Nonlinear H∞-control and estimation of optimal H∞-gain

The structure of nonlinear H_∞-controller and the estimation of optimal H_∞-gain are investigated in this paper. The essential problem boils down to the existence of semipositive solution to the Hamilton-Jacobi-Isaacs inequality. Further, the solvability of this first-order fully nonlinear differential inequality is discussed. We look for one kind of special semipositive radial solution to the Hamilton-Jacobi-Isaacs inequality. An explicit estimation of optimal H_∞-gain and explicit formulas of semipositive radial solutions to the Hamilton-Jacobi-Isaacs inequality are obtained. The results are quite simple and intuitive. They even shed a new insight on the linear H_∞-theory.     

A Piecewise Envelope Approach to H ∞ Control of Nonlinear Systems

This paper provides a computable approach for the H _∞ control synthesis of nonlinear systems using the piecewise envelope method. Firstly, the computation method of piecewise envelope is presented such that the required piecewise linear differential inclusions (PWLDIs) can be obtained by solving a linear optimization problem. Then, we propose the H _∞ control design method for PWLDIs using piecewise Lyapunov function theory, all the synthesis conditions are formulated as the feasibility of linear matrix inequalities, hence computationally tractable. Because of the inclusion relationship, the H _∞ controllers designed for PWLDIs are also effective for original nonlinear systems. The validity of the proposed approach is tested by application to control a wheeled mobile robot.     

Dissipativity and nonlinear systems with finite power gain

In this paper we present a generalization of the concept of dissipativity to include systems which exhibit finite power gain. Included in this is a generalization of supply rate, available storage, and the integral and differential forms of the appropriate dissipation inequality. We show that under certain conditions, systems which are power dissipative are also stable in the sense that the trajectory is eventually bounded. © 1998 John Wiley & Sons, Ltd.     

High gain observer for a class of non-triangular systems

This paper presents a high gain observer for a class of MIMO nonlinear systems involving some uncertainties. The latter is particularly composed of cascade subsystems where each subsystem is associated with a subset of the output variables, and assumes a triangular dependence on its own state variables and may depend on the state variables of all other subsystems. The main contribution consists in extending the available results to allow more interconnections between the subsystems. Of fundamental interest, it is shown that the underlying observation error exponentially converges to zero in the absence of uncertainties. Moreover, the observation error can be made as small as desired by properly specifying the observer design parameter in the case where uncertainties are considered.     

Some applications of small gain theorem to interconnected systems

Based on small gain theorem, decentralized control of interconnected systems is studied. Linear matrix inequality conditions are given for stabilizability of interconnected systems. Interconnections can be used to improve system robustness in the sense of small gain theorem. An example is given to illustrate the results. Finally, interconnected systems composed of symmetric subsystems are studied.     

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.     

A simpler formula for the singular values of a certain Hankel operator

This paper deals with the problem of computing the singular values and vectors of a Hankel operator with symbol m^*W where m ε H^∞ is arbitrary inner and W ε H^∞ is rational. A simplified version of the formula given in [6] is obtained for computing the singular values of the Hankel operator. This result is applied to the (one-block) H^∞ optimal control of SISO stable infinite dimensional plants and rational weights. Using this new formula a simple expression is derived for the H^∞ optimal controller whose structure was observed in [9].     

On computing the worst-case peak gain of linear systems

Based on the bounds due to Doyle and Boyd, we present simple upper and lower bounds for the l¹-norm of the ‘tail’ of the impulse response of finite-dimensional discrete-time linear time-invariant systems. Using these bounds, we may in turn compute the l^∞-gain of these systems to any desired accuracy. By combining these bounds with results due to Khammash and Pearson, we derive upper and lower bounds for the worst-case l^∞-gain of discrete-time systems with diagonal perturbations.     

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.     

Discovering inequality conditions in the analytic solution of optimization problems

Necessary and sufficient conditions for the problem of maximizing or minimizing a function subject to inequality constraints are given by a set of equalities and inequalities known as the Kuhn-Tucker conditions. These conditions can provide an analytic solution to the optimization problem if the artificial variables known as Lagrange multipliers can be eliminated. However, this is tedious to do by hand. This paper develops a computer program to assist in the solution process which combines symbolic computation and automated reasoning techniques. The program may also be useful for other problems involving algebraic reasoning with inequalities which employ general functions or symbolic parameters.     

Nonlinear robust H∞ control via dynamic output feedback

The problem on robust H_∞ control for a class of nonlinear systems with parameter uncertainty is studied. Sufficient conditions for the existence of the dynamic output feedback controller are obtained. Under these conditions, the closed-loop systems have robust H_∞-performance. A numerical example is given to illustrate the design of a robust controller using the proposed approach.     

New stability criteria of singular systems with time-varying delay via free-matrix-based integral inequality

This paper concerns the stability problem of singular systems with time-varying delay. First, the singular system with time-varying delay is transformed into the neutral system with time-varying delay. Second, a more proper Lyapunov–Krasovskii functional (LKF) is constructed by adding some integral terms to quadratic forms. Then, to obtain less conservative conditions, the free-matrix-based integral inequality is adopted to estimate the derivative of LKF. As a result, some delay-dependent stability criteria are given in terms of linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness and superiority of the proposed method.     

Remarks on the adaptive control of linear plants with unknown high-frequency gain

Adaptive control of first-order linear systems of the form $\text{x}\text{?}\text{= ax + bu}$ with the sign of b unknown is considered. It is shown that stabilizing adaptive controllers exist that have globally bounded gain and prespecified terminal dynamics.     

A new H∞ design methodology for the control of flexible structures: Application to a sight system

We present an Hscr;^∞ controller design methodology for flexible systems. We illustrate our approach using an example drawn from an industrial application. The proposed method avoids undesirable (lightly damped) pole-zero cancellations and yields a low-order controller, using a new formulation of the standard problem.     

Robust stabilization of nonlinear plants—an L2 approach

This paper extends the theory of H_∞ control for linear plants to input affine nonlinear plants without special output structure. This result is used to develop solutions to the robust stabilization problem for several classes of uncertain nonlinear plants.     

Robust H∞ control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers

This paper considers the problem of robust H_∞ control for uncertain discrete systems with time-varying delays. The system under consideration is subject to time-varying norm-bounded parameter uncertainties in both the state and measured output matrices. Attention is focused on the design of a full-order exponential stable dynamic output feedback controller which guarantees the exponential stability of the closed-loop system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties. In terms of a linear matrix inequality (LMI), a sufficient condition for the solvability of this problem is presented, which is dependent on the size of the delay. When this LMI is feasible, the explicit expression of the desired output feedback controller is also given. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.     

Robust stabilization of nonlinear systems via stable kernel representations with L2-gain bounded uncertainty

The approach to robust stabilization of linear systems using normalized left coprime factorizations with _∞ bounded uncertainty is generalized to nonlinear systems. A nonlinear perturbation model is derived, based on the concept of a stable kernel representation of nonlinear systems. The robust stabilization problem is then translated into a nonlinear disturbance feedforward _∞ optimal control problem, whose solution depends on the solvability of a single Hamilton-Jacobi equation.