正倒向随机微分方程与一类线性二次随机最优控制问题

讨论一类正倒向随机微分方程解的存在唯一性及其对应的一类线性二次随机最优控制问题,利用单调性方法证明了一类特殊的正倒向随机微分方程解的存在唯一性定理,利用该结果研究一类耦合了一个倒向随机微分方程的线性随机控制系统广义最优指标随机控制问题,得到由正倒向随机微分方程的解所表示的唯一最优控制的显式表达式,并得到精确的线性反馈及其对应的Riccati方程.     

Numerical computation of cross‐coupled algebraic Riccati equations related to H2/H∞ control problem for singularly perturbed systems

In this paper, we present a numerical algorithm to the cross‐coupled algebraic Riccati equations(CARE) related to H₂/H_∞ control problems for singularly perturbed systems (SPS) by means of Newton's method. The resulting algorithm can be widely used to solve Nash game problems and robust control problems because the CARE is solvable even if the quadratic term has an indefinite sign. We prove that the resulting iterative algorithm has the property of the quadratic convergence. Using the solution of the CARE, we construct the high‐order approximate H₂/H_∞ controller. Copyright © 2004 John Wiley & Sons, Ltd.     

Robust performance control of uncertain nonlinear systems: A self-tuning approach

The problem of output regulation with a guaranteed H^∞ performance, besides robust stability, for a class of feedback linearizable nonlinear systems via self-tuning controllers is investigated. The H^∞ performance consists of the desired disturbance attenuation and internal finite L²-gain stability. We show that, under the perturbations of matched parametric uncertainties, the sufficient condition for the existence of a self-tuning controller reduces to the solvability of a single Hamilton-Jacobi-Isaacs inequality (or algebraic Riccati equation) which indicates that the design can be performed as if the system uncertainties were absent. Under certain situations, the sufficient condition is also necessary. Once a solution of this nonlinear differential inequality (or algebraic equation) is available, a desired self-tuning controller with gradient-type parameter estimator can easily be constructed. The present work falls into the category of singular nonlinear H^∞ control since the desired H^∞ performance does not require any penalty on control input variables. The results also provide an immediate application to H^∞ self-tuning model reference control in linear systems under full state measurement.     

一类带随机跳跃的完全耦合的线性二次随机控制问题

研究一类带随机跳跃的完全耦合的线性二次随机控制问题. 得到了最优控制的显式解, 并可以证明最优控制是唯一的. 引入了一类推广的黎卡提方程并讨论了其可解性. 利用这一类推广的黎卡提方程的解, 得到了上述带随机跳跃的最优控制问题的线性状态反馈调节器.     

带Poisson跳的线性二次随机微分博弈及其在鲁棒控制中的应用

研究了一类带Poisson跳扩散过程的线性二次随机微分博弈,包括非零和博弈的Nash均衡策略与零和博弈的鞍点均衡策略问题．利用微分博弈的最大值原理,得到Nash均衡策略的存在条件等价于两个交叉耦合的矩阵Riccati方程存在解,鞍点均衡策略的存在条件等价于一个矩阵Riccati方程存在解的结论,并给出了均衡策略的显式表达及最优性能泛函值．最后,将所得结果应用于现代鲁棒控制中的随机H₂/H_∞控制与随机H_∞控制问题,得到了鲁棒控制策略的存在条件及显式表达,并验证所得结果在金融市场投资组合优化问题中的应用．     

Fixed-order H∞ compensator design

A method is presented for the construction of fixed-order compensators to provide H_∞ norm constraint for linear control systems with exogenous disturbances. The method is based on the celebrated bounded-real lemma that predicates the H_∞ norm constraint via a Riccati inequality. The synthesis of fixed-order controllers whose dimensions are less than the order of a given plant, is demonstrated by a set of sufficient conditions along with a numerical algorithm.     

A first principles solution to the non-singular H∞ control problem

This paper presents an elementary solution to the non-singular H^∞ control problem. In this control problem, the underlying linear system satisfies a set of assumptions which ensures that the solution can be obtained by solving just two algebraic Riccati equations of the game type. This leads to the central solution to the H^∞ control problem. The solution presented in this paper uses only elementary ideas beginning with the Bounded Real Lemma.     

H∞ estimation for uncertain systems

This paper deals with the problem of H_∞ estimation for linear systems with a certain type of time-varying norm-bounded parameter uncertainty in both the state and output matrices. We address the problem of designing an asymptotically stable estimator that guarantees a prescribed level of H_∞ noise attenuation for all admissible parameter uncertainties. Both an interpolation theory approach and a Riccati equation approach are proposed to solve the estimation problem, with each method having its own advantages. The first approach seems more numerically attractive whilst the second one provides a simple structure for the estimator with its solution given in terms of two algebraic Riccati equations and a parameterization of a class of suitable H_∞ estimators. The Riccati equation approach also pinpoints the ‘worst-case’ uncertainty.     

H2/H∞ filtering theory and an aerospace application

In many filtering problems of practical interest, some of the noise signals satisfy the assumptions of H₂ (Kalman-Bucy) filtering, while others can be more accurately modelled as signals with bounded variance (hence more amenable to an H_∞ filtering approach). These problems may be addressed by considering a mixed H₂/H_∞ filtering problem. In this paper we present a novel theory which solves the mixed problem exactly and in a computationally efficient way. The applicability of the theory is illustrated by designing a filter to estimate the states of an aircraft flying through a downburst.     

Min-max certainty equivalence principle and differential games

This paper presents a version of the certainty equivalence principle, usable for nonlinear, variable end-time, partial observation zero-sum differential games, which states that under the unicity of the solution to the auxiliary problem, optimal controllers can be derived from the solution of the related perfect observation game. An example is provided where in one region, the new extended result holds, giving an optimal control, and in another region, the unicity condition if is not met, leading indeed to a non-certainty equivalent optimal controller.     

Decentralized linear–quadratic–Gaussian control of networked control systems with asymmetric information

The decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain.     

A non-recursive method for solving the general discrete-time riccati equations related to the H∞ control problem

In this paper we propose a nonrecursive method for solving the general discrete-time algebraic Riccati equation related to the H_∞ control problem (H_∞-DARE). We have achieved this by casting the problem of solving a given H_∞-DARE to the problem of solving an auxiliary continuous-time algebraic Riccati equation associated with the H_∞ control problem (H_∞-CARE) for which the well known nonrecursive methods of solving are available. The advantages of our approach are: it reduces the computation involved in the recursive algorithms while giving much more accurate solutions, and it readily provides the properties of the general H_∞-DARE.     

A transfer matrix framework approach to the synthesis of H∞ controllers

This paper presents a transfer matrix framework approach to the now well-known Riccati solution to the nonsingular H^∞ control problem. The approach taken is a frequency-domain one based on the concepts of coprime factorization and J-lossless transfer matrices that lead to the development of relatively simple method for obtaining the state-space solutions. Using two dual pairs of coupled coprime factorizations of certain chain scattering-matrices which describe the problem, we derive an identity, analogous to the Bezout identity for all stabilizing controllers, to generate all proper real-rational H^∞ controllers. We also point out the connections between algebraic Riccati solutions and J-lossless matrices.