Research Article Quasipolynomial Approach to Simultaneous Robust Control of Time-Delay Systems Nikolaj SemeniI, 1 Andrej Sarjaš, 2 Amor Chowdhury, 1 and Rajko SveIko 2 1 Margento R&D, Gosposvetska c. 84, 2000 Maribor, Slovenia 2 Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova Ulica 17, 2000 Maribor, Slovenia Correspondence should be addressed to Nikolaj Semeniˇ c; semenic@gmail.com Received 6 June 2014; Revised 12 August 2014; Accepted 17 August 2014; Published 11 September 2014 Academic Editor: Zhan Shu Copyright © 2014 Nikolaj Semeniˇ c et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A control law for retarded time-delay systems is considered, concerning infnite closed-loop spectrum assignment. An algebraic method for spectrum assignment is presented with a unique optimization algorithm for minimization of spectral abscissa and efective shaping of the chains of infnitely many closed-loop poles. Uncertainty of plant delays of a certain structure is considered in a sense of a robust simultaneous stabilization. Robust performance is achieved using mixed sensitivity design, which is incorporated into the addressed control law. 1. Introduction Time-delay systems are an important and well established topic in modern control theory [1–3]. Its diverse applications, for instance, in dynamics of fuids, internal combustion, heating systems, regenerative chatter in metal cutting, and networked control systems, led to the development of various complex approaches to system and controller synthesis [4–8]. Infnite dimensional spectrum of such systems might cause difculties in appropriate spectrum assignment using stan- dard control laws, which consequently means that stabiliza- tion cannot be always achieved. Due to the complexity of the spectrum, much interest has been shown in the development of control laws based on the computation of rightmost poles, since such algorithms ensure results with high precision [9–12]. In principle, with a limited number of controller parameters only a few poles can be placed to desired positions in the complex plane [13]. As it has been shown in [8], some controller parameters might be used to directly assign a few dominant poles to an arbitrary position in the complex plane. Te remaining controller parameters are then used to shif the chains of infnitely many system poles as far to the lefof the dominant poles as possible. Another pole-placement-based technique has been introduced in [14] for retarded systems and in [15] for neutral systems. Te method is based on continuous modifcations applied to controller parameters in order to shif the rightmost or unstable poles to the lef half plane in a quasicontinuous way, while monitoring other eigenvalues. A direct optimization approach [16, 17], used to minimize the spectral abscissa, has been introduced in [18]. Alternative eigenvalue based control laws might be con- sidered, which incorporate diferent algebraic approaches [19]. General SISO time-delay plants treated in the form of a transfer function and its admissible coprime-inner/outer factorizations, derived by using numerical algorithms for computation of closed-loop poles, were discussed in [20]. In addition, an optimal H ∞ controller was designed. In this paper, a control law for retarded time-delay sys- tems is considered, which results in a controller in the form of a transfer function as a quotient of quasipolynomials. An individual quasipolynomial in the numerator and the denom- inator consists of several delayed terms that result in a closed- loop quasipolynomial of a certain structure of sequentially solvable polynomial Diophantine equations. Te structure of the controller is examined and derived in the case of infnite solutions of individual polynomial equations. As has been Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 930697, 10 pages http://dx.doi.org/10.1155/2014/930697