# A Time-Varying Gain Design Method for State Feedback Control of Upper Triangular Nonlinear Systems

Authors: Yanmin Yin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, a time-varying gain design method is used to investigate the state feedback control problem of upper triangular nonlinear systems. Firstly, the nonlinear term recognizes an incremental rate relying on the unknown constant and the function with respect to time. Then, a time-varying gain design method is utilized to construct a state feedback controller. With the help of a suitable coordinate transformation and a Lyapunov function, one obtains that all the signals of the closed-loop system converge to zero. Finally, two numerical examples are presented to display the effectiveness of the time-varying gain design method.


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## Body

## 1. Introduction

Many physical models can be described by nonlinear systems [1–4]. Therefore, the control problem of these physical models can be transformed into the control problem of nonlinear systems [5]. Compared with linear systems, the behavior of nonlinear systems is more diverse [6–10]. The research of control algorithms is generally developed for the specific type of nonlinear systems [11, 12].In general, many results about nonlinear systems have focused on nonlinear systems with triangular structures, that is, lower triangular nonlinear systems [13, 14] and upper triangular nonlinear systems [15]. The common method for studying lower triangular nonlinear systems is the backstepping design method [16], and the common method for considering upper triangular nonlinear systems is the forwarding design method [17]. Although, based on the iterative design algorithm, these methods can effectively deal with strong nonlinearities, the design procedure is more complicated. In the past few decades, the gain design method is a very effective tool to deal with the control problem of upper triangular nonlinear systems [18].Based on the coordinate transformation, the time-varying gain design method is an effective strategy for dealing with the uncertain parameter of upper triangular nonlinear systems [19]. By introducing a time-varying function in the controller, it can effectively deal with the nonlinear terms of upper triangular nonlinear systems [20]. Compared with the commonly adaptive control strategy, the time-varying gain design method is more concise, the calculation process is less, and a lot of calculation work is reduced. Furthermore, the time-varying gain design method does not require too many design parameters and avoids complicated calculation process.This paper uses the time-varying gain design method to study the control problem of upper triangular nonlinear systems. The nonlinear characteristics of the system considering here are more obvious, that is, the unknown constant and the function with respect to time are allowed in the nonlinear terms. Compared with the assumption about the nonlinear terms in [19, 20], the assumption in this paper is more general. Thus, a time-varying gain design method is introduced to achieve the control goals of the concerned system.


## 2. Preliminaries

In this paper, we consider a class of nonlinear systems in the following form:(1)ξ˙1t=ξ2t+ψ1t,ξt,ut,ξ˙2t=ξ3t+ψ2t,ξt,ut,⋮ξ˙n−1t=ξnt+ψn−1t,ξt,ut,ξ˙nt=ut,where ξt=ξ1t,ξ2t,…,ξntT∈ℝn is the state and ut∈ℝ is the input; the uncertain continuous functions ψit,ξt,ut:ℝ+×ℝn×ℝ⟶ℝ, i=1,2,…,n−1, satisfy the following growth condition.Assumption 1.
For allt,ξt,ut∈ℝ+×ℝn×ℝ, the following inequalities hold:(2)ψit,ξt,ut≤θ1+tcξi+2t+ξi+3t+⋯+ξnt+ut,i=1,2,…,n−1,where θ is an unknown positive constant and c is a known constant which satisfies 0≤c<1.Remark 1.
Assumption1 is a reasonable condition. According to Assumption 1, we can know that the nonlinear terms of system (1) can include the unknown constant and the function with respect to time. Therefore, compared with the assumption in [19, 20], Assumption 1 is more general, and the nonlinear characteristics of system (1) are more diverse. The controller designed in this paper is effective for a class of nonlinear systems as long as the nonlinear terms satisfy (2).
The control goal of this paper is to construct a state feedback controller such that all the signals of the closed-loop system converge to zero. As long as the context does not cause confusion, the parameters of the function can be simplified.


## 3. Main Results

Theorem 1.
When the constantc satisfies 0<c<0.5, all the signals of system (1) can converge to zero by the following controller:(3)u=−β1t+1γnξ1−β2t+1γn−1ξ2−⋯−βnt+1γξn,where βi, i=1,2,…,n, are coefficients of the Hurwitz polynomial ϕρ=ρn+βnρn−1+⋯+β2ρ+β1 and γ=2c.Proof.
Letγ=2c. One presents the coordinate transformations as follows:(4)ϵi=t+1γiξi,i=1,2,…,n.
Based on (1) and (4), it is obtained that(5)ϵ˙i=1t+1γϵi+1+t+1γiψi+γit+1ϵi,i=1,2,…,n,where ϵn+1=u.
Letting(6)u=−1t+1γn+1β1ϵ1+β2ϵ2+⋯+βnϵn,where βi, i=1,2,…,n, are given in (3), the following equation is satisfied:(7)ϵ˙=1t+1γΦϵ+γt+1Λϵ+Ψ,where ϵ=ϵ1,ϵ2,…,ϵnT, Λ=diag1,2,…,n, and(8)Φ=01⋯0⋮⋮⋱000⋯1−β1−β2⋯−βn,Ψ=t+1γψ1⋮t+1n−1γψn−10.
Becauseβi, i=1,2,…,n, are coefficients of the Hurwitz polynomial ϕρ, there is a positive definite matrix Γ satisfying ΓΦ+ΦTΓ≤−I [21]. Letting Vϵ=ϵTΓϵ, one gets(9)V˙ϵ|7≤−1t+1γϵ2+2γΛΓt+1ϵ2+2ϵTΓΨ.
By Assumption1 and (4), one has(10)t+1iγψit,ξ,u≤θ1+δt+11.5γϵ1+ϵ2+⋯+ϵn≤nθ1+δt+11.5γϵ.
It follows from (10) that(11)2ϵTΓΨ≤2ϵ⋅Γ⋅Ψ≤2Γnn−1θ1+δt+11.5γϵ2.
Substituting (11) into (9), one gets(12)V˙ϵ|7≤−1t+1γϵ2+2γΛΓt+1ϵ2+2Γnn−1θ1+δt+11.5γϵ2.
Sinceγ=2c<1, there is a finite time T such that(13)−1t+1γ+2γΛΓt+1+2Γnn−1θ1+δt+11.5γ<0,t≥T.
Therefore, it is verified from (12) that(14)V˙<−ϵ2,fort≥T.
Based on (14) and the definition of V, the states ϵi, i=1,2,…,n, converge to zero. By (6), the controller u converges to zero. From (4), one has that the states ξi, i=1,2,…,n, converge to zero.Remark 2.
From the proof of Theorem1, we can see that the constant c is a key design parameter. As long as the parameter c satisfies the condition 0<c<0.5, one guarantees that equation (13) holds, and then, one handles the effects of the unknown parameter θ and the function on time in the nonlinear terms. The parameters in controller (3) consist of two parts. One is the parameter γ, which only needs to be satisfied by γ=2c. The other part is the Hurwitz polynomial coefficients βi, i=1,2,…,n, which are also relatively easy to choose. Therefore, the parameters in controller (3) are better selected, which avoid excessive calculation process.Theorem 2.
In the case ofc=0, the states of system (1) can converge to zero by the following controller:(15)u=−β1ξ1t+10.5n−β2ξ2t+10.5n−1−⋯−βnξnt+10.5,where βi, i=1,2,…,n, are coefficients of the Hurwitz polynomial ϕρ=ρn+βnρn−1+⋯+β2ρ+β1.Proof.
The proof procedure is similar to the proof procedure of Theorem1. One chooses γ=0.5 in (4), and then, controller (15) is designed. In order to avert repetition, the detailed proof is omitted.Theorem 3.
When the constantc satisfies 0.5≤c<1, all the signals of system (1) can converge to zero by the following controller:(16)u=−β1t+1nξ1−β2t+1n−1ξ2−⋯−βnt+1ξn,where βi, i=1,2,…,n, are coefficients of the Hurwitz polynomial ϕρ=ρn+βnρn−1+⋯+β2ρ+β1.Proof.
One presents the coordinate transformations as follows:(17)ϵi=t+1iξi,i=1,2,…,n.
Based on (16) and (17), it is obtained that(18)ϵ˙i=1t+1ϵi+1+t+1iψi+it+1ϵi,i=1,2,…,n.
Letting(19)u=−1t+1n+1β1ϵ1+β2ϵ2+⋯+βnϵn,where βi, i=1,2,…,n, are given in (3), the following equation is satisfied:(20)ϵ˙=1t+1Φϵ+1t+1Λϵ+Ψ,where ϵ=ϵ1,ϵ2,…,ϵnT, Λ=diag1,2,…,n, and(21)Φ=01⋯0⋮⋮⋱000⋯1−β1−β2⋯−βn,Ψ=t+1ψ1⋮t+1n−1ψn−10.
Becauseβi, i=1,2,…,n, are coefficients of the Hurwitz polynomial ϕρ, there is a positive definite matrix Γ satisfying ΓΦ+Λ+Φ+ΛTΓ≤−I [21]. Letting Vϵ=ϵTΓϵ, one gets(22)V˙ϵ|7≤−1t+1ϵ2+2ϵTΓΨ.
By Assumption1 and (4), one has(23)t+1iψit,ξ,u≤θ1+δt+12−cϵ1+ϵ2+⋯+ϵn≤nθ1+δt+12−cϵ.
It follows from (10) that(24)2ϵTΓΨ≤2ϵ⋅Γ⋅Ψ≤2Γnn−1θ1+δt+12−cϵ2.
Substituting (11) into (9), one gets(25)V˙ϵ|7≤−1t+1ϵ2+2Γnn−1θ1+δt+12−cϵ2.
Sincec<1, there is a finite time T such that(26)−1t+1+2Γnn−1θ1+δt+12−c<0,t≥T.
Therefore, it is verified from (12) that(27)V˙<−ϵ2,fort≥T.
Based on (27) and the definition of V, the states ϵi, i=1,2,…,n, converge to zero. By (20), the controller u converges to zero. From (17), one has that the states ξi, i=1,2,…,n, converge to zero.Remark 3.
In this paper, with the help of the Lyapunov function, a new control strategy is proposed for upper triangular nonlinear systems, and state feedback controllers (3), (15), and (16) are designed such that all the signals of the closed-loop system converge to zero. In Theorem 2, when c=0, one can choose γ=0.5. Then, controller (15) can ensure the convergence performance of the states. In fact, when c=0, as long as the constant γ<1 is selected, the effectiveness of controllers (15) and (16) can be ensured.


## 4. Simulation Examples

Example 1.
One considers the following nonlinear system:(28)ξ˙1=ξ2+0.11+t0.2ξ3+u,ξ˙2=ξ3+0.11+t0.2u,ξ˙n=u.
One gets that system (28) satisfies Assumption 1 with θ=0.1 and c=0.2. Let γ=0.4, β1=0.3, β2=1.2, and β3=0.7. Based on Theorem 1, a state feedback controller for system (28) is designed as(29)u=−3ξ110t+11.2−6ξ25t+10.8−7ξ310t+10.4.
The initial condition is chosen asξ10=0.5, ξ20=0.8, and ξ30=0.8. From Figures 1–3, one has that the states ξ1, ξ2, and ξ3 of system (28) converge to zero. The controller u is shown in Figure 4.Figure 1
Trajectory ofξ1t.Figure 2
Trajectory ofξ2t.Figure 3
Trajectory ofξ3t.Figure 4
Trajectory ofut.Example 2.
A practical example about the resonant circuit system is investigated as follows [22](30)i˙η1=−vτη1−r1η1iη2+r12η1sinvτ,v˙τ=iη2τ−12τsinvτ,i˙η2=−r2η2iη2+μη2.
In system (30), the meaning of the parameters is found in [22]. Following the coordinate transformation in [22], one gets(31)ξ˙1=ξ2+2ξ3,ξ˙2=ξ3,ξ˙n=u.
One gets that system (30) satisfies Assumption 1 with θ=2 and c=0. Let γ=0.5, β1=0.3, β2=1.2, and β3=0.7. Based on Theorem 2, a state feedback controller for system (30) is designed as(32)ut=−β1ξ1t+11.5−β2ξ2t+1−β3ξ3t+10.5.
The initial condition is chosen asξ10=−0.5, ξ20=−0.3, and ξ30=−0.5. From Figures 5–7, one has that the states ξ1, ξ2, and ξ3 of system (30) converge to zero. The controller u is shown in Figure 8.Figure 5
Trajectory ofξ1t.Figure 6
Trajectory ofξ2t.Figure 7
Trajectory ofξ3t.Figure 8
Trajectory ofut.Example 3.
One considers the following nonlinear system:(33)ξ˙1=ξ2+0.51+t0.9ξ3+u,ξ˙2=ξ3+0.51+t0.9u,ξ˙n=u.
One gets that system (33) satisfies Assumption 1 with θ=0.5 and c=0.9. Let β1=5, β2=10, and β3=5. Based on Theorem 1, a state feedback controller for system (33) is designed as(34)u=−5ξ1t+13−10ξ2t+12−5ξ3t+1,
The initial condition is chosen asξ10=0.4, ξ20=0.4, and ξ30=0.3. From Figures 9–11, one has that the states ξ1, ξ2, and ξ3 of system (33) converge to zero. The controller u is shown in Figure 12.Figure 9
Trajectory ofξ1t.Figure 10
Trajectory ofξ2t.Figure 11
Trajectory ofξ3t.Figure 12
Trajectory ofut.Remark 4.
In the two simulation examples, one can see that the parameters in controllers (29), (32), and (34) are relatively easy to select, which do not require a large amount of calculation to deal with the unknown parameter in systems (28), (30), and (33), respectively. In addition, controllers (29), (32), and (34) are effective for a class of nonlinear systems as long as the nonlinear terms satisfy (2). Compared with the example of the results [18, 22], the unknown constant and the function with respect to time are allowed in the nonlinear terms. Compared with the commonly adaptive control strategy, the time-varying gain design method is more concise, the calculation process is less, and a lot of calculation work is reduced.


## 5. Conclusion

This paper has investigated the state feedback control problem of upper triangular nonlinear systems. One has assumed that the nonlinear term recognizes an incremental rate relying on the unknown constant and the function with respect to time. A time-varying gain design method has been used to construct a state feedback controller. With the help of a Lyapunov function, one has obtained that all the signals of the closed-loop system have converged to zero. Finally, two numerical examples have been presented to illustrate the effectiveness of the time-varying gain design method.



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