Michal Kvasnica

Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances

Piecewise affine (PWA) systems are useful models for describing non-linear and hybrid systems. One of the key problems in designing controllers for these systems is the inherent computational complexity of controller synthesis and analysis. These problems are amplified in the presence of state and input constraints and additive but bounded disturbances. In this paper, we exploit set invariance and parametric programming to devise an efficient robust time optimal control scheme. Specifically, the state is driven into the maximal robust invariant set /spl Omega//sub /spl infin// in minimum time. We show how to compute /spl Omega//spl tilde//sub /spl infin// and derive conditions for finite time computation.

Automatic code generation for real-time implementation of Model Predictive Control

Model Predictive Control (MPC) is a proven control concept with many applications in the process industry. Popularity of the framework is mainly due to its ability to optimize behavior of the process while respecting physical and economical constraints. The major challenge of implementing MPC in real time on low-cost hardware is the inherent computational complexity. To address this goal, it is proposed to solve a given MPC problem using parametric programming, which encodes the optimal control moves as a lookup table. A great advantage being that such tables can then be processed even with low computational resources and therefore allow MPC to be deployed to low cost control devices. In the paper we present a unique software tool which allows MPC problems to be designed with low human effort, and is capable to automatically generate real-time executable code for various target platforms.

Stabilizing polynomial approximation of explicit MPC

A given explicit piecewise affine representation of an MPC feedback law is approximated by a single polynomial, computed using linear programming. This polynomial state feedback control law guarantees closed-loop stability and constraint satisfaction. The polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage.

Low-complexity polynomial approximation of explicit MPC via linear programming

This paper addresses the issue of the practical implementation of Model Predictive Controllers (MPC) to processes with short sampling times. Given an explicit solution to an MPC problem, the main idea is to approximate the optimal control law defined over state space regions by a single polynomial of pre-specified degree which, when applied as a state-feedback, guarantees closed-loop stability, constraint satisfaction, and a bounded performance decay. It is shown how to search for such a polynomial by solving a single linear program.

Polynomial Approximation of Closed-form MPC for Piecewise Affine Systems

Abstract This paper addresses the issue of the practical implementation of closed-form Model Predictive Controllers (MPC) to processes with very short sampling times. Such questions come in consideration when the solution to MPC problems is expressed in a so-called parametric or closed-form fashion. The underlying idea of this paper is to approximate the optimal control law defined over state space regions by a higher degree polynomial which then guarantees closed-loop stability, constraint satisfaction, and a bounded performance decay. The advantage of the proposed scheme lies in faster controller evaluation and lower storage demand compared to currently available techniques.

Performance-lossless complexity reduction in Explicit MPC

The idea of Explicit Model Predictive Control (MPC) is to find the optimal control input as an explicit Piecewise Affine (PWA) function of the initial conditions. The function, however, is often too complex to be processed by a typical control hardware setup in real time. Therefore the paper proposes a novel method of replacing a generic continuous PWA function by a different function of significantly lower complexity in such a way that optimal closed-loop performance, stability and constraint satisfaction are preserved. The idea is based on eliminating a significant portion of the regions of the PWA function over which the function attains a saturated value. An extensive case study is presented which confirms that a significant reduction of complexity is achieved in general.

Model predictive control of a CSTR: A hybrid modeling approach

This paper presents a case study of model predictive control (MPC) applied to a continuous stirred tank reactor (CSTR). It is proposed to approximate nonlinear behavior of a plant by several local linear models, enabling a piecewise affine (PWA) description of the model used to predict and optimize future evolution of the reactor behavior. Main advantage of the PWA model over traditional approaches based on single linearization is a significant increase of model accuracy which leads to a better control quality. It is also illustrated that, by adopting the PWA modeling framework, MPC strategy can be implemented using significantly less computational power compared to nonlinear MPC setups.

Explicit stochastic MPC approach to building temperature control

In this paper we show how to synthesize explicit representations of Model Predictive Control (MPC) feedback laws that maintain temperatures in a building within of a comfortable range while taking into account random evolution of external disturbances. The upside of such an explicit MPC solution stems from the fact that optimal control input can be obtained on-line by a mere function evaluation. This task can be accomplished quickly even on cheap hardware. To account for random disturbances, our formulation assumes probabilistic version of thermal comfort constraints. We illustrate how a finite-sampling approach can be used to convert probabilistic bounds into deterministic constraints. To reduce complexity, and to allow for synthesis of explicit feedbacks in reasonable time, we furthermore propose to prune the set of samples depending on activity of constraints. Performance of the stochastic explicit MPC controller is then compared against best-case and worst-case scenarios.

A memory-efficient representation of explicit MPC solutions

Amount of memory needed to describe explicit model predictive control (MPC) solutions is an often neglected, but a very important factor which decides whether it will be possible to implement such a control strategy on a selected control platform. We show how to exploit geometric properties of explicit MPC controllers to obtain their memory-efficient representation. The three-layer procedure first identifies similarities between polytopic regions in form of an affine transformation. If such amapping exists, certain regions can be represented using less data. The second layer then applies data de-duplication to identify and remove repeating sequences of data. Regions are then described by integer pointers to such a unique set. Finally, Huffman encoding is applied to compress such integer pointers using prefix-free variable-length bit encoding. Reduction in memory is traded for an increase in evaluation time, which is quantified for each layer. Main advantage of the overall procedure is that it can be applied on top of most existing complexity reduction schemes available in the literature.

Efficient computation of controller partitions in multi-parametric programming

The off-line solution to optimal control problems for linear or piecewise-affine systems with constraints has garnered much attention because the on-line implementation can be realized with a simple look-up table. Specifically, multi-parametric programming techniques can be utilized to compute a piecewise-affine feedback law off-line. Even though the computation is performed off-line, the necessary computation time may easily become excessive for larger problems. This paper identifies the computation of minimal representations of polytopes as a key driver for complexity and presents an efficient method for reducing the associated computation cost. The method utilizes bounding-boxes and ray-shooting to discard redundant hyper-planes efficiently. A case study demonstrates the computational advantages.