Global output tracking by state feedback for high-order nonlinear systems with time-varying delays

Abstract This paper focuses on the problem of global practical output tracking for a class of high-order non-linear systems with time-varying delays (via state feedback). Under mild growth conditions on the system nonlinearities involving time-varying delays, we construct a state feedback controller with an adjustable scaling gain. With the aid of a Lyapunov–Krasovskii functional, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by the growth conditions and make the tracking error arbitrarily small while all the states of the closed-loop system remain to be bounded. Finally, a simulation example is given to illustrate the effectiveness of the tracking controller.


PUBLIC INTEREST STATEMENT
Modern control theory occupies one of the leading places in the technical sciences and at the same time belongs to one of the branches of applied mathematics, which is closely related to computer technology. Control theory based on mathematical models allows you to study dynamic processes in automatic systems, to establish the structure and parameters of the components of the system to give the real control process the desired properties and specified quality. It is the foundation for special disciplines that solve the problems of automation of management and control of technological processes, design of servo systems and regulators, automatic monitoring of production and the environment, the creation of automatic machines and robotic systems. It is well known that the creation of a new model of a robot and, moreover, a robot technical system (RTS) is associated with organizational issues of the interaction of four interdependent functional elements, which can be designated as: mechanisms, energy, electronics, programs (algorithms).
Problems of practical output tracking of nonlinear systems are the most challenging and hot issues for the field of nonlinear control and it has drawn increasing attention during past decades. A number of interesting results have been achieved over the past years, see (Alimhan & Inaba, 2008a, 2008bAlimhan & Otsuka, 2011;Alimhan, Otsuka, Adamov, & Kalimoldayev, 2015;Alimhan, Otsuka, Kalimoldayev, & Adamov, 2016;Gong & Qian, 2005Lin & Pongvuthithum, 2003;Qian & Lin, 2002;Sun & Liu, 2008;Zhai & Fei, 2011), as well as the references therein. However, the aforementioned results do not consider the effect of time delay. It is well known that time-delay phenomena exist in many practical systems. Due to the presence of time delay in systems, it often significant effect on system performance and may induce instability, oscillation and so on. Therefore, the study of the problems of global control design of time-delay nonlinear systems has important practical significance. However, due to there being no unified method being applicable to nonlinear control design, this problem has not been fully investigated and there are many significant problems which remain unsolved. In recent years, by using the Lyapunov-Krasovskii method to deal with the timedelay, control theory, and techniques for stabilization problem of time-delay nonlinear systems were greatly developed and advanced methods have been made; see, for instance, (Chai, 2013;Gao & Wu, 2015;Gao, Wu, & Yuan, 2016;Gao, Yuan, & Wu, 2013;Sun, Liu, & Xie, 2011;Sun, Xie, & Liu, 2013;Zhang, Lin, & Lin, 2017;Zhang, Zhang, & Gao, 2014) and reference therein. In the case when the nonlinearities contain time-delay, for the output tracking problems, some interesting results also have been obtained (Alimhan, Otsuka, Kalimoldayev, & Tasbolatuly, 2019;Jia, Xu, Chen, Li, & Zou, 2015;Jia, Xu, & Ma, 2016;Yan & Song, 2014). However, the contributions only considered special cases such as p i equal one or constant timedelay for the system (1) when the case p i greater one. When the system under consideration is time-varying delays non-linear, the problem becomes more complicated and remain unsolved. This motivates the research in this paper.
In this paper, under mild conditions on the system nonlinearities involving time-varying delay, we will be to solve the aforementioned problem with the aid of the basis of the homogeneous domination technique (Chai, 2013;Polendo & Qian, 2007, 2006) and a homogeneous Lyapunov-Krasovskii functional. The main contributions of this paper are summarized as follows: (i) By comparison with the existing results in Jia et al., , 2016, due to the appearance of high-order terms, how to construct an appropriate Lyapunov-Krasovskii functional for system (1) is a nontrivial work. In this paper, we constructing a new Lyapunov-Krasovskii functional and using the adding a power integrator technique, a number of difficulties emerged in design and analysis are overcome. (ii) This note extended the results in (Alimhan et al., 2019) to time-varying delay cases.

Practical output tracking for high-order nonlinear systems
The objective of the paper is to construct an appropriate controller such that the output of system (1) practically tracks a reference signal y r ðtÞ. That is, for any pre-given tolerance ε > 0 to design a state feedback controller of the form uðtÞ ¼ gðxðtÞ; y r ðtÞÞ; (2) such that for the all initial condition (i) All the trajectories of the closed-loop system (1) with state controller (2) are well defined and globally bounded on ½0; þ1Þ: (ii) There exists a finite time T > 0, such that yðtÞ À y r ðtÞ j j< ε; " t ! T > 0: In this section, we show that under the following three assumptions, the practical output tracking problem can be solved by state feedback for high-order nonlinear systems with timevarying delays (1).
Assumption2. The time-delays d i ðtÞ are differentiable and satisfies 0 d i ðtÞ d i ; d 0 i ðtÞ # i < 1, for constants d i and # i , i ¼ 1; . . . ; n.
Assumption3. The reference signal y r ðtÞ and its derivative are bounded, that is, there is a constant D > 0 such that y r ðtÞ j j D and _ y r ðtÞ j j D.
Our main purpose are dealt with the practical output tracking problem by delay-independent state feedback for high-order time-varying delays nonlinear systems (1) under Assumptions 1-3. To this end, we introduce the following coordinate transformation.
Proposition1. For the system (9), Suppose there exists a state feedback controller of the form v ¼ Àβ with a positive definite, C 1 and radially unbounded Lyapunov function, Such that . . . ; n are positive constants. Then, the closed-loop system (9) and (10) is globally asymptotically stable.
Next, we use the homogeneous domination approach to design a global tracking controller for the system (1) which can be described in the following main theorem in this paper.
Theorem 1. Under Assumptions 1-3, the global practical output tracking problem of the high-order time-varying delays nonlinear system (1) can be solved by the state feedback controller u ¼ L κnþ1 v in (7) and (10).

Proof
By (10), it is not difficult to prove that u preserves the equilibrium at the origin.
By the definitions of r i 's and σ, we easily see that u ¼ L κnþ1 v is a continuous function of z and u ¼ 0 for z ¼ 0. This together with Assumption1 implies that the solutions of z system is defined on a time interval [0, t M ], where t M > 0 may be a finite constant or +∞, and u preserves the equilibrium at the origin.
In what follows, we define the following notations Then, the closed-loop system (7)-(10) can be written as the following compact form by the same notations (6) and (13), Introducing the dilation weight Δ ¼ ðr 1 ; . . . ; r n Þ, from Definition A1, it be not difficult to prove that V n is homogeneous of degree 2σ-τ with respect to the weight Δ.
Therefore, using the same Lyapunov function (11) and by Lemma A2 and Lemma A3, it can be concluded that where m 1 > 0 is constant.
By (8), Assumption 1 and L > 1, it can be found constants δ i > 0 and 0 < γ i 1 such that and noting that for i ¼ 1; . . . ; n, by Lemma A2, @V n =@z i is homogeneous of degree 2σ À τ À r i , and by Substituting (18) into (15) yields By Lemma A4, there exists a constant m 3 > 0 such that which yields Next, we construct a Lyapunov-Krasovskii functional as follows: VðzðtÞÞ ¼ V n ðzðtÞÞ þ UðzðtÞÞ; where λ is a positive parameter to be determined later. Because V n ðzðtÞÞ is positive definite, C 1 , radially unbounded and by Lemma 4.3 in (Khalil, 1996), there exist two class K 1 functions α 1 and α 2 , such that α 1 ð zðtÞ j jÞ V n ðzðtÞÞ α 2 ð zðtÞ j jÞ According to the homogeneous theory, there are positive constants η 1 and η 2 such that where W(z(t)) is a positive definite function, whose homogeneous degree is 2σ. Therefore, the following inequality holds α 1 ð zðtÞ j jÞ WðzðtÞÞ α 2 ð zðtÞ j jÞ with two class K 1 functions α 1 and α 2 .
This completes the proof of our main Theorem.
Remark2. It should be noted that the proposed controller can only work well when the whole state vector is measurable. Therefore, a natural and more interesting problem is how to design feedback output tracking controller for the time-varying delay nonlinear systems studied in the paper if only partial state vector being measurable, which is now under our further investigation. Although (Alimhan & Inaba, 2008a, 2008bGong & Qian, 2007;Sun & Liu, 2008;Zhai & Fei, 2011) studies global practical tracking problems by output feedback, it does not include the time delay. In addition, in recent years, many results on nonlinear fuzzy systems have been achieved (Chadli & Borne, 2013;Chadli & Guerra, 2012;Chadli, Maquin, & Ragot, 2002;Khalil, 1996), and so forth. An important problem is whether the results in this paper can be extended to nonlinear fuzzy systems.

An illustrative example
This section gives a numerical example to illustrate the effectiveness of Theorem 1.
(ii) When the scaling gain L is chosen as L ¼ 300, the tracking error obtained is about 0.075 as shown in Figure 2.

Conclusion
In this paper, we extend the result in (Alimhan et al., 2019) to solve the global practical tracking problem for a class of high-order nonlinear time-varying delays systems by state feedback. Under some mild-growth conditions, we first construct a state feedback controller with an adjustable scaling gain. Then, With the aid of a Lyapunov-Krasovskii functional, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by the growth conditions and make the tracking error arbitrarily small while all the states of the closed-loop system remain to be bounded.

Funding
The work has been performed under grant projects of "Development of technologies for multilingual automatic speech recognition using deep neural networks" AP05131207 (2018-2020) at the Institute of Information and Computational Technologies CS MES Republic of Kazakhstan.