Relating H2 and H∞-norm bounds forsampled-data systems

We relate the H_∞ and H₂ norms for multi-input/multi-output sampled-data feedback control systems, where a continuous-time plant is controlled by a digital compensator with hold and sampler. Upper bounds on both H₂ and H_∞ norms are obtained based on fundamental relations derived by two different approaches, namely the hybrid state-space approach and the fast sampling and lifting approach     

Discrete Fourier transform and H∞ approximation

It is shown that uniform rational approximation of nonrational transfer functions can always be obtained by means of the discrete Fourier transform (DFT) as long as such approximants exist. Based on this fact, it is permissible to apply the fast Fourier transform (FFT) algorithm in carrying out rational approximations without being apprehensive of convergence. The DFT is used to obtain traditional approximations for transfer functions of infinite-dimensional systems. Justification is provided for using the DFT in such approximations. It is established that whenever a stable transfer function can be approximated uniformly on the right half-plane by a rational function, its approximants can always be recognized by means of a DFT     

H∞-control of discrete-time nonlinear systems

This paper presents an explicit solution to the problem of disturbance attenuation with internal stability via full information feedback, state feedback, and dynamic output feedback, respectively, for discrete-time nonlinear systems. The H_∞-control theory is first developed for affine systems and then extended to general nonlinear systems based on the concepts of dissipation inequality, differential game, and LaSalle's invariance principle in discrete time. A substantial difficulty that V(A(x)+B(x)u+E(x)w) [respectively, V(f(x,u,w))] is no longer quadratic in [_w^u] arising in the case of discrete-time nonlinear systems has been surmounted in the paper. In the case of a linear system, we show how the results reduce to the well-known ones recently proposed in the literature     

Controller approximation: approaches for preserving H∞ performance

Investigates the design of reduced-order controllers using an H _∞ framework. Given a stabilizing controller which satisfies a prespecified level of closed-loop H_∞ performance, sufficient conditions are derived for another controller to be stabilizing and satisfy the same level of H_∞, performance. Such controllers are said to be (P,γ)-admissible, where P is the model of the plant under consideration and γ is the required level of prespecified H_∞ performance. The conditions are expressed as norm bounds on particular frequency-weighted errors, where the weights are selected to make a specific transfer function a contraction. The design of reduced-order (P,γ)-admissible controllers is then formulated as a frequency-weighted model reduction problem. It is advantageous for the required weights to be large in some sense. Solutions which minimize either the trace, or the determinant, of the inverse weights are characterized. We show that the procedure for minimizing the determinant of the inverse weights always gives a direction where the weights are the best possible. To conclude, we demonstrate by way of a numerical example, that when used in conjunction with a combined model reduction/convex optimization scheme, the proposed design procedures are effective in substantially reducing controller complexity     

Optimal solution and approximation to thel1/H∞ control problem

This paper addresses the l₁/H_∞ optimal control problem for a system described by linear time-invariant finite dimensional discrete-time equations. It is shown that a solution to this problem exists and can be approximated arbitrarily by real-rational transfer matrices. Perhaps more interesting from a computational point of view, a bound on the order of a δ-suboptimal solution is also given     

H∞ control of uncertain dynamical fuzzydiscrete-time systems

A new kind of dynamical fuzzy model is proposed to represent discrete-time complex systems which include both linguistic information and system uncertainties. A new stability analysis and control system design approach is then developed for this kind of dynamical fuzzy model. Furthermore, a constructive algorithm is developed to obtain the H(infinity) feedback control law. An example is given to illustrate the application of the method.     

H∞ state estimation for linear periodic systems

This note deals with the H_∞ state estimation problem for linear periodic systems. The question addressed is the design of an unbiased linear periodic and asymptotically stable estimator that achieves a prescribed H_∞ performance on an infinite horizon. Necessary and sufficient conditions for the existence of a periodic estimator have been derived. Asymptotic properties of the finite horizon estimation problem when the time-horizon tends to infinity are also investigated     

H∞ and positive-real control for linear neutraldelay systems

This note is concerned with the H_∞ and positive-real control problems for linear neutral delay systems. The purpose of H_∞ control is the design of a memoryless state feedback controller which stabilizes the neutral delay system and reduces the H_∞ norm of the closed-loop transfer function from the disturbance to the controlled output to a prescribed level, while the purpose of positive-real control is to design a memoryless state feedback controller such that the resulting closed-loop system is stable and the closed-loop transfer function is extended strictly positive real. Sufficient conditions for the existence of the desired controllers are given in terms of a linear matrix inequality (LMI). When this LMI is feasible, the expected memoryless state feedback controllers can be easily constructed via convex optimization     

Reliable H∞ control for affine nonlinear systems

This paper addresses the reliable H_∞-control problems for affine nonlinear systems. Based on the Hamilton-Jacobi inequality approach developed in the H∞-control problems for affine nonlinear systems, a method for the design of reliable nonlinear control systems is presented. The resulting nonlinear control systems are reliable in that they provide guaranteed local asymptotic stability and H_∞ performance not only when all control components are operational, but also in the case of some component outages within a prespecified subset of control components. A numerical example is also given     

H2 and H∞ control forinfinite-dimensional systems with jumps

We consider a semigroup model with jumps in the state that covers distributed parameter systems with impulse control or sampled-data distributed parameter systems with control realized through zero-order or first-order hold. We then introduce the H₂ and H_∞ problems for this system and give the solutions in terms of the solutions of Riccati equations with jumps     

H∞ control via measurement feedback for generalnonlinear systems

This paper shows how the problem of (local) disturbance attenuation via measurement feedback, with internal stability, can be solved for a nonlinear system of rather general structure. The solution of the problem is shown to be related to the existence of solutions of a pair of Hamilton-Jacobi inequalities in n independent variables, which are associated with state-feedback and, respectively, output-injection design     

H2 and H∞ robust filtering for convexbounded uncertain systems

Investigates robust filtering design problems in H₂ and H_∞ spaces for continuous-time systems subjected to parameter uncertainty belonging to a convex bounded-polyhedral domain. It is shown that, by a suitable change of variables, both designs can be converted into convex programming problems written in terms of linear matrix inequalities. The results generalize the ones available in the literature to date in several directions. First, all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured. Then, assuming the order of the uncertain system is known, the optimal guaranteed performance H₂ and H_∞ filters are proven to be of the same order as the order of the system. A numerical example illustrate the theoretical results     

Time-domain H∞ identification

The problem of parameter identification, for single-input, single-output ARX systems, is considered. Recent results in H_∞-nonlinear filtering are used to formulate a nonlinear H_∞ time-domain prediction-error-modeling (PEM) identification method. The performance of the new method is guaranteed by a preassigned bound on the ratio between the energy of the prediction error of the obtained model and the energy of the exogenous disturbances. The potential usefulness of the H_∞ time-domain identification method is illustrated by a numerical example     

Low-rank matrix decomposition in L1-norm by dynamic systems

Low-rank matrix approximation is used in many applications of computer vision, and is frequently implemented by singular value decomposition under L₂-norm sense. To resist outliers and handle matrix with missing entries, a few methods have been proposed for low-rank matrix approximation in L₁ norm. However, the methods suffer from computational efficiency or optimization capability. Thus, in this paper we propose a solution using dynamic system to perform low-rank approximation under L₁-norm sense. From the state vector of the system, two low-rank matrices are distilled, and the product of the two low-rank matrices approximates to the given measurement matrix with missing entries, in L₁ norm. With the evolution of the system, the approximation accuracy improves step by step. The system involves a parameter, whose influences on the computational time and the final optimized two low-rank matrices are theoretically studied and experimentally valuated. The efficiency and approximation accuracy of the proposed algorithm are demonstrated by a large number of numerical tests on synthetic data and by two real datasets. Compared with state-of-the-art algorithms, the newly proposed one is competitive.     

H∞ strong stabilization

This note presents a technique for designing stable H_∞ controllers. Similar to some methods in the existing literature, the proposed method also uses the parameterization of all suboptimal H _∞ controllers so that the stable H_∞ design problem can be (conservatively) converted into another 2-block standard H_∞ problem. However, a weighting function is introduced in this method to alleviate the conservativeness of the previous formulations. It is further shown that the resulting high-order controller can be significantly reduced by a two-step reduction algorithm. Numerical examples are presented to demonstrate the effectiveness of the proposed method     

Mixed l1/H∞ control of MIMO systems viaconvex optimization

We present a methodology for designing mixed l₁/H_∞ controllers for MIMO systems. These controllers allow for minimizing the worst case peak output due to persistent disturbances, while at the same time satisfying an H_∞-norm constraint upon a given closed loop transfer function. Therefore, they are of particular interest for applications dealing with multiple performance specifications given in terms of the worst case peak values, both in the time and frequency domains. The main results of the paper show that: 1) contrary to the H₂/H_∞ case, the l₁/H_∞ problem admits a solution in l₁; and 2) rational suboptimal controllers can be obtained by solving a sequence of problems, each one consisting of a finite-dimensional convex optimization and a four-block H_∞ problem. Moreover, this sequence of controllers converges in the l₁ topology to an optimum     

Robust H∞ performance for lineardelay-differential systems with time-varying uncertainties

The present paper concerns robust H∞ performance for linear delay-differential systems which involve an uncertain time delay and time-varying norm-bounded parameter uncertainties. Based on the Lyapunov functional, a simple criterion is proposed which assures the pseudo-quadratic stability as a well as an H_∞-norm bound. The criterion is given in the form of a linear matrix inequality which is affine or convex in additional scalar parameters. A simple criterion is presented to evaluate the extent of the performance robustness     

Stabilizing receding horizon H∞ controls forlinear continuous time-varying systems

In this note, new matrix inequality conditions on the terminal weighting matrices are proposed for linear continuous time-varying systems. Under these conditions, nonincreasing and nondecreasing monotonicities of the saddle point value of a dynamic game are shown to be guaranteed. It is proved that the proposed terminal inequality conditions ensure the closed-loop stability of the receding horizon H _∞ control (RHHC). The stabilizing RHHC guarantees the H _∞ norm bound of the closed-loop system. The proposed terminal inequality conditions for the monotonicity of the saddle point value and the closed-loop stability include most well-known existing terminal conditions as special cases. The results for time-invariant systems are obtained correspondingly from those in the time-varying case     

On robust H∞ control for nonlinear discrete andsampled-data systems

The linearization technique for H_∞ control for nonlinear discrete and sampled-data systems is developed in a unified framework under minimal assumptions. It is shown directly, without using any Riccati or Hamilton-Jacobi-Isaacs equation, that H_∞ control for a linearized system is also H_∞ control for a nonlinear original one     

H∞-norm and invariant manifolds of systems with state delays

The problem of finding bounds on the H_∞-norm of systems with a finite number of point delays and distributed delay is considered. Sufficient conditions for the system to possess an H_∞-norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial differential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stable manifold of the associated Hamiltonian system. For small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the Hamiltonian system. This leads to an algebraic, finite-dimensional, criterion for systems with small delays.