robust adaptive control for unmanned helicopter with stochastic disturbance

doi: 10.1016/j.procs.2017.01.212 ScienceDirect 2016 IEEE International Symposium on Robotics and Intelligent Sensors, IRIS 2016, 17-20 December 2016, Tokyo, Japan Robust Adaptive Contro

Procedia Computer Science 105 ( 2017 ) 209 – 214

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

ScienceDirect

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

 2016 The Authors Published by Elsevier B.V.

Peer-review under responsibility of organizing committee of the 2016 IEEE International Symposium on Robotics and Intelligent Sensors (IRIS 2016).

Keywords: Unmanned helicopter; Stochastic disturbance; Adaptive control; Output constraint.

1 Introduction

In the recent years, the growing demand for advanced unmanned helicopter systems has inspired significant re-search and development for the flight controller design1,2 A structure robust linear control approach was proposed for unmmaned helicopters3 In 4, an adaptive attitude control method was investigated for the unmanned helicopter, which considered a kind of input nonlinearity In 5 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 helicopters7

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 disturbance9 Among the existing control methods, backstepping technique has been widely adopted for the stochastic nonlinear systems10,11 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

© 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).

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 2 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 4 Finally, Section 5 draws the conclusion of this paper

2 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 as5

d Θ = H (Θ) Ωdt

I m dΩ =−Ω × I m Ω + Gτ − Q T a1− Q M a2+ ΔMdt + hΩdwΩ (1)

whereΘ = φ, θ, ψT

andΩ = p , q, rT

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 I m ∈ R3×3denotes the

inertial moment matrix Q M and Q T 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 ∈ R3denotes 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 14: The desired trajectoriesΘd(t)= [φd, θd, ψd]T are the known bounded sufficiently smooth functions

of time, with bounded and continuous first derivative

Assumption 21: All state vectors are measurable

Assumption 35: The roll angleφ satisfies inequality constraint −π/2 < φ < π/2, and the pitch angle ψ satisfies inequality constraint−π/2 < θ < π/2

Assumption 46: The uncertain functionΔM is bounded ΔM ≤ Δ ¯ M, andΔ ¯M> 0 represents unknown constant

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

d χ (t) = f (χ (t)) dt + h (χ (t)) dw (t) (2) whereχ (t) ∈ R n represents the state, and w denotes the Wiener process Continuous functions f : R n → R nand

h : R n → R n satisfy f (0) = 0 and h (0) = 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) ∈ C2, associated with the stochastic system (2), the infinitesimal generator L can

be defined as follows10:

LV (χ) = ∂V

∂χf (χ) +1

2tr

h T(χ)∂ 2V

Lemma 110: Consider the stochastic system (2) Suppose that there exist an C2function V : R n → R+, class K∞

functions b1(|χ|), b2(|χ|), and two constants η > 0, ϑ > 0, such that, for all χ ∈ R n and for all t > t0, the inequalities

b1(|χ|) ≤ V (χ) ≤ b2(|χ|)

holds Then, there exists a unique solution of stochastic system (2) for eachχ0∈ R nand it satisfies

E

V (χ)≤ V (χ0)e −ηt+ϑ

Furthermore, if the stochastic system (2) satisfies the inequality (5), the states are semi-globally uniformly ultimately bounded in mean square

Lemma 211: For any vectors a , b ∈ R n , there exists an inequality a T b≤ (μp /p) a p + (1/qμ q)b q, whereμ > 0,

p > 1, q > 1 and (p − 1) (b − 1) = 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

3 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 errorδΘ∈ R3as follows:

whereΘdis the reference input vector A performance functionρiis chosen as12

ρi(t)= (ρi0− ρi∞)e −lt+ ρi∞, i = 1, 2, 3 (7)

where l> 0 The constant ρi0 indicates the amplitude boundary of the ideal tracking error The choice of l will

deter-mine the convergence rate ofδΘi Therefore, the appropriate choice of performance function and design parameters

specifies the bounds on the system output trajectory

An error transformation is defined as13

Θi = T−1δ

Θi

ρi, ¯αi(t), αi(t)

= 1

2lnαi+δΘi ρi

¯

αi−δΘi ρi , i = 1, 2, 3 (8) where ¯αiandαiare the positive constants The time derivative of Θibecomes

˙Θi= Θi

∂δΘi ρi 1ρi

˙

δΘi− Θi

∂δΘi ρi δ

Θi

ρ 2

i

˙

ρi=1 2

¯

αi +a i



a i+δΘi ρi α ¯i−δΘi ρi 1ρi



H i(Θ)Ωi− ˙Θdi



−1 2

¯

αi +a i



a i+δΘi ρi α ¯i−δΘi ρi δ

Θi

ρ 2

i

˙

where H i(Θ) indicates the ith row vector of H(Θ) In order to facilitate the controller design, we define MΘi , N Θi,

i= 1, 2, 3 as follows

M Θi= α ¯i +a i

2ρi



a i+δΘi ρi α ¯i−δΘi ρi > 0, N Θi= δΘi

ρ 2

i

˙

Further, we have MΘ= diag {MΘ1, MΘ2, MΘ3}, NΘ= diag {NΘ1, NΘ2, NΘ3} Invoking (9) and (10), we obtain

d Θ= MΘ



H(Θ)Ω − ˙Θd − NΘ



According to the Assumption 3, we know H is the invertible matrix Therefore, the immediate control is chosen

Ωd = H−1(Θ)( ˙Θd − M−1

where ˙Θd is the time derivative of reference trajectoryΘd , and KΘ ∈ R3×3 is the constant positive definite matrix.

Then, the time derivative of Θbecomes

d Θ= (MΘH(Θ)(Ω − Ωd)− KΘ Θ)dt (13) Consider the Lyapunov functional candidate

VΘ= 1

4

 T

Θ Θ

2

(14) Substituting (13) into (14), we obtain

LVΘ T

Θ Θ TΘKΘ Θ TΘ Θ TΘMΘH(Θ)(Ω − Ωd) (15)

According to Young’s inequality, we have T

Θ Θ TΘMΘH(Θ)(Ω − Ωd)≤3

4μΘ4 Θ4+MΘ  4H(Θ)4

4 μ 4

Θ Ω − Ωd4,μΘ> 0

is a design parameter Invoking (15), then we obtain

LVΘ T

Θ Θ TΘKΘ Θ+3

4μΘ4 Θ4+MΘ  4H(Θ)4

4 μ 4

Step 2: Define the angular velocity tracking errorδΩ∈ R3as follows:

Considering (1) and (17), we have

I m dδΩ=−Ω × I m Ω + Gτ − Q T a1− Q M a2+ ΔM − ˙Ω d



Now, we design the ideal moment control input as

τ = G−1

−KΩδΩ+ Ω × I m Ω + Q T a1+ Q M a2−MΘ  4H(Θ)4

4μ 4

Θ I m−1δΩ+ ˙Ωd − ˆW T

Ω Ω

(19)

where KΩ = K T

Ω > 0, KΩ ∈ R3 ×3is the designed matrix G and I m are invertible matrix SΩis the Gaussian function

vector, ˆWΩis the estimate value of WΩ∗ The updated law of the RBFNN can be obtained as

˙

WΩ= ΓΩ



δΩδT

ΩI mδΩ Ω− σΩΓ−1

ΩWˆΩ



(20) whereΓΩ= ΓT

Ω> 0, σΩ> 0 are the design parameters Consider the Lyapunov functional candidate

VΩ=1

4



δT

ΩI mδΩ

2

+1

2tr

˜

W T

ΩΓ−1

ΩW˜Ω



(21) where ˜WΩ represents the neural weight estimate error, which is defined as ˜WΩ = W∗

Ω − ˆWΩ According to I ˆto

differentiation rule, we have

LVΩ= δT

ΩI mδΩδT

Ω



−KΩδΩ−MΘ  4H(Θ)4

4 μ 4

Θ I m−1δΩ+ ΔM − ˆW T

Ω Ω

+3 2



δT

ΩI m I mδΩ



tr

h (Ω) h T

Ω(Ω)

−trW˜T

ΩΓ−1

By using the Young’s inequality, we obtain

δT

ΩI m I mδΩtr

h (Ω) hT

Ω(Ω)≤ χΩ+I m 4

χ Ω



δT

ΩδΩ2tr

h (Ω) hT

where χΩ > 0 is a design parameter As we know, ΔM and hΩ(Ω) are unknown nonlinear functions Thus, for approximatingΔM and hΩ(Ω), it can be defined as ΔM +3I m 4

2 χ Ω tr

h (Ω) h T

Ω(Ω)2δΩ= W ∗T

Ω S (Ω) + ε∗

Ω Invoking (23),

we have

LVΩ≤ δT

ΩI mδΩδT

Ω



−KΩδΩ−MΘ  4H(Θ)4

4μ 4

Θ I m−1δΩ+ ˜W T

Ω Ω+ ε∗ Ω

+3

2χΩ− trW˜T

ΩΓ−1

ΩW˙ˆΩ

(24)

Consider the following factδT

ΩI mδΩδT

Ωε∗

Ω≤ 3

4μΩ4I m4

δΩ4+ 1 4μ 4 Ω

∗ Ω

4

,μΩ> 0 is a design parameter Then we obtain

LVΩ≤ −δT

ΩI mδΩδT

ΩKΩδΩ−MΘ  4H(Θ)4

4 μ 4

ΩδΩδT

ΩδΩ+ δT

ΩI mδΩδT

ΩW˜ΩT Ω

−trW˜T

ΩΓ−1

ΩW˙ˆΩ +3

4μΩ4I m4

δΩ4+ 1

4 μ 4 Ω

∗ Ω 4

+3

2χΩ

(25)

Substituting (20) into (25) yields

LVΩ≤ −δT

ΩI mδΩδT

ΩKΩδΩ−MΘ  4H(Θ)4

4 μ 4

ΩδΩδT

ΩδΩ− σΩW˜ΩTΓ−1

ΩWˆΩ+3

4μΩ4I m4

δΩ4+ 1

4 μ 4 Ω

∗ Ω 4

+3

2χΩ (26) Consider the following fact 2σΩW˜ΩTΓ−1

ΩWˆΩ≥ σΩλ−1

max(ΓΩ) WΩ 2− σΩλ−1

max(ΓΩ) WΩ∗ 2, according to (26), we obtain

LVΩ≤ −δT

ΩI mδΩδT

ΩKΩδΩ−MΘ  4H(Θ)4

4μ 4

ΩδΩδT

ΩδΩ− σΩλ−1

max(ΓΩ) WΩ 2+ σΩλ−1

max(ΓΩ) WΩ∗ 2

+3

4μΩ4I m4

δΩ4+ 1 4μ 4 ∗ Ω 4

+3

2χΩ

(27)

Up to now, we define a Lyapunov function candidate VΣto 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 asymp-totically 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

Differentiating VΣand considering (16), (27) and (28), we obtain

LVΣ= LVΘ+ LVΩ

≤ −λmin(KΘ)−3

4μΘ4 Θ4−λ min(KΩ)

λ max(I m) −3

4μΩ4I m4

δΩ4

−σΩλ−1

max(ΓΩ) WΩ 2+ σΩλ−1

max(ΓΩ) WΩ∗ 2+ 1

4μ 4 Ω

∗ Ω 4

+3

2χΩ

(29)

In order to simplify the description, the equation can be defined as

where and C are given by  := minλmin(KΘ)−3

4μΘ4, λ min(KΩ)

λ max (m) −3

4μΩ4I m4

, σΩλ−1 max(ΓΩ)

, C := σΩλ−1

max(ΓΩ) WΩ∗ 2+

1

4μ 4

Ω

∗

Ω

4

+3

2χΩ Furthermore, the corresponding design parameters KΘ, KΩ,μΘ,μΩ,σΩ,ΓΩare chosen such that

λmin(KΘ)−3

4μΘ4 > 0, λmin (KΩ)

λ max(I m) −3

4μΩ4I m4

According to (30) and (31), we have

From (32) and Lemma 1, we know that the tracking errors Θ,δΩand approximation error ˜WΩare bounded

More-over, according to the definition of the transformed performance (8), we know that the boundedness of transformed error Θcan guarantee the boundedness of tracking errorδΘ This concludes the proof

4 Simulation Results

Now, we consider an uncertain nonlinear unmanned helicopter system, and the main parameters are given in5 In this simulation, the system uncertaintyΔM is set ΔM = 0.1Ω × I m 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) = 0 For the output constrain condition, the bound of

output tracking errors is set asδΘ ≤ 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 = 1 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 trajectoryΘd

to a neighborhood of zero The control input vectorτ is shown in Fig.1(d)

5 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

(a) (b)

Fig 1 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)

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