Global Output Control for a Class of Inherently Higher-Order Nonlinear Time-Delay Systems Based on Homogeneous Domination Approach

This paper addresses the problem of global output feedback stabilization for a class of inherently higher-order uncertain nonlinear systems subject to time-delay. By using the homogeneous domination approach, we construct a homogeneous output feedback controller with an adjustable scaling gain. With the aid of a homogeneous Lyapunov-Krasovskii functional, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by homogeneous growth conditions and render the closed-loop system globally asymptotically stable. In addition, we also show that the proposed approach is applicable for time-delay systems under nontriangular growth conditions.

It has been known that the problem of global output feedback stabilization for uncertain nonlinear systems is very challenging compared to the state feedback case.In the past decade, global stabilization by output feedback domination method has been proved to be achievable for a series of nonlinear systems.For the system of a five-spot pattern reservoir, a nonlinear reduced-order model is identified and an asymptotically stabilizing controller is proposed based on the circle criterion in [1].With the help of linear feedback domination design [2], some interesting results have been established under a linear growth condition [2] and under a higher-order growth condition [3].Recently, the homogeneous domination approach proposed in [3] has been used as a universal tool to solve the problem of global output feedback stabilization for inherently nonlinear systems.As a consequence, fruitful research results have been achieved in [3][4][5][6][7][8][9].
However, the aforementioned results have not considered the time-delay effect which is actually very common in state, input, and output due to the time consumed in sensing, information transmitting, and controller computing.In the case when the nonlinearities contain time-delay, some interesting results have been obtained.For instance, in [10], the global asymptotic stability analysis problem is investigated for a class of stochastic bidirectional associative memory (BAM) networks with mixed time-delays and parameter uncertainties.The paper [11] investigated the state estimation problem for a class of discrete time-delay nonlinear complex networks with randomly occurring phenomena from the sensor measurements.In [12], an adaptive approach was employed to design a state feedback controller to globally stabilize a class of upper-triangular systems with time-delay.The work [13] relaxed the growth condition imposed in [12] by employing a dynamic gain.In [14], a state feedback stabilizer was constructed for a class of time-delay higher-order nonlinear systems.In [15], the problem of state-feedback stabilization for a class of lower-triangular stochastic timedelay nonlinear systems was investigated.
In the case when only output is available, the problem of output feedback stabilization is more challenging and fewer results have been achieved for nonlinear systems with timedelay.For a linear system with time-delay in the input, the problem of output feedback stabilization of was solved in [16,17].For nonlinear systems (1) subject to time-delay and uncertainties, the problem of output feedback stabilization has not been extensively investigated.
In this paper, we aim to tackle the problem by using the output feedback domination approach.First, based on homogeneous domination approach [3,9], output feedback controllers are constructed to globally stabilize higherorder nonlinear time-delay systems whose nonlinearities are bounded by homogeneous growth conditions.Then, we construct a Lyapunov-Krasovskii functional and use it to choose an appropriate scaling gain in the output feedback controller to guarantee the closed-loop system globally asymptotic stability.In addition, homogeneous output stabilization controllers are extended to the nontriangular time-delay systems.The simulation results show the effectiveness of the proposed method.

Homogeneous Output Feedback Controllers of Inherently Nonlinear Systems
In this section, we show that under a lower-triangular homogeneous growth condition, the nonlinear time-delay system (1) can be globally stabilized by a homogeneous output feedback controller.First, we consider the nonlinear continuous functions . ., , which satisfy the following higher-order growth condition.
Remark 2. When  > 0, (2) is a higher-order growth condition which is actually homogeneous (see Definition A.1 in the appendix) with the following dilation (For simplicity, in this paper we assume  = / with an even integer  and an odd integer .Therefore,   is a ratio of two odd integers.) First, we construct a output feedback stabilizer for the following linear system where V ∈ R,  ∈ R are the control input and system output, respectively.Using the approach in [5], we can design for (7) a homogeneous output feedback stabilizer, which can be described in the following lemma.
It can be verified that the closed-loop system (7)-( 9) is homogeneous according to Definition A.1 in the appendix.As a matter of fact, by defining the compact notation  := ( 1 , . . .,   ,  2 , . . .,   )  , the closed-loop system (7)-( 9) can be rewritten as the following compact form: where where Proof.The output feedback controller is constructed by introducing a scaling gain into the output feedback controller obtained in Lemma 3. First, we define a change of coordinates as where the constant gain  ≥ 1 will be determined later.Under (14), system (1) can be rewritten as The observer ( 9) can be rewritten as follows: where the same   ,  = 1, . . .,  as in (9).Now, the closed-loop system ( 15) and ( 16) can be rewritten as where the vector field () has the exactly same construction of (10).Therefore, we adopt the same Lyapunov function  used in (12), whose derivative along ( 17) is According to Assumption for a constant ]  > 0, since it can be seen that by definition Substituting ( 13) and ( 19) into (18) yields By Lemma A.5 in the appendix, there exists a constant  3 > 0 such that which yields Construct a candidate Lyapunov functional as follows: where  is a positive constant.Let  =  3 ∑  =1 l  1−]  , so it follows from ( 23) and (24) that Hence, by choosing a large enough  as  > max{(((1 , where ] = min =1,..., {]  }, the right-hand side of (25) is negative definite, that is, there exists a constant  > 0, such that As a conclusion, we know that the system described by (1) under Assumption 1 can be globally stabilized by the output controller (8).Proof of Theorem 6.The proof is very similar to that of Theorem 4. We can use the exactly same observer (4) and control law (3).Although the nonlinear function is not in the triangular form, Assumption 5 will lead directly to (19) by using the change of coordinates (14).Then, the global stabilization can be concluded with an appropriate choice of gain , which is similar to that in (25).The detailed proof is omitted here for brevity.

Examples and Simulations
Consider the following inherently nonlinear time-delay system.
The computer simulation results of the closed-loop systems are given in Figures 1 and 2 with the following initial functions

Conclusion
In this paper, we have studied the problem of global output feedback stabilization for a class of higher-order timedelay nonlinear systems under a homogeneous condition.First, homogeneous output feedback controllers have been constructed with adjustable scaling gains.Then, with the help of a homogeneous Lyapunov-Krasovskii functional, we've redesigned the homogeneous domination approach to tune the scaling gain for the overall stability of the closedloop systems.The output feedback controllers proposed in this paper are memoryless and, therefore, can be easily implemented in practice.

Appendix
This appendix collects the definition of homogeneous function and several useful lemmas.