robust adaptive control for unmanned helicopter with stochastic disturbance

Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 105 (2017) 209 – 214 2016 IEEE International Symposium on Robotics and Intelligent Sensors, IRIS 2016, 17-20 December 2016, Tokyo, Japan Robust Adaptive Control for Unmanned Helicopter with Stochastic Disturbance Rong Lia , Qingxian Wua,∗, Mou Chena a College of Automation and Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210000, P.R China Abstract In this paper, the problem of robust adaptive control is concerned for a class of small-scale unmanned helicopter systems in the presence of system uncertainty, stochastic disturbance and output constraint The adaptive neural network approximator is introduced to handle the unknown system function Meanwhile, a prescribed performance function is employed to deal with output constraint It is proved that the proposed control method is able to guarantee the ultimately bounded convergence of all closed-loop system signals in mean square via Lyapunov stability theory The effectiveness of the developed robust controller are illustrated and confirmed by numerical simulations for a class of unmanned helicopter systems c 2017 2016Published The Authors Published Elsevier B.V.access article under the CC BY-NC-ND license © by Elsevier B.V.by This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of organizing committee of the 2016 IEEE International Symposium on Robotics and Intelligent Peer-review under responsibility of organizing committee of the 2016 IEEE International Symposium on Robotics and Intelligent Sensors (IRIS 2016) Sensors(IRIS 2016) Keywords: Unmanned helicopter; Stochastic disturbance; Adaptive control; Output constraint Introduction In the recent years, the growing demand for advanced unmanned helicopter systems has inspired significant research and development for the flight controller design 1,2 A structure robust linear control approach was proposed for unmmaned helicopters In 4, an adaptive attitude control method was investigated for the unmanned helicopter, which considered a kind of input nonlinearity In and 6, the adaptive tracking control was developed for a class of model-scaled unmanned helicopters A disturbance observer based robust nonlinear tracking control approach was proposed for unmanned helicopters It is well known that the stochastic disturbance often exists in the unmanned helicopter systems In 8, by using the Kalman estimator, a robust control approach was developed for a class of linear systems with stochastic disturbance A class of mean-square H∞ filter was proposed for the linear system with stochastic disturbance Among the existing control methods, backstepping technique has been widely adopted for the stochastic nonlinear systems 10,11 However, there are few existing research results for the unmanned helicopter control systems with stochastic disturbance and ∗ Qingxian Wu E-mail address: wuqingxian@nuaa.edu.cn 1877-0509 © 2017 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of organizing committee of the 2016 IEEE International Symposium on Robotics and Intelligent Sensors(IRIS 2016) doi:10.1016/j.procs.2017.01.212 210 Rong Li et al / Procedia Computer Science 105 (2017) 209 – 214 output constraint In this paper, we will study a robust control scheme for unmanned helicopter with the stochastic disturbance by using radical basis function neural networks (RBFNNs) Motivated by the above observation, the prescribed performance based robust adaptive control scheme is developed for the unmanned helicopters with stochastic disturbance and output constraints The remainder of this paper is organized as follows Section presents the problem formulation and preliminaries In Section 3, a prescribed performance based nonlinear control approach is developed for the unmanned helicopter system, and the closed-loop system stability is rigorously illustrate by using Lyapunov synthesis Simulation results are given to demonstrate the effectiveness of the developed control scheme in Section Finally, Section draws the conclusion of this paper Problem statement and preliminaries This section aims to briefly review the complete nonlinear stochastic dynamic model of unmanned helicopters and introduce some preliminary knowledge 2.1 Helicopter Modeling The rigid-body attitude dynamics of an unmanned helicopter with stochastic disturbance can be expressed as dΘ = H (Θ) Ωdt Im dΩ = −Ω × Im Ω + Gτ − QT a1 − Q M a2 + ΔM dt + hΩ dwΩ (1) where Θ = φ, θ, ψ T and Ω = p, q, r T denote the attitude angle and angular rate, respectively a1 = [1, 1, 0]T , a2 = [0, 0, 1]T , g represents the gravitational acceleration, m ∈ R indicates the total mass, and Im ∈ R3×3 denotes the inertial moment matrix Q M and QT respectively denote the main rotor anti-torque and tail rotor anti-torque G ∈ R3×3 is the control gain matrix related to the control torque τ ∈ R3 denotes control torque on the helicopter ΔM indicates the parameter uncertainty, hΩ (Ω) ∈ R3 , are unknown nonlinear functions wΩ ∈ R are the independent standard Brownian motion H (Θ) stands for the transformation matrix To proceed with the design of the robust adaptive control for the small-scale helicopter system (1) with stochastic disturbance, we make the following assumptions Assumption : The desired trajectories Θd (t) = [φd , θd , ψd ]T are the known bounded sufficiently smooth functions of time, with bounded and continuous first derivative Assumption : All state vectors are measurable Assumption : The roll angle φ satisfies inequality constraint −π/2 < φ < π/2, and the pitch angle ψ satisfies inequality constraint −π/2 < θ < π/2 ¯ and Δ M ¯ > represents unknown constant Assumption : The uncertain function ΔM is bounded ΔM ≤ Δ M, 2.2 Stochastic Nonlinear System To develop the robust adaptive control for the unmanned helicopter with stochastic disturbance, consider a class of stochastic differential equation as follows dχ (t) = f (χ (t)) dt + h (χ (t)) dw (t) (2) where χ (t) ∈ Rn represents the state, and w denotes the Wiener process Continuous functions f : Rn → Rn and h : Rn → Rn satisfy f (0) = and h (0) = To analyze the stability of the nonlinear stochastic system (2), we need the following definition and lemmas Definition 1: For any give V (χ) ∈ C , associated with the stochastic system (2), the infinitesimal generator L can be defined as follows 10 : LV (χ) = ∂V ∂χ f (χ) + 12 tr hT (χ) ∂∂χV2 h (χ) (3) Lemma 10 : Consider the stochastic system (2) Suppose that there exist an C function V : Rn → R+ , class K∞ functions b1 (|χ|), b2 (|χ|), and two constants η > 0, ϑ > 0, such that, for all χ ∈ Rn and for all t > t0 , the inequalities b1 (|χ|) ≤ V (χ) ≤ b2 (|χ|) LV (χ) ≤ −ηV (χ) + ϑ (4) 211 Rong Li et al / Procedia Computer Science 105 (2017) 209 – 214 holds Then, there exists a unique solution of stochastic system (2) for each χ0 ∈ Rn and it satisfies E V (χ) ≤ V (χ0 ) e−ηt + ϑη , ∀t > t0 (5) Furthermore, if the stochastic system (2) satisfies the inequality (5), the states are semi-globally uniformly ultimately bounded in mean square Lemma 11 : For any vectors a, b ∈ Rn , there exists an inequality aT b ≤ (μ p /p) a p + (1/qμq ) b q , where μ > 0, p > 1, q > and (p − 1) (b − 1) = In this paper, the control objective is to design a prescribed performance function based robust adaptive control for the unmanned helicopter with system uncertainty and stochastic disturbance Adaptive Controller Design In this section, a prescribed performance based adaptive control scheme is proposed for the unmanned helicopter system Step 1: Firstly, define the angular tracking error δΘ ∈ R3 as follows: δΘ = Θ − Θd (6) where Θd is the reference input vector A performance function ρi is chosen as 12 ρi (t) = (ρi0 − ρi∞ ) e−lt + ρi∞ , i = 1, 2, (7) where l > The constant ρi0 indicates the amplitude boundary of the ideal tracking error The choice of l will determine the convergence rate of δΘi Therefore, the appropriate choice of performance function and design parameters specifies the bounds on the system output trajectory An error transformation is defined as 13 Θi δΘi ¯i ρi , α = T −1 (t) , αi (t) = ln δΘi ρi δ α¯ i − ρΘi i αi + , i = 1, 2, where α¯i and αi are the positive constants The time derivative of ˙ Θi = ∂ ∂ Θi δΘi ρi ˙ ρi δΘi − ∂ ∂ Θi δΘi ρi δΘi ρ˙ ρ2i i = α¯ i +ai 1 a + δΘi α¯ − δΘi ρi i i ρ ρ i (8) Θi becomes ˙ di − Hi (Θ)Ωi − Θ i α¯ i +ai δΘi ˙i a + δΘi α¯ − δΘi ρ2i ρ i i ρ ρ i (9) i where Hi (Θ) indicates the ith row vector of H(Θ) In order to facilitate the controller design, we define MΘi , NΘi , i = 1, 2, as follows MΘi = α¯ i +ai 2ρi + δΘi ρi α¯ i − δΘi ρi > 0, δΘi ρ˙ ρ2i i NΘi = (10) Further, we have MΘ = diag {MΘ1 , MΘ2 , MΘ3 }, NΘ = diag {NΘ1 , NΘ2 , NΘ3 } Invoking (9) and (10), we obtain d Θ ˙ d − NΘ dt = MΘ H(Θ)Ω − Θ (11) According to the Assumption 3, we know H is the invertible matrix Therefore, the immediate control is chosen ˙ d − M −1 KΘ Ωd = H −1 (Θ)(Θ Θ Θ + NΘ ) (12) ˙ d is the time derivative of reference trajectory Θd , and KΘ ∈ R3×3 is the constant positive definite matrix where Θ Then, the time derivative of Θ becomes d Θ = (MΘ H(Θ)(Ω − Ωd ) − KΘ Θ ) dt (13) Consider the Lyapunov functional candidate VΘ = T Θ (14) Θ Substituting (13) into (14), we obtain LVΘ = − T Θ Θ T Θ KΘ Θ + T Θ Θ T Θ MΘ H(Θ)(Ω − Ωd ) (15) 212 Rong Li et al / Procedia Computer Science 105 (2017) 209 – 214 According to Young’s inequality, we have TΘ Θ TΘ MΘ H(Θ)(Ω − Ωd ) ≤ 34 μΘ3 is a design parameter Invoking (15), then we obtain LVΘ ≤ − T Θ T Θ KΘ Θ Θ + 34 μΘ3 Θ + MΘ H(Θ) 4μ4Θ Ω − Ωd Θ MΘ + H(Θ) 4μ4Θ Ω − Ω d , μΘ > (16) Step 2: Define the angular velocity tracking error δΩ ∈ R3 as follows: δΩ = Ω − Ωd (17) Considering (1) and (17), we have ˙ d dt + hΩ dwΩ Im dδΩ = −Ω × Im Ω + Gτ − QT a1 − Q M a2 + ΔM − Ω (18) Now, we design the ideal moment control input as τ = G−1 −KΩ δΩ + Ω × Im Ω + QT a1 + Q M a2 − MΘ H(Θ) 4μ4Θ ˙d −W ˆ TSΩ Im−1 δΩ + Ω Ω (19) where KΩ = KΩT > 0, KΩ ∈ R3×3 is the designed matrix G and Im are invertible matrix S Ω is the Gaussian function ˆ Ω is the estimate value of W ∗ The updated law of the RBFNN can be obtained as vector, W Ω ˆΩ ˆ˙Ω = ΓΩ δΩ δT Im δΩ S Ω − σΩ Γ−1 W W Ω Ω (20) where ΓΩ = ΓTΩ > 0, σΩ > are the design parameters Consider the Lyapunov functional candidate VΩ = δTΩ Im δΩ ˜Ω ˜ T Γ−1 W + 12 tr W Ω Ω (21) ˜ Ω = W∗ − W ˆ Ω According to I tˆo ˜ Ω represents the neural weight estimate error, which is defined as W where W Ω differentiation rule, we have LVΩ = δTΩ Im δΩ δTΩ −KΩ δΩ − ˙ˆ ˜ T Γ−1 W −tr W Ω Ω MΘ H(Θ) 4μ4Θ ˆ TSΩ + Im−1 δΩ + ΔM − W Ω δTΩ Im Im δΩ tr hΩ (Ω) hTΩ (Ω) (22) Ω By using the Young’s inequality, we obtain δTΩ Im Im δΩ tr hΩ (Ω) hTΩ (Ω) ≤ χΩ + Im χΩ δTΩ δΩ tr hΩ (Ω) hTΩ (Ω) (23) where χΩ > is a design parameter As we know, ΔM and hΩ (Ω) are unknown nonlinear functions Thus, for approximating ΔM and hΩ (Ω), it can be defined as ΔM + 32χImΩ tr hΩ (Ω) hTΩ (Ω) δΩ = WΩ∗T S (Ω) + ε∗Ω Invoking (23), we have LVΩ ≤ δTΩ Im δΩ δTΩ −KΩ δΩ − MΘ H(Θ) 4μ4Θ ˙ˆ ˜ T S Ω + ε∗ + χΩ − tr W ˜ T Γ−1 W Im−1 δΩ + W Ω Ω Ω Ω Ω Consider the following fact δTΩ Im δΩ δTΩ ε∗Ω ≤ 34 μΩ3 Im 4 (24) δΩ + 4μ14 ε∗Ω , μΩ > is a design parameter Then we obtain Ω ˜ TSΩ LVΩ ≤ −δTΩ Im δΩ δTΩ KΩ δΩ − MΘ 4μH(Θ) δTΩ δΩ δTΩ δΩ + δTΩ Im δΩ δTΩ W Ω Θ 4 ˙ˆ + μ I δ + ε∗ + χ ˜ T Γ−1 W −tr W Ω m Ω Ω Ω Ω Ω Ω 4μ (25) Ω Substituting (20) into (25) yields LVΩ ≤ −δTΩ Im δΩ δTΩ KΩ δΩ − MΘ H(Θ) 4μ4Θ 4 ˜ T Γ−1 W ˆ Ω + μ Im δTΩ δΩ δTΩ δΩ − σΩ W Ω Ω Ω ˜ ˜ T Γ−1 W ˆ Ω ≥ σΩ λ−1 Consider the following fact 2σΩ W max (ΓΩ ) WΩ Ω Ω LVΩ ≤ −δTΩ Im δΩ δTΩ KΩ δΩ − + 34 μΩ3 Im δΩ + MΘ 4μ4Ω H(Θ) T δΩ δΩ δTΩ δΩ 4μ4Θ ε∗Ω + 32 χΩ δΩ + 4μ4Ω ε∗Ω + 32 χΩ (26) ∗ − σΩ λ−1 max (ΓΩ ) WΩ , according to (26), we obtain ˜ − σΩ λ−1 max (ΓΩ ) WΩ ∗ + σΩ λ−1 max (ΓΩ ) WΩ (27) 213 Rong Li et al / Procedia Computer Science 105 (2017) 209 – 214 Up to now, we define a Lyapunov function candidate VΣ to prove the closed-loop system stability, and summarize the above robust adaptive control design procedure by the following theorem Theorem 1: Consider the MIMO nonlinear unmanned helicopter system (1) in the presence of system uncertainty and stochastic disturbance The adaptive laws of RBFNNs are designed as (20) The adaptive prescribed performance control laws are designed as (12) and (19) Then, the origin of the closed-loop system is globally uniformly asymptotically stable Furthermore, by choosing appropriate design parameters, the trajectory tracking errors converge to an arbitrarily small neighborhood of the origin Proof For considering the convergence of closed-loop state tracking errors, the Lyapunov function candidate for closed-loop control system can be chosen as VΣ = VΘ + VΩ (28) Differentiating VΣ and considering (16), (27) and (28), we obtain LVΣ = LVΘ + LVΩ ≤ − λmin (KΘ ) − 34 μΘ3 −σΩ λ−1 max ˜Ω (ΓΩ ) W Θ + 4 λmin (KΩ ) λmax (Im ) − σΩ λ−1 max − 34 μΩ3 Im (ΓΩ ) WΩ∗ + 4μ4Ω δΩ 4 ε∗Ω (29) + χΩ In order to simplify the description, the equation can be defined as LVΣ = VΣ + C (30) where and C are given by := λmin (KΘ ) − 34 μΘ , 4μ4Ω ε∗Ω λmin (KΩ ) λmax (Im ) 4 − 34 μΩ Im , ∗ , C := σΩ λ−1 σΩ λ−1 max (ΓΩ ) WΩ + max (ΓΩ ) + 32 χΩ Furthermore, the corresponding design parameters KΘ , KΩ , μΘ , μΩ , σΩ , ΓΩ are chosen such that λmin (KΘ ) − 34 μΘ3 > 0, λmin (KΩ ) λmax (Im ) − 34 μΩ3 Im >0 (31) According to (30) and (31), we have E (VΣ ) ≤ VΣ (0) e− t + C (32) ˜ Ω are bounded MoreFrom (32) and Lemma 1, we know that the tracking errors Θ , δΩ and approximation error W over, according to the definition of the transformed performance (8), we know that the boundedness of transformed error Θ can guarantee the boundedness of tracking error δΘ This concludes the proof Simulation Results Now, we consider an uncertain nonlinear unmanned helicopter system, and the main parameters are given in In this simulation, the system uncertainty ΔM is set ΔM = 0.1Ω × Im Ω, and unknown function is chosen as hΩ (Ω) = Ω The reference signals are chosen φ = 0.5 sin(t), θ = 0.5 sin(t) and ψ = 0.5 sin(t) The initial condition is set as φ(0) = 0.1, θ(0) = 0.1, ψ(0) = 0.1, p(0) = 0, q(0) = 0, r(0) = For the output constrain condition, the bound of output tracking errors is set as δΘ ≤ 0.4, so the bound of output is set as Θ ≤ 0.9 We apply the control (19) with design parameters KΘ = diag{10, 10, 10}, KΩ = diag{10, 10, 10}, μΘ = 1, μΩ = 1, σΩ = 8, ΓΩ = diag{0.1, 0.1, 0.1}, α¯ = 0.4, α = 0.4, ρ = The simulation results are shown in Figs.1 Figs.1(a),(b),(c) show the output tracking performance It can be seen that the output error remains within the compact set and tracks the desired trajectory Θd to a neighborhood of zero The control input vector τ is shown in Fig.1(d) Conclusion In this paper, a prescribed performance method based robust adaptive control scheme has been studied for the unmanned helicopter nonlinear system with stochastic disturbance Closed-loop system stability and tracking control performance have been illustrated and analyzed based on the rigorous Lyapunov synthesis Finally, simulation results of unmanned helicopter system have been presented to confirm the effectiveness of the proposed robust adaptive control approach 214 Rong Li et al / Procedia Computer Science 105 (2017) 209 – 214 (a) (b) (c) (d) Fig The results of attitude tracking for unmanned helicopter Acknowledgment This research is supported by National Nature Science Foundation of China (No 61573184), 333 Talents Project in Jiangsu Province (No BRA2015359), the Six Talents Peak Project of Jiangsu Province (No 2012-XXRJ-010) and the Fundamental Research Funds for the Central Universities (No NE2016101) References Cai G, Chen B M, Dong X Design and implementation of a robust and nonlinear flight control system for an unmanned helicopter Mechatronics 2011;21(5): 803-820 Liu C, Chen W H, Andrews J Tracking control of 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guaranteed transient and steady state tracking error bounds for strict feedback systems Automatica 2009;45(2): 532-538  ... the control objective is to design a prescribed performance function based robust adaptive control for the unmanned helicopter with system uncertainty and stochastic disturbance Adaptive Controller... is bounded ΔM ≤ Δ M, 2.2 Stochastic Nonlinear System To develop the robust adaptive control for the unmanned helicopter with stochastic disturbance, consider a class of stochastic differential equation... Brownian motion H (Θ) stands for the transformation matrix To proceed with the design of the robust adaptive control for the small-scale helicopter system (1) with stochastic disturbance, we make the