Timm Faulwasser

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.

Nonlinear Model Predictive Control for Constrained Output Path Following

We consider the tracking of geometric paths in output spaces of nonlinear systems subject to input and state constraints without pre-specified timing requirements, commonly referred to as constrained output path-following problems. Specifically, we propose a predictive control approach to constrained path-following problems with and without velocity assignments. We present sufficient convergence conditions based on terminal regions and end penalties. Furthermore, we analyze the geometric nature of constrained output path-following problems and thereby provide insight into the computation of suitable terminal control laws and terminal regions. We draw upon an example from robotics to illustrate our findings.

Turnpike and dissipativity properties in dynamic real-time optimization and economic MPC

We investigate the turnpike and dissipativity properties of continuous-time optimal control problems. These properties play a key role in the analysis and design of schemes for dynamic real-time optimization and economic model predictive control. We show in a continuous-time setting that dissipativity of a system with respect to a steady state implies the existence of a turnpike at this steady state and optimal stationary operation at this steady state. Furthermore, we investigate the converse statements: We show that the existence of a turnpike at a steady state implies (a) that this steady state is the optimal steady state; and (b) that over an infinite horizon the system is optimally operated at this steady state. We draw upon a numerical example to illustrate our findings.

Optimal Exact Path-Following for Constrained Differentially Flat Systems

We propose a dynamic optimization approach to calculate optimal feedforward controls for exact path-following problems of differentially flat systems. Besides the derivation of a small dimensional optimal control problem, we provide easily checkable conditions on the existence of inputs guaranteeing that a given path is exactly followable in the presence of constraints on states and inputs. Our approach is based on the projection of the feedforward controlled, nonlinear MIMO dynamics along a geometric path onto a linear single-input system in Brunovsky normal form. The presented results indicate how the computation of admissible trajectories for set-point changes can be simplified by relying on steady state consistent paths. The set-point change of a Van de Vusse reactor is considered as an example.

Nonlinear Model Predictive Path-Following Control

In the frame of this work, the problem of following parametrized reference paths via nonlinear model predictive control is considered. It is shown how the use of parametrized paths introduces new degrees of freedom into the controller design. Sufficient stability conditions for the proposed model predictive path-following control are presented. The method proposed is evaluated via simulations of an autonomous mobil robot.

On the design of economic NMPC based on approximate turnpike properties

We discuss the design of sampled-data economic nonlinear model predictive control schemes for continuous-time systems based on turnpike properties. In a recent paper we have shown that an exact turnpike property allows establishing finite-time convergence of the NMPC scheme to the optimal steady state, and also recursive feasibility, without using terminal penalties or terminal constraints. Herein, we extend our previous results to the more general case of approximate turnpikes. We establish sufficient conditions, based on a dissipativity assumption, that guarantee (i) convergence to a neighborhood of the optimal steady state, and (ii) recursive feasibility in the presence of state constraints. The proposed conditions do not rely on terminal regions or terminal penalties. A key step in our developments is the use of a storage function as a penalty on the initial condition in the NMPC scheme. We draw upon the example of a chemical reactor to illustrate our findings.

Constrained Output Path-Following for Nonlinear Systems Using Predictive Control

We address the problem of steering the output of a nonlinear system along a given parametrized reference path, taking input and state constraints into account. Such problems are known as output path-following problems, which typically arise in vehicle and robot control and also in process control. We propose an efficient, continuous time, sampled-data state feedback, predictive control scheme which guarantees convergence to an output path in the presence of state and input constraints.

On turnpike and dissipativity properties of continuous-time optimal control problems

This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a continuous-time setting that if along optimal trajectories a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. Finally, we establish novel converse turnpike results, i.e.,we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. We draw upon a numerical example to illustrate our findings.

On the Use of Second-Order Modifiers for Real-Time Optimization

We consider the real-time optimization of static plants and propose a generalized version of the modifier-adaptation strategy that relies on second-order adaptation of the cost and constraint functions. We show that second-order adaptation allows checking whether a (local) plant optimum is reached upon convergence. A sufficient convergence condition that is applicable to first- and second-order modifier-adaptation schemes is proposed. We also discuss how second-order updates can lead to SQP-like model-free RTO schemes. The approach is illustrated via the simulated example of a continuous reactor.

Constrained reachability and trajectory generation for flat systems

We consider the problem of trajectory generation for constrained differentially flat systems. Based on the topological properties of the set of admissible steady state values of a flat output we derive conditions which allow for an a priori verification of the feasibility of constrained set-point changes. We propose to utilize this relation to generate feasible reference trajectories. To this end, we suggest to split the trajectory generation problem into two stages: (a) the planning of geometric reference paths in the flat output space combined with (b) an assignment of a dynamic reference motion to these paths. This assignment is based on a reduced optimal control problem. The unique feature of the approach is that due to the specific construction of the reference paths the optimal control problem to be solved is guaranteed to be feasible. To illustrate our results we consider a Van de Vusse reactor as an example.