Pavel Zítek

Mapping Based Algorithm for Large-Scale Computation of Quasi-Polynomial Zeros

A method for computing all zeros of a retarded quasi-polynomial that are located in a large region of the complex plane is presented. The method is based on mapping the quasi-polynomial and on utilizing asymptotic properties of the chains of zeros. First, the asymptotic exponentials of the chains are determined based on the distribution diagram of the quasi-polynomial. Secondly, large regions free of zeros are defined. Finally, the zeros are located as the intersection points of the zero-level curves of the real and imaginary parts of the quasi-polynomial, which are evaluated over the areas of the region outside those free of zeros.

An observer design for linear time-delay systems

An observer design is proposed for linear systems with time delays. The key to the design is to find a generalized coordinate change such that, in the new coordinates, all the time-delay terms are injected by the systems output. The existence of such a coordinate change is guaranteed by a rank condition on the observability matrix. The novelty of the proposed design is clearly reflected in the multiple-output case, where a dimensional expansion in the coordinate change could become necessary, and hence is allowed.

Quasipolynomial mapping based rootfinder for analysis of time delay systems

Abstract An original method for computing poles and zeros of time delay systems based on mapping the characteristic functions (quasipolynomials) of the system is presented. The method, which locates all the quasipolynomial roots in a bounded region, is based on mapping the real and imaginary parts of the quasipolynomial in the complex plane using the level curve tracing algorithm.

Anisochronic internal model control of time-delay systems

Abstract This paper is concerned with the robust control of time-delay systems with both state and control delays. The Smith predictor is not a suitable control scheme for this class of systems and special approaches developed hitherto are not practically applicable due to their poor robustness and complicated design and structure. However, a suitable controller design method that is both robust and practicably applicable can be based on the extension of the internal model control scheme. This paper describes the design of internal model controllers for this class of time-delay systems in both one degree-of-freedom and two degrees-of-freedom arrangement. A common feature of the proposed controllers is the application of both delayors and integrators in their final implementation, resulting in a better compensation of infinite order process dynamics. The practical applicability of this approach is demonstrated on a small scale heating plant with several internal delays and nonlinear steady state behaviour.

Algebraic design of anisochronic controllers for time delay systems

The prime objective of the paper is to overcome the conventional restriction to the field of rational functions as far as the algebraic design of control systems is concerned. By adopting the affine parameterization approach from algebraic control synthesis for the controllers given as the so-called RQ-meromorphic functions an algebraic solution to time-delay system control design is opened where neither an introduction of 2D transfer functions nor the assumption of commensurate delays is needed. The proposed design results in controllers combining both the integrators and delayers due to this property referred to as anisochronic.

QPmR - Quasi-Polynomial Root-Finder: Algorithm Update and Examples

An updated QPmR algorithm implementation for computation and analysis of the spectrum of quasi-polynomials is presented. The objective is to compute all the zeros of a quasi-polynomial located in a given region of the complex plane. The root-finding task is based on mapping the quasi-polynomial in the complex plane. Consequently, utilizing spectrum distribution diagram of the quasi-polynomial, the asymptotic exponentials of the retarded chains are determined. If the quasi-polynomial is of neutral type, the spectrum of associated exponential polynomial is assessed, supplemented by determining the safe upper bound of its spectrum. Next to the outline of the computational tools involved in QPmR, its Matlab implementation is presented. Finally, the algorithm is demonstrated by three examples.

Strong Stability of Neutral Equations with an Arbitrary Delay Dependency Structure

The stability theory for linear neutral equations subjected to delay perturbations is addressed. It is assumed that the delays cannot necessarily vary independently of each other, but depend on a possibly smaller number of independent parameters. As a main result, necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum, which take into account the precise dependency structure of the delays. In the derivation of the stability theory, results from realization theory and determinantal representations of multivariable polynomials play an important role. The observations and results obtained in the paper are first illustrated and validated with a numerical example. Next, the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed.

Frequency-domain synthesis of hereditary control systems via anisochronic state space

A method of control feedback design for systems with hereditary properties is presented which is based on the concept of anisochronic state space. A new kind of state feedback control is proposed using as state variables the system accumulation attributes only, while the other delay dynamics are represented by functional terms. Assigning the control parameters is approached as a placement of characteristic function zeros; however, owing to the infinite set of these zeros in the hereditary system, only the dominant zeros are placed. For this purpose a new technique applying the conformal mapping of transcendental characteristic function of the hereditary system has been developed.

Anisochronic state observers for hereditary systems

The anisochronic observers result from a functional extension of the classical state space bringing a significantly reduced number of needed state variables, as compared with the standard approach. The anisochronic state variables can be selected as measurable outputs in most cases and their availability provides significant advantage in process monitoring. Observability condition is presented for anisochronic process models with arbitrary sort of delays. Since hereditary systems are considered a typical result of control system integration the issue of decomposition on separate units is followed in observer design. As a special aim, the observer design delay cancellation in the observation process is solved using a functional extension of Ackermanns formula for this purpose.

Control System Design Based on a Universal First Order Model with Time Delays

An original modelling approach for SISO systems is presented, based on a first order model with more than one delay in its structure. By means of this model it is possible truly to hit off the properties of systems which are conventionally described by higher order models. The identification method making use of a relay feedback test combined with transient responses of the system has proved to be suitable for assessing the model parameters. With respect to its plain structure the model is well suited to be applied in the framework of an internal model control scheme (IMC). The resultant control algorithm with only one optional parameter is very simple and can easily be implemented, for example by means of a programmable controller (PLC).