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.     

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.     

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 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.     

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.     

Adaptive PSO-LS-wavelet H<Subscript>∞</Subscript> control for two-wheeled self-balancing scooter

The current study is concerned with adaptive Particle Swarm Optimization Least Square Wavelet H_∞ for a two-wheel self-balancing scooter that provides a platform in order to balance itself and transport the driver in accordance to its natural lean. In order to keep the rider close to the upright position over smooth and non-smooth surfaces, providing a stable control system is the main challenge for the aforementioned vehicle. For this purpose, H_∞ is combined with adaptive algorithm, Least Square Support Vector Machine (LS-SVM) and Particle Swarm Optimization (PSO) to construct the adaptive control. The most important feature of the proposed control strategy is its inherent robustness and ability to handle the nonlinear behavior of the system. Simulations results indicated that the introduced motion control architecture is capable of providing appropriate control actions to achieve both position control and trajectory tracking satisfactorily.     

<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.     

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.     

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.     

The B<Subscript>4</Subscript>-valued propositional logic with unary logical connectives ~<Subscript>1</Subscript> / ~<Subscript>2</Subscript> /¬

A B ₄-valued propositional logic will be proposed in this paper which there are three unary logical connectives ~₁, ~₂, ¬ and two binary logical connectives ∧, ∨, and a Gentzen-typed deduction system will be given so that the system is sound and complete with B ₄-valued semantics, where B ₄ is a Boolean algebra.     

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.     

<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.     

Approximate Pattern Matching with the <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> Metrics

Given an alphabet Σ={1,2,…,|Σ|} text string T∈Σ ⁿ and a pattern string P∈Σ ^m , for each i=1,2,…,n−m+1 define L _p (i) as the p-norm distance when the pattern is aligned below the text and starts at position i of the text. The problem of pattern matching with L _p distance is to compute L _p (i) for every i=1,2,…,n−m+1. We discuss the problem for d=1,2,∞. First, in the case of L ₁ matching (pattern matching with an L ₁ distance) we show a reduction of the string matching with mismatches problem to the L ₁ matching problem and we present an algorithm that approximates the L ₁ matching up to a factor of 1+ε, which has an O(\frac1e²nlogmlog|S|)O(\frac{1}{\varepsilon^{2}}n\log m\log|\Sigma|) run time. Then, the L ₂ matching problem (pattern matching with an L ₂ distance) is solved with a simple O(nlog m) time algorithm. Finally, we provide an algorithm that approximates the L _∞ matching up to a factor of 1+ε with a run time of O(\frac1enlogmlog|S|)O(\frac{1}{\varepsilon}n\log m\log|\Sigma|) . We also generalize the problem of String Matching with mismatches to have weighted mismatches and present an O(nlog ⁴ m) algorithm that approximates the results of this problem up to a factor of O(log m) in the case that the weight function is a metric.     

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.     

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 <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.     

Delay-dependent <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for jumping delayed systems with two Markov processes

This paper addresses the H _∞ control problem for jumping delayed systems with two separable Markov processes. The objective is to design a controller such that the resulting closed-loop system is exponentially mean-square stable with a given decay rate and satisfies a prescribed H _∞ performance level. A decay-rate-dependent condition is obtained for the existence of admissible controllers in term of linear matrix inequalities. Two numerical examples are presented to demonstrate the effectiveness of the proposed method.     

Dynamic output feedback robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control of uncertain switched nonlinear systems

The problem of robust H _∞ control problem for a class of switched nonlinear systems with parameter uncertainty is studied by using single Lyapunov function method. The problem under switch-dependent and switch-independent dynamic output feedbacks is solved respectively. Sufficient conditions for solvability of the problem are obtained. The Lyapunov function and the corresponding switching law are explicitly constructed. A numerical example is given to demonstrate the validity of the proposed approach.     

Simultaneous <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> stabilization for chemical reaction systems based on orthogonal complement space

This paper presents a simultaneous H₂/H_∞ stabilization problem for the chemical reaction systems which can be modeled as a finite collection of subsystems. A single dynamic output feedback controller which simultaneously stabilizes the multiple subsystems and captures the mixed H₂/H_∞ control performance is designed. To ensure that the stability condition, the H₂ characterization and the H_∞ characterization can be enforced within a unified matrix inequality framework, a novel technique based on orthogonal complement space is developed. Within such a framework, the controller gain is parameterized by the introduction of a common free positive definite matrix, which is independent of the multiple Lyapunov matrices. An iterative linear matrix inequality (ILMI) algorithm using Matlab Yalmip toolbox is established to deal with the proposed framework. Simulation results of a typical chemical reaction system are exploited to show the validity of the proposed methodology.