Sezai Emre Tuna

Model predictive control: for want of a local control Lyapunov function, all is not lost

We present stability results for unconstrained discrete-time nonlinear systems controlled using finite-horizon model predictive control (MPC) algorithms that do not require the terminal cost to be a local control Lyapunov function. The two key assumptions we make are that the value function is bounded by a K/sub /spl infin// function of a state measure related to the distance of the state to the target set and that this measure is detectable from the stage cost. We show that these assumptions are sufficient to guarantee closed-loop asymptotic stability that is semiglobal and practical in the horizon length and robust to small perturbations. If the assumptions hold with linear (or locally linear) K/sub /spl infin// functions, then the stability will be global (or semiglobal) for long enough horizon lengths. In the global case, we give an explicit formula for a sufficiently long horizon length. We relate the upper bound assumption to exponential and asymptotic controllability. Using terminal and stage costs that are controllable to zero with respect to a state measure, we can guarantee the required upper bound, but we also require that the state measure be detectable from the stage cost to ensure stability. While such costs and state measures may not be easy to construct in general, we explore a class of systems, called homogeneous systems, for which it is straightforward to choose them. In fact, we show for homogeneous systems that the associated K/sub /spl infin// functions are linear, thereby guaranteeing global asymptotic stability. We discuss two examples found elsewhere in the MPC literature, including the discrete-time nonholonomic integrator, to demonstrate our methods. For these systems, we give a new result: They can be globally asymptotically stabilized by a finite-horizon MPC algorithm that has guaranteed robustness. We also show that stable linear systems with control constraints can be globally exponentially stabilized using finite-horizon MPC without requiring the terminal cost to be a global control Lyapunov function.

Examples when nonlinear model predictive control is nonrobust

We consider nominal robustness of model predictive control for discrete-time nonlinear systems. We show, by examples, that when the optimization problem involves state constraints, or terminal constraints coupled with short optimization horizons, the asymptotic stability of the closed loop may have absolutely no robustness. That is to say, it is possible for arbitrarily small disturbances to keep the closed loop strictly inside the interior of the feasibility region of the optimization problem and, at the same time, far from the desired set point. This phenomenon does not occur when using model predictive control for linear systems with convex constraint sets. We emphasize that a necessary condition for the absence of nominal robustness in nonlinear model predictive control is that the value function and feedback law are discontinuous at some point(s) in the interior of the feasibility region.

Nominally Robust Model Predictive Control With State Constraints

In this paper, we present robust stability results for constrained discrete-time nonlinear systems using a finite-horizon model predictive control (MPC) algorithm for which we do not require the terminal cost to have any particular properties. We introduce a definition that attempts to characterize the robustness properties of the MPC optimization problem. We assume the systems under consideration satisfy this definition (for which we give sufficient conditions) and make two further assumptions. These are that the value function is bounded by a Kinfin function of a state measure (related to the distance from the state to some target set) and that this measure is detectable from the stage cost used in the MPC algorithm. We show that these assumptions lead to stability that is robust to sufficiently small disturbances. While in general the results are semiglobal and practical, when the detectability and upper bound assumptions are satisfied with linear Kinfin functions, the stability and robustness are either semiglobal or global (with respect to the feasible set). We discuss algorithms employing terminal inequality constraints and also provide a specific example of an algorithm that employs a terminal equality constraint.

Discrete-time certainty equivalence output feedback: allowing discontinuous control laws including those from model predictive control

We present certainty equivalence output feedback results for discrete-time nonlinear systems that employ possibly discontinuous control laws in the feedback loop. Coupling assumptions of nominal robustness with uniform observability or detectability assumptions, we assert nominally robust stability for output feedback closed loops. We further show that model predictive control (MPC) can be used to generate a feedback control law that is robustly globally asymptotically stabilizing when used in a certainty equivalence output feedback closed loop. Allowing for discontinuous feedback control laws is important for systems employing MPC, since the method can, and sometimes necessarily does, result in discontinuous control laws.

Nominally robust model predictive control with state constraints

We present robust stabilization results for constrained, discrete-time, nonlinear systems using a finite-horizon model predictive control (MPC) algorithm that does not require any particular properties for the terminal cost. We introduce a property that characterizes the robustness properties of the MPC optimization problem. Assuming the system has this property (for which we give sufficient conditions), we make two further key assumptions. These are that the value function is bounded by a K/sub /spl infin// function of a state measure (related to the distance of the state to some target set) and that this measure is detectable from the stage cost used in the MPC algorithm. We show that these assumptions lead to stability that is robust to sufficiently small disturbances and measurement noise. While in general the results are semiglobal practical, when the detectability and upper bound assumptions are satisfied with linear K/sub /spl infin// functions, the stability and robustness is global with respect to the feasible set. We discuss algorithms employing terminal equality or inequality constraints. We provide two examples, one involving a terminal equality constraint and the other involving a nonrobustness-inducing state constraint.

Model predictive control when a local control Lyapunov function is not available

This paper presents closed-loop stability results for the control of unconstrained nonlinear systems using the model predictive control methodology with semidefinite costs. The results do not require the use of a local control Lyapunov function as the terminal cost. The key assumptions are that the value function is bounded by a K/sub /spl infin// function of some measure of the state and that this measure is detectable through the stage cost. Sufficient conditions to yield semiglobal practical (and global) MPC stability results are given. In each case, a minimum horizon (uniform for global results) is determined for which the MPC method will result in the stabilization of a desired set.

BINOCULAR: a system monitoring framework

Recent advances in hardware technology facilitate applications requiring a large number of sensor devices, where each sensor device has computational, storage, and communication capabilities. However these sensors are subject to certain constraints such as limited power, high communication cost, low computation capability, presence of noise in readings and low bandwidth. Since sensor devices are powered by ordinary batteries, power is a limiting resource in sensor networks and power consumption is dominated by communication. In order to reduce power consumption, we propose to use a linear model of temporal, spatial and spatio-temporal correlations among sensor readings. With this model, readings of all sensors can be estimated using the readings of a few sensors by using linear observers and multiple queries can be answered more efficiently. Since a small set of sensors are accessed for query processing, communication is significantly reduced. Furthermore, the proposed technique can also be beneficial at filtering out the noise which directly affects the accuracy of query results.

Examples of zero robustness in constrained model predictive control

Nominal robustness of model predictive control for nonlinear systems is considered. It is shown, by examples, that when the optimization problem involves state constraints, or terminal constraints coupled with short optimization horizons, the asymptotic stability of the closed loop may have absolutely no robustness. Namely, it is possible for arbitrarily small disturbances to keep the closed loop strictly inside the interior of the feasibility region of the optimization problem and, at the same time, far from the desired set point. This phenomenon does not occur when using model predictive control for linear systems with convex constraint sets. It is emphasized that a necessary condition for the absence of nominal robustness in nonlinear model predictive control is that the value function and feedback law are discontinuous at some point(s) in the interior of the feasibility region.

Discrete-time homogeneous Lyapunov functions for homogeneous difference inclusions

In this paper, we consider homogeneity (of discrete-time systems) with respect to generalized dilations, which define a broader class of operators than dilations. The notion of generalized dilations allows us to deal with the stability of attractors that are more general than a single point, which may be unbounded sets. We study homogeneous difference inclusions where every solution passed through a homogeneous measure function is bounded from above by a class-KL estimate in terms of time and the initial state passed through the measure function. We show that for such inclusions, under some generic assumptions, there exist a continuous Lyapunov function that is homogeneous of arbitrary degree and smooth everywhere possibly except at the attractor.

Generalized dilations and numerically solving discrete-time homogeneous optimization problems

We introduce generalized dilations, a broader class of operators than that of dilations, and consider homogeneity with respect to this new class of dilations. For discrete-time systems that are asymptotically controllable and homogeneous (with degree zero) we propose a method to numerically approximate any homogeneous value function (solution to an infinite horizon optimization problem) to arbitrary accuracy. We also show that the method can be used to generate an offline computed stabilizing feedback law.