Stochastic problems in <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> and <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

We consider stochastic control systems subjected simultaneously to stochastic and determinate perturbations. Stochastic perturbations are assumed to be state-multiplicative stochastic processes, while determinate perturbations can be any processes with finite energy on infinite time interval. The results of the determinate H _∞-theory are compared to their stochastic analogs. The determinate and stochastic theories are linked together by the lemma that establishes the equivalence between the stability and boundness of the ‖L‖_∞ < γ norm of the perturbation operator L, from one side, and the solvability of certain linear matrix inequalities (LMIs), from the other side. As soon as the stochastic version of the lemma is proven, the γ-controller analysis and design problems are solved, in general, identically in the frame of the united LMI methodology.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> DOF control of stochastic systems with time-varying delay and <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> disturbance

This paper considers the problem of H _∞ dynamic output feedback (DOF) control for a class of stochastic systems with time-varying delay and L _∞ disturbance. A new delay-dependent sufficient condition for the existence of the DOF controller is derived and the controller design method is given in the form of bilinear matrix inequalities (BMIs). Moreover, a variable step size path-following algorithm is proposed to solve the BMI problem. Numerical examples are given to illustrate the effectiveness of the proposed methods.     

Discontinuous <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of underactuated mechanical systems with friction and backlash

Nonlinear H _∞-control is extended to discontinuous mechanical systems with degree of underactuation one, where nonlinear phenomena such as Coulomb friction and backlash are considered. The problem in question is to design a feedback controller via output measurements so as to obtain the closed-loop system in which all trajectories are locally ultimate bounded, and the underactuated link is regulated to a desired position while also attenuating the influence of external perturbations and nonlinear phenomena. It is considered that positions are the only measurements available for feedback in the system. Performance issues of the discontinuous H _∞-regulation controller are illustrated in an experimental study made for a rectilinear plant with friction modified to have a gap in the point of contact between bodies.     

Robust finite-time <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of stochastic jump systems

This paper provides the stochastic finite-time stabilization and H _∞ control problem of Markov jump systems with norm-bounded uncertainties and state delays that possess randomly jumping parameters. The transition of the jumping parameters is governed by a finite-state Markov process. The finite-time H _∞ controller via state feedback is provided to guarantee the stochastic finite-time bounded-ness and stochastic finite-time stabilization of the resulting closed-loop system for all admissible uncertainties and unknown time-delays. The control criterion is formulated in the form of linear matrix inequalities and the designed finite-time stabilization controller is described as an optimization one. Simulation results illustrate the effectiveness of the developed approaches.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for a class of nonlinear networked control systems

In this paper, the mixed H ₂/H _∞ control problem is investigated for a class of nonlinear discrete-time networked control systems with random network-induced delays, stochastic packet dropouts and probabilistic sensor faults. The packet dropouts process is modeled as a homogeneous Markov chains taking values in a finite state space. Network-induced delays occur in a random way with known upper bound. A set of stochastic variables are exploited to describe sensor faults with different probabilistic density functions. By using a delay-dependent Lyapunov functional, a mode-dependent mixed H ₂/H _∞ controller is designed to guarantee both stochastic stability of the closed-loop system and the prescribed H2, H¥ control performances. Sufficient conditions for the existence of the mixed H ₂/H _∞ controller are presented in terms of a series of LMIs. If these LMIs are feasible, then the modedependent mixed H ₂/H _∞ controller can be obtained. A numerical example is given to demonstrate the effectiveness of the developed method.     

Generalized <Emphasis Type="Italic">H</Emphasis> <Subscript>∞</Subscript>-optimal control of continuous-discrete linear plant

For the linear continuous nonstationary plants whose states are measured at fixed time instants, the notion of perturbation damping level was introduced as an index of joint impact of the initial and external perturbations on the worst-case objective output and terminal state. The H_∞-optimal laws of control were formulated as the state feedbacks minimizing the level of feedback damping of the closed-loop system.     

Design of optimal and robust control with <Emphasis Type="Italic">H</Emphasis> <Subscript>∞</Subscript>/<Emphasis Type="Italic">γ</Emphasis> <Subscript>0</Subscript> performance criterion

For a double-input single-output system, this paper defines a disturbance attenuation level (called H_∞/γ₀ norm) as the maximum-value L₂ norm of the output under an unknown disturbance with a bounded L₂ norm supplied to the first input and an impulsive disturbance in the form of the product of an unknown vector and the delta function supplied the second input, where the squared L₂ norm of the former disturbance plus the quadratic form of the impulsive disturbance vector does not exceed 1. Weight matrix choice in the H_∞/γ₀ norm yields a trade-off between the attenuation level of the L₂ disturbance and the attenuation level of the impulsive disturbance in corresponding channels. For the uncertain systems with dynamic or parametric uncertainty in the feedback loop, a robust H_∞/γ₀ norm is introduced that includes the robust H_∞ and γ₀ norms as special cases. All these characteristics or their upper bounds in the uncertain system are expressed via solutions of linear matrix inequalities. This gives a uniform approach for designing optimal and robust control laws with the H_∞/γ₀, H_∞ and γ₀ performance criteria.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for linear infinite-dimensional systems

In this paper, we consider mixed H ₂/H _∞ control problems for linear infinite-dimensional systems. The first part considers the state feedback control for the H ₂/H _∞ control problems of linear infinite-dimensional systems. The cost horizon can be infinite or finite time. The solutions of the H ₂/H _∞ control problem for linear infinitedimensional systems are presented in terms of the solutions of the coupled operator Riccati equations and coupled differential operator Riccati equations. The second part addresses the observer-based H ₂/H _∞ control of linear infinite-dimensional systems with infinite horizon and finite horizon costs. The solutions for the observer-based H ₂/H _∞ control problem of linear infinite-dimensional systems are represented in terms of the solutions of coupled operator Riccati equations. The first-order partial differential system examples are presented for illustration. In particular, for these examples, the Riccati equations are represented in terms of the coefficients of first-order partial differential systems.     

Design of robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-controller for energetic boiler plant

A problem of robust controller design for control of a boiler plant is solved using an H _∞ loop-shaping method with constraints imposed on regional pole placement of transfer functions of the closed-loop system, and linear matrix inequalities tools. Requirements to the closed-loop system are formulated both in the form of frequency constraints on singular values of transfer function of the open-loop system, and in the form of pole placement constraints for the transfer function of the closed-loop system as a given region on the complex plane. A reduction of the full-order H _∞-controller obtained in the design procedure is performed.     

Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> consensus control of uncertain multi-agent systems with time delays

This paper is devoted to the robust H _∞ consensus control of multi-agent systems with model parameter uncertainties and external disturbances. In particular, switching networks of multiple agents with general linear dynamics are considered, and uncertain communication delays are also taken into account. It shows that a sufficient condition in terms of linear matrix inequalities (LMIs) is derived for the robust consensus performance with a given H _∞ disturbance attenuation level, and meanwhile the unknown feedback matrix of the proposed distributed state feedback protocol is also determined. A numerical example is included to validate the theoretical results.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filtering for a class of high-speed sampling uncertain systems

The problem of mixed H2/H∞ filtering for polytopic Delta operator systems is investigated. The aim is to design a linear asymptotically stable filter which guarantees that the filtering error system has different performances in different filtering channels. Based on a parameter-dependent Lyapunov function, a new mixed H2/H∞ performance criterion is presented. Upon this performance criterion, a sufficient condition for the full-order mixed H2/H∞ filter is derived in terms of linear matrix inequalities. The filter can be obtained from the solution of a convex optimization problem. The proposed filter design procedure is less conservative than the strategy based on the quadratic stability notion. A numerical example is given to illustrate the feasibility of the proposed approach.     

Synthesis and modeling of the <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-system of magnetic control of the plasma in the tokamak-reactor

The article deals with the development of a two-contour system of magnetic control of the position, current, and shape of the plasma in the tokamak-reactor. The H _∞-theory of control is used for the synthesis of a scalar and a multidimensional feedback controller. The controllers are synthesized on the basis of the multidimensional linear model DINA-L of an object (plasma in the tokamak). The linear model DINA-L is obtained from the nonlinear model implemented by the plasmophysical DINA code for conditions of the international thermonuclear experimental reactor (ITER). The numerical modeling of a closed control system is carried out on the linear DINA-L model and the nonlinear DINA model of an object at disturbances of the type of small disruptions. The modeling results for both of the cases were superimposed on each other, which showed their good coincidence at the acceptable quality of control of the synthesized system.     

FIR-type robust <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of discrete linear time-invariant polytopic systems via memory state-feedback control laws

In this paper, robust H ₂ and H _∞ control problems for discrete linear time-invariant (LTI) systems with polytopic uncertainties are addressed. The so-called finite impulse response (FIR) controller incorporating the states over several samples from the past to the present is adopted to design robust control laws with improved performances. For the closed-loop stability, parameter-dependent quadratic Lyapunov functions (PD-QLFs) are employed. Sufficient controller synthesis conditions are derived in the form of linear matrix inequalities (LMIs). Finally, examples are given to demonstrate the usefulness of the proposed methods.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of networked control systems with packet disordering with an improved prediction-based method

This paper is concerned with H _∞ control of networked control systems (NCSs) with both network-induced delay and packet disordering taken into account. A switching-based method depends on different cases of prediction numbers is presented to model the NCSs as a switched system. With the application of an improved prediciton-based method to compensate for the time delay and packet disordering, H _∞ controller is designed by solving the linear matrix inequalities (LMIs). The proposed method can be applied variously due to all kinds of prediciton numbers of the consective disordering packet have been considered, which means that our techniques are less conservative than existing literatures. Finally numerical examples and simulations are used to illustrate the effectiveness and validity of the proposed switching-based method and the improved prediction-based H _∞ controller design synthesis.     

Exponential <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for singular systems with time-varying delay

This paper studies the exponential admissibility and H _∞ control problems for a class of singular systems with time-varying delay in state. Firstly, an exponential admissibility criterion is obtained based on linear matrix inequalities (LMIs). It is worth mentioning that the derivative of the time-varying delay does not need to be smaller than one. Based on the proposed condition, a new delay-dependent H _∞ controller is also given, which guarantees the admissibility and the H _∞ performance γ. Numerical examples are given to illustrate the effectiveness of the proposed method.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of singular systems via delta operator method

This paper studies the problem of state feedback H _∞ control for singular systems through delta operator approach. A necessary and sufficient condition is presented such that a singular delta operator system is admissible with a prescribed H _∞ performance, which can provide a unified framework of the existing H _∞ performance analysis results for both continuous case and discrete case. The existence condition and explicit expression of a desirable H _∞ controller are also obtained for singular delta operator systems. The proposed design method can be used for both singular continuous systems and singular discrete systems directly. The corresponding design procedures, which simplify the classical approaches, are discussed and presented. All obtained conditions in this paper are in the form of strict linear matrix inequalities whose feasible solutions can be found by standard linear programming method. Numerical examples are provided to illustrate the effectiveness of the theoretical results obtained in this paper.     

Algorithms for solving a class of problems of <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-optimization of control systems

Problems of synthesis of feedbacks providing H _∞-norm-maximal suppression of disturbances, acting on a linear stationary SISO system with account of limitedness of control resources, are considered. Two methods for solving are proposed. The first one is based on the Nevanlinna-Pick interpolation, and the second one is constructed on the ground of guaranteeing singularities of controllers. Within the framework of these approaches, simple computation algorithms that do not require solving Riccati equations or linear matrix inequalities are generated. The application of the developed algorithms is illustrated by an example.     

Adaptive robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of time delay systems with unknown uncertainty bounds

The problem of adaptive robust H _∞ control for uncertain systems with multiple delays is considered in this paper. The essential requirement for the uncertainties is that they satisfy matching conditions and are norm-bounded, but the bounds are not necessarily known. The objective is to design adaptive robust H _∞ state-feedback controller such that the resulting closed-loop system is asymptotically stable and a prescribed H _∞ performance level of the closed loop system for disturbance attenuation is guaranteed. To solve this problem, a new sufficient condition is presented in terms of a linear matrix inequality (LMI). The effectiveness of proposed method is verified by its application to an unstable system.     

Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> state feedback control of NCSs with Poisson noise and successive packet dropouts

This paper examines various constraints of networked control systems (NCSs) such as network-induced random delays, successive packet dropouts and Poisson noise. Time delays are represented as modes of Markov chain and successive packet dropouts are modeled using Poisson probability distribution. For each delay-mode, a separate Poisson distribution is used with the help of an indicator function. Poisson noise is incorporated in the design to cater sudden network link failures and power shutdowns. After modeling these constraints, a stability criterion is proposed by using Lyapunov-Krasovskii functional. On the basis of the stability criteria, sufficient conditions for the existence of a robust H _∞ state feedback controller are given in terms of bilinear matrix inequalities (BMIs). Later, BMIs are converted into quasi-convex linear matrix inequalities (LMIs) and are solved by using a cone complementarity linearizing algorithm. The effectiveness of the proposed scheme is elaborated with the help of two simulation examples. Moreover, the effects of successive packet dropouts and Poisson noise on H _∞ performance are analyzed.     

Finite time control for one class of nonlinear systems with the <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> performance criterion

This work considers nonautonomous control systems with uncertain nonlinearities subjected to the impact of external disturbances which are continuous functions bounded by the L ₂-norm. Based on the method of matrix comparison systems and technique of differential linear matrix inequalities, it is proposed to solve problems of finite time stability and boundedness with respect to the given sets, as well as suppress the disturbances and initial deviations using the state feedback with an estimation of the performance by the H _∞-criterion.