Generalized Hermite-Biehler Theorem Research Articles

A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices

The classical Hermite–Biehler theorem describes the zero configuration of a complex linear combination of two real polynomials whose zeros are real, simple, and strictly interlace. We provide the full characterization of the zero configuration for the case when this interlacing is broken at exactly one location. We apply this result to solve the direct and inverse spectral problem for non-Hermitian rank-one multiplicative perturbations and rank-two additive perturbations of finite Hermitian and Jacobi matrices.

Stabilizing Parametric Region of Multiloop PID Controllers for Multivariable Systems Based on Equivalent Transfer Function

The aim of this paper is to determine the stabilizing PID parametric region for multivariable systems. Firstly, a general equivalent transfer function parameterization method is proposed to construct the multiloop equivalent process for multivariable systems. Then, based on the equivalent single loops, a model-based method is presented to derive the stabilizing PID parametric region by using the generalized Hermite-Biehler theorem. By sweeping over the entire ranges of feasible proportional gains and determining the stabilizing regions in the space of integral and derivative gains, the complete set of stabilizing PID controllers can be determined. The robustness of the design procedure against the approximation in getting the SISO plants is analyzed. Finally, simulation of a practical model is carried out to illustrate the effectiveness of the proposed technique.

Optimal PI Controller Tuning Based on ITAE Criterion for Low-Order Systems with Large Time Delay

For low order process with large time delay, a kind of optimal PI controller tuning method is proposed based on generalized Hermite-Biehler theorem and Genetic Algorithm. Firstly, the stable region of PI controller is obtained by using the generalized Hermite-Biehler theorem. Then the optimum parameters are selected from this region based on ITAE criterion and genetic algorithm. A tuning formula is obtained by nonlinear fitting of optimization result, which has the capability to cover the variety of normalized time delays up to 100. Simulation of Monte-Carlo stochastic experiment indicates that the proposed method has good performance robustness when parameter uncertainty occurs, compared with other four PI tuning methods.

Stabilizing of Subspaces Based on DPGA and Chaos Genetic Algorithm for Optimizing State Feedback Controller

The main purpose of the paper is to optimize state feedback parameters using intelligent method, GA, Hermite‐Biehler, and chaos algorithm. GA is implemented for local search but it has some deficiencies such as trapping into a local minimum and slow convergence, so the combination of Hermite‐Biehler and chaos algorithm has been added to GA to avoid its deficiencies. Dividing search space is usually done by distributed population genetic algorithm (DPGA). Moreover, using generalized Hermite‐Biehler Theorem can find the domain of parameters. In order to speed up the convergence at the first step, Hermite‐Biehler method finds some intervals for controller, in the next step the GA will be added, and, finally, chaos disturbance will help the algorithm to reach a global minimum. Therefore, the proposed method can optimize the parameters of the state feedback controller.

SYNTHESIS OF FIXED ORDER STABILIZING CONTROLLERS USING FREQUENCY RESPONSE MEASUREMENTS

In this paper, a method is proposed for synthesizing sets of stabilizing controllers of strictly proper, delay-free, Single Input, Single Output Linear Time Invariant (LTI) plants directly from their empirical frequency response data and from some coarse information about them. The coarse information that is required is the following: the number of non minimum phase zeros of the plant and the frequency range beyond which the phase response of the LTI plant does not change appreciably and the amplitude response goes to zero. It is assumed that the LTI plant does not have purely imaginary zeros or poles. The method of synthesizing stabilizing controllers involves the use of generalized Hermite-Biehler theorem for rational functions for counting the roots and the use of recently developed Sum-of- Squares techniques for checking the non-negativity of a polynomial in an interval through the Markov-Lucaks theorem. The method does not require an explicit analytical model of the plant that must be stabilized or the order of the plant, rather, it only requires an empirical frequency response data of the plant. The method also allows for measurement errors in the frequency response of the plant.

A new method for the computation of all stabilizing controllers of a given order

A new method is given for computing the set of all stabilizing controllers of a given order for linear, time invariant, scalar plants. The method is based on a generalized Hermite–Biehler theorem and the successive application of a modified constant gain stabilizing algorithm to subsidiary plants. It is applicable to both continuous and discrete time systems.

Robust phase margin, robust gain margin and Nyquist envelope of an interval plant family

In this paper, robust gain margin, robust phase margin and Nyquist envelope of interval systems are studied. It is first presented that the robust gain and phase margins of a control system with an interval plant family are achieved at one of the Kharitonov plant by using the interlacing theorem and virtual compensator concept. Then, using the generalized Hermite–Biehler theorem, it is shown that the outer boundary of the Nyquist envelope of a stable interval plant is covered by the Nyquist plots of the 16 Kharitonov plants family. Examples are given to illustrate the method presented.