H∞ prediction and unconstrained H∞ predictive control: multi‐input–multi‐output case

Through the combination of the sequential spectral factorization and the coprime factorization, a k‐step ahead MIMO H_∞ (cumulative minimax) predictor is derived which is stable for the unstable noise model. This predictor and the modified internal model of the reference signal are embedded into the H_∞ optimization framework, yielding a single degree of freedom multi‐input–multi‐output H_∞ predictive controller that provides stochastic disturbance rejection and asymptotic tracking of the reference signals described by the internal model. It is shown that for a plant/disturbance model, that represents a large class of systems, the inclusion of the H_∞ predictor into the H_∞ control algorithm introduces a performance/robustness tuning knob: an increase of the prediction horizon enforces a more conservative control effort and, correspondingly, results in deterioration of the transient and the steady‐state (tracking error variance) performance, but guarantees large robustness margin, while the decrease of the prediction horizon results in a more aggressive control signal and better transient and steady‐state performance, but smaller robustness margin. Copyright © 2001 John Wiley & Sons, Ltd.     

Three‐dimensional nonlinear H∞ guidance law

This paper proposes a novel three‐dimensional missile guidance law design based on nonlinear H_∞ control. The complete nonlinear kinematics of pursuit–evasion motion is considered in the three‐dimensional spherical co‐ordinates system; neither linearization nor small angle assumption is made here. The nonlinear H_∞ guidance law is expressed in a simple form by solving the associated Hamilton–Jacobi partial differential inequality analytically. Unlike adaptive guidance laws, the implement of the proposed robust H_∞ guidance law does not require the information of target acceleration, while ensuring acceptable interceptive performance for arbitrary target with finite acceleration. The resulting pursuit–evasion trajectories for both the H_∞‐guided missile and the worst‐case target are determined in closed form, and the performance robustness against variations in target acceleration, in engagement condition, and in control loop gain, is verified by numerical simulations. Copyright © 2001 John Wiley & Sons, Ltd.     

RECEDING HORIZON H∞ CONTROL FOR SYSTEMS WITH A STATE‐DELAY

This paper proposes the receding horizon H_∞ control (RHHC) for linear systems with a state‐delay. We first proposes a new cost function for a finite horizon dynamic game problem. The proposed cost function includes two terminal weighting terms, each of which is parameterized by a positive definite matrix, called a terminal weighting matrix. Secondly, we derive the RHHC from the solution to the finite dynamic game problem. Thirdly, we propose an LMI condition under which the saddle point value satisfies the nonincreasing monotonicity. Finally, we show the asymptotic stability and H_∞‐norm boundedness of the closed‐loop system controlled by the proposed RHHC. Through a numerical example, we show that the proposed RHHC is stabilizing and satisfies the infinite horizon H_∞‐norm bound.     

H∞ boundary control for a class of nonlinear stochastic parabolic distributed parameter systems

This paper addresses the problem of H_∞ boundary control for a class of nonlinear stochastic distributed parameter systems expressed by parabolic stochastic partial differential equations (SPDEs) of Itô type. A simple but effective H_∞ boundary static output feedback (SOF) control scheme with collocated boundary measurement is introduced to ensure the local exponential stability in the mean square sense with an H_∞ performance. By using the semigroup theory, the disturbance‐free closed‐loop well‐posedness analysis is first given. Then, based on the SPDE model, a general linear matrix inequality based H_∞ boundary SOF control design is provided via Lyapunov technique and infinite‐dimensional infinitesimal operator, such that the disturbance‐free closed‐loop system is locally exponentially stable in the mean square sense and the H_∞ performance of disturbance attenuation can also be achieved in the presence of disturbances. Finally, simulation results on a stochastic Fisher‐Kolmogorov‐Petrovsky‐Piscounov equation illustrate the effectiveness of the proposed method.     

Robust H∞ control of uncertain singular systems based on equivalent‐input‐disturbance approach

In this paper, a robust H_∞ control problem is considered for an uncertain singular system. An active disturbance rejection method called equivalent input disturbance (EID) is used to reduce the influence of exogenous disturbances and uncertainties on the system. At the first, there exists an EID, which can produces the same effect on the system as disturbances and uncertainties do in the control channel according to the EID concept. Then, an EID estimator is constructed to estimate the influence of EID on the system. Finally, based on Lyapunov stability theory, a static output feedback‐based robust H_∞ controller combined with EID estimate is designed, guaranteeing that closed‐loop system is admissible (regular, impulse‐free, and stable) with a prescribed H_∞ performance level. Compared with traditional H_∞ control method, H_∞ control based on EID method improve the control performance of the system. A numerical example demonstrates the validity of the method.     

H∞ feedback‐control theory in biochemical systems

In this paper we study the possible optimality of biochemical pathways in the H_∞ sense. We start by presenting simple linearized models of single enzymatic reaction systems, where we apply classical and modern tools of feedback‐control theory. We then apply the results obtained by our analysis to a linearly unbranched enzyme pathway system, where we explore the effect of a negative feedback loop internally exerted on the system by a self‐product of the pathway. We then probe the sensitivity of the enzymatic system to variations in certain variables and we deal with the problem of assessing the optimality of the static‐output feedback control, in the H_∞ sense, inherent to the closed‐loop system. In this point we demonstrate the applicability of our results via a theoretical example that provides an open‐loop and closed‐loop analysis of a four‐block enzymatic system. We then apply the various tools we developed to the optimal analysis of the Threonine synthesis pathway which is regulated by three feedback loops. We demonstrate that this pathway is optimal in the H_∞ sense, in the face of considerable uncertainties in the various enzyme concentrations of the pathway. Copyright © 2007 John Wiley & Sons, Ltd.     

Quadratic stabilizability and H∞ control of linear discrete‐time stochastic uncertain systems

This paper considers quadratic stabilizability and H_∞ feedback control for stochastic discrete‐time uncertain systems with state‐ and control‐dependent noise. Specifically, the uncertain parameters considered are norm‐bounded and external disturbance is an l²‐square summable stochastic process. Firstly, both quadratic stability and quadratic stabilization criteria are presented in the form of linear matrix inequalities (LMIs). Then we design the robust H_∞ state and output feedback H_∞ controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H_∞ controllable. Sufficient conditions for the existence of the desired robust H_∞ controllers are obtained via LMIs. Finally, some examples are supplied to illustrate the effectiveness of our results.     

Delay‐dependent adaptive reliable H∞ control of linear time‐varying delay systems

This paper considers the problem of delay‐dependent adaptive reliable H_∞ controller design against actuator faults for linear time‐varying delay systems. Based on the online estimation of eventual faults, the parameters of adaptive reliable H_∞ controller are updating automatically to compensate the fault effects on the system. A new delay‐dependent reliable H_∞ controller is established using a linear matrix inequality technique and an adaptive method, which guarantees the stability and adaptive H_∞ performance of closed‐loop systems in normal and faulty cases. A numerical example and its simulation results illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

A mixed PI/VI design method for nonlinear H∞ Control

This paper proposes a new mixed policy iteration and value iteration (PI/VI) design method for nonlinear H_∞ control based on the theories of polynomial optimization and Lasserre's hierarchy. The design of a mixed PI/VI controller can be carried out in four steps: firstly, initialize design parameters and expand nonlinear system matrices; secondly, obtain a polynomial matrix inequality for policy improvement; thirdly, obtain the Lasserre's hierarchy of a global polynomial optimization problem for value improvement; fourthly, perform the mixed PI/VI algorithm to approximate the optimal nonlinear H_∞ control law. The novelty of this work lies in that the problem of designing a nonlinear H_∞ controller is translated into a polynomial global optimization problem, which can be solved by Lasserre's hierarchy directly, and then, the mixed PI/VI algorithm is presented to approximate the optimal nonlinear H_∞ control law by updating global optimizers iteratively. The main results of this paper consist of the mixed PI/VI algorithm and the related three theorems, which guarantee robust stability and performance of the closed‐loop nonlinear system. Numerical simulations show that the mixed PI/VI algorithm converges very fast and achieves good robust stability and performance in transient behavior, disturbance rejection, and enlarging the domain of attraction of the close‐loop system.     

Event‐triggered observer‐based robust H∞ control for networked control systems with unknown disturbance

In this article, the event‐triggered robust H_∞ control is studied for a class of uncertain networked control systems (NCSs) subject to unknown state and variable disturbance. First, aiming to decrease the unnecessary transmissions of sampled data, an efficient adaptive event‐triggered scheme (AETS) is presented, which can reflect the full real‐time variation of addressed NCSs and help to reduce the conservativeness. Second, based on the triggered output signals and disturbance model, two effective observers are, respectively, exploited to estimate the state and disturbance, which are further utilized to reject the disturbance and design the controller. By using the overall closed‐loop system and selecting an augmented Lyapunov‐Krasovskii functional, two sufficient conditions on jointly designing the adaptive event scheme, observers, and controller are established via linear matrix inequality forms, which can guarantee the global exponential stability and ensure H_∞ performance. Finally, some simulations and comparisons in a numerical example are provided to demonstrate the effectiveness of the derived results.     

Delay‐ and Packet‐Disordering‐Dependent H∞ Output Tracking Control for Networked Control Systems

This paper is concerned with the problem of H_∞ output tracking control for networked control systems (NCSs) with network‐induced delay and packet disordering. Different from the results in existing literature, the controller design in this paper is both delay‐ and packet‐disordering‐dependent. Based on the different cases of consecutive predictions, the networked output tracking system is modeled into a switched system. Moreover, by the corresponding switching‐based Lyapunov functional approach, a linear matrix inequality (LMI)‐based procedure is proposed for designing state‐feedback controllers, which guarantees that the output of the closed‐loop NCSs tracks the output of a given reference model well in the H_∞ sense. In addition, the proposed method can be applied variously due to all kinds of prediction numbers of the consecutive disordering packet have been considered, and the designed controller is based on the prediction case in the last transmission interval, which brings about less conservatism. Finally numerical examples and simulations are used to illustrate the effectiveness and validity of the proposed switching‐based method and the delay‐ and packet‐disordering‐dependent H_∞ output tracking controller design.     

State feedback H∞ control for quantized discrete‐time systems

This paper is concerned with the quantized state feedback H_∞ control problem for discrete‐time linear time‐invariant systems. The quantizer considered here is dynamic and composed of an adjustable “zoom” parameter and a static quantizer. Static quantizer ranges are with practical significance and fully considered here. A quantized H_∞ controller design strategy is proposed with taking quantizer errors into account, where an iterative linear matrix inequality (LMI) based optimization algorithm is developed to minimize static quantizer ranges with meeting H_∞ performance requirement for quantized closed‐loop systems. An example is presented to illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust H∞ control for uncertain singular systems with state delay

This paper addresses the problem of robust H_∞ control for uncertain continuous singular systems with state delay. The singular system under consideration involves state time delay and time‐invariant norm‐bounded uncertainty. Based on the linear matrix inequality (LMI) approach, we design a memoryless state feedback controller law, which guarantees that, for all admissible uncertainties, the resulting closed‐loop system is not only regular, impulse free and stable, but also meets an H_∞‐norm bound constraint on disturbance attenuation. A numerical example is provided to demonstrate the applicability of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

H∞ fuzzy control design of discrete‐time nonlinear active fault‐tolerant control systems

This paper is concerned with the problem of H_∞ fuzzy controller synthesis for a class of discrete‐time nonlinear active fault‐tolerant control systems (AFTCSs) in a stochastic setting. The Takagi and Sugeno (T–S) fuzzy model is employed to exactly represent a nonlinear AFTCS. For this AFTCS, two random processes with Markovian transition characteristics are introduced to model the failure process of system components and the fault detection and isolation (FDI) decision process used to reconfigure the control law, respectively. The random behavior of the FDI process is conditioned on the state of the failure process. A non‐parallel distributed compensation (non‐PDC) scheme is adopted for the design of the fault‐tolerant control laws. The resulting closed‐loop fuzzy system is the one with two Markovian jump parameters. Based on a stochastic fuzzy Lyapunov function (FLF), sufficient conditions for the stochastic stability and H_∞ disturbance attenuation of the closed‐loop fuzzy system are first derived. A linear matrix inequality (LMI) approach to the fuzzy control design is then developed. Moreover, a suboptimal fault‐tolerant H_∞ fuzzy controller is given in the sense of minimizing the level of disturbance attenuation. Finally, a simulation example is presented to illustrate the effectiveness of the proposed design method. Copyright © 2008 John Wiley & Sons, Ltd.     

Structured H∞‐control of infinite‐dimensional systems

We develop a novel frequency‐based H_∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H_∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H_∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H_∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H_∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H_∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set.     

Quantized H∞ Control for Networked Systems with Communication Constraints

The problem of quantized H_∞ control for networked control systems (NCSs) subject to time‐varying delay and multiple packet dropouts is investigated in this paper. Both the control input and the measurement output signals are quantized before being transmitted and the quantized errors are described as sector bound uncertainties. The measurement channel and the control channel packet dropouts are considered simultaneously, and the stochastic variables satisfying Bernoulli random binary distribution are utilized to model the random multiple packet dropouts. Sufficient conditions for the existence of an observer‐based controller are established to ensure the exponential mean‐square stablility of the closed‐loop system and achieve the optimal H_∞ disturbance attenuation level. By using a globally convergent algorithm involving convex optimization, the nonconvex feasibility can be solved successfully. Finally, a numerical example is given to illustrate the effectiveness and applicability of the proposed method.     

Reliable H∞ Control for a Class of Nonlinear Networked Control Systems with Random Delays

In this paper, the problem of reliable H_∞ controller design is studied for a class of nonlinear networked control systems. A novel model is presented, which contains random transmission delays, faults of the sensor and actuator. The sensor‐to‐controller and controller‐to‐actuator transmission delays with upper bounds are considered, simultaneously. The working conditions of the sensor and actuator are formulated as two independent Markov chains, which take matrix values in finite sets, respectively. The resulting closed‐loop system is converted into a Markov switching system. On the basis of the cone complementary linearization algorithm, a mode‐dependent reliable controller is constructed such that the closed‐loop system is stochastically stable and attains the prescribed H_∞ disturbance attenuation level. Finally, a numerical example is given to show the effectiveness of the developed technique.     

H∞ ALMOST DISTURBANCE DECOUPLING FOR MIMO BILINEAR SYSTEMS

The H_∞ almost disturbance decoupling problem is considered. In this paper, a nonlinear design is proposed to find a state feedback controller for bilinear systems. The closed‐loop system is internally stable and achieves disturbance attenuation in nonlinear H_∞ sense. We defined a special form of Lyapunov function, which is constructed in terms of one or a set of positive definite constant matrices. If, except of the origin of system, the corresponding polynomial of the positive definite matrix (or several polynomials relevant to the positive definite constant matrices) has (have) no zero on a given subset of state space, then we can construct a controller to solve our problem. It is found that the controller structure could be complicated, but is feasible in computation and may require optimization technique to search the solution. We consider both SIMO and MIMO cases with illustrated examples.     

H∞ finite‐horizon filtering for complex networks with state saturations: The weighted try‐once‐discard protocol

This paper addresses the finite‐horizon H_∞ filtering problem for a kind of discrete state‐saturated time‐varying complex networks subjected to the weighted try‐once‐discard (WTOD) protocol. Under the WTOD protocol, only the measurement signal from one sensor node is allowed to be transmitted to the filter at each time point, where such a node is selected based on a certain quadratic selection principle. The main purpose of this paper is to design an H_∞ filter that guarantees the disturbance attenuation level on a given finite time‐horizon for the underlying complex network subject to both state saturations and WTOD protocols. By using the convex hull approach, sufficient conditions are first obtained to ensure the existence for the desired filter to achieve the H_∞ performance specification by means of a few recursive matrix inequalities. Then, based on the obtained results, the filter parameters are designed, which cope effectively with both state saturations and communication protocols. Finally, a numerical simulation is employed to demonstrate the validity of the developed filter algorithm.     

ROBUST H∞ CONTROL FOR UNCERTAIN STOCHASTIC SYSTEMS WITH MARKOVIAN SWITCHING AND TIME‐VARYING DELAYS

This paper is concerned with the problem of robust H_∞ control for uncertain stochastic systems with Markovian jump parameters and time‐varying state delays. A linear matrix inequality approach is developed and state feedback controllers are designed, which guarantee mean square asymptotic stability of the closed‐loop system and a prescribed H_∞ performance level for all modes and admissible uncertainties. A numerical example is provided to demonstrate the application of the proposed method.