Unconstrained H∞ predictive control with H∞ prediction: single‐input–single‐output case

Does the replacement of the quadratic (H₂) predictor by the worst‐case (H_∞, or cumulative minimax) predictor robustify the predictive control laws? The present work provides a partial answer to this question, positive for the examples considered, representative of three broad classes of systems. The H_∞ prediction is demonstrated to be a powerful and convenient tool for frequency shaping of the gain of the closed‐loop complementary sensitivity function, capable of robustifying the closed loop for systems with different stability properties. The H_∞‐optimal k‐step ahead predictor is derived for an unstable single‐input–single‐ output CARMA model. A BIBO unstable filter for the disturbance rejection is obtained using the internal model principle and included into the closed loop, and the H_∞ predictor is applied to the combination of this filter with the plant. The sum over a finite horizon of the current and the predicted tracking error and control signal power spectral densities (PSDs) is decomposed into two parts, one induced by the worst‐case predicted disturbance and the other—by the known future reference input. A two degrees of freedom algorithm, referred to as the multi‐step closed‐loop polynomial H_∞ predictive control law, is obtained that minimizes the peaks of the PSD of the first part and the integral on the unit circle of the PSD of the second. It is demonstrated on several systems that H_∞ prediction introduces a very intuitive tuning knob in the form of the prediction horizon capable of setting a trade‐off between the steady‐state disturbance rejection perfor mance in terms of the output error variance and the closed‐loop robustness, however the efficacy of the knob strongly depends on the stability properties of the system and its inverse. The trade‐off becomes less pronounced or completely disappears when the H_∞ predictor is replaced by the quadratic one. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal‐switched extended H∞ filter for nonlinear systems with stochastic uncertainties

The extended H_∞ filter (EH_∞F) is a conservative solution with infinite‐horizon robustness for the state estimation problem regarding nonlinear systems with stochastic uncertainties, which leads to excessive costs in terms of filtering optimality and reduces the estimation precision, particularly when uncertainties related to external disturbances and noise appear intermittently. In order to restore the filtering optimality lost due to the conservativeness of the EH_∞F design, we developed an optimal‐switched (OS) filtering mechanism based on the standard EH_∞F to obtain an optimal‐switched extended H_∞ filter (OS‐EH_∞F). The OS mechanism has an error‐tolerant switched (ETS) structure, which switches the filtering mode between optimal and H_∞ robust by setting a switching threshold with redundancy to uncertainties, and a robustness‐optimality cost function (ROCF) is introduced to determine the threshold and optimize the ETS structure online. The ROCF is the weighted sum of the quantified filtering robustness and optimality. When a weight is given, the proposed OS‐EH_∞F can obtain the optimal state estimates while maintaining the filtering robustness at an invariant ratio. A simulation example of space target tracking has demonstrated the superior estimation performance of the OS‐EH_∞F compared with some other typical filters, thereby verifying the effectiveness of using the weight to evaluate the estimation result of the filters.     

Consensus problem of high‐order multi‐agent systems with external disturbances: An H∞ analysis approach

This paper is devoted to the output consensus problem of directed networks of multiple high‐order agents with external disturbances, and proposes a distributed protocol using the neighbors' measured outputs. By defining an appropriate controlled output and conducting a model transformation in two steps, consensus performance analysis of the multi‐agent system under the proposed protocol is transformed into a normal H_∞ problem. Then using H_∞ theory of linear systems, conditions are derived to ensure the consensus performance with a prescribed H_∞ index for networks with fixed and switching topologies, respectively. A numerical example of the formation control application is included to validate the theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite horizon H∞ with parameter variations

In this paper we consider the finite horizon H_∞ performance robustness problem with parameter variations. Assuming a nominal model, an iterative procedure is given to synthesize a suboptimal H_∞ controller, which yields the required performance even under parameter variations. As a byproduct, an expression for the variation of performance due to parameter variations is given for this specific controller. An example which illustrates the methodology is worked out under parameter uncertainties.     

Robust H∞ Tracking Control of Uncertain Robotic Systems with Periodic Disturbances

This paper addresses the problem of designing robust tracking control for a large class of uncertain robotic systems. A more general model of the external disturbance is employed in the sense that the external disturbance can be expressed as the sum of a modeled disturbance and an unmodeled disturbance, for example, any periodic disturbance can be expressed in this general form. An adaptive neural network system is constructed to approximate the behavior of unknown robot dynamics. An adaptive control algorithm is designed to estimate the behavior of the modeled disturbance, and in turn the robust H_∞ control algorithm is required to attenuate the effects of the unmodeled disturbance only. Consequently, an intelligent adaptive/robust tracking control scheme is constructed such that an H_∞ tracking control is achieved in the sense that all the states and signals of the closed‐loop system are bounded and the effect due to the unmodeled disturbance on the tracking error can be attenuated to any preassigned level. Finally, simulations are provided to demonstrate the effectiveness and performance of the proposed control algorithm.     

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.     

Robust nonlinear ship course‐keeping control by H∞ I/O linearization and μ‐synthesis

In this paper, the H_∞ input/output (I/O) linearization formulation is applied to design an inner‐loop nonlinear controller for a nonlinear ship course‐keeping control problem. Due to the ship motion dynamics are non‐minimum phase, it is impossible to use the ordinary feedback I/O linearization to resolve. Hence, the technique of H_∞ I/O linearization is proposed to obtain a nonlinear H_∞ controller such that the compensated nonlinear system approximates the linear reference model in I/O behaviour. Then a μ‐synthesis method is employed to design an outer‐loop robust controller to address tracking, regulation, and robustness issues. The time responses of the tracking signals for the closed‐loop system reveal that the overall robust nonlinear controller is able to provide robust stability and robust performance for the plant uncertainties and state measurement errors. Copyright © 2002 John Wiley & Sons, Ltd.     

ADAPTIVE FUZZY‐BASED MIXED H2/H∞ TRACKING CONTROL DESIGN IN UNCERTAIN ROBOTIC SYSTEMS

In this study, an adaptive fuzzy‐based mixed H₂/H_∞ tracking control design is developed in robotic systems under unknown or uncertain plant parameters and external disturbances. The mixed H₂/H_∞ control design has the advantage of both H₂ optimal control performance and H_∞ robust control performance and the fuzzy adaptive control scheme is used to compensate for the plant uncertainties. By virtue of the skew‐symmetric property in the robotic systems and adequate choice of state variable transformation, sufficient conditions are developed for the adaptive fuzzy‐based mixed H₂/H_∞ tracking control problems in terms of a pair of coupled algebraic equations instead of a pair of coupled differential equations. The proposed methods are simple and the coupled algebraic equations can be solved analytically. Simulation results indicate that the desired performance of the proposed adaptive fuzzy‐based mixed H₂/H_∞ tracking control schemes for the uncertain robotic systems can be achieved.     

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.     

H∞ loop transfer recovery synthesis of discrete-time systems

This paper addresses an H_∞ disturbance attenuation problem of discrete-time systems, and presents a new discrete-time H_∞ loop transfer recovery (LTR) synthesis method. The method involves two design steps. First, an H_∞ state-feedback controller is designed to satisfy the design specifications, such as stability robustness and disturbance attenuation. Subsequently, the properties of the H_∞ state-feedback are reobtained by designing full/minimal order observers. A numerical examples is included to illustrate the presented method. © 1998 John Wiley & Sons, Ltd.     

Guaranteed H∞ robustness bounds for Wiener filtering and prediction

The paper deals with special classes of H_∞ estimation problems, where the signal to be estimated coincides with the uncorrupted measured output. Explicit bounds on the difference between nominal and actual H_∞ performance are obtained by means of elementary algebraic manipulations. These bounds are new in continuous‐time filtering and discrete‐time one‐step ahead prediction. As for discrete‐time filtering, the paper provides new proofs that are alternative to existing derivations based on the Krein spaces formalism. In particular, some remarkable H_∞ robustness properties of Kalman filters and predictors are highlighted. The usefulness of these results for improving the estimator design under a mixed H₂/H_∞ viewpoint is also discussed. The dualization of the analysis allows one to evaluate guaranteed H_∞ robustness bounds for state‐feedback regulators of systems affected by actuator disturbances. 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.     

Computation of infimal H∞ norm over a finite horizon

In this paper we develop necessary conditions for a minimax problem involving control and exogenous inputs. The problem can be regarded as a finite horizon version of the H_∞ optimal control problem. We consider problems involving generalized cost functional and non-zero initial conditions. A criterion for the evaluation of the performance index is given in these cases. Our computational experience shows that the finite horizon performance is useful in computing the infimal H_∞ norm in the infinite horizon case, as the final time becomes large. Also, expressions are derived for the variation in performance in terms of system parameter variations. These linear expressions are useful in the evaluation of the robustness of the proposed control strategy.     

MIXED H2/H∞ ADAPTIVE TRACKING CONTROL DESIGN FOR UNCERTAIN CONSTRAINED ROBOTS

In this study, a PID‐type controller incorporating an adaptive learning scheme for the mixed H₂/H_∞ tracking performance is developed for constrained robots under unknown or uncertain plant parameters and external disturbances. The mixed H₂/H_∞ control design has the advantage of both H₂ optimal control performance and H_∞ robust control performance and the adaptive control scheme is used to compensate the plant uncertainties. By virtue of the skew‐symmetric property of the constrained robotic systems and an adequate choice of state variable transformation, sufficient conditions are developed for the adaptive mixed H₂/H_∞ tracking control problems in terms of a pair of coupled algebraic equations instead of a pair of coupled nonlinear differential equations. The proposed methods are simple and the coupled algebraic equations can be solved analytically. Simulation results indicate that the desired performance of the proposed adaptive mixed H₂/H_∞ tracking control schemes for the uncertain constrained robotic systems can be achieved.     

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.     

A design of disturbance observer in standard H∞ control framework

The disturbance observer (DOB)‐based controller is widely used to estimate and suppress disturbance in motion control system. Because the low‐pass filter (Q‐filter) in DOB decides the performances of disturbance suppression, noise rejection, and robust stability against system uncertainties, design of Q‐filter is the principal task in DOB construction. This paper presents a systematic scheme for Q‐filter design based on H_∞ norm optimization. Cost function for optimization is proposed by considering performance and relative order condition of the Q‐filter. The norm minimization problem is then transformed to a standard H_∞ control problem. Furthermore, the relationship between performance and frequency weighting functions is investigated based on which selection of weighting functions is presented. Simulation results validate the global optimality and systematicness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

REGIONAL STABILITY AND H∞ PERFORMANCE CONTROL OF AN INPUT‐SATURATED INDUCTION MOTOR VIA LMI APPROACH

In this paper, the robust speed control of an induction motor is investigated, and a parameter‐dependent model is addressed. The proposed scheme includes the feedback of states and the integral of the tracking error. In practice, considering the specification of the drive circuit, the saturation phenomenon of control signals is addressed. It is noted that the saturated signal is represented as a perturbed signal, which can be further investigated to determine the achieved robustness. Regional stability and H_∞ performance are discussed in this paper, where the derivations of feedback gains are formulated in terms of linear matrix inequalities (LMIs). To evaluate the control performances, a DSP‐based scheme for an induction motor was set up. Experimental results, associated with versatile command profiles and load conditions, show that the proposed system achieves satisfactory performance from the viewpoint of tracking accuracy and disturbance rejection.     

H∞ tracking control for linear discrete‐time systems via reinforcement learning

In this paper, the H_∞ tracking control of linear discrete‐time systems is studied via reinforcement learning. By defining an improved value function, the tracking game algebraic Riccati equation with a discount factor is obtained, which is solved by iteration learning algorithms. In particular, Q‐learning based on value iteration is presented for H_∞ tracking control, which does not require the system model information and the initial allowable control policy. In addition, to improve the practicability of algorithm, the convergence analysis of proposed algorithm with a discount factor is given. Finally, the feasibility of proposed algorithms is verified by simulation examples.     

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.