A Nonlinear Robust Controller Design of Lower-Triangular Systems Based on Dissipation Theory

Abstract

Based on dissipation theory, a novel robust control is proposed for the lower-triangular nonlinear systems, which include strict-feedback systems and high-order lower-triangular systems. Some important concepts in dissipation theory are integrated into the recursive design, which are used to dominate the uncertain disturbance and construct the robust controller. The gotten controller renders the closed-loop system finite-gain 𝐿₂ stable in the presence of disturbance and asymptotically stable in the absence of disturbance. Especially, the controller has its advantage in regulating large disturbance. Finally, one example and one application are given to show the effectiveness of the design method. Moreover, by comparing with another robust controller, the characteristic of the proposed controller is illustrated in the simulations.

1. Introduction

Over the last decade, the lower-triangular systems are researched widely in the field of nonlinear systems [1–4]. This class of systems is not only important theoretically, but a lot of practical systems can conform to or be transformed into its form, such as the power generators [5], aircraft control system [6], and mobile robots [7]. It is well known that backstepping design [8, 9] is proposed to design the strict-feedback systems, which hold the simplest triangular structure. For different types of triangular systems, the condition for existence of control laws is investigated in [10, 11], and the explicit constructions of controllers are provided in [12, 13]. Then, a power integrator technique is given to design the high-order lower-triangular systems, which are neither feedback linearizable nor affine in the control input [4, 14]. Moreover, the robust control problem of lower-triangular systems has attracted a lot of attention [15–18] for overcoming the hurdle of disturbance, which possibly comes from model simplification, external disturbance, or other unknown factors.

In this paper, based on dissipation theory, a novel design method is proposed to solve the robust control problem of lower-triangular nonlinear systems in the presence of uncertain disturbance. This approach constructs the storage function and designs the controller recursively, which ensures that the closed-loop system satisfies the dissipation inequality having 𝐿₂-gain performance. The technique is firstly used in the strict-feedback system. Then, it is extended to the high-order lower-triangular system. Finally, one example of lower-triangular system and one application of synchronous generator system are given to illustrate the effectiveness of the robust control.

For the lower-triangular system with uncertain disturbance, this work uses the frame of dissipation property theory to solve the problem of disturbance rejection. First of all, the appropriate function of supply rate is chosen, which is used to gather the disturbance inputs in the recursive design. Next, the storage function and robust controller are constructed recursively to guarantee that the result at each step satisfies the dissipation property. During the design, by using feedback domination technique [4, 14], the disturbed parts are dominated by the functions of system states and disturbances. Then, the function of disturbances is regarded as the input of supply rate, which satisfies the dissipation inequality ensuring finite-gain 𝐿₂ stability. The effect of this method can be summarized as two aspects. Firstly, the restriction of uncertain disturbance is relaxed, and the only assumption of uncertain disturbance is bounded. So, no information about disturbance is required in the gotten control law, which means that certain controller can regulate uncertain disturbance. Secondly, compared with some previous robust controllers only applicable for small disturbances, the proposed robust controller is effective for all cases of bounded disturbance, especially for large disturbance.

The structure of this paper is as follows: Section 2 describes the problem and offers key lemma; Section 3 gives the robust controller design for strict-feedback systems; Section 4 extends the design method to high-order lower-triangular systems; Section 5 provides an example and application to illustrate the effectiveness of design approach. The conclusion is contained in Section 6.

2. Problem Formulation and Key Lemma

In this paper, we focus on the problem of constructing robust controller for the lower-triangular system, which is described by the following differential equations: 𝑥̇₁=𝑓₁𝑥₁+𝑥^𝑝1₂+𝜔₁(𝑡)𝑥̇₂=𝑓₂𝑋₂+𝑥^𝑝2₃+𝜔₂(𝑡)⋮𝑥̇_𝑛=𝑓_𝑛𝑋_𝑛+𝑢^𝑝𝑛+𝜔_𝑛(𝑡)𝑦=𝑥₁, (1) where for 𝑖=1,2,…,𝑛, the state 𝑋_𝑖=(𝑥₁,…,𝑥_𝑖)^𝑇∈ℝ^𝑖, the input 𝑢∈ℝ, the output 𝑦∈ℝ, the uncertain disturbance vector 𝜔=[𝜔₁,𝜔₂,…,𝜔_𝑛]^𝑇, 𝜔_𝑖(𝑡) is unknown nonlinear function, which denotes external disturbance with unknown bound, 𝑓_𝑖∶ℝ^𝑖↦ℝ is a smooth function, and 𝑓_𝑖(0)=0. About the system (1), one hypothesis is given as follows.

Assumption 1. 𝑝_𝑖≥1 (𝑖=1,2,…,𝑛) is odd integer, and 𝑝₁ is maximum of them.
Compared with the corresponding assumption in [4, 14], Assumption 1 relaxes the condition of𝑝_𝑖. Next, the main definitions of dissipation theory are given as follows.

Definition 2 (see [1]). The general systems are described in the following form: 𝑥̇(𝑡)=𝑓𝑥,𝑢₀𝑦₀(𝑡)=ℎ𝑥,𝑢₀, (2) where the state 𝑥∈ℝ^𝑛, the input 𝑢₀∈ℝ^𝑝, and the output 𝑦₀∈ℝ^𝑞. Let the function𝑠(𝑢₀,𝑦₀)∶ℝ^𝑝×ℝ^𝑞↦ℝ if there exists 𝑉(𝑥)≥0, 𝑉∶ℝ^𝑛↦ℝ⁺, such that 𝑉(𝑥(𝜏))≤𝑉(𝑥(0))+^𝜏₀𝑠𝑢₀,𝑦₀𝑑𝑡; (3) for any 𝑥(0) and 𝜏, the system (3) is dissipative about 𝑠(𝑢₀,𝑦₀). And 𝑠(𝑢₀,𝑦₀) is the supply rate, and 𝑉(𝑥) is the storage function.

Remark 3. The supply rate 𝑠(𝑢₀,𝑦₀) in Definition 2 is selectable. And there are some optional functions. The most commonly used are two options: one is𝑠(𝑢₀,𝑦₀)=𝑢^𝑇₀𝑦₀, and the other is𝑠(𝑢₀,𝑦₀)=Υ²‖𝑢₀‖²−‖𝑦₀‖²,Υ>0. In this paper, when the uncertain disturbance 𝜔=[𝜔₁,𝜔₂,…,𝜔_𝑛]^𝑇 is regarded as the input of the system (3), the latter is chosen as the supply rate. And if the closed-loop system is dissipative, we have 𝑉(𝑥(𝜏))−𝑉(𝑥(0))≤^𝜏₀Υ²‖𝜔‖²−𝑦₀²𝑑𝑡. (4)

That is, the gain of the system is not more than Υ for the uncertain disturbance.

Definition 4 (see [1]). A mapping 𝐻∶𝐿^𝑚_𝑒→𝐿^𝑞_𝑒 is finite-gain 𝐿 stable if there exist nonnegative constants Υ_𝐿 and 𝛽_𝐿, such that (𝐻𝑣)_𝜏𝐿≤Υ_𝐿𝑣_𝜏𝐿+𝛽_𝐿, (5) for 𝑣∈𝐿^𝑚_𝑒 and 𝜏∈[0,∞).
Young’s inequality is an important tool used in the recursive design, and Lemma 6 is one direct consequence of Young’s inequality.

Lemma 5 (Young’s inequality). For any two vectors 𝑥 and𝑦, the following holds 𝑥^𝑇𝜀𝑦≤^𝑝𝑝‖𝑥‖^𝑝+1𝑞𝜀^𝑞𝑦^𝑞, (6) where 𝜀>0 and the constants 𝑝>1 and 𝑞>1 satisfy (𝑝−1)(𝑞−1)=1.

Lemma 6 (see [19]). For real numbers 𝑎≥0, 𝑏≥0, and 𝑚≥1, the following inequality holds: 𝑎𝑎≤𝑏+𝑚^𝑚𝑚−1𝑏^𝑚−1. (7)

3. Robust Controller Design of Strict-Feedback Systems

The strict-feedback system is described by the following equations 𝑥̇₁=𝑓₁𝑥₁+𝑥₂+𝜔₁(𝑡)𝑥̇₂=𝑓₂𝑋₂+𝑥₃+𝜔₂(𝑡)⋮𝑥̇_𝑛=𝑓_𝑛𝑋_𝑛+𝑢+𝜔_𝑛(𝑡)𝑦=𝑥₁. (8)

Based on the dissipation theory, the novel robust controller design is proposed step by step. For𝑖=1,2,…,𝑛, firstly set the design parameter𝐶_𝑖>0.

Step 1. Let𝑧₁=𝑥₁. The storage function is constructed as𝑉₁=𝑧²₁/2, whose derivative is 𝑉̇₁=𝑧₁𝑧̇₁=𝑧₁𝑓₁𝑥₁+𝑥₂+𝑧₁𝜔₁. (9)

For the above equation, according to the technique of adding one power integrator [4, 14], by using Young’s inequality (6) with 𝑝=𝑞=2 and𝜀=√𝑛/𝛾, the term 𝑧₁𝜔₁ is dominated by(𝑛/2𝛾²)𝑧²₁+(𝛾²/2𝑛)𝜔²₁, where the disturbance attenuation coefficient𝛾>0. Taking it into formula (9) yields 𝑉̇₁≤𝑧₁𝑥₂+𝑓₁𝑥₁+12+𝑛2𝛾²𝑧₁+𝛾²𝜔2𝑛²¹−𝑧²₁2. (10)

Design the smooth virtual control as 𝛼₂=−𝐶₁𝑧₁−𝑓₁𝑥₁−12𝑛1+𝛾²𝑧₁. (11)

Taking controller (11) into formula (10) results in 𝑉̇₁≤−𝐶₁𝑧²₁+𝑧₁𝑥₂−𝛼₂+𝛾²𝜔2𝑛²¹−12𝑧²₁. (12)

Step k
Through 𝑘−1 steps, a group of virtual controllers are𝑧₁=𝑥₁,𝑧₂=𝑥₂−𝛼₂,…,𝑧_𝑘−1=𝑥_𝑘−1−𝛼_𝑘−1, and the storage function is 𝑉_𝑘−1=∑^𝑘−1_𝑖=1(𝑧²_𝑖/2), whose derivative is 𝑉̇_𝑘−1≤−^𝑘−1_𝑖=1𝐶_𝑖𝑧²_𝑖+𝑧_𝑘−1𝑥_𝑘−𝛼_𝑘+𝛾²2^𝑘−1_𝑗=1𝑘−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2. (13)
Let 𝑧_𝑘=𝑥_𝑘−𝛼_𝑘, and the storage function is changed into𝑉_𝑘=𝑉_𝑘−1+𝑧²_𝑘/2; its derivative is obtained as 𝑉̇_𝑘=𝑉̇_𝑘−1+𝑧_𝑘𝑧̇_𝑘≤−^𝑘−1_𝑖=1𝐶_𝑖𝑧²_𝑖+𝛾²2^𝑘−1_𝑗=1𝑘−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2+𝑧_𝑘𝜔_𝑘−𝑧_𝑘^𝑘−1_𝑚=1𝜕𝛼_𝑘𝜕𝑥_𝑚𝜔_𝑚+𝑧_𝑘𝑓_𝑘𝑋_𝑘+𝑥_𝑘+1+𝑧_𝑘−1−^𝑘−1_𝑚=1𝜕𝛼_𝑘𝜕𝑥_𝑚𝑓_𝑚𝑋_𝑚+𝑥_𝑚+1. (14)

By using Young’s inequality, we have 𝑉̇_𝑘≤−^𝑘−1_𝑖=1𝐶_𝑖𝑧²_𝑖+𝛾²2^𝑘−1_𝑗=1𝑘−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2+𝛾²2()𝜔𝑛+1−𝑘^2𝑘+𝛾²2^𝑘−1_𝑚=1𝜔²_𝑚𝑛+1−𝑚+𝑧_𝑘𝑓_𝑘𝑋_𝑘+𝑥_𝑘+1+𝑧_𝑘−1+𝑛+1−𝑘2𝛾²𝑧_𝑘+𝑧_𝑘2𝛾^2𝑘−1_𝑚=1(𝑛+1−𝑚)𝜕𝛼_𝑘𝜕𝑥_𝑚²−^𝑘−1_𝑚=1𝜕𝛼_𝑘𝜕𝑥_𝑚𝑓_𝑚𝑋_𝑚+𝑥_𝑚+1. (15)

In this step, the smooth virtual controller is taken as follows: 𝛼_𝑘+1=−𝐶_𝑘𝑧_𝑘−𝑓_𝑘𝑋_𝑘−𝑧_𝑘−1−𝑛+1−𝑘2𝛾²𝑧_𝑘−𝑧_𝑘2𝛾^2𝑘−1_𝑚=1(𝑛+1−𝑚)𝜕𝛼_𝑘𝜕𝑥_𝑚²+^𝑘−1_𝑚=1𝜕𝛼_𝑘𝜕𝑥_𝑚𝑓_𝑚𝑋_𝑚+𝑥_𝑚+1. (16)

Under the action of the above controller, formula (15) is turned into 𝑉̇_𝑘≤−^𝑘_𝑖=1𝐶_𝑖𝑧²_𝑖+𝑧_𝑘𝑥_𝑘+1−𝛼_𝑘+1+𝛾²2^𝑘_𝑗=1𝑘+1−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2. (17)

Step n. Let 𝑧_𝑛=𝑥_𝑛−𝛼_𝑛, and the storage function is 𝑉=𝑉_𝑛=𝑉_𝑛−1+𝑧²_𝑛/2=(1/2)∑^𝑛_𝑖=1𝑧²_𝑖. The nonlinear robust controller is designed as 𝑢=−𝐶_𝑛𝑧_𝑛−𝑓_𝑛𝑋_𝑛−𝑧_𝑛−1−𝑧_𝑛2𝛾²−𝑧_𝑛2𝛾^2𝑛−1_𝑚=1(𝑛+1−𝑚)𝜕𝛼_𝑛𝜕𝑥_𝑚²+^𝑛−1_𝑚=1𝜕𝛼_𝑛𝜕𝑥_𝑚𝑓_𝑚𝑋_𝑚+𝑥_𝑚+1. (18)

Now, the derivative of the storage function is 𝑉̇_𝑛≤−^𝑛_𝑖=1𝐶_𝑖𝑧²_𝑖+𝛾²2^𝑛_𝑗=1𝜔²_𝑗−𝑧²₁2. (19)

From system (8), it is known that the uncertain disturbance vector 𝜔=[𝜔₁,𝜔₂,…,𝜔_𝑛]^𝑇 and the output 𝑦=𝑥₁=𝑧₁. So, we have 𝑉̇_𝑛≤−^𝑛_𝑖=1𝐶_𝑖𝑧²_𝑖+𝛾²2‖𝜔‖²₂−12𝑦²₂≤12𝛾²‖𝜔‖²₂−𝑦²₂. (20)

Based on dissipation theory, the robust controller (18) makes the closed-loop system dissipative with the uncertain disturbance, and the supply rate 𝑠(𝜔,𝑦)=𝛾²‖𝜔‖²₂−‖𝑦‖²₂. Furthermore, the finite-gain 𝐿₂ stability of the closed-loop system is concluded in the following theorem.

Theorem 7. Consider the strict-feedback system with disturbance (8). There exists the smooth robust controller (18), such that the closed-loop system is finite-gain 𝐿₂ stable and the 𝐿₂ gain is no more than𝛾. Moreover, when the disturbance input𝜔=[𝜔₁,𝜔₂,…,𝜔_𝑛]^𝑇=0, the closed-loop system is asymptotically stable.

Proof. From (20), we have𝑉̇=𝑉̇_𝑛≤(1/2)(𝛾²‖𝜔‖²₂−‖𝑦‖²₂). Integrating it yields the following inequality: 𝑉(𝑥(𝜏(𝑥(0≤1))−𝑉))2^𝜏₀𝛾²‖𝜔‖²₂−𝑦²₂𝑑𝑡. (21)
Due to 𝑉(𝑥)≥0, we obtain ^𝜏₀𝑦²₂𝑑𝑡≤𝛾^2𝜏₀‖𝜔‖²₂𝑑𝑡+2𝑉(𝑥(0)). (22)
Taking the square roots and using the inequality 𝑎²+𝑏²≤𝑎+𝑏 for nonnegative numbers 𝑎 and𝑏, one obtains 𝑦_𝜏𝐿2≤𝛾𝜔_𝜏𝐿2+2𝑉(𝑥(0)). (23)
Therefore, based on Definition 4, It is known that the system is finite-gain 𝐿₂ stable and the gain from disturbance input to system output is no more than𝛾.
Moreover, when 𝜔=0, from (20) we have 𝑉̇=𝑉̇_𝑛≤𝛾²2‖𝜔‖²−12𝑦²1≤−2𝑦²≤0, (24) and the storage function 𝑉=(1/2)∑^𝑛_𝑖=1𝑧²_𝑖 is positive definite. According to the invariance principle [1], we need to find 𝑆={𝑋_𝑛∈ℝ^𝑛∣𝑉̇=0}. Note that 𝑉̇=0⟹𝑦=0⟹𝑥₁=0. (25)
Hence, 𝑆={𝑋_𝑛∈ℝ^𝑛∣𝑥₁=0}. Let 𝑋_𝑛(𝑡) be a solution that belongs identically to 𝑆. From the system (8), we can deduce that 𝑥₁≡0⟹𝑥̇₁≡0⟹𝑥₂≡0⟹𝑥̇₂≡0⟹⋯⟹𝑥_𝑛≡0. (26)
Therefore, the only solution that can stay identically in 𝑆 is the trivial solution 𝑋_𝑛(𝑡)≡0. Thus, the closed-loop system is asymptotically stable in the case of 𝜔=0.

Remark 8. About the proposed method, there are the following opinions.(1)This method integrates the idea of dissipation property into the recursive design. In the frame of dissipation theory, by using the feedback domination technology, the unknown disturbance is decoupled from the known state and gathered together to satisfy the dissipation inequality, which leads to the result of disturbance rejection and finite-gain 𝐿₂ stability.(2)In the previous research of lower-triangular system (1), some assumption is needed for the uncertain disturbance, such as boundary condition and functional constraint. In this paper, the assumption is relaxed to be bounded, which can ensure that the closed-loop system is finite-gain 𝐿₂ stable. On the other hand, just for there is no constraint about the uncertain disturbance, the result is only finite-gain 𝐿₂ stable.(3)It is interesting to note that the external disturbance 𝜔_𝑖 can be generalized to more general uncertainties Δ_𝑖(𝑋_𝑖,𝑡), which satisfies the condition |Δ_𝑖(𝑋_𝑖,𝑡)|≤𝜔_𝑖(𝑡)𝜙_𝑖(𝑋_𝑖), with 𝜙_𝑖(𝑋_𝑖) being a known smooth function and 𝜔_𝑖(𝑡) being the unknown bound. The proposed method is still applicable for the more general case.

4. Robust Controller Design of Lower-Triangular Systems

The robust control law of lower-triangular systems (1) is designed in this section. Similarly, for 𝑖=1,2,…,𝑛, the design parameter 𝐶_𝑖>0.

Step 1. Let𝑧₁=𝑥₁. Due to Assumption 1, the storage function is constructed as𝑉₁=𝑧^{𝑝1−𝑝1}₁⁺²/(𝑝₁−𝑝₁+2)=𝑧²₁/2, and its derivative is 𝑉̇₁=𝑧₁𝑧̇₁=𝑧₁𝑓₁𝑥₁+𝑥^𝑝1₂+𝑧₁𝜔₁. (27)

Similar to the design of strict-feedback systems, one obtains: 𝑉̇₁≤𝑧₁𝑥^𝑝1₂+𝑓₁𝑥₁+12+𝑛2𝛾²𝑧₁+𝛾²𝜔2𝑛²¹−𝑧²₁2. (28)

Owing to (7), for any positive real number 𝜎>0, let 𝑏=𝜎, 𝑎=|𝑧₁[𝑓₁+((1/2)+(𝑛/2𝛾²))𝑧₁]|, and 𝑚=𝑝₁+1. We have 𝑧₁𝑓₁+12+𝑛2𝛾²𝑧₁≤𝜎+𝑧^𝑝1₁⁺¹𝜌₁𝑧₁, (29) where 𝜌₁(𝑧₁)=(1/(𝑝₁+1))[𝑝₁/(𝑝₁+1)𝜎]^𝑝1(𝑓₁+𝑧₁/2+𝑛𝑧₁/2𝛾²)^𝑝1+1≥0.

Substituting (29) into (28) yields 𝑉̇₁≤𝑧₁𝑥^𝑝1₂+𝑧^𝑝1₁⁺¹𝜌₁𝑧₁+𝛾²𝜔2𝑛²¹−𝑧²₁2+𝜎. (30)

Paying attention to Assumption 1, we design the smooth virtual control as 𝛼₂=−𝑧₁𝐶₁+1+𝜌₁𝑧₁^1/𝑝1. (31)

Taking the controller (31) into (30) results in 𝑉̇₁≤−𝐶₁+1𝑧^𝑝1₁⁺¹+𝑧₁𝑥^𝑝1₂−𝛼^𝑝1₂+𝛾²𝜔2𝑛²¹−𝑧²₁2+𝜎. (32)

Step k
Through 𝑘−1 steps, a group of virtual controllers are 𝑧₁=𝑥₁, 𝑧₂=𝑥₂−𝛼₂,…,𝑧_𝑘−1=𝑥_𝑘−1−𝛼_𝑘−1, and the storage function is 𝑉_𝑘−1=∑^𝑘−1_𝑖=1(𝑧^{𝑝1−𝑝𝑖}_𝑖⁺²/(𝑝₁−𝑝_𝑖+2)), whose derivative is 𝑉̇_𝑘−1≤−^𝑘−1_𝑖=1𝐶_𝑖𝑧^𝑝1_𝑖⁺¹−𝑧^𝑝1+1_𝑘−1+𝑧^{𝑝1−𝑝𝑘−1+1}_𝑘−1𝑥^𝑝𝑘−1_𝑘−𝛼^𝑝𝑘−1_𝑘+𝛾²2^𝑘−1_𝑗=1𝑘−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2+(𝑘−1)𝜎. (33)
Let 𝑧_𝑘=𝑥_𝑘−𝛼_𝑘, and consider the storage function 𝑉_𝑘=𝑉_𝑘−1+𝑧^{𝑝1−𝑝𝑘}_𝑘⁺²/(𝑝₁−𝑝_𝑘+2); its derivative is obtained as 𝑉̇_𝑘=𝑉̇_𝑘−1+𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹𝑧̇_𝑘≤−^𝑘−1_𝑖=1𝐶_𝑖𝑧^𝑝1_𝑖⁺¹−𝑧^𝑝1+1_𝑘−1+𝑧^{𝑝1−𝑝𝑘−1+1}_𝑘−1𝑥^𝑝𝑘−1_𝑘−𝛼^𝑝𝑘−1_𝑘+𝛾²2^𝑘_𝑗=1𝑘+1−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2+(𝑘−1)𝜎+𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹𝑥^𝑝𝑘_𝑘+1+𝑓_𝑘𝑋_𝑘+𝑛+1−𝑘2𝛾²𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹+𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹2𝛾^2𝑘−1_𝑚=1(𝑛+1−𝑚)𝜕𝛼_𝑘𝜕𝑥_𝑚²−^𝑘−1_𝑚=1𝜕𝛼_𝑘𝜕𝑥_𝑚𝑓_𝑚+𝑥^𝑝𝑚_𝑚+1. (34)

Due to Young’s inequality, there exists the smooth function 𝜌_𝑘(𝑧₁,…,𝑧_𝑘), such that 𝑧^{𝑝1−𝑝𝑘−1+1}_𝑘−1𝑥^𝑝𝑘−1_𝑘−𝛼^𝑝𝑘−1_𝑘≤𝑧^𝑝1+1_𝑘−1+𝑧^𝑝1_𝑘⁺¹𝜌_𝑘𝑧₁,…,𝑧_𝑘, (35) where 𝜌_𝑘𝑧₁,…,𝑧_𝑘=𝑝_𝑘−1𝑝₁2+1^{𝑝𝑘−1−2}𝑝_𝑘−1^{(𝑝1+1)/𝑝𝑘−1}×2𝑝₁−𝑝_𝑘−1+1𝑝₁+1^{(𝑝1−𝑝𝑘−1+1)/𝑝𝑘−1}+1𝑝₁+11+2^{𝑝𝑘−1−2}𝑝_𝑘−1×𝐶_𝑘−1+1+𝜌_𝑘−1^{(𝑝𝑘−1−1)/𝑝𝑘−1𝑝1+1}×2𝑝₁𝑝₁+1^𝑝1, (36) and the construction of 𝜌_𝑘(𝑧₁,…,𝑧_𝑘) can refer to [19].

Then, we define a smooth function as follows: 𝐷_𝑘𝑧₁,…,𝑧_𝑘=𝑓_𝑘𝑋_𝑘+𝑛+1−𝑘2𝛾²𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹+𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹2𝛾^2𝑘−1_𝑚=1(𝑛+1−𝑚)𝜕𝛼_𝑘𝜕𝑥_𝑚²−^𝑘−1_𝑚=1𝜕𝛼_𝑘𝜕𝑥_𝑚𝑓_𝑚+𝑥^𝑝𝑚_𝑚+1. (37)

Similar to Step 1, we obtain 𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹𝐷_𝑘𝑧₁,…,𝑧_𝑘≤𝜎+𝑧^𝑝1_𝑘⁺¹𝜌_𝑘𝑧₁,…,𝑧_𝑘. (38)

Taking (35) and (38) into (34) yields 𝑉̇_𝑘≤−^𝑘−1_𝑖=1𝐶_𝑖𝑧^𝑝1_𝑖⁺¹+𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹𝑥^𝑝𝑘_𝑘+1+𝑧^𝑝1_𝑘⁺¹𝜌_𝑘(⋅)+𝜌_𝑘(⋅)+𝛾²2^𝑘_𝑗=1𝑘+1−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2+𝑘𝜎. (39)

Design the following virtual control law 𝛼_𝑘+1=−𝑧_𝑘𝐶_𝑘+1+𝜌_𝑘𝑧₁,…,𝑧_𝑘+𝜌_𝑘𝑧₁,…,𝑧_𝑘^1/𝑝𝑘, (40) which renders 𝑉̇_𝑘≤−^𝑘_𝑖=1𝐶_𝑖𝑧^𝑝1_𝑖⁺¹−𝑧^𝑝1_𝑘⁺¹+𝑧^{𝑝1−𝑝𝑘}_𝑘⁺¹𝑥^𝑝𝑘_𝑘+1−𝛼^𝑝𝑘_𝑘+1+𝛾²2^𝑘_𝑗=1𝑘+1−𝑗𝜔𝑛+1−𝑗^2𝑗−𝑧²₁2+𝑘𝜎. (41)

Step n. In the end, for the storage function, 𝑉=𝑉_𝑛=∑^𝑛_𝑖=1(𝑧^{𝑝1−𝑝𝑖}_𝑖⁺²/(𝑝₁−𝑝_𝑖+2)), where 𝑧_𝑛=𝑥_𝑛−𝛼_𝑛. There exists the smooth controller 𝑢𝑧₁,…,𝑧_𝑛=−𝑧_𝑛𝐶_𝑛+𝜌_𝑛𝑧₁,…,𝑧_𝑛+𝜌_𝑛𝑧₁,…,𝑧_𝑛^1/𝑝𝑛, (42) such that 𝑉̇_𝑛≤−^𝑛_𝑖=1𝐶_𝑖𝑧^𝑝1_𝑖⁺¹+𝛾²2^𝑛_𝑗=1𝜔²_𝑗−𝑧²₁2+𝑛𝜎. (43)
Similar to Section 3, we finish this section with the following theorem.

Theorem 9. Consider the lower-triangular system with disturbance (1). There exists the smooth robust controller (42), such that the closed-loop system is finite-gain 𝐿₂ stable and the 𝐿₂ gain is no more than𝛾.

Proof. The proof of this theorem is similar to Theorem 7, which is omitted.

5. Example and Application

Example 1. Consider the stabilization problem of a lower-triangular system as follows: 𝑥̇₁=𝑥²₁+𝑥³₂+𝜔₁𝑥̇₂=𝑥₁𝑥²₂+𝑥₂𝑒^𝑥2+𝑢³+𝜔₂𝑦=𝑥₁, (44) where𝜔₁ and 𝜔₂ are bounded unknown disturbance.

Firstly, let 𝑧₁=𝑥₁, and construct the storage function𝑉₁=𝑧²₁/2. According to the design steps, the virtual control is obtained as 𝛼₂=−𝑧₁𝐶₁+1+𝜌₁𝑧₁^1/3, (45) where𝜌₁(𝑧₁)=(27/256𝜎³)(𝑥²₁+𝑥₁/2+𝑥₁/𝛾²)⁴, and the positive parameters 𝜎 and 𝛾 can be adjusted according to the requirements.

Secondly, let 𝑧₂=𝑥₂−𝛼₂ and 𝑉₂=𝑉₁+𝑧²₂/2. Based on the design process, the robust controller is taken as 𝑢=−𝑧₂𝐶₂+𝜌₂𝑧₁,𝑧₂+𝜌₂𝑧₁,𝑧₂^1/3, (46) where 𝜌₂𝑧₁,𝑧₂=1434𝜎³𝑥₁𝑥²₂+𝑥₂𝑒^𝑥2+𝑧₂𝛾²𝜕𝛼₂𝜕𝑥₁²−𝜕𝛼₂𝜕𝑥₁𝑥²₁+𝑥₂⁴𝜌₂𝑧₁,𝑧₂=92³√3+279𝐶32¹+1+𝜌₁𝑧₁^2/34. (47)

According to Theorem 9, it is known that the solutions of the system are globally bounded and the closed-loop system is finite-gain 𝐿₂ stable.

In the simulation, the system parameters are given as follows:𝐶₁=1, 𝐶₂=1, 𝜎=2, and 𝛾=2. When the disturbances are 𝜔₁=5, 𝜔₂=4sin𝑡, the state curves are shown as Figure 1, and the control law is in Figure 2. From them, we know that the closed-loop system is finite-gain 𝐿₂ stable and the gain is no more than𝛾=2.

Figure 1 

State response curves.

Figure 2 

Curve of control law.

Example 2. Consider one machine connected to an infinite bus system which is shown in Figure 3. The dynamic model of synchronous generator can be described as follows: 𝛿̇=𝜔−𝜔₀̇𝐷𝜔=−𝐻𝜔−𝜔₀+𝜔₀𝐻𝑃_𝑚−𝑃_𝑒+𝜔₁𝑃̇_𝑚=−𝑃_𝑚𝑇_𝐻∑+𝑃_𝑚0𝑇_𝐻∑+𝐶𝑇_𝐻∑𝜇+𝜔₂, (48) where 𝑃_𝑒=𝐸^′_𝑞𝑉_𝑠𝑋^′_𝑑∑𝑉sin𝛿+^2𝑠2𝑋^′_𝑑∑−𝑋_𝑞∑𝑋^′_𝑑∑𝑋_𝑞∑sin2𝛿, (49)𝛿 is the power angle; 𝛿₀ is the operating point of power angle; 𝜔 is the relative speed; 𝜔₀ is the synchronous machine speed; 𝑃_𝑚 is the mechanical input power; 𝑃_𝑚0 is the operating point of mechanical input power; 𝑃_𝑒 is the electromagnetic power; 𝜇 is the steam-valving controller; the uncertain disturbances 𝜔₁ and 𝜔₂ are bounded.
Let (𝛿₀,𝜔₀,𝑃_𝑚0) be the operating point, and define state variables by 𝑥₁=𝛿−𝛿₀, 𝑥₂=𝜔−𝜔₀, and 𝑥₃=𝑃_𝑚−𝑃_𝑚0; then, system (48) and (49) is represented by 𝑥̇₁=𝑥₂𝑥̇₂𝐷=−𝐻𝑥₂−𝜔₀𝐻Δ𝑃_𝑒+𝜔₀𝐻𝑥₃+𝜔₁𝑥̇₃1=−𝑇_𝐻∑𝑥₃+𝐶𝑇_𝐻∑𝜇+𝜔₂, (50) where Δ𝑃_𝑒=𝐸^′_𝑞𝑉_𝑠𝑋^′_𝑑∑sin𝛿₀+𝑥₁−sin𝛿₀+𝑉²_𝑠2𝑋^′_𝑑∑−𝑋_𝑞∑𝑋^′_𝑑∑𝑋_𝑞∑sin2𝛿₀+𝑥₁−sin2𝛿₀. (51)
From the model (50) and (51), we know that this is a strict-feedback system. According to the design process in Section 3, the robust controller is obtained as follows: 𝜇=𝑇_𝐻∑−𝐶₃𝑧₃+𝑥₃𝑇_𝐻∑−𝑧₂−𝑧₂2𝛾²×1+3𝜕𝛼₃𝜕𝑥₁²+2𝜕𝛼₃𝜕𝑥₂²+𝜕𝛼₃𝜕𝑥₁𝑥₂+𝜕𝛼₃𝜕𝑥₂−𝐷𝐻𝑥₂−𝜔₀𝐻Δ𝑃_𝑒+𝑥₃, (52) where 𝑧₁=𝑥₁, 𝑧₂=𝑥₂−𝛼₂, 𝑧₃=𝑥₃−𝛼₃, 𝛼₂=−(𝐶₁+1/2+3/2𝛾²)𝑧₁, 𝛼₃=𝐻𝜔₀−𝐶₂𝑧₂+𝐷𝐻𝑥₂+𝜔₀𝐻Δ𝑃_𝑒−𝑧₁−3𝑧₂2𝛾²𝜕𝛼₂𝜕𝑥₁²−𝑧₂𝛾²+𝜕𝛼₂𝜕𝑥₁𝑥₂. (53)
In order to verify the viability and effectiveness of the control law (52), the computer simulation is performed with the following parameters: 𝐷=8, 𝐻=2, 𝐶=1, 𝑉_𝑠=1, 𝑇_𝐻∑=0.35, and 𝑃_𝑚0=0.87455 (see Figures 4, 5, 6, and 7).

Figure 3 

One-machine to infinite-bus circuit.

Figure 4 

State curves with disturbances.

Figure 5 

Control law with disturbances.

Figure 6 

State curves without disturbances.

Figure 7 

Control law without disturbances.

In the presence of external disturbances 𝜔₁=10 and 𝜔₂=15cos𝑡, the state responses and control law are shown in Figures 4 and 5, respectively. In the absence of disturbance, the simulation results are given in Figures 6 and 7. It is obvious that the proposed controller is effective for bounded disturbance and the closed-loop system can achieve finite-gain 𝐿₂ stable. And, with the reduction of disturbance, the 𝐿₂ gain decreases. At last, when the disturbance vanishes, the closed-loop system is asymptotically stable as shown in Figure 6.

Moreover, in order to demonstrate the superiority of the proposed controller, we compare it with another robust controller designed based on Hamiltonian function method [20].

Firstly, Hamilton energy function is constructed as 𝐸𝐻=^′𝑞𝑉_𝑠𝑋^′_𝑑∑()+𝑉1−cos𝛿^2𝑠2𝑋_𝑞∑−𝑋^′_𝑑∑𝑋^′_𝑑∑𝑋_𝑞∑cos²𝛿()+𝐻Δ𝜔²2𝜔₀+12𝑃_𝑚−𝑃_𝑚0²+𝑃_𝑚0(𝜋−𝛿). (54)

Then, the model (48) in Hamilton form is rewritten into 𝛿̇Δ𝜔̇𝑃̇_𝑚=0𝜔₀𝐻0−𝜔₀𝐻−𝐷𝜔₀𝐻²𝜔₀𝐻100−𝑇_𝐻∑00𝐶∇𝐻+𝑇_𝐻∑0𝜇+𝜔₁𝜔₂. (55)

Based on Hamiltonian function method, the robust controller is designed as 𝜇=𝜇₀−Γ𝐺^𝑇𝑇∇𝐻=−^𝐻∑𝐶𝐶Δ𝜔−Γ𝑇_𝐻∑𝑃_𝑚−𝑃_𝑚0, (56) where 𝜇₀=−(𝑇_𝐻∑/𝐶)Δ𝜔 is pre-feedback control and Γ∈ℝ is design parameter.

Next, two methods are compared by the simulations. Figures 8, 10, and 12 are the controlled results with Hamiltonian function method; Figures 9, 11, and 13 are the controlled results with dissipation theory method.

Figure 8 

Response curves of Hamiltonian function method without disturbances.

Figure 9 

Response curves of dissipation theory method without disturbances.

Figure 10 

Response curves of Hamiltonian function method with small disturbances.

Figure 11 

Response curves of dissipation theory method with small disturbances.

Figure 12 

Response curves of Hamiltonian function method with large disturbances.

Figure 13 

Response curves of dissipation theory method with large disturbances.

In the absence of disturbance, from Figures 8 and 9, it is known that both of them are effective for the generator system. From the dynamic process, it is seen that overshooting of Hamiltonian function method is less than that of dissipation theory method, and settling time of Hamiltonian function method is longer than that of dissipation. Then, in the presence of small disturbances with𝜔₁=1, 𝜔₂=1, the simulation results are shown in Figures 10 and 11. It is shown that both of closed-loop systems are stable and the robust controller is effective. Next, with the enlargement of disturbances (𝜔₁=10, 𝜔₂=10), the response curves are shown in Figures 12 and 13, where the controlled system of Hamiltonian function method is unstable and the power angle is out of control, while that of dissipation theory method is still stable, and the closed-loop system is 𝐿₂ stable.

In summary, based on Hamiltonian function method, the design process is concise, and the gotten controller (56) is simple, which is effective for the generator system in the presence of no disturbance or small disturbance. By using dissipation theory method, the construction of robust controller is relatively complex. However, it can guarantee that the closed-loop system is 𝐿₂ stable in all cases involving no disturbance, small disturbance, or large disturbance. Therefore, the proposed robust controller is more effective for the system with uncertain disturbance, especially for large disturbance.

6. Conclusions

In lower-triangular nonlinear systems, strict-feedback systems and high-order lower-triangular systems are two classes of main mathematical models. In this paper, the dissipation-based nonlinear controller is proposed to solve the robust control problem of these two systems. Uncertain disturbance is dominated by the supply rate, and the stability analysis is based on the storage function. The design method integrates the energy supply, energy storage, and energy dissipation into the recursive construction of robust controller. The simulations illustrate that the gotten controller is effective and has the advantage in regulating large disturbance. In the future research, the dissipation-based idea can be expanded to the robust controller design for more nonlinear systems.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant no. 51007019).