M Mircea Lazar

Stabilizing Model Predictive Control of Hybrid Systems

In this note, we investigate the stability of hybrid systems in closed-loop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1, infin-norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theory

Min-max Model Predictive Control of Nonlinear Systems: A Unifying Overview on Stability

Min-max model predictive control (MPC) is one of the few techniques suitable for robust stabilization of uncertain nonlinear systems subject to constraints. Stability issues as well as robustness have been recently studied and some novel contributions on this topic have appeared in the literature. In this survey, we distill from an extensive literature a general framework for synthesizing min-max MPC schemes with ana priori robust stability guarantee. First, we introduce a general predictionmodel that covers a wide class of uncertainties, which includes bounded disturbances as well as state and input dependent disturbances (uncertainties). Second, we extend the notion of regional input-to-state stability (ISS) in order to fit the considered class of uncertainties. Then, we establish that the standard min-max approach can only guarantee practical stability. We concentrate our attention on two different solutions for solving this problem. The first one is based on a particular design of the stage cost of the performance index, which leads to aH∞ strategy, while the second one is based on a dual-mode strategy. Under fairly mild assumptions both controllers guarantee ISS of the resulting closed-loop system.Moreover, it is shown that the nonlinear auxiliary control law introduced in [29] to solve theH∞ problem can be used, for nonlinear systems affine in control, in all the proposed min-max schemes and also in presence of state-independent disturbances. A simulation example illustrates the techniques surveyed in this article.

On Input-to-State Stability of Min-max Nonlinear Model Predictive Control

In this paper we consider discrete-time nonlinear systems that are affected, possibly simultaneously, by parametric uncertainties and other disturbance inputs. The min–max model predictive control (MPC) methodology is employed to obtain a controller that robustly steers the state of the system towards a desired equilibrium. The aim is to provide a priori sufficient conditions for robust stability of the resulting closed-loop system using the input-to-state stability (ISS) framework. First, we show that only input-to-state practical stability can be ensured in general for closed-loop min–max MPC systems; and we provide explicit bounds on the evolution of the closed-loop system state. Then, we derive new conditions for guaranteeing ISS of min–max MPC closed-loop systems, using a dual-mode approach. An example illustrates the presented theory.

Event Based State Estimation With Time Synchronous Updates

To reduce the amount of data transfer in networked systems, measurements are usually taken only when an event occurs rather than at each synchronous sample instant. However, this complicates estimation problems considerably, especially in the situation when no measurement is received anymore. The goal of this paper is therefore to develop a state estimator that can successfully cope with event based measurements and attains an asymptotically bounded error-covariance matrix. To that extent, a general mathematical description of event sampling is proposed. This description is used to set up a state estimator with a hybrid update, i.e., when an event occurs the estimated state is updated using the measurement, while at synchronous instants the update is based on knowledge that the sensor value lies within a bounded subset of the measurement space. Furthermore, to minimize computational complexity of the estimator, the algorithm is implemented using a sum of Gaussians approach. The benefits of this implementation are demonstrated by an illustrative example of state estimation with event sampling.

Technical communique: On polytopic inclusions as a modeling framework for systems with time-varying delays

One of the important issues in networked control systems is the appropriate handling of the nonlinearities arising from uncertain time-varying delays. In this paper, using the Cayley-Hamilton theorem, we develop a novel method for creating discrete-time models of linear systems with time-varying input delays based on polytopic inclusions. The proposed method is compared with existing approaches in terms of conservativeness, scalability and suitability for controller synthesis.

Brief paper: Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions

This article presents a novel model predictive control (MPC) scheme that achieves input-to-state stabilization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability (ISS) is guaranteed when an optimal solution of the MPC optimization problem is attained. Special attention is paid to the effect that sub-optimal solutions have on ISS of the closed-loop system. This issue is of interest as firstly, the infimum of MPC optimization problems does not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the deviation of the predictive control law from the optimum and the resulting deterioration of the ISS property of the closed-loop system. By imposing stronger conditions on the sub-optimal solutions, ISS can even be attained in this case.

Flexible control Lyapunov functions

A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative. In this paper we propose a novel idea that improves the design of CLFs in terms of flexibility, i.e. the CLF is permitted to be locally non-monotone along the closed-loop trajectory. The focus is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly reduced compared to classical CLFs, which makes flexible CLFs more suitable for stabilization of constrained discrete-time nonlinear systems and real-time control.

On Event Based State Estimation

To reduce the amount of data transfer in networked control systems and wireless sensor networks, measurements are usually taken only when an event occurs, rather than at each synchronous sampling instant. However, this complicates estimation and control problems considerably. The goal of this paper is to develop a state estimation algorithm that can successfully cope with event based measurements. Firstly, we propose a general methodology for defining event based sampling. Secondly, we develop a state estimator with a hybrid update, i.e. when an event occurs the estimated state is updated using measurements; otherwise the update makes use of the knowledge that the monitored variable is within a bounded set that defines the event. A sum of Gaussians approach is employed to obtain a computationally tractable algorithm.

Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.

State fusion with unknown correlation: Ellipsoidal intersection

Some crucial challenges of estimation over sensor networks are reaching consensus on the estimates of different systems in the network and separating the mutual information of two estimates from their exclusive information. Current fusion methods of two estimates tend to bypass the mutual information and directly optimize the fused estimate. Moreover, both the mean and covariance of the fused estimate are fully determined by optimizing the covariance only. In contrast to that, this paper proposes a novel fusion method in which the mutual information results in an additional estimate, which defines a mutual mean and covariance. Both variables are derived from the two initial estimates. The mutual covariance is used to optimize the fused covariance, while the mutual mean optimizes the fused mean. An example of decentralized state estimation, where the proposed fusion method is applied, shows a reduction in estimation error compared to the existing alternatives.