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.     

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.     

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.     

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> control for sochastic time-delayed Markovian switching systems with partly known transition rates and input saturation

The paper is concerned with the problem of H _∞ control for stochastic time-delayed Markovian switching systems with partly known transition rates and input saturation. By employing more appropriate Lyapunov- Krasovskii functional, a state feedback controller is designed to guarantee stochastic stability of the corresponding closed-loop system with H _∞ performance. A linear matrix inequality approach is employed to obtain the controller gain matrix. Two illustrative examples are provided to show the potential of the proposed techniques.     

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.     

Dynamic output-feedback <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for active half-vehicle suspension systems with time-varying input delay

This paper addresses the new output-feedback H _∞ control problem for active half-vehicle suspension systems with time-varying input delay. By introducing multi-objective synthesis, a new dynamic output-feedback H _∞ controller is designed such that the closed-loop suspension system is asymptotically stable with guaranteed robust performance in the H _∞ sense. The proposed controller is formulated in terms of linear matrix inequality (LMI) based on the auxiliary function-based integral inequality method and the reciprocally convex approach. A new delay-dependent sufficient condition for the desired controller offers a wider range of control input delay. Numerical examples are provided to validate the effectiveness of the proposed design 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.     

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.     

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.     

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.     

Finite-time <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control of Cascade Nonlinear Switched Systems under State-dependent Switching

In this paper, we investigate the finite-time H_∞ control problem for the class of cascade nonlinear switched systems consisting of two parts which are also switched systems using a state-dependent switching method. The state feedback controller and the switching law are designed respectively, which guarantees that the corresponding closed-loop system is finite-time bounded and has a prescribed H_∞ performance index; the corresponding sliding motion problems are also considered. Sufficient conditions for the solvability of the problem are obtained. A numerical example is given to demonstrate the validity of the proposed approach.     

Robust sliding mode-<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control approach for a class of nonlinear systems affected by unmatched uncertainties using a poly-quadratic Lyapunov function

This paper proposes a robust sliding mode-H _∞ control design methodology for a class of nonlinear systems with unmatched parametric uncertainty and external disturbance. The design procedure combines the high robustness of the sliding mode control (SMC) with the H _∞ norm performance. First, based on linear matrix inequalities (LMI) technique and multiple Lyapunov functions approach, the sliding surface design problem is formulated as a H _∞ state-feedback control for a reduced uncertain nonlinear system with polytopic representation. Then, a sliding mode controller that drives the system states to the sliding surface in finite time and maintains a sliding mode is constructed. Finally, a comparative study is done to prove the effectiveness of the results.     

Fuzzy <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> FIR Filtering for T–S Fuzzy Systems with Quantization and Packet Dropout

This paper proposes a new fuzzy H_∞ finite impulse response (FIR) filter with quantization and packet dropout for Takagi–Sugeno (T–S) fuzzy systems with external disturbance. The measurements are quantized by a logarithmic quantizer and then transmitted from the plant to the filter imperfectly due to random packet loss described by the Bernoulli random process. The proposed fuzzy H_∞ FIR filter is in the form of fuzzy-basis-independent linear matrix inequalities (LMIs) that guarantee H_∞ performance. Two simulation examples are given to illustrate the effectiveness and robustness of the proposed fuzzy H_∞ FIR filter.     

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.     

Non-fragile control of fuzzy affine dynamic systems via piecewise Lyapunov functions

This paper addresses the robust H_∞ static output feedback (SOF) controller design problem for a class of uncertain fuzzy affine systems that are robust against both the plant parameter perturbations and controller gain variations. More specifically, the purpose is to synthesize a non-fragile piecewise affine SOF controller guaranteeing the stability of the resulting closed-loop fuzzy affine dynamic system with certainH_∞ performance index. Based on piecewise quadratic Lyapunov functions and applying some convexification procedures, two different approaches are proposed to solve the robust and non-fragile piecewise affine SOF controller synthesis problem. It is shown that the piecewise affine controller gains can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, simulation examples are given to illustrate the effectiveness of the proposed methods.     

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.     

Robust Finite-time <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control of Linear Time-varying Delay Systems with Bounded Control via Riccati Equations

In this paper, we will present new results on robust finite-time H_∞ control for linear time-varying systems with both time-varying delay and bounded control. Delay-dependent sufficient conditions for robust finite-time stabilization and H_∞ control are first established to guarantee finite-time stability of the closed-loop system via solving Riccati differential equations. Applications to finite-time H_∞ control to a class of linear autonomous time-delay systems with bounded control are also discussed in this paper. Numerical examples are given to illustrate the effectiveness of the proposed method.     

A new extended LMI-based robust gain scheduled state feedback <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript> controller design

This paper proposes an improved robust H ₂ state feedback control synthesis for the Linear Parameter Varying (LPV) systems by attaining the affine quadratic stability. In place of standard H ₂ computation in the literature, a new H ₂ computation based on extended Linear Matrix Inequality (LMI) is improved by means of the slack variable, where it is obtained by separation Lyapunov matrix from system matrix. State feedback H ₂ synthesis is improved for the systems, and is more effective and less conservative than the common ones in the literature. Therefore, the less conservative results are obtained for gain scheduling controller design for LPV systems. The numerical examples are presented to show the superiority of the proposed controller design.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> suboptimal tracking controller design for a class of nonlinear systems

In this paper, a new technique is proposed to solve the H _∞ tracking problem for a broad class of nonlinear systems. Towards this end, based on a discounted cost function, a nonlinear two-player zero-sum differential (NTPZSD) game is defined. Then, the problem is converted to another NTPZSD game without any discount factor in its corresponding cost function. A state-dependent Riccati equation (SDRE) technique is applied to the latter NTPZSD game in order to find its approximate solution which leads to obtain a feedback-feedforward control law for the original game. It is proved that the tracking error between the system state and its desired trajectory converges asymptotically to zero under mild conditions on the discount factor. The proposed H _∞ tracking controller is applied to two nonlinear systems (the Vander Pol’s oscillator and the insulin-glucose regulatory system of type I diabetic patients). Simulation results demonstrate that the proposed H _∞ tracking controller is so effective to solve the problem of tracking time-varying desired trajectories in nonlinear dynamical systems.