Nonlinear system modeling and robust predictive control based on RBF-ARX model

An integrated modeling and robust model predictive control (MPC) approach is proposed for a class of nonlinear systems with unknown steady state. First, the nonlinear system is identified off-line by RBF-ARX model possessing linear ARX model structure and state-dependent Gaussian RBF neural network type coefficients. On the basis of the RBF-ARX model, a combination of a local linearization model and a polytopic uncertain linear parameter-varying (LPV) model are built to approximate the present and the future system's nonlinear behavior, respectively. Subsequently, based on the approximate models, a min–max robust MPC algorithm with input constraint is designed for the output-tracking control of the nonlinear system with unknown steady state. The closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and the feasibility of the linear matrix inequalities (LMIs). Simulation study to a NO_x decomposition process illustrates the effectiveness of the modeling and robust MPC approaches proposed in this paper.     

RBF-ARX model-based MPC strategies with application to a water tank system

A hybrid pseudo-linear RBF-ARX model that combines Gaussian radial basis function (RBF) networks and linear ARX model structure is utilized for representing the dynamic behavior of a class of smooth nonlinear and non-stationary systems. This model is locally linear at each working point and globally nonlinear within whole working range. Based on the structural characteristics of the RBF-ARX model, three receding horizon predictive control (RBF-ARX-MPC) strategies are designed: (1) the RBF-ARX-MPC algorithm based on single-point linearization (MPC-SPL); (2) the RBF-ARX-MPC algorithm based on multi-point linearization (MPC-MPL); and (3) the RBF-ARX-MPC algorithm based on globally nonlinear optimization (MPC-GNO). In the MPC-SPL, the future multi-step-ahead predictive output of the system is obtained based on the local linearization of the RBF-ARX model at only current working-point, while in the MPC-MPL the future long-term output prediction is obtained according to the future local characteristics from previous online optimization results of the RBF-ARX model based MPC. In the MPC-GNO, the globally nonlinear characteristics of the RBF-ARX model are fully used for online getting control variables of the MPC. Real-time control experiments for the three type MPCs are carried out on a water tank system, which are also compared with a classical PID control and a traditional linear ARX model-based MPC. The results verify that the modeling method and the model-based predictive control strategies are realizable and effective for the nonlinear and unstable system. Moreover, it is also shown that the MPC-GNO can obtain better control performance but need more computation time compared to the other MPCs, which makes it possible to be applied into some slowly varying processes.     

RBF-ARX模型在液位系统建模中的应用

针对单容液位系统紊流时的非线性特征,采用RBF-ARX模型对单容液位系统进行离线动态特性建模的研究;分别在液位高中低三个工作点建立了其局部线性ARX模型,它们的单位阶跃响应存在巨大差异,证实了整个系统具有较强的非线性;讨论了RBF-ARX模型结构的选取,模型参数辨识,RBF参数优化等问题;模型的预测输出和仿真结果,证实了RBF-ARX模型在非线性系统建模和辨识中的有效性.     

基于RBF-ARX模型的预测控制在液位系统中的应用

针对单容液位系统紊流时的非线性特征,研究了基于RBF-ARX模型预测控制策略控制单容液位系统;讨论RBF-ARX模型结构的选取,模型参数辨识,RBF参数优化,基于RBF-ARX模型的预测控制策略等问题;模型的仿真结果,证实了RBF-ARX模型在非线性系统建模和辨识中的有效性;同基于全局线性ARX模型的预测控制器和PID控制器相比较,基于此模型的预测控制取得了优异的控制效果。     

Distributed model predictive control for polytopic uncertain systems subject to actuator saturation

In this paper, we present a distributed model predictive control (MPC) algorithm for polytopic uncertain systems subject to actuator saturation. The global system is decomposed into several subsystems. A set invariance condition for polytopic uncertain system with input saturation is identified and a min–max distributed MPC strategy is proposed. The distributed MPC controller is designed by solving a linear matrix inequalities (LMIs) optimization problem. An iterative algorithm is developed for making coordination among subsystems. Case studies are carried out to illustrate the effectiveness of the proposed algorithm.     

Robust MPC for systems with output feedback and input saturation

In this work, it is proposed an MPC control algorithm with proved robust stability for systems with model uncertainty and output feedback. It is assumed that the operating strategy is such that system inputs may become saturated at transient or steady state. The developed strategy aims at the case in which the controller performs in the output-tracking scheme following an optimal set point that is provided by an upper optimization layer of the plant control structure. In this case, the optimal operating point usually lies at the boundary of the region where the input is defined. Assuming that the system remains stabilizable in the presence of input saturation, the design of the robust controller is performed off-line and an on-line implementation strategy is proposed. At each sampling step, a sub optimal control law is obtained by combining control configurations that correspond to particular subsets of available manipulated inputs. Stability of the closed-loop system is forced by considering in the off-line step of the controller design, a state contracting restriction for the closed-loop system. To produce an offset free controller and to attend the case of unknown steady state, the method is developed for a state-space model in the incremental form. The method is illustrated with simulation examples extracted from the process industry.     

RBF-ARX模型在倒立摆系统建模中的应用

针对直线一级倒立摆控制系统的非线性特性,采用RBF-ARX模型对倒立摆系统的全局非线性动态特性进行建模.讨论了RBF-ARX模型结构的选取,模型参数辨识,RBF参数优化等问题.并且分别比较了该倒立摆系统的RBF-ARX模型与全局线性ARX模型,以及将RBF-ARX在某一工作点局部线性化后的模型与局部线性ARX模型的预测输出和模型误差,验证了RBF-ARX模型在倒立摆系统建模和辨识中的有效性.     

Nonlinear system modeling and predictive control using the RBF nets-based quasi-linear ARX model

On the basis of the single-input single-output (SISO) RBF-ARX model proposed in previous works [Peng, H., et al. (2003b). Stability analysis of the RBF-ARX model based nonlinear predictive control. In Proceedings of the ECC2003; Peng, H., et al. (2003c). Modeling and control of nonlinear nitrogen oxide decomposition process. In Proceedings of the CDC’03; Peng, H., et al. (2004). RBF-ARX model based nonlinear system modeling and predictive control with application to a NO_x decomposition process. Control Engineering Practice, 12, 191–203; Peng, H., et al. (2007). Nonlinear predictive control using neural nets-based local linearization ARX model—Stability and industrial application. IEEE Transactions on Control Systems Technology, 15, 130–143] the multi-input multi-output (MIMO) RBF-ARX model and its state-space representation are derived to describe the dynamics of a class of multivariable nonlinear systems whose working-point varies with time and which may be linearized around the working-point. The proposed MIMO RBF-ARX model has a basic regression-model structure that is analogous to the linear ARX model structure, and the elements of its regression matrices are composed of Gaussian radial basis function (RBF) neural networks that are dependent on the working-point state of the current system. An off-line estimation approach to parameters and orders of the MIMO RBF-ARX model is presented, and, on the basis of the estimated MIMO RBF-ARX model, a predictive control strategy is designed to control the underlying nonlinear system. A case study on a simulator of a thermal power plant is also given to illustrate the effectiveness of the nonlinear modeling and control method proposed in this paper.     

LMI-based robust model predictive control and its application to an industrial CSTR problem

In this paper, robust model predictive control (MPC) is studied for a class of uncertain linear systems with structured time-varying uncertainties. This general class of uncertain systems is useful for nonlinear plant modeling in many circumstances. The controller design is characterizing as an optimization problem of the “worst-case” objective function over infinite moving horizon, subject to input and output constraints. A sufficient state-feedback synthesis condition is provided in the form of linear matrix inequality (LMI) optimizations, and will be solved on-line. The stability of such a control scheme is determined by the feasibility of the optimization problem. To demonstrate its usefulness, this robust MPC technique is applied to an industrial continuous stirred tank reactor (CSTR) problem with explicit input and output constraints. Its relative merits to conventional MPC approaches are also discussed.     

Robust MPC for Linear Systems with Structured Time‐Varying Uncertainties and Saturating Actuator

In this paper, a synthesis of model predictive control (MPC) algorithm is presented for uncertain systems subject to structured time‐varying uncertainties and actuator saturation. The system matrices are not exactly known, but are affine functions of a time varying parameter vector. To deal with the nonlinear actuator saturation, a saturated linear feedback control law is expressed into a convex hull of a group of auxiliary linear feedback laws. At each time instant, a state feedback law is designed to ensure the robust stability of the closed‐loop system. The robust MPC controller design problem is formulated into solving a minimization problem of a worst‐case performance index with respect to model uncertainties. The design of controller is then cast into solving a feasibility of linear matrix inequality (LMI) optimization problem. Then, the result is further extended to saturation dependent robust MPC approach by introducing additional variables. A saturation dependent quadratic function is used to reduce the conservatism of controller design. To show the effectiveness, the proposed robust MPC algorithms are applied to a continuous‐time stirred tank reactor (CSTR) process.     

一种新的ARX模型在磁悬浮系统建模中的应用

采用一种新的ARX模型（RBF-ARX模型）对磁悬浮系统进行离线建模,讨论了RBF-ARX模型的原理、结构的选取、模型参数辨识和RBF参数优化等问题。文章分别采用不同的序列作为状态变量,分别建立RBF-ARX模型,并分析了各模型的性能及可靠性。模型的预测输出和仿真结果,证实了RBF-ARX模型在非线性系统建模和辨识中的有效性。通过与ARX模型的比较,证明了RBF-ARX模型在非线性系统建模中效果更好。     

Robust output feedback model predictive control using off-line linear matrix inequalities

A fundamental question about model predictive control (MPC) is its robustness to model uncertainty. In this paper, we present a robust constrained output feedback MPC algorithm that can stabilize plants with both polytopic uncertainty and norm-bound uncertainty. The design procedure involves off-line design of a robust constrained state feedback MPC law and a state estimator using linear matrix inequalities (LMIs). Since we employ an off-line approach for the controller design which gives a sequence of explicit control laws, we are able to analyze the robust stabilizability of the combined control laws and estimator, and by adjusting the design parameters, guarantee robust stability of the closed-loop system in the presence of constraints. The algorithm is illustrated with two examples.     

A convex optimization approach to robust iterative learning control for linear systems with time-varying parametric uncertainties

In this paper, we present a new robust iterative learning control (ILC) design for a class of linear systems in the presence of time-varying parametric uncertainties and additive input/output disturbances. The system model is described by the Markov matrix as an affine function of parametric uncertainties. The robust ILC design is formulated as a min–max problem using a quadratic performance criterion subject to constraints of the control input update. Then, we propose a novel methodology to find a suboptimal solution of the min–max optimization problem. First, we derive an upper bound of the worst-case performance. As a result, the min–max problem is relaxed to become a minimization problem in the form of a quadratic program. Next, the robust ILC design is cast into a convex optimization over linear matrix inequalities (LMIs) which can be easily solved using off-the-shelf optimization solvers. The convergences of the control input and the error are proved. Finally, the robust ILC algorithm is applied to a physical model of a flexible link. The simulation results reveal the effectiveness of the proposed algorithm.     

Synthesis of Mixed Objective Output Feedback Robust Model Predictive Control

Aiming at the constrained polytopic uncertain system with energy‐bounded disturbance and unmeasurable states, a novel synthesis scheme to design the output feedback robust model predictive control(MPC)is put forward by using mixed H₂/H_∞ design approach. The proposed scheme involves an offline design of a robust state observer using linear matrix inequalities(LMIs)and an online output feedback robust MPC algorithm using the estimated states in which the desired mixed objective robust output feedback controllers are cast into efficiently tractable LMI‐based convex optimization problems. In addition, the closed‐loop stability and the recursive feasibility of the proposed robust MPC are guaranteed through an appropriate reformulation of the estimation error bound (EEB). A numerical example subject to input constraints illustrates the effectiveness of the proposed controller.     

Theoretical analysis of unconstrained nonlinear model predictive control

Model predictive control (MPC) is a well-established controller design strategy for linear process models. Because many chemical and biological processes exhibit significant nonlinear behaviour, several MPC techniques based on nonlinear process models have recently been proposed. The most significant difference between these techniques is the computational approach used to solve the nonlinear model predictive control (NMPC) optimization problem. Consequently, analysis of NMPC techniques is often connected to the computational approach employed. In this paper, a theoretical analysis of unconstrained NMPC is presented that is independent of the computational approach. A nonlinear discrete-time, state-space model is used to predict the effects of future inputs on future process outputs. It is shown that model inverse, pole-placement, and steady-state controllers can be obtained by suitable selection of the control and prediction horizons. Moreover, the NMPC optimization problem can be modified to yield nonlinear internal model control (NIMC). The computational requirements of NIMC are considerably less than NMPC, but the NIMC approach is currently restricted to nonlinear models with well-defined and stable inverses. The NIMC controller is shown to provide superior servo and regulatory performance to a linear IMC controller for a continuous stirred tank reactor.     

RBF-ARX模型在串级系统中的应用

针对液位串级系统的非线性特征,采用RBF-ARX模型对液位串级系统的非线性动态特性进行建模,讨论了RBF—ARX模型结构的选取,模型参数辨识,RBF参数优化等问题。采用了不同的序列作为状态向量,分别建立了液位串级系统的训练数据和测试数据的RBF—ARX模型,分析了各模型的可靠性。模型的预测输出和仿真结果表明,RBF—ARX模型在非线性系统建模和辨识中是有效的。     

RBF-ARX模型在三容水箱液位控制系统建模中的应用

针对三容水箱液位系统的非线性，采用RBF-ARX模型对三容液位系统进行了离线动态特性建模的研究。着重讨论了RBF-ARX模型结构的选取、模型参数辨识、RBF参数优化等问题。RBF-ARX模型与ARX模型的一步预测输出比较的结果证实了RBF-ARX模型在非线性系统建模中的优越性。     

Design of robust gain-scheduled PI controllers for nonlinear processes

Gain-scheduling has proven to be a successful design methodology in many engineering applications. However, in the absence of a sound theoretical analysis, these designs come with no guarantees of robust stability, performance or even nominal stability of the overall gain-scheduled deign.This paper presents such an analysis for one type of nonlinear gain-scheduled control system based on the process input for nonlinear chemical processes. A methodology is also proposed for the design and optimization of the robust gain-scheduled PI controller. Conditions which guarantee robust stability and performance are formulated as a finite set of linear matrix inequalities (LMIs) and hence, the resulting problem is numerically tractable. Issues of modeling error and input-saturation are explicitly incorporated into the analysis. A simulation study of a nonlinear continuous stirred tank reactor (CSTR) process indicates that this approach can produce efficient sub-optimal robust gain-scheduled controllers.