Fixed-time attitude tracking control for spacecraft based on a fixed-time extended state observer

This paper deals with the problem of fixed-time attitude tracking control for spacecraft subject to model uncertainties and external disturbances.Firstly,by usi...     

Practical time-varying formation tracking for high-order nonlinear multi-agent systems based on the distributed extended state observer

Practical time-varying formation tracking analysis and design problems for high-order nonlinear multi-agent systems are investigated, where the time-varying formation tracking error is controlled within an arbitrarily small bound. The states of followers form a predefined time-varying formation while tracking the state of the leader with unknown control input. Besides, the dynamics of each agent has heterogeneous nonlinearity and disturbance. First, a distributed extended state observer is constructed to estimate the follower's nonlinearity and disturbance, and the leader's unknown control input simultaneously. A protocol based on the distributed extended state observer is proposed. Second, sufficient conditions for the multi-agent systems to achieve the practical time-varying formation tracking under the protocol are presented by using the Lyapunov stability theory. Third, an approach is derived to design the proposed protocol by solving a linear matrix inequality. Finally, numerical simulation results are given to illustrate the effectiveness of the theoretical results.     

A strong tracking extended Kalman observer for nonlineardiscrete-time systems

The authors show how the extended Kalman filter, used as an observer for nonlinear discrete-time systems or extended Kalman observer (EKO), becomes a useful state estimator when the arbitrary matrices, namely R_k and Q_k, are adequately chosen. As a first step, we use the linearization technique given by Boutayed et al. (1997), which consists of introducing unknown diagonal matrices to take the approximation errors into account. It is shown that the decreasing Lyapunov function condition leads to a linear matrix inequality (LMI) problem, which points out the connection between a good convergence behavior of the EKO and the instrumental matrices R_k and Q _k. In order to satisfy the obtained LMI, a particular design of Q_k is given. High performances of the proposed technique are shown through numerical examples under the worst conditions     

A fast convergent extended Kalman observer for nonlinear discrete-time systems

A new extended Kalman observer for nonlinear discrete-time systems is derived. As always, stability and convergence rate depend directly on the matrices Q k and R k and the initial state estimation error, the new observer has made improvement on these two aspects. A new criterion for the design of matrices Q k and R k enables the observer to enlarge the convergent domain, an off-line-trained neural network can significantly reduce the initial state estimation error without increasing on-line computation burden. The integration of the techniques results in a stable and fast convergent observer. The observer performance is demonstrated with the estimation of flux and angular speed of an induction motor.     

Transfer characteristics of extremely fast state feedback and observer systems

Extremely fast state feedback and observer systems involve differentiating operation on the inputs. The conflicting dependences of the transient and disturbance behaviour on the chosen eigenvalues are investigated for linear time-invariant single-input-single-output systems in the limit case of eigenvalues having infinitely large negative real parts. The limiting procedure of the infinitely fast state feedback transient yields precisely the inverse controllability map of the system. A dual relation exists for observers and the observability map     

Control design based on a linear state function observer

A control design method based on a linear state function observer is proposed. The method is a semi-inverse design procedure in that the control law is not designed before the observer system, but is a result of the observer design. However, the observer design is not completely independent of the control design, but seeks to yield a feedback signal that is close to a prescribed control law. First, the observer design problem is considered as the reconstruction of a linear function of the state vector. The linear state function to be reconstructed is the given control law. Then, based on the derivation for linear state function observers, the observer design is formulated as a parameter optimization problem. The optimization objective is to generate a matrix that is close to the given feedback gain matrix. Based on that matrix, the form of the observer and a new control law can be determined. The semi-inverse design procedure can yield a reduced-order observer with dimension considerably smaller than that of the system. Two numerical examples are used to demonstrate the proposed design procedure.     

On the convergence of an extended state observer for nonlinear systems with uncertainty

The extended state observer first proposed by Jingqing Han in [J.Q. Han, A class of extended state observers for uncertain systems, Control Decis. 10 (1) (1995) 85-88 (in Chinese)] is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering. Unfortunately, there is no rigorous proof of convergence to date. In this paper, we attempt to tackle this long unsolved extraordinary problem. The main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance, for which the effect of total disturbance error is eliminated by the high-gain.     

Stabilization of uncertain linear systems via observer based state feedback

A unified approach to the stabilization problem using reduced order observer based state feedback is presented for uncertain discrete and continuous systems. The main result states that, for the system class specified, the insensitive observer of Breinl and Müller (1982) may be used together with any state feedback gain matrix designed for quadratic stabilization to asymptotically stabilize the uncertain system via reduced order observer based state feedback. Intermediary results give sufficient conditions for the asymptotic stability of time-varying one-way connected subsystems.     

Optimal filtering for linear state delay systems

In this note, the optimal filtering problem for linear systems with state delay over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. As a result, the optimal estimate equation similar to the traditional Kalman-Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman-Bucy filter. The resulting system of equations for determining the error variance consists of a set of equations, whose number is specified by the ratio between the current filtering horizon and the delay value in the state equation and increases as the filtering horizon tends to infinity. In the example, performance of the designed optimal filter for linear systems with state delay is verified against the best Kalman-Bucy filter available for linear systems without delays and two versions of the extended Kalman-Bucy filter for time-delay systems.     

Optimal preview-based stable-inversion for output tracking of nonminimum-phase linear systems

In this article, a new approach to output tracking of nonminimum-phase systems is proposed. The proposed technique extends the preview-based stable-inversion method to optimally utilize finite-preview (in time) of the future desired output trajectory to find the feedforward input (called the inverse input) for achieving precision output tracking for nonminimum-phase systems. It has been shown that having a large enough preview time is critical to ensure the precision in the preview-based output tracking. The available preview time, however, can be limited due to the physical constraints, and more generally, the associated cost and/or hardware limits. Therefore, we propose obtaining the optimal preview-based inverse input by minimizing, within the preview time window, the predicted tracking error (under the preview-based inverse input) relative to the input energy. A simulation study on a piezoelectric actuator model is used to illustrate the proposed technique.     

Optimal state reconstruction algorithms for linear discrete-time systems

This paper deals with optimal time-invariant reconstruction of the state of a linear time-invariant discrete-time system from output measurements. The problem is analysed in two settings, depending on whether or not the present output measurement is available for the estimation of the present state. The results prove complete separation of observer and controller design for the optimal dynamic output feedback control with respect to a quadratic cost.     

Optimal linear control systems with incomplete state measurements

Optimal control laws usually require the complete measurement of the plant state. However, in practice one often has available only a small number of measurements. A procedure is developed that leads to a dynamic feedback control law which is a function of any given set of measurements. The resulting closed-loop system is optimal for all initial states of the system in the sense of minimizing a quadratic performance index. The order of the controller depends upon the observability properties of the plant. The development is extended to time-variable problems.     

Optimal simultaneous stabilization of linear single-input systems via linear state feedback control

Simultaneous stabilization of a collection of linear time-invariant systems is considered. The aim is to develop a practical method for the design of a single feedback controller such that all the systems involved are stabilized with good transient responses. Hurwitz's necessary and sufficient conditions are used as the required set of constraints for simultaneous stability. For transient behaviour, we introduce the usual quadratic objective function as performance index for each system. Their sum is the objective function for the collection of systems and the problem is to minimize it by choosing a feedback gain vector subject to boundedness and the Hurwitz stability constraints. A computational technique is proposed for solving this problem. The results obtained give good transient behaviour. For illustration, two numerical examples are solved.     

A frequency domain approach to minimal-order observer design for several linear functions of the state

A design procedure for minimal-order observers which is applicable to linear, observable, time-invariant systems is developed based on certain polynomial matrices associated with the transfer functions of the plant. The approach is to select the observer dynamics so that the feedback control law is satisfied with a minimal-order system. The method is a frequency domain technique but is significantly different from earlier results. Necessary and sufficient conditions for construction of a minimal-order observer to observe several linear functions of the state are given.     

自适应窗宽的均值迁移图像跟踪算法

针对传统均值迁移跟踪算法中核函数窗宽固定使之无法满足图像中运动目标尺寸变化需要的问题，在分析目标特征零阶矩特点基础上，以其变化比作为观测量，以目标面积的变化比作为状态量，利用卡尔曼滤波器对未来帧中目标面积的变化进行预测进而获得同目标尺度变化相适应的核函数窗宽。该算法通过窗宽自适应变化，提高了跟踪精度，增强了跟踪稳定性，同时仍保证了跟踪的实时性。实验结果证明了该方法的有效性。     

Optimal control for linear systems with state equality constraints

This paper deals with the optimal control problem for linear systems with linear state equality constraints. For deterministic linear systems, first we find various existence conditions for constraining state feedback control and determine all constraining feedback gains, from which the optimal feedback gain is derived by reducing the dimension of the control input space. For systems with stochastic noises, it is shown that the same gain used for constraining the deterministic system also optimally constrains the expectation of states inside the constraint subspace and minimizes the expectation of the squared constraint error. We compare and discuss performance differences between unconstrained (using penalty method), projected, and constrained controllers for both deterministic and stochastic systems. Finally, numerical examples are used to demonstrate the performance difference of the three controllers.     

Linear extended state observer based sliding mode disturbance decoupling control for nonlinear multivariable systems with uncertainty

In this paper, the sliding mode dynamic disturbance decoupling tracking control method based on the linear extended state observer (LESO) is proposed for a class of square multivariable nonlinear uncertain system. The model plant contains the known linear dynamics, the unknown nonlinear dynamics and the internal and external disturbances, and the various input-output pairs are interacted. The system states are not available for measurement. An improved LESO is developed. The leading feature which is different from the typical ESO lies in that its extended state does not contain the known linear dynamics. The improved LESO can guarantee the error variables to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters. So this ball radius can be arbitrarily as small as desired by tuning design parameters. And we give a simple method by which the gain parameters of LESO can be computed easily. This estimation to the total disturbance of the original system is introduced into the sliding mode control design to complete disturbance rejection and decoupling. Rigorous stability analysis shows that the system output can track the desired signal closely. Finally, a class of mass–spring–damper system is taken to make the numerical simulation analysis to illustrate the effectiveness of the proposed control method.     

Optimal control of the delay linear systems with allowance for the terminal state constraints

Consideration was given to the linear problem of optimal control of one type of the delay systems where the delay appears in one equation of the system mathematical model. The terminal states of the system are bounded, the optimal control is realized by the discrete control actions obeying the geometrical constraints. Consideration was given to two types of solutions—program and positional. A dual method of calculation of the optimal programs was presented. Described was an algorithm of the optimal controller generating in real time the current values of the positional solution (optimal feedback). The results obtained were illustrated by the example of control of a system with the fourth-order delay.     

Algebraic separation in realizing a linear state feedback control law by means of an adaptive observer

Based on a novel adaptive observer, which does not require signal boundedness in its stability proof, an algebraic separation property of linear state feedback control and adaptive state observation is established. This means, whenever a linear, stabilizing state feedback control law is realized with the state replaced by the state estimate of the given stable adaptive observer, then the resulting nonlinear control system is also globally asymptotically Lyapunov stable with respect to the initial state and parameter observation error of the adaptive observer. In particular, no assumptions on the system dynamics nor on the speed of the adaptation are made.