Composite Antidisturbance Control for a Class of Nonlinear Stochastic Systems via Disturbance Observer

Abstract

The stabilization problem is investigated in this paper for a class of nonlinear systems with disturbances. The disturbances are supposed to be classified into two types. One type in the input channel is generated by an exogenous system, which can represent the constant or harmonic signals with unknown phase and magnitude. The other type is stochastic disturbance. Two kinds of nonlinear dynamics in the plants are considered, respectively, which correspond to the known and unknown functions. By integrating the disturbance observers with conventional control method, the first type of disturbances can be estimated and rejected. Simultaneously, the desired dynamic performances can be guaranteed. An example is given to show the effectiveness of the proposed scheme.

1. Introduction

Though the stochastic stabilization theory emerged in the 1960s [1], the progress has been slow. This was mainly due to a fundamental theory obstacle in the Lyapunov analysis; the Itô differentiation introduces not only the gradient but also the Hessian term of the Lyapunov function. Along with the advances in differential geometric theory [2] and the discovery of a simple constructive formula for Lyapunov stabilization [3], the stochastic stabilization problem was reexamined and some constructive results have been achieved. The stabilization of nonlinear stochastic systems was considered in the work of Florchinger [4–7], who, among other things, extended the concept of control Lyapunov functions and Sontag’s stabilization formula to stochastic setting. Pan and Başer [8] solved the stabilization problem for a class of strict-feedback systems representative of stabilization results for deterministic systems. Deng and Krstić [9] developed a simpler control strategy for strict-feedback systems and then extended the results on inverse optimal stabilization for general systems to the stochastic case. The authors of [10] considered the dissipativity analysis and dissipativity-based sliding mode control for a class of continuous-time switched stochastic systems and [11] designed multistep predictive controller for a class of Markov jump convex polyhedron linear parameter time-varying systems with both constraints on input and output. The adaptive neural tracking control problem was the concern in [12] for a class of strict-feedback stochastic nonlinear systems with unknown dead zone. However, most of these results were focused on systems that only have one kind of disturbance-stochastic disturbance. In [13], Hinrichsen proposed the stochastic robust control method for systems with deterministic and stochastic disturbances, which enabled us to deal with a broader class of systems. However, only stability of the nominal system in the absence of deterministic disturbances was the concern in this approach, which means that the stability cannot be guaranteed in the presence of both deterministic and stochastic disturbances.

Disturbance-observer-based control (DOBC) approach, which is based on the idea of feed-forward compensation, appeared in the late 1980s and has attracted considerable attention in control theory literatures [14–17]. The controller design of this method can be accomplished in two steps. First, a disturbance observer is designed to estimate the deterministic disturbance and then compensate it. Second, feedback controller is designed to stabilize the nominal system without disturbance. DOBC approach has its roots in many mechanical applications in the last two decades, in particular for linear systems [18–20]. Recently, some attempts have been made to establish theoretic justification of these DOBC applications and extend DOBC from linear systems to nonlinear systems [21, 22]. Besides, [23] designed output feedback controller for a class of Markovian jump repeated scalar nonlinear systems and [24] investigated the problem of composite DOBC and 𝐻_∞ control for Markovian jump systems with nonlinearity and multiple disturbances.

This paper considers the application of DOBC approach to a class of nonlinear stochastic systems. The nonlinear dynamics are described by known and unknown nonlinear functions, respectively. And apart from the stochastic noises, the deterministic disturbance is supposed to be generated by an exogenous system as investigated in [18, 19], which is not confined to be bounded in norm [25]. By using the disturbance estimation, the DOBC strategy can be integrated with the conventional stabilization controllers to reject the deterministic disturbance and globally stabilize the closed-loop systems in probability. Finally, simulations on an A4D aircraft model show the effectiveness of the proposed approaches.

2. Problem Statement

The following MIMO stochastic system with nonlinearity is described as 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐹₀𝑓(𝑥(𝑡),𝑡)+𝐵[𝑢(𝑡)+𝑣(𝑡)]𝑑𝑡+𝐴₀𝑥(𝑡)𝑑𝜔, (1) where 𝑥(𝑡)∈𝑅^𝑛, 𝑢(𝑡)∈𝑅^𝑚 and 𝑣(𝑡)∈𝑅^𝑚, are the state, control input, and disturbance, respectively. 𝑓(𝑥(𝑡),𝑡) is a nonlinear vector function satisfying bounded condition as described in Assumption 1. 𝜔 is an 𝑟-dimensional standard Brownian motion defined on a complete probability space (Ω,ℱ,𝒫) with Ω being a sample space, ℱ being a 𝜎-field, and 𝒫 being the probability measure. 𝐴, 𝐵, 𝐹₀, and 𝐴₀ are given system matrices with corresponding dimensions.

Assumption 1. For any 𝑥_𝑗(𝑡)∈𝑅^𝑛, 𝑗=1,2, nonlinear function 𝑓(𝑥(𝑡),𝑡) satisfy 𝑓()𝑓𝑥0,𝑡=0,₁(𝑡),𝑡−𝑓𝑥₂(𝑡),𝑡≤𝑈𝑥₁(𝑡)−𝑥₂(𝑡), (2) where 𝑈 is a given constant weighting matrix.

Assumption 2. The disturbance 𝑣(𝑡) in the control input path is supposed to be generated by the following exogenous systems: 𝜛̇(𝑡)=𝑊𝜛(𝑡),𝑣(𝑡)=𝑉𝜛(𝑡), (3) where 𝑊∈𝑅^𝑟×𝑟, 𝑉∈𝑅^𝑚×𝑟 are known constant weighting matrices.

Remark 3. In fact, many kinds of disturbances in engineering can be described by Assumption 2, for example, unknown constant and harmonics with unknown phase and magnitude.

The following assumption is the necessary condition for the DOBC problem.

Assumption 4. (𝐴,𝐵) is controllable and (𝑊,𝐵𝑉) is observable.

In this paper, we suppose system states are available, which means that only the estimation of disturbance needs to be focused on. In this situation, the objective of DOBC is to design an observer for system (1) to estimate the unknown disturbance 𝑣(𝑡), and then construct a composite controller with the disturbance estimation and a conventional controller so that the disturbance can be rejected and the stability in probability of the resulting composite system can be guaranteed.

3. DOBC for the Case with Known Nonlinearity

In this section, the nonlinearity function 𝑓(𝑥(𝑡),𝑡) is supposed to be given and Assumptions 1, 2, and 4 hold. Due to the fact that the states of system are available, the disturbance observer is designed as 𝑣(𝑡)=𝑉𝜛(𝑡),𝜛(𝑡)=𝑠(𝑡)−𝐿𝑥(𝑡), (4) where 𝜛(𝑡) is the estimation of 𝜛(𝑡) and 𝑠(𝑡) is the auxiliary variable generated by (𝑡)=𝑠(𝑡)(𝑡(𝑡)𝑑𝑠(𝑊+𝐿𝐵𝑉)(−𝐿𝑥))+𝐿𝐴𝑥+𝐹₀𝑓(𝑥,𝑡)+𝐵𝑢(𝑡)𝑑𝑡. (5)

Denote the estimation error as 𝑒_𝜛(𝑡)=𝜛(𝑡)−𝜛(𝑡). Based on (1), (3), (4), and (5), the error dynamics satisfy 𝑑𝑒_𝜛(𝑡)=(𝑊+𝐿𝐵𝑉)𝑒_𝜛𝑑𝑡+𝐿𝐴₀𝑥(𝑡)𝑑𝜔. (6)

The objective of disturbance rejection can be achieved by designing the observer gain 𝐿 such that (6) satisfies the desired stability in probability.

For DOBC strategy, the controller is usually selected as [22, 26–28] 𝑢(𝑡)=−𝑣(𝑡)+𝐾𝑥(𝑡), (7) where 𝑣(𝑡) is to compensate the disturbance in control input path and 𝐾 is the conventional feedback gain needed to be determined later.

By substituting (7) into (1), the closed loop system can be written in the following form: 𝑑𝑥(𝑡)=(𝐴+𝐵𝐾)𝑥(𝑡)+𝐹₀𝑓(𝑥(𝑡),𝑡)+𝐵𝑉𝑒_𝜛(𝑡)𝑑𝑡+𝐴₀𝑥(𝑡)𝑑𝜔. (8)

Combining (8) with (6), the composite system is described as 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐹𝑓(𝑥,𝑡)𝑑𝑡+𝐴₀𝑥(𝑡)𝑑𝜔, (9) where 𝑥(𝑡)=𝑥(𝑡)𝑒_𝜛(𝑡),,𝐹𝐴=𝐴+𝐵𝐾𝐵𝑉0𝑊+𝐿𝐵𝑉𝐹=⁰0,𝐴₀=𝐴₀0𝐿𝐴₀0. (10)

In the following, the objective is to design gain 𝐿 and 𝐾 such that system (9) is asymptotical stabilization in probability. For the convenience of research, the following lemma is presented.

Lemma 5 (see [29, 30]). For a given stochastic system 𝑑𝑥(𝑡)=𝑔(𝑥,𝑡)𝑑𝑡+ℎ(𝑥,𝑡)𝑑𝜔,𝑥𝑡₀=𝑥₀, (11) if there exists function 𝑉(𝑥,𝑡)∈𝐶^1,2, 𝜇₁(⋅),𝜇₂(⋅)∈𝒦_∞, constants 𝑐₁>0, and a nonnegative 𝑀(𝑥,𝑡), such that 𝜇₁(𝑥)≤𝑉(𝑥,𝑡)≤𝜇₂(𝑥),ℒ𝑉≤−𝑐₁𝑀(𝑥,𝑡), (12) then, one has the following.(i)The equilibrium 𝑥=0 is globally stable in probability and the solution 𝑥(𝑡) satisfied 𝑃{lim_𝑡→∞𝑀(𝑥(𝑡))=0}=1, when 𝑔(0,𝑡)=0, ℎ(0,𝑡)=0, and 𝑀(𝑥,𝑡)=𝑀(𝑥) is continuous.(ii)The equilibrium 𝑥=0 is globally asymptotically stable in probability, when 𝑔(0,𝑡)=0, ℎ(0,𝑡)=0, and 𝑀(𝑥,𝑡) is positive definite.Here, the differential operator ℒ for differentiable function 𝑉(𝑥,𝑡) is defined as ℒ𝑉=𝜕𝑉𝑔()+1𝜕𝑥𝑥,𝑡2𝑇𝑟ℎ(𝑥,𝑡)^𝑇𝜕²𝑉𝜕𝑥²ℎ(𝑥,𝑡). (13)

Theorem 6. Consider system (1) with disturbance (3) under Assumptions 1, 2, and 4. For some 𝜆>0, if there exist 𝑃₁=𝑃^𝑇₁>0, 𝑅₁ and constant 𝛽>0 satisfying 𝑠𝑦𝑚𝑃₁𝑊+𝑅₁𝐵𝑉𝑉^𝑇𝐵^𝑇𝐵𝑉−𝛽²𝐼<0 (14) and 𝑄₂=𝑄^𝑇₂>0, 𝑅₂ satisfying Ξ𝑄₂𝐴^𝑇₀𝑄₂𝐴^𝑇₀𝑅^𝑇₁𝑄₂𝑈^𝑇𝐴₀𝑄₂−𝑄₂𝑅00₁𝐴₀𝑄₂0−𝑃₁0𝑈𝑄₂00−𝜆²𝐼<0, (15) where Ξ=𝑠𝑦𝑚(𝐴𝑄₂+𝐵𝑅₂)+𝜆²𝐹₀𝐹^𝑇₀+𝛽²𝐼, then the closed loop system (8) under DOBC law (7) with gain 𝐾=𝑅₂𝑄₂⁻¹ and observer (4) with gain 𝐿=𝑃₁⁻¹𝑅₁ are global asymptotical stabilization in probability.

Proof. Define Σ₁𝑒_𝜛,𝑡=𝑒^𝑇_𝜛𝑃₁𝑒_𝜛,Σ₂(𝑥,𝑡)=𝑥^𝑇𝑃₂1𝑥+𝜆^2𝑡0‖𝑈𝑥(𝑡)‖²−𝑓(𝑥,𝑡)²𝑑𝜏, (16) where 𝑃₂⁻¹=𝑄₂.
It is noted that for all 𝑥 and 𝑒_𝜛, Σ₁≥0, Σ₂≥0. In addition, along with (6), (8), the Itó differential of Σ₁,Σ₂ satisfies ℒΣ₁=𝑒^𝑇_𝜛𝑃₁(𝑊+𝐿𝐵𝑉)+(𝑊+𝐿𝐵𝑉)^𝑇𝑃₁𝑒_𝜛+𝑥^𝑇𝐴^𝑇₀𝐿^𝑇𝑃₁𝐿𝐴₀𝑥,ℒΣ₂=𝑥^𝑇𝑃₂(𝐴+𝐵𝐾)+(𝐴+𝐵𝐾)^𝑇𝑃₂𝑥+2𝑥^𝑇𝑃₂𝐹₀𝑓+2𝑥^𝑇𝑃₂𝐵𝑉𝑒_𝜛+1𝜆²𝑥^𝑇𝑈^𝑇𝑈𝑥+𝑥^𝑇𝐴^𝑇₀𝑃₂𝐴₀1𝑥−𝜆²𝑓^𝑇𝑓. (17) Via Young inequality, we get 2𝑥^𝑇𝑃₂𝐹₀𝑓≤𝜆²𝑥^𝑇𝑃₂𝐹₀𝐹^𝑇₀𝑃₂1𝑥+𝜆²𝑓^𝑇𝑓,2𝑥^𝑇𝑃₂𝐵𝑉𝑒_𝜛≤𝛽²𝑥^𝑇𝑃₂𝑃₂1𝑥+𝛽²𝑒^𝑇_𝜛𝑉^𝑇𝐵^𝑇𝐵𝑉𝑒_𝜛. (18) Then we have ℒΣ₂≤𝑥^𝑇sym𝑃₂(𝐴+𝐵𝐾)+𝐴^𝑇₀𝑃₂𝐴₀+1𝜆²𝑈^𝑇𝑈+𝜆²𝑃₂𝐹₀𝐹^𝑇₀𝑃₂+𝛽²𝑃₂𝑃₂1𝑥+𝛽²𝑒^𝑇_𝜛𝑉^𝑇𝐵^𝑇𝐵𝑉𝑒_𝜛. (19)
A Lyapunov function candidate for (9) is chosen as Σ(𝑥,𝑒_𝜛,𝑡)=Σ₁(𝑒_𝜛,𝑡)+Σ₂(𝑥,𝑡); hence, it is easy to get ℒΣ≤𝑒^𝑇_𝜛sym𝑃₁(+1𝑊+𝐿𝐵𝑉)𝛽²𝑉^𝑇𝐵^𝑇𝐵𝑉𝑒_𝜛+𝑥^𝑇sym𝑃₂(𝐴+𝐵𝐾)+𝐴^𝑇₀𝑃₂𝐴₀+1𝜆²𝑈^𝑇𝑈+𝜆²𝑃₂𝐹₀𝐹^𝑇₀𝑃₂+𝛽²𝑃₂𝑃₂+𝐴^𝑇₀𝐿^𝑇𝑃₁𝐿𝐴₀𝑥=𝑒^𝑇_𝜛Π₁𝑒_𝜛+𝑥^𝑇Π₂𝑥, (20) where Π₁=sym𝑃₁(+1𝑊+𝐿𝐵𝑉)𝛽²𝑉^𝑇𝐵^𝑇Π𝐵𝑉,₂=sym𝑃₂(𝐴+𝐵𝐾)+𝐴^𝑇₀𝑃₂𝐴₀+1𝜆²𝑈^𝑇𝑈+𝜆²𝑃₂𝐹₀𝐹^𝑇₀𝑃₂+𝛽²𝑃₂𝑃₂+𝐴^𝑇₀𝐿^𝑇𝑃₁𝐿𝐴₀. (21)
Using Lemma 5, it can be verified that system (9) is global asymptotical stabilization in probability if Π₁<0 and Π₂<0 hold.
Based on Schur complement, Π₁<0 is equivalent to Π₁₀<0, where Π₁₀=sym𝑃₁𝑊+𝑅₁𝐵𝑉𝑉^𝑇𝐵^𝑇𝐵𝑉−𝛽²𝐼, (22) and Π₂<0 is equivalent to Π₂₁<0, where Π₂₁=Ξ₀𝐴^𝑇₀𝐴^𝑇₀𝐿^𝑇𝑃₁𝑈^𝑇∗−𝑃₂⁻¹00∗∗−𝑃₁0∗∗∗−𝜆²𝐼 (23) and Ξ₀=sym[𝑃₂(𝐴+𝐵𝐾)]+𝜆²𝑃₂𝐹₀𝐹^𝑇₀𝑃₂+𝛽²𝑃₂𝑃₂. In addition, ∗ represents the corresponding elements in the symmetric matrix.
Π₂₁ is premultiplied and postmultiplied simultaneously by diag{𝑄₂,𝐼,𝐼,𝐼}; then it is equivalent to Π₂₀, where Π₂₀=Ξ𝑄₂𝐴^𝑇₀𝑄₂𝐴^𝑇₀𝑅^𝑇₁𝑄₂𝑈^𝑇∗−𝑄₂00∗∗−𝑃₁0∗∗∗−𝜆²𝐼 (24) and Ξ=sym[(𝐴+𝐵𝐾)𝑄₂]+𝜆²𝐹₀𝐹^𝑇₀+𝛽²𝐼.
Thus, (14), (15) can be obtained.
On the other hand, (14), (15) hold, meaning that there exist 𝛼₁>0, 𝛼₂>2 such that Π₁₀<−𝛼₁𝐼, Π₂₀<−𝛼₂𝐼; that is, Π₁<−𝛼₁𝐼, Π₂<−𝛼₂𝐼. Hence, we have ℒΣ≤𝑒^𝑇_𝜛Π₁𝑒_𝜛+𝑥^𝑇Π₂𝑥≤−𝛼₁𝑒_𝜛²−𝛼₂‖𝑥‖²≤−min𝛼₁,𝛼₂‖𝑥‖²+𝑒_𝜛²=−min𝛼₁,𝛼₂‖𝑥‖². (25) Therefore, the closed-loop system (9) is global asymptotical stabilization in probability when the control gain is selected as 𝐾=𝑅₂𝑄₂⁻¹ and observer gain is selected as 𝐿=𝑃₁⁻¹𝑅₁. The proof is completed.

4. DOBC for the Case with Unknown Nonlinearity

In this section, the nonlinear function 𝑓(𝑥(𝑡),𝑡) is supposed to be unknown, which means disturbance observer should be designed different from Section 3. In this case, the disturbance observer can be constructed as 𝑣(𝑡)𝜛(𝑡),𝜛(𝑡)(𝑡)(𝑡),(𝑡)=(𝑠(𝑡)(𝑡((𝑡)(𝑡=𝑉=𝑠−𝐿𝑥𝑑𝑠𝑊+𝐿𝐵𝑉)(−𝐿𝑥))+𝐿𝐴𝑥+𝐵𝑢))𝑑𝑡. (26)

Compared with (6), the estimation error 𝑒_𝜛(𝑡)=𝜛(𝑡)−𝜛(𝑡) satisfies 𝑑𝑒_𝜛(𝑡)=(𝑊+𝐿𝐵𝑉)𝑒_𝜛+𝐿𝐹₀𝑓(𝑥,𝑡)𝑑𝑡+𝐿𝐴₀𝑥(𝑡)𝑑𝜔. (27)

Thus, the composite system combined (8) with (27) is given by 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐹𝑓(𝑥,𝑡)𝑑𝑡+𝐴₀𝑥(𝑡)𝑑𝜔, (28) where 𝑥(𝑡)=𝑥(𝑡)𝑒_𝜛(𝑡),,𝐹𝐴=𝐴+𝐵𝐾𝐵𝑉0𝑊+𝐿𝐵𝑉𝐹=⁰𝐿𝐹₀,𝐴₀=𝐴₀0𝐿𝐴₀0. (29)

The objective of this section is similar to Section 3, that is, to design gain 𝐿 and 𝐾 such that system (28) is asymptotically stable in probability.

Theorem 7. Consider system (1) with disturbance (3) under Assumptions 1, 2, and 4. For some 𝜆₁>0, 𝜆₂>0, if there exist 𝑃₁=𝑃^𝑇₁>0, 𝑅₁ and constant 𝛽>0 satisfying 𝑠𝑦𝑚𝑃₁𝑊+𝑅₁𝐵𝑉𝑅₁𝐹₀𝑉^𝑇𝐵^𝑇∗−𝜆₁⁻²𝐼0∗∗−𝛽²𝐼<0 (30) and 𝑄₂=𝑄^𝑇₂>0, 𝑅₂ satisfying Ξ𝑄₂𝐴^𝑇₀𝑄₂𝐴^𝑇₀𝑅^𝑇₁𝑄₂𝑈^𝑇𝑄₂𝑈^𝑇∗−𝑄₂000∗∗−𝑃₁00∗∗∗−𝜆²₁𝐼0∗∗∗∗−𝜆²₂𝐼<0, (31) where Ξ=𝑠𝑦𝑚(𝐴𝑄₂+𝐵𝑅₂)+𝜆²₂𝐹₀𝐹^𝑇₀+𝛽²𝐼, then the closed loop system (28) under DOBC law (7) with gain 𝐾=𝑅₂𝑄₂⁻¹ and observer (26) with gain 𝐿=𝑃₁⁻¹𝑅₁ is global asymptotical stabilization in probability.

Proof. Let Σ₁𝑒_𝜛,𝑡=𝑒^𝑇_𝜛𝑃₁𝑒_𝜛+1𝜆²₁^𝑡0‖𝑈𝑥(𝑡)‖²−𝑓(𝑥,𝑡)²𝑑𝜏,Σ₂(𝑥,𝑡)=𝑥^𝑇𝑃₂1𝑥+𝜆²₂^𝑡0‖𝑈𝑥(𝑡)‖²−𝑓(𝑥,𝑡)²𝑑𝜏, (32) where 𝑃₂⁻¹=𝑄₂. And a Lyapunov function candidate for (28) is chosen as Σ(𝑥,𝑒_𝜛,𝑡)=Σ₁(𝑒_𝜛,𝑡)+Σ₂(𝑥,𝑡). The following proof procedure can be given similarly to that of the proof for Theorem 6.

5. Simulation Example

In [21], DOBC strategy was employed to a deterministic system of A4D aircraft with disturbance, and better system performance was obtained than some previous results [31]. However, stochastic noise should be considered when higher precision of system performance was required. In this section, the stochastic system that represents the longitudinal dynamics of A4D aircraft is considered, which is described as follows: 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐹₀𝑓(𝑥,𝑡)+𝐵[𝑢(𝑡)+𝑣(𝑡)]𝑑𝑡+𝐴₀𝑥(𝑡)𝑑𝜔 (33) with the following coefficient: ,𝐹𝐴=−0.060532.37032.2−0.00014−1.47510−0.0111−34.72−2.79300010₀=00000,𝐴60,𝐵=−0.1064−33.8₀=.100000.300.200000000.5 (34)

The parameter matrices for disturbance 𝑣(𝑡) that are described by (3) are given by 𝑊=05−50,𝑉=2510. (35)

Case 1 (with known nonlinearity). In this case, the nonlinear dynamic is supposed to be denoted by 𝑓(𝑥,𝑡)=sin(10𝜋𝑡)𝑥₂(𝑡) in the simulation. In order to satisfy Assumption 1, 𝑈 is selected as diag{0,1,0,0}. The initial value of state is taken to be 𝑥(0)=[2,−2,3,0] and 𝜆 is selected as 20. Based on Theorem 6, it can be solved that ,.𝐿=0−0.00010.001900−0.00010.00120𝐾=0.29522.40530.38054.8333 (36)

When the stochastic robust control strategy is applied to (33), which was first given in [13], it can be shown from Figure 1 that asymptotical stabilization in probability cannot be guaranteed in the presence of disturbance 𝑣(𝑡). Figures 2 and 3 show the system response and estimation error of system disturbance for the case with known nonlinearity, respectively. The simulation results show that asymptotical stabilization can be achieved using the method proposed in this paper and that the proposed disturbance observer is fine and effective.

Figure 1 

System performance using stochastic robust control strategy.

Figure 2 

System performance using DOBC strategy for known nonlinearity.

Figure 3 

Estimation error of disturbance for known nonlinearity.

Case 2 (with unknown nonlinearity). When nonlinear term 𝑓(𝑥,𝑡) is unknown, we assume 𝑓(𝑥,𝑡)=𝑟(𝑡)𝑥₂(𝑡) in simulation, where 𝑟(𝑡) is supposed to be stochastic input that obeys uniform distribution. Based on Theorem 7, it can be solved that ,.𝐿=00.00010.0011000.00010.00060𝐾=0.22090.96950.25363.1946 (37)

It is clear from Figure 4 that all states of system converge to zero and estimation errors of disturbance also converge to zero as shown in Figure 5. As has been shown above, we can see that asymptotical stabilization in probability is guaranteed and satisfied performance of the closed-loop control systems is achieved.

Figure 4 

System performance using DOBC strategy for unknown nonlinearity.

Figure 5 

Estimation error of disturbance for unknown nonlinearity.

6. Conclusion

In this paper, the DOBC approach is investigated for a class of nonlinear systems with deterministic and stochastic disturbances. Feasible design procedures are proposed under different conditions to estimate and reject deterministic disturbance for the plants with known and unknown nonlinearity. Based on the estimation of disturbances, the composite control laws can guarantee the composite closed-loop systems to be global asymptotical stabilization in probability in the presence of disturbances. Simulation for an aircraft model shows the efficiency of the proposed algorithms.

Acknowledgments

This work was supported partly by the National Natural Science Foundation of China under Grants 61374131, 60925012, and 61121003 and the National Basic Research Program of China (973 Program) under Grant 2012CB720003.