Moritz Schulze Darup

MathePraxis – connecting first-year mathematics with engineering applications

First-year engineering students often complain about their mathematics courses as the significance of the difficult and abstract calculus to their field of study remains unclear. We report on the project MathePraxis, a feasibility study which was designed as a means to give first-year students some impression about the use of mathematics in real practice. We aim to increase the motivation and retention rates among engineering students by connecting the contents of the first-year mathematics lectures with practical applications. We developed three projects, two of which are described in this article: an inverted pendulum considered as a model for the automated control within a Segway and a study on the optimal design of a ribbed cooler. In this article, we briefly present the mathematical content of the projects and report on their implementation.

Approximate explicit NMPC with guaranteed stability ensured by a simple auxiliary controller

We investigate two methods for the calculation of suboptimal explicit solutions to nonlinear MPC problems and show that these two methods can be combined for guaranteed stability and good performance. The first method calculates an explicit piecewise constant (PWC) control law and a corresponding positively invariant set that is represented by a hyperrectangular partition in the state space. The explicit PWC law provides a suboptimal solution to the nonlinear MPC problem, but asymptotical stability of the closed-loop system can be guaranteed. A second explicit controller is constructed by solving the nonlinear MPC problem for a representative set of initial conditions and interpolating these pointwise solutions nonlinearly. Note that the PWC law provides feasible initial solutions to the nonlinear MPC problem and therefore can be used to speed up the construction of the second controller significantly. The PWC control law and the explicit nonlinear control law can be combined for guaranteed asymptotical stability (by virtue of the PWC control law) and good performance (from the nonlinear control law). We claim the hybrid controller is an interesting alternative, because its domain of attraction is typically larger than that of the nonlinear controller alone.

Optimal and suboptimal event-triggering in linear model predictive control

We present two event-triggered MPC laws that do not require to solve a quadratic program (QP) in every time step but only upon certain events. We prove one of the control laws results in exactly the same closed-loop behavior as classical MPC. The second control law requires even fewer QPs per time. It is suboptimal w.r.t. the MPC cost function, but still results in asymptotically stable closed-loop behavior. We illustrate the event-triggered MPC laws with two examples.

A stabilizing control scheme for linear systems on controlled invariant sets

Abstract We present a new stabilizing control scheme for linear discrete-time systems with input and state constraints. Essentially, we seek a controller that is able to steer all initial states within a controlled invariant set towards the origin without violating the constraints. The control law builds on a predictive control scheme. We show that a prediction horizon of n steps, where n denotes the dimension of the system, is sufficient to solve the described control task. The proposed controller is related to but different from established feedback laws associated with λ -contractive sets. In fact, the new control scheme successfully stabilizes systems, where classical λ -contractive control laws fail.

Explicit feasible initialization for nonlinear MPC with guaranteed stability

We present a method for the computation of control invariant (c.i.) sets and a simple suboptimal explicit controller for a large class of constrained nonlinear discrete time systems. The explicit controller provides, for any point in the c.i. set, a finite sequence of input values that drive the system to the origin. These input sequences may serve as feasible initializations for the nonlinear program (NLP) associated to nonlinear model predictive control (NMPC). The proposed method is a straightforward extension of an existing method for the computation of c.i. sets, which we augment by a mechanism to record feasible control actions. In contrast to existing explicit NMPC approaches, the explicit control law is calculated without solving the underlying NLP. The method is computationally expensive, but most of the computational effort can be moved offline, and the evaluation of the resulting explicit controller is quite fast.

Computation of the Largest Constraint Admissible Set for Linear Continuous-Time Systems with State and Input Constraints

Abstract We present a new method for the approximation of the largest constraint admissible set (CAS) for linear continuous-time systems with state and input constraints. The CAS is the set of initial states for which the controlled system does not violate the input or state constraints. The presented approach is based on a suitable discretization of the continuous-time system. In fact, we will show that CAS in the continuous-time case can be computed analogously to the discrete-time case, given an appropriate sampling time was chosen. We stress that the computation of CAS for continuous-time systems is considerably more difficult than for discrete-time systems, since one has to guarantee that the system does not violate the constraints in between the sampling instances.

Accurate approximation of the largest null-controllable set for single-input bilinear systems

We present a method for the accurate approximation of the largest null-controllable set N∞ for constrained bilinear systems. It is central to the presented approach that a simple quantitative measure of the accuracy of approximation can be determined. This measure can be used as a termination criterion for an iterative approximation of N∞ with step sets. If the termination criterion is met, the proposed method results in an inner approximation of N∞ that covers a requested percentage of N∞.

Improved Automatic Computation of Hessian Matrix Spectral Bounds

This paper presents a fast and powerful method for the computation of eigenvalue bounds for Hessian matrices

Towards Encrypted MPC for Linear Constrained Systems

\nabla^2{\varphi(x)}

Fast computation of Lipschitz constants on hyperrectangles using sparse codelists

of nonlinear twice continuously differentiable functions