation of the algorithm will lead to the construction of the Lyapunov function level set in the form of a piece-wise smooth (smooth, if additional conditions are met) closed surface of dimension equal to the dimension of the original system. It is shown that the modified algorithm significantly improves the stability boundary estimates obtained with frequency methods. Unlike the algorithm for constructing piecewise linear Lyapunov functions, the running time of the proposed algorithm for constructing the Lyapunov spline functions does not tend to infinity as the system approaches the stability boundary. This circumstance makes it possible to use a modified algorithm to determine the stability of systems that are close to the stability boundary. An estimate of the accuracy of determining the stability area using an example of a third-order system is shown. Specific recommendations on the algorithm initial conditions choice are given.

Алгоритм определения полных областей устойчивости нестационарных нелинейных систем

The paper proposes a numerical algorithm for constructing piecewise linear Lyapunov functions for investigating the absolute stability of nonlinear nonstationary systems. Such functions define necessary and sufficient conditions for the stability of nonlinear nonstationary systems satisfying sector constraints. In the case of asymptotic stability of the system, the implementation of the algorithm will lead to the construction of the Lyapunov function level set in the form of a polyhedron of dimension equal to the dimension of the original system. Such a polyhedron can be used for constructing a piecewise linear Lyapunov function. The number of faces of the polyhedron increases as the system approaches to the stability boundary in the parameter space, which can lead to unacceptable time costs for calculations. The analysis of specific systems of the 2nd and 3rd order and the results of comparison with classical methods are given. Specific recommendations on the algorithm initial conditions choice are given.

Calculation of absolute stability regions for time-varying electromechanical systems

The paper proposes a numerical method of constructing piecewise-linear Lyapunov functions for absolute stability of nonlinear time-varying systems researching. There are examples and results of comparison with classical methods.

Structural Stability of Biochemical Networks: Quadratic vs. Polyhedral Lyapunov Functions

Due to the intrinsic uncertainty and variability affecting biochemical reaction networks, it is fundamental to assess their structural stability, i.e., to establish if all the networks having a given structure are stable independent of specific parameter values. For basic motifs in biochemical networks, we show that stability cannot be structurally proved by quadratic Lyapunov functions. However, structural stability of these motifs can be shown resorting to piecewise-linear Lyapunov functions, based on the results by Blanchini and Giordano (2014), who provide a theoretical framework and efficient numerical methods to evaluate structural stability of biochemical reaction networks with monotone reaction rates.

Robust constrained stabilization of boost DC–DC converters through bifurcation analysis

This paper proposes a new methodology for designing robust affine state-feedback control laws, so that wide-range safe and efficient operation of switched-mode DC–DC boost converters is guaranteed. Several undesirable nonlinear phenomena such as unstable attractors and subharmonic oscillations are avoided through bifurcation analysis based on the bilinear averaged model of the converter. The control design procedure also relies on constrained stabilization principles and the generation of safety domains using piecewise linear Lyapunov functions, so that robustness to supply voltage and output load variations is ensured, while input saturation is avoided and additional state constraints are also respected. The technique has been numerically and experimentally validated. • We combine constrained stabilization and bifurcation theory for DC–DC converters. • We demonstrate that performance deterioration due to multiple equilibria is possible. • A thorough investigation of the bifurcation phenomena is presented. • We prove that the averaged model can unfold the core nonlinear system׳s behavior. • A new control design procedure for boost DC–DC converters is proposed.

Piecewise-linear Lyapunov functions for structural stability of biochemical networks

We consider the problem of assessing structural stability of biochemical reaction networks with monotone reaction rates, namely of establishing if all the networks with a certain structure are stable regardless of specific parameter values. We investigate stability by absorbing the network equations in a linear differential inclusion and seeking for a polyhedral Lyapunov function proper to the considered network structure. A numerical recursive procedure is devised to test stability. For a wide class of mono- and bimolecular reaction networks, which we name unitary, the procedure is shown to be very efficient since, due to the particular structure of the problem, it requires iterations in the space of integer-valued matrices. We also consider a similar, less conservative procedure that allows us to test, even when the Lyapunov function cannot be found, whether the system evolution is structurally bounded. In this case, we absorb the equations in a positive linear differential inclusion. To show the effectiveness of the proposed procedure, we report the outcomes of both a stability and a boundedness test, for many non-trivial biochemical reaction networks, and we analyze well established models in the literature.

Stabilizing Model Predictive Control of Stochastic Constrained Linear Systems

This paper investigates stochastic stabilization procedures based on quadratic and piecewise linear Lyapunov functions for discrete-time linear systems affected by multiplicative disturbances and subject to linear constraints on inputs and states. A stochastic model predictive control (SMPC) design approach is proposed to optimize closed-loop performance while enforcing constraints. Conditions for stochastic convergence and robust constraints fulfillment of the closed-loop system are enforced by solving linear matrix inequality problems off line. Performance is optimized on line using multistage stochastic optimization based on enumeration of scenarios, that amounts to solving a quadratic program subject to either quadratic or linear constraints. In the latter case, an explicit form is computable to ease the implementation of the proposed SMPC law. The approach can deal with a very general class of stochastic disturbance processes with discrete probability distribution. The effectiveness of the proposed SMPC formulation is shown on a numerical example and compared to traditional MPC schemes.

Existence of piecewise linear Lyapunov functions in arbitrary dimensions

Lyapunov functions are an important tool to determine the basin of attraction of exponentially stable equilibria in dynamical systems. In Marinosson (2002), a method to construct Lyapunov functions was presented, using finite differences on finite elements and thus transforming the construction problem into a linear programming problem. In Hafstein (2004), it was shown that this method always succeeds in constructing a Lyapunov function, except for a small, given neighbourhood of the equilibrium. For two-dimensional systems, this local problem was overcome by choosing a fan-like triangulation around the equilibrium. In Giesl/Hafstein (2010) the existence of a piecewise linear Lyapunov function was shown, and in Giesl/Hafstein (2012) it was shown that the above method with a fan-like triangulation always succeeds in constructing a Lyapunov function, without any local exception. However, the previous papers only considered two-dimensional systems. This paper generalises the existence of piecewise linear Lyapunov functions to arbitrary dimensions.

Stabilisation of discrete-time switched positive linear systems via time- and state-dependent switching laws

This study investigates the stabilisation problem of discrete-time switched positive linear systems by means of piecewise linear copositive Lyapunov functions. Two stabilisation strategies are designed under time- and state-dependent switching cases, respectively. The former case aims at determining an upper bound of the minimum dwell time to guarantee that the underlying system is stable for any switching signal with dwell time greater than this bound. The latter case is focused on deriving a state-dependent switching law stabilising the underlying system from the solution of a family of so-called linear copositive Lyapunov–Metzler inequalities. In each case, a sufficient stabilisation condition is given first, then based on which an associated guaranteed cost is further proposed. A practical system derived from the distributed power control in communication networks is given to illustrate the effectiveness and applicability of the theoretical results.

A Convex Condition for Robust Stability Analysis via Polyhedral Lyapunov Functions

In this paper we study the robustness analysis problem for linear continuous-time systems subject to parametric time-varying uncertainties making use of piecewise linear (polyhedral) Lyapunov functions. A given class of Lyapunov functions is said to be “universal” for the uncertain system under consideration if the robust stability of the system is equivalent to the existence of a Lyapunov function belonging to the class. In the literature it has been shown that the class of polyhedral functions is universal, while, for instance, the class of quadratic functions is not. This fact justifies the effort of developing efficient algorithms for the construction of polyhedral Lyapunov functions. In this context, we provide a low computational cost procedure, based on a novel convex condition, for the construction of a polyhedral Lyapunov function. In the section on the numerical examples, we consider some benchmark problems for the robust stability analysis and we show that the proposed low computational cost approach, though only sufficient, is less conservative than all the other approaches presented so far in the literature.

Padé Discretization for Linear Systems With Polyhedral Lyapunov Functions

This technical note has been motivated by the need to assess the preservation of polyhedral Lyapunov functions for stable continuous-time linear systems under numerical discretization of the transition matrix. This problem arises when discretizing linear systems in such a manner as to preserve a certain type of stability of the discrete time approximation. Our main contribution is to show that a continuous-time system and its Padé discretization (of any order and sampling) always share at least one common piecewise linear (polyhedral) Lyapunov function.

Constrained switching stabilization of linear uncertain switched systems using piecewise linear Lyapunov functions

The problem of switching stabilizability of linear uncertain switched systems with unstable modes has recently attracted increased interest. For a given collection of unstable linear uncertain subsystems, the problem is to determine asymptotically stabilizing switching signals or to ascertain the absence of such laws. In this paper, we deal with the switching stabilizability problem for a class of autonomous continuous-time switched linear systems with time-varying parametric uncertainties in the presence of additional state constraints. With a special emphasis on two-dimensional systems we extend previous relevant results in the literature, and we propose a computational technique that generates piecewise linear Lyapunov functions of a convex or non-convex nature. The algorithm is graph-based and consists of determining controlled invariant polytopes induced by systematic conic decompositions of the state-space and corresponding state-dependent switching control actions. Further extension of this computational tool to higher dimensional systems is also discussed.

Stability and instability conditions using polyhedral Lyapunov functions

In this article, the authors study the robustness analysis problem for linear continuous-time systems subject to parametric time-varying uncertainties making use of piecewise linear (polyhedral) Lyapunov functions. A given class of Lyapunov functions is said to be ‘universal’ for the uncertain system under consideration if the search of a Lyapunov function that proves the robust stability of the system can be restricted, without conservatism, to the elements of the class. In the literature, it has been shown that the class of polyhedral functions is universal, while, for instance, the class of quadratic Lyapunov functions is not. This fact justifies the effort of developing efficient algorithms for the construction of optimal polyhedral Lyapunov functions. In this context, the authors provide a novel procedure that enables to construct, in the general n-dimensional case, a polyhedral Lyapunov function to prove the robust stability of a given system. Some numerical examples are included, where the authors show the effectiveness of the proposed approach comparing it with other approaches proposed in the literature.

Stability Criteria for Switched and Hybrid Systems

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.