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Explicit MPC represents one of the fastest ways of real-time MPC implementation. As the explicit MPC policy is optimization-free in real-time control, its efficiency is determined by solving a point location problem. This paper proposes the novel concept of accelerating explicit MPC that significantly speeds up the real-time evaluation of the point location problem. The introduced strategy has two layers: (i) an offline phase determines a smart order of the regions to be explored, and (ii) an online phase removes further regions to be explored on the fly based on the current value of the value function. The main advantage of layer (i) is that the order is evaluated offline, therefore, it does not increase the real-time implementation of explicit MPC. The implementation of layer (ii) slightly increases the real-time evaluation but leads to further speed-up of the point location problem. As the proposed layers are based just on the evaluation of some appropriate value function, the main benefit is that these layers are fully applicable also for higher-dimensional systems. Although the accelerated explicit MPC variant does not reduce the worst-case time of solving the point location problem, an extensive case study demonstrates the efficiency of the proposed strategy.

This paper addresses the problem of construction of the tunable robust positive invariant sets in the framework of convex-lifting-based robust control. Moreover, the paper introduces a switching control law minimizing the necessity to compute control input by solving a linear programming, that is computationally more expensive than a pre-computed linear control law. Simultaneously, the control performance is improved by introducing another robust positive invariant subset with more aggressive control law. The robust stability guarantee is proven. Two case studies investigate the benefits of the proposed strategy.

Optimal control of a CSTR represents a challenging task. The proposed paper discusses two issues. The first one addresses control of pH in a chemical vessel, where the reaction between sodium hydroxide and acetic acid occurs. The objective here is to improve control performance of a well tuned PI controller. It will be shown that this can be achieved by introducing a reference governor scheme. The second problem, elaborated in this paper, is the implementation of the reference governor paradigm. Concretely, we aim to design a fast and cheap MPC-based feedback controller. To achieve these goals, we exploit the region-less explicit technique, which efficiently reduces memory footprint issues of standard explicit MPC schemes. Such MPC-based reference governor was employed to control pH in the chemical vessel. Its control performance is compared with conventional PI controller. Finally, comparison of implementation requirements of region-less and region-based explicit techniques is investigated.

We show how to design and solve, in a computationally tractable fashion, optimization-based reference governors for systems where multiple inner loops are closed by separate model predictive control (MPC) strategies. These individual controllers, however, do not have the knowledge of higher-level coupling constraints and therefore require coordination. We show that such a coordination can be achieved by formulating and solving a suitable bilevel optimization problem that optimizes the references for the inner controllers. We review an approach to deal with such problems, which relies on the Karush-Kuhn-Tucker optimality conditions. The presented approach is firstly commented, with hints to enhance its practicality, and then applied to an illustrative example.

This paper deals with education of graduate and undergraduate students in the field of classical and advanced controllers. A pilot scale binary distillation column is utilized to serve the teaching purposes. The emphasis of the paper is to practically teach identification, and widely used classical and advanced control schemes, such as Proportional-Integral-Derivative (PID) and Model Predictive Control (MPC), over a plant. MATLAB/Simulink-based Human Machine Interface (HMI) is used by students to implement these control strategies. Performance results of both controllers subject to reference tracking, constraints and disturbance handling are presented. Developed directives and control schemes can be used for effective process control education of chemical/process engineers.

This paper presents a development and educational application of robotic traffic simulator. Setup presented in his work consists of ten laboratory-scale vehicles designed for simulations of various traffic situations as well as the evaluation of advanced control scenarios. These can be the control of traffic fluency, such as congestion movement, vehicle group acceleration, breaking, obstacle avoidance and other situations known from everyday traffic. Further, the paper describes the technical realization of simulator from both, the hardware and software point of view. Moreover, the applicability of solution is discussed over the various situations, which can be solved in educational as well as the scientific matter. The educational value of developed traffic simulator is demonstrated on the case study, where an optimal control strategy using the Model Predictive Control was designed and evaluated by a master’s degree student.

This chapter explains the synthesis of explicit MPC feedback laws that allow for real-time implementation on hardware with limited computational and storage properties. Four methods are introduced. The first one replaces the potentially complex explicit MPC controller by a simpler feedback law by exploiting the geometry of explicit solutions. The second method reduces the storage footprint of explicit MPC by a complete elimination of critical regions, replaced by a direct evaluation of optimality conditions. The common denominator of both methods is that they preserve optimality while considerably reducing the complexity. The third method trades lower complexity for suboptimality while simultaneously minimizing the performance loss. Finally, a method for designing stabilizing explicit MPC controllers for control of nonlinear systems is introduced.

In this paper we show how to synthesize simple explicit MPC controllers based on approximate dynamic programming. Here, a given MPC optimization problem over a finite horizon is solved iteratively as a series of problems of size one. The optimal cost function of each subproblem is approximated by a quadratic function that serves as a cost-to-go function for the subsequent iteration. The approximation is designed in such a way that closed-loop stability and recursive feasibility is maintained. Specifically, we show how to employ sum-of-squares relaxations to enforce that the approximate cost-to-go function is bounded from below and from above for all points of its domain. By resorting to quadratic approximations, the complexity of the resulting explicit MPC controller is considerably reduced both in terms of memory as well as the on-line computations. The procedure is applied to control an inverted pendulum and experimental data are presented to demonstrate viability of such an approach.

Software tools play an important dissemination role by bringing cutting-edge theoretical algorithms into the hands of researchers and practitioners. This paper introduces a new robust analysis and control module of the Multi-Parametric toolbox, which is one of the most successful open-source tools in the field. We discuss how robust reachable and invariant sets can be computed using a convenient user interface. Such sets play an important role in many control-oriented tasks, such as in design of recursively feasible optimization-based control laws. Moreover, the new module also allows to synthesize robustly stabilizing linear controller and, more importantly, offers robust Model Predictive Control (MPC) synthesis features. The main aim of the paper is illustrate how complex tasks can be implemented using simple operators such that more sophisticated algorithms could be developed easily by the research community.

This paper deals the with design of a control strategy which will effectively control the level of pH in a neutralization reactor in the whole range of pH. The process consists of a continuously stirred tank, where aqueous solutions streams of acetic acid and of sodium hydroxide are mixed together. The main challenge of a successful control strategy for this process arises mainly from its non-linear behavior. The paper will show how to handle such non-linearity efficiently by introducing an augmented output and an optimization based control. Simulation results will be given to demonstrate the behavior of the proposed control strategy. Comparison between an optimal based controller and a simple PI controller is discussed.

Summary We consider the problem of synthesizing simple explicit model predictive control feedback laws that provide closed-loop stability and recursive satisfaction of state and input constraints. The approach is based on replacing a complex optimal feedback law by a simpler controller whose parameters are tuned, off-line, to minimize the reduction of the performance. The tuning consists of two steps. In the first step, we devise a simpler polyhedral partition by solving a parametric optimization problem. In the second step, we then optimize parameters of local affine feedbacks by minimizing the integrated squared error between the original controller and its simpler counterpart. We show that such a problem can be formulated as a convex optimization problem. Moreover, we illustrate that conditions of closed-loop stability and recursive satisfaction of constraints can be included as a set of linear constraints. Efficiency of the method is demonstrated on two examples. Copyright © 2014 John Wiley & Sons, Ltd.

We show that explicit MPC solutions admit a closed-form solution which does not require the storage of critical regions. Therefore significant amount of memory can be saved. In fact, not even the construction of such regions is required. Instead, all possible optimal active sets are first extensively enumerated. Then, for each optimal , only the analytical expressions of primal and dual variables are stored. Optimality of a particular if checked by verifying primal and dual feasibility conditions, which are unique for all candidate sets. We show that the required memory storage can be further reduced by only storing the factors for the dual variables.

This paper reviews the Multi-Parametric Toolbox 3, a new version of the easy-to-use software tool for design, verification, and implementation of optimization-based controllers. Specifically, we introduce advanced building blocks which allow to synthesize and analyze explicit representations of model predictive controllers of low real-time implementation complexity. These building blocks include, but are not limited to, integration of functions over polytopes, computational geometry operations, as well as procedures to analyze invariance and closed-loop stability. We show how to combine these building blocks as to create sophisticated algorithms which lead to well-performing, yet simple controllers which adhere to prescribed requirements.

Over the last decade, anesthesia research community witnessed numerous advances in controllers and their implementation platforms to control the depth of anesthesia (DoA) in a patient undergoing surgery. Today's operating theaters are extremely complex and crowded. New surgical techniques bring new medical technologies and more devices in the operation rooms, which often results in complex configurations, computer based control, and cable clutter. In an effort to reduce hardware size and to the improve quality control of anesthesia, we present a field programmable gate array (FPGA) based explicit model predictive control (EMPC) scheme which can take into account the control and state constraints that naturally arise in anesthesia. Real-time implementation of model predictive control (MPC), mainly requires solving an optimization problem at regular time intervals. We propose an FPGA-based EMPC-on-a-chip algorithm with customized 32-bit floating-point addition, substation, and multiplication algorithms. Simulation results with four compartmental PK-PD model, input constraints and a variable bispectral index (BIS) set-point are presented. The real-time simulation results are achieved with Xilinx's Vertex 4 XC4VLX25-10FF668 FPGA.

This paper shows how explicit model predictive control (MPC) strategies can be implemented in Python. They use a pre-calculated map between state measurements and control inputs to simplify and accelerate the calculation of optimal control inputs. By shifting majority of the computational effort off-line, the concept of explicit MPC offers a significantly faster and cheaper implementation of model predictive control. We show how explicit MPC feedbacks are designed and exported to a self-contained Python code that can be easily merged with existing applications. Two examples are provided to illustrate the procedure. One considers the design of an artificial player for a videogame. The second one tackles the problem of quadrocopter control.

Explicit Model Predictive Control (MPC) is an attractive control strategy, especially when one aims at a fast, computationally less demanding implementation of MPC. Although leading to a fast implementation of optimization-based control, the main downside of explicit MPC is its high complexity in terms of memory occupancy, which often limits practical applicability of such a control methodology. Therefore in this paper we propose to obtain simple explicit MPC controllers that provide guarantees of recursive satisfaction of input and state constraints. The task is accomplished by optimizing, off-line, the parameters of the feedback law such that an integrated square error between the optimal, but complex controller and its simpler replacement is minimized. We show that the task can be formulated as a quadratic optimization problem which always yields an admissible solution. In this way, suboptimality of simple feedbacks with respect to their complex optimal counterparts is significantly mitigated.

Explicit Model Predictive Control (MPC) is an attractive control strategy, especially when one aims at a fast, computationally less demanding implementation of MPC. However the major obstacle that prevents a successful application of explicit MPC controllers lies in the increased memory occupancy. This is a major limitation of the approach when aiming at implementing MPC in control hardware that has restricted amount of memory storage. Therefore in this paper we propose to obtain a much more simpler representation of explicit MPC solutions that occupy less memory. We propose to achieve this goal by constructing a simpler, albeit suboptimal, representation of the explicit MPC optimizer. This task is accomplished by first synthesizing a simpler explicit optimizer as a piecewise affine function that maps state measurements onto the predicted sequence of control inputs. Subsequently, parameters of such a function are refined as to achieve better performance. We show that such a function approximation problem is always feasible. Efficacy of the proposed procedure is demonstrated on several examples.

We investigate how the performance of explicit MPC feedback laws is affected by rounding-based quantization of the control commands. Specifically, we address the problem of providing a rigorous certificate that a given quantized piecewise affine explicit MPC feedback is bounded from below and from above by specific functions. These functions are constructed as to reflect typical control requirements, such as recursive feasibility and closed-loop stability. We show how to obtain an analytical form of the quantized MPC feedback and how to provide the certificate by solving a set of linear programs.