Florian David Brunner

Robust self-triggered MPC for constrained linear systems

In this paper we propose a robust self-triggered model predictive control algorithm for linear systems with additive bounded disturbances and hard constraints on the inputs and state. In self-triggered control, at every sampling instant the time until the next sampling instant is computed online based on the current state of the system. The goal is to achieve a low average sampling rate, thereby minimizing communication in the control system and possibly reducing the number of control updates as is required in sparse control applications. Naturally, and intentionally, our approach leads to long spans of time in which the plant is controlled in an open-loop fashion. Especially for unstable plants or large disturbances this necessitates taking into account the disturbance characteristics in the design of the control law in order to prevent constraint violation in the closed-loop system. We use constraint tightening methods as proposed in Tube Model Predictive Control to guarantee robust constraint satisfaction. The self-triggered controller is shown to stabilize a robust invariant set in the state space for the closed-loop system.

Robust self-triggered model predictive control for constrained discrete-time LTI systems based on homothetic tubes

In this paper we present a robust self-triggered model predictive control (MPC) scheme for discrete-time linear time-invariant systems subject to input and state constraints and additive disturbances. In self-triggered model predictive control, at every sampling instant an optimization problem based on the current state of the system is solved in order to determine the input applied to the system until the next sampling instant, as well as the next sampling instant itself. This leads to inter-sampling times that depend on the trajectory of the system. By maximizing the inter-sampling time, the amount of communication in the control system is reduced. In order to guarantee robust constraint satisfaction, Tube MPC methods are employed. Specifically, in order to account for the uncertainty in the system, homothetic sets are used in the prediction of the future evolution of the system. The proposed controller is shown to stabilize a closed and bounded set including the origin in its interior.

Feedback design for multi-agent systems: A saddle point approach

In this paper we propose a dynamic state feedback controller for an input affine nonlinear system that asymptotically stabilizes a point in the output space that is implicitly given as the solution to a convex optimization problem. The construction of the feedback law is based on saddle point flows for convex optimization problems and a backstepping technique. An explicit solution of the optimization problem is not needed for the controller design. We show how the design approach can be applied to multi-agent systems, yielding a decentralized controller. For a particular example, we extend the controller to the output feedback case.

Resource-aware set-valued estimation for discrete-time linear systems

In this paper, we propose a self-triggered estimator for discrete-time linear systems subject to unknown but bounded disturbances affecting both the system states and outputs. The proposed self-triggered estimator is a set-valued estimator that employs rollout techniques to reduce the communication between the sensors and the estimator with respect to a periodic sampling strategy. Moreover, at each time instant it provides an estimate of the plant state and a guaranteed bound on the difference between the true plant state and the estimate.

Model Predictive Control for Nonlinear Time-Delay Systems without Terminal Constraint

Abstract In this work we present new results on model predictive control (MPC) for nonlinear time-delay systems. In the first part we derive a novel scheme for determining a suitable terminal cost and terminal region based on the Jacobi linearization of the nonlinear system. The main advantage of the proposed scheme compared to previous results is that the terminal region is defined as a sublevel of the terminal cost functional without any restrictive requirements on the sampling time of the MPC. Based on this result, we present an MPC scheme without terminal constraint in the second part. The result extends existing results for delay-free systems and guarantees asymptotic stability of the closed-loop. The main difficulty in the derivation is to show that the integral over the stage cost has a lower bound if the state is outside of a certain region. This is directly satisfied for delay-free finite-dimensional systems, but requires additional arguments for time-delay systems.

An explicit solution to constrained stabilization via polytopic tubes

This paper proposes a method to obtain stabilizing controllers for constrained linear systems with assigned sets of initial conditions. The controller synthesis method is based on invariant tubes and works for linear time-invariant systems and for linear systems with multiplicative uncertainties. Given a compact initial condition set, a sequence of sets and an associated sequence of control laws is computed such that the initial condition set is contained in the first set of the sequence and every state in any set of the sequence is controlled to the next set in the sequence while satisfying state and input constraints. Assumptions on the parameterizations of the sets and the control laws are given that guarantee recursive feasibility of the tube synthesis problem and convergence of the closed-loop trajectories. For a particular type of parameterization it is shown that these assumptions are satisfied. Numerical simulations are presented that illustrate the developed synthesis method.

Stochastic thresholds in event-triggered control: A consistent policy for quadratic control

We propose an event-triggered control scheme for discrete-time linear systems subject to Gaussian white noise disturbances. The event-conditions are given in terms of the deviation between the actual system state and the state of a nominal undisturbed system whose state is identical to the real system state at the event times. In order to ensure that the conditional distribution of the deviation between the two systems, under the condition that no event occurs, remains a normal distribution, we employ thresholds that are themselves random variables. This allows us to: (i) provide expressions for the probability mass function of the times between events and, in turn, arbitrarily select this function; (ii) synthesize controllers associated with the proposed transmissions scheduler that are optimal in terms of an average quadratic cost. In particular, these two properties allow us to show that our event-triggered scheme is consistent in the sense that it outperforms (in a quadratic cost sense) traditional periodic control for the same average transmission rate and does not generate transmissions in the absence of disturbances. We demonstrate the effectiveness of our scheme in a numerical example and describe a way to solve the non-convex optimization problem arising in the approach.

Robust Event-Triggered MPC With Guaranteed Asymptotic Bound and Average Sampling Rate

We propose a robust event-triggered model predictive control (MPC) scheme for linear time-invariant discrete-time systems subject to bounded additive stochastic disturbances and hard constraints on the input and state. For given probability distributions of the disturbances acting on the system, we design event conditions such that the average frequency of communication between the controller and the actuator in the closed-loop system attains a given value. We employ Tube MPC methods to guarantee robust constraint satisfaction and a robust asymptotic bound on the system state. Moreover, we show that instead of a given periodically updated Tube MPC scheme, an appropriate event-triggered MPC scheme can be applied, with the same guarantees on constraints and region of attraction, but with a reduced number of average communications.

Enhancing output feedback MPC for linear discrete-time systems with set-valued moving horizon estimation

We propose a novel output feedback model predictive control scheme for linear discrete-time systems incorporating a set-valued estimator based on a fixed finite number of recent measurements. Recursive feasibility is established by basing predictions that are farther in the future on fewer measurements. The resulting optimization problem is convex with linear constraints. We demonstrate in a numerical example that the proposed model predictive control scheme allows an enlargement of the feasible set beyond what is possible with earlier schemes using linear estimators.

A tube-based approach to nonlinear explicit MPC

We propose an approximate explicit model predictive control (MPC) scheme based on ideas in robust MPC. For every subset in a partial partition of the state space we solve a robust MPC problem offline, where the (only) uncertainty is the initial condition which may be any state in this particular subset. As a consequence, the control law defined by the solution of the MPC problem is valid for any state in the subset. The online computational effort reduces to a point location problem and application of the pre-computed MPC solution.