English

We present a novel numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. Our proposed approach constructs a continuous piecewise affine (CPA) function given a suitable partition of the state space, called a triangulation, and values at the vertices of the triangulation. The vertex values are obtained from a Lyapunov function in a classical converse Lyapunov theorem and verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Furthermore, by refining the triangulation, we show that it is always possible to construct a CPA Lyapunov function. Numerical examples are presented demonstrating the effectiveness of the proposed method.

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.

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

Invariant set estimation for piecewise affine dynamical systems using piecewise affine barrier function

The online computational burden of linear model predictive control (MPC) can be moved offline by using multi-parametric programming, so called explicit MPC. The explicit MPC is a piecewise affine (PWA) function defined over a polyhedral subdivision of the set of feasible states. The online evaluation of such a control law needs to determine the polyhedral region in which the current state lies. This procedure is called the point location problem and its computational complexity is challenging. In this paper a new flexible algorithm is proposed which enables the designer to tradeoff between time and storage complexities. Utilizing the concept of hash tables and the associate hash functions the proposed method is modified to solve an aggregated point location problem in processing complexity independent of the number of polyhedral regions while the storage needs remains tractable. The effectiveness of this approach is supported by several numerical examples.

Mathematical modeling of biological regulatory networks provides valuable insights into the structural and dynamical properties of the underlying systems. While dynamic models based on differential equations provide quantitative information on the biological systems, qualitative models that rely on the logical interactions among the components provide coarse-grained descriptions useful for systems whose mechanistic underpinnings remain incompletely understood. The middle ground class of piecewise affine differential equation models was proven informative for systems with partial knowledge of kinetic parameters. In this work we provide a comparison of the dynamic characteristics of these three approaches applied on several biological regulatory network motifs. Specifically, we compare the attractors and state transitions in asynchronous Boolean, piecewise affine and Hill-type continuous models. Our study shows that while the fixed points of asynchronous Boolean models are observed in continuous Hill-type and piecewise affine models, these models may exhibit different attractors under certain conditions. Overall, qualitative models are suitable for systems with limited knowledge of quantitative information. On the other hand, when practical, using quantitative models can provide detailed information about additional real-valued attractors not present in the qualitative models.

This paper presents a new algorithm for piecewise affine (PWA) approximation of nonlinear systems. Such an approximation is very important to enable a reduction in the complexity of models of nonlinear systems while keeping the global validity of the models. The paper builds on previous work on PWA approximation methods, in particular on the work done by Casselman and Rodrigues, known as the Set of Linearization Points (SLP) PWA approximation. The proposed extension method can be used to approximate any continuous function of one variable by a PWA function. The algorithm is based on the points at which the linearization lines intersect with each other. The method assumes that a desired approximation error and one linearization point are given. The algorithm then performs several linearizations. It is shown that the new linearization points are optimal in the sense of decreasing the error between the exact function and the approximation. The main advantages of this methodology compared to previous approaches are the reduction of the number of pieces of the PWA function, the guarantee that the approximation is continuous, elimination of the numerical optimization to find the point of maximum error, and that the derivative of the approximation and the derivative of the exact function are equal at all linearization points.

This paper proposes a new Piecewise Affine Proportional-Integral-Derivative (PA-PID) controller for cart position tracking and hook and load sway angle suppression of a double pendulum overhead crane (DPOC) system. A Safe Experimentation Dynamics (SED) algorithm is utilized as a model-free optimization tool to find the optimal PA-PID controller parameters such that the integral square of error and input are reduced. The essential feature of the PA-PID controller is that the parameters of proportional, integral and derivative gains are adaptive to the error variations according to the Piecewise Affine (PA) function. Moreover, the proposed PA function is expected to produce better control accuracy than the conventional PID controller. In order to justify the efficacy of the PA-PID controller, a widely known nonlinear model of DPOC plant is considered. The performances of the proposed controller are observed based on the integral square of error and input, and the responses of the cart position, the hook and load sway angle and the control input. The numerical results verify that the proposed PA-PID controller yields higher control accuracy than the conventional PID controller of DPOC system.

Coexhausters are families of convex compact sets that allow one to represent the approximation of the increment of a function at a given point in the form of minmax or maxmin of affine functions. We demonstrate that this representation can be used to define a piecewise affine function and therefore coexhausters are a natural technique for studying the problem of finding a global minimum of piecewise affine functions. All the conditions and methods in the current study were obtained by means of coexhausters theory. Firstly, we apply coexhauster based conditions to state and prove necessary and sufficient conditions for a piecewise affine function to be bounded from below. Secondly, we use coexhausters to construct a simple method which allows one to get the minimum value of the studied function and the corresponding set of all its global minimizers. Illustrative numerical examples are provided throughout the paper.

In traditional vehicle trajectory tracking processes, the dynamic model of the vehicle may not accurately represent complex and nonlinear vehicle behaviors. Moreover, conventional control methods may perform poorly when dealing with system uncertainties and disturbances, facing challenges in real-time computation. To address these issues, this paper proposes an autonomous driving control method based on control-affine feedforward neural network (CAFNN) and fast tube model predictive control (tube-MPC). This method utilizes CAFNN for system dynamic identification, replacing traditional mathematical modeling with data-driven neural network pattern recognition to more accurately describe the vehicle's nonlinear dynamic characteristics. On this basis, the proposed tube-MPC structure is divided into two parts: nominal MPC and sliding mode control (SMC). The nominal MPC controller associates the MPC problem with a linear complementarity problem (LCP) using a ramp function, enabling rapid computation of the quadratic programming (QP) solution through piecewise affine (PWA) functions; the auxiliary SMC controller employs multi-power sliding mode reaching laws to enhance the system's robustness against external disturbances and model uncertainties. This control strategy demonstrates high accuracy and stability in vehicle trajectory tracking under complex road conditions, providing strong support for the advancement of autonomous driving technology.

Among various notions of critical points available for nonsmooth functions, the approach via the ‘weak slope’ has considerable appeal. This property is less restrictive than that based on the ‘strong’ slope of DeGiorgi–Marino–Tosques, but remains purely metric in nature. Within variational analysis, this class of critical points is intermediate between those associated with the Clarke and the limiting subdifferentials. However, recognizing such points for concrete functions seems challenging. We present a basic topological characterization for the simplest nontrivial case: piecewise affine (and, more generally, ‘definable’) functions of two variables.

We study model predictive control (MPC) schemes for discrete-time linear time-invariant systems with constraints on inputs and states, that can be formulated using a linear program (LP). In particular, we focus our attention on performance criteria based on a mixed 1 -norm, namely, 1-norm with respect to time and -norm with respect to space. First we provide a method to compute the terminal weight so that closed-loop stability is achieved. We then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it. The piecewise affine form allows to eliminate online LP, as the computation associated with MPC becomes a simple function evaluation. Besides practical advantages, the availability of the explicit structure of the MPC controller provides an insight into the type of control action in different regions of the state space, and highlights possible conditions of degeneracies of the LP, such as multiple optima.

A relevant challenge in hybrid electric vehicles (HEVs) and full EVs is the torque control of externally excited synchronous machines (EESMs). Effective torque control requires an efficient solution to the state reference generation problem, which is a nonlinear nonconvex optimization problem. The goal of this paper is to develop a state reference generation algorithm based on the gridding of the state and output spaces. First, an approximation defined over a cubic partition of the torque function with a piecewise affine (PWA) function is made. As a result, the state reference generation problem is reduced in each cube to solving a convex optimization problem. Moreover, this approach provides guarantees about the error bound introduced by the state reference generation procedure for the full operational state-space. To illustrate the effectiveness and robustness of the proposed algorithm, several real-time results obtained on an industrial hardware-in-the-loop (HIL) test-bench are presented. The obtained results show significant improvement compared with existing state-of-the-art reference generation methods.

This paper is concerned with the design of an explicit model predictive controller (eMPC) for attitude stabilization using on-off thrusters subject to switching-time constraints. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sup> In the proposed approach, the control actions are specified by a sequence of integer values in the set {-1, 0, +1}, i.e the thrusters can be fully opened in one of two possible directions (-1 or +1) or remain closed (0). Switching-time constraints are taken into account by imposing restrictions on the minimum number of consecutive equal elements in the control sequence. The explicit solution to the constrained MPC problem is computed by solving a multi-parametric mixed integer linear program (mpMILP), where the parameters are the components of the state vector. The solution to the mpMILP is a piecewise affine (PWA) function, which can be evaluated at each sampling time to obtain the optimal control law. Since the on-line computation effort is restricted to a table-lookup, the controller can be implemented using a simple computational hardware.

A duality-based convex optimization approach to L 2 -gain control of piecewise affine slab differential inclusions