Markus J. Kögel

Fast predictive control of linear systems combining Nesterov's gradient method and the method of multipliers

The fast, tailored solution of linear, predictive control problems is important, yet challenging. We present an algorithm based on the fast gradient method and the method of multipliers for model predictive control of linear, discrete-time, time-invariant, systems with box constraints. The algorithm uses the augmented Lagrangian method to handle equality constraints, so it can takes advantage of the sparsity of the problem. We present different schemes to update the multipliers. An example illustrates the performance of the algorithm, which is competitive with other tailored solution methods.

Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements

This paper presents a new observer design for Lipschitz nonlinear continuous-time systems with nonuniformly sampled measurements. Based on recent results in sampled-data control of linear continuous-time systems, linear matrix inequality (LMI) conditions are established to guarantee global stability of the estimation error dynamics and to design the observer matrix. The applicability of the proposed observer is demonstrated via two examples, that are the flexible joint robotic arm and Chuas circuit.

Implementation aspects of model predictive control for embedded systems

We discuss implementation related aspects of model predictive control schemes on embedded platforms. Exemplarily, we focus on fast gradient methods and present results from an implementation on a low-cost microcontroller. We show that input quantization in actuators should be exploited in order to determine a suboptimality level of the online optimization that requires a low number of algorithm iterations and might not significantly degrade the performance of the real system. As a case study we consider a Segway-like robot, modeled by a linear time-invariant system with 8 states and 2 inputs subject to box input constraints. The test system runs with a sampling period of 4 ms and uses a horizons up to 20 steps in a hard real-time system with limited CPU time and memory.

A fast gradient method for embedded linear predictive control

Abstract This work considers the fast solution of model predictive control problems for linear systems with input constraints and a quadratic cost criterion. If the resulting optimization problem arising from the model predictive control is solved online using the Fast Gradient method one needs to determine the gradient of the cost function. We propose a method, tailored for embedded control purposes, that efficiently calculates the gradient taking the underlying structure of the system into account. Moreover, we discuss how the stability of the plant influences the required number of iterations to obtain a solution within a prescribed accuracy.

Robust Nonlinear Model Predictive Control with Constraint Satisfaction: A Relaxation-based Approach

Abstract A nonlinear model predictive control scheme guaranteeing robust constraint satisfaction is presented. The scheme is applicable to polynomial or rational systems and guarantees that state, terminal, and output constraints are robustly satisfied despite uncertain and bounded disturbances, parameters, and state measurements or estimates. In addition, for a suitably chosen terminal set, feasibility of the underlying optimization problem at a time instance guarantees that the constraints are robustly satisfied for all future time instances. The proposed scheme utilizes a semi-infinite optimization problem reformulated as a bilevel optimization problem: The outer program determines an input minimizing a performance index for a nominal nonlinear system, while several inner programs certify robust constraint satisfaction. We use convex relaxations to deal with the nonlinear dynamics in the inner programs efficiently. A simulation example is presented to demonstrate the approach.

μAO-MPC: A free code generation tool for embedded real-time linear model predictive control

Implementing linear model predictive controllers in embedded systems with limited computational resources is still challenging. Recently, several code generation tools have been developed that produce highly efficient library-free optimization algorithms. We present a tool that focuses on controller performance and hardware with low computational resources. The underlying optimization algorithm has been explicitly developed for real-time embedded applications, and is based on an augmented Lagrangian method together with Nesterovs gradient method. The tool provides offline methods that allow the generation of online controllers that have low computational requirements and quickly reach good performance. We demonstrate the capabilities of the software, and the performance of the generated controllers with two examples.

Fast predictive control of linear, time-invariant systems using an algorithm based on the fast gradient method and augmented Lagrange multipliers

We present an algorithm based on the fast gradient method and augmented Lagrange multipliers for model predictive control of linear, discrete-time, time-invariant, systems with constraints. In particular, the algorithm solves the underlying quadratic program in the so-called condensed form and takes advantage of the problem structure. At the end, we illustrate the performance of the algorithm, which is competitive with tailored interior-point methods, by an example.

Optimal and optimal-linear control over lossy, distributed networks

Abstract We consider an optimal control problem for networked control systems, where the loop is closed via a lossy, distributed network with an acknowledgment mechanism. The network is distributed in the sense that there are different sets of sensors and actuators that each communicate individually with the controller. We assume that all packets, i.e., the measurement packets, the control packets and the acknowledgment packets are sent over the lossy network and thus are subject to loss. We derive suboptimal controllers with respect to a quadratic cost criterion for the general case and optimal controllers for the case that all states are perfectly measured over a single link. Additionally, we present stability criteria for both cases.

Robust output feedback predictive control with self-triggered measurements

In many applications it is desired to limit the amount of sensor measurements for example to save energy. Developing control strategies that only perform measurements if necessary to achieve the required control performance is thus desirable. This work presents a combination of a robust predictive controller with self-triggered estimation for the control of linear, constrained systems subject to additive noise. We assume that for some sensors the maximum number of measurements within a specific time-span is limited, but the actual time-instants at which the measurements are taken can vary. We propose an output feedback control scheme, which optimizes the applied input as well as the measurement instances to improve closed loop performance. Conditions to robustly guarantee closed loop properties such as constraint satisfaction, recursive feasibility or stability are presented. The results are illustrated by an example.

Event-based reduced-attention predictive control for nonlinear uncertain systems

Event-based control is an alternative to traditional control where new measurements are sampled only if critical events occur. This not only allows to reduce the control effort but it satisfies nowadays application requirements, such for example reduction of information exchange, computational power, or energy consumption. The work in this field is, however, still sparse and only a few results are available. Properly choosing an event-detection logic can considerably improve the overall systems performance. We propose a control algorithm which makes use of a model-based triggering strategy to reduce the control effort (reduced-attention control), while guaranteeing robustness against bounded additive perturbations for nonlinear continuous time systems. In particular, we derive conditions which guarantee that asymptotic stability of the nominal system implies practical stability of the real one in a neighborhood of the origin. A continuous stirred tank reactor is used as a benchmark problem to show the effectiveness of the presented algorithm.