The explicit linear quadratic regulator for constrained systems

Model Predictive Control (MPC), the dominant advanced control approach in industry over the past twenty-five years, is presented comprehensively in this unique book. With a simple, unified approach, and with attention to real-time implementation, it covers predictive control theory including the stability, feasibility, and robustness of MPC controllers. The theory of explicit MPC, where the nonlinear optimal feedback controller can be calculated efficiently, is presented in the context of linear systems with linear constraints, switched linear systems, and, more generally, linear hybrid systems. Drawing upon years of practical experience and using numerous examples and illustrative applications, the authors discuss the techniques required to design predictive control laws, including algorithms for polyhedral manipulations, mathematical and multiparametric programming and how to validate the theoretical properties and to implement predictive control policies. The most important algorithms feature in an accompanying free online MATLAB toolbox, which allows easy access to sample solutions. Predictive Control for Linear and Hybrid Systems is an ideal reference for graduate, postgraduate and advanced control practitioners interested in theory and/or implementation aspects of predictive control.

Predictive Control for Linear and Hybrid Systems

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

This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity equations, the boundary and continuity conditions, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper.

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Optimal control of switched systems based on parameterization of the switching instants

Three decades have passed since milestone publications by several industrialists spawned a flurry of research and industrial / commercial activities on model predictive control (MPC). This article reviews major developments and achievements during the three decades and attempts to put a perspective on them. The first decade is characterized by the fast-growing industrial adoption of the technology, primarily in the refining and petrochemical sectors, which sparked much interest and also confusion among the academicians. The second decade saw a number of significant advances in understanding the MPC from a control theoretician’s viewpoint, which included state-space interpretations / formulations and stability proofs. These theoretical triumphs contributed to the makings of the second generation of commercial software, which was significantly enhanced in generality and rigor. The third decade’s main focus has been on the development of “fast MPC,” a term chosen to collectively describe the various efforts to bring orders-of-magnitude improvement in the efficiency of the on-line computation so that the technology can be applied to systems requiring very fast sampling rates. Throughout the three decades of the development, theory and practice supported each other quite effectively, a primary reason for the fast and steady rise of the technology.

Model predictive control: Review of the three decades of development

A wheel slip controller is developed and experimentally tested in a car equipped with electromechanical brake actuators and a brake-by-wire system. A gain scheduling approach is taken, where the vehicle speed is viewed as a slowly time-varying parameter and the model is linearized about the nominal wheel slip. Gain matrices for the different operating conditions are designed using an LQR approach. The stability and robustness of the controller are studied via Lyapunov theory, frequency analysis, and experiments using a test vehicle.

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Gain-scheduled wheel slip control in automotive brake systems

Explicit sub-optimal linear quadratic regulation with state and input constraints

In a conventional antilock brake system (ABS), the wheel slip will oscillate around a "critical slip" within some given thresholds. This oscillation will have as side effects a noticeable vibration for the driver and limitations in ABS performance. Thus, the actual friction force between tyre and road will oscillate around a "maximum" point. The level of complexity present in current production ABS systems has serious limitations for further development and analysis.This thesis looks at the analysis and design of an ABS controller using a continuously adjustable electromechanical actuator where the ABS aims to control the slip of the wheel to arbitrary setpoints provided by a higher level control system such as the electronic stability program (ESP). Thus, maximum friction force can be obtained together with a vibration free braking.This thesis contributes to stability and robustness analysis of a nonlinear ABS controller with respect to uncertainty in the road/tyre friction using Lyapunov theory, frequency analysis and experiments with a test vehicle. A communication delay between the ABS controller and the electromechanical actuator together with the actuator dynamics introduce phase losses and the effect of these performance limitations are also analysed.This thesis contributes to model-based nonlinear wheel slip controller design, as an explicit gain scheduled LQR design method was used for controller design.Full-scale results are presented for a Mercedes car (E220) equipped with a brake-by-wire system and electromechanical actuators for various test scenarios, which show that high performance and robustness are achieved. The test scenarios consist of straight-line braking on different road surfaces (ice, snow, dry asphalt, wet asphalt and inhomogeneous asphalt/plastic coated surface) and a single experiment for braking in a turn on dry asphalt.The main results of this dissertation have been published in international journals, at international conferences and as a book chapter.

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Wheel Slip Control in ABS Brakes using Gain Scheduled Optimal Control with Constraints

A wheel slip controller for ABS brakes is formulated using an explicit constrained LQR design. The controller gain matrices are designed and scheduled on the vehicle speed based on local linearizations. A Lyapunov function for the nonlinear control system is dervied using the Riccati equation solution in order to prove stability and robustness with respect to uncertainty in the road/tyre friction characteristic. Experimental results from a test vehicle with electromechanical brake actuators and brake-by-wire show that high performance and robustness are achieved.

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Idar Petersen

Papers

Gain-scheduled wheel slip control in automotive brake systems

Explicit sub-optimal linear quadratic regulation with state and input constraints

Wheel Slip Control in ABS Brakes using Gain Scheduled Optimal Control with Constraints

Wheel slip control in ABS brakes using gain scheduled constrained LQR

Gain-scheduled control of a solar power plant