Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Output Dynamic Observer-Based Control Design of Uncertain Neutral Systems

In this paper, the robust H _∞ control problem of output dynamic observer-based control for a class of uncertain neutral systems is considered. The linear matrix inequality optimization approach is used to design the new H _∞ output dynamic controls. Three classes of H _∞ observer-based controls are proposed. The minimal H _∞-norm bound and the maximal perturbed bound are given. Based on the result of this paper, the constraint of matrix equality is not necessary for designing the H _∞ observer-based controls. A numerical example is given to stress the usefulness of the proposed results. Communicated by C. T. Leondes The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 94-2213-E-507-002.     

A solution to anisotropic suboptimal filtering problem by convex optimization

This paper considers a robust filtering problem for a linear discrete time invariant system with measured and estimated outputs. The system is exposed to random disturbances with imprecisely known distributions generated by an unknown stable shaping filter from the Gaussian white noise. The stochastic uncertainty of the input disturbance is measured by the mean anisotropy functional. The estimation error is quantified by the anisotropic norm which is a stochastic analogue of the H _∞ norm. A sufficient condition for an estimator to exist and ensure that the error is less than a given threshold value is derived in form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities. The suboptimal problem setting results to a set of the estimators ensuring the anisotropic norm of the error to be strictly bounded thereby providing some additional degree of freedom to impose some additional constraints on the estimator performance specification.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Observer-Based Control for a Class of Uncertain Neutral Time-Delay Systems via LMI Optimization Approach

In this paper, the H_∞ observer-based control for a class of uncertain neutral time-delay systems is considered. The linear matrix inequality (LMI) optimization approach is used to design the H_∞ robust control with disturbance attenuation. Two classes of observer-based controls are proposed and their H_∞-norm bounds are given. The control and observer gains are given from the LMI feasible solutions. A numerical example is given to illustrate the results. The research reported here was supported by the National Science Council of Taiwan under Grant NSC 93-2213-E-214-020.     

LMI Approach to Robust Delay-Dependent Mixed H 2/H ∞ Controller of Uncertain Neutral Systems with Discrete and Distributed Time-Varying Delays

The design of a robust mixed H ₂/H _∞ controller for a class of uncertain neutral systems with discrete, distributed, and input time-varying delays is considered. More precisely, the proposed robust mixed H ₂/H _∞ controller minimizes an upper bound of the H ₂ performance measure, while guaranteeing an H _∞ norm bound constraint. Based on the Lyapunov-Krasovskii functional theory, a delay-dependent criterion is derived for the existence of a desired mixed H ₂/H _∞ controller, which can be constructed easily via feasible linear matrix inequalities (LMIs). Furthermore, a convex optimization problem satisfying some LMI constraints is formulated to obtain a suboptimal robust mixed H ₂/H _∞ controller achieving the minimization of an upper bound of the closed-loop H ₂ performance measure. Finally, a numerical example is illustrated to show the usefulness of the obtained design method.The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 94-2213-E-507-002.     

Robust H ∞ Filtering for Uncertain Time-Delay Systems: Matrix Inequality Approach

In this paper, a new robust H _∞ filtering problem for uncertain time-delay systems is considered. Based on the Lyapunov method, a design criterion of the robust H _∞ filter, in which the filtering process remains asymptotically stable for all admissible uncertainties and the transfer function from the disturbance inputs to error state outputs satisfies the prespecified H _∞ norm upper bound constraint, is derived in terms of matrix inequalities. The inequalities can be solved easily by efficient convex optimization algorithms. A numerical example is included to illustrate the validity of the proposed design approach.     

Parameter-dependent <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control for Time-varying Delay Polytopic Systems

This paper addresses the robust stabilization and H _∞ control problem for a class of linear polytopic systems with continuously distributed delays. The control objective is to design a robust H _∞ controller that satisfies some exponential stability constraints on the closed-loop poles. Using improved parameter-dependent Lyapunov Krasovskii functionals, new delay-dependent conditions for the robust H _∞ control are established in terms of linear matrix inequalities.     

Delay-Dependent Nonfragile <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Observer-Based Control for Neutral Systems with Time Delays in the State and Control Input

The problem of the delay-dependent nonfragile H _∞ observer-based control for a class of neutral systems with time delays is investigated. The additive gain variations under consideration are contained in both the controller gain and the observer gain. Novel delay-dependent criteria are derived to guarantee the stability of the nonfragile H _∞ observer-based control system using the Lyapunov function approach combined with linear matrix inequalities (LMI). The controller and observer gains are given from the LMI feasible solutions. Based on the result of this paper, the constraint of matrix equality is not necessary for designing a nonfragile H _∞ observer-based control. The computer software Matlab can be applied to solve the proposed problems. Finally, a numerical example is given illustrating the design of the nonfragile H _∞ observer-based control. The research was supported by the National Science Council of Taiwan, ROC under Grant NSC 96-2221-E-507-003.     

Robust Chaotic Synchronization for a Class of Disturbed Nonlinear Systems with Multiple Equilibria

This study concerns with the robust H _∞ synchronization problem for a class of nonlinear feedback control systems, which are subject to a vector-valued periodic nonlinearity in the feedback path. Under such synchronization configuration, the master system is assumed to be subject to an energy bounded input disturbance, and the slave one is under control. Sufficient conditions for controller design are proposed in terms of linear matrix inequalities by respectively utilizing the output feedback control and the dynamic output control strategies, such that the master system robustly synchronizes the slave one with a guaranteed H _∞ performance. The derived methods can be applied to the robust H _∞ synchronization of many practical systems, and effectiveness of the obtained results are demonstrated through a concrete example of phase-locked loops (PLL).     

Robust H ∞ control for singular systems with time-varying uncertainties

In this paper, we investigate the problem of robust H _∞ control for singular systems with polytopic time-varying parameter uncertainties. By introducing the notion of generalized quadratic H _∞ performance, the relationship between the existence of a robust H _∞ dynamic state feedback controller and that of a robust H _∞ static state feedback controller is given. By using matrix inequalities, the existence conditions of robust H _∞ static state feedback and dynamic output feedback controllers are derived. Moreover, the design methods for such controllers are provided in terms of the solutions of matrix inequalities. An example is also presented to demonstrate the validity of the proposed methods. __________ Translated from Journal of Northeastern University (Natural Science), 2004, 25(2): 110–113     

Delay-Dependent Robust Stabilization and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control for Nonlinear Stochastic Systems with Markovian Jump Parameters and Interval Time-Varying Delays

This paper considers the problems of delay-dependent robust stabilization and H _∞ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays. Based on the Lyapunov method and introducing some appropriate free-weighting matrices, sufficient conditions for the solvability of above problems have been investigated in terms of linear matrix inequalities (LMIs). Furthermore, the desired state feedback controller has also been designed by solving these LMIs. Finally, a numerical example is provided to demonstrate the potential of the proposed techniques.     

Noncommutative Burkholder/Rosenthal inequalities II: Applications

We show norm estimates for the sum of independent random variables in noncommutative L _p -spaces for 1 < p < ∞, following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. As applications, we derive an equivalence for the p-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative L _p for 2 < p < ∞. The first author is partially supported by the National Science Foundation DMS-0301116. The second author is partially supported by the Agence Nationale de Recherche 06-BLAN-0015.     

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H _∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H _∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 87–104, 2004.     

The Iyengar inequality,revisited

Detailed analysis shows that the famous Iyengar inequality actually says that the Trapezoidal formula is a central algorithm for approximating integrals over an appropriate interval for the class of functions whose derivatives are bounded by a positive number K in L _∞-sense. The inherent nonlinearity from central algorithms reflects the importance of the Iyengar inequality and thus makes familiar linear methods malfunction when one tries to generalize it. It is shown that the generalization depends on a nonlinear system of equations satisfied by a set of free nodes of a perfect spline. Explicit constructions are obtained in the spirit of the Iyengar inequality for the class of functions whose rth (r≤4) derivatives are bounded by a positive number K in L _∞-sense because a closed solution to the nonlinear system is only available for r≤4. Connections with computational mathematics, especially with best interpolation and best quadrature, are discussed. Numerical experiments are also included. AMS subject classification (2000)  65D30, 41A17     

Robust generalized <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-suboptimal control

We consider an optimal perturbation damping problem taking into account not only an external perturbation with bounded L ₂-norm and an initial perturbation caused by unknown initial conditions in the system but also unknown bounded parametric perturbations. We synthesize a robust generalized H _∞-suboptimal control minimizing the upper bound, expressed in terms of solutions of linear matrix inequalities, for the perturbation damping level under uncertainty in the closed system.     

Delay-dependent parameter-dependent <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filtering for a class of LPV delayed systems using PPDQ functions

The problem of delay-dependent parameter-dependent H _∞ filtering for state estimation of a class of LPV systems with multiple constant time delays in the state vector is investigated. With the help of polynomially parameter-dependent quadratic (PPDQ) functions and a suitable change of variables, the required sufficient conditions with high precision are established in terms of delay-dependent parameter-independent linear matrix inequalities (LMIs) for the existence of the desired H _∞ filters. The explicit expression of the delay-dependent parameter-dependent H _∞ filters is derived to satisfy both asymptotic stability and a prescribed level of disturbance attenuation. Accordingly, the designed filter will have the ability to track the plant states in the presence of external disturbances. Two numerical examples are provided to demonstrate the validity of the proposed design approach. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝⁿ be a bounded convex domain with C ² boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H _k ^p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥_p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H _k ^p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.     

On quasiconformal deformations of the universal hyperbolic solenoid

We investigate the Teichmüller metric and the complex structure on the Teichmüller space (H _∞) of the universal hyperbolic solenoid H _∞. In particular, a version of the Reich-Strebel inequality for H _∞ is obtained. As a consequence, we show that the Teichmüller type Beltrami coefficients determine unique geodesics in (H _∞), and we compute the infinitesimal form of the Teichmüller metric. In addition, we show that a Beltrami coefficient is Teichmüller extremal if and only if it is infinitesimally extremal. Finally, we show that the Kobayashi metric on (H _∞) equals the Teichmüller metric.     

Nonparametric density estimation for nonmixing approximable Stochastic Processes

The author deals with nonparametric density estimation for stochastic processes which satisfy the L _∞-approximability property. He considers a Parzen–Rosenblatt estimator of the density for general stationary L _∞-approximable processes. He states conditions under which it is consistent and investigates its rate of convergence. Finally, he applies his results to general nonmixing linear processes and nonmixing nonlinear autoregressive processes.     

Norm optimization problem for linear operators in classical Banach spaces

The main result of the paper shows that, for 1 < p < ∞ and 1 ≤ q < ∞, a linear operator T: ℓ _p → ℓ _q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p = 1). For 1 < p ≠ q < ∞, as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T: ℓ _p → ℓ _q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p = q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.