A comparison of nonlinear observers for output feedback model-based control of seeded batch crystallization processes

This paper presents a comparative analysis of various nonlinear estimation techniques when applied for output feedback model-based control of batch crystallization processes. Several nonlinear observers, namely an extended Luenberger observer, an extended Kalman filter, an unscented Kalman filter, an ensemble Kalman filer and a moving horizon estimator are used for closed-loop control of a semi-industrial fed-batch crystallizer. The performance of the nonlinear observers is evaluated in terms of their closed-loop behavior as well as their ability to cope with model imperfections and process uncertainties such as measurement errors and uncertain initial conditions. The simulation results suggest that the extended Kalman filter and the unscented Kalman filter provide accurate state estimates that ensure adequate fulfillment of the control objective. The results also confirm that adopting a time-varying process noise covariance matrix further enhances the estimation accuracy of the latter observers at the expense of a slight increase in their computational burden. This tuning method is particularly suited for batch processes as the state variables often vary significantly along the batch run. It is observed that model imperfections and process uncertainties are largely detrimental to the accuracy of state estimates. The degradation in the closed-loop control performance arisen from inadequate state estimation is effectively suppressed by the inclusion of a disturbance model into the observers.     

Optimal observers for continuous time linear stochastic systems

In the past, a major objection to application of the well known solution to the linear stochastic optimal control problem has been that the complexity, or dimension, of the solution exceeds that which is necessary for satisfactory feedback control. For a system described by n first order linear differential equations excited by white noise, the solution requires a filter of dimension n. If non white noise sources excite the system, an even higher dimensional filter is required. This paper presents a technique which alleviates the above objection. A recursive algorithm similar to the Kalman filter algorithm is presented which permits design of a reduced order linear estimator to replace the well known Kalman filter. The new estimator, called an observer, is stochastically optimal subject to its reduced order dimensionality constraint, but its performance is not as good as a full Kalman filter. The observer algorithm is general in that it applies to time variable, multivariable systems. A minimum order = n ? m₁ exists for the observer. Here n = state dimension and m₁ = dimension of the non white measurement. Non white noise sources of any order q may exist and need not contribute to the dimension of the optimal observer. Optimal observers of all orders between n ? m₁ and n ? m₁ + q may be designed, the latter case being a Kalman filter. Detailed examples are given to illustrate the theory.     

Comments on ‘ Dynamic feedback methodology for interconnected control systems ’ and ‘ Dynamic multilevel optimization for a class of non-linear systems ’

The paper considers the Kalman-Bucy filter for a linear system when the measurement noise covariance matrix is singular. It is shown that the problem of infimizing the square of a linear functional of the state estimation error is the dual of the optimal singular linear regulator problem. Furthermore there is an optimal reduced-order Kalman-Bucy filter for minimization of the trace of the state error covariance matrix, when all extremal controls for a dual regulator have finite order of singularity, and no Luenberger observer is needed. The proof is constructive. Necessary and sufficient conditions for the existence of a reduced-order optimal estimator are derived.     

Direct solution to the general reduced-order stochastic observation problem

The stochastic optimal state observation problem is considered for a general linear, continuous, time-invariant system with unmeasurable stationary inputs and measurement outputs that may be, at least in part, perfect. A general solution to the problem is obtained by processing the perfect measurements through a specific differentiation-transformation scheme in order to extract the maximum accurate information on the system states. Using this information the original system is transformed to a new reduced-order model whose measurements are corrupted by a white noise of non-singular intensity matrix. A minimum-order full-state estimator to the original system is then constructed by combining the outputs of any full-order observer to the reduced-order model and the perfect combinations of the system states that were derived by the differentiation-transformation scheme. A solution to the general singular Kalman filtering problem is then obtained by minimizing the variance of the estimation error of the observer to the reduced-order model.     

State estimate recovery via nonlinear transforms during periods of destabilisation initiated by measurement uncertainty

A parallel approach is being proposed for the construction of a nonlinear-based state estimator for closed-loop systems failing the Hurwitz criteria during finite time intervals. The instability originates from sensor errors, and it is assumed that a Kalman–Bucy/Luenberger estimator is being processed in closed loop. It is shown that a bounded nonlinear state estimator can be constructed via nonlinear state transforms over a finite time interval, and makes use of the control input and the measured output of the destabilised closed-loop system. The filter presented provides an interesting new approach for the correction of state estimation due to sensor bias. In cases where the closed-loop plant remains stable, this approach is useful for dealing with pure sensor bias without the need for high gain filtering. The original uncontrolled open-loop system is assumed stable as a preemptive criterion throughout this study.     

Reduced-order estimation Part 1. Filtering

This paper presents a method for designing an ‘optimum’ unbiased reduced-order filter. For the proposed approach to work, the order of the filter must be greater than a certain minimum determined by the number of independent observations of the system available. The filler is much like a Luenberger observer for the state to be estimated, but with parameters optimized with respect to the noises in the system. A reduced-order innovation process is proposed that has properties similar to those of the full-order innovation process when the reduced filter is optimized. The approach offers the possibility of significant reduction in real-time computational requirements compared with the full-order filter, though at the cost of some loss of performance. The algorithm for the reduced-order filter is simple to implement— quite similar to that of the Kalman filter. An example is presented to compare the performance of the proposed method with the full-order Kalman filter.     

Output feedback second‐order sliding mode control of the cart on a beam system

We propose an output feedback second‐order sliding mode controller to stabilize the cart on a beam system. A second‐order sliding mode controller is designed using a Lyapunov function‐based switching surface and finite‐time controllers, while the state estimator is designed based on the Luenberger‐like observer. The proposed observer extends the applicability of Luenberger‐like observer to nonlinear systems that are not input–output linearizable, but can be approximately input–output linearized. The approximation is based on the physical property of the system, wherein certain terms in the total energy are neglected. Extensive numerical simulations validate the robustness of the proposed controller to parametric uncertainties using estimated states. Copyright © 2009 John Wiley & Sons, Ltd.     

High order sliding mode observer for linear systems with unbounded unknown inputs

A global observer is designed for strongly detectable systems with unbounded unknown inputs. The design of the observer is based on three steps. First, the system is extended taking the unknown inputs (and possibly some of their derivatives) as a new state; then, using a global high-order sliding mode differentiator, a new output of the system is generated in order to fulfil, what we will call, the Hautus condition, which finally allows decomposing the system, in new coordinates, into two subsystems; the first one being unaffected directly by the unknown inputs, and the state vector of the second subsystem is obtained directly from the original system output. Such decomposition permits designing of a Luenberger observer for the first subsystem, which satisfies the Hautus condition, i.e. all the outputs have relative degree one w.r.t. the unknown inputs. This procedure enables one to estimate the state and the unknown inputs using the least number of differentiations possible. Simulations are given in order to show the effectiveness of the proposed observer.     

迭代无味卡尔曼滤波器

通过对无味卡尔曼滤波器(Unscented Kalman filter,UKF)的误差进行分析,提出了迭代UKF(IUKF)算法.该基本思路是用测量更新后的状态估计去重新对状态量和观测量的一步预测,然后再次应用LMMSE估计子估计状态量的均值和协方差阵,如此多次迭代后的滤波估计输出具有更高的精度和更小的方差,故滤波器表现出更好的一致性.Monte Carlo仿真表明,IUKF主要应用于观测噪声较小的场合,其中的迭代只需进行2～3次即可.     

The extended Luenberger observer for nonlinear systems

For nonlinear single-input single-output systems , the relationships for a state transformation into the nonlinear observer canonical form are developed. It is possible to dimension a nonlinear observer by an eigenvalue assignment without solving the nonlinear partial differential equations for the transformation, if the transformed nonlinearities are linearized about the reconstructed state. With reference to the extended Kalman filter algorithm, this nonlinear observer design is called the extended Luenberger observer.     

Stability of a class of discrete-time nonlinear recursive observers

This work provides a framework for nominal and robust stability analysis for a class of discrete-time nonlinear recursive observers (DNRO). Given that the system has linear output mapping, local observability and Jacobian matrices satisfying certain conditions, the nominal and robust stability of the DNRO is defined by the property of estimation error dynamics and is analyzed using Lyapunov theory. Moreover, a simultaneous state and parameter estimation scheme is shown to be Input-to-State Stable (ISS), and adaptively reduce plant-model mismatch on-line. Three design strategies of the DNRO that satisfy the stability results are given as examples, including the widely used extended Kalman filter, extended Luenberger observer, and the fixed gain observer.     

A robust controller for trajectory tracking of a DC motor pendulum system

This work presents a solution to the output feedback trajectory tracking problem for an uncertain DC motor pendulum system under the effect of an unknown bounded disturbance. The proposed algorithm uses a Proportional Derivative (PD) controller plus a novel on-line estimator of the unknown disturbance. The disturbance estimator is obtained by coupling a standard second-order Luenberger observer with a third-order sliding modes differentiator. The Luenberger observer provides estimates of the motor angular position and velocity. Moreover, an ideal disturbance estimator in terms of the Luenberger observer error and its first and second time derivatives is obtained from the observer error formulae; these time derivatives are not available from measurements. Subsequently, the sliding modes third-order differentiator allows obtaining estimates of these time derivatives in finite time. The estimates replace the real values of the first and second time derivatives in the ideal disturbance estimator thus producing a practical disturbance estimator, and also permit obtaining an estimate of the motor angular velocity. A depart from previous approaches is the fact that the disturbance is not directly estimated by the Luenberger observer or the third-order differentiator. Numerical simulations and real-time experiments validate the effectiveness of the proposed approach.     

Application of Kalman filters to the regulation of dead time processes

A single-input single-output control algorithm for a process with dead time and dead time uncertainty is described. The process dynamics consist of first-order mixing and pure delay with the pure delay being dominant. The process is disturbed at the upstream end by a disturbance sequence consisting of white noise passed through a first-order shaping filter. The process output is subject to white measurement noise. A discrete Kalman filter is used to produce state estimates for the disturbance mixing and dead time states which are updated from the process output residual error. In order to handle dead time uncertainty of up to a priori established limits, the residual error is passed through a dynamic dead band whose magnitude is a function of the dead time states of a separate process dynamic model driven by the process input. The dead band eliminates dead time error components from the residual. Control is achieved by state feedback from the upstream Kalman filter state estimate. The algorithm is in use on paper machine and bleach plant control applications and gives near minimum variance performance when properly tuned.     

Observer-based predictive controller design with network-enhanced time-delay compensation

State feedback control is very attractive due to the precise computation of the gain matrix, but the implementation of a state feedback controller is possible only when all state variables are directly measurable. This condition is almost impossible to accomplish due to the excess number of required sensors or unavailability of states for measurement in most of the practical situations. Hence, the need for an estimator or observer is obvious to estimate all the state variables by observing the input and the output of the controlled system. As such, the goal of this paper is to provide a control design methodology based on a Luenberger observer design that can assure the closed-loop performances of a vehicle drivetrain with backlash, while compensating the network-enhanced time-varying delays. This goal is achieved in a sequential manner: firstly, a piecewise linear model of two inertias drivetrain, which takes into consideration the backlash nonlinearity and the network-enhanced time-varying delay effects is derived; then, a Luenberger observer which estimates the state variables is synthesized and the robust full state-feedback predictive controller based on flexible control Lyapunov functions is designed to explicitly take into account the bounds of the disturbances caused by time-varying delays and to guarantee also the input-to-state stability of the system in a non-conservative way. The full state-feedback predictive control strategy based on the Luenberger observer design was experimentally tested on a vehicle drivetrain emulator controlled through controller area network, with the aim of minimizing the backlash effects while compensating the network-enhanced delays.     

Robust linear output feedback controller for autonomous landing of a quadrotor on a ship deck

Ship deck landing control of a quadrotor requires certain robustness with respect to ship heave motion. Typical systems only provide relative height, therefore do not have relative heave rate information. In this paper, a linear output feedback control consisting of a full state feedback controller and a Luenberger observer is formulated. Invariant ellipsoid method is used to formulate an estimation of a bound on the response of a linear output feedback-controlled system subjected to external disturbances and measurement noise. The gains that result in a minimum bound are optimized using a gradient descent iterative approach proposed in this paper where the invariant ellipsoid condition is linearized into a tractable LMI condition. This approach is applied to a simulation of a quadrotor landing on a ship deck and results are compared with other gains. The gains selected using the proposed approach exhibits improved robustness to external disturbances and measurement noise.     

基于自适应固定滞后卡尔曼平滑器的状态观测器在假手上的应用

针对以电位计为角度传感器的假手系统,提出了一种基于自适应固定滞后卡尔曼平滑器的状态观测器以观测手指的当前位置、速度和加速度信息.首先,分析了卡尔曼滤波器滤除电位计热噪声并观测速度与加速度的合理性,进而建立了其系统的离散状态转移矩阵.其次,相比卡尔曼滤波器,卡尔曼平滑器在参数相同的情况下具有更好的平滑效果,据此提出一种基于固定滞后卡尔曼平滑器的状态观测器,并通过引入渐消因子以提高动态响应特性.同时给出了一种将本文算法滞后特性降至一个控制周期的有效实现方式.最后,在HIT-V仿人假手实验平台上进行了实验验证.实验结果表明,相比对原始数据直接进行差分,该方法将速度噪声降低了20倍以上,加速度噪声降低了10 000倍以上.相比标准卡尔曼滤波器和固定滞后卡尔曼平滑器,该方法在动态响应方面具有更好的效果.     

曲线焊缝跟踪的视觉伺服协调控制

为实现工业机器人自动跟踪曲线焊缝，提出了协调焊枪运动和视觉跟踪的视觉伺服控制方法．建立了特征点的数学模型，并在此基础上确定机器人运动的旋转轴．设计了一种双层结构的模糊视觉伺服控制器，通过动态确定控制量有效范围来保证图像特征存在于视场中．为准确确定有效范围，设计了带模型动态补偿的Kalman滤波器．曲线焊缝的自动跟踪实验验证了所提方法的有效性．     

A dynamic observer for a synchronous machine†

The closed-loop optimal control law obtained for a linear time-invariant system always requires the entire state vector to be available for direct measurement. It is seldom that all the state variables of a physical system are available for measurement. A compatible dynamic observer of the Luenberger type is designed to reconstruct the entire state vector for a synchronous machine-infinite bus system. The performance of the system with the observer cascaded to the optimal controller is compared with the performance of the system when all the state variables are available for feedback. The transfer matrix relating the output and input is derived.     

Modified stochastic Luenberger observers

A modified stochastic Luenberger observer (MSLO) structure is proposed to recover the optimal performance of the coventional SLO for obtaining full-state estimates in linear discrete-time stochastic systems. The optimal MSLO (OMSLO) which gives the MMSE estimates is derived by using the general two-stage Kalman filter. A reduced-order form of the OMSLO is also proposed for systems having singular measurement noises. The connection between the OMSLO and the optimal minimal-order observer of Leondes and Novak is also shown.