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> 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.     

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.     

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.     

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.     

A Nash Game Approach to Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Model Predictive Control: Part 3 – Output Feedback Case

In this paper, the state-feedback Nash game based mixed H₂/H_∞ design^{[1, 2]} has been extended for output feedback case. The algorithm is applied to control bioreactor system with a Laguerre-Wavelet Network (LWN)^{[3, 4]} model of the bioreactor. This is achieved by using the LWN model as a deviation model and by successively linearising the deviation model along the state trajectory. For reducing the approximation error and to improve the controller performance, symbolic derivation algorithm, viz., automatic differentiation is employed. A cautionary note is also given on the fragility of the output feedback mixed H₂/H_∞ model predictive controller^{[4, 5]} due to its sensitivity to its own parametric changes.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> digital redesign for LTI systems

The so-called digital redesign (DR) is a sampled-data (SD) controller design method where an analogue controller is designed firstly, and then transformed to an approximately equivalent digital controller in the sense of state-matching. In this approach, the SD controller is designed by reducing the discrepancy between the discrete-time (DT) counterpart of the closed-loop SD control system and the continuous-time (CT) closed-loop system. In this paper, we develop a DR strategy for CT linear time-invariant systems. More specifically, H _∞ norm of the error dynamic system between the CT and DT plants is minimized for the optimal state-matching performance at every sampling point. The design problem is formulated as linear matrix inequalities which can be efficiently solved by using convex optimization techniques. Finally, an example is given to illustrate the effectiveness of the proposed method.     

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.     

<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.     

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.     

Model reference robust adaptive <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> controller design

In this paper, a novel model reference robust adaptive H _∞ controller is designed, which not only guarantees asymptotically stability, but also optimizes adaptive H _∞ performance by estimating the optimal unknown control parameters. Furthermore, a novel state feedback framework is introduced, which depends on the optimal unknown parameter estimations, to guarantee that the defined cost function is minimized. At the same time, the minimal adaptive H _∞ performance index is obtained.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control Design Using Dynamic Neural Networks

In this paper, a new approach is investigated for adaptive dynamic neural network-based H _∞ control, which is designed for a class of non-linear systems with unknown uncertainties. Currently, non-linear systems with unknown uncertainties are commonly used to efficiently and accurately express the real practical control process. Therefore, it is of critical importance but a great challenge and still at its early age to design a stable and robust controller for such a process. In the proposed research, dynamic neural networks were constructed to precisely approximate the non-linear system with unknown uncertainties first, a non-linear state feedback H _∞ control law was designed next, then an adaptive weighting adjustment mechanism for dynamic neural networks was developed to achieve H _∞ regulation performance, and last a recurrent neural network was employed as a neuro-solver to efficiently and numerically solve the standard LMI problem so as to obtain the appropriate control gains. Finally, case studies further verify the feasibility and efficiency of the proposed research.     

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.     

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.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filtering for discrete-time systems with time-varying delay

The problem of H _∞ filtering for discrete-time systems with time-varying delay in measurement is investigated in this paper. First, under the assumption that the time-varying delay is of a known upper bound, the delayed measurement is re-described as the one with multiple state delays. Then the proposed H _∞ filtering problem is transformed into one for systems with multiple measurement channels that contain the same state information as the original measurement and each channel has a single constant delay. Finally, based on the reorganized innovation analysis approach in Krein space, a necessary and sufficient condition for the existence of an H _∞ filter which guarantees a prescribed attenuation level is derived. The solution to the H _∞ filtering is given in terms of the solutions to Riccati and matrix difference equations.     

Scalar Field Cosmology in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>,<Emphasis Type="Italic">T</Emphasis>) Gravity with Λ

We study the behavior of cosmological parameters, massive and massless scalar fields (normal or phantom) with a scalar potential in f(R, T) theory of gravity for a flat Friedmann-Robertson-Walker (FRW) universe. To get exact solutions to the modified field equations, we use the f(R, T) = R + 2f(T) model by Harko et al. (T. Harko et al., Phys. Rev. D 84, 024020 (2011)), where R is the Ricci scalar and T is the trace of the energy momentum tensor. Our cosmological parameter solutions agree with the recent observational data. Finally, we discuss our results with various graphics.     

Adaptive Reliable <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control of Uncertain Affine Nonlinear Systems

For general input affine nonlinear systems, robust reliable control designs are commonly available that compensate the actuator faults in pure outage mode. In this paper, a more general and complex problem is considered and an adaptive reliable H_∞ controller is designed for a class of uncertain input affine nonlinear systems in the presence of actuators fault. The key element of the work is the introduction of a novel adaptive mechanism that estimates the faults which are modeled as an outage or loss of effectiveness and stabilizes the overall system. Incorporating with the parameter projection algorithm and the solution of Hamilton-Jacobi-Inequality (HJI), the proposed method combines adaptive reliable control and robust H_∞ control techniques. A numerical approach is developed based on the Taylor series expansion for solving the HJI. Various simulation examples are given to illustrate the effectiveness of the proposed adaptive reliable H_∞ control scheme over the conventional H_∞ control and reliable H_∞ control method.     

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.     

<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.