A RBF-ARX model-based robust MPC for tracking control without steady state knowledge

A RBF-ARX modeling and robust model predictive control (MPC) approach to achieving output-tracking control of the nonlinear system with unknown steady-state knowledge is proposed. On the basis of the RBF-ARX model with considering the system time delay, a local linearization state-space model is obtained to represent the current behavior of the nonlinear system, and a polytopic uncertain linear parameter varying (LPV) state-space model is built to represent the future system’s nonlinear behavior. Based on the two models, a quasi-min–max MPC algorithm with constraint is designed for output-tracking control of the nonlinear system with unknown steady state knowledge. The optimization problem of the quasi-min–max MPC algorithm is finally converted to the convex linear matrix inequalities (LMIs) optimization problem. Closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and feasibility of the LMIs. Two examples, i.e. the modeling and control of a continuously stirred tank reactor (CSTR) and a two tank system demonstrate the effectiveness of the RBF-ARX modeling and robust MPC approach.     

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.     

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.     

Robust tracking design based on adaptive fuzzy control of uncertain nonlinear MIMO systems with time delayed states

It is proposed here to use a robust tracking design based on adaptive fuzzy control technique to control a class of multi-input-multi-output (MIMO) nonlinear systems with time delayed uncertainty in which each uncertainty is assumed to be bounded by an unknown gain. This technique will overcome modeling inaccuracies, such as drag and friction losses, effect of time delayed uncertainty, as well as parameter uncertainties. The proposed control law is based on indirect adaptive fuzzy control. A fuzzy model is used to approximate the dynamics of the nonlinear MIMO system; then, two on-line estimation schemes are developed to overcome the nonlinearities and identify the gains of the delayed state uncertainties, simultaneously. The advantage of employing an adaptive fuzzy system is the use of linear analytical results instead of estimating nonlinear system functions with an online update law. The adaptive fuzzy scheme uses a Variable Structure (VS) scheme to resolve the system uncertainties, time delayed uncertainty and the external disturbances such that H_∞ tracking performance is achieved. The control laws are derived based on a Lyapunov criterion and the Riccati-inequality such that all states of the system are uniformly ultimately bounded (UUB). Therefore, the effect can be reduced to any prescribed level to achieve H _∞ tracking performance. A two-connected inverted pendulums system on carts and a two-degree-of-freedom mass-spring-damper system are used to validate the performance of the proposed fuzzy technique for the control of MIMO nonlinear systems.     

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 economic model predictive control based on a periodicity constraint

This paper proposes robust economic model predictive control based on a periodicity constraint for linear systems subject to unknown‐but‐bounded additive disturbances. In this economic MPC design, a periodic steady‐state trajectory is not required and thus assumed unknown, which precludes the use of enforcing terminal state constraints as in other standard economic formulations. Instead, based on the desired periodicity of system operation, we optimize the economic performance over a set of periodic trajectories that include the current state. To achieve robust constraint satisfaction, we use a tube‐based technique in the economic MPC formulation. The mismatches between the nominal model and the closed‐loop system with perturbations are limited using a local control law. With the proposed robust tube‐based strategy, recursive feasibility is guaranteed. Moreover, under a convexity assumption, the closed‐loop convergence of the closed‐loop system is analyzed, and an optimality certificate is provided to check if the closed‐loop trajectory reaches a neighborhood of the optimal nominal periodic steady trajectory using Karush‐Kuhn‐Tucker optimality conditions. Finally, through numerical examples, we show the effectiveness of the proposed approach.     

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.     

Dynamic Output Feedback Robust MPC With one Free Control Move for LPV Model With Bounded Disturbance

For a linear parameter‐varying (LPV) model which is a convex combination of several linear time invariant sub‐models, this paper considers the case when the combining coefficients are unknown (except being nonnegative and their sum being one). For this model with norm‐bounded unknown disturbance, an output feedback robust model predictive control (MPC) is proposed by parameterizing the infinite horizon control moves and estimated states into one free control move, one free estimated state (i.e., one control move and one estimated state as degrees of freedom for optimization) and a dynamic output feedback law. This is the first endeavour to apply the free control move and free estimated state in the output feedback MPC for this model. The algorithm is shown to be recursively feasible and the system state is guaranteed to converge to the neighborhood of the equilibrium point. A numerical example verifies the effectiveness of the proposed algorithm.     

Affine and predictive control policies for a class of nonlinear systems

The input-state linear horizon (ISLH) for a nonlinear discrete-time system is defined as the smallest number of time steps it takes the system input to appear nonlinearly in the state variable. In this paper, we employ the latter concept and show that the class of constraint admissible N-step affine state-feedback policies is equivalent to the associated class of constraint admissible disturbance-feedback policies, provided that N is less than the system’s ISLH. The result generalizes a recent result in [Goulart, P. J., Kerrigan, E. C., Maciejowski, J. M. (2006). Optimization over state feedback policies for robust control with constraints. Automatica, 42(4), 523-533] and is significant because it enables one: (i) to determine a constraint admissible state-feedback policy by employing well-known convex optimization techniques; and (ii) to guarantee robust recursive feasibility of a class of model predictive control (MPC) policies by imposing a suitable terminal constraint. In particular, we propose an input-to-state stabilizing MPC policy for a class of nonlinear systems with bounded disturbance inputs and mixed polytopic constraints on the state and the control input. At each time step, the proposed MPC policy requires the solution of a single convex quadratic programme parameterized by the current system state.     

Robustness design of nonlinear dynamic systems via fuzzy linearcontrol

This study introduces a fuzzy linear control design method for nonlinear systems with optimal H^∞ robustness performance. First, the Takagi and Sugeno fuzzy linear model (1985) is employed to approximate a nonlinear system. Next, based on the fuzzy linear model, a fuzzy controller is developed to stabilize the nonlinear system, and at the same time the effect of external disturbance on control performance is attenuated to a minimum level. Thus based on the fuzzy linear model, H^∞ performance design can be achieved in nonlinear control systems. In the proposed fuzzy linear control method, the fuzzy linear model provides rough control to approximate the nonlinear control system, while the H^∞ scheme provides precise control to achieve the optimal robustness performance. Linear matrix inequality (LMI) techniques are employed to solve this robust fuzzy control problem. In the case that state variables are unavailable, a fuzzy observer-based H^∞ control is also proposed to achieve a robust optimization design for nonlinear systems. A simulation example is given to illustrate the performance of the proposed design method     

Robust model predictive control for constrained continuous-time nonlinear systems

In this paper, a robust model predictive control (MPC) is designed for a class of constrained continuous-time nonlinear systems with bounded additive disturbances. The robust MPC consists of a nonlinear feedback control and a continuous-time model-based dual-mode MPC. The nonlinear feedback control guarantees the actual trajectory being contained in a tube centred at the nominal trajectory. The dual-mode MPC is designed to ensure asymptotic convergence of the nominal trajectory to zero. This paper extends current results on discrete-time model-based tube MPC and linear system model-based tube MPC to continuous-time nonlinear model-based tube MPC. The feasibility and robustness of the proposed robust MPC have been demonstrated by theoretical analysis and applications to a cart-damper springer system and a one-link robot manipulator.     

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.     

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.     

Robust model predictive control of integrating processes

Here, it is studied the control of integrating systems in the presence of model uncertainty. For this kind of system, a method is proposed to overcome one of the major barriers to the practical implementation of the existing robust MPC approaches: the assumption that the steady state of the true plant is known. To deal with unknown steady states, the controller incorporates a state-space model in the incremental form, which is a model framework frequently adopted by MPC packages. In this case, it is shown that for integrating systems, minimizing the integrating states at steady state is not sufficient to guarantee the stability of the uncertain plant. It is proposed a modified cost function that allows the controller to stabilize a family of plants, even when the steady state is not at the origin. To compute the control law, a Min-Max problem is solved with model uncertainty assumed to be of polytopic type. The application of the proposed controller is illustrated with the simulation of an industrial multivariable system. For this example, the effect of the new tuning parameters is discussed.     

Output feedback model predictive control for LPV systems based on quasi-min–max algorithm

This paper proposes a robust output feedback model predictive control (MPC) scheme for linear parameter varying (LPV) systems based on a quasi-min–max algorithm. This approach involves an off-line design of a robust state observer for LPV systems using linear matrix inequality (LMI) and an on-line robust output feedback MPC algorithm using the estimated state. The proposed MPC method for LPV systems is applicable for a variety of systems with constraints and guarantees the robust stability of the output feedback systems. A numerical example for an LPV system subject to input constraints is given to demonstrate its effectiveness.     

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.     

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。     

Robust distributed model predictive control of linear systems with structured time-varying uncertainties

In this work, synthesis of robust distributed model predictive control (MPC) is presented for a class of linear systems subject to structured time-varying uncertainties. By decomposing a global system into smaller dimensional subsystems, a set of distributed MPC controllers, instead of a centralised controller, are designed. To ensure the robust stability of the closed-loop system with respect to model uncertainties, distributed state feedback laws are obtained by solving a min–max optimisation problem. The design of robust distributed MPC is then transformed into solving a minimisation optimisation problem with linear matrix inequality constraints. An iterative online algorithm with adjustable maximum iteration is proposed to coordinate the distributed controllers to achieve a global performance. The simulation results show the effectiveness of the proposed robust distributed MPC algorithm.     

持续有界扰动下的非线性H∞鲁棒预测控制

针对未知但有界的持续扰动, 提出了一种约束非线性 H∞ 鲁棒预测控制策略. 首先, 引入离散系统的输入状态稳定性概念; 其次, 采用仿射输入定义预测控制的控制律, 并给出相应终端约束集的估计解法. 进一步, 得到预测控制闭环系统的鲁棒稳定性结论. 最后, 数值仿真验证了上述策略的有效性.