Affine Lyapunov Function Research Articles

Stability Analysis of Hybrid Integrator-Gain Systems using Linear Programming

Stability and performance of hybrid integrator-gain systems (HIGS) are mostly analyzed using linear matrix inequalities (LMIs) to construct continuous piecewise quadratic (CPQ) Lyapunov functions. To create additional methods for the analysis of HIGS and other conewise linear systems, a method based on linear programming (LP) for constructing continuous piecewise affine (CPA) Lyapunov functions is investigated. In this paper, it is shown how linear programming can be used to prove input-to-state stability and calculate upper bounds on the L1-gain and H1-norm. The numerical efficiency of this CPA (LP) method will be compared to that of the CPQ (LMI) method on a numerical example involving HIGS.

Discontinuous Lyapunov functions for a class of piecewise affine systems

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

Construction of continuous and piecewise affine Lyapunov functions via a finite-time converse

Abstract: A novel numerical Massera-type approach for the computation of Lyapunov functions (LFs) for nonlinear continuous-time systems is presented. The construction is enabled by verifying a finite-time decrease condition for a candidate function, which is allowed to be any K∞ function of the norm of the state, and applying a converse Lyapunov theorem. In the construction we make use of approximated system trajectories and we obtain a continuous and piecewise affine (CPA) LF. By optimization, the obtained CPA LF is verified and an estimate of the domain of attraction is obtained. Several examples are presented for illustration and demonstration of the effectiveness of the proposed approach.

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise affine function can be parameterized by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterized function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.

Computing continuous and piecewise affine lyapunov functions for nonlinear systems

We present a numerical technique for the computation of a Lyapunov functionfor nonlinear systems with an asymptotically stable equilibrium point. The proposed approachconstructs a partition of the state space, called a triangulation, and then computes values atthe vertices of the triangulation using a Lyapunov function from a classical converse Lyapunovtheorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuousand Piecewise Affine (CPA) function.Verification that the obtained CPA function is a Lyapunov function is shownto be equivalent to verification of several simple inequalities.Numerical examples are presented demonstrating different aspects of the proposed method.

Stability of Prioritized Scheduling Policies in Manufacturing Systems With Setup Times

We study the dynamic scheduling of a multi-item machine with setup times and static priorities. We consider a class of base-stock policies for this system and seek sharp stability conditions that allow us to respect the item priorities over a large region of the decision space. We show that requiring the existence of a linear Lyapunov function that decreases between production runs reduces to an intuitive stability condition, which can nevertheless be too conservative for this policy class. We then sharpen this result by showing that, if the original condition is satisfied for the highest-priority $N\!\!-\!\!1$ part types, the system with $N$ items is stable. Finally, we develop stability conditions based on affine Lyapunov functions.

Computation of Lyapunov functions for nonlinear discrete time systems by linear programming

Given an autonomous discrete time system with an equilibrium at the origin and a hypercube containing the origin, we state a linear programming problem, of which any feasible solution parameterizes a continuous and piecewise affine (CPA) Lyapunov function for the system. The linear programming problem depends on a triangulation of the hypercube. We prove that if the equilibrium at the origin is exponentially stable, the hypercube is a subset of its basin of attraction, and the triangulation fulfils certain properties, then such a linear programming problem possesses a feasible solution. We present an algorithm that generates such linear programming problems for a system, using more and more refined triangulations of the hypercube. In each step the algorithm checks the feasibility of the linear programming problem. This results in an algorithm that is always able to compute a Lyapunov function for a discrete time system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibrium's basin of attraction. The system is assumed to have a right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type such as linear, piecewise affine and so on. Our approach is a non-trivial adaptation of the CPA method to compute Lyapunov functions for continuous time systems to discrete time systems.

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibriumʼs basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

Construction of Lyapunov functions for nonlinear planar systems by linear programming

Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions (Giesl and Hafstein, 2010 [7]). Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in Marinosson (2002) [17] and further improved in Hafstein (2007) [10]. The algorithm constructs a linear programming problem for the system at hand, and any feasible solution to this problem parameterizes a Lyapunov function for the system. We prove that the algorithm always succeeds in constructing a Lyapunov function if the system possesses an exponentially stable equilibrium. The size of the region of the Lyapunov function is only limited by the region of attraction of the equilibrium and it includes the equilibrium.

Robust ℋ ∞ filter design for singular systems with time-varying uncertainties

This article deals with the design of robust ℋ∞ filter for singular systems in general form, subject to bounded time-varying uncertainties. The aim is to design linear admissible descriptor filters that ensure a guaranteed bound on the ℒ2 gain of the operator from the noise signals to the estimation error signal, irrespective of the time-varying uncertain parameters. A strict linear matrix inequality based method is proposed to the robust ℋ∞ filter design employing affine Lyapunov functions. A numerical example is presented to show the efficiency of the proposed method.

Computing Lyapunov functions for strongly asymptotically stable differential inclusions

We present a numerical algorithm for computing Lyapunov functions for a class of strongly asymptotically stable nonlinear differential inclusions which includes switched systems and systems with uncertain parameters. The method relies on techniques from nonsmooth analysis and linear programming and leads to a piecewise affine Lyapunov function. We provide a thorough analysis of the method and present two numerical examples.

Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.

Robust saturation controller for linear time-invariant system with structured real parameter uncertainties

This paper is focused on a robust saturation controller for the linear time-invariant (LTI) system involving both actuator's saturation and structured real parameter uncertainties. The controller suggested in this paper can analytically prescribe the upper and lower bounds of parameter uncertainties, and guarantee the closed-loop robust stability of the system in the presence of actuator's saturation. The suboptimal bang–bang control method is extended to LTI system with parameter uncertainties. Based on affine quadratic stability and multi-convexity concept, the robust optimal bang–bang controller is newly derived by minimizing the time derivative of affine Lyapunov function subjected to the limit of control force. Since this controller is a gain-scheduled type, it requires the exact knowledge of uncertain parameters. Another robust saturation controller with a fixed gain is proposed and the linear matrix inequality ( LMI)- based sufficient existence conditions for a fixed-gain controller are derived. The effectiveness and the availability of the proposed controller are investigated by a practical numerical example. Through numerical simulations, it is confirmed that the proposed robust saturation controller is robustly stable with respect to parameter uncertainties over the prescribed range defined by the upper and lower bounds.

Stability of a PI-Controlled Nonlinear System

This paper studies two methods to reduce the conservatism of the stability analysis of a PI-controlled nonlinear process: 1-gain scheduling and 2-parameter affine Lyapunov functions. The use of a gain-scheduled PI controller was found to allow for a significant increase in the proportional gain compared to the fixed controller case, resulting in more aggressive control. In contrast, the use of the parameter-affine Lyapunov function lead to only a marginal reduction of conservatism of the stability test compared to the case where a parameter independent function is used.