Robust stabilization of nonlinear systems by H∞ state feedback

This paper is concerned with robust stabilization of nonlinear systems with unstructured uncertainty via state feedback. First, a robust stability condition is given for a closed loop system which is composed of a nonlinear nominal system and an unstructured uncertainty. Second, based on the obtained robust stability condition, a sufficient condition for robust stabilization by state feedback is given in terms of the solvability of some H_∞ state feedback control.     

Robust H∞ control for uncertain discrete-time systems with circular pole constraints

This paper investigates the problem of robust H_∞ control for uncertain discrete-time systems with circular pole constraints. The system under consideration is subject to norm-bounded time-invariant uncertainties in both the state and input matrices. The problem we address is to design state feedback controllers such that the closed poles are located within a prespecified circular region, and the H_∞ norm of the closed-loop transfer function is strictly less than a given positive scalar for all admissible uncertainties. By introducing the notion of quadratic d stabilizability with an H_∞ norm-bound, the problem is solved. Necessary and sufficient conditions for quadratic d stabilizability with an H_∞ norm-bound are derived. Our results can be regarded as extensions of existing results on robust H_∞ control and robust pole assignment of uncertain systems.     

Relating H2 and H∞ bounds for finite-dimensional systems

For a linear time invariant system, the infinity-norm of the transfer function can be used as a measure of the gain of the system. This notion of system gain is ideally suited to the frequency domain design techniques such as H_∞ optimal control. Another measure of the gain of a system is the H₂ norm, which is often associated with the LQG optimal control problem. The only known connection between these two norms is that, for discrete time transfer functions, the H₂ norm is bounded by the H_∞ norm. It is shown in this paper that, given precise or certain partial knowledge of the poles of the transfer function, it is possible to obtain an upper bound of the H_∞ norm as a function of the H₂ norm, both in the continuous and discrete time cases. It is also shown that, in continuous time, the H₂ norm can be bounded by a function of the H_∞ norm and the bandwidth of the system.     

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.     

H∞ filtering for a class of uncertain nonlinear systems

This paper investigates the problem of H_∞ filtering for a class of uncertain continuous-time nonlinear systems with real time-varying parameter uncertainty and unknown initial state. We develop an infinite horizon H_∞ filtering methodology which provides both robust stability and a guaranteed H_∞ performance for the filtering error irrespective of the parameter uncertainty.     

H∞ controller synthesis for pendulum-like systems

This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H_∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satisfied, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs).     

H∞ control design in bilinear systems: a tensor formal series approach

In this paper, the H_∞ disturbance attenuation problem of bilinear system is discussed. Dynamic game theory is used to solve this bilinear minimax problem. The solvability of H_∞ disturbance attenuation in bilinear system is also discussed. The techniques of tensor products and formal power series are employed to solve the nonlinear Bellman-Isaac differential equation. Furthermore, the convergence for the tensor formal series approach of this H_∞ control problem is discussed, and the radius of convergence for this H_∞ control to be well defined is also obtained.     

H∞ control for a class of structured time-delay systems

In this paper, we design an H_∞ controller for a class of lower-triangular time-delay systems. Backstepping is applied to construct an explicit feedback controller, and the closed-loop system maintains internal stability and an L₂-gain from the disturbance input to the output. The design is delay-dependent. Simulations on an example system demonstrate the good performance of the proposed design.     

Static output feedback stabilisation with H∞ performance for a class of plants

In this paper, the problem of static output feedback control of a linear system is considered. The existence of a static output feedback control law is given in terms of the solvability of two coupled Lyapunov inequalities which result in a non-linear optimisation problem. However, using state-coordinate and congruence transformations and by imposing a block-diagonal structure on the Lyapunov matrix, we will see that the determination of a static output feedback gain reduces, for a specific class of plants, to finding the solution of a system of linear matrix inequalities. The class of plants considered is those which are minimum phase with a full row rank Markov parameter. The method is extended to incorporate H_∞ performance objectives. This results in a sub-optimal static H_∞ control law found by non-iterative means. The simplicity of the method is demonstrated by a numerical example.     

On stability, -gain and control for switched systems

This paper addresses the issues of stability, L₂-gain analysis and H_∞ control for switched systems via multiple Lyapunov function methods. A concept of general Lyapunov-like functions is presented. A necessary and sufficient condition for stability of switched systems is given in terms of multiple generalized Lyapunov-like functions, which enables derivation of improved stability tests, an L₂-gain characterization and a design method for stabilizing switching laws. A solution to the H_∞ control problem for switched systems is also provided.     

Quadratic stabilisation with H∞-norm bound of non-linear discrete-time uncertain systems with bounded control

The paper addresses the problem of quadratic stabilisability with H_∞-norm bound of uncertain discrete-time control-affine systems by norm-bounded controls. Both structured parameter uncertainties and unstructured exogenous disturbances are taken into account. The given definition of quadratic stabilisability is a generalisation of that used for linear systems so far. A necessary condition of the stabilisability is formulated. A state feedback control satisfying an a priori constraint is proposed for the solution of the formulated H_∞ problem. The proposed method may be applicable even in such cases when the linearisation technique cannot be used.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

H∞ filtering for linear periodic systems with parameter uncertainty

This paper focuses on H_∞ filtering for a class of linear periodic systems with a certain type of norm-bounded time-varying parameter uncertainty which appears in both the state and output matrices. The problem addressed is the design of a linear periodic estimator that guarantees both the quadratic stability and and prescribed H_∞ performance on infinite horizon for the estimation error for all admissible parameter uncertainties. A solution to this problem is obtained via a Riccati equation approach.     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

Robust H∞ control for linear discrete-time systems with norm-bounded time-varying uncertainty

The problem of robust H_∞ analysis and synthesis for linear discrete-time systems with norm-bounded time-varying uncertainty is studied in this paper. It will be shown that this problem is equivalent to the problem of H_∞ analysis and synthesis of an auxiliary system. The necessary and sufficient conditions for the equivalency are proved. Thus the original problem can be solved by existing H_∞ control methods.     

output feedback control for uncertain stochastic systems with time-varying delays

This paper deals with the problem of H_∞ output feedback control for uncertain stochastic systems with time-varying delays. The parameter uncertainties are assumed to be time-varying norm-bounded. The aim is the design of a full-order dynamic output feedback controller ensuring robust exponential mean-square stability and a prescribed H_∞ performance level for the resulting closed-loop system, irrespective of the uncertainties. A sufficient condition for the existence of such an output feedback controller is obtained and the expression of desired controllers is given.     

Robust control of uncertain differential linear repetitive processes

For two-dimensional (2-D) systems, information propagates in two independent directions. 2-D systems are known to have both system-theoretical and applications interest, and the so-called linear repetitive processes (LRPs) are a distinct class of 2-D discrete linear systems. This paper is concerned with the problem of L₂–L_∞ (energy to peak) control for uncertain differential LRPs, where the parameter uncertainties are assumed to be norm-bounded. For an unstable LRP, our attention is focused on the design of an L₂–L_∞ static state feedback controller and an L₂–L_∞ dynamic output feedback controller, both of which guarantee the corresponding closed-loop LRPs to be stable along the pass and have a prescribed L₂–L_∞ performance. Sufficient conditions for the existence of such L₂–L_∞ controllers are proposed in terms of linear matrix inequalities (LMIs). The desired L₂–L_∞ dynamic output feedback controller can be found by solving a convex optimization problem. A numerical example is provided to demonstrate the effectiveness of the proposed controller design procedures.     

On the gap between H2 and entropy performance measures in H∞ control design

Recent papers have considered the problem of minimizing an entropy functional subject to an H_∞ performance constraint. Since the entropy is an upper bound for the H₂ cost, there remains a gap between entropy minimization and H₂ minimization. In this paper we consider a generalized cost functional involving both H₂ and entropy aspects. This approach thus provides a means for optimizing H₂ performance within H_∞ control design.     

Bounded real lemma and robust control of 2-D singular Roesser models

This paper discusses the problem of robust H_∞ control for linear discrete time two-dimensional (2-D) singular Roesser models (2-D SRM) with time-invariant norm-bounded parameter uncertainties. The purpose is the design of static output feedback controllers such that the resulting closed-loop system is acceptable, jump modes free, stable and satisfies a prescribed H_∞ performance level for all admissible uncertainties. A version of bounded realness of 2-D SRM is established in terms of linear matrix inequalities. Based on this, a sufficient condition for the solvability of the robust H_∞ control problem is solved, and a desired output feedback controller can be constructed by solving a set of matrix inequalities. A numerical example is provided to demonstrate the applicability of the proposed approach.