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Multiparametric programming and control has received a lot of attention in the past twenty years with significant advances reported in the open literature. Most existing algorithms for multiparametric programming typically suffer from a computational slowdown with respect to the number of parameters, constraints and variables resulting from the large expansion of active set combinations that must be considered. In this work, we present a novel parallel combinatorial algorithm for the solution of multiparametric Quadratic programs (mpQP) and multiparametric Linear Programs (mpLP) (i), generating better exploration rules that greatly reduces the number of active set combinations that must be considered and (ii) a high-performance parallelization scheme for this algorithm that scales well with the number of cores. The proposed algorithm is validated on numerous computational examples, including scaling analysis of the parallel algorithm up to 48 CPU cores and generating optimal Pareto fronts for the Markowitz Portfolio Selection Problem. In large problem instances, speedups of over a factor of 500 are observed. [ABSTRACT FROM AUTHOR]

The hierarchical decision making in process industries has been traditionally viewed as having a common objective, such as the overall cost, which needs to be optimized. However, a more appropriate approach is to formulate and solve hierarchical optimization and control problems. The solution algorithms for hierarchical optimization problems have been reported in the literature. The idea is to recast each optimization sub-problem in the hierarchy into a multiparametric programming problem, considering the variables of upper-level problems as unknown parameters. In this paper, explicit Model Predictive Control (MPC) and hierarchical optimization techniques, employing multiparametric programming, are combined for hierarchical MPC. The solution algorithm for hierarchical MPC is described in detail. Note that the solution to a hierarchical MPC problem is challenging, even for the simplest case of linear-quadratic objectives. Closed-loop simulations of a thermal mixing process, under two different hierarchical MPC formulations, are performed and the control performance is studied.

A strategy that calculates an explicit state feedback policy to regulate constrained uncertain discrete‐time uncertain linear systems is presented. We consider uncertain processes, affected by box‐bounded multiplicative uncertainty as well as bounded additive uncertainty with linear state and inputs constraints. The proposed method includes (i) the calculation of a terminal set constraint and (ii) the robust reformulation of state constraints in the prediction horizon. These features allow the derivation of the desired policy by solving a single multiparametric quadratic programming problem that guarantees feasible operation in the presence of uncertainty. Additionally, we employ variable and constraint elimination approaches to enhance the computational performance of the strategy. We demonstrate the steps and benefits of these developments with a numerical example and a chemical engineering case study. [ABSTRACT FROM AUTHOR]

Climate control in buildings involves multiple conflicting objectives, such as energy consumption and occupant comfort, which have to be considered simultaneously during the operation of the climate control system. In this work, the Multi-Objective Model Predictive Control (MOMPC) solution method is further developed through the multiparametric programming approach (mpMOMPC). The MOMPC optimal control problem is reformulated according to the ϵ -constraint method, and the ϵ vector is treated as unknown parameters to generate the control law expressions offline. This reduces online calculations to point location followed by function evaluation, enabling the controller to be implemented through a chip or low-cost hardware. To demonstrate the potential and versatility of the developed mpMOMPC algorithm, three case studies are conducted. Numerical simulation results show that the extreme-value case is the same as the rule-based MPC case and the preference function case results in maximum energy reduction by 20.1% compared to the rule-based MPC case. [Display omitted] • Building climate control problem involves optimizing multiple conflicting objectives. • Multi-Objective Model Predictive Control strategy is employed for climate control. • The control algorithm is further developed through multiparametric programming. • Online controller calculations can be performed on a chip without needing a computer. [ABSTRACT FROM AUTHOR]

Model Predictive Control (MPC) is a wide spread advanced process control methodology for optimization based control of multi-input and multi-output processes systems. Typically, a surrogate model of the process dynamics is utilized to predict the future states of a process as a function of input actions and an initial state. The predictive model is often a linear model, such as a state space model, due to the computational burden of the resulting optimization problem when utilizing nonlinear models. Recently, rectified linear unit (ReLU) based neural networks (NN) were shown to be mixed integer linear representable, thus allowing their incorporation into mixed integer programming (MIP) frameworks. However, the resulting MIP-based MPC problems are often computationally intractable to solve in real-time. The computational intractability of the reformulated NN-based optimization models is typically addressed in the literature by applying some form of bounds tightening approach. However, this in itself may have a large computational cost. In this work, a novel bound tightening procedure based on a multiparametric (MP) programming formulation of the corresponding MIP reformulated MPC optimization problems is proposed. Which tightening only needs to be computed and applied once-and-offline, thereby significantly improving the computational performance of the MPC in real-time. Some aspects of the effect of regularization during NN regression on the computational difficulty of these optimization problems are also investigated in conjunction with the proposed a priori bounds-tightening approach. The proposed method is compared to the base case without the parametric tightening procedure, as well as NN regularization through two optimal control case studies: (1) A ReLU NN-based MPC of an unstable nonlinear chemostat and, (2) a ReLU NN-based MPC of a nonlinear continuously stirred tank reactor (CSTR). Significant reductions in average time of 99.96% and 91.90% are observed, for the chemostat NN based MPC and CSTR NN based MPC, respectively. • Multiparametric programming to accelerate NN based MPCs. • Analysis of regularization on variable bounds effect on MILP reformulation. • Two case studies displaying the proposed methodology on solve time. [ABSTRACT FROM AUTHOR]

An investor has to carefully select the location and size of new generation units it intends to build, since adding capacity in a market affects the profit from units this investor may already own. To capture this closed-loop characteristic, strategic investment (SI) of generation can be posed as a bilevel optimization. By analytically studying a small market, we first show that its objective function can be non-convex and discontinuous. Realizing that existing mixed-integer problem formulations become impractical for larger markets and number of instances, this work put forth two SI solvers: a grid search to handle setups where the candidate investment locations are few, and a stochastic gradient descent approach for otherwise. Both solvers leverage powerful results of multiparametric programming (MPP), each in a unique way. The grid search entails finding the primal/dual solutions for a large number of optimal power flow (OPF) problems, which nonetheless can be efficiently computed several at once thanks to the properties of MPP. The same properties facilitate the rapid calculation of gradients in a mini-batch fashion, thus accelerating the implementation of a stochastic (sub)-gradient descent search. Tests on the IEEE 30- and 118-bus systems using real-world data corroborate the advantages of the novel solvers. [ABSTRACT FROM AUTHOR]

Model predictive control (MPC) represents an optimal strategy where constraints on inputs, outputs and system states can be implemented as part of the control law that takes the form of a mathematical program. MPC of linear systems with a quadratic objective function results in a quadratic program (QP) that needs to be solved online at each sampling instant. Multiparametric programming methods that attempt an explicit solution to QPs have been successfully used in context of MPC and is termed as explicit MPC (eMPC). eMPC for linear systems results in a piecewise affine-in-state feedback control law and is determined offline. During online implementation, the control law is selected from among the different pieces based on the real-time value of the states. An ability to verify the controller output over all possible state realizations of the feasible state-space is important in any critical application including health and aerospace and is a unique feature of eMPC. Since state feedback MPC requires full state information, it is always used in conjunction with a state estimator such as a Kalman Filter. Further, fault tolerant control methodologies depend on state and parameter estimation to detect and diagnose faults followed by compensation. Conventionally, the state filtering step is performed prior to MPC and thus the interaction between the estimator and controller remains hidden and can be analyzed only via numerical simulations. In this work, we propose using multiparametric programming to find the explicit solution of the joint estimator-MPC problem. In particular, the eMPC control law now depends linearly on the joint information of states and measurements. This allows explicitly obtaining the sensitivity of the MPC control law to the measurement as well as the estimator parameters. The proposed explicit solution of the joint estimator-MPC problem is demonstrated on a SISO 2-tank system. The effect of the estimator gain on the size of the feasible region is delineated. [ABSTRACT FROM AUTHOR]

This paper investigates large fluctuations of locational marginal prices (LMPs) in wholesale energy markets caused by volatile renewable generation profiles. Specifically, we study events of the form ..., where LMP is the vector of LMPs at the n power grid nodes, and α-, α+ ∈ ℝn are vectors of price thresholds specifying undesirable price occurrences. By exploiting the structure of the supply-demand matching mechanism in power grids, we look at LMPs as deterministic piecewise affine, possibly discontinuous functions of the stochastic input process, modelling uncontrollable renewable generation. We use techniques from large deviations theory to identify the most likely ways for extreme price spikes to happen, and to rank the nodes of the power grid in terms of their likelihood of experiencing a price spike. Our results are derived in the case of Gaussian fluctuations, and are validated numerically on the IEEE 14-bus test case. [ABSTRACT FROM AUTHOR]

Explicit model predictive control is an established methodology for the offline determination of the optimal control policy for linear discrete time-invariant systems with linear constraints. Nevertheless, nonlinearities in the form of quadratic constraints naturally appear in process models or are imposed for stability purposes in model predictive control formulations. In this manuscript, we present the theoretical developments and propose an algorithm for the exact solution of explicit nonlinear model predictive control problems with convex quadratic constraints. Our approach is based on a second-order Taylor approximation of Fiacco's Basic Sensitivity Theorem, which allows for the existence and the analytic derivation of the optimal control actions. The complete exploration of the parameter space is founded on an active set strategy, which employs a pruning criterion to eliminate infeasible active sets. Based on that, the optimal map of solutions is constructed along with the corresponding control actions. The proposed strategy is applied to an explicit nonlinear model predictive control problem with an ellipsoidal terminal set, and comparisons with approximate solutions are drawn to demonstrate the benefits of the presented approach. Furthermore, as a practical application, the optimal operation of a chemostat in the presence of disturbances is exhibited. • The exact solution of multiparametric/explicit quadratically constrained MPC problems. • Active set-based methodology for the complete exploration of the parameter space. • Piecewise nonlinear explicit control actions and nonconvex critical regions. • Smaller number of critical regions compared to approximate solutions. [ABSTRACT FROM AUTHOR]

Interconnection studies for distributed energy resources (DERs) can currently take months since they entail simulating a large number of power flow scenarios. If DERs are to be actively controlled, probabilistic hosting capacity analysis (PHCA) studies become more time-consuming since they require solving multiple optimal power flow (OPF) tasks. PHCA is expedited here by leveraging the powerful tool of multiparametric programming (MPP). Using an approximate grid model, optimal DER setpoints are decided by a quadratic program, which depends on analysis and uncertain parameters in a possibly nonlinear fashion. By reformulating this program, feasible and infeasible OPF instances alike are handled in a unified way to uniquely reveal the location, frequency, and severity of feeder constraint violations. The effect of voltage regulators is also captured by novel approximate models. Upon properly extending MPP to PHCA, we were able to find the exact minimizers for 518,400 OPF instances on the IEEE 123-bus feeder by solving only 6,905 of them, and 86,400 instances on a 1,160-bus feeder by solving only 2,111 instances. This accelerated PHCA by a factor of 10. Thus, a utility can promptly infer grid statistics using real-world data without a probabilistic characterization of uncertain parameters. [ABSTRACT FROM AUTHOR]

We present a novel parameter space exploration algorithm for three classes of multiparametric problems, namely linear (mpLP), quadratic (mpQP), and mixed-integer linear (mpMILP). We construct subsets of the parameter space in the form of simplices through Delaunay triangulation to facilitate identification of the optimal partitions that describe the solution space. The presented exploration strategy prioritizes identifying volumetrically larger critical regions compared to existing methods. We demonstrate the exploration algorithm on an illustrative example, and compare the volumetrically identified parameter space against existing solvers on randomly generated problems in all three classes. [ABSTRACT FROM AUTHOR]

The inevitable presence of uncertain parameters in critical applications of process optimization can lead to undesirable or infeasible solutions. For this reason, optimization under parametric uncertainty was, and continues to be a core area of research within Process Systems Engineering. Multiparametric programming is a strategy that offers a holistic perspective for the solution of this class of mathematical programming problems. Specifically, multiparametric programming theory enables the derivation of the optimal solution as a function of the uncertain parameters, explicitly revealing the impact of uncertainty in optimal decision-making. By taking advantage of such a relationship, new breakthroughs in the solution of challenging formulations with uncertainty have been created. Apart from that, researchers have utilized multiparametric programming techniques to solve deterministic classes of problems, by treating specific elements of the optimization program as uncertain parameters. In the past years, there has been a significant number of publications in the literature involving multiparametric programming. The present review article covers recent theoretical, algorithmic, and application developments in multiparametric programming. Additionally, several areas for potential contributions in this field are discussed, highlighting the benefits of multiparametric programming in future research efforts.

In this paper, we present a strategy for the exact solution of multiparametric quadratically constrained quadratic programs (mpQCQPs). Specifically, we focus on multiparametric optimization problems with a convex quadratic objective function, quadratic inequality and linear equality constraints, described by constant matrices. The proposed approach is founded on the expansion of the Basic Sensitivity Theorem to a second-order Taylor approximation, which enables the derivation of the exact parametric solution of mpQCQPs. We utilize an active set strategy to implicitly explore the parameter space, based on which (i) the complete map of parametric solutions for convex mpQCQPs is constructed, and (ii) the determination of the optimal parametric solution for every feasible parameter realization reduces to a nonlinear function evaluation. Based on the presented results, we utilize the second-order approximation to the Basic Sensitivity Theorem to expand to the case of nonconvex quadratic constraints, by employing the Fritz John necessary conditions. Example problems are provided to illustrate the algorithmic steps of the proposed approach. [ABSTRACT FROM AUTHOR]

We consider combinatorial optimization problems with uncertainty in the cost vector. Recently, a novel approach was developed to deal with such uncertainties: instead of a single one robust solution, obtained by solving a min max problem, the authors consider a set of solutions obtained by solving a min max min problem. In this new approach, the set of solutions is computed once and we can choose the best one in real time each time a cost vector occurs yielding better solutions compared to the min max approach. In this paper, we apply the new approach to the absolute and relative regret cases. Algorithms to solve the min max min robust (relative) regret problems are presented with computational experiments. [ABSTRACT FROM AUTHOR]

Summary: In this article, a real‐time nonlinear model predictive idle speed controller based on multiparametric programming is designed for an SI engine. Idle speed is a crucial recurring condition in urban vehicles demanding proper control to avoid stall. As will be seen, the nonlinear model predictive control (NMPC) system designed, besides complying with the predefined constraints, demonstrates a far better performance than the prevalent industrial controllers and even conventional linear MPC controllers. More importantly, a new special structure combining offline nonlinear MPC and classical controller is employed to provide both robustness and fast response. Not only is the computational burden of the controller within that of the ordinary ECU controllers, it is also able to readily damp a disturbance of 20 N·m in less than 2.5 seconds and easily deal with parameter uncertainties. The control system also regulates engine gas pedal release, and converges to the set point with a settling time of less than 3 seconds and minimum fluctuations. [ABSTRACT FROM AUTHOR]

In this paper, the design of a piecewise affine proportional integral (PWA-PI) controller algorithm based on invariant set and multiparametric programming for constrained systems is proposed. We implemented the algorithm in a programmable logic controller (PLC) to control an industrial constrained level plant and analyze its behavior. Structured text routines were programmed and validated while controlling two systems with PLC. The results show that the constraints on the error, integral of the error, system output and control action are respected because PWA-PI controllers are tuned from the solution of an optimization problem. The evaluated performance indexes (such as mean square error, Goodhart, overshoot and settling time) show that PWA-PI can be adjusted for better performance than proportional integral (PI) controller tuned by Ziegler–Nichols (Z–N) rules. In the analyzed cases, a settling time of 108 s was obtained, whereas PI controller with Z–N rules presented a 179 s settling time. All of the analyzed performance indexes that we used to evaluate both controllers show PWA-PI as a better controller for constrained systems. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Parameter estimates are obtained as an explicit function of measurements. • Effect of step size on results obtained is presented. • Implicit Euler's method gives complex parametric expressions. • Implicit Euler's method gives accurate parameter estimates. Abstract This work presents a study that aims to compare two discretization methods for solving parameter estimation using multiparametric programming. In our earlier work, parameter estimation using multiparametric programming was presented where model parameters were obtained as an explicit function of measurements. In this method, the nonlinear ordinary equations (ODEs) model was discretized by using explicit Euler's method to obtain algebraic equations. Then, a square system of parametric nonlinear algebraic equations was obtained by formulating optimality condition. These equations were then solved symbolically to obtain model parameters as an explicit function of measurements. Thus, the online computation burden of solving optimization problems for parameter estimation is replaced by simple function evaluations. In this work, we use implicit Euler's method for discretization of nonlinear ODEs model and compare with the explicit Euler's method for parameter estimation using multiparametric programming. Complexity of explicit parametric functions, accuracy of parameter estimates and effect of step size are discussed. [ABSTRACT FROM AUTHOR]

Highlights • Affine approximate models have significant impact on the multiparametric solution. • Suitable error metrics provide confidence in the developed closed loop solutions. • Knowing which approximate model to use and when is in general non trivial. Abstract Incorporating a high fidelity model that accurately describes a dynamical system in a model predictive control study may often lead to an intractable formulation where the use of model approximation is required. This study examines system identification, time series modeling, and linearization in the context of multiparametric model predictive control with the use of key error metrics including: (i) a novel comparison of key features of the feasible space and objective function in the optimization formulation, (ii) integral time absolute error, (iii) error distribution analysis, and (iv) step response profiles. Two examples are used as a basis for this study: a tank system which highlights the techniques used and a Continuously Stirred Tank Reactor (CSTR). [ABSTRACT FROM AUTHOR]

Summary: We develop an approximate multiparametric convex programming approach with its application to control constrained linear parameter‐varying systems. Recently, the application of the real‐time model predictive control (MPC) for various engineering systems has been significantly increased by using the multiparametric convex programming tool, known as explicit MPC approach. The main idea of explicit MPC is to move the major parts of the computations to offline phase and to provide an explicit piecewise affine solution of the constrained MPC problem, which is defined over a set of convex polyhedral partitions. In the proposed method, the idea of convex programming and partitioning is applied for linear parameter‐varying control systems. The feasible space of the time‐varying parameters is divided into simplices in which approximate solutions are calculated such that the approximation error is kept limited by solving sequences of linear programs. The approximate optimal solution within each simplex is obtained by linear interpolation of the optimal solutions in the simplex vertices, and then multiparametric programming tool is utilized to compute an explicit state feedback solution of linear quadratic optimal control problem for simplex vertices subject to state and input constraints. The proposed method is illustrated by a numerical example and the simulation results show the advantages of this approach. [ABSTRACT FROM AUTHOR]

The short-term forecasting of real-time locational marginal price (LMP) and network congestion is considered from a system operator perspective. A new probabilistic forecasting technique is proposed based on a multiparametric programming formulation that partitions the uncertainty parameter space into critical regions from which the conditional probability distribution of the real-time LMP/congestion is obtained. The proposed method incorporates load/generation forecast, time varying operation constraints, and contingency models. By shifting the computation associated with multiparametric programs offline, the online computational cost is significantly reduced. An online simulation technique by generating critical regions dynamically is also proposed, which results in several orders of magnitude improvement in the computational cost over standard Monte Carlo methods. [ABSTRACT FROM AUTHOR]

Power networks are gateways to transfer power from generators to end-users. Often, it is assumed that the transfer occurs freely without any limiting factors. However, power flows over a network can be limited by predetermined limits that may come from physical reasons, such as line capacity or Kirchhoff’s laws. When flow is constrained by these limits, this is called congestion, which reduces the energy efficiency and splits the price for electricity across the congested lines. One promising, cost-effective way to relieve the impact of the congestion is demand-side management (DSM). However, it is unclear how much DSM can impact congestion and where it can control the demand. This paper proposes a new DSM mechanism based on locational willingness-to-pay (WTP) centered around income statistics; utilizes a state-space tool to determine the possibility to alter prices by DSM; and formulates a convex optimization problem to decide the DSM. The proposed methodology is tested on IEEE (Institute of Electrical and Electronics Engineers) systems with two commonly used objectives: cost minimization and social welfare maximization.

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 [ABSTRACT FROM AUTHOR]

This paper presents a solution method for parametric linear complementarity problems (PLCP) that relies on an enumeration technique to discover all feasible bases. The enumeration procedure is based on evaluating all possible combinations of active constraints and testing for feasibility. Although the enumeration approach is known to grow exponentially in the number of constraints, the formulation of the PLCP allows incorporation of cheap rank tests to quickly prune the infeasible directions in the exploration. The motivation for the development of the enumeration based PLCP solver is that it represents a direct method to solve parametric linear and quadratic optimization problems as well as their mixed-integer counterparts. These types of problems often arise in the field of model predictive control for linear and hybrid systems. The enumeration based PLCP solver offers another alternative to compute explicit solutions in the field of hybrid model predictive control that can be extremely effective in some important cases. [ABSTRACT FROM AUTHOR]

• Proposed parameter transformation yields mp-QP problems with special topology properties. • New solution strategy enabled by parameter transformation. • Solution delivered as two compact statements, or optionally, in the classic explicit format. • Significant improvements on speed of calculation and size of offline solutions. • New framework for the analysis and calculation of multiparametric problems. An innovative paradigm is presented in this paper for convex multiparametric quadratic programming (mp-QP), and made possible by a convenient transformation of the vector of parameters, and expansion of the corresponding parameter space. Solutions of mp-QP problems are now presented in a highly compact form, in place of deriving explicit expressions for all critical regions and optimizer functions, as is the standard practice in the field. This represents two significant advantages over the state-of-the art mp-QP algorithms: firstly, this paradigm offers a much less expensive path for solution calculation; secondly, these compact solutions require minimal storage requirements. This is particularly significant for mp-QP problems including many inequality constraints, which can now be addressed with minimal computational burden. Compact solutions may then be explored to obtain the explicit solutions for the original mp-QP problems if required, where the original parameter dependence and bounds are recovered. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of transmission power control within a network of resource-constrained wireless sensors that operate within a particular ambient healthcare environment. Sensor data transmitted to a remote base station within the network arrive subject to node location, orientation, and movement. Power is optimally allocated to all channels using a novel resource efficient algorithm. The proposed algorithm is based on a computationally efficient min–max model predictive controller that uses an uncertain linear state-space model of the tracking error that is estimated via local received signal strength feedback. An explicit solution for the power controller is computed offline using a multiparametric quadratic solver. It is shown that the proposed design leads to a robust control law that can be implemented quite readily on a commercial sensor node platform where computational and memory resources are extremely limited. The design is validated using a fully IEEE 802.15.4 compliant testbed using Tmote Sky sensor nodes mounted on fully autonomous MIABOT Pro miniature mobile robots. A repeatable representative selection of scaled ambulatory scenarios is presented that is quite typical of the data that will be generated in this space. The experimental results illustrate that the algorithm performs optimal power assignments, thereby ensuring a balance between energy consumption and a particular outage-based quality of service requirement while robustly compensating for disturbance uncertainties such as channel fading, interference, quantization error, noise, and nonlinear effects. [ABSTRACT FROM PUBLISHER]

In recent years, the embedding approach for solving switched optimal control problems has been developed in a series of papers. However, the embedding approach, which advantageously converts the hybrid optimal control problem to a classical nonlinear optimization, has not been extensively compared with alternative solution approaches. The goal of this paper is thus to compare the embedding approach with multiparametric programming, mixed-integer programming [mixed integer programming (MIP), commercial (CPLEX)], and gradient-descent-based methods in the context of five recently published examples: 1) a spring-mass system; 2) moving-target tracking for a mobile robot; 3) two-tank filling; dc–dc boost converter; and 5) skid-steered vehicle. A sixth example, an autonomous switched 11-region linear system, is used to compare a hybrid minimum principle method and traditional numerical programming. For a given performance index (PI) for each case, cost and solution times are presented. It is shown that there are numerical advantages of the embedding approach: lower PI cost (except in some instances when autonomous switches are present), generally faster solution time, and convergence to a solution when other methods may fail. In addition, the embedding method requires no ad hoc assumptions (e.g., predetermined mode sequences) or specialized control models. Theoretical advantages of the embedding approach over the other methods are also described; guaranteed existence of a solution under mild conditions, convexity of the embedded hybrid optimization problem (under the customary conditions on the PI), solvability with traditional techniques (e.g., sequential quadratic programming) avoiding the combinatorial complexity in the number of modes/discrete variables of MIP, applicability to affine nonlinear systems, and no need to explicitly assign discrete/mode variables to autonomous switches. Finally, common misconceptions regarding the embedding approach are addressed, including whether it uses an average value control model (no), whether it is necessary to tweak the algorithm to obtain bang-bang solutions (no), whether it requires infinite switching to implement embedded solution (no), and whether it has real-time capability (yes). [ABSTRACT FROM PUBLISHER]

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We address short-term batch process scheduling problems contaminated with uncertainty in the data. The mixed integer linear programming (MILP) scheduling model, based on the formulation of Ierapetritou and Floudas, Ind Eng Chem Res. 1998; 37(11):4341-4359, contains parameter dependencies at multiple locations, yielding a general multiparametric (mp) MILP problem. A proactive scheduling policy is obtained by solving the partially robust counterpart formulation. The counterpart model may remain a multiparametric problem, yet it is immunized against uncertainty in the entries of the constraint matrix and against all parameters whose values are not available at the time of decision making. We extend our previous work on the approximate solution of mp-MILP problems by embedding different uncertainty sets (box, ellipsoidal and budget parameter regulated uncertainty), and by incorporating information about the availability of uncertain data in the construction of the partially robust scheduling model. For any parameter realization, the corresponding schedule is then obtained through function evaluation. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4184-4211, 2013 [ABSTRACT FROM AUTHOR]

Abstract: This work presents a methodology to derive explicit multiparametric controllers for nonlinear systems, combining model approximation techniques and multiparametric model predictive control (mp-MPC) algorithms. Particular emphasis is given to an approach that applies a nonlinear model reduction technique, based on balancing of empirical gramians, which generates a reduced order model suitable for nonlinear mp-MPC algorithms. This approach is compared with a recently proposed method that uses a meta-modelling based model approximation technique which can be directly combined with standard multiparametric programming algorithms. The methodology is illustrated for two nonlinear models, of a distillation column and a train of CSTRs, respectively. [Copyright &y& Elsevier]

Abstract: This paper proposes an explicit solution to the model predictive control of linear systems subject to non-convex polyhedral constraints. These constraints are modeled as the union of a finite number of convex polyhedra. The algorithm is based on calculating the explicit solution to a modified problem with linear constraints defined as the convex hull of the original ones and classifying its regions by their relation with the regions of the explicit solution to the original problem. Some of the regions are divided, and a procedure based on sum-of-squares programming is designed to determine which of the possible solutions are in fact optimal. Finally, the online algorithm is shown to be better in terms of computational cost and memory requirements than an algorithm based on obtaining and comparing the solutions of the problem using as constraints the polyhedra whose union forms the non-convex regions, both theoretically and by the results of an example. [Copyright &y& Elsevier]

Abstract: Multiparametric (mp) programming pre-computes optimal solutions offline which are functions of parameters whose values become apparent online. This makes it particularly well suited for applications that need a rapid solution of online optimization problems. In this work, we propose a novel approach to multiparametric programming problems based on an enumeration of active sets and use it to obtain a parametric solution for a convex quadratic program (QP). To avoid the combinatorial explosion of the enumeration procedure, an active set pruning criterion is presented that makes the enumeration implicit. The method guarantees that all regions of the partition are critical regions without any artificial cuts, and further that no region of the parameter space is left unexplored. [Copyright &y& Elsevier]

This paper considers the optimal reference-tracking control of an electropneumatic clutch actuator for heavy duty trucks for which there exists an experimentally validated model. To control both supply to and exhaust from the clutch actuator chamber, one pair of on/ off valves is used. The fast dynamics of the actuator and the presence of state and input constraints makes it appropriate to design an explicit nonlinear model predictive controller (NMPC). In this paper, two different types of explicit NMPC controllers for optimal reference-tracking control of the electropneumatic clutch actuator are designed (one of the controllers applies a pulsewidth modulation (PWM) scheme, while the other controller allows the valves to be only fully opened or fully closed). The closed-loop performance of the two controllers is verified by simulating with the clutch actuator model. The controllers are compared with respect to control performance, real-time computational complexity, and storage requirements. A performance comparison with a sliding-mode controller and a PID controller shows that the explicit NMPC with PWM provides the highest reference-tracking quality. An appealing feature of explicit NMPC is that the performance improvement is feasible to implement with low computational complexity even for fast mechatronic systems. [ABSTRACT FROM PUBLISHER]

discrete-time hybrid model of a permanent magnet synchronous motor (PMSM) with saturation in voltage and current is formulated. The controller design with incorporated constraints is achieved in a systematic way from modeling to control synthesis and implementation. The Hybrid System Description Language is used to obtain a mixedlogical dynamical (MLD) model. Based on the MLD model, a model predictive controller is designed for an optimal speed regulation of the motor. For reducing computation complexity and computation time, the MPC controller is converted to its equivalent explicit piecewise affine form by multiparametric programming. Simulations and experiments show that good and robust control performance is achieved by the hybrid model predictive controller as compared with the linear quadratic regulator (LQR) and the PID controller. [ABSTRACT FROM AUTHOR]

Abstract: This work introduces two algorithms for the solution of pure integer and mixed-integer bilevel programming problems by multiparametric programming techniques. The first algorithm addresses the integer case of the bilevel programming problem where integer variables of the outer optimization problem appear in linear or polynomial form in the inner problem. The algorithm employs global optimization techniques to convexify nonlinear terms generated by a reformulation linearization technique (RLT). A continuous multiparametric programming algorithm is then used to solve the reformulated convex inner problem. The second algorithm addresses the mixed-integer case of the bilevel programming problem where integer and continuous variables of the outer problem appear in linear or polynomial forms in the inner problem. The algorithm relies on the use of global multiparametric mixed-integer programming techniques at the inner optimization level. In both algorithms, the multiparametric solutions obtained are embedded in the outer problem to form a set of single-level (M)(I)(N)LP problems – which are then solved to global optimality using standard fixed-point (global) optimization methods. Numerical examples drawn from the open literature are presented to illustrate the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP( q( θ), M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present recent developments in multiparametric nonlinear programming. For the case of convex problems, we highlight key issues regarding the full characterization of the parametric solution space and we discuss, through an illustrative example problem, four alternative state-of-the-art multiparametric nonlinear programming algorithms. We also identify a number of main challenges for the non-convex case and highlight future research directions. [Copyright &y& Elsevier]

Abstract: This paper proposes a novel model predictive control (MPC) scheme based on multiobjective optimization. At each sampling time, the MPC control action is chosen among the set of Pareto optimal solutions based on a time-varying, state-dependent decision criterion. Compared to standard single-objective MPC formulations, such a criterion allows one to take into account several, often irreconcilable, control specifications, such as high bandwidth (closed-loop promptness) when the state vector is far away from the equilibrium and low bandwidth (good noise rejection properties) near the equilibrium. After recasting the optimization problem associated with the multiobjective MPC controller as a multiparametric multiobjective linear or quadratic program, we show that it is possible to compute each Pareto optimal solution as an explicit piecewise affine function of the state vector and of the vector of weights to be assigned to the different objectives in order to get that particular Pareto optimal solution. Furthermore, we provide conditions for selecting Pareto optimal solutions so that the MPC control loop is asymptotically stable, and show the effectiveness of the approach in simulation examples. [Copyright &y& Elsevier]

The multiparametric 0-1-Integer Programming (0-1-IP) problem relative to the objective function is a family of 0-1-IP problems which are related by having identical constraint matrix and righthand-side vector. In this paper we present an algorithm to perform a complete multiparametric analysis relative to a generalized min max objective function such that the min sum and min max are particular cases. [ABSTRACT FROM AUTHOR]