Robust stability of nonlinear model predictive control with extended Kalman filter and target setting

This work deals with the closed‐loop robust stability of nonlinear model predictive control (NMPC) coupled with an extended Kalman filter (EKF). First, we point out the gaps between the practical formulations and theoretical research. Then, we show that the estimation error dynamics of an EKF are input‐to‐state stable (ISS) in the presence of nonvanishing perturbations. Moreover, a target setting optimization problem is proposed to solve the target state corresponding to the desired set points, which are used in the objective function in NMPC formulation. Thus, the objective function is a Lyapunov function candidate, and the input‐to‐state practical stability (ISpS) of the closed‐loop system can be established. Moreover, we see that the stability property deteriorates because of the estimation error. Simulation results of the proposed scheme are presented.Copyright © 2012 John Wiley & Sons, Ltd.     

Product property and production rate control of styrene polymerization

A multivariable multi-rate nonlinear model predictive control (NMPC) strategy is applied to styrene polymerization. The NMPC algorithm incorporates a multi-rate Extended Kalman Filter (EKF) to handle state variable and parameter estimation. A fundamental model is developed for the styrene polymerization CSTR, and control of polymer properties such as number average molecular weight (NAMW) and polydispersity is considered. These properties characterize the final polymer distribution and are strong indicators of the polymer qualities of interest. Production rate control is also demonstrated. Temperature measurements are available frequently while laboratory measurements of concentration and molecular weight distribution are available infrequently with substantial time delays between sampling and analysis. Observability analysis of the augmented system provides guidelines for the design of the augmented disturbance model for use in estimation using the multi-rate EKF. The observability analysis links measurement sets and corresponding observable disturbance models, and shows that measurements of moments of the polymer distribution are essential for good estimation and control. The CSTR is operated at an open-loop unstable steady state. Control simulations are performed under conditions of plant-model structural mismatch and in the presence of parameter uncertainty and disturbances, and the proposed multi-rate NMPC algorithm is shown to provide superior performance compared to linear multi-rate and nonlinear single-rate MPC algorithms. The major contributions of this work are the development of the multi-rate estimator and the measurement design study based on the observability analysis.     

State estimation and nonlinear predictive control of autonomous hybrid system using derivative free state estimators

In this work, we develop a state estimation scheme for nonlinear autonomous hybrid systems, which are subjected to stochastic state disturbances and measurement noise, using derivative free state estimators. In particular, we propose the use of ensemble Kalman filters (EnKF), which belong to the class of particle filters, and unscented Kalman filters (UKF) to carry out estimation of state variables of autonomous hybrid system. We then proceed to develop novel nonlinear model predictive control (NMPC) schemes using these derivative free estimators for better control of autonomous hybrid systems. A salient feature of the proposed NMPC schemes is that the future trajectory predictions are based on stochastic simulations, which explicitly account for the uncertainty in predictions arising from the uncertainties in the initial state and the unmeasured disturbances. The efficacy of the proposed state estimation based control scheme is demonstrated by conducting simulation studies on a benchmark three-tank hybrid system. Analysis of the simulation results reveals that EnKF and UKF based NMPC strategies is well suited for effective control of nonlinear autonomous three-tank hybrid system.     

一类不确定非线性系统的鲁棒输出反馈控制*

本文考虑一类具有输出反馈标准形的非线性系统的输出反馈控制问题，为了给出其不同于自适应控制的解决办法，设计出了新的鲁棒输出反馈控制方案。该方案不需辨识系统未知参数，也不需知识未知参数向量的上界，就能保证所有控制系统信号的全局有界性，并且顺参数的适当选择，可以使系统跟踪误差达到任意小。仿真结果表明可以获得良好的跟踪效果。     

Explicit output-feedback nonlinear predictive control based on black-box models

Nonlinear model predictive control (NMPC) algorithms are based on various nonlinear models. A number of on-line optimization approaches for output-feedback NMPC based on various black-box models can be found in the literature. However, NMPC involving on-line optimization is computationally very demanding. On the other hand, an explicit solution to the NMPC problem would allow efficient on-line computations as well as verifiability of the implementation. This paper applies an approximate multi-parametric nonlinear programming approach to explicitly solve output-feedback NMPC problems for constrained nonlinear systems described by black-box models. In particular, neural network models are used and the optimal regulation problem is considered. A dual-mode control strategy is employed in order to achieve an offset-free closed-loop response in the presence of bounded disturbances and/or model errors. The approach is applied to design an explicit NMPC for regulation of a pH maintaining system. The verification of the NMPC controller performance is based on simulation experiments.     

Robust stability of economically oriented infinite horizon NMPC that include cyclic processes

Nonlinear model predictive control (NMPC) with economic objective attracts growing interest. In our previous work [1], nominal stability of economically oriented NMPC for cyclic processes was proved by introducing a transformed system, and an infinite horizon NMPC formulation with discount factors was proposed. Moreover, the nominal stability property for economically oriented NMPC was analyzed in [2] for a class of systems satisfying strong duality. In this study, we extend the previous stability analysis in [1] to a general infinite horizon NMPC formulation with economic objectives. Instead of the strong duality assumption, we require the stage cost to be strongly convex, which is easier to check for a general nonlinear system. In addition, robust stability of this NMPC controller is also analyzed based on the Input-to-State Stability (ISS) framework. A simulated nonlinear double tank system subject to periodic change in electricity price is presented to illustrate the stability property. Finally, an industrial size air separation unit case study with periodic electricity cost is presented.     

Anti-windup via constrained regulation with observers

A method is presented for the output-feedback control of discrete-time linear systems with hard constraints on state and control variables. Prior work has shown that optimal controllers for constrained systems take the form of a nonlinear feedback law acting on a set-valued state estimate. In this paper, conventional state estimation schemes are used. A nonlinear control law is derived which views the state estimation error as a disturbance. The resulting control law is then used in conjunction with the conventional observer, rather than set-valued observer, to achieve the desired constrained regulation. The significantly reduced real-time computations come at the cost of restricting the controller structure and thereby introducing possible conservatism in the achievable performance. The results are specialized to the problem of anti-windup for systems with control saturations. A “measurement governor” scheme is introduced that alters plant measurements in such a way to improve performance in the presence of controller saturations.     

非线性离散随机时变系统比例微分滤波

扩展卡尔曼滤波（EKF）是从极小化状态估计误差的方差得到的，没有考虑状态误差的变化率，因而对非线性时变系统EKF估计方法惯性作用 较大，从而产生估计滞后，提出了非线性离散随机系统比例微分滤波（PDF），PDF联合考虑极小化状态估计误差和状态误差变化率的方差，克服了EKF对非线性时变系统估计滞后的缺点，估计具有适时性，提出了高估计的精度，仿真例子证明了所提出的估计方法的有效性。     

Robust adaptive output-feedback control for a class of nonlinear systems with general uncertainties

This paper investigates the robust adaptive output-feedback control for a class of nonlinear systems with general uncertainties and unknown parameters. First, a stable state observer is constructed and the system state is observed, and then the adaptive output-feedback controller is constructively designed for tracking the given reference signal. It is proven that the constructed controller is robust to the uncertainties of both the unknown parameters and the system states. These results show that the global stability of the resulting closed-loop systems has been guaranteed and the ε-tracking problem has been solved. Meanwhile, it is also proven that the tracking error tends to a ‘steady state’ at the negative exponential attenuating rate. Simulation examples show that the tracking effects of the designed adaptive control systems are good, and the control quantities used in the simulation examples are always within the range of the admissible control.     

Robust unscented Kalman filter with adaptation of process and measurement noise covariances

Unscented Kalman filter (UKF) has been extensively used for state estimation of nonlinear stochastic systems, which suffers from performance degradation and even divergence when the noise distribution used in the UKF and the truth in a real system are mismatched. For state estimation of nonlinear stochastic systems with non-Gaussian measurement noise, the Masreliez–Martin extended Kalman filter (EKF) gives better state estimates in relation to the standard EKF. However, the process noise and the measurement noise covariance matrices should be known, which is impractical in applications. This paper presents a robust Masreliez–Martin UKF which can provide reliable state estimates in the presence of both unknown process noise and measurement noise covariance matrices. Two numerical examples involving relative navigation of spacecrafts demonstrate that the proposed filter can provide improved state estimation performance over existing robust filtering approaches. Vision-aided robot arm tracking experiments are also provided to show the effectiveness of the proposed approach.