Enhanced Disturbance-Observer-Based Control for a Class of Time-Delay System with Uncertain Sinusoidal Disturbances

Abstract

This paper is concerned with disturbance-observer-based control (DOBC) for a class of time-delay systems with uncertain sinusoidal disturbances. The disturbances are decomposed as precise and uncertain parts using nonlinear disturbance observer (DO) after appropriate coordinate transformation. And then the two parts can be compensated by corresponding controller, respectively, such that the classic DOBC method is extended to uncertain disturbance rejection. One novel feature of the proposed method is that even if the precise disturbance parameters are inaccessible, the merits of DOBC can be inherited. By integrating the disturbance observers with feedback control laws with time delay, the disturbances can be rejected and the desired dynamic performances can be guaranteed. Finally, simulations for a flight control system are given to demonstrate the effectiveness of the results.

1. Introduction

Dynamical systems with time delays [1–3] widely exist in many systems, such as hydraulic processes, chemical systems, and temperature processes. In addition, the presence of exogenous disturbances is inevitable in engineering control systems; a complex system may suffer various of disturbances due to inherent physical property including sensor measurement noise, control error, and structural vibrations. As the phenomenons mentioned above are often primary sources of instability and performance degradation, it is an impendence thing to design control strategy for time-delay systems characteristic with antidisturbance performance. Some researchers [4–6] have contributed on this subject recently, in [4, 5] the 𝐻_∞ control is adopted to attenuate the influences from disturbances to a desired level, for systems with the bounded disturbances. In [6] a reduced-order observer is structured for the estimation of the modeled disturbance; simultaneously, 𝐻_∞ scheme can attenuate norm bounded signals. Due to the increasing complexity of the controlled plants and environment, it makes the higher demand for system accuracy, reliability, and real-time performance.

Disturbance-observer-based control (DOBC) is a prevalent anti-disturbance control strategy, which has a simple structure and is easily implemental in engineering (see surveys [7] and references therein). If the priori characteristics of disturbance to be estimated can be obtained, DOBC can be implemented where the disturbance compensation dynamic property within a composite system can be analyzed [8–11]. Originating from [9], a hierarchical control strategy is established in [6, 10, 11] aiming at multiple disturbances in multiinput multioutput (MIMO) nonlinear system; the outcome shows that the strategy has high precision together with strong robustness. The literature mentioned above shows that the DOBC is feasible for more complex structure and can avoid heavy computation, such as resolve of partial differential equations (PDEs) compared with output regulation theory. However, the main limitation of the classical DOBC is that the precise characteristic parameters of disturbance must be available. Moreover, failure in modeling for disturbance accurately may lead to severe deterioration of closed loop system performance, even to instability. It has not been reported that DOBC is presented for time-delay systems subject to uncertain disturbances.

In DOBC [9–11, 13], the disturbance is seen as extended state, correspondingly an extended state observer; that is, disturbance observer (DO) can be constructed to estimate the disturbance. Once we have no access of the precise disturbance model, no effective observer can be constructed directly to estimate the disturbance as the matching condition [12] is not satisfied. It still remains challenging work to extend the DOBC to the general case, in which the disturbance dynamic model has parametric uncertainty The aim of this paper is to provide a novel approach to estimate and reject the uncertain disturbances, such that the merits of DOBC can be inherited. We first construct an auxiliary observer and then decompose the disturbances into a known precise function, an uncertain nonlinear function, and a decaying vector defined by the auxiliary observer. Corresponding disturbance rejection strategy can be implemented to deal with the uncertain disturbance after the sophisticated design with lower conservativeness compared with the literature mentioned above.

The organization of the problem is given below. Section 2 gives the problem formulation. In Section 3, the formulation for the uncertain disturbance estimation with time delay is introduced. In Section 4, by using the auxiliary vector, DOBC combined with adaptive controller is designed to reject the disturbance and globally stabilize the closed-loop systems. In Section 5, the proposed method is applied to an A4D aircraft model; simulations show the effectiveness of the proposed approaches. Section 6 provides conclusions.

2. Formulation of the Problem

The following continuous time-delay system with uncertain perturbation is considered: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐹𝑓(𝑥(𝑡),𝑡)+𝐴_𝑑𝑥(𝑡−𝜏)+𝐵[𝑢(𝑡)+𝑑(𝑡)], (1) where 𝑥(𝑡)∈𝑅^𝑛, 𝑢(𝑡)∈𝑅^𝑚 are the state and the control input, respectively. 𝐴∈𝑅^𝑛×𝑛, 𝐵∈𝑅^𝑛×𝑚, and 𝐴_𝑑∈𝑅^𝑛×𝑛 are the coefficient matrices, satisfying rank(𝐵)=𝑚. 𝐹 is the corresponding weighting matrix, 𝑓(𝑥(𝑡),𝑡) is nonlinear function which is supposed to satisfy bounded conditions described as Assumption 1. 𝑑(𝑡) is a vector of sinusoidal disturbance and 𝜏 is the delay time. Such a model can also represent a wider class of time-delay system compared with papers [6, 9, 14].

Assumption 1. For any 𝑥_𝑗(𝑡)∈𝑅^𝑛, 𝑗=1,2 nonlinear functions 𝑓(𝑥,𝑡) satisfy 𝑓()𝑓𝑥0,𝑡=0,₁(𝑡),𝑡−𝑓𝑥₂(𝑡),𝑡≤𝑈𝑥₁(𝑡)−𝑥₂(𝑡), (2) where 𝑈 is the given constant weighting matrix.
Similar to the output regulation theory, DOBC strategy [9, 15], each unknown external disturbance 𝑑_𝑖 (𝑖=1,2,…,𝑚) is supposed to be generated by an exogenous system described by 𝑤̇_𝑖=Γ_𝑖𝑤_𝑖,𝑑_𝑖=𝑉_𝑖𝑤_𝑖, (3) where (Γ_𝑖,𝑉_𝑖) is uniformly observable. To show the main ideology of our paper, suppose 𝑤_𝑖∈𝑅² and the linear uncertain matrix Γ_𝑖∈𝑅^2×2. For sake of simplicity, (Γ_𝑖,𝑉_𝑖) has observable canonical form, which can be expressed as follows: Γ_𝑖=01−𝑊_𝑖0,𝑉_𝑖=10,𝑖=1,…,𝑚, (4) where 𝑊_𝑖 is parameter characteristics related to disturbance frequency; different from the present work, we consider 𝑊_𝑖 to be uncertain constant values, for the sake of simplicity, denote that 𝑊𝑊=⋮¹𝑊_𝑛Θ,Θ=⋮¹Θ_𝑛Ξ,Ξ=⋮¹Ξ_𝑛, (5) where Θ_𝑖 and Ξ_𝑖 represent precise and unknown part of 𝑊_𝑖, respectively, that is, 𝑊_𝑖=Θ_𝑖+Ξ_𝑖. In application, many kinds of disturbances in engineering can be described by this model, for example, the control of aircraft control [9], magnetic bearing control [16], robotic systems [14], and so forth.
In the conventional DOBC strategy [8, 9, 11, 14, 17], the Γ_𝑖 in disturbance must be known in advance. This condition is so strict for reason that the disturbances acting on a system are difficult to be modeled precisely in general. Up to now, there is no related method discussing the uncertain disturbances estimation problem subject to time delay. This is the major hurdle that mostly impedes the further research and application in DOBC and other disturbance rejection research.
In this paper, we will derive the relation between the uncertain parameters Γ and 𝑑(𝑡), according to which the exogenous disturbance may be expressed as nonlinear functions including precise part and uncertain part. The control problem considered will be solved by means of DOBC combined with adaptive control (DOBC + adaptive) such that the proposed controller can achieve arbitrary disturbance attenuation.

3. Nonlinear Disturbance Observer

The disturbance parameters are inaccessible in this state time-delay system (1), so it is difficult to construct the disturbance observer with traditional ways directly as in [9, 14]. In this section, we first design the auxiliary observer for nonlinear vector 𝜉 with time delay. After an appropriate coordinate transformation, the disturbance 𝑑 may be formulated as a parametric uncertain function. According to (2), 𝑑 in (1) can be expressed as follows: 𝑤̇=Γ𝑤,𝑑=𝑉𝑤, (6) where 𝑤∈𝑅^2𝑚, Γ∈𝑅^2𝑚×2𝑚, and ΓΓ=¹000⋱000Γ_𝑛𝑉,𝑉=¹000⋱000𝑉_𝑛. (7) In this section, we suppose that 𝑓(𝑥(𝑡),𝑡) is given and Assumption 1 holds. When all states of the system are available, it is unnecessary to estimate the states, then only the estimation of the disturbance need to be concerned. Construct an auxiliary MIMO nonlinear system as follows: (𝑡)̇𝜈(𝑡)𝐵̌(𝑥(𝑡))𝜉=𝜈+𝜓,𝜈=𝐺+𝜓−𝐿𝐴𝑥+𝐹𝑓,𝑡+𝐴_𝑑𝑥()(𝑡,̌(𝑡),𝑡−𝜏+𝐵𝑢)𝜓=𝐿𝐵𝑥 (8) where 𝜉𝜉=¹𝜉₂⋮𝜉_𝑛,𝜉_𝑖=𝜉_𝑖1𝜉_𝑖2. (9)𝐺 and 𝐿 are given constant matrices in form of 𝐺𝐺=¹000⋱000𝐺_𝑛𝐿,𝐿=¹000⋱000𝐿_𝑛, (10) where 𝐿_𝑖=01,𝐺_𝑖=01−𝑔_𝑖1−𝑔_𝑖2, (11)𝐺_𝑖 is Hurwitz by selection of 𝑔_𝑖1 and 𝑔_𝑖2. Considering rank(𝐵)=𝑚, there exists pseudoinverse 𝐵̌ such that 𝐵̌𝐵=𝐼, so system (8) can be transformed as 𝜉̇=𝐺𝜉+𝐿𝑑. (12) Comparing (6) with (12) yields 𝜉̇𝑤̇=𝜉𝑤𝐺𝐿𝑉0Γ. (13)

Lemma 2. For system (6), if 𝐺_𝑖 and 𝐿_𝑖 have form of (11) and guarantee 𝑔_𝑖1−𝑊_𝑖²+𝑔²_𝑖2𝑊_𝑖≠0 (14) in global region of 𝑊_𝑖, then there exists an invertible constant matrix Π_𝑖 such that 𝜉̇_𝑖+Π_𝑖𝑤̇_𝑖=𝐺_𝑖𝜉_𝑖+Π_𝑖𝑤_𝑖. (15)

Proof. Considering an invertible matrix 𝑃_𝑖⁻¹=𝐼^2×2Π_𝑖0𝐼^2×2, (16) where Π_𝑖=^{−𝑔𝑖1−𝑊𝑖𝑔𝑖2−𝑊𝑖𝑔𝑖2−𝑔𝑖1−𝑊𝑖}𝑔_𝑖1−𝑊_𝑖²+𝑔²_𝑖2𝑊_𝑖, (17) it is obvious that if (14) is satisfied, then Π_𝑖 is invertible in global region of 𝑊_𝑖; furthermore it can be derived that Π_𝑖⁻¹=−𝑔_𝑖1+𝑊_𝑖−𝑔_𝑖2𝑊_𝑖𝑔_𝑖2−𝑔_𝑖1+𝑊_𝑖. (18) According to (13), notice that 𝜉̇_𝑖𝑤̇_𝑖=𝐺_𝑖𝑉_𝑖𝐿_𝑖0Γ_𝑖𝜉_𝑖𝑤_𝑖, (19) we can define the following coordinate transformation 𝜉_𝑖𝑤_𝑖=𝑃_𝑖⁻¹𝜉_𝑖𝑤_𝑖. (20) Combining (16) with (20) yields ̇𝜉_𝑖𝑤̇‌_𝑖=𝑃_𝑖⁻¹𝐺_𝑖𝑉_𝑖𝐿_𝑖0Γ_𝑖𝑃_𝑖𝜉_𝑖𝑤_𝑖. (21) After calculation, it can be verified that 𝑃_𝑖⁻¹𝐺_𝑖𝐿_𝑖𝑉_𝑖0Γ_𝑖𝑃_𝑖=𝐺_𝑖00Γ_𝑖. (22) Thus (15) can be got directly following (21) and (22).

Based on Lemma 2, we can give another form of 𝑑_𝑖 as 𝑑_𝑖(𝑡)=𝑉_𝑖𝑤_𝑖(𝑡)=−𝑉_𝑖(𝑡)𝜉_𝑖(𝑡)+𝑉_𝑖(𝑡)𝜉_𝑖(𝑡), (23) where 𝑤_𝑖(𝑡)=−Π_𝑖⁻¹𝜉_𝑖(𝑡)+Π_𝑖⁻¹𝜉_𝑖(𝑡),𝑉_𝑖=𝑉_𝑖Π_𝑖⁻¹,𝜉_𝑖=𝜉_𝑖+Π_𝑖𝑤_𝑖 (24) and satisfy ̇‌𝜉_𝑖=𝐺_𝑖𝜉_𝑖. (25) Similarly, 𝑑 can be rewritten as 𝑑=𝑉𝜉+𝑉𝜉, (26) where 𝑉𝑉=¹Π₁⁻¹000⋱000𝑉_𝑚Π_𝑚⁻¹,𝜉=𝜉₁⋮𝜉_𝑚, (27) according to (10) and (25), 𝜉 satisfies ̇‌𝜉=𝐺𝜉. (28) From Lemma 2, we have given another form of 𝑑 through auxiliary observer 𝜉, and can construct observer of 𝑑 as 𝑑=𝑉𝜉, (29) thus the proposed method will exhibit classic DOBC property.

Notice that 𝑉 in observer 𝑑 cannot be implemented directly as the 𝑊_𝑖 is uncertain. To show it clearly, we divide 𝑑 into two parts. One part is a precise value that can be predicted and the other is uncertain constant parameter multiplied by a known nonlinear term. For the sake of simplicity, denote 𝑉𝑉𝜉=¹000⋱000𝑉_𝑛−Π₁⁻¹𝜉₁−Π₂⁻¹𝜉₂⋮−Π_𝑛⁻¹𝜉_𝑛=−𝑔₁₁+Θ₁₁𝜉₁₁+Θ₁₂−𝑔₁₂𝜉₁₂⋮−𝑔_𝑚1+Θ_𝑚1𝜉_𝑚1+Θ_𝑚2−𝑔_𝑚2𝜉_𝑛2+𝜉₁₁⋱𝜉_𝑚1Ξ₁⋮Ξ_𝑛. (30)

A notable property of (29) is that uncertain sinusoidal 𝑑 can be expressed in form of parametric uncertainty. So, we need not estimate the upper bounds of 𝑑 as in [18–21].

4. DOBC with Stability Analysis

After substituting (26) into system (1), we have 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐹𝑓(𝑥(𝑡),𝑡)+𝐴_𝑑𝑥()𝑢(𝑡)+𝑡−𝜏+𝐵𝑉𝜉+𝑉𝜉. (31) For the plants with known nonlinearity, the DOBC+adaptive strategy can be designed by using the separation principle as follows: 𝑢=𝐾𝑥+𝑢_𝑜+𝑢_𝑎, (32) where 𝐾 is the conventional control gain for stabilization, 𝑢_𝑜 and 𝑢_𝑎 are used to reject and attenuate the disturbances known and uncertain parts, respectively, and according to (30) we select 𝑢_𝑜=−−𝑔₁₁+Θ₁₁𝜉₁₁+Θ₁₂−𝑔₁₂𝜉₁₂⋮−𝑔_𝑚1+Θ_𝑚1𝜉_𝑚1+Θ_𝑚2−𝑔_𝑚2𝜉_𝑚2. (33)

Similar to [22], adaptive controller 𝑢_𝑎 is used to compensate unknown part of disturbance 𝑑 which satisfies 𝑢_𝑎=−ΘΞ (34) where 𝜉Θ=¹¹⋱𝜉_𝑛1 (35) and Ξ is estimation of Ξ. At last the dynamic system (31) may be rewritten as follows: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐹𝑓(𝑥(𝑡),𝑡)+𝐴_𝑑𝑥(𝑡−𝑑)+𝐵𝐾𝑥+ΘΞ−Ξ+𝑉𝜉. (36)

At this stage, our objective is to find 𝐾 such that the closed-loop system (31) with 𝑢=𝐾𝑥 is asymptotically stable. For the sake of simplifying descriptions, we denote sym(𝑀)∶=𝑀+𝑀^𝑇 and 𝑁₁=𝑃₁(𝐴+𝐵𝐾)+(𝐴+𝐵𝐾)^𝑇𝑃₁. (37)

Theorem 3. For given 𝜆>0, if (14) can be guaranteed and there exist 𝑄₁>0,𝑃₁>0, 𝐸=𝑆⁻¹>0 and 𝑅₁ satisfying Ω=Sym𝐴𝑄₁+𝐵𝑅₁𝐴_𝑑𝐸𝐹₁𝑄₁𝑈𝑄₁1∗−𝐸000∗∗−𝜆²₁1𝐼00∗∗∗−𝐸0∗∗∗∗∗−𝜆²₁𝐼<0, (38) then under DOBC law (32) and adaptive dynamic Ξ̇=𝛾₁Θ^𝑇𝐵^𝑇𝑃𝑥,𝛾₁>0 (39) the closed-loop system (36) with gain 𝐾=𝑅₁𝑄₁⁻¹ is asymptotically stable.

Proof. Denote 𝑉(𝑥(𝑡),𝜉(𝑡),Ξ(𝑡),𝑡)=𝑉₁(𝑥(𝑡),𝑡)+𝑉₂(𝜉(𝑡),𝑡)+𝑉₃(Ξ(𝑡),𝑡), where 𝑉₁(𝑥(𝑡),𝑡)=𝑥(𝑡)^𝑇(𝑡)+1𝑃𝑥𝜆^2𝑡0‖𝑈𝑥(𝜎)‖²−𝑓(𝑥(𝜎),𝜎)²+𝑑𝜎_{𝑡𝑡−𝜏}𝑥^𝑇(𝜎)(𝜎)𝑉𝑆𝑥𝑑𝜎,₂𝜉(𝑡),𝑡=𝛾₂𝜉(𝑡)^𝑇𝑃₂𝜉(𝑡),𝛾₂𝑉>0₃(Ξ(𝑡),𝑡)=𝛾₁⁻¹Ξ^𝑇Ξ. (40) Along with the trajectories of (36) and (39), firstly it can be verified that 𝑉̇₁(𝑥(𝑡),𝑡)+𝑉̇₃=𝑥̇(𝑡)Ξ,𝑡_𝑇𝑃𝑥(𝑡)+𝑥(𝑡)^𝑇𝑃𝑥̇(𝑡)+1𝜆²‖𝑈𝑥(𝑡)‖²−𝑓(𝑥(𝑡),𝑡)²+𝑥^𝑇(𝑡)𝑆𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑆𝑥(𝑡−𝜏)+2𝑥^𝑇(𝑡)𝑃𝐵𝑉𝜉≤𝐴𝑥(𝑡)+𝐹𝑓(𝑥(𝑡),𝑡)+𝐴_𝑑𝑥(𝑡−𝜏)^𝑇𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝐴𝑥(𝑡)+𝐹𝑓(𝑥(𝑡),𝑡)+𝐴_𝑑𝑥()+1𝑡−𝜏𝜆²𝑥^𝑇(𝑡)𝑈^𝑇(𝑡)−1𝑈𝑥𝜆²𝑓^𝑇(𝑥(𝑡),𝑡)𝑓(𝑥(𝑡),𝑡)+𝑥^𝑇(𝑡)𝑆𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑆𝑥(𝑡−𝜏)+2𝑥^𝑇𝑃𝐵𝑉𝜉=𝑥(𝑡)𝑥()𝑓(𝑥(𝑡))𝑡−𝜏,𝑡^𝑇Ω₁𝑥(𝑡)𝑥()𝑓(𝑥(𝑡))𝑡−𝜏,𝑡+2𝑥^𝑇𝑃𝐵𝑉𝜉, (41) where Ω₁=𝑁₁1+𝑆+𝜆²₁𝑈^𝑇₁𝑈₁𝑃𝐴_𝑑𝑃𝐹₁1∗−𝑆0∗∗−𝜆²₁𝐼. (42) Premultiplied and postmultiplied simultaneously by diag {𝑄,𝐼,𝐼}, Ω₁<0 is equivalent to Ω₂<0, where Ω₂=Sym𝐴𝑄₁+𝐵𝑅₁+𝑄₁𝑆𝑄₁+1𝜆²₁𝑄₁𝑈^𝑇₁𝑈₁𝑄₁𝐴_𝑑𝐹₁1∗−𝑆0∗∗−𝜆²₁𝐼. (43) Based on Schur complement, it can be seen that Ω₂<0 is equivalent to Ω₃<0 and Ω₃=Sym𝐴𝑄₁+𝐵𝑅₁𝐴_𝑑𝐸𝐹₁𝑄₁𝑈₁𝑄₁1∗−𝑆000∗∗−𝜆²₁𝐼00∗∗∗−𝑆⁻¹01∗∗∗∗∗−𝜆²₁𝐼. (44) Premultiplied and postmultiplied simultaneously by diag {𝐼,𝐸,𝐼,𝐼,𝐼}, Ω₃<0 is equivalent to Ω<0. That is to say that if (38) upholds, there exists constant 𝛼₁>0 such that 𝑉̇₁(𝑥(𝑡),𝑡)+𝑉̇₃Ξ,𝑡<−𝛼₁𝑥(𝑡)𝑥()𝑓(𝑥(𝑡))𝑡−𝜏,𝑡²+2𝑥^𝑇𝑃𝐵𝑉𝜉. (45) Together with the definition of 𝐺 in (10), we can find that 𝛼₂>0 satisfies 𝐺^𝑇𝑃₂+𝑃₂𝐺+𝛼₂<0. (46) Furthermore, there exists 𝛾₃>0 depending on 𝑃₁ such that for any 𝑥 and 𝜉2𝑥^𝑇𝑃₁𝐵𝑉𝜉≤𝛾₃‖𝑥‖𝜉. (47) After substituting (45), (46), and (49), derivative along Lyapunov function candidate is given by 𝑉̇𝑥,𝜉,Ξ≤−𝛼₁𝑥(𝑡)𝑥()𝑓(𝑥(𝑡))𝑡−𝜏,𝑡²+𝛾₃‖𝑥‖𝜉−𝛼₂𝛾₂𝜉²≤−𝛼₁𝑥(𝑡)²+𝛾₃‖𝑥‖𝜉−𝛼₂𝛾₂𝜉². (48) The right part of the above inequality can be regarded as a polynomial with respect to two variables ‖𝑥‖ and ‖𝜉‖. Thus for all ‖𝑥‖ and ‖𝜉‖, 𝑉̇(𝑥,𝜉,Ξ)≤0 holds if there exists a group of parameters 𝛾_𝑖(𝑖=2,3) satisfying 2√𝛼₁𝛼₂𝛾₂≥𝛾₃. (49)

The disturbance-observer-based control design procedure can be summarized as follows.

Step  1. Select weighting matrices 𝐺 and 𝐿 with form of (10) and (11), apply 𝐺 and 𝐿 into (8) to calculate auxiliary vector.

Step  2. According to (26), give another form of disturbance represented by auxiliary vector.

Step  3. Design time-delay feedback controller 𝐾𝑋 based on Theorem 3, apply controller into (32), then DOBC + adaptive control can be realized.

5. Simulation

To show the efficiency of the proposed scheme, let us consider the continuous time-delay models under the proposed DOBC + adaptive scheme. The longitudinal dynamics of A4D aircraft at a flight condition of 15000 ft altitude and 0.9 Mach can be given by (1). The meaning and significance of the parameters are the same as in [6, 9], where 𝑥₁(𝑡) is the forward velocity (𝑓𝑡/𝑠), 𝑥₂(𝑡) is the angle of attack (rad), 𝑥₃(𝑡) is the pitching velocity (rad/𝑠), 𝑥₄(𝑡) is the pitching angle (rad), and 𝑢(𝑡) is elevator deflection (deg) and coefficient matrices: ,00,𝐴𝐴=−0.060532.37032.2−0.00014−1.47510−0.0111−34.72−2.79300010𝐵=−1.1064−33.8_𝑑=,000.0.090.030.031.030.0020.040.07−0.0034−0.004−0.040.03−0.001−0.0004−0.0140.0080.03𝐹=50 (50) Similar to [9], it is supposed that nonlinearity and/or uncertainty 𝑓(𝑥(𝑡),𝑡)=sin(2𝜋5𝑡)𝑥₂(𝑡), state delay time 𝜏=2, and set 𝑈=diag{0100} guarantee ‖𝑓(𝑥(𝑡),𝑡)‖≤‖𝑈𝑓(𝑥(𝑡),𝑡)‖. Paper [9, 14] pointed out that if the frequency is perturbed, the pure DOBC approach will be unavailable because the disturbances cannot be rejected accurately. In order to investigate further, it has been considered that uncertainties exist in such an exogenous model for the disturbance in (3). That is, 𝑑=25sin1+Ξ𝑡, (51) where frequency perturbations Ξ₁₁=Ξ₂₁=4. Set 𝐺=01−4−4,𝛾₁=10000. (52) It is noted that the selection of 𝜆 is tradeoff and we select 𝜆=0.8, and the initial value of the disturbance is 0. When the full states can be measured, applying the approach in Theorem 3, when DOBC law is applied in system (31), the corresponding parameter in (32) can be gotten that 𝐾=9.9316530.50331.7402552.7417. (53)

Figure 1 plots the estimation error of the uncertain disturbances with traditional DO [9, 11, 14]. The results show that if system suffers uncertain disturbance, it may bring large disturbance estimation error, and the control performance is deteriorated (Figure 2). Figures 3 and 4 demonstrate the system performance using the proposed DOBC + adaptive schemes, obviously the system output converges to zero with sufficiently small steady error. The results show that although there exists uncertainty in the disturbance parameters, the disturbance rejection performance is improved and enhanced system responses can be achieved.

Figure 1 

Estimation error using traditional DO.

Figure 2 

System output using traditional DOBC.

Figure 3 

Estimation error using DO proposed in this paper.

Figure 4 

System output using DOBC + adaptive.

6. Conclusion

The DOBC strategy is extended to the uncertain disturbance rejection problem by combining with adaptive control. We first construct the auxiliary observer for 𝑑(𝑡) with uncertain parameters, and then the exogenous disturbance may be divided into two parts. One part is a precise term which can be compensated by a feed-forward controller, and the other can be expressed as uncertain constant parameter multiplied by a known nonlinear term; an adaptive controller is adopted to compensate the effect of the second part. Simulations on an aircraft model demonstrate the advantages of the proposed scheme. However, if there exist multiple disturbances and unmodeled dynamics in the system as in [10, 11], the situation turns to be more complicated, and further research is required in the future.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 61203049 and Scholar Doctor Priming Foundation (20122048).