Singular H∞ control for nonlinear systems

This paper presents a solution to the singular H_∞ control problem via state feedback for a class of nonlinear systems. It is shown that the problem of almost disturbance decoupling with stability plays a fundamental role in the solution of the considered problem. We also point out when the singular problem can be reduced to a regular one or solved via standard H_∞ technique. We must stress that the solution of the singular problem is obtained without making any approximation of it by means of regular problems. © 1997 John Wiley & Sons, Ltd.     

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     

H∞ control for a class of cascade switched nonlinear systems

In this paper, we investigate the H_∞ control problem for a class of cascade switched nonlinear systems consisting of two nonlinear parts which are also switched systems using the multiple Lyapunov function method. Firstly, we design the state feedback controller and the switching law, which guarantees that the corresponding closed‐loop system is globally asymptotically stable and has a prescribed H_∞ performance level. This method is suitable for a case where none of the switched subsystems is asymptotically stable. Then, as an application, we study the hybrid H_∞ control problem for a class of nonlinear cascade systems. Finally, an example is given to illustrate the feasibility of our results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

A new approach for the design of robust H_∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H_∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.     

On H∞-control of discrete-time nonlinear systems

Following recent works on continuous-time nonlinear H_∞-control, where connections with game theory and passivity conditions have been set, the present paper studies the corresponding problem for discrete-time systems. The paper describes sufficient conditions for the existence and the construction of a feedback law which imposes a prescribed level of disturbance attenuation with internal stability. Both cases of state feedback and measurement feedback are considered.     

Robust control of constrained systems via convex optimization

A successful controller design paradigm must take into account both model uncertainty and design specifications. Model uncertainty can be successfully addressed using ??^∞ robust control theory. However, this framework cannot directly accommodate the realistic case where in addition to robustness considerations the system is subject to both time- and frequency-domain specifications, such as bounds on the control action. In this paper we propose a design procedure, based upon the use of convex optimization, that takes explicitly into account both time- and frequency-domain specifications. The main result of the paper is a new framework to address problems having both control and output constraints and model uncertainty. Additionally, the paper serves as a brief tutorial on the issues involved in addressing design problems with multiple design specifications via convex optimization.     

Robust H∞ nonlinear control via measurement feedback

This paper deals with the problem of finite-time-horizon robust H_∞ control via measurement feedback, for affine nonlinear systems with nonlinear time-varying parameter uncertainty. The problem addressed is the design of a control law, which processes the measured output and guarantees a prescribed level of closed-loop disturbance attenuation. Conditions for the existence of such a controller are obtained by solving an auxiliary control problem for a related system which is obtained from the original one by converting the parameter uncertainty into exogenous bounded energy signals. This approach allows us to apply the recently developed H_∞ nonlinear control techniques to solve the robust control problem. The problem is investigated in both the continuous- and discrete-time cases. The results are demonstrated by a simple example. © 1997 by John Wiley & Sons, Ltd.     

Robust static output feedback design for polynomial nonlinear systems

A computational scheme of solving the nonlinear static output feedback design problems for a class of polynomial nonlinear systems is investigated in this paper. Sufficient conditions to achieve the closed‐loop stability with or without H_∞ performance are presented as state‐dependent matrix inequalities, which provides an effective way for the application of the new sum of squares programming technique to obtain computationally tractable solutions. By introducing additional matrix variables, we succeed in eliminating the coupling between system matrices and the Lyapunov matrix. The proposed methodology is also extended to the synthesis for the parameter‐dependent polynomial systems. Robust polynomial output feedback controller is designed in an efficient computational manner. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

Reduction of H∞ state feedback control problems for the MIMO servo systems

We deal with H_∞ state feedback control problem for the multi‐input‐multi‐output (MIMO) servo system and discuss the advantages of the facial reduction (FR) to the resulting linear matrix inequality (LMI) problems. In fact, as far as our usual setting, the dual of the LMI problem is not strictly feasible because the generalized plant has always stable invariant zeros. Thus FR is available to such LMI problems, and we can reduce and simplify the original LMI problem to a smaller‐size LMI problem. As a result, we observe that the numerical performance of the SDP solvers is improved. Also, as a by‐product, we obtain the best performance index of the reduced LMI problem with a closed‐form expression. This helps the H_∞ performance limitation analysis. Another contribution is to reveal that the resulting LMI problem obtained from H_∞ control problem has a finite optimal value, but no optimal solutions under an additional assumption. This is also confirmed in the numerical experiment of this paper. FR also plays an essential role in this analysis.     

PID controller optimization for H∞ control by linear programming

The constraints on the PID gains, which are derived from the H_∞ norm performance index by discretization of the frequency, are convex or concave depending on frequencies. This problem is a non‐convex problem, and a new method of approximating these constraints as adequate linear inequalities is proposed. Then, the optimal solution can be efficiently and successfully searched for by applying linear programming iteratively. This method is compared with methods based on barrier function and linear matrix inequality. Copyright © 2000 John Wiley & Sons, Ltd.     

Event‐triggered output feedback H∞ control for networked control systems

This paper investigates event‐triggered output feedback H_∞ control for a networked control system. Transmitted through a network under an event‐triggered scheme, the sample outputs of the plant are used to drive the dynamical output feedback controller to generate a new control signal in the discrete‐time domain. The discrete‐time control signals are also transmitted through the network to drive the plant. As a result of two types of transmission delays, the controlled plant and the dynamical output feedback controller are driven by the discrete‐time outputs and control signals at different instants of time. An interval decomposition method is introduced to place the controlled plant and the output feedback controller into the same updated time interval but with updated signals at different instants. Based on a proper Lyapunov‐Krasovskii functional, sufficient conditions are derived to ensure H_∞ performance for the controlled plant. Finally, numerical simulations are used to demonstrate the practical utility of the proposed method.     

Robust performance control of uncertain nonlinear systems: A self-tuning approach

The problem of output regulation with a guaranteed H^∞ performance, besides robust stability, for a class of feedback linearizable nonlinear systems via self-tuning controllers is investigated. The H^∞ performance consists of the desired disturbance attenuation and internal finite L²-gain stability. We show that, under the perturbations of matched parametric uncertainties, the sufficient condition for the existence of a self-tuning controller reduces to the solvability of a single Hamilton-Jacobi-Isaacs inequality (or algebraic Riccati equation) which indicates that the design can be performed as if the system uncertainties were absent. Under certain situations, the sufficient condition is also necessary. Once a solution of this nonlinear differential inequality (or algebraic equation) is available, a desired self-tuning controller with gradient-type parameter estimator can easily be constructed. The present work falls into the category of singular nonlinear H^∞ control since the desired H^∞ performance does not require any penalty on control input variables. The results also provide an immediate application to H^∞ self-tuning model reference control in linear systems under full state measurement.     

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.     

Robust passivity‐based H∞ control for uncertain switched nonlinear systems

In this paper, we investigate the H_∞ control problem for uncertain switched nonlinear systems with passive and non‐passive subsystems. For any given average dwell time, any given passivity rate and any given disturbance attenuation level, we design feedback controllers of subsystems, which may depend on the pre‐given constants, to solve the H_∞ control problem for the uncertain switched nonlinear systems for all admissible uncertainties. For linear systems, the exponential small‐time norm‐observability is shown to be preserved under disturbance. Two examples are provided to demonstrate the effectiveness of the proposed design method. Copyright © 2016 John Wiley & Sons, Ltd.     

H∞ tracking of linear continuous‐time systems with stochastic uncertainties and preview

The problem of finite‐horizon H_∞ tracking for linear continuous time‐invariant systems with stochastic parameter uncertainties is investigated for both, the state‐feedback and the output‐feedback control problems. We consider three tracking patterns depending on the nature of the reference signal i.e. whether it is perfectly known in advance, measured on line or previewed in a fixed time‐interval ahead. The stochastic uncertainties appear in both the dynamic and measurement matrices of the system. In the state‐feedback case, for each of the above three cases a game theory approach is applied where, given a specific reference signal, the controller plays against nature which chooses the initial condition and the energy‐bounded disturbance. The problems are solved using the expected value of the standard performance index over the stochastic parameters, where, in the state‐feedback case, necessary and sufficient conditions are found for the existence of a saddle‐point equilibrium. The corresponding infinite‐horizon time‐invariant tracking problem is also solved for the latter case, where a dissipativity approach is considered. The output‐feedback control problem is solved as a max–min problem for the three tracking patterns, where necessary and sufficient condition are obtained for the solution. The theory developed is demonstrated by a simple example where we compare our solution with an alternative solution which models the tracking signal as a disturbance. Copyright © 2004 John Wiley & Sons, Ltd.     

Integrated fault estimation and fault‐tolerant control for uncertain Lipschitz nonlinear systems

This paper proposes an integrated fault estimation and fault‐tolerant control (FTC) design for Lipschitz non‐linear systems subject to uncertainty, disturbance, and actuator/sensor faults. A non‐linear unknown input observer without rank requirement is developed to estimate the system state and fault simultaneously, and based on these estimates an adaptive sliding mode FTC system is constructed. The observer and controller gains are obtained together via H_∞ optimization with a single‐step linear matrix inequality (LMI) formulation so as to achieve overall optimal FTC system design. A single‐link manipulator example is given to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

The discrete-time H∞ control problem with measurement feedback

This paper is concerned with the discrete-time H_∞ control problem with measurement feedback. We extend previous results by having weaker assumptions on the system parameters. We also show explicitly the structure of H_∞ controllers. Finally, we show that it is in certain cases possible, without loss of performance, to reduce the dynamical order of the controllers.     

H∞ control and filtering with initial uncertainty for infinite dimensional systems

The H_∞-control problem with a non-zero initial condition for infinite dimensional systems is considered The initial conditions are assumed to be in some subspace. First the H_∞ problem with full information is considered and necessary and sufficient conditions for the norm of an input-output operator to be less than a given number are obtained, The characterization of all admissible controllers is also given. This result is then used to solve the general H^∞ control problem and the filtering problem with initial uncertainty. The filtering problem on finite horizon involves the estimate of the state at final time. The set of all suboptimal filters is given both on finite and infinite horizons.