Robust moving horizon estimation based output feedback economic model predictive control

In this work, we develop an economic model predictive control scheme for a class of nonlinear systems with bounded process and measurement noise. In order to achieve fast convergence of the state estimates to the actual system state as well as the robustness of the observer to measurement and process noise, a deterministic (high-gain) observer is first applied for a small time period with continuous output measurements to drive the estimation error to a small value; after this initial small time period, a robust moving horizon estimation scheme is used on-line to provide more accurate and smoother state estimates. In the design of the robust moving horizon estimation scheme, the deterministic observer is used to calculate reference estimates and confidence regions that contain the actual system state. Within the confidence regions, the moving horizon estimation scheme is allowed to optimize its estimates. The output feedback economic model predictive controller is designed via Lyapunov techniques based on state estimates provided by the deterministic observer and the moving horizon estimation scheme. The stability of the closed-loop system is analyzed rigorously and conditions that ensure the closed-loop stability are derived. Extensive simulations based on a chemical process example illustrate the effectiveness of the proposed approach.     

压电陶瓷驱动平台自适应输出反馈控制

压电陶瓷驱动平台的精度和动态特性主要取决于所设计的控制器是否可以有效地补偿压电陶瓷固有的迟滞特性. 针对这一问题, 提出了一种基于神经网络 (Neural network, NN)的自适应输出反馈控制策略. 为了避免压电陶瓷速度测量噪声的影响, 采用高增益观测器对压电陶瓷平台的速度状态进行估计; 为了克服压电陶瓷的迟滞非线性特征, 采用神经网络动态补偿策略; 针对神经网络逼近误差和观测器估计误差, 控制器设计中增加了鲁棒控制项. 最后应用Lyapunov 稳定性理论证明了所提出的控制器的收敛性问题. 仿真实验表明了所提控制方法的有效性.     

On the performance of high-gain observers with gain adaptation under measurement noise

We address the problem of state observation for a system whose dynamics may involve poorly known, perhaps even nonlocally Lipschitz functions and whose output measurement may be corrupted by noise. It is known that one way to cope with all these uncertainties and noise is to use a high-gain observer with a gain adapted on-line. The proposed method, while presented for a particular case, relies on a “generic” analysis tool based on the study of differential inequalities involving quadratic functions of the error system in two coordinate frames plus the gain adaptation law. We establish that, for bounded system solutions, the estimated state and the gain are bounded. Moreover, we provide an upper bound for the mean value of the error signals as a function of the observer parameters. Since due to perturbations the gain adaptation law may drive the observer/plant interconnection to nearby boundary of its stability region, oscillatory behavior may emerge. To overcome this issue, we suggest an adaptive procedure based on a space averaging technique involving several copies of the observer.     

Input observer convergence and robustness: Application to compression ratio estimation

The focus of this paper is the analysis of a high-gain observer for estimating un-steady inputs, including guarantees for: (1) robustness to measurement uncertainty, and (2) transient upper bound on estimator error. A method for selecting the observer gain based on measurement uncertainty, acceptable steady-state errors, and desired estimation error convergence rates, is also described. This strategy is demonstrated in practice for internal combustion (IC) engine effective compression ratio (ECR) estimation. Experimental results are shown to be consistent with analytical guarantees—convergence within 4 engine cycles, and a steady-state error less than 0.5 ECR, in the presence of 10% measurement error.     

A suite of robust controllers for the manipulation of microscale objects.

A suite of novel robust controllers is introduced for the pickup operation of microscale objects in a microelectromechanical system (MEMS). In MEMS, adhesive, surface tension, friction, and van der Waals forces are dominant. Moreover, these forces are typically unknown. The proposed robust controller overcomes the unknown contact dynamics and ensures its performance in the presence of actuator constraints by assuming that the upper bounds on these forces are known. On the other hand, for the robust adaptive critic-based neural network (NN) controller, the unknown dynamic forces are estimated online. It consists of an action NN for compensating the unknown system dynamics and a critic NN for approximating a certain strategic utility function and tuning the action NN weights. By using the Lyapunov approach, the uniform ultimate boundedness of the closed-loop manipulation error is shown for all the controllers for the pickup task. To imitate a practical system, a few system states are considered to be unavailable due to the presence of measurement noise. An output feedback version of the adaptive NN controller is proposed by exploiting the separation principle through a high-gain observer design. The problem of measurement noise is also overcome by constructing a reference system. Simulation results are presented and compared to substantiate the theoretical conclusions.     

Robust near-optimal output feedback control of non-linear systems

This work proposes a robust near-optimal non-linear output feedback controller design for a broad class of non-linear systems with time-varying bounded uncertain variables. Both vanishing and non-vanishing uncertainties are considered. Under the assumptions of input-to-state stable (ISS) inverse dynamics and vanishing uncertainty, a robust dynamic output feedback controller is constructed through combination of a high-gain observer with a robust optimal state feedback controller synthesized via Lyapunov's direct method and the inverse optimal approach. The controller enforces exponential stability and robust asymptotic output tracking with arbitrary degree of attenuation of the effect of the uncertain variables on the output of the closed-loop system, for initial conditions and uncertainty in arbitrarily large compact sets, provided that the observer gain is sufficiently large. Utilizing the inverse optimal control approach and singular perturbation techniques, the controller is shown to be near-optimal in the sense that its performance can be made arbitrarily close to the optimal performance of the robust optimal state feedback controller on the infinite time-interval by selecting the observer gain to be sufficiently large. For systems with non-vanishing uncertainties, the same controller is shown to ensure boundedness of the states, uncertainty attenuation and near-optimality on a finite time-interval. The developed controller is successfully applied to a chemical reactor example.     

A high-gain scaling technique for adaptive output feedback control of feedforward systems

In this note, we propose an adaptive output feedback control design technique for feedforward systems based on our recent results on dynamic high-gain scaling techniques for controller design for strict-feedback systems. The system is allowed to contain uncertain functions of all the states and the input as long as the uncertainties satisfy certain bounds. Unknown parameters are allowed in the bounds assumed on the uncertain functions. If the uncertain functions involve the input, then the output-dependent functions in the bounds on the uncertain functions need to be polynomially bounded. It is also shown that if the uncertain functions can be bounded by a function independent of the input, then the polynomial boundedness requirement can be relaxed. The designed controllers have a very simple structure being essentially a linear feedback with state-dependent dynamic gains and do not involve any saturations or recursive computations. The observer utilized to estimate the unmeasured states is similar to a Luenberger observer with dynamic observer gains. The Lyapunov functions are quadratic in the state estimates, the observer errors, and the parameter estimation error. The stability analysis is based on our recent results on uniform solvability of coupled state-dependent Lyapunov equations. The controller design provides strong robustness properties both with respect to uncertain parameters in the system model and additive disturbances. This robustness is the key to the output feedback controller design. Global asymptotic results are obtained.     

A bounded-error approach to simultaneous state and actuator fault estimation for a class of nonlinear systems

This paper proposes an approach for the joint state and fault estimation for a class of uncertain nonlinear systems with simultaneous unknown input and actuator faults. This is achieved by designing an unknown input observer combined with a set-membership estimation in the presence of disturbances and measurement noise. The observer is designed using quadratic boundedness approach that is used to overbound the estimation error. Sufficient conditions for the existence and stability of the proposed state and actuator fault estimator are expressed in the form of linear matrix inequalities (LMIs). Simulation results for a quadruple-tank system show the effectiveness of the proposed approach.     

Cascade Integral Predictors and Feedback Control for Nonlinear Systems with Unknown Time-varying Input-delays

In this paper, we consider the problem of predictor design for nonlinear systems in the presence of unknown time-varying input-delays. A cascade integral high-gain predictor is proposed to estimate the future state. With a distinctive structure, the predictor can handle unknown delays and eliminate the “peaking phenomenon” during the transient period. Then, a predictor-based output feedback control is designed to guarantee the boundedness of system states. Lyapunov-Krasovskii functional and perturbation theories are used to prove the convergence of the estimation error and the closed-loop system. Finally, simulation results illustrate the superior performance of the cascade integral predictor compared to the standard high-gain predictor.

     

Flatness-based linear output feedback control for disturbance rejection and tracking tasks on a Chua's circuit

A linear output feedback controller is developed for trajectory tracking problems defined on a modified version of Chua's circuit. The circuit modification considers the introduction of a flat input, i.e. a suitable external control input channel guided by (a) the induction of the flatness property on a measurable output signal of the circuit and (b) the physical viability of the control input. A linear active disturbance rejection control based on a high-gain linear disturbance observer, is implemented on a laboratory prototype. We show that the state-dependent disturbance can be approximately, but arbitrarily closely, estimated through a linear high-gain observer, called a generalised proportional integral (GPI) observer, which contains a linear combination of a sufficient number of extra iterated integrals of the output estimation error. Experimental results are presented in the output reference trajectory tracking of a signal generated by an unrelated chaotic system of the Lorenz type. Laboratory experiments illustrate the proposed linear methodology for effectively controlling chaos.     

Adaptive neural network control of nonlinear systems by state andoutput feedback

This paper presents a novel control method for a general class of nonlinear systems using neural networks (NNs). Firstly, under the conditions of the system output and its time derivatives being available for feedback, an adaptive state feedback NN controller is developed. When only the output is measurable, by using a high-gain observer to estimate the derivatives of the system output, an adaptive output feedback NN controller is proposed. The closed-loop system is proven to be semi-globally uniformly ultimately bounded (SGUUB). In addition, if the approximation accuracy of the neural networks is high enough and the observer gain is chosen sufficiently large, an arbitrarily small tracking error can be achieved. Simulation results verify the effectiveness of the newly designed scheme and the theoretical discussions.     

Adaptive nonlinear output feedback tracking with a partialhigh-gain observer and backstepping

An adaptive output feedback controller for nonlinear systems with nonlinearities depending on the first r (1⩽r⩽n) derivatives of the output is proposed. The derivatives are estimated with a partial state high-gain observer, and the remaining states are handled using a backstepping method. Compared with methods based on full state high-gain observer, this approach improves robustness with respect to measurement noise and avoids overparametrization. Semiglobal tracking is proven under the assumption that the regressor is persistently exciting     

Nonlinear observer design via passivation of error dynamics

We present a new design scheme of nonlinear state observers (global, full order, asymptotic observers) through passivation of the error dynamics. In order to consider passivity of the error dynamics for the observer problem, we place a conceptual input and output on the generalized error dynamics which also includes the plant, and the strictness of passivity is extended with respect to a set in which the estimation error becomes zero. Then, output feedback passivation for the error dynamics will lead to the construction of a state observer. It is also shown that a nonlinear observer is generally vulnerable to measurement disturbance, in the sense that even an arbitrarily small measurement disturbance can lead to a blowup of the error state. However, due to the passivity of the error dynamics, the proposed nonlinear injection gain can be easily modified for the observer to be robust to measurement disturbances.     

Variable high-gain disturbance observer design with online adaption of observer gains embedded in numerical integration

In this paper, a variable gain design approach for the high-gain disturbance observer, called Proportional-Integral-Observer (PI-Observer), is proposed to solve the problem of choosing suitable observer gains. The high-gain PI-Observer is successfully applied to estimate unknown inputs of systems together with the system states. It is known that reasonable estimations of unknown inputs can only be derived using high observer gains. On the other hand, extremely large gains will cause serious problems with respect to measurements noise and unmodeled dynamics. According to the analysis of the estimation quality regarding to the factors which influence the estimation results, the optimal level of observer gains is changing during the estimation, an online adaption for the observer gains is therefore developed. The designed PI-Observer, called Advanced PI-Observer (API-Observer), will use changing observer gains from the adaption algorithm, which is proved to give stable estimation error dynamics. Simulation results from an elastic beam example are shown to illustrate the implementation of the API-Observer.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Output feedback control of a class of feedforward nonlinear systems in the presence of sensor noise

We propose an output feedback controller for a class of feedforward nonlinear systems under sensor noise. The sensor noise is any signal whose DC component is finite, which covers not only deterministic signals but also random signals including many practical noises. We introduce a notion of virtual state, then propose a measurement output feedback controller that utilizes a gain scaling factor. The gain scaling factor is commonly employed by the observer and controller. Through analysis, we show that all system states and output remain to be bounded in the presence of sensor noise, and the bound of states except output can be made arbitrarily small. Moreover, if the DC component of sensor noise is zero, the ultimate bound of the states and output can be made arbitrarily small by increasing the gain scaling factor in the presence of sensor noise. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive regulation of uncertain nonlinear systems by output feedback: A universal control approach

The problem of global state regulation via output feedback is investigated for uncertain nonlinear systems. The class of uncertain systems under consideration is assumed to be dominated by a bounding system which is linear growth in the unmeasurable states but can be a polynomial function of the system output, with unknown growth rates. To achieve global state regulation in the presence of parametric uncertainty, we propose a non-identifier based output feedback control scheme by employing the idea of universal control integrated with the design of a linear high-gain observer, whose gains are composed of two components, both of them are not constant and need to be dynamically updated. In particular, we explicitly design a universal output feedback controller which globally regulates all the states of the uncertain system while maintaining global boundedness of the closed-loop system.     

Adaptive output feedback control of uncertain nonlinear systems using single-hidden-layer neural networks

We consider adaptive output feedback control of uncertain nonlinear systems, in which both the dynamics and the dimension of the regulated system may be unknown. However, the relative degree of the regulated output is assumed to be known. Given a smooth reference trajectory, the problem is to design a controller that forces the system measurement to track it with bounded errors. The classical approach requires a state observer. Finding a good observer for an uncertain nonlinear system is not an obvious task. We argue that it is sufficient to build an observer for the output tracking error. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. The theoretical results are illustrated in the design of a controller for a fourth-order nonlinear system of relative degree two and a high-bandwidth attitude command system for a model R-50 helicopter.     

Adaptive λ-tracking for a class of infinite-dimensional systems

For a class of high-gain stabilizable multivariable linear infinite-dimensional systems we present an adaptive control law which achieves approximate asymptotic tracking in the sense that the tracking error tends asymptotically to a ball centred at 0 and of arbitrary prescribed radius λ>0. This control strategy, called λ-tracking, combines proportional error feedback with a simple nonlinear adaptation of the feedback gain. It does not involve any parameter estimation algorithms, nor is it based on the internal model principle. The class of reference signals is W^1,∞, the Sobolev space of absolutely continuous functions which are bounded and have essentially bounded derivative. The control strategy is robust with respect to output measurement noise in W^1,∞ and bounded input disturbances. We apply our results to retarded systems and integrodifferential systems.     

Optimal selection of the forgetting matrix into an iterative learning control algorithm

A recursive optimal algorithm, based on minimizing the input error covariance matrix, is derived to generate the optimal forgetting matrix and the learning gain matrix of a P-type iterative learning control (ILC) for linear discrete-time varying systems with arbitrary relative degree. This note shows that a forgetting matrix is neither needed for boundedness of trajectories nor for output tracking. In particular, it is shown that, in the presence of random disturbances, the optimal forgetting matrix is zero for all learning iterations. In addition, the resultant optimal learning gain guarantees boundedness of trajectories as well as uniform output tracking in presence of measurement noise for arbitrary relative degree.