**Streszczenie**

This paper investigates the problem of global practical output tracking by state feedback for a class of uncertain high-order nonlinear timedelay systems. Further, we design a homogeneous state feedback controller with an adjustable scaling gain, under mild conditions on the system nonlinearities involving time delay. Through the use of a homogeneous Lyapunov-Krasovskii functional method, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by homogeneous growth conditions and render the tracking error can be made arbitrarily small while all the states of the closed-loop system remain to be bounded.

**Słowa kluczowe:**

*global practical output tracking, nonlinear time delay systems, state feedback controller, Lyapunov-Krasovskii functional*

**Abstract**

W artykule opisano problem globalnego praktycznego śledzenia wyjścia za pomocą sprzężenia zwrotnego od stanu dla klasy niepewnych nieliniowych układów opóźniających wysokiego rzędu. Ponadto zaprojektowany został jednorodny kontroler sprzężenia od stanu z regulowanym wzmocnieniem skali, przy łagodnej nieliniowości i opóźnieniu czasowym. Dzięki zastosowaniu jednorodnej metody funkcjonalnej Lapunowa-Krasowskiego, wzmocnienie skali jest przystosowane do dominacji nieliniowości opóźnienia czasowego ograniczonej przez jednorodne warunki wzrostu i powoduje, że błąd śledzenia może być dowolnie mały, podczas gdy wszystkie stany systemu ze sprzężeniem zwrotnym pozostają ograniczonye.

**Keywords:**

*globalne śledzenie wyjścia, nieliniowe systemy z opóźnieniem czasowym, kontroler sprzężenia zwrotnego od stanu, funkcjonał Lapunowa-Krasowskiego*

The control design is one of the most relevant topics in nonlinear system theory, so a number of researchers have paid particular attention to it, for example, it can be seen in references [1-11]. The fundamental problem is to construct a feedback control law making the controlled output track a given reference signal as much as possible. The problem of global practical tracking for nonlinear state feedback systems was solved by the method of adding a power integrator [3,4] and using the idea of universal control [1,2]. However, the above results do not take into account time delays and their impact on the system as a whole. For example, in three-dimensional systems, delay is determined by the fact that the signals propagate at a finite speed and they need time to overcome distances [12]. Delay of the reaction to the signal and feedback with delay are inherent in many physical [13], chemical [14], climatic [15] and biological [16] objects and processes. In the study of systems with delay, it is important to know the values of time delays, the value of which largely determines the dynamics and properties of the system. Since time-delay exists widely in many practical systems such as electrical networks, microwave oscillator, and hydraulic systems, etc., and usually makes the considered system instable, to achieve some control objectives such as stabilization and trajectory tracking, the influence of time delay phenomenon should be considered. In view of these facts, it is meaningful and necessary to study control problems of accidental nonlinear systems with unknown parameters and time-delays. In recent years, by employing the Lyapunov-Krasovskii method to deal with the time-delay, control theory, and techniques for stabilization problem of time-delay nonlinear systems were greatly developed and advanced methods have been made; see, for instance, [17-21] and reference therein. Compared with study the stabilization problem [...]

## Prenumerata

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