Robust memory state feedback model predictive control for discrete-time uncertain state delayed systems

In this paper, we propose a memory state feedback model predictive control (MPC) law for a discrete-time uncertain state delayed system with input constraints. The model uncertainty is assumed to be polytopic, and the delay is assumed to be unknown, but with a known upper bound. We derive a sufficient condition for cost monotonicity in terms of LMI, which can be easily solved by an efficient convex optimization algorithm. A delayed state dependent quadratic function with an estimated delay index is considered for incorporating MPC problem formulation. The MPC problem is formulated to minimize the upper bound of infinite horizon cost that satisfies the sufficient conditions. Therefore, a less conservative sufficient conditions in terms of linear matrix inequality (LMI) can be derived to design a more robust MPC algorithm. A numerical example is included to illustrate the effectiveness of the proposed method.     

Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model

In this paper, a model predictive control (MPC) scheme for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities, arising in the context of transport-reaction processes, is proposed. A spatial operator of a parabolic PDE system is characterized by a spectrum that can be partitioned into a finite slow and an infinite fast complement. In this view, first, Galerkin method is used to derive a set of finite dimensional slow ordinary differential equation (ODE) system that captures the dominant dynamics of the initial PDE system. Then, a Multilayer Neural Network (MNN) is employed to parameterize the unknown nonlinearities in the resulting finite dimensional ODE model. Finally, a Galerkin/neural-network-based ODE model is used to predict future states in the MPC algorithm. The proposed controller is applied to stabilize an unstable steady-state of the temperature profile of a catalytic rod subject to input and state constraints.     

HIV dynamics: Analysis and robust multirate MPC-based treatment schedules

Analysis and control of human immunodeficiency virus (HIV) infection have attracted the interests of mathematicians and control engineers during the recent years. Several mathematical models exist and adequately explain the interaction of the HIV infection and the immune system up to the stage of clinical latency, as well as viral suppression and immune system recovery after treatment therapy. However, none of these models can completely exhibit all that is observed clinically and account the full course of infection. Besides model inaccuracies that HIV models suffer from, some disturbances/uncertainties from different sources may arise in the modelling. In this paper we study the basic properties of a 6-dimensional HIV model that describes the interaction of HIV with two target cells, CD4⁺ T cells and macrophages. The disturbances are modelled in the HIV model as additive bounded disturbances. Highly Active AntiRetroviral Therapy (HAART) is used. The control input is defined to be dependent on the drug dose and drug efficiency. We developed treatment schedules for HIV infected patients by using robust multirate Model Predictive Control (MPC)-based method. The MPC is constructed on the basis of the approximate discrete-time model of the nominal model. We established a set of conditions, which guarantee that the multirate MPC practically stabilizes the exact discrete-time model with disturbances. The proposed method is applied to the stabilization of the uninfected steady state of the HIV model. The results of simulations show that, after initiation of HAART with a strong dosage, the viral load drops quickly and it can be kept under a suitable level with mild dosage of HAART. Moreover, the immune system is recovered with some fluctuations due to the presence of disturbances.     

Nonlinear MPC: the Impact of Sampling on Closed Loop Stability

In this paper we consider model predictive control (MPC) schemes without stabilizing terminal constraints and/or costs for continuous time systems. While the estimates on the required prediction horizon length such that asymptotic stability of the MPC closed loop is guaranteed yield, in general, satisfactory results their applicability is limited due to the fact that the respective proofs assume that the input function can be switched arbitrarily often on compact time intervals. We present a technique which allows to determine a suitable discretization accuracy such that the obtained performance bound is preserved while the control signal is only switched at the sampling instants of the corresponding sampled data system. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extension of SPH to simulate non-isothermal free surface flows during the injection molding process

This article presents an extension of smoothed particle hydrodynamics (SPH) to non-isothermal free surface flows during the injection molding process. Specifically, we use the method presented by Xu and Yu, Appl. Math. Model. 48 (2017) pp. 384–409, in which the corrected kernel gradient is implemented to increase the computational accuracy and the Rusanov flux is introduced into the continuity equation to alleviate large and random pressure oscillations. To model non-isothermal free surface flows, a working SPH discretization of the temperature equation is derived. An enhanced treatment of the wall boundary is further developed, which can model arbitrary-shaped mold walls. The proposed SPH method is first validated by solving non-isothermal Couette flow and non-isothermal injection molding of a circular disc with a core and comparing the SPH results with those obtained by other numerical methods or experiments. We then extend the numerical method to non-isothermal injection molding of F-shaped and N-shaped cavities. The convergence of the method is examined with several different particle sizes. The effects of the operating conditions (e.g., injection temperature, temperature of the mold wall, and injection velocity) on the flow behavior are analyzed. All the results illustrate that the present SPH method is a powerful computational tool for simulations of non-isothermal free surface flows during the injection molding process.     

New Robust Model Predictive Control for Uncertain Systems with Input Constraints Using Relaxation Matrices

In this paper, we propose a new robust model predictive control (MPC) method for time-varying uncertain systems with input constraints. We formulate the problem as a minimization of the worst-case finite-horizon cost function subject to a new sufficient condition for cost monotonicity. The proposed MPC technique uses relaxation matrices to derive a less conservative terminal inequality condition. The relaxation matrices improve feasibility and system performance. The optimization problem is solved by semidefinite programming involving linear matrix inequalities (LMIs). A numerical example shows the effectiveness of the proposed method. The authors thank the associate editor and two anonymous referees for careful reading and useful suggestions.     

Analysis of the filling capability to the microstructures in micro-injection molding

Micro-injection molding is an important fabrication process for polymer plastics with micro-features. In micro-injection molding of products with microstructures, the ability for the polymer melt to flow into the microstructures is a crucial factor for successful molding. An analytical model in micro-injection molding is constructed in this research. It has been reported that most of the filling in microstructure is done during the packing pressure. In this analytical model, the temperature of the polymer melt near the entrance of the microstructure at the end of mold filling is estimated first. With the temperature, we can calculate the injection distance into the microstructures of the mold insert during the packing stage. The model is compared with experimental results, and shows the feasibility. The experiment uses the LIGA-like lithography process to define the micro-feature and a micro-electroforming method to form the mold insert with the replicated micro-feature. The injection distance into the microstructures predicted by this analytical model shows reasonable result as compared to the experimental measurement.     

Model predictive control of nonlinear hybrid systems with discrete inputs employing a hybrid fuzzy model

The paper deals with model predictive control (MPC) of nonlinear hybrid systems with discrete inputs based on reachability analysis. In order to implement a MPC algorithm, a model of the process that we are dealing with is needed. In the paper, a hybrid fuzzy modelling approach is proposed. The hybrid system hierarchy is explained and the Takagi–Sugeno fuzzy formulation for hybrid fuzzy modelling purposes is tackled. An efficient method of identification of the hybrid fuzzy model is also discussed.

An algorithm that is–due to its MPC nature–suitable for controlling a wide spectrum of systems (provided that they have discrete inputs only) is presented.

The benefits of the algorithm employing a hybrid fuzzy model are verified on a batch reactor example. The results suggest that by suitably determining the cost function, satisfactory control can be attained, even when dealing with complex hybrid–nonlinear–stiff systems such as the batch reactor.

Finally, a comparison between MPC employing a hybrid linear model and a hybrid fuzzy model is carried out. It has been established that the latter approach clearly outperforms the approach where a linear model is used.     

LMI Approach to Robust Model Predictive Control

This paper introduces a new approach to robust model predictive control (MPC) based on conservative approximations to semi-infinite optimization using linear matrix inequalities (LMIs). The method applies to problems with convex quadratic costs, linear and convex quadratic constraints, and linear predictive models with bounded uncertainty. If the MPC optimization problem is feasible at the initial control step (the first application of the MPC optimization), it is shown that the MPC optimization problems will be feasible at all future time steps and that the controlled system will be closed-loop stable. The method is illustrated with a solenoid control example. The authors thank the anonymous reviewers for suggestions that improved the presentation of this work. The work was supported in part by the EPRI/DoD Complex Interactive Networks/Systems Initiative under Contract EPRI-W08333-05 and by the US Army Research Office Contract DAAD19-01-1-0485.     

Optimal Open-Loop Feedback Control of Robots under uncertainty

The aim of this presentation is to construct an optimal open-loop feedback controller for robots, which takes into account stochastic uncertainties. This way, optimal regulators being insensitive with respect to random parameter variations can be obtained. Usually, a precomputed feedback control is based on exactly known or estimated model parameters. However, in practice, often exact informations about model parameters, e.g. the payload mass, are not given. Supposing now that the probability distribution of the random parameter variation is known, in the following, stochastic optimisation methods will be applied in order to obtain robust open-loop feedback control. Taking into account stochastic parameter variations, the method works with expected cost functions evaluating the primary control expenses and the tracking error. The expectation of the total costs has then to be minimized. Corresponding to Model Predictive Control (MPC), here a sliding horizon is considered. This means that, instead of minimizing an integral from a starting time point t₀ to the final time t_f, the future time range [t; t+T], with a small enough positive time unit T, will be taken into account. The resulting optimal regulator problem under stochastic uncertainty will be solved by using the Hamiltonian of the problem. After the computation of a H-minimal control, the related stochastic two-point boundary value problem is then solved in order to find a robust optimal open-loop feedback control. The performance of the method will be demonstrated by a numerical example, which will be the control of robot under random variations of the payload mass. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Terminal constrained robust hybrid iterative learning model predictive control for complex time-delayed batch processes

This work mainly addresses terminal constrained robust hybrid iterative learning model predictive control against time delay and uncertainties in a class of complex batch processes with input and output constraints. In this work, an equivalently novel extended two-dimensional switched system is first constructed to represent the process model by introducing state difference, output error and new relaxation variable information. Then, a hybrid predictive updating controller is proposed and an optimal performance index function including terminal constraints is designed. Under the condition that the switching signal meets certain conditions, the solvable problem of model predictive control is realized by Lyapunov stability theory. Meanwhile, the design scheme of controller parameters is also given. In addition, the robust constraint set is adopted to overcome the disadvantage that the traditional asymptotic stability cannot converge to the origin when it involves disturbances, such that the system state converges to the constraint set and meets its expected value. Finally, the effectiveness of the proposed algorithm is verified by controlling the speed and pressure parameters of the injection molding process.     

Reduced-order model predictive control of a fish schooling model

We study the problem of model predictive control (MPC) for the fish schooling model proposed by Gautrais et al. (2008). The high nonlinearity of the model attributed to its attraction/alignment/repulsion law suggests the need to use MPC for controlling the fish schooling’s motion. However, for large schools, the hybrid nature of the law can make it numerically demanding to perform finite-horizon optimizations in MPC. Therefore, this paper proposes reducing the fish schooling model for numerically efficient MPC; the reduction is based on using the weighted average of the directions of individual fish in the school. We analytically show how using the normalized eigenvector centrality of the alignment-interaction network can yield a better reduction by comparing reduction errors. We confirm this finding on the weight and numerical efficiency of the MPC with the reduced-order model by numerical simulations. The proposed reduction allows us to control a school with up to 500 individuals. Further, we confirm that reduction with the normalized eigenvector centrality allows us to improve the control accuracy by factor of five when compared to that using constant weights.     

Theoretical and algorithmic advances in multi-parametric programming and control

This paper presents an overview of recent theoretical and algorithmic advances, and applications in the areas of multi-parametric programming and explicit/multi-parametric model predictive control (mp-MPC). In multi-parametric programming, advances include areas such as nonlinear multi-parametric programming (mp-NLP), bi-level programming, dynamic programming and global optimization for multi-parametric mixed-integer linear programming problems (mp-MILPs). In multi-parametric/explicit MPC (mp-MPC), advances include areas such as robust multi-parametric control, multi-parametric nonlinear MPC (mp-NMPC) and model reduction in mp-MPC. A comprehensive framework for multi-parametric programming and control is also presented. Recent applications include a hydrogen storage device, a fuel cell power generation system, an unmanned autonomous vehicle (UAV) and a hybrid pressure swing adsorption (PSA) system.     

Multivariable RBF-ARX model-based robust MPC approach and application to thermal power plant

For a class of smooth nonlinear multivariable systems whose working-points vary with time and the future working-points knowledge are unknown, a combination of a local linearization and a polytopic uncertain linear parameter-varying (LPV) state-space model is built to approximate the present and the future system’s nonlinear behavior, respectively. The combination models are constructed on the basis of a matrix polynomial multi-input multi-output (MIMO) RBF-ARX model identified offline for representing the underlying nonlinear system. A min–max robust MPC strategy is designed to achieve the systems’ output-tracking control based on the approximate models proposed. The closed loop stability of the MPC algorithm is guaranteed by the use of time-varying parameter-dependent Lyapunov function and the feasibility of the linear matrix inequalities (LMIs). The effectiveness of the modeling and control methods proposed in this paper is illustrated by a case study of a thermal power plant simulator.     

Analysis of filling distance in cylindrical microfeatures for microinjection molding

In the injection molding of plastic components with cylindrical microfeatures, the ability for the polymer melt to flow into the microchannels is a crucial factor for successful molding. Penetration distance of the polymer melt into the microstructure depends on several factors as the melt flow rate and the cooling rate in the microfeatures, which depends on the materials and geometric dimensions. In this study, a simplified analytical model was constructed to estimate the filling distance into the cylindrical microchannels. The effects of the mold temperature, injection rate, heat transfer coefficient, and microchannel dimension on the filling distance were investigated. The filling distance decreases dramatically with respect to the decrease of the channel radius. In molding of plastic components with cylindrical microfeatures as those analyzed in this study, decrease of the part thickness could also increase the filling distance in the microfeatures.     

A Fokker–Planck control framework for multidimensional stochastic processes

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model.     

Min-max optimal control of systems approximated by finite-dimensional models: Nonquadratic cost functional

An open-loop control problem with nonquadratic performance criterion for a dynamical system, described by an abstract linear equation of evolution and approximated by a finite-dimensional model, is solved. A min-max approach is taken: the value of the cost functional in theworst-case output error between the system and the model, under the assumption that the norm of this output error is estimated by the norm of the input, is minimized.The form of the cost functional reproduces itself under maximization, so that the min-max control problem, when only thedistance between the model and the system is given, has the same features and proprieties of the control problem when the system is thorougly known.Existence and uniqueness theorems for the optimal control are proven, using the spectral proprieties of the model transfer function; and, in the case of a time-invariant model, the min-max control computation is reduced to the solution of a constant-coefficients Sturm-Liouville problem followed by the search for the zeros of a very simple numerical function.This work was made within the Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche.     

基于拟线性化方法的非线性系统闭环反馈控制保辛算法

提出了一种求解非线性系统闭环反馈控制问题的保辛算法．首先,通过拟线性化方法将非线性系统最优控制问题转化为线性非齐次Hamilton系统两端边值问题的迭代格式求解．然后,通过作用量变分原理与生成函数构造了保辛的数值算法,且该算法保持了原Hamilton系统的辛几何性质．最后,通过时间步的递进完成状态与控制变量的更新,进而达到闭环控制的目的．数值算例表明:保辛算法具有较高的计算精度和较快的收敛速度．此外,将闭环反馈控制与开环控制分别应用于驱动小车上的倒立摆控制系统中,结果表明:在存在初始偏差的情况下,开环控制会导致稳定控制任务的失败,而闭环反馈控制能够在一段时间后消除初始偏差的影响,并使系统达到稳定状态．     

Dual-Sampling-Rate Moving-Horizon Control of a Class of Linear Systems with Input Saturation and Plant Uncertainty

Moving-horizon control is a type of sampled-data feedback control in which the control over each sampling interval is determined by the solution of an open-loop optimal control problem. We develop a dual-sampling-rate moving-horizon control scheme for a class of linear, continuous-time plants with strict input saturation constraints in the presence of plant uncertainty and input disturbances. Our control scheme has two components: a slow-sampling moving-horizon controller for a nominal plant and a fast-sampling state-feedback controller whose function is to force the actual plant to emulate the nominal plant. The design of the moving-horizon controller takes into account the nonnegligible computation time required to compute the optimal control trajectory.We prove the local stability of the resulting feedback system and illustrate its performance with simulations. In these simulations, our dual-sampling-rate controller exhibits performance that is considerably superior to its single-sampling-rate moving-horizon controller counterpart.     

Control-oriented modeling of servo-pump driven injection molding machines in the filling and packing phase

Servo-valves or variable displacement pumps are typically used to control conventional hydraulic injection molding machines (IMMs). Recent developments in electrical drive technology allow to utilize servo-motor driven pumps instead, which is beneficial due to their higher energy efficiency. Their dynamic behavior, however, is significantly different compared to the conventional setup. Thus, currently used mathematical models and control concepts cannot be directly applied. This paper presents a computationally efficient and scalable mathematical model of the injection process for these servo-pump driven IMMs. A first-principles model of the injection machine is combined with a phenomenological model describing the injection process, i.e. the compression of the melt and the polymer flow into the mold. The proposed model is tailored to real-time applications and serves as an ideal basis for the design of model-based control strategies. The feasibility of the proposed model is demonstrated by a number of different experiments. They confirm a high model accuracy over the whole operating range for different mold geometries.