Discrete-time Piecewise Affine Systems Research Articles

Data-driven control for discrete-time piecewise affine systems

This paper studies the problem of data-driven control for discrete-time piecewise affine (PWA) systems. Based on a sequel of sampled control inputs and states with satisfactory rank conditions, the closed loop dynamics of PWA systems can be represented and therefore the explicit information of system matrices is not needed in system stability analysis and controller synthesis processes. First, via S-procedure, system stability analysis results have been obtained, and then, two novel data-driven controller design methods in the form of linear matrix inequalities and equations have been proposed such that the resulting closed loop system is asymptotically stable. For PWA systems with unknown state space partition information, an online data-driven control method has been further proposed to guarantee the asymptotic stability of the resulting closed loop systems. Finally, simulation studies of two examples have been given to validate the effectiveness of the proposed data-driven controller design approaches.

Event-triggered control for discrete-time piecewise affine systems

This work addresses the event-triggered control (ETC) of discrete-time piecewise affine systems. We propose a method to design a triggering strategy relying on an implicit representation of piecewise affine systems. Thanks to this implicit representation based on ramp functions, we propose a partition-dependent piecewise quadratic functions to define the trigger criterion and use a piecewise quadratic Lyapunov function candidate to derive conditions to certify the global exponential stability of the origin under the ETC strategy. Since the stability conditions can be expressed as linear matrix inequalities constraints, we propose a convex optimization solution to design the triggering function parameters and to compute the Lyapunov function to ensure the closed-loop stability and a reduction on the control updates. The approach is illustrated by numerical examples.

An implicit representation for the analysis of piecewise affine discrete-time systems

This paper presents a new framework for stability assessment of discrete-time piecewise affine systems. An implicit representation for piecewise functions based on ramp nonlinearities is proposed. Instead of the usual sector inequalities adopted in the study of Lurie systems, we directly exploit properties of the ramp function, which are given by a particular set of identities and inequalities. These properties are then used to derive stability conditions associated to piecewise quadratic Lyapunov functions. These functions are implicitly defined from quadratic forms involving ramp functions. These conditions can be cast as LMIs and their numerical solution is illustrated by examples highlighting the advantages of the proposed method over existing stability analysis methods for PWA systems.

Robust Action Governor for Discrete-Time Piecewise Affine Systems With Additive Disturbances

In this letter, we introduce an extension of the Action Governor (AG), which is an add-on, supervisory scheme to a nominal control loop to enforce safety-related requirements. This extension enables AG design based on discrete-time piecewise affine (PWA) system models with additive set-bounded disturbance inputs and non-convex exclusion-zone avoidance constraints. We establish theoretical properties and computational approaches for this robust version of AG, and illustrate its application to an autonomous vehicle control problem.

Event-triggered feedback control for discrete-time piecewise-affine systems subject to input saturation and bounded disturbance

The problem of event-triggered control for discrete-time piecewise-affine systems subject to input saturation and bounded disturbance is investigated in this paper. An LMI-based event-triggered controller is designed to guarantee the local asymptotic stability by introducing auxiliary feedback matrices to handle input saturation. In addition, the region of attraction is well estimated. Meanwhile, by synthesising the feedback control gains and the given event-triggering condition, an optimisation problem for maximising the region of attraction is formulated in LMIs. For input-saturated piecewise-affine systems with bounded disturbance, our control objective is to reduce the influence of disturbance while reducing communication in order to guarantee that the system is uniformly ultimately bounded. The effectiveness and advantages of the proposed approaches are demonstrated by simulations.

Event-triggered H ∞ control for discrete-time piecewise affine systems with norm-bounded uncertainties

This paper is concerned with the problem of designing event-triggered controllers for discrete-time piecewise-affine (PWA) systems with norm-bounded uncertainties. Event-triggered scheme transmits signals only when the event-triggering condition is violated. Due to its advantage in saving communication resources, it is utilised to design the state feedback controllers for piecewise-affine systems. With the utilisation of a piecewise Lyapunov function combined with the S-procedure and some matrix inequality convexifying techniques, sufficient conditions for novel event-triggered controllers are formulated in the form of linear matrix inequalities (LMIs). The resulting closed-loop system stability is ensured with a guaranteed disturbance attenuation level while the number of transmitted signals is reduced. A numerical example is given to show the effectiveness and merits of our method.

Robust bounded feasibility verification of piecewise affine systems via reachability computations

We address robust bounded feasibility verification for a discrete-time PieceWise Affine (PWA) system whose evolution can be influenced by some input. The aim is to determine an input sequence that makes the system satisfy a certain property within a finite time horizon, while maximizing the amount of perturbation that can be applied to the input without violating the given property. This is important to assess the robustness of the solution to numerical errors. We focus on the case of a property expressed in terms of the output of the system taking values in a certain spec set, and propose a verification method resting on reachability computations. The idea is to determine the set of states that the system can reach through all its possible evolutions, with the input taking values in its whole range, and check if the computed reach sets intersect with the one corresponding to the output spec set. If this is the case, then, an input sequence driving the output to the spec set exists and can be determined together with its robustness level by solving a linear optimization program.

Event-triggered feedback control for discrete-time piecewise affine systems subject to input saturation

The problem of event-triggered control is investigated for discrete-time piecewise affine systems subject to input saturation. Linear matrix inequality (LMI) based on local stability criterion is formed by introducing a nonlinear dead-zone function which represent saturation characteristics. Unique state feedback control gains can be obtained by solving the LMI problem under the event-triggering strategy. The domain of attraction is estimated and maximized with the controller synthesis via ellipsoidal approximations. The effectiveness of controller is illustrated by simulations of numerical examples.

Event-triggered feedback control for discrete-time piecewise-affine systems

ABSTRACTIn this paper, the problem of event-triggered control for discrete-time piecewise-affine system is investigated. An optimisation problem for minimising the upper bound of quadratic cost function for quantifying the control performance is formulated in the form of linear matrix inequalities (LMIs). The unique state-feedback control gain under the event-triggered strategy can be obtained by solving the LMIs optimisation problems. Comparisons by the simulations are given to demonstrate the advantages of the proposed approach, where the number of control input updates is reduced, while the control performance is still guaranteed.

Near-optimal control with adaptive receding horizon for discrete-time piecewise affine systems

We consider the infinite-horizon optimal control of discrete-time, Lipschitz continuous piecewise affine systems with a single input. Stage costs are discounted, bounded, and use a 1 or ∞-norm. Rather than using the usual fixed-horizon approach from model-predictive control, we tailor an adaptive-horizon method called optimistic planning for continuous actions (OPC) to solve the piecewise affine control problem in receding horizon. The main advantage is the ability to solve problems requiring arbitrarily long horizons. Furthermore, we introduce a novel extension that provides guarantees on the closed-loop performance, by reusing data (“learning”) across different steps. This extension is general and works for a large class of nonlinear dynamics. In experiments with piecewise affine systems, OPC improves performance compared to a fixed-horizon approach, while the data-reuse approach yields further improvements.

A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains

ABSTRACTThis paper analyses stability of discrete-time piecewise-affine systems, defined on possibly non-invariant domains, taking into account the possible presence of multiple dynamics in each of the polytopic regions of the system. An algorithm based on linear programming is proposed, in order to prove exponential stability of the origin and to find a positively invariant estimate of its region of attraction. The results are based on the definition of a piecewise-affine Lyapunov function, which is in general discontinuous on the boundaries of the regions. The proposed method is proven to lead to feasible solutions in a broader range of cases as compared to a previously proposed approach. Two numerical examples are shown, among which a case where the proposed method is applied to a closed-loop system, to which model predictive control was applied without a-priori guarantee of stability.

Stability Analysis by means of Discrete Abstraction. Application to Voltage Stability of Distributed Generators

This paper proposes a stability analysis method for discrete-time piecewise affine systems based on the reachability study of the equivalent discrete abstraction. The finite discrete abstraction eases reachability analysis but does not allow to conclude on system instability. To be able to conclude, this abstraction is refined by bisimulation calculation until the system is found to be unstable or the discrete abstraction is found to be equivalent to the system. This method allows to conclude both on the stability and the instability of the system, avoiding the undecidable situations. This approach is illustrated on voltage stability of a distribution feeder hosting distributed generators equipped with piecewise affine control laws.

Language-Guided Controller Synthesis for Linear Systems

This paper considers the problem of controlling discrete-time linear systems from specifications given as formulas of syntactically co-safe linear temporal logic over linear predicates in the state variables. A systematic procedure is developed for the automatic computation of sets of initial states and feedback controllers such that all the resulting trajectories of the closed-loop system satisfy the given specifications. The procedure is based on the iterative construction and refinement of an automaton that enforces the satisfaction of the formula. Linear programming based approaches are proposed to compute the polytope-to-polytope controllers that label the transitions of the automaton. Extensions to discrete-time piecewise affine systems and specifications given as formulas of full linear temporal logic are included. The algorithms developed in this paper were implemented as a software package that is available for download. Their application and effectiveness are demonstrated for several case studies.

Symbolic Models and Control of Discrete-Time Piecewise Affine Systems: An Approximate Simulation Approach

Symbolic models have been recently used as a sound mathematical formalism for the formal verification and control design of purely continuous and hybrid systems. In this note we propose a sequence of symbolic models that approximates a discrete-time Piecewise Affine (PWA) system in the sense of approximate simulation and converges to the PWA system in the so-called simulation metric. Symbolic control design is then addressed with specifications expressed in terms of non-deterministic finite automata. A sequence of symbolic control strategies is derived which converges, in the sense of simulation metric, to the maximal controller solving the given specification on the PWA system.

Global input-to-state stabilization with quantized feedback for discrete-time piecewise affine systems with time delays

In this paper, global input-to-state stabilization with quantized feedback for discrete-time piecewise affine systems (PWA) with time delays are considered. Both feedback with time delays and feedback without time delays are considered. Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhin and Lyapunov-Krasovskii methods are adopted. The theorems for global input-to-state stabilization with quantized feedback of discrete PWA systems with time delays are shown.

Robust stabilization and l 2-gain control of uncertain discrete-time constrained piecewise-affine systems

In this paper, the robust stabilization and l 2 gain control are investigated for a class of uncertain discrete-time piecewise-affine systems with input saturation and state constraints. By introducing some auxiliary feedback matrices, the LMI-based local stabilization condition is firstly proposed, where the designed controller can be saturated, and meanwhile, the domain of attraction is well estimated. Under energy bounded disturbance, the disturbance tolerance condition is proposed under which all trajectories starting from the origin will remain inside a bounded state region, then the l 2-gain controller is designed in the restricted region. Finally, numerical examples and simulations illustrate the effectiveness and the reduced conservativeness of the proposed results.

Convex relaxations for L2-gain analysis of piecewise affine/polynomial systems

This paper proposes some sufficient conditions based on the computation of polynomial and piecewise polynomial storage functions for -gain analysis of discrete-time piecewise affine or piecewise polynomial systems. The computation of such storage functions is performed by means of convex optimisation techniques via the sum-of-squares decomposition of multivariate polynomials.

Passive fault-tolerant control of discrete time piecewise affine systems against actuator faults

In this article, we propose a new method for passive fault-tolerant control of discrete time piecewise affine systems. Actuator faults are considered. A reliable piecewise linear quadratic regulator state feedback is designed such that it can tolerate actuator faults. A sufficient condition for the existence of a passive fault-tolerant controller is derived and formulated as the feasibility of a set of linear matrix inequalities (LMIs). The upper bound on the performance cost can be minimised using a convex optimisation problem with LMI constraints which can be solved efficiently. The approach is illustrated on a numerical example and a two degree of freedom helicopter.

Temporal Logic Control of Discrete-Time Piecewise Affine Systems

We present a computational framework for automatic synthesis of a feedback control strategy for a discrete-time piecewise affine (PWA) system from a specification given as a linear temporal logic (LTL) formula over an arbitrary set of linear predicates in the system's state variables. Our approach consists of two main steps. First, by defining appropriate partitions for its state and input spaces, we construct a finite abstraction of the PWA system in the form of a control transition system. Second, by leveraging ideas and techniques from LTL model checking and Rabin games, we develop an algorithm to generate a control strategy for the finite abstraction. While provably correct and robust to state measurements and small perturbations in the applied inputs, the overall procedure is conservative and expensive. The proposed algorithms have been implemented as a software package and made available for download. Illustrative examples are included.

Robust observer-based control for uncertain discrete-time piecewise affine systems

The main contribution of this paper is to present a novel robust observer-based controller design method for discrete-time piecewise affine systems with norm-bounded uncertainties. The key ideas are to construct a piecewise-quadratic Lyapunov function to guarantee the stability of the closed-loop systems, approximate polytopic operating regions by ellipsoids, and use the singular value decomposition technique to treat the constraint of matrix equality. It is shown that the suggested control method can be formulated as linear matrix inequalities that are numerically feasible with commercially available software. A numerical example is also given to verify the proposed approach.