Libor Pekar

On delay (in)dependent stability for TDS

A preliminary-study contribution to delay dependent (DDS) and independent (DIS) stability of control systems with multiple delays is focused in this paper. We first introduce some basic facts about the spectral properties of such systems and a recently developed DIS method is concisely described. Then our simple and applicable numerical gridding iterative DDS algorithm based on the finite dimensional approximation of the characteristic quasipolynomial by means of the Taylor series expansion and the Regula-Falsi method is presented. The methodologies are used and verified by a simulation example of control a delayed feedback system with an unstable plant with two delays. Since it is shown that the system is not DIS, the DDS feature with respect to delay deviations is investigated via the proposed algorithm. Numerical experiments indicate a very good correspondence between the DDS algorithm results and the searching of crossing delays by using the QuasiPolynomial matrix Rootfinder.

Implementation of a New Quasi-Optimal Controller Tuning Algorithm for Time-Delay Systems

Systems and models with dead time or aftereffect, also called hereditary, anisochronic or time-delay systems (TDS), belonging to the class of infinite dimensional systems have been largely studied during last decades due to their interesting and important theoretical and practical features. A wide spectrum of systems in natural sciences, economics, pure informatics etc., both real-life and theoretical, is affected by delays which can have various forms; to name just a few the reader is referred e.g. to (Gorecki et al., 1989; Marshall et al., 1992; Kolmanovskii & Myshkis, 1999; Richard, 2003; Michiels & Niculescu, 2008; Pekař et al., 2009) and references herein. Linear time-invariant dynamic systems with distributed or lumped delays (LTI-TDS) in a single-input single-output (SISO) case can be represented by a set of functional differential equations (Hale & Verduyn Lunel, 1993) or by the Laplace transfer function as a ratio of so-called quasipolynomials (El’sgol’ts & Norkin, 1973) in one complex variable s, rather than polynomials which are usual in system and control theory. Quasipolynomials are formed as linear combinations of products of s-powers and exponential terms. Hence, the Laplace transform of LTI-TDS is no longer rational and socalled meromorphic functions have to be introduced. A significant feature of LTI-TDS is (in contrast to undelayed systems ) its infinite spectrum and transfer function poles decide except some cases of distributed delays, see e.g. (Loiseau, 2000) about the asymptotic stability as in the case of polynomials. It is a well-known fact that delay can significantly deteriorate the quality of feedback control performance, namely stability and periodicity. Therefore, design a suitable control law for such systems is a challenging task solved by various techniques and approaches; a plentiful enumeration of them can be found e.g. in (Richard, 2003). Every controller design naturally requires and presumes a controlled plant model in an appropriate form. A huge set of approaches uses the Laplace transfer function; however, it is inconvenient to utilize a ratio of quasipolynomials especially while natural requirements of internal (impulse-free modes) and asymptotic stability of the feedback loop and the feasibility and causality of the controller are to be fulfilled. The meromorphic description can be extended to the fractional description, to satisfy requirements above, so that quasipolynomials are factorized into proper and stable meromorphic functions. The ring of stable and proper quasipolynomial (RQ)

TFC robust control design for time delay systems — Analysis and example

This paper is focused on the use of the ring of quasipolynomial meromorphic functions (RMS) for algebraic control of time delay systems (TDS) in a robust sense. Unlike many other contributions, we intend to utilize Two-Feedback-Controllers (TFC) control system structure instead of a simple negative control feedback loop, which involves making some analysis first. Tunable controller parameters are set such that requirements of robust stability and robust performance are satisfied. The theoretically described procedure is verified by simulations in Matlab/Simulink environment for temperature control of a laboratory circuit heating plant containing internal (state) delays. The appliance provides unordinary step responses, which makes it difficult to be modeled, identified and controlled in general. The obtained results show the applicability of the controller design methodology.

Design of controllers for delayed integration processes using RMS ring

This contribution deals with design of controllers for integrating processes with time delay. In contrast to many other methods, the proposed method is not based on the time delay approximation. A control structure combining standard 1DOF and 2DOF control structures is considered. The control design is performed in the RMS ring of retarded quasipolynomial (RQ) meromorphic functions - an algebraic method based on the solution of the Bezout equation with Youla-Kucera parameterization is presented. Final controllers are of so-called anisochronic type and ensure feedback loop stability, tracking of the step reference and load disturbance attenuation. Among many others tuning methods, the dominant pole assignment method is adopted.

A Simple DDS Algorithm For TDS: An Example.

This paper is aimed at the presentation and simulation verification of a novel simple and fast delay dependent stability (DDS) testing algorithm for linear timeinvariant time delay systems (LTI TDS). The algorithm can be used for systems with multiple delays and/or those with many controller parameters. Value ranges of delays and tunable parameters decide about the (exponential) stability of LTI TDS and the goal is to determine such ranges that might not be convex. Our numerical gridding iterative DDS algorithm is based on the finite dimensional approximation of the characteristic quasipolynomial by means of the Taylor series expansion or a Pade-like approach followed by the Regula-Falsi result enhancement. Stability regions are sought through the determination of crossing delays (or parameters) which cause the system stability switching. Numerical simulation experiments performed in MATLAB environment prove a sufficient accuracy and the practical usability of the proposed DDS algorithm. INTRODUCTION Delays in system dynamics constitute one of decisive factors on system stability. They can appear as a natural consequence of plant internal retarded feedbacks (Pekař et al. 2009; Zitek 1983), or more frequently, due to delayed control feedback systems. During recent decades, many approaches and methods on the decision about stability of linear time-invariant time delay systems (LTI TDS) with fixed parameters and delays and have been published, see e.g. (Gu et al. 2003; Michiels and Niculescu 2007; Richard 2003). However, delays and/or controller parameters may vary or can be undetermined, therefore there it emerges the task of the decision about the possible parameters or delays ranges which keeps a LTI TDS stable. We decide between delay-independent stability (DIS) (Delice and Sipahi 2010; Ergenc 2010) which is satisfied if the system is stable for any delay values vector, and delay-dependent stability (DDS) investigating admissible ranges of delays inside which the system remains stable (Olgac et al. 2007; Shao et al. 2013). Both the problems are usually studied via heavy mathematical tools, such as Ljaponov-Krasovskii matrices. A powerful idea is the searching of crossing delays (or verifying their existence) that make the system switching to stability/instability due to root continuity property via, for instance, the so-called Rekasius transformation (Rekasius 1980) which represents one of practically applicable means for DDS. The goal of this paper is to investigate and present a simple and computationally fast (by using an advanced polynomial-root finding computer program function) algorithm determining the crossing delays by using the iterative polynomial approximation of the characteristic quasipolynomial. The well-know Taylor series expansion constitutes the main tool for this task; furthermore, the Regula-Falsi principle, utilized consequently, makes the estimation more accurate. The methodology can also be applied for the determination of controller parameters which keeps the feedback system stable with fixed (nominal) delays. A rather detailed simulation example performed in MATLAB and Simulink environment provides the reader with the demonstration and verification of the algorithm and it proves a very good accuracy. LTI TDS AND ITS STABILITY The aim of this section is to briefly introduce a LTI TDS model, its exponential stability and some spectral properties. LTI TDS Model Let the system be governed by transfer function ( ) ( ) ( ) τ τ , / , s a s b s G = , where ( ) τ , s a , ( ) τ , s b are (retarded) quasipolynomials in ∈ s  of the form ( ) ( ) ∑ ∑ ∑ − = = = − + = 1 0 1 1 , exp , n i hj L k k k ij i ij n i s s q s s q τ λ τ (1) where ∈ ij q , ∈ k ij , λ 0 and [ ] L τ τ ,.. 1 = τ stands for independent delays. Proceedings 29th European Conference on Modelling and Simulation ©ECMS Valeri M. Mladenov, Petia Georgieva, Grisha Spasov, Galidiya Petrova (Editors) ISBN: 978-0-9932440-0-1 / ISBN: 978-0-9932440-1-8 (CD) Spectral Properties and Stability The spectrum of a such system is infinite, and if there are no common zeros of the numerator and denominator, roots ( ) 0 , : = τ σ σ a agree with system poles (characteristic values). Definition 1. The spectral abscissa is ( ) ( ) ( ) 0 , : Re max : = = τ τ τ σ σ α m a Property 1. Isolated poles behave continuously and smoothly with respect to τ on ; however, the function ( ) τ α may be nonsmooth or even non-Lipschitz at a finite number of points (Vanbiervliet et al. 2008). Definition 2. The system (1) is exponentially stable if ( ) 0 < τ α with a fixed τ . In the light of Property 1 and Definition 2, the stability can switch if the rightmost pole crosses the imaginary axis at crossing frequencies ( ) { } 0 , j : : 0 = ∈ = + τ Ω ω ω a  for some corresponding crossing delays τ . Definition 3. The system (1) is DIS if and only if ( ) 0 , ≠ τ s a for any ∈ s 0+ and + ∈ 0  τ . It is DDS if and only if ( ) 0 , ≠ τ s a for all ∈ s 0+ and some open and bounded sets ∞ ∈ + \ 0 L i  τ , ,... 2 , 1 = i Hence, if the system is DIS, then ( ) 0 0 < α . Lemma 1. A quasipolynomial ( ) τ , s q has roots ω j = s if and only if the polynomial ( ) ( ) s T s T s i i i s q + − → − 1 1 exp , ˆ τ τ (2) has the same roots for some [ ] L T T T ,..., , : 2 1 = T ,  ∈ i T . Transformation (2) is called the exact Rekasius transformation (Rekasius 1980) and it is widely used to determinate the set { } ∈ l l l , ,T Ω , Ul l Ω Ω = , and, consequently, Ω ,{ } l τ for DDS; however, such algorithms are mostly computationally heavy (Ergenc 2010). DDS ALGORITHM Now we intent to present our novel simple iterative gridding algorithm that estimates crossing delays by means of a rational approximation yet without the Rekasius transformation. The successive calculation of the well-known Taylor series expansion (in the neighborhood of the current leading root) followed by the (linear) Regula-Falsi zero point estimation is used to determine sets Ω ,{ } l τ . Polynomial roots then can be easily and quickly computed by means of a standard software tool, e.g. in MATLAB, for fixed delay values. The gridding procedure ensures a sufficiently fast yet accurate crossing delays approximation even if the number of polynomial zeros seeking calculations is high. Preliminaries For a given characteristic quasipolynomial ( ) τ , s a with fixed τ let us introduce the approximate polynomial ( ) ( ) ∑ = = n i i i s a s a ˆ 0 ˆ , ˆ τ τ (3) of the appropriate order, for which it holds that ( ) ( ) μ ,.., 1 , 0 , , ˆ ,

Control of time delay systems in a robust sense using two controllers

The use of the ring of quasipolynomial meromorphic functions for algebraic control design for time delay systems in a robust sense is the aim of this paper. Unlike the habitual simple feedback loop, Two-Feedback-Controllers control system structure is intended to be utilized, which involves making some robust analysis first. Tunable controller parameters are set such that requirements of robust stability and robust performance are satisfied. The theoretically described methodology is verified by simulations in Matlab/Simulink environment followed by real experiments with temperature control of a laboratory circuit heating plant containing internal as well as input-output delays. The obtained results promise the applicability of the controller design approach.

On Finite-Dimensional Transformations of Anisochronic Controllers Designed by Algebraic Means: A User Interface

© 2012 Pekař et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. On Finite-Dimensional Transformations of Anisochronic Controllers Designed by Algebraic Means: A User Interface

Compromising controller parameters setting for a delayed thermal process

The primary goal of this contribution is to present an original idea of a suboptimal controller parameters setting that intends to achieve a compromise between various requirements on the control response performance. Performance (quality) measures include integral and absolute criteria. The idea is demonstrated and applied to a robust control of a thermal process that shows internal delays. As the subsidiary objective, the reader is acquainted with a concise summary of the robust control design for the delayed model based on the algebraic principle over a special ring. The obtained results are demonstrated not only by means of computer simulations but via laboratory measurements as well.

Robust Stabilization of a Heating-Cooling System by Using Two Feedback Controllers: a Numerical-Graphical Analysis

This contribution is intended to present a numerical-graphical method for analyzing the robust stability of a control loop which contains two feedback controllers and a heating-cooling system with a heat exchanger. The controlled plant is described by the anisochronic model with internal delays and the parameters of this model are supposed to vary within given intervals. The applied technique for robust stability analysis is based on the numerical calculations of the value sets in combination with the graphical test via the zero exclusion condition.

Delay systems with meromorphic functions design

The paper is focused on systems with delay terms at the left (and the right) side of differential equations. Analysis and synthesis of delay systems can be conveniently studied through a special ring of RQ-meromorphic functions. The control methodology is based on a solution of Diophantine equations in this ring. Final controllers result in the Smith predictor like structure. Controller parameters are tuned through a pole-placement problem as a desired multiple root of the characteristic closed loop equation. The methodology is illustrated by a stable second order transfer function with a dead-time term. Then the paper brings an autotuning method as a combination of biased-relay feedback estimation and the proposed algebraic control design. The developed approach is illustrated by examples in the Matlab and Simulink environment.