Petter Tøndel

Brief An algorithm for multi-parametric quadratic programming and explicit MPC solutions

Explicit solutions to constrained linear model predictive control problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the components of the state vector. We study the properties of the polyhedral partition of the state space induced by the multi-parametric piecewise affine solution and propose a new mp-QP solver. Compared to existing algorithms, our approach adopts a different exploration strategy for subdividing the parameter space, avoiding unnecessary partitioning and QP problem solving, with a significant improvement of efficiency.

Technical Communique: Evaluation of piecewise affine control via binary search tree

We present an algorithm for generating a binary search tree that allows efficient computation of piecewise affine (PWA) functions defined on a polyhedral partition. This is useful for PWA control approaches, such as explicit model predictive control, as it allows the controller to be implemented online with small computational effort. The computation time is logarithmic in the number of regions in the PWA partition.

An algorithm for multi-parametric quadratic programming and explicit MPC solutions

Explicit solutions to constrained linear model-predictive control (MPC) problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the components of the state vector. We study the properties of the polyhedral partition of the state space induced by the multi-parametric piecewise linear solution and propose a new mp-QP solver. Compared to existing algorithms, our approach adopts a different exploration strategy for subdividing the parameter space, avoiding unnecessary partitioning and QP problem solving, with a significant improvement in efficiency.

Hardware Synthesis of Explicit Model Predictive Controllers

The general solution to constrained linear and piecewise linear model predictive control (MPC) has recently been explicitly characterized in terms of piecewise-linear (PWL) state feedback control. This means that a PWL controller can be precomputed using parametric programming, and the exact explicit MPC implementation amounts to the evaluation of a PWL function in the control unit. It has recently been shown that PWL function evaluation can be accelerated by searching a binary tree data structure, leading to highly efficient, accurate, and verifiable software implementation in low-cost embedded control units. In this work, we report hardware synthesis results for this type of PWL control, and show that explicit MPC solutions can be implemented in an application specific integrated circuit (ASIC) with about 20 000 gates, leading to computation times in the microsecond scale. This opens the way for the use of highly advanced control designs such as constrained MPC in small-scale industrial and consumer electronics application areas that are characterized by fast sampling or low cost, including mechatronics, microelectromechanical systems (MEMS), automotive control, power electronics, and acoustics. The main limitation of the approach is that the memory requirements increase rapidly with the problem dimensions

Control allocation for yaw stabilization in automotive vehicles using multiparametric nonlinear programming

We investigate the use of a nonlinear control allocation scheme for automotive vehicles. Such a scheme is useful in e.g. yaw or roll stabilization of the vehicle. The control allocation allows a modularization of the control task, such that a higher level control system specifies a desired moment to work on the vehicle, while the control allocation distributes this moment among the individual wheels by commanding appropriate wheel slips. The control allocation problem is defined as a nonlinear optimization problem, to which an explicit piecewise linear approximate solution function is computed off-line. Such a solution function can computationally efficiently be implemented in real time with at most a few hundred arithmetic operations per sample. Yaw stabilization of the vehicle yaw dynamics is used as an example of use of the control allocation. Simulations show that the controller stabilizes the vehicle in an extreme manoeuvre where the vehicle yaw dynamics otherwise becomes unstable.

Efficient Optimal Constrained Control Allocation via Multiparametric Programming

Constrained control allocation is studied, and it is shown how an explicit piecewise linear representation of the optimal solution can be computed numerically using multiparametric quadratic programming. Practical benefits of the approach include simple and efficient real-time implementation that permits software verifiability. Furthermore, it is shown how to handle control deficiency, reconfigurability, and flexibility to incorporate, for example, rate constraints. The algorithm is demonstrated on several overactuated aircraft control configurations, and the computational complexity is compared to other explicit approaches from the literature. The applicability of the method is further demonstrated using overactuated marine vessel dynamic position experiments on a scale model in a basin.

Spacecraft attitude control using explicit model predictive control

In this paper, an explicit model predictive controller for the attitude of a satellite is designed. Explicit solutions to constrained linear MPC problems can be computed by solving multi-parametric quadratic programs (mpQP), where the parameters are the components of the state vector. The solution to the mpQP is a piecewise affine (PWA) function, which can be evaluated at each sample to obtain the optimal control law. The on-line computation effort is restricted to a table-lookup, and the controller can be implemented on inexpensive hardware as fixed-point arithmetics can be used. This is useful for systems with limited power and CPU resources. An example of such systems is micro-satellites, which is the focus of this paper. In particular, the explicit MPC (eMPC) approach is applied to the SSETI/ESEO micro-satellite, initiated by the European Space Agency (esa). The theoretical results are supported by simulations.

Technical communique: On the facet-to-facet property of solutions to convex parametric quadratic programs

In some of the recently developed algorithms for convex parametric quadratic programs it is implicitly assumed that the intersection of the closures of two adjacent critical regions is a facet of both closures; this will be referred to as the facet-to-facet property. It is shown by an example, whose solution is unique, that the facet-to-facet property does not hold in general. Consequently, some existing algorithms cannot guarantee that the entire parameter space will be explored. A simple modification, applicable to several existing algorithms, is presented for the purpose of overcoming this problem. Numerical results indicate that, compared to the original algorithms for parametric quadratic programs, the proposed method has lower computational complexity for problems whose solutions consist of a large number of critical regions.

COMPLEXITY REDUCTION IN EXPLICIT LINEAR MODEL PREDICTIVE CONTROL

Abstract Explicit piecewise linear (PWL) state feedback laws solving constrained linear model predictive control (MPC) problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the elements of the state vector. This allows MPC to be implemented via a PWL function evaluation without real-time optimization. The main drawback of this approach is dramatic increase in the number of regions in the state space partition as the number of states, inputs and constraints increases. Here we study two approaches to complexity reduction. First, we consider input trajectory parameterization which significantly reduces the number of regions. Second, we develop a search tree that allows PWL function evaluation to be implemented in real time with low computational complexity.

Computational aspects of approximate explicit nonlinear model predictive control

It has recently been shown that the feedback solution to linear and quadratic constrained Model Predictive Control (MPC) problems has an explicit representation as a piecewise linear (PWL) state feedback. For nonlinear MPC the prospects of explicit solutions are even higher than for linear MPC, since the benefits of computational efficiency and verifiability are even more important. Preliminary studies on approximate explicit PWL solutions of convex nonlinear MPC problems, based on multi-parametric Nonlinear Programming (mp-NLP) ideas show that sub-optimal PWL controllers of practical complexity can indeed be computed off-line. However, for non-convex problems there is a need to investigate practical computational methods that not necessarily lead to guaranteed properties, but when combined with verification and analysis methods will give a practical tool for development and implementation of explicit NMPC. The present paper focuses on the development of such methods. As a case study, the application of the developed approaches to compressor surge control is considered.