Predictive Control Of Hybrid Systems Research Articles

A Piecewise Linear Regression and Classification Algorithm With Application to Learning and Model Predictive Control of Hybrid Systems

This paper proposes an algorithm for solving multivariate regression and classification problems using piecewise linear predictors over a polyhedral partition of the feature space. The resulting algorithm that we call PARC (Piecewise Affine Regression and Classification) alternates between (<i>i</i>) solving ridge regression problems for numeric targets, softmax regression problems for categorical targets, and either softmax regression or cluster centroid computation for piecewise linear separation, and (<i>ii</i>) assigning the training points to different clusters on the basis of a criterion that balances prediction accuracy and piecewise-linear separability. We prove that PARC is a block-coordinate descent algorithm that minimizes a suitably constructed objective function and that it converges in a finite number of steps. The algorithm is used to learn hybrid numerical/categorical (HYNC) dynamical models from data that contain real and discrete labeled values. The resulting model has a piecewise linear structure that is particularly useful to formulate model predictive control problems and solve them by mixed-integer programming. A Python implementation of PARC is available at <uri>http://cse.lab.imtlucca.it/~bemporad/parc</uri>.

Warm Start of Mixed-Integer Programs for Model Predictive Control of Hybrid Systems

In hybrid model predictive control (MPC), a mixed-integer quadratic program (MIQP) is solved at each sampling time to compute the optimal control action. Although these optimizations are generally very demanding, in MPC, we expect consecutive problem instances to be nearly identical. This article addresses the question of how computations performed at one time step can be reused to accelerate (warm start) the solution of subsequent MIQPs. Reoptimization is not a rare practice in integer programming: for small variations of certain problem data, the branch-and-bound algorithm allows an efficient reuse of its search tree and the dual bounds of its leaf nodes. In this article, we extend these ideas to the receding-horizon settings of MPC. The warm-start algorithm we propose copes naturally with arbitrary model errors, has a negligible computational cost, and frequently enables an a priori pruning of most of the search space. Theoretical considerations and experimental evidence show that the proposed method tends to reduce the combinatorial complexity of the hybrid MPC problem to that of a one-step look-ahead optimization, greatly easing the online computation burden.

LMI-based robust mixed-integer model predictive control for hybrid systems

ABSTRACT This paper proposes a linear matrix inequality (LMI) approach to mixed-integer model predictive control (MPC) of uncertain hybrid systems with binary and real valued control inputs. The stability condition of the hybrid system is obtained by using the Lyapunov function and then a sufficient state feedback control law is achieved so that guarantees the closed-loop stability and also minimises an infinite horizon performance index. The primal optimisation problem is convex, therefore, a convex relaxation is investigated by introducing the Lagrange dual function. The real and binary control inputs are obtained by solving the dual of the dual problem in the framework of LMIs. The performance and effectiveness of the proposed MPC approach and the duality gap of the convex relaxation are studied through simulation results of a hybrid system with mixed real and binary inputs.

Dealing with Complexity through Advanced Control Techniques

The control of complex dynamic systems, both in their behavior and in their mission, goes through the implementation of multi-loop control architectures based on information about the system internal state and from the environment, as well as on the mission plan state. This results in systems that are becoming increasingly autonomous, for which requirements in terms of safety and reliability, as well as expected performance, are increasingly high. Research works developed at ONERA in the field of control for autonomous systems cover all levels of the control architectures, which are basically structured with respect to temporal aspects, as well as the level of abstraction that they entail for the system dynamic. We will consider them in this paper by increasing level. We will discuss the advances achieved recently in the robust control techniques of uncertain dynamic systems generally implemented at the lower control level and we will discuss their extensions to consider input and output constraints, as well as the hybrid nature of most of the systems considered. To design task level control primitives that take place just above the previous control loops, we will introduce sensor-based robust and non-linear control techniques. These are based on information on the environment extracted from exteroceptive sensors, to adapt system behavior to uncertainties and perturbations. Multi-sensor and/or multi-objective controls will be discussed in this particular context. We will also present several recent results in the field of trajectory tracking based on visual navigation techniques in complex environments, which combine objectives and constraints within the same control architecture. We will discuss how model predictive control (MPC) techniques and advanced optimization techniques can be used for solving the resulting control problems. In addition, we will discuss several ongoing developments of these methods by exploiting distributed model predictive control techniques (DMPC) and predictive control of hybrid systems. Finally, integration with the control architectures at the upper level of reactive, predictive and distributed planning capabilities will be proposed to accommodate time constraints and uncertainties in decision.

A Hierarchical Algorithm for Integrated Scheduling and Control With Applications to Power Systems

The contribution of this paper is a hierarchical algorithm for integrated scheduling and control via model predictive control of hybrid systems. The controlled system is a linear system composed of continuous control, state, and output variables. Binary variables occur as scheduling decisions in the optimal control problem (OCP). The scheduling decisions are made on a slow time scale compared with the system dynamics. This gives rise to a temporal separation of the scheduling and control variables in the OCP. Accordingly, the proposed hierarchical algorithm consists of two optimization levels. The upper level (scheduling level) solves a mixed-integer linear program (MILP) with a low frequency. The lower level (control level) solves an LP with a high frequency. The main advantage of the proposed approach is that it requires online solution of an LP rather than an MILP. Simulations based on a power portfolio case study show that the hierarchical algorithm reduces the computation to solve the OCP by several orders of magnitude. The improvement in computation time is achieved without a significant increase in the overall cost of operation.

Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming

Hybrid systems are dynamical systems characterized by the simultaneous presence of discrete and continuous variables. Model‐based control of such systems is computationally demanding. To this effect, explicit controllers which provide control inputs as a set of functions of the state variables have been derived, using multiparametric programming mainly for the linear systems. Hybrid polynomial systems are considered resulting in a Mixed Integer Polynomial Programming problem. Treating the initial state of the system as a set of bounded parameters, the problem is reformulated as a multiparametric Mixed Integer Polynomial optimization (mp‐MIPOPT) problem. A novel algorithm for mp‐MIPOPT problems is proposed and the exact explicit control law for polynomial hybrid systems is computed. The key idea is the computation of the analytical solution of the optimality conditions while the binary variables are treated as relaxed parameters. Finally, using symbolic calculations exact nonconvex critical regions are computed. © 2016 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 62: 3441–3460, 2016

Embedded optimization for mixed logical dynamical systems

Predictive control of hybrid systems is currently considered prohibitive using embedded computing platforms. To overcome this limitation for mixed logical dynamical systems of small to medium size, we propose to use (1) a standard branch-and-bound approach combined with a fast embedded interior point solver, (2) pre-processing heuristics, run once and offline, to significantly reduce the number of subproblems to be solved, and (3) relaxations of the original MPC problem that allow a trade off between computation time and closed-loop performance. A problem-specific ANSI C implementation of the proposed method can be automatically generated, and has a fixed memory footprint and a code size that is insignificantly larger than that of the subproblem solver. Two extensive numerical studies are presented, where problems with up to 60 binary variables are solved in less than 0.2s with a performance deterioration of below 2% when compared to an optimal MPC scheme.

A dynamic programming based approach for explicit model predictive control of hybrid systems

This work presents an algorithm for explicit model predictive control of hybrid systems based on recent developments in constrained dynamic programming and multi-parametric programming. By using the proposed approach, suitable for problems with linear cost function, the original model predictive control formulation is disassembled into a set of smaller problems, which can be efficiently solved using multi-parametric mixed-integer programming algorithms. It is also shown how the methodology is applied in the context of explicit robust model predictive control of hybrid systems, where model uncertainty is taken into account. The proposed developments are demonstrated through a numerical example where the methodology is applied to the optimal control of a piece-wise affine system with linear cost function.

A novel adaptive fuzzy predictive control for hybrid systems with mixed inputs

This paper proposes a new adaptive nonlinear model predictive control (NMPC) methodology for a class of hybrid systems with mixed inputs. For this purpose, an online fuzzy identification approach is presented to recursively estimate an evolving Takagi–Sugeno (eTS) model for the hybrid systems based on a potential clustering scheme. A receding horizon adaptive NMPC is then devised on the basis of the online identified eTS fuzzy model. The nonlinear MPC optimization problem is solved by a genetic algorithm (GA). Diverse sets of test scenarios have been conducted to comparatively demonstrate the robust performance of the proposed adaptive NMPC methodology on the challenging start-up operation of a hybrid continuous stirred tank reactor (CSTR) benchmark problem.

Deterministic finite automata representation for model predictive control of hybrid systems

As is well known, the computational complexity in the mixed integer programming (MIP) problem is one of the main issues in model predictive control (MPC) of hybrid systems such as mixed logical dynamical systems. Thus several efficient MIP solvers such as multi-parametric MIP solvers have been extensively developed to cope with this problem. On the other hand, as an alternative approach to this issue, this paper addresses how a deterministic finite automaton, which is a part of a hybrid system, should be expressed to efficiently solve the MIP problem to which the MPC problem is reduced. More specifically, a modeling method to represent a deterministic finite automaton in the form of a linear state equation with a smaller set of binary input variables and binary linear inequalities is proposed. After a motivating example is described, a derivation procedure of a linear state equation with linear inequalities representing a deterministic finite automaton is proposed as three steps; modeling via an implicit system, coordinate transformation to a linear state equation, and state feedback binarization. Various significant properties on the proposed modeling are also presented throughout the proofs on the derivation procedure.

Analytical non-linear model predictive control for hybrid systems with discrete inputs only

Model predictive control (MPC) is known as an efficient technique for controlling different industrial processes. However, the computation load remains the main challenge facing the real-time application of the MPC technique especially for complex systems, like, for example, hybrid systems with discrete or binary variables. In this study the authors propose an analytical non-linear model predictive control (NMPC) technique for hybrid systems with discrete inputs only. The proposed controller has lower computation complexity compared to other techniques presented in the literature; as a result real-time implementation is turned to be possible for several systems. The proposed analytical NMPC controller can be applied efficiently for different classes of hybrid systems: switching systems, linear hybrid systems, non-linear hybrid systems and constrained systems. The proposed technique is validated through several examples representing different classes of hybrid systems with discrete inputs.

Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions

This article presents a novel model predictive control (MPC) scheme that achieves input-to-state stabilization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability (ISS) is guaranteed when an optimal solution of the MPC optimization problem is attained. Special attention is paid to the effect that sub-optimal solutions have on ISS of the closed-loop system. This issue is of interest as firstly, the infimum of MPC optimization problems does not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the deviation of the predictive control law from the optimum and the resulting deterioration of the ISS property of the closed-loop system. By imposing stronger conditions on the sub-optimal solutions, ISS can even be attained in this case.

Graph-Switching Based Modeling of Mode Transition Constraints for Model Predictive Control of Hybrid Systems

Graph-Switching Based Modeling of Mode Transition Constraints for Model Predictive Control of Hybrid Systems

Predictive Control of Hybrid Systems: Stability results for sub-optimal solutions

This article presents a novel model predictive control (MPC) scheme that achieves input-to-state stabilization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability (ISS) is guaranteed when an optimal solution of the MPC optimization problem is attained. Special attention is paid to the effect that sub-optimal solutions have on ISS of the closed-loop system. This issue is of interest as firstly, the infimum of MPC optimization problems does not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the deviation of the predictive control law from the optimum (called the optimality margin) and the resulting deterioration of the ISS property of the closed-loop system (called the ISS margin).

Model-based Predictive Control of Hybrid Systems: A Probabilistic Neural-network Approach to Real-time Control

This paper proposes an approach for reducing the computational complexity of a model-predictive-control strategy for discrete-time hybrid systems with discrete inputs only. Existing solutions are based on dynamic programming and multi-parametric programming approaches, while the one proposed in this paper is based on a modified version of performance-driven reachability analyses. The algorithm abstracts the behaviour of the hybrid system by building a 'tree of evolution'. The nodes of the tree represent the reachable states of a process, and the branches correspond to input combinations leading to designated states. A cost-function value is associated with each node and based on this value the exploration of the tree is driven. For any initial state, an input sequence is thus obtained, driving the system optimally over a finite horizon. According to the model predictive strategy, only the first input is actually applied to the system. The number of possible discrete input combinations is finite and the feasible set of the states of the system may be partitioned according to the optimization results. In the proposed approach, the partitioning is performed offline and a probabilistic neural network (PNN) is then trained by the set of points at the borders of the state-space partitions. The trained PNN is used as a system-state-based control-law classifier. Thus, the online computational effort is minimized and the control can be implemented in real time.

Predictive control for hybrid systems. Implications of polyhedral pre-computations

The present paper deals with the predictive control laws for hybrid systems. The modelling formalism used will be the Mixed Logical Dynamical (MLD) which offers the advantage of a compact expression of the dynamics in terms of linear equalities and inequalities on the logical and continuous-time states and inputs. Being an optimization-based control technique, the predictive control needs an efficient implementation scheme in order to be effective in real time. Several studies assess the importance of the prediction horizon and the terminal constraints due to their implications in the structure of the associated optimal control problem. Lately it has been shown that as long as the constraints remain linear, the polyhedral computations can serve as tools for the migration of the on-line computational effort to off-line explicit constructions in terms of explicit solutions which can avoid a costly on-line optimum seeking and thus pushing the application of predictive laws to even higher sampling rates. This paper reviews the on-line optimization techniques proposed for the predictive control of hybrid systems based on mixed integer optimization problems. Further, the explicit solutions are analyzed using a parameterized polyhedron approach.

Stabilizing Model Predictive Control of Hybrid Systems

In this note, we investigate the stability of hybrid systems in closed-loop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1, infin-norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theory

ハイブリッドシステムのモデル予測制御における離散ダイナミクスモデリング

ハイブリッドシステムのモデル予測制御における離散ダイナミクスモデリング

AN EFFICIENT ALGORITHM FOR PREDICTIVE CONTROL OF PIECEWISE AFFINE SYSTEMS WITH MIXED INPUTS

This paper presents a mixed optimization algorithm devoted to predictive control of hybrid systems belonging to the PieceWise Affine (PWA) class with mixed (i.e. continuous and discrete) inputs. By using the particular structure of the optimization problem, the number and the dimension of the subproblems to solve in order to find the optimum are significantly reduced. This approach is applied to a classical case study in the field of hybrid systems: the control of water levels of a three tank system.

Predictive Control of Hybrid Systems Under a Multi-MLD Formalism

The Mixed Logical Dynamical (MLD) formalism has proved to be an efficient modelling framework for hybrid systems described by dynamics, logic and constraints. Furthermore, it allows formulating and solving practical problems such as control, using for example predictive strategies. However, its main drawback remains the computation load due to the complexity ofthe MIQPs to be solved.To overcome this problem, this paper presents a multi-MLD model, considering split state space regions inside which restricted simpler size MLD models take into account only variables variations that may occur. This approach enables to considerably reduce the computation time, to become more convenient for real time implementation even with small sampling time. This strategy is applied in simulation to the control of a three tanks benchmark.