## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

We consider the problem of controlling nonlinear systems which are modeled as a set of piecewise linear (PL) or affine systems using a model predictive control (MPC). The paper reviews recent results on the analysis and control of PL systems, which can model a wide range of practically relevant nonlinear systems. Using techniques from the theory of linear matrix inequalities (LMIs), we develop a multiple model MPC technique involving a sequence of local state feedback matrices, which minimize an upper bound on the ‘worst-case’ objective function. The resulting problem, which utilizes a single quadratic Lyapunov function and multiple local state-feedback matrices, can be cast as a convex optimization problem involving LMIs. Several extensions of this technique involving approximating the local regions by ellipsoids or polytopes, and their respective advantages and disadvantages, are discussed.

To read the full-text of this research,

you can request a copy directly from the authors.

... However, although a nonlinear model predictive controller is applied to the nonlinear systems directly, the incoming optimization is usually non-convex and the computing effort for the corresponding nonlinear programming becomes very large, making it hard to meet the real-time control requirement [1]. As a simple and effective method, model predictive control with multiple model scheme has been applied for such nonlinear processes, because the processes can be described by a set of linear submodels near different operating points (equilibriums), although the overall process behaviour is still nonlinear [2] [3] [4] [5]. Furthermore, many means have been applied to obtain the multiple model descriptions for nonlinear processes with large operating regions, such as fuzzy satisfactory clustering (FSC) [5]. ...

... Then, it is possible to obtain the prediction of dynamics of nonlinear processes . It has been noticed that each piecewise linear submodel is an effective representative in a given region for some nonlinear processes with several operating points [3] [4]. Therefore it is not unreasonable to set up logic rules to represent such switching situations. ...

... For a nonlinear process with several operating points/equilibriums (x i , y i , u i ), linearization of the nonlinear process around these operating points will give rise to a bank of piecewise linear models. More analytical discussion and verification of the piecewise linearization techniques around the equilibrium can be found in [3] [4]. Now, suppose there exist S outputs in the output space Y , which satisfies y min ≤ y 0 ≤ · · · ≤ y S−1 ≤ y max . ...

In this study a procedure to design multiple model switching pre- dictive controllers (MMSPC) is proposed for the nonlinear dynamic processes with large operation regions. To facilitate the MMSPC design, a general mixed logic dynamic system (MLDS) model is proposed for approximating the nonlinear processes. A major con- tribution of this study is to integrate a number of techniques to form a novel procedure, and therefore to make multistep state and output predictions effectively realizable within the frame of multiple model switching control. A case for support is presented to demonstrate the efficiency of the design procedure.

... Approximation and control of nonlinear functions by Piecewise linear (PWL) functions has undergone a wealth of theoretical development as evidenced by growing list of research articles dedicated to the subject e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], etc. The reasons suggesting it worthwhile to investigate PWL systems are that PWL systems are simple to implement and offer ease of theoretical analysis and calculation. ...

... Nevertheless there exist research articles e.g. [15,16,17] which deal with specific systems for their respective control issues. The reason, why PWL models are not widely applied, lies in that the conventional representations of PWL functions are concerned with too many parameters which occur in the functions expression and the domain partitions. ...

Due to the inherent nonlinear nature of real world systems, one of the most popular ways to deal with nonlinear systems is to find a feedback linearizing input and then deal with the system by using the rich literature on linear control methods. As an alternative, the nonlinear terms in the system model can be piecewise linearized and then the system can be controlled by using various linear control approaches. The purpose of this paper is to show that under certain conditions, piecewise linearization (PWL) outperforms feedback linearization. In this note, a systematic procedure is outlined for approximating convex separable nonlinear systems with a continuous piecewise linear function and, to emphasize upon the basic idea of this paper, a widely employed nonlinear vehicle following model is used as an example. For this particular model, it is shown that the approximation scheme is optimal with respect to the number of local linear models and the inherent problem of increased dimensionality of PWL systems is not significant. As an extension of our previous work, it is also shown that parametric uncertainties can be deeply investigated, which is not possible in the case of feedback linearization. A simulation study is carried out to show that the proposed system not only guarantees asymptotic tracking of the desired trajectories, but also ensures safety and ride comfort under the constraints of physical limitations inherent in the system. Various issues of vehicle following, e.g. convergence of error in the inter-vehicle spacing, velocity following, control saturation and parametric uncertainties are addressed in this paper. The performance analysis reveals that this new strategy yields promising results.

... In this method , a multi-model system is used to approximate the nonlinear process in a wide operating range , and each local model is valid only in a certain region. Prasad [ 2 ] Some other literature like Ozkan [ 6 ] and Porfirio [ 7 ] designed MMPC controller for worst-case process model , but they also neglected the local validity of local models , which was unreasonable for wide range of operating condi2 ...

... Local model (6) can be further rewritten as ...

Because model switching system is a typical form of Takagj-Sugeno(T-S) model which is an universal approximator of continuous
nonlinear systems, we describe the model switching system as mixed logical dynamical (MLD) system and use it in model predictive
control (MPC) in this paper. Considering that each local model is only valid in each local region, we add local constraints
to local models. The stability of proposed multi-model predictive control (MMPC) algorithm is analyzed, and the performance
of MMPC is also demonstrated on an inulti-multi-output(MIMO) simulated pH neutralization process.

... fast) modes may be relevant when the model is used for predicting long-term effects (e.g. [31] ). In contrast, balanced truncation that aims at approximating high frequency modes may be relevant for problems such as circuit design (e.g. ...

In this article we study balanced model reduction of linear control systems using the singular perturbation approximation. Balanced model reduction techniques have been successfully applied to systems with homogeneous initial conditions, with one of their most important features being a priori L 2 and H ∞ bounds for the approximation error. The main focus of this work is to derive an L 2 error bound for the singular perturbation approximation for system with inhomogeneous initial conditions, extending related work on balanced truncation. This L 2 error bound measures the difference between the input-output maps of the original and of the reduced initial value systems. The advantages and flexibility of this approach are demonstrated with a variety of numerical examples.

... Piecewise linearization [22,23] around operating points has been widely studied to simplify controller-designed procedures when plants are subject to mild nonlinear dynamics. It should be mentioned that a group of piecewise linear models can be admitted as a linear model, with varying order and parameters in different operating intervals. ...

Nonlinear rational systems/models, also known as total nonlinear dynamic systems/models, in an expression of a ratio of two polynomials, have roots in describing general engineering plants and chemical reaction processes. The major challenge issue in the control of such a system is the control input embedded in its denominator polynomials. With extensive searching, it could not find any systematic approach in designing this class of control systems directly from its model structure. This study expands the U-model-based approach to establish a platform for the first layer of feedback control and the second layer of adaptive control of the nonlinear rational systems, which, in principle, separates control system design (without involving a plant model) and controller output determination (with solving inversion of the plant U-model). This procedure makes it possible to achieve closed-loop control of nonlinear systems with linear performance (transient response and steady-state accuracy). For the conditions using the approach, this study presents the associated stability and convergence analyses. Simulation studies are performed to show off the characteristics of the developed procedure in numerical tests and to give the general guidelines for applications.

... • Multi-model Uncertainty: Equally acceptable class of models, a countable set of models likely to represent the true system, see [32] as an example, ...

In this paper, we discuss the model predictive control algorithms that are tailored for uncertain systems. Robustness notions with respect to both deterministic (or set based) and stochastic uncertainties are discussed and contributions are reviewed in the model predictive control literature. We present, classify and compare different notions of the robustness properties of state of the art algorithms, while a substantial emphasis is given to the closed-loop performance and computational complexity properties. Furthermore, connections between (i) the theory of risk and (ii) robust optimization research areas and robust model predictive control are discussed. Lastly, we provide a comparison of current robust model predictive control algorithms via simulation examples illustrating closed loop performance and computational complexity features.

... In the first step, sub-controllers are designed for each submodel on the basis of various linear control techniques. By far, the most popular control methodology for PWA systems is the multiple model predictive control [24][25][26][27][28][29][30][31]. Different from the conventional model predictive control for a single model, where the control signal is computed by minimizing a cost function that penalizes the future output tracking error and the variation in control signal, the switching among different local submodels and their corresponding controllers also need to be taken into consideration. ...

The multiple model approach provides a powerful tool for identification and control of nonlinear systems. Among different multiple model structures, the piecewise affine (PWA) models have drawn most of the attention in the past two decades. However, there are two major issues for the PWA model-based identification and control: the curse of dimensionality and the computational complexity. To resolve these two issues, we propose a novel multiple model approach in this paper. Different from PWA models in which all dimensions of the regressor space are engaged in the partitioning, the key idea of the proposed multiple model architecture is to partition only the range of the control input u(k) at time k (the instant of interest in the control problem) into several intervals and identify a local model that is linear in u(k) within each interval. On the basis of Taylor's theorem, a theoretical upper bound for the approximation error of the model structure can also be obtained. With the proposed multiple model architecture, a switching control algorithm is derived to control nonlinear systems on the basis of the weighted one-step-ahead predictive control method and constrained optimization techniques. In addition, the upper bound for the tracking error using this switching control strategy is also analyzed rigorously under certain assumptions. Finally, both simulation studies and experimental results demonstrate the effectiveness of the proposed multiple model architecture and switching control algorithm.

... Second, optimal control laws are synthesized which allow to track a desired trajectory, considering constraints on the control vector using the theory of MPC. This approach consists of minimizing an upper bound of the objective function over an infinite horizon [19,23]. The quadratic cost minimization under inequality constraints is formulated as a linear convex programming subject to linear matrix inequalities (LMIs). ...

In this paper, we propose a method to deal with the control of an electroplating line without stopping the production. The considered system is running in repetitive functioning mode, and it is modelled by a P-time event graph. The main objective is to switch the number of parallel resources to maintain a resource while the production is assumed to be always running. So, two functioning modes appears: normal and maintenance modes. Each mode is described by a state space model written in the standard algebra and a global model corresponding to a switching system which is a class of hybrid systems is obtained. Using model predictive control technique, stabilizing state feedback gains are computed over an infinite horizon respecting the time constraints while maintenance is performed.

... Any large changes in the operating points and capacity will require re-linearization around the new nominal conditions to ensure the robustness of the controller. In such cases, the piecewise linearized ( € Ozkan et al., 2000;Shafiee et al., 2008) and multiple models ( € Ozkan et al., 2003;Porf ırio et al., 2003;Chen et al., 2009) predictive control techniques can be applied. By approximating a nonlinear system as a family of affine systems, the analysis of the nonlinear system can be transformed into an analysis of several linear systems. ...

Distillation processes presents many challenging control problems, such as nonlinear dynamic behavior, uncertain and time varying parameters, and unmeasured disturbances. This chapter addresses advanced control strategies for DWC, which can make the nonlinear process control much more practical. A dynamic model of a DWC is presented and used on an industrial case study (BTX separation) to illustrate the performance of various advanced control strategies, such as linear quadratic Gaussian (LQG), generic model control (GMC), H∞ loop shaping design procedure (LSDP), multivariable controller μ-synthesis (DK iteration procedure), and model predictive control (MPC). While PI control structures are also able to control the DWC, significantly shorter settling times and lower overshooting can be achieved using MIMO controllers. A very practical scheme based on combination of MPC and PID controllers is also proposed to overcome the disadvantages of individual structures.

... Generalized Predictive Control (GPC), a computer control method developed with the Adaptive Control, has been successfully used in industrial process control. In recent years, for nonlinear predictive control system, many foreign scholars have proposed model predictive control method based on piecewise linear [3][4][5][6]. But domestic scholars use linear method or hierarchical optimization method, or directly use Hammersteina model, Wiener model, Volterra model, fuzzy reasoning and neural network as a predictive model for nonlinear model predictive control. ...

According to the nonlinear and parameters time-varying characteristics of stripper temperature control system, the PVC stripping process Generalized Predictive Control based on implicit algorithm is proposed. Firstly, supporting vector machine is adopted to dynamically modelize for the stripper temperature; Secondly, combining with real-time model linearized of nonlinear model, a predictive model is linearized for real-time online correction. Then, the implicit algorithm is used for optimal control law. Finally, the simulation results show that the algorithm has excellent validity and robustness of temperature control of the stripper.

... The outputs of the linearized model may be written as follows, = k k y Cx (10) Note that this model describes all locations of the nonlinear hybrid system in the vicinity of a single operating point. Similar linearized discrete-time models may obtained at different operating points (Ozkan et al., 2000) characterized by the continuous states and continuous inputs (x c , u c ). These models are then combined using a weighting scheme such as Bayesian weighting (Schott and Bequette, 1997;Nandola and Bhartiya, 2008) to reconstitute the original nonlinear model. ...

This paper presents an efficient optimization algorithm for mixed integer nonlinear programming (MINLP) problem resulting from multiple partially linearized (MPL) model based control of nonlinear hybrid dynamical system (NHDS). The algorithm uses structural information of the canonical MPL framework and derives comparatively easier quadratic programming (QP) primal problem as well as an MILP master problem for generalized outer approximation (GOA) algorithm, a decomposition based solution strategy for MINLP. Computational efficiency of the algorithm over the branch and bound strategy is demonstrated using a simulated benchmark three-spherical tank system.

... Multiple model approximation has been extensively used for the modeling and model based control of continuous state, nonlinear systems [11,12,13,14,15,16,17]. The basic idea is use of a number of linear models to adequately represent the nonlinear space. ...

Many applications in chemical engineering often exhibit a switching character due to the presence of discrete modes in the course of their operation. First principles models of such systems constructed using process simulators are far too complex for use in online applications, especially in model-based control. For such systems, numerous control-relevant modeling approaches have been reported in the literature such as mixed logic dynamical (MLD) models [1] and piece wise affine (PWA) [2] models among others. These models describe the evolution of states in each discrete mode using linear equations. Fewer control-relevant models have been reported that address the nonlinear behavior of switched systems. To model nonlinear hybrid systems, Nandola and Bhartiya [3] proposed a multiple linear model approach wherein multiple linear models are used to describe the dynamic behavior in each mode of the hybrid system. However, no guidelines were provided to select the number of models necessary in each mode and their region of validity. In this work, we address these lacunae by presenting a systematic multiple model approach to describe nonlinear switched systems. The method involves a trajectory based linearization and employs a model bank with a set of local linear models for each discrete operational mode. The model bank is generated by linearizing the first principles model across a carefully designed trajectory based on accuracy of multi-step ahead predictions. The numerous models thus obtained are clustered using the gap metric as the distance measure and representative models are selected. The selected linear models are aggregated using Bayesian or Fuzzy approaches to obtain the global model for the nonlinear switched system. A simulation case study of spherical two-tank system and an experimental case study of a benchmark problem consisting of three tanks are used to validate the proposed modeling strategy.

... The outputs of the linearized model may be written as follows, = k k y Cx (10) Note that this model describes all locations of the nonlinear hybrid system in the vicinity of a single operating point. Similar linearized discrete-time models may obtained at different operating points (Ozkan et al., 2000) characterized by the continuous states and continuous inputs (x c , u c ). These models are then combined using a weighting scheme such as Bayesian weighting (Schott and Bequette, 1997;Nandola and Bhartiya, 2008) to reconstitute the original nonlinear model. ...

This paper presents an efficient optimization algorithm suitable for online solution of mixed integer nonlinear programs resulting from the model predictive control (MPC) of nonlinear hybrid systems. The system model is based on a recently proposed multiple partially linear (MPL) modeling scheme. The algorithm, based on generalized outer approximation (GOA), uses structural information of the canonical MPL framework as well as analytical expressions for the objective function and constraints of a relatively simple primal problem as well as the master problem. Specifically, the primal problem of GOA reduces to a quadratic program when MPL models are used in MPC. Computational efficiency of the algorithm over the branch and bound strategy is demonstrated using a simulated benchmark three-spherical tank system and a hydraulic process plant.

... Any large changes in the operating points and capacity will require the re-linearization around the new nominal conditions in order to ensure the robustness of the controller. In such cases, the piecewise linearized (Ö zkan et al., 2000;Shafiee et al., 2008) and multiple models (Ö zkan et al., 2003;Porfírio et al., 2003;Chen et al., 2009) predictive control techniques can be applied. By approximating a nonlinear system as a family of affine systems, the analysis of the nonlinear system can be transformed into an analysis of several linear systems. ...

Dividing-wall column (DWC) is one of the best examples of process intensification, as it can bring significant reduction in the capital invested as well as savings in the operating costs. Conventional ternary separations progressed from the (in-)direct sequences to thermally coupled columns such as Petlyuk configuration, and later to the DWC compact design that integrates the two distillation columns into a single shell. Nevertheless, this integration leads also to changes in the control and operating mode due to the higher number of degrees of freedom. In this work we explore the dynamic optimization and advanced control strategies based on model predictive control (MPC), coupled or not with PID. These structures were enhanced by adding an extra loop controlling the heavy component in the top of the feed side of the column, using the liquid split as manipulated variable, thus implicitly achieving energy minimization. To allow a fair comparison with previously published references, this work considers as a case-study the industrially relevant separation of the mixture benzene–toluene–xylene (BTX) in a DWC. The results show that MPC leads to a significant increase in performance, as compared to previously reported conventional PID controllers within a multi-loop framework. Moreover, the optimization employed by the MPC efficiently accommodates the goal of minimum energy requirements – possible due to the addition of an extra loop – in a transient state. The practical benefits of coupling MPC with PID controllers are also clearly demonstrated.

... The multiple linear models concept has been used in the recent years for modeling of nonlinear systems. Also, multiple linear model based approaches for controller design has attracted the process control community and a plethora of multiple model control schemes has been proposed in the control literature [7,13,5,6]. The method by Arkun et al. [2] uses the nonlinear first principles model to obtain the linear state space models at different operating points. ...

Model predictive control (MPC) schemes are now widely used in process industries for the control of key unit operations. Linear model predictive control (LMPC) schemes which make use of linear dynamic model for prediction, limit their applicability to a narrow range of operation (or) to systems which exhibit mildly nonlinear dynamics.In this paper, a nonlinear observer based model predictive controller (NMPC) for nonlinear system has been proposed. An approach to design NMPC based on fuzzy Kalman filter (FKF) and augmented state fuzzy Kalman filter (ASFKF) has been presented. The efficacy of the proposed NMPC schemes have been demonstrated by conducting simulation studies on the continuous stirred tank reactor (CSTR). The analysis of the extensive dynamic simulation studies revealed that, the NMPC schemes formulated produces satisfactory performance for both servo and regulatory problems. Simulation results also include an inferential control case, where the reactor concentration is not measured but estimated from temperature measurement and used in the NMPC based on FKF and ASFKF formulations.

... We illustrate the application of this algorithm on a low-order continuous stirred tank reactor (CSTR) model with an exothermic ÿrst-order irreversible reaction ( Ozkan, Kothare, & Georgakis, 2000). The example serves to familiarize the reader with the multi-model MPC algorithm discussed in Section 3 before demonstrating its applicability on a high-dimensional complex polymerization reactor. ...

We study the control of a solution copolymerization reactor using a model predictive control algorithm based on multiple piecewise linear models. The control algorithm is a receding horizon scheme with a quasi-infinite horizon objective function which has finite and infinite horizon cost components and uses multiple linear models in its predictions. The finite horizon cost consists of free input variables that direct the system towards a terminal region which contains the desired operating point. The infinite horizon cost has an upper bound and takes the system to the final operating point. Simulation results on an industrial scale methyl methacrylate vinyl acetate solution copolymerization reactor model demonstrate the ability of the algorithm to rapidly transition the process between different operating points.

In this work, a novel iterative least squares method is proposed to approximate nonlinear functions with continuous piecewise linear (CPWL) functions, where the continuity is guaranteed by constrained least squares. A typical least-squares-based method for CPWL fitting consists of two major steps: to perform least squares for a fixed set of breakpoints (i.e. for a specific partition of function domain), and to update breakpoints to reduce the overall fitting error. The proposed method aims to improve the existing CPWL method, the one with canonical representation, by modifying both major steps. Instead of performing ordinary least squares with canonical representation, partitioned least squares and constrained least squares are successively performed to reduce the computational complexity. For the update of breakpoints, an iterative procedure is proposed, where gradient descent with momentum method is employed to improve the convergence characteristics. The advantages of proposed method are illustrated using illustrative examples.

In this work, the authors have designed and implemented a Takagi-Sugeno fuzzy model based control scheme (TS-MBC) for the continuous stirred tank reactor (CSTR) process. The proposed control scheme can be viewed as an alternative to the nonlinear model based control scheme proposed by Panda & Prakash¹. In this work the nonlinear process is represented using a bank of linear discrete state space models combined using fuzzy membership function. The output of the proposed control scheme is computed in two steps. The bank of linear models weighted using fuzzy membership function is used to compute the value of the controller output in the first step. In the second step, the controller output computed in the first step is updated based on the measurement. The performance of the proposed control scheme has been compared with that of a nonlinear model based control scheme (NMBC) and nonlinear PID (NPID).

An optimization-based multi-layer operability framework is introduced for process design of nonlinear energy systems that are challenged by complexity and highly constrained environments. In the first layer of this framework, an MILP-based iterative algorithm considers the minimization of footprint and achievement of process intensification targets. Then, in the second layer, an operability analysis is performed to incorporate into the approach key features for optimality and feasibility accounting for the system operation with changeable input conditions. The outcome of the framework consists of a set of modular designs, considering both the aspects of size and process operability. For this study, the nonlinear system is represented by multiple linearized models, resulting in low computational expense and efficient quantification of operability regions. The developed framework is applied to a membrane reactor for direct methane aromatization conversion to hydrogen and benzene. Subsystems of dimensionalities 2 x 2 and 3 x 3 (design inputs x outputs) are considered in the first layer to obtain a modular design region. The possible modular designs inside this region are then ranked according to an operability index obtained from an additional 3 x 3 (operational inputs x outputs) mapping. This step analyzes the effect of operational inputs, producing a mapping of total dimensionality 6 x 3 (inputs x outputs). The application of the developed framework generates two candidate designs for system modularity, the most operable design and the optimal design with respect to process intensification in terms of footprint minimization. The developed framework thus provides guidelines for obtaining modular designs that simultaneously consider process intensification and operability aspects.

This article presents a new Extended Prediction Self-Adaptive Control (EPSAC) algorithm based on the Just-in-Time Learning (JITL) method. In the proposed JITL-based EPSAC design, linearization of the process model is achieved by a set of local state-space models, each of which can be independently and simultaneously identified by the JITL method along the base trajectory. For the end-product quality control for a simulated semi-batch pH-shift reactive crystallization process where shrinking prediction and control horizons are essential, the proposed EPSAC algorithm not only simplifies the control weight tuning but also provides better and more robust closed-loop control performance than its previous counterpart.

This paper presents an efficient optimization algorithm for mixed integer nonlinear programming (MINLP) problem resulting from multiple partially linearized (MPL) model based control of nonlinear hybrid dynamical system (NHDS). The algorithm uses structural information of the canonical MPL framework and derives comparatively easier quadratic programming (QP) primal problem as well as an MILP master problem for generalized outer approximation (GOA) algorithm, a decomposition based solution strategy for MINLP. Computational efficiency of the algorithm over the branch and bound strategy is demonstrated using a simulated benchmark three-spherical tank system. Copyright © 2007 International Federation of Automatic Control All Rights Reserved.

Process Analytical Technology (PAT) is an area of intense research and interest currently. The interest in and applications for PAT span many industries: petrochemicals, bulk chemicals, food, pharmaceuticals and biopharmaceuticals amongst others. Adoption in the biopharmaceutical industry is in its infancy but is being driven by both regulatory demand and the business case. Ultimately, both motivations stem from the fact that effective application of PAT to bioprocesses increases process understanding and process control, mitigating the risk of substandard drug products to both the manufacturer and the patient. In order to realise the value that PAT can offer, all aspects of the PAT system must be considered and appropriately chosen. These include the PAT instrument, data analysis techniques, control strategies and algorithms and process optimization. It is only by the clear definition of the objective for the PAT system and the selection of suitable elements that the value may be realised. This chapter will discuss the instruments, techniques and strategies of relevance to animal cell culture currently.

This chapter presents a new approach for deriving the explicit model-based control law for hybrid and continuous time linear dynamic systems via parametric programming. Our method first formulates an open-loop receding horizon optimal control problem and then recasts it as a multiparametric mixed integer quadratic program (mp-MIQP) in the case of hybrid systems and as a multiparametric dynamic optimization (mp-DO) problem in the case of continuous time dynamics. The solution of the parametric programs (see Chapter 4 of Volume 1 and Chapter 4 of Volume 2 of this book series) derives off-line an explicit parametric controller for the pertinent plant before any actual process implementation occurs. The key features of our novel approach are demonstrated via mathematical, and chemical and biomedical process examples.

Two integrated multi-linear model predictive control (MLMPC) algorithms are proposed for nonlinear chemical processes. The gap metric and the gap metric stability margin are employed to select local linear models and design local MPC controllers. Thus the local stability and desired closed-loop performance can be incorporated into the model bank selection process. After that, a gap-metric-based weighting method is used to combine the local MPC controllers into a global MLMPC controller for the nonlinear process. Therefore, the local model selection, the local controller design, and the local controller combination are all completed according to the gap-metric-based criteria. Close connections are established among the three key elements of the multi-linear model predictive control approach. Thereby the design of a MLMPC controller is more systematical, which is found to improve the accuracy and robust performance of a MLMPC controller. Since the gap metric does not consider constraints and the use of linear models in the multi-model approach may not lead to a globally stable control systems, an additional simulation-based criterion is employed to evaluate the overall closed-loop performance. A SISO and a MIMO CSTR processes are studied to demonstrate the effectiveness of the proposed algorithms.

In this study a control-oriented model is proposed to represent a wide range of non-linear discrete-time dynamic plants. As a testimony to the efficiency of the model structure for control system design, a pole placement controller is designed for non-linear discrete-time plants. Mathematically the solution of the controller output is converted into resolving a polynomial equation in the current control term u(t), which significantly reduces the difficulties encountered in non-linear control system synthesis and computational complexities. The integrated procedure provides a straightforward methodology to use in linear control system design techniques when designing non-linear control systems. For a demonstration of the effectiveness of the proposed methodology used to deal with practical problems, pole placement controllers are designed for three non-linear plants, including the Hammerstein model, a laboratory-scale liquid level system and a continuous stirred tank reactor. The simulation results are presented with graphical illustrations.

This paper studies the explicit model predictive control for piecewise linear (PWL) systems with the output sampling period several times larger than the input updating period. First, based on dynamic programming, the model predictive control optimization problem is decomposed into multi-stage optimization problems with one-step optimal horizon. Further, optimization problem of each stage is separated into sub-problems according to the sub-models of the piecewise linear system and the form of objective function. After that, by utilizing multi-parametric quadratic programming (MP-QP) technique and comparing the solutions of all the sub-problems, the optimal explicit control laws are obtained. Besides, the maximal positively invariant set of the piecewise linear system is chosen as the terminal constraint set of the optimization problem such that the stability can be ensured. The numerical example shows that the proposed explicit model predictive control can reduce online computation and satisfy the demand of the multi-rate piecewise linear system with fast updating speed of input.

This paper deals with the application of neural networks to design intelligent nonlinear predictive controllers. Predictive controllers are now widely used in many industrial applications. They have been used for linear systems in early applications and then some methods based on predictive control theory were proposed to govern the dynamics of nonlinear systems. In this paper, we will make use of multi-layer perceptron neurofuzzy models with Locally Linear Model Tree (LoLiMoT) learning algorithm as a part of intelligent predictive control system, which has shown excellent performance in identifying of nonlinear systems. The nonlinear dynamics of the system is identified using the neural network based method and then the identified model is used as a part of predictive control algorithm. The proposed method is used to solve the control problems in some benchmark systems. As a first study, the viscosity control in a Continuous Stirred Tank Reactor (CSTR) plant is considered. The mathematical model of the plant is used to generate the input output data set and then the dynamic behavior of the system is identified using a proper multi-layer perceptron neural network, which is used in the predictive control loop. Also, the predictive control based on the locally linear neurofuzzy model is applied to temperature control of an electrically heated micro heat exchanger. The dynamic behavior of the heat exchanger is identified based on some experimental data of the real plant. Comparing the identification results obtained by the neurofuzzy model with those of some linear models such as ARX and BJ, confirms the superior performance for the locally linear neurofuzzy model. Then, the predictive control is applied to the identified model to obtain a satisfactory performance in the output temperature that should track a desired reference signal. As another application, the algorithm is applied to temperature control of a solution polymerization methyl methacrylate in a batch reactor. The results show also somehow satisfactory performance for this highly nonlinear system. All the simulation results reveal the effectiveness of the proposed intelligent control strategy.

The control of ultra-supercritical (USC) power unit is a difficult issue for its characteristic of the nonlinearity, large dead time and coupling of the unit. In this paper, model predictive control (MPC) based on multi-model and double layered optimization is introduced for coordinated control of USC unit. The linear programming (LP) combined with quadratic programming (QP) is used in steady optimization for computation of the ideal value of dynamic optimization. Three inputs (i.e. valve opening, coal flow and feedwater flow) are employed to control three outputs (i.e. load, main steam temperature and main steam pressure). The step response models for the dynamic matrix control (DMC) are constructed using the three inputs and the three outputs. Piecewise models are built at selected operation points. Double-layered multi-model predictive controller is implemented in simulation with satisfactory performance.

This paper describes the modeling of a hyperfast switching Peltier cooling system. Linear output-error models are applied to model the influence of input variables on the output of the system. From the frequency response analysis of the collected data, it is observed that there is an uncertain time delay in one of the channels. By adding a variable time delay term into the model, the error at the output can be significantly reduced in the validation set.

Asymmetrical processes are common nonlinear systems, where the switching between two different operating modes depends on whether the system input or output is increasing or decreasing. The existing intelligent control methods for asymmetrical system can't explain the relationship between continuous and discrete part of the system. With the idea of hybrid systems, a supervisory control method is developed to distinguish asymmetry phenomena of the thermal process of a furnace. In order to ensure the stability of arriving at the setting point, a supervisory controller modelled by the extended Controlled Petri nets is designed to track errors and the direction of the output. Then the system can be switched to an appropriate mode, where the tracking determines the switch. Test result verifies the validity of this method.

For constrained piecewise linear (PWL) systems, the possible existing model uncertainty will bring the difficulties to the design approaches of model predictive control (MPC) based on mixed integer programming (MIP). This paper combines the robust method and hybrid method to design the MPC for PWL systems with structured uncertainty. For the proposed approach, as the system model is known at current time, a free control move is optimized to be the current control input. Meanwhile, the MPC controller uses a sequence of feedback control laws as the future control actions, where each feedback control law in the sequence corresponds to each partitions and the arbitrary switching technique is adopted to tackle all the possible switching. Furthermore, to reduce the online computational burden of MPC, the segmented design procedure is suggested by utilizing the characteristics of the proposed approach. Then, an offline design algorithm is proposed, and the reserved degree of freedom can be online used to optimize the control input with lower computational burden.

It is widely assumed that the fission gases and the volatile fission products play little or no role in influencing the chemical state of irradiated UO2 + x. We show this to be incorrect when even a small quantity of these fission products is retained within the fuel grains. Prior to intragranular release most of the gases and volatiles occupy neutral trivacancy sites in the host lattice. However, these complexes can bind oxygen to be converted into fission product atoms residing in cation vacancies or (cation + one anion) vacancy complexes, and this results in reduction of the oxide. Here we analyse this behaviour and its effect on the oxygen potential and cation volume self-diffusion coefficient.

This paper presents an Hinfin state feedback controller design method for discrete-time piecewise affine (PWA) systems based on a piecewise quadratic Lyapunov function, and the partition information of the PWA systems is taken into account to reduce the design conservatism. Each polytopic operating region is outer approximated by a union of ellipsoids, then the minimum problem of the Hinfin performance index for PWA systems can be cast as a convex optimization problem involving linear matrix inequalities, which can be solved very efficiently. A simulation example is also given to illustrate the advantage of the proposed approach.

In this paper, a model predictive control is proposed for the hybrid systems with constrained control input based on piecewise affine (PWA) model. The constrained control input of the considered systems is solved in terms of linear matrix inequalities (LMIs). An additional terminal ellipsoid is introduced to ensure the states converging to the equilibrium faster. The control law is obtained by convex optimization involving LMIs. The simulation results verify the effectiveness of the proposed method. It is shown that, based on LMI constraints, it is easy to get the model predictive control for the hybrid systems and is suitable for practical application

In this paper, a novel stable observer-based integral model predictive controller using piecewise Lyapunov functions is proposed for constrained nonlinear systems. The main idea is to design integrator based state feedback control laws that minimize the worst-case objective function based on fuzzy model prediction, and then to design observer based output feedback controller. It is expected that satisfactory transient control performance without any steady-state offset can be achieved. The asymptotic stability of the resulting closed-loop predictive control system is established by solving a set of linear matrix inequalities. Simulations on a highly nonlinear benchmark system are finally presented to demonstrate the tracking performance of the proposed output feedback fuzzy predictive controllers.

One of the important operations in nuclear power plants is load-following in which imbalance of axial power distribution induces xenon oscillations. These oscillations must be maintained within acceptable limits otherwise the nuclear power plant could become unstable. Therefore, bounded xenon oscillation considered to be a constraint for the load-following operation. In this paper, a robust nonlinear model predictive control for the load-following operation problem is proposed that ensures xenon oscillations are kept bounded within acceptable limits. The proposed controller uses constant axial offset (AO) strategy to maintain xenon oscillations to be bounded. The constant AO is a robust state constraint for load-following problem. The controller imposes restricted state constraints on the predicted trajectory during optimization which guarantees robust satisfaction of state constraints without restoring to a min–max optimization problem. Simulation results show that the proposed controller for the load-following operation is so effective so that the xenon oscillations kept bounded in the given region.

Abstract A novel adaptive controller, suitable for linear and non-linear systems was developed. The controller is a discrete algorithm suitable for computer implementation,and is based on gradient descent adaptation rules. Traditi onal recursive least squares based algorithms suffer from performance,deterioration due tothe continuous reduction of a covariance matrix used for adaptation. When this covariance matrix becomes too small, recursive least squares algorithms respond slow to changes in model parameters. Gradient descent adaptation was used to avoid the performance deteriorationwith time associated with regression based adaptation such as Recursive Least Squares methods. Stability was proven with Lyapunov stability theory, using anerror filter designed to fulfill stability requirements. Similarities between the proposed controller with PIcontrol have been found.

The paper presents the algorithm in robust constrained model predictive control for uncertain piecewise affine with time-invariant state-delay system using saturated linear feedback controller. Polytopic uncertain piecewise affine with time-invariant state-delay system is considered, and polytopic description for saturated function is used in the formulation. The robust performance index is presented based on the modified Lyapunov-Krasovskii function. An infinite horizon worst-case performance function is used as online optimization objective subject to inputs constraint. The optimization objective is cast as a Linear Matrix Inequalities optimization problem. The feasibility of piecewise affine control law design at the initial sampling time implies the sufficient condition of robust stability of the closed-loop system. The simulation example illustrates that the improved algorithm is suitable and effective.

In this paper, the authors have designed local internal model controllers on the basis of multiple-linear discrete transfer function models and the weighted sum of the output from the local internal model controllers (Non-linear Internal Model Controller) has been used to control the nonlinear process. The effectiveness of the proposed control schemes has been demonstrated on a pH process. From the extensive simulation studies, we have shown that the proposed non-linear internal model controller provides satisfactory servo as well as regulatory performances.

This paper proposes a novel synthesis technique for robust predictive control of constrained nonlinear systems based on linear
matrix inequalities (LMIs) formalism. Local discrete-time polytopic models are exploited for prediction of the system behavior.
This design strategy can be applied to nonlinear systems provided a suitable embedding is available. The devised procedure
guarantees constraint satisfaction and asymptotic stability. The proposed result extends previous works by handling less conservative
input constraints and exploiting the different local descriptions of nonlinearity and uncertainty. The multi-model prediction
together with the modified input constraints show significant improvements in terms of closed-loop performance and estimation
of the feasibility domain.
KeywordsControl of constrained nonlinear systems-LMIs-model predictive control

This study develops two efficient static output feedback (SOF) control approaches for a class of constrained non-linear processes represented by uncertain piecewise affine models. The parameter uncertainties in the piecewise affine models are assumed to be time varying and norm bounded. With the aid of the congruence transformation, degenerate ellipsoids-based S-procedure, and some bounding inequalities, the piecewise SOF controller can be easily obtained by solving a convex semi-definite programming problem subject to some linear matrix inequalities. The resulting closed-loop system stability can be guaranteed as well as the H _{∞} control performance, and the stabilising design is less conservative owing to the piecewise quadratic Lyapunov functions. The performance of the proposed control approaches are demonstrated by simulation results on a non-linear benchmark plant.

This paper provides a solution to the problem of robust model predictive control (MPC) for piecewise linear (PWL) systems with constraints and persistent, unknown but bounded disturbances. The robust MPC policy is to minimize a quadratic performance index with terminal cost for the nominal PWL systems and with tightened systems constraints and tightened terminal set constraints for all admissible disturbances. Off-line, pre-determined feedback gains for PWL systems are introduced and are applied to generate a candidate correction policy to show how constraints in the MPC optimization are restricted against persistent bounded disturbances. On-line, at each sampling time, the resultant optimization problem is a mixed integer quadratic programming (MIQP) of similar complexity to that required in MPC of nominal PWL systems. The robust predictive controller obtained in this way guarantees robust feasibility, constraints satisfaction and robust convergence for all admissible disturbances. Simulation results demonstrate the effectiveness of the proposed approach.

This paper presents modeling and control of nonlinear hybrid systems using multiple linearized models. Each linearized model is a local representation of all locations of the hybrid system. These models are then combined using Bayes theorem to describe the nonlinear hybrid system. The multiple models, which consist of continuous as well as discrete variables, are used for synthesis of a model predictive control (MPC) law. The discrete-time equivalent of the model predicts the hybrid system behavior over the prediction horizon. The MPC formulation takes on a similar form as that used for control of a continuous variable system. Although implementation of the control law requires solution of an online mixed integer nonlinear program, the optimization problem has a fixed structure with certain computational advantages. We demonstrate performance and computational efficiency of the modeling and control scheme using simulations on a benchmark three-spherical tank system and a hydraulic process plant.

The solid oxide fuel cell (SOFC) is widely accepted for clean and distributed power generation use, but critical operation problems often occur when the stand-alone fuel cell is directly connected to the electricity grid or the dc electric user. In order to address these problems, in this paper, a data-driven fuzzy modeling method is employed to identify the dynamic model of an integrated SOFC/capacitor system. A novel offset-free input-to-state stable fuzzy predictive controller is developed based on the obtained fuzzy model. Both the rapid power load following and safe SOFC operation requirements are taken into account in the design of the closed-loop control system. Simulations are also given to demonstrate the load following control performance of the proposed fuzzy predictive control strategy for the SOFC/capacitor power system.

This paper discusses universal learning network (ULN) and its application to control a class of nonlinear systems with long time delay. Two control architectures, model predictive control based on ULN model and single neuron PID (SN-PID) controller based on ULN predictor, are designed to control pH neutralization process, respectively. In addition, to verify the performance of ULN, two comparisons are also made. One is the generalization ability between ULN and back-propagation (BP) network, the other comparison is ULN predictor and Smith predictor, in which the same controller is used. Simulation results prove the applicability and effectiveness of the ULN model. The special architecture and its learning algorithm give ULN more representing abilities to model and control complicated nonlinear systems with long time delay.

. This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach. Keywords. Piecewise linear systems, Lyapunov stability, linear matrix inequalities. 1. Introduction Construction of Lyapunov functions is one of the most fundamental problems in systems theory. The most direct application is stability analysis, but analogous problems appear more or less implicitly also in performance analysis, controller synthesis and system identification. Consequently, methods for constructing Lyapunov functions for general nonlinear systems is of great theoretical and practical interest. The objective of this paper is to develop a uniform and compu...

In this paper a method for nonlinear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the state-space called clusters the plant is assumed to be an element in a polytope which vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to be linear systems. The clusters cover the region of interest in the state-space. An affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single nonconvex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed-loop. The feasibility problem is attacked by a branch-and-bound based global approach. If the feasibility check is successful, the Liapunov matrix and the piecewise affine state-feedback are given directly by the feasible solution. Control constraints are shown to be representable by LMIs or BMIs, and an application of the control design method to robustify constrained nonlinear model predictive control is presented. Also, the control design method is applied to a simple example.

We consider analysis and controller synthesis of piecewise-linear
systems. The method is based on constructing quadratic and
piecewise-quadratic Lyapunov functions that prove stability and
performance for the system. It is shown that proving stability and
performance, or designing (state-feedback) controllers, can be cast, as
convex optimization problems involving linear matrix inequalities that
can be solved very efficiently. A couple of simple examples are included
to demonstrate applications of the methods described

Poor control of the steam generator water level in the secondary
circuit of a nuclear power plant can lead to frequent reactor shutdowns.
Such shutdowns are caused by violation of safety limits on the water
level and are common at low operating power where the plant exhibits
strong nonminimum phase characteristics and flow measurements are
unreliable. There is, therefore, a need to systematically investigate
the problem of controlling the water level in the steam generator in
order to prevent such costly reactor shutdowns. The paper presents a
framework for addressing this problem based on an extension of the
standard linear model predictive control algorithm to linear parameter
varying systems

The primary disadvantage of current design techniques for model predictive control (MPC) is their inability to deal explicitly with plant model uncertainty. In this paper, we present a new approach for robust MPC synthesis which allows explicit incorporation of the description of plant uncertainty in the problem formulation. The uncertainty is expressed both in the time domain and the frequency domain. The goal is to design, at each time step, a statefeedback control law which minimizes a "worst-case" infinite horizon objective function, subject to constraints on the control input and plant output. Using standard techniques, the problem of minimizing an upper bound on the "worst-case" objective function, subject to input and output constraints, is reduced to a convex optimization involving linear matrix inequalities (LMIs). It is shown that the feasible receding horizon state-feedback control design robustly stabilizes the set of uncertain plants under consideration. Several extensions...