Recursive Stochastic H2/H∞ Control Problem for Delay Systems Involving Continuous and Impulse Controls

In this paper, we discuss the recursive stochastic H₂/H_∞ control problem of delay systems with random coefficients involving both continuous and impulse controls. By virtue of a new type of forward backward stochastic differential equations, a necessary and sufficient condition for the existence of a unique solution to the control problem under consideration is derived. The existence and uniqueness of the forward backward stochastic differential equations are also be proved.     

A new static output feedback approach to the suboptimal mixed H2/H∞ problem

In this paper we propose a new approach to solve the static output feedback suboptimal mixed H₂/H_∞ control problem using a state fixed‐structure feedback design. We formulate the static output feedback problem as a constrained static state feedback problem and obtain three coupled design equations: one Riccati equation, one Lyapunov equation, and a gain equation. We will prove the equivalence of the proposed solution to the existing solution. A very simple iterative algorithm is then presented to solve the design equations for the stabilizing output feedback gain that minimizes an upper bound of H₂ norm while satisfying the H_∞ disturbance attenuation requirement. A unique feature of the new approach is that it admits the Kalman gain as an initial stabilizing gain to start the above iterative solution procedure, which is computationally attractive and advantageous compared to the direct approach, as the latter has to deal with the difficult algorithm initialization problem. Some illustrative numerical examples are given to demonstrate the effectiveness of the approach. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Application Of H2/H∞ Technique To Aircraft Landing Control

The paper describes the automatic control of an aircraft in the longitudinal plane during landing, taking into account sensor errors and disturbances. Aircraft auto‐landing is achieved by combining H₂ and H_∞ control techniques, thereby obtaining a robust H₂/H_∞ controller. It provides good precision tracking and robust stability with respect to the uncertainties caused by different disturbances and noise‐type signals. The weights of the H₂ and H_∞ control techniques within the robust H₂/H_∞ controller are adjusted so that the aircraft accurately tracks the desired trajectory during the two main stages of the landing process. The theoretical results are validated by numerical simulations for the landing of a Boeing; the results are very good (Federal Aviation Administration accuracy requirements for Category III are met) and prove the robustness of the new auto‐landing system even in the presence of disturbances and sensor errors.     

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.     

Mixed H2/H∞ filtering

In paper we consider the problem of finding a filter or estimator that minimizes a mixed H₂/H_∞ filtering cost on the transfer matrix from a given noise input to the filtering error subject to an H_∞ constraint on the transfer matrix from a second noise input to the filtering error. This problem can be interpreted and motivated in many different ways; for instance, as a problem of optimal filtering in the presence of noise with fixed and known spectral characteristics subject to a bound on the filtering error due to a second noise source whose spectral characteristics are unknown. It is shown that one can come arbitrarily close to the optimal mixed H₂/H_∞ filtering cost using a standard Kalman-Luenberger estimator. Moreover, the problem of finding suitable Kalman-Luenberger estimator gains can be converted into a convex optimization problem involving affine symmetric matrix inequalities.     

H2 and H∞ almost disturbance decoupling problem with internal stability

For both continuous-time and discrete-time systems, we establish the necessary and sufficient conditions under which an H₂ or H_∞ almost disturbance decoupling problem with internal stability is solvable by proper controllers and/or strictly proper controllers.     

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.     

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.     

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.     

Robust H2/H∞ control for time‐delay systems with polytopic uncertainty

In this paper, new separated H_∞ and H₂ performance criteria are derived for a class of time‐delay systems. When used in robust performance analysis and synthesis for real polytopic uncertainty and in multiobjective controller synthesis, they can partially rule out the technical restriction of using a single Lyapunov function, and therefore, lead to potentially less conservative linear matrix inequality (LMI) characterizations. Based on the criteria, robust multiobjective H₂/H_∞ controller is designed for time‐delay systems with polytopic uncertainty. All the conditions are given in terms of LMIs. Numerical examples are given to illustrate the proposed method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞ preview tracking in output feedback setting

The performance of preview control is investigated in terms of H^∞‐criterion. In the output feedback setting, an H^∞ preview control problem is discussed and the analytic solution is characterized by introducing matrix Riccati equations. The strength and the limitation of the H^∞ preview performance are illustrated with numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.     

A zero compensation approach to singular H2 and H∞ problems

In this work we analyse singular H² and H^∞ problems for which the usual Riccati equations become ill-posed owing to the existence of plant zeros at infinity. We adopt a two-step approach to the analysis. First we replace the usual Riccati equations with two generalized eigenproblems; these problems are always well-posed. Next we extract those structural elements which pertain to the troublesome plant zeros. We do this by introducing pre-compensators which cancel the offending zeros. In so doing, we temporarily relax the controller properness constraint that is traditionally imposed in H² and H^∞ problems by allowing pole-zero cancellations between the plant and controller at infinity. Since no significant added complexity of analysis results, we also treat the case of singularity due to finite jω-axis plant zeros by relaxing the internal stability requirement and allowing finite jω-axis pole-zero cancellations. The resultant theory allows us to specify necessary and sufficient conditions for the existence of solutions to singular H² and H^∞ problems. The existence conditions and the resultant control laws are expressed directly in terms of the eigenvalues and eigenvectors of two Hamiltonian matrices associated with the problem. The theory also gives some insight into the character of the subset of all proper, internally stabilizing solutions, including whether this set is nonempty. An example is included.     

ADAPTIVE FUZZY‐BASED MIXED H2/H∞ TRACKING CONTROL DESIGN IN UNCERTAIN ROBOTIC SYSTEMS

In this study, an adaptive fuzzy‐based mixed H₂/H_∞ tracking control design is developed in robotic systems under unknown or uncertain plant parameters and external disturbances. The mixed H₂/H_∞ control design has the advantage of both H₂ optimal control performance and H_∞ robust control performance and the fuzzy adaptive control scheme is used to compensate for the plant uncertainties. By virtue of the skew‐symmetric property in the robotic systems and adequate choice of state variable transformation, sufficient conditions are developed for the adaptive fuzzy‐based mixed H₂/H_∞ tracking control problems in terms of a pair of coupled algebraic equations instead of a pair of coupled differential equations. The proposed methods are simple and the coupled algebraic equations can be solved analytically. Simulation results indicate that the desired performance of the proposed adaptive fuzzy‐based mixed H₂/H_∞ tracking control schemes for the uncertain robotic systems can be achieved.     

Mixed H∞/H2 design of digital phase‐locked loops with polytopic‐type uncertainties

A robust H_∞ control method is applied to the design of loop filters for digital phase locked loop carrier phase tracking. The proposed method successfully copes with large S‐curve slope uncertainty and with a significant decision delay in the closed‐loop that may stem from the decoder and/or the equalizer there. The design problem is transformed into a state‐feedback control problem where phase and gain‐margins should be guaranteed in spite of the uncertainty. Of all the loop filters that achieve the required margins the one that minimizes an upper‐bound on the effect of the phase and the measurement noise signals is derived. Copyright © 2002 John Wiley & Sons, Ltd.     

MIXED H2/H∞ ADAPTIVE TRACKING CONTROL DESIGN FOR UNCERTAIN CONSTRAINED ROBOTS

In this study, a PID‐type controller incorporating an adaptive learning scheme for the mixed H₂/H_∞ tracking performance is developed for constrained robots under unknown or uncertain plant parameters and external disturbances. The mixed H₂/H_∞ control design has the advantage of both H₂ optimal control performance and H_∞ robust control performance and the adaptive control scheme is used to compensate the plant uncertainties. By virtue of the skew‐symmetric property of the constrained robotic systems and an adequate choice of state variable transformation, sufficient conditions are developed for the adaptive mixed H₂/H_∞ tracking control problems in terms of a pair of coupled algebraic equations instead of a pair of coupled nonlinear differential equations. The proposed methods are simple and the coupled algebraic equations can be solved analytically. Simulation results indicate that the desired performance of the proposed adaptive mixed H₂/H_∞ tracking control schemes for the uncertain constrained robotic systems can be achieved.     

H∞ Control for Continuous‐Time Mean‐Field Stochastic Systems

An H_∞‐type control is considered for mean‐field stochastic differential equations (SDEs) in this paper. A stochastic bounded real lemma (SBRL) is proved for mean‐field stochastic continuous‐time systems with state‐ and disturbance‐dependent noise. Based on SBRL, a sufficient condition is given for the existence of a stabilizing H_∞ controller in terms of coupled nonlinear matrix inequalities.     

H∞ generalized minimum sensitivity control design

This paper presents a two-stage approach to the H_∞ generalized sensitivity minimization problem (H_∞-GSM) for SISO, continuous-time plants. Besides some possible advantages in the control law evaluation with respect to alternative polynomial methods, the proposed approach provides a direct link with the solution of an underlying generalized minimum variance (H₂-GMV) problem and allows one to identify the class of the joint H₂/H_∞-GMV (equalizing H₂-GMV) optimal controllers.