We consider the characterization and computation of H-infinity norms for a class of time-delay systems. It is well known that in the finite dimensional case the H-infinity norm of a transfer function can be computed using the connections between the corresponding singular value curves and the imaginary axis eigenvalues of a Hamiltonian matrix, leading to the established level set methods. We show a similar connection between the transfer function of a time-delay system and the imaginary axis eigenvalues of an infinite dimensional linear operator $\mathcal{L}_ξ^N$. Based on this result, we propose a predictor-corrector algorithm for the computation of the H-infinity norm. In the prediction step, a finite-dimensional approximation of the problem, induced by a spectral discretization of the operator $\mathcal{L}_ξ$, and an adaptation of the algorithms for finite-dimensional systems, allow to obtain an approximation of the H-infinity norm of the transfer function of the time-delay system. In the next step the approximate results are corrected to the desired accuracy by solving a set of nonlinear equations which are obtained from the reformulation of the eigenvalue problem for the linear infinite-dimensional operator $\mathcal{L}_ξ^N$ as a finite dimensional nonlinear eigenvalue problem. These equations can be interpreted as characterizations of peak values in the singular value plot. The effects of the discretization in the predictor step are fully characterized and the choice of the number of discretization points is discussed. The paper concludes with a numerical example and the presentation of the results of extensive benchmarking.
H-infinity controllers are frequently used in control theory due to their robust performance and stabilization. Classical H-infinity controller synthesis methods for finite dimensional LTI MIMO plants result in high-order controllers for high-order plants whereas low-order controllers are desired in practice. We design fixed-order H-infinity controllers for a class of time-delay systems based on a non-smooth, non-convex optimization method and a recently developed numerical method for H-infinity norm computations.
We analyze and design H-infinity controllers for general time-delay systems with time-delays in systems' state, inputs and outputs. We allow the designer to choose the order of the controller and to introduce constant time-delays in the controller. The closed-loop system of the plant and the controller is modeled by a system of delay differential algebraic equations (DDAEs). The advantage of the DDAE modeling framework is that any interconnection of systems and controllers prone to various types of delays can be dealt with in a systematic way, without using any elimination technique. We presen…
This manuscript discusses a scalable controller synthesis method for networked systems with a large number of identical subsystems based on the H-infinity control framework. The dynamics of the individual subsystems are described by identical linear time-invariant delay differential equations and the effect of transport and communication delay is explicitly taken into account. The presented method is based on the result that, under a particular assumption on the graph describing the interconnections between the subsystems, the H-infinity norm of the overall system is upper bounded by the robus…
We present SSD, Software for Systems with Delays, a de novo MATLAB package for the analysis and model reduction of retarded time delay systems (RTDS). Underneath, our delay system object bridges RTDS representation and Linear Fractional Transformation (LFT) representation of MATLAB. This allows seamless use of many available visualizations of MATLAB. In addition, we implemented a set of key functionalities such as H2 norm and system gramian computations, balanced realization and reduction by direct integral definitions and utilizing sparse computation. As a theoretical contribution, we extend …
The finite time stability (FnTSta) theory of delayed systems has not been set up until now. In this paper, we propose a two-phases-method (2PM), to achieve this object. In the first phase, we prove that the time for norm of system error evolving from initial values to 1 is finite; then in the second phase, we prove that the time for norm of the error evolving from 1 to 0 is also finite, thus FnTSta is obtained. Considering the cost and complexity of controller, we use only two simple terms to realize this aim. For the proposed 2PM, time delays can be the same or asynchronous, bounded or unboun…
This paper presents a characterization of distributed controllers subject to delay constraints induced by a strongly connected communication graph that achieve a prescribed closed loop $\mathcal{H}_\infty$ norm. Inspired by the solution to the $\mathcal{H}_2$ problem subject to delays, we exploit the fact that the communication graph is strongly connected to decompose the controller into a local finite impulse response component and a global but delayed infinite impulse response component. This allows us to reduce the control synthesis problem to a linear matrix inequality feasibility test.
We develop a novel frequency-based H-infinity control method for a large class of infinite-dimensional Linear-Time-Invariant systems in transfer function form. Major benefits of our approach is that reduction or identification techniques are not needed thereby avoiding possible distortions. It can exploit either transfer function models or input/output frequency response data when available. It computes simple practical controllers of any size and structure. We use a non-smooth trust-region method to compute arbitrarily structured locally optimal H-infinity controllers for a frequency sampled …
Low-Rank Matrix Approximation in the Infinity Norm
Department of Mathematics and Operational Research, University of Mons（蒙斯大学数学与运筹学系）；National Research University Higher School of Economics（俄罗斯国家研究大学高等经济学院）
In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a p-norm of eigenvalues of the Dirichlet-Laplace operator. The extremal case of the infinity norm, where we minimize the largest fundamental eigenvalue of each cell, is one of our main interests. We propose three numerical algorithms which approximate the optimal configurations and we obtain tight upper bounds for the energy, which are better than the ones given by theoretical results. A thorough comparison of the results obtained by the three methods is given. We also i…
