Equivalent optimal compensation problem in the delta domain for systems with white stochastic parameters

The equivalent representation in the delta domain of the digital optimal compensation problem is provided and computed in this article. This problem concerns finding digital optimal full and reduced-order output feedback controllers for linear time-varying and time-invariant systems with white stochastic parameters. It can subsequently be solved in the delta domain using the strengthened optimal projection equations that we recently formulated in this domain as well. If the sampling rate becomes high, stating and solving the problem in the delta domain becomes necessary because the conventional discrete-time problem formulation and solution become ill-conditioned. In this article, by means of several numerical examples and compensator implementations, we demonstrate this phenomenon. To compute and quantify the improved performance when the sampling rate becomes high, a new delta-domain algorithm is developed. This algorithm computes the performance of arbitrary digital compensators for linear systems with white stochastic parameters. The principle application concerns nonconservative robust digital perturbation feedback control of nonlinear systems with high sampling rates.     

Compensatability and optimal compensation of systems with white parameters in the delta domain

Using the delta operator, the strengthened discrete-time optimal projection equations for optimal reduced-order compensation of systems with white stochastic parameters are formulated in the delta domain. The delta domain unifies discrete time and continuous time. Moreover, when formulated in this domain, the efficiency and numerical conditioning of algorithms improves when the sampling rate is high. Exploiting the unification, important theoretical results, algorithms and compensatability tests concerning finite and infinite horizon optimal compensation of systems with white stochastic parameters are carried over from discrete time to continuous time. Among others, we consider the finite-horizon time-varying compensation problem for systems with white stochastic parameters and the property mean-square compensatability (ms-compensatability) that determines whether a system with white stochastic parameters can be stabilised by means of a compensator. In continuous time, both of these appear to be new. This also holds for the associated numerical algorithms and tests to verify ms-compensatability. They are illustrated with three numerical examples that reveal several interesting theoretical and numerical issues. A fourth example illustrates the improvement of both the efficiency and numerical conditioning of the algorithms. This is of vital practical importance for digital control system design when the sampling rate is high.     

Linear quadratic regulation for discrete-time systems with state delays and multiplicative noise

In this paper, the linear quadratic regulation problem for discrete-time systems with state delays and multiplicative noise is considered. The necessary and sufficient condition for the problem admitting a unique solution is given. Under this condition, the optimal feedback control and the optimal cost are presented via a set of coupled difference equations. Our approach is based on the maximum principle. The key technique is to establish relations between the costate and the state.     

On a general optimal algorithm for multirate output feedbackcontrollers for linear stochastic periodic systems

A modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed. By combining the discrete-time linear quadratic regulation (LQR) control problem and the discrete-time stochastic linear quadratic regulation (SLQR) control problem to obtain an extended linear quadratic regulation (ELQR) control problem, one derives a general optimal algorithm to balance the advantages of the optimal transient response of the LQR control problem and the optimal steady-state regulation of the SLQR control problem. In general, the solution of this algorithm is obtained by solving a set of coupled matrix equations. Special cases for which the coupled matrix equations can be reduced to a discrete-time algebraic Riccati equation are discussed. A reducable case is the optimal algorithm derived by H.M. Al-Rahmani and G.F. Franklin (1990), where the system has complete state information and the discrete-time quadratic performance index is transformed from a continuous-time one     

Polynomial approach to quadratic tracking in discrete linear systems

A new solution to the quadratic optimal tracking problem in linear multivariable discrete-time systems is presented. The approach is based on polynomial input-output models. The optimal control law consists of a plant output feedback and a reference feedforward. It is obtained by performing spectral factorization and then solving two matrix polynomial equations. The design procedure is simple and well suited for systems with inaccessible states.     

Discrete-time optimal control for stochastic nonlinear polynomial systems

This paper presents a solution to the discrete-time optimal control problem for stochastic nonlinear polynomial systems over linear observations and a quadratic criterion. The solution is obtained in two steps: the optimal control algorithm is developed for nonlinear polynomial systems by considering complete information when generating a control law. Then, the state estimate equations for discrete-time stochastic nonlinear polynomial system over linear observations are employed. The closed-form solution is finally obtained substituting the state estimates into the obtained control law. The designed optimal control algorithm can be applied to both distributed and lumped systems. To show effectiveness of the proposed controller, an illustrative example is presented for a second degree polynomial system. The obtained results are compared to the optimal control for the linearized system.     

具有乘性噪声的线性离散时间随机控制系统综述

具有乘性噪声的随机不确定系统的控制问题有着广泛的应用背景. 本文概述了具有乘性噪声的线性离散时间随机系统的稳定性分析、均方镇定、最优控制以及最优估计问题和相关结论. 同时, 本文研究了具有状态与控制乘性噪声的线性多变量离散时间系统的均方镇定和最优控制问题, 分析了这两个问题之间的联系, 并讨论了最优状态反馈控制器的设计算法.     

Block pulse function approach for the identification of Hammerstein model non-linear systems

Two recursive algorithms based on block pulse functions are presented for identifying continuous Hammerstein models of non-linear systems with (i) a state space model and (ii) an input–output model. Since the continuous non-linear systems are transformed approximately into the corresponding difference equations via block pulse functions, these recursive estimation algorithms can easily be obtained using a derivation similar to that of the discrete-time models expressed by difference equations. Both algorithms derived here are simple and straightforward, and can easily be implemented on-line. As discussed in this paper, these algorithms can also be extended to the identification of certain continuous non-linear systems with a feedback loop or with time delays. The illustrative examples show that these recursive algorithms give satisfactory results for the identification problems of certain continuous non-linear systems.     

Parallel Algorithms for LQ Optimal Control of Discrete-Time Periodic Linear Systems

This paper analyzes the performance of two parallel algorithms for solving the linear-quadratic optimal control problem arising in discrete-time periodic linear systems. The algorithms perform a sequence of orthogonal reordering transformations on formal matrix products associated with the periodic linear system and then employ the so-called matrix disk function to solve the resulting discrete-time periodic algebraic Riccati equations needed to determine the optimal periodic feedback. We parallelize these solvers using two different approaches, based on a coarse-grain and a medium-grain distribution of the computational load. The experimental results report the high performance and scalability of the parallel algorithms on a Beowulf cluster.     

Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems

In this paper we consider the stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The performance criterion is assumed to be formed by a linear combination of a quadratic part and a linear part in the state and control variables. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a necessary and sufficient condition under which the problem is well posed and a state feedback solution can be derived from a set of coupled generalized Riccati difference equations interconnected with a set of coupled linear recursive equations. For the case in which the quadratic-term matrices are non-negative, this necessary and sufficient condition can be written in a more explicit way. The results are applied to a problem of portfolio optimization.     

Constrained Optimal Control of Hybrid Systems With a Linear Performance Index

We consider the constrained finite and infinite time optimal control problem for the class of discrete-time linear hybrid systems. When a linear performance index is used the finite and infinite time optimal solution is a piecewise affine state feedback control law. In this paper, we present algorithms that compute the optimal solution to both problems in a computationally efficient manner and with guaranteed convergence and error bounds. Both algorithms combine a dynamic programming exploration strategy with multiparametric linear programming and basic polyhedral manipulation     

Minimal and non-minimal optimal fixed-order compensators for time-varying discrete-time systems

Using the minimality property of finite-horizon time-varying compensators, established in this paper, and the Moore-Penrose pseudo-inverse instead of the standard inverse, strengthened discrete-time optimal projection equations (SDOPE) and associated boundary conditions are derived for finite-horizon fixed-order LQG compensation. They constitute a two-point boundary value problem explicit in the LQG problem parameters which is equivalent to first-order necessary optimality conditions and which is suitable for numerical solution. The minimality property implies that minimal compensators have time-varying dimensions and that the finite-horizon optimal full-order compensator is not minimal. The use of the Moore-Penrose pseudo-inverse is further exploited to reveal that the optimal projection approach can be generalised, but only to partially include non-minimal compensators. Furthermore, the structure of the space of optimal compensators with arbitrary dimensions is revealed to a large extent. Max-min compensator dimensions are introduced and their significance in solving numerically the two-point boundary value problem is explained. The numerical solution is presented in a recently published companion paper, which relies on the results of this paper.     

Computation of optimal output feedback gains for linear multivariable systems

For linear systems with quadratic performance indices, it is shown that the optimal output feedback gains can be computed using gradient techniques. Unlike previous algorithms, this approach avoids the solution of nonlinear matrix equations while appearing to ensure convergence. Computational results for a fourth-order system are presented.     

一类离散时间非齐次马尔可夫跳跃系统最优控制

本文研究了一类离散时间非齐次马尔可夫跳跃线性系统的线型二次高斯(linear quadratic Gaussian,LQG)问题,其中系统模态转移概率矩阵随时间随机变化,其变化特性由一高阶马尔可夫链描述.对于该系统的LQG问题,文中首先给出了线性最优滤波器,得到最优状态估计;其次,验证分离定理成立,并利用利用动态规划方法设计了系统最优控制器;最后,数值仿真结果验证了所设计控制器的有效性.     

Descent algorithms for optimal periodic output feedback control

Periodic output feedback is investigated in the context of linear-quadratic regulation for finite-dimensional time-invariant linear systems. Discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control. The optimal solution is obtained in two steps by separating the continuous-time from the discrete-time structure. First, the optimal pole placement problem under periodic output feedback is solved explicitly under the assumption that the behavior at the sample times has been specified in terms of a gain matrix G. Then the minimum value, which depends on G, is substituted into the overall objective. This results in a finite-dimensional nonlinear programming problem over all admissible gain matrices G. The solution defines the optimal periodic output feedback control via the formulas of the optimal pole placement problem. A steepest descent and a direct iterative method for solving this problem are formulated and compared. Numerical examples show that the performance using periodic output feedback is almost equivalent to that using optimal continuous-state feedback     

Discrete-time controller for stochastic nonlinear polynomial systems with Poisson noises

This paper presents a solution to the optimal control problem for discrete-time stochastic nonlinear polynomial systems confused with white Poisson noises subject to a quadratic criterion. The solution is obtained in the following way: a nonlinear optimal controller is first developed for polynomial systems, considering the state vector completely available for control design. Then, based on the solution of the state estimation problem for polynomial systems with white Poisson noises, the state estimate vector is used in the control law to obtain a closed-form solution. Performance of this controller is compared to that of the controller employing the extended Kalman filter and the linear-quadratic regulator and the controller designed for polynomial systems confused with white Gaussian noises.     

Parameter identification and control of linear discrete-time systems

The parameter-adaptive self-organizing control of linear discrete-time systems is considered by designing dynamic feedback controllers which depend on the estimates of the parameters provided by an appropriate identifier. Two stochastic approximation algorithms for consistent identification of feedback systems are investigated and a condition of identifiability is presented. Then two controllers, one based on "overall" and another based on "per-interval" optimization, both depending on the output of the identifier, are discussed and their evaluations relative to the optimal are compared in illustrative examples.     

离散双输出反馈控制的最优协调性合成设计

本文以离散二次型最优控制理论为基础，研究了线性时不变系统在双输出反馈控制情况下控制作用的协调性合成设计问题，构造了一个数值迭代算法来求解相应的最侉 控制规律。通过产例计算和对比分析，既证明了该方法的收敛性和有效性又说明了这种控制问题的一些本质特征。     

Output feedback disk pole assignment for systems with positive realuncertainty

In this paper, the problem of pole assignment in a disk by output feedback for continuous or discrete-time uncertain systems is addressed. A necessary and sufficient condition for quadratic d stabilizability by output feedback is presented. This condition is expressed in terms of two parameter-dependent Riccati equations whose solutions satisfy two extra conditions. An output d stabilization algorithm is derived and a controller formula given. Some related problems are also discussed     

Model matching for discrete-time periodic systems with time-varying relative degree and order

This paper extends the well-known solution for the linear time invariant model matching problem to discrete-time periodic systems with time-varying relative degree and order. It is shown that a key step to the design of a periodic output feedback controller is to compute the stable inverse of the periodic system. Using input–output equations, this problem is solved and model matching is achieved with system internal stability.