The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise

The optimal projection equations obtained in [2,3] for reduced-order, discrete-time state estimation are generalized to include the effects of state- and measurement-dependent noise to provide a model of parameter uncertainty. In contrast to the single matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order discrete-time estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by both an oblique projection and stochastic effects.     

The optimal projection equations for reduced-order state estimation

First-order necessary conditions for optimal, steady-state, reduced-order state estimation for a linear, time-invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form. In contrast to the lone matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains. This coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by "full-order" estimator design or full-order estimator design followed by estimator-reduction techniques. The results given here complement recently obtained results which characterize the optimal reduced-order model by means of a pair of coupled modified Lyapunov equations [7] and the optimal fixed-order dynamic compensator by means of a coupled system of two modified Riceati equations and two modified Lyapunov equations [6].     

The optimal reduced-order estimator for systems with singularmeasurement noise

The optimal reduced-order estimator is completely characterized by necessary conditions, resulting from the optimal projection equations. The solution consists of one Riccati equation and two Lyapunov equations coupled by two projections. Explicit expressions for all of the estimator parameters are given. The relation between the reduced-order singular estimator and the full-order optimal singular estimator (which is of reduced order itself) is investigated. It is shown that under certain conditions the optimal estimator is recovered from the reduced-order estimator     

Unified optimal projection equations for simultaneous reduced-order,robust modelling,estimation and control

An optimal design problem which unifies reduced-order modelling, estimation and control problems is stated. Necessary conditions for optimality are obtained in the form of a coupled system of modified Riccati and Lyapunov equations. The results permit treatment of several new problems, such as reduced-order dynamic compensation with partially known disturbances and unified reduced-order control and estimation. Upon appropriate specialization, results obtained previously for the individual problems of reduced-order modelling, estimation and control are recovered. An additional feature is the inclusion of parameter uncertainty bounds so that the necessary conditions for an auxiliary minimization problem serve as sufficient conditions for simultaneous robust, reduced-order modelling, estimation and control.     

Optimal observers for systems with colored noises

The problem of optimal full-order observers for continuous-time linear systems with colored process and measurement noises is considered. In such cases, optimal estimation of the state involves augmenting the system, thus a higher-order observer is required. The structure of a full-order observer is assumed and necessary conditions for the optimal observer are derived. The conditions are given for the general case where the intensity of the white-noise component of the measurement noise may be singular. The solution consists of a modified Riccati equation and a Lyapunov equation coupled by two projection matrices in the singular case and one projection matrix in the nonsingular case     

A periodic fixed-structure approach to multirate control

We develop an approach to designing reduced-order multirate controllers. A discrete-time model that accounts for the multirate timing sequence of measurements is presented and is shown to have periodically time-varying dynamics. Using discrete-time stability theory, the optimal projection approach to fixed-order (i.e. full- and reduced-order) dynamic compensation is generalized to obtain reduced-order periodic controllers that account for the multirate architecture. It is shown that the optimal reduced-order controller is characterized by means of a periodically time-varying system of equations consisting of coupled Riccati and Lyapunov equations. In addition, the multirate static output-feedback control problem is considered. For both problems, the design equations are presented in a concise, unified manner to facilitate their accessibility for developing numerical algorithms for practical applications     

Study of the discrete singularly perturbed linear-quadratic control problem by a bilinear transformation

This paper presents a new approach in the study of the linear quadratic control problem of singularly perturbed discrete systems. By applying a bilinear transformation, the algebraic discrete Riccati equation is converted into a continuous one, which can be solved by using the reduced-order recursive method already documented in the control literature. This method produces the reduced-order near-optimal solution up to an arbitrary order of accuracy and reduces the size of required computations. The method is very suitable for parallel programming. A real world example, an F-8 aircraft, demonstrates the efficiency of the proposed method.     

A new approach to robust guaranteed cost control for uncertain multimodeling systems

The guaranteed cost control problem for multimodeling systems with norm bounded uncertainty is investigated. The main contribution in this paper is that a new ?-independent controller is derived by solving the reduced-order slow and fast algebraic Riccati equations (AREs) whose dimension is smaller than the dimension of full-order multiparameter algebraic Riccati equation (MARE). It is shown that if these AREs have a positive definite stabilizing solution then the closed-loop system is quadratically stable and has the cost bound.     

Exact slow-fast decomposition of a class of non-linear singularly perturbed optimal control problems via invariant manifolds

We study a Hamilton-Jacobi partial differential equation, arising in an optimal control problem for an affine non-linear singularly perturbed system. This equation is solvable iff there exists a special invariant manifold of the corresponding Hamiltonian system. We obtain exact slow-fast decomposition of the Hamiltonian system and of the special invariant manifold into slow and fast components. We get sufficient conditions for the solvability of the Hamiltonian-Jacobi equation in terms of the reduced-order slow submanifold, or, in the hyperbolic case, in terms of a reduced-order slow Riccati equation. On the basis of this decomposition we construct asymptotic expansions of the optimal state-feedback, optimal trajectory and optimal open-loop control in powers of a small parameter.     

LQG control with an H∞ performance bound:a Riccati equation approach

An LQG (linear quadratic Gaussian) control-design problem involving a constraint on H^∞ disturbance attenuation is considered. The H^∞ performance constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on L₂ performance. In contrast to the pair of separated Riccati equations of standard LQG theory, the H^∞-constrained gains are given by a coupled system of three modified Riccati equations. The coupling illustrates the breakdown of the separation principle for the H^∞ -constrained problem. Both full- and reduced-order design problems are considered with an H^∞ attenuation constraint involving both state and control variables. An algorithm is developed for the full-order design problem and illustrative numerical results are given     

A note on observers for Lipschitz nonlinear systems

This note deals with the design of reduced-order observers for Lipschitz nonlinear systems. It shows that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. A design method of the reduced-order observer that is dependent on the solution of the Riccati equation is then presented and an example is given to illustrate effects of the design method.     

Near-optimal tracking control for discrete-time systems with delayed input

This paper considers the near-optimal tracking control problem for discrete-time systems with delayed input. Using a variable transformation, the system with delayed input is transformed into a non-delayed system, and the quadratic performance index of the optimal tracking control is transformed into a relevant format. The optimal tracking control law is constructed by the solution of a Riccati matrix equation and a Stein matrix equation. A reduced-order observer is constructed to solve the physically realizable problem of the feedforward compensator and a near-optimal tracking control is obtained. Simulation results demonstrate the effectiveness of the optimal tracking control law.     

Robust stability and performance via fixed-order dynamic compensation with guaranteed cost bounds

A feedback control-design problem involving structured plant parameter uncertainties is considered. Two robust control-design issues are addressed. The Robust Stability Problem involves deterministic bounded structured parameter variations, while the Robust Performance Problem includes, in addition, a quadratic performance criterion averaged over stochastic disturbances and maximized over the admissible parameter variations. The optimal projection approach to fixed-order, dynamic compensation is merged with the guaranteed cost control approach to robust stability and performance to obtain a theory of full- and reduced-order robust control design. The principle result is a sufficient condition for characterizing dynamic controllers of fixed dimension which are guaranteed to provide both robust stability and performance. The sufficient conditions involve a system of modified Riccati and Lyapunov equations coupled by an oblique projection and the uncertainty bounds. The full-order result involves a system of two modified Riccati equations and two modified Lyapunov equations coupled by the uncertainty bounds. The coupling illustrates the breakdown of the separation principle for LQG control with structured plant parameter variations. Supported in part by the Air Force Office of Scientific Research under Contract F49620-86-C-0002.     

组合导航系统滤波器截断误差抑制方法

组合导航系统作为重要的定位和姿态测量的技术手段,其基本设计思想是将GPS和SINS等导航设备输出的信息经过滤波器进行最优估计。但在采用Riccati方程更新协方差矩阵和计算Kalman增益过程中,截断误差随着迭代次数的增大而累积,破坏协方差矩阵的正定性和对称性,降低滤波器计算的数值稳定性,严重时导致组合系统故障发散。本文建立了Riccati方程一阶误差模型,从理论上分析截断误差对滤波器估计性能的影响,引入基于Bierman算法和Thorton算法的Kalman滤波器进行更新方法,解决了截断误差引起的滤波器数值稳定性的问题。通过强实时半物理仿真系统验证表明,相比于基于Kalman滤波器的系统,基于Bierman-Thorton算法的组合导航系统有更强的数值稳定性和较高的导航精度。     

线性系统的动态输出反馈最优扰动抑制

首先通过求解Riccati方程和Sylvester方程,推导出前馈反馈最优扰动抑制控制律．然后构建了能同时预估状态和扰动的降维观测器,解决了前馈控制的物理不可实现问题．进而结合降维观测器和前馈反馈最优控制律,提出了一种动态输出反馈扰动抑制控制器的设计算法．最后通过仿真实例表明本文提出的控制算法的可行性．     

非线性系统降维观测器设计

对Lipschitz非线性系统降维观测器进行了讨论.首先考虑了降维观测器存在之条件,然后给出了此条件下的降维观测器的具体设计方法,该设计方法基于与非线性函数的Lipschitz常数有关的代数Riccati不等式的求解.最后,针对一个具体实际控制模型的仿真,验证了文中所提出之方法的实用性.     

The recursive reduced-order numerical solution of the singularlyperturbed matrix differential Riccati equation

Under stability-observability conditions imposed on a singularly perturbed system, an efficient numerical method for solving the corresponding matrix differential Riccati equation is obtained in terms of the reduced-order problems. The order reduction is achieved via the use of the Chang transformation applied to the Hamiltonian matrix of a singularly perturbed linear-quadratic control problem. An efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the Chang transformation     

Linearized reduced-order filtering

A reduced-order version of extended Kalman filtering is presented in which both the filtering equation and the associated Riccati equation have been reduced in dimension to allow for real-time processing. The procedure for designing the reduced-order filter is similar to that for designing the extended Kalman filter, the same approximations being applied. One technique useful for limiting the computational burden in a linearized filter design problems is presented and illustrated by an example. The primary limitation of the result is that the nonlinearity must be in terms of the vector to be estimated     

无线通讯网络功率和流量的预联合控制

To mitigate the loop delay in distributed wireless networks, a predictive power and rate control scheme is proposed for the system model that also accounts for the congestion levels and input delay instead of state-delayed in a network. A measurement feedback control problem with input delay is formulated by minimizing the energy of the difference between the actual and the desired signal-to-interference-plus-noise ratio (SNR) levels, as well as the energy of the control sequence. To solve this problem, we present two Riccati equations for the control and the estimation for the time delay systems. A complete analytical optimal controller is obtained by using the separation principle and solving two Riccati equations, where one is backward equation for stochastic linear quadratic regulation and the other is the standard filtering Riccati equation. Simulation results illustrate the performance of the proposed power and the rate control scheme.     

Actuator amplitude saturation control for systems with exogenous disturbances

We have developed fixed-order (i.e. full- and reduced-order) controllers for systems with actuator amplitude constraints and exogenous bounded energy L 2 disturbances. The actuator amplitude saturation and disturbance rejection constraints were embedded within an optimization problem by constructing a Riccati equation whose solution guaranteed closed-loop global asymptotic stability in the face of sector-bounded input nonlinearities and non-expansivity (gain boundedness) of the input-output system energy. The efficacy of the proposed framework is demonstrated via a numerical example.