Technical Report: Control of Nonlinear Systems with Explicit-MPC-like Controllers

This paper describes synthesis of controllers involving Quadratic Programming (QP) optimization problems for control of nonlinear systems. The QP structure allows an implementation of the controller as a piecewise affine function, pre-computed offline, which is a technique extensively studied in the field of explicit model predictive control (EMPC). The nonlinear systems being controlled are assumed to be described by polynomial functions and the synthesis also generates a polynomial Lyapunov function for the closed-loop system involving the obtained controller. The synthesis is based on a sum-of-squares (SOS) stability verification for polynomial discrete-time systems, described in continuous-time in this paper. The presented synthesis method allows a design of EMPC controllers with closed-loop stability guarantees without relying on a terminal cost and/or constraint, and even without using the prediction horizon concept to formulate the control optimization problem. In particular, for a specified QP structure the method directly searches for the stabilizing coefficients in the cost and/or the constraint set. The method involves two phases, where the first searches for stabilizing controllers by minimizing a polynomial slack function introduced to the SOS stability condition and the second phase optimizes some user-specified performance criteria. The two phases are formulated as optimization problems which can be tackled by using a black-box optimization technique such as Bayesian optimization, which is used in this paper. The synthesis is demonstrated on a numerical example involving a bilinear model of a permanent magnet synchronous machine (PMSM), where in order to demonstrate the modeling flexibility of the proposed synthesis method a QP-based controller for speed regulation of PMSM is synthesized that is robust to parametric uncertainty coming from the temperature-dependent stator resistance of the PMSM.

Low-Complexity Optimization-Based Control: Design, Methods and Applications

Ivan Pejcic

Optimization-based controllers are advanced control systems whose mechanism of determining control inputs requires the solution of a mathematical optimization problem. In this thesis, several contributions related to the computational effort required for optimization-based controller execution are provided. The content of the thesis is divided into three parts: The first part provides methods capable of performing automatic controller tuning for constrained control of nonlinear systems. Given a specified controller structure, the presented methods are able to perform an offline tuning of the controller parameters such that some user-specified performance metric is optimized while imposing stability guarantees on the obtained closed-loop system. The methods are characterized by a broad flexibility that allows their application to many control schemes that are widely popular in practice, but also to novel user-specified control schemes that are convenient from a computational or some other point of view. The controller tuning is formulated as an optimization problem that can be tackled by black-box optimization techniques such as Bayesian optimization. The methods are demonstrated by application examples involving speed control of a permanent magnet synchronous machine and position control of a mechanical gyroscopic system. The second part provides an accelerated version of the alternating direction method of multipliers (ADMM) optimization algorithm derived by using a recently proposed accelerated Douglas-Rachford (DR) splitting. The obtained method is an accelerated ADMM version that replaces the internal proximal point convergence mechanism of the classical ADMM by the accelerated gradient method applied on a specially constructed scaled DR envelope function. The form of the accelerated ADMM is derived and conditions are provided under which the underlying accelerated DR splitting is validly addressing the Fenchel dual problem. The third part describes a model predictive control scheme for power electronics control which involves a combination of the integral of squared predicted tracking error as the controller's cost function together with offline computed optimal steady-state voltage signals. These offline computed optimal steady-state signals are in the power electronics community referred to as Optimized Pulse Patterns (OPPs). The method is presented by considering an industrial case study involving a grid-tied converter with LC filter. After introducing an optimal control problem based on OPPs, low computational complexity approximate versions are provided. The resulting approximate controller versions are addressed by using memory storage of the dynamic behavior of the system, leading to controller forms whose execution can be performed on embedded hardware.

EPFL2019

Predictive Control of Fast Unstable and Nonminimum-phase Nonlinear Systems

Dominique Bonvin, Kahina Guemghar, Philippe Müllhaupt

Predictive Control of Unstable Nonminimum-phase Systems K. Guemghar, B. Srinivasan, Ph. Mullhaupt, D. Bonvin ´ Institut d’Automatique, Ecole Polytechnique F´d´rale de Lausanne, ee CH-1015 Lausanne, Switzerland. Predictive control is a very eﬀective approach for tackling problems with constraints and nonlinear dynamics, especially when the analytical computation of the control law is diﬃcult. Standard predictive control involves predicting the system behavior over a prediction horizon and calculating the input that minimizes a criterion expressing the system behavior in the future. Only the ﬁrst part of the computed input is applied to the system, and this procedure is repeated with the advent of each new measurement. This methodology is widely used in the process industry where system dynamics are sufﬁciently slow to permit its implementation. In contrast, applications of predictive control to fast unstable dynamic systems are rather limited. Apart from computational considerations, a fundamental limitation arises from the accuracy of that prediction, which can be quite poor due to accumulation of numerical errors if the prediction horizon is large. In this paper, an upper bound on the prediction horizon based on the location of the unstable pole(s) of the linearized system will be provided. The problem becomes more acute when nonminimum-phase systems are considered. Nonminimum phase implies that the system starts in a direction opposite to its reference (inverse response). To control such systems, it is reasonable to predict the maneuvers, thus making predictive control a natural strategy. However, a large prediction horizon is required since it is necessary to look beyond the inverse response. In this context, a lower bound on the prediction horizon based on the location of the unstable zero(s) of the linearized system will be provided. The two bounds mentioned above may lead to the situation where there exist no value of the prediction horizon that can stabilize a given unstable nonmiminum-phase system. For such a case, a combination of tools from diﬀerential geometry and predictive control is proposed in this paper. The envisaged procedure has a cascade structure and is outlined below: 1. Using the input-output linearization technique, the nonlinear system is transformed into a linear subsystem and internal dynamics. However, since the original nonlinear system is nonminimum-phase, the internal dynamics are unstable. 2. The linear subsystem is made arbitrarily fast by using a stabilizing high-gain linear feedback (inner-loop). 3. A predictive control scheme is then used to stabilize the slow internal dynamics (outerloop) by manipulating the reference of the inner loop. The stability of the proposed procedure is analyzed using a singular perturbation approach. The results will be illustrated in simulation on a system consisting of an inverted pendulum on a cart. 1

2002

Predictive Control of Fast Unstable and Nonminimum-phase Nonlinear Systems

Dominique Bonvin, Kahina Guemghar, Philippe Müllhaupt

2002

Low-Complexity Optimization-Based Control: Design, Methods and Applications

Ivan Pejcic

EPFL2019