Adaptive fuzzy tracking control for non-affine nonlinear yaw channel of unmanned aerial vehicle helicopter

Abstract

The article deals with the problem of the yaw control of the unmanned aerial vehicle helicopter which is non-affine nonlinear by use of a novel projection-based adaptive fuzzy control approach. First, principle model and control design model of the yaw channel of the unmanned aerial vehicle helicopter are described. Then, a dynamic approximation technique is introduced to approach the non-affine model of the unmanned aerial vehicle helicopter into an affine model with variable parameters, which is applied to facilitate the design of nonlinear control scheme. Next, in the proposed controller, fuzzy controller is designed to deal with the unknown uncertainties and disturbances. Meanwhile, the projection-based adaptive law applied in fuzzy controller bounds the parameter estimation function and can also guarantee the robustness of the designed control scheme against uncertain disturbances. Moreover, the convergence and stability of the designed controller are proved by Lyapunov stability theory. Finally, the simulation results of the yaw channel of an unmanned aerial vehicle helicopter are performed to illustrate that the designed controller has good tracking performance, stability, and robustness under the condition of the time-vary uncertain disturbances.

Keywords

Unmanned aerial vehicle helicopter yaw control non-affine nonlinear systems fuzzy control adaptive control

Introduction

An unmanned aerial vehicle helicopter (UAVH) is a kind of autonomous control aircraft, which has significant advantages in the condition of takeoff, hovering, vertical landing, low-altitude flight and multi-attitude flight.¹ In the past decades, UAVHs are widely used in the civilian and military applications, such as power lines inspection, disaster rescue, agricultural, surveillance missions, wildlife monitoring, national defense, and so on, due to their versatility and maneuverability.^2

–5 However, the control of UAVHs is a challenging task, because the dynamics of UAVHs are highly nonlinear, inherently unstable, intensely coupled multivariable.^5

–9 Since the uncertainties of the parameters and model caused by the complicated dynamics of UAVHs affect the control performance, advanced control strategies are applied in the research of UAVHs to improve their reliability, stability, and robustness, which is still a hotspot in the control field.

At present, the flight control technology is mainly divided into two categories: linear control techniques such as proportional–integral–derivative (PID) control,¹⁰ H_∞ control,^11,12 and H₂ control¹³; nonlinear control techniques such as fuzzy control, ^14

–18 feedback linearization control,¹⁹ backstepping control,^20
–22 sliding model control,^1,4,23 and neural network control.^5,24
–26 Linear control method is very suitable for practical applications, since its design process is simple and the controller is easy to be implemented, which is also the reason for UAVHs dominated by linear control methods.^{7

–13,24} However, the linear control techniques cannot guarantee the global approximation of the model, which may lead to an undesirable response and even instability for the tracking control when the system is far from equilibrium point. Compared with linear control technology, nonlinear control techniques can maintain the stabilization of the system in a large scale, which have been researched in the control system of UAVHs.^{14
–16,19

–23} However, the nonlinear model of the UAVH is too complicated and generally non-affine, so the practical application of nonlinear controller is still an open and challenging research field. Therefore, the key motivation for this article is not only to simplify the yaw channel model of UAVHs for the nonlinear controller design but also guarantee the basic nonlinear characteristics of UAVHs to ensure the designed controller to be globally stable.

Due to the rapid development of nonlinear control theory, fuzzy control is an effective nonlinear control techniques applied in UAVHs.^14

–17,27 It has been known that the fuzzy control is extremely useful to deal with the problem of uncertainties and unknown disturbances in nonlinear control system. Fuzzy controller is a mathematical system for analyzing the analog input by use of logical variables.²⁸ In the fuzzy controller, solutions to the control problem can be realized by understandable terms and IF/THEN rules of the controller can be designed by human experience. However, the nonlinear, high order, time variation, and random disturbance of the system will cause the imperfection of the control rule and also influence the control performance, so fuzzy controller is usually combined with other control methods to achieve the optimal control effect, such as fuzzy PID controller,⁶ backstepping adaptive fuzzy control,²⁷ adaptive fuzzy sliding mode control,¹⁶ fuzzy neural network control,¹⁷ a fuzzy adaptive consensus method combined with prescribed performance control,²⁹ and so on. However, to the best of the authors’ knowledge, the existing controllers are mostly proposed for the linear model system or affine nonlinear model system^30
–32 and general control method cannot control the yaw system well due to the non-affine nonlinearity.

In this article, an adaptive fuzzy controller via projection operator is designed for the yaw channel of the UAVH which is non-affine nonlinear. At the same time, the designed control system can maintain asymptotic stability by the proof of stability using the Lyapunov stability theory. The reminder of the article is organized as follows: the yaw channel of the UAVH and its simplified model are described in section “Model of the yaw channel of the UAVH.” Section “Design of controller for yaw channel” presents a dynamic approximation method to approach the non-affine model of UAVH into an affine nonlinear model and a projection-based adaptive fuzzy controller is designed for tracking desired yaw angle of the UAVH. In section “Simulation results,” simulation results are provided to prove the validity of the designed control scheme. In “Conclusion” section, some conclusions are drawn and future works are given.

Model of the yaw channel of the UAVH

The yaw channel plays a significant part in motion of UAVHs, and the control of a small UAVH is also an extremely complicated problem,^7,8 because the torque of the small UAVHs combined with yaw channel is exceedingly sensitive. So, a more accurate model feature is indispensable to obtain better control performance of yaw channel of UAVHs. In the following paper, a higher fidelity mathematical principle model (PM) is applied specifically for the simulation of the UAVH, and a controller design model (CDM) with reduced complexity is applied for the design of controller and stability analysis.

Principle model

The PM of yaw channel of the UAVH used in this article is referenced by Leishman.¹⁸ The framework of the PM of an UAVH displayed in Figure 1 is built by use of the rigid body motion equation of the UAVH fuselage.

Figure 1.

Helicopter and its frame.

The UAVH is subjected to a lot of aerodynamic forces and moments in the normal flight, which can be calculated by summing up all parts of the UAVH including horizontal stabilizer, vertical fin, fuselage, tail rotor, and main rotor. Therefore, the dynamic model of the yaw channel is given by

\begin{array}{l} \dot{θ} = γ \\ I_{z z} \dot{γ} = N_{m r} + N_{t r} + N_{f u s} + N_{h s} + N_{v f} \end{array}

where θ represents the yaw angle of the UAVH, γ represents angular rate of the UAVH, I_zz represents the inertia around z-axis, N_vf represents the torque of the vertical fin, N_hs represents the torque of horizontal, N_fus represents the torque of fuselage, N_tr represents the torque of tail rotor, and N_mr represents the torque of main rotor.

The key forces and torque of the UAVH are determined by tail rotor and main rotor in condition of low speed flight and hovering.¹⁸ Therefore, the rest part of the UAVH can be simplified and the dynamic expression of the yaw channel is reformulated as

\begin{array}{l} \dot{θ} = γ \\ I_{z z} \dot{γ} = - N_{m r} + F_{t r} l_{t r} + z_{1} γ + z_{2} θ \end{array}

where z₁ and z₂ are damping constants, l_tr represents the distance from tail rotor to z-axis, and F_tr represents the thrust of tail rotor.

Then, the torque caused by main rotor can be calculated by use of the blade element method as following

N_{m r} = \int_{R_{0}}^{R} (\frac{ρ Ω^{2} r^{2} C_{l} c φ}{2} + \frac{ρ Ω^{2} r^{2} C_{d} c}{2}) r d r

with $C_{d} \approx C_{d 0} + C_{d 1} α + C_{d 2} α^{2}, C_{l} = a α, and φ = υ_{1} / (Ω r)$ , where Ω represents the velocity of main rotor, φ represents the angle of attack of blade element, c represents the inflow angle, α represents the chord of blade, r represents the velocity radial distance, a represents the slope of lift curve, ρ represents the density of air, and υ₁ represents the induced speed.

And we can calculate the equation (3) by use of the mathematical software Maple⁹ as following

\begin{array}{l} N_{m r} = & \frac{1}{8} C_{d 2} ρ c Ω^{2} (R^{4} - R_{0}^{4}) θ_{m r}^{2} + [8 C_{d 2} Ω \sqrt{ρ π R^{2} (2 C_{1} θ_{m r} + C_{2}^{2} - C_{2} \sqrt{C_{2}^{2} + 4 C_{2} θ_{m r}})} (R_{0}^{3} - R^{3}) + \\ 4 a Ω \sqrt{ρ π R^{2} (2 C_{1} θ_{m r} + C_{2}^{2} - C_{2} \sqrt{C_{2}^{2} + 4 C_{1} θ_{m r}})} (R^{3} - R_{0}^{3}) + 6 C_{d 2} C_{1} (R^{2} - R_{0}^{2}) + \\ 6 a C_{1} (R_{0}^{2} - R^{2}) + 6 C_{d 1} ρ π Ω^{2} R^{2} (R^{4} - R_{0}^{4})] \frac{c θ_{m r}}{48 π R^{2}} [3 C_{d 2} C_{2} \sqrt{C_{2}^{2} + 4 C_{1} θ_{m r}} (R_{0}^{2} - R^{2}) + \\ 3 a C_{2} \sqrt{C_{2}^{2} + 4 C_{1} θ} (R^{2} - R_{0}^{2}) + 4 C_{d 1} Ω \sqrt{ρ π R^{2} (2 C_{1} θ_{m r} + C_{2}^{2} - C_{2} \sqrt{C_{2}^{2} + 4 C_{1} θ_{m r}})} (R_{0}^{3} - R^{3}) + \\ 6 C_{d 0} ρ π Ω^{2} R^{2} (R^{4} - R_{0}^{4}) + 3 a C_{2}^{2} (R_{0}^{2} - R^{2}) + 3 C_{d 2} C_{2}^{2} (R^{2} - R_{0}^{2})] \frac{c}{48 π R^{2}} . \end{array}

with $C_{1} = \frac{1}{6} ρ a b c Ω^{2} (R^{3} - R_{0}^{3}) and C_{2} = \frac{1}{8} ρ a b c Ω \sqrt{2 / ρ π R^{2}} (R^{2} - R_{0}^{2})$ , where b, R, and θ_mr are the number and radial of the rotor and pitch angle of main rotor, respectively.

Similarly, we can describe the force caused by the tail rotor as

F_{t r} = \frac{1}{2} ρ a_{t r} b_{t r} c_{t r} Ω_{t r}^{2} \int_{R_{t r 0}}^{R_{t r}} (θ_{t r} r_{t r}^{2} - \frac{υ_{t r 1}}{Ω_{t r}} r) d r_{t r}

υ_{t r 1} = \sqrt{\frac{T_{t r}}{2 ρ A_{t r}}}

Substituting equation (6) into equation (5), we have

\begin{array}{l} F_{t r} & = \frac{1}{2} ρ a_{t r} b_{t r} c_{t r} Ω_{t r}^{2} \int_{R_{t r 0}}^{R_{t r}} (θ_{t r} r_{t r}^{2} - \sqrt{\frac{T_{t r}}{2 ρ A_{t r}}} \frac{r}{Ω_{t r}}) d r_{t r} \\ = C_{3} θ_{t r} + \frac{1}{2} C_{4} (C_{4} + \sqrt{C_{4}^{2} + 4 C_{3} θ_{t r}}) \end{array}

with $C_{3} = \frac{1}{6} ρ a_{t r} b_{t r} c_{t r} Ω_{t r}^{2} (R_{t r}^{3} - R_{t r 0}^{3}) and C_{4} = \frac{1}{8} ρ a_{t r} b_{t r} c_{t r} Ω_{t r} \sqrt{2 / ρ π R_{t r}^{2}} (R_{t r}^{2} - R_{t r 0}^{2})$ , where a_tr represents the slope of lift curve, b_tr represents the number of rotor, c_tr represents the chord of blade, υ_tr represents the induced velocity of tail rotor, r_tr represents the radial distance, θ_tr represents the pitch angle, Ω_tr represents the velocity of tail rotor, and A_tr represents the area of tail rotor disk.

Likewise, we can describe the force caused by the main rotor as:

\begin{array}{l} F_{m r} = C_{1} θ_{m r} + \frac{1}{2} C_{2} (C_{2} + \sqrt{C_{2}^{2} + 4 C_{1} θ_{m r}}) \end{array}

Controller design model

We can clearly find that the coupling relationship exists between tail rotor thrust F_tr and main rotor torque N_mr in PM. Equations (4) and (7) further display that the PM is too hard to be applied for controller design. So, a simplified model that is used for controller design and stability analysis has been established to replace the dynamic equation expressed by equations (4) and (7). Although the result of nonlinear yaw dynamic model of the UAVH is still very complicated, the advantage of this simplified model which preserves the relevant dynamic characteristics of the PM is that it can be analyzed theoretically.

The relationship between θ_mr and N_mr can be approximated with quadratic polynomial by drawing the pitch angle versus torque.⁷

\begin{array}{l} N_{m r} = c_{N_{2}} θ_{m r}^{2} + c_{N_{1}} θ_{m r} + c_{N_{0}} \end{array}

where $c_{N_{2}}, c_{N_{1}}, and c_{N_{0}}$ are mainly determined by the speed of main rotor Ω_mr and the shape of the blades. So again, we can approximate the lift of tail rotor F_tr as

\begin{array}{l} F_{t r} = c_{F_{2}} θ_{t r}^{2} + c_{F_{1}} θ_{t r} + c_{F_{0}} \end{array}

where $c_{F_{2}}, c_{F_{1}}, and c_{F_{0}}$ are mainly determined by the velocity of tail rotor Ω_tr and the shape of blades.

Combining equations (2), (9), and (10), the non-affine CDM can be calculated as following

\begin{array}{l} \dot{θ} & = γ \\ I_{z z} \dot{γ} & = z_{1} r + z_{2} θ - (c_{N_{2}} θ_{m r}^{2} + c_{N_{1}} θ_{m r} + c_{N_{0}}) + (c_{F_{2}} θ_{t r}^{2} + c_{F_{1}} θ_{t r} + c_{F_{0}}) l_{t r} \end{array}

where z₁ and z₂ are damping constants, and we can learn from the equation (11) that: (1) There are coupling relationships between the tail rotor force and mainly rotor torque. (2) The yaw channel of the UAVH is represented by two-order time-variant system which is nonlinear in the control input. (3) The input nonlinearity mainly determined by the velocity of tail rotor and main rotor and main rotor collective.

Design of controller for yaw channel

In this section, a dynamic approximation approach is designed to approximate the non-affine system of yaw channel of the UAVH into an affine nonlinear form, then fuzzy control scheme is applied to handle the unknown uncertainties and disturbances in the UAVH. By use of the projection-based adaptive law, the parameters estimation function can be bounded, and it can also guarantee the robustness of the controller against the uncertain disturbances. Meanwhile, Lyapunov stability theory is used to prove that the control system can be maintained asymptotically stable.

Non-affine nonlinear approximation

It is an extremely difficult task to control the system with non-affine nonlinear characteristics in the control input vector.³³ Especially in tracking control, the traditional linearization make the system into a time-varying linearized model under the condition of the stabilization only around the stable state point, so it is quite hard for the design of controller to track the desired trajectory accurately.³³ Therefore, a systematic control approach is needed to solve the condition of the nonlinear model with non-affine characteristics, which can make the nonlinear models more appropriate for tracking the desired trajectories accurately.

First, in general, the non-affine nonlinear mathematical model is considered as

\begin{array}{l} {\dot{x}}_{i} = x_{i + 1}, i = 1, 2, \dots, n - 1 \\ {\dot{x}}_{n} = f (x, u) + d (t) \\ y = x_{1} \end{array}

where u ∈ ℜ represents the system input vector, y ∈ ℜ represents the system output, $x = {[x_{1}, \dots, x_{n}]}^{T} \in ℜ^{n}$ represents the system state vector, $d (t)$ represents the unknown exterior disturbances, and $f (x, u)$ represents a known nonlinear function.

The Taylor expansion for known smooth nonlinear function $f (x, u)$ in the non-affine nonlinear system in equation (12) with respect to $u (t) around u (t - τ)$ is expressed as follows

\begin{array}{l} {\dot{x}}_{i} = x_{i + 1} \\ {\dot{x}}_{n} = f (x, u (t - τ)) + f_{d} (x, u (t - τ)) Δ u + R_{n} + d \end{array}

where $f_{d} (x, u (t - τ)) = \frac{\partial f (x, u)}{\partial u} |_{u = u (t - τ)} and Δ u = u (t) - u (t - τ)$ , and Taylor remainder $R_{n} = \frac{f_{n n} Δ u^{2}}{2}$ is restricted to

\begin{array}{l} | R_{n} | \leq \frac{r_{n} Δ u^{2}}{2} \end{array}

with $f_{n n} = \frac{\partial^{2} f (x, u)}{\partial^{2} u} |_{u = ξ}$ , where ξ is a point between $u (t) and u (t - τ)$ , and make $0 \leq | f_{n n} | \leq r_{n}$ , where r_n is defined as a finite positive number. The variable τ > 0 is the updating input, which can be selected as the sampling time. When the function of the known smooth nonlinear system $f (x, u)$ changes fast, a good option of variable τ is the sampling time, because the higher variable τ can cause imprecise approximation.

Then, we can rewrite equation (13) as following

\begin{array}{l} {\dot{x}}_{i} = x_{i + 1} \\ {\dot{x}}_{n} = f_{n} (x, u (t - τ)) + f_{d} (x, u (t - τ)) u + d_{ξ} (t) \end{array}

where $f_{n} (x, u (t - τ)) = f (x, u (t - τ)) - f_{d} (x, u (t - τ)) u (t - τ) and d_{ξ} (t) = R_{n} + d (t)$ which is assumed that $| d_{ξ} (t) | \leq d_{M} (t)$ . Moreover, in order to obtain the approximation model, the control input vector u should meet that $0 < | \frac{\partial f}{\partial u} | \leq β and | Δ u | \in [0, δ]$ with respect to finite positive numbers δ and β. Meanwhile, for the sake of making the approximation error of model (14) bounded, $| Δ u |$ should not be too large. Furthermore, in many practical flight systems, $| Δ u (t) | \in [0, δ]$ is the physics limitation of actuators, since the outputs and states cannot change too quickly due to the inertia of the system.

According to the equation (14), we can find that $u (t - τ)$ is in the vicinity of u, and the τ should not be chosen too high to diminish the accuracy of the affine approximation model. Although the smaller is the τ the more accurate is the approximation model, τ = 0 is impossible to be implemented to obtain the best affine approximation model, because u is the control law which is needed to be processed. So, the further improvement of the abovementioned method is used to get the accurate time-varying trim point which is expressed as following

\dot{μ} = - σ μ + σ u

From the equation (15), we can draw that $\lim_{σ \to \infty} μ = u$ , and the σ → ∞ is just an expression in the mathematical sense. Generally speaking, $σ \in [5, 50]$ .

In order to facilitate the following variables and functions, $u (t - τ)$ is redefined as $μ (t), f_{n} (x, μ) + d_{ξ} (t)$ is redefined as $f_{n d} (x, μ)$ , so equation (14) is reformulated as following

\begin{array}{l} {\dot{x}}_{i} = x_{i + 1} \\ {\dot{x}}_{n} = f_{n d} (x, μ) + f_{d} (x, μ) u \end{array}

From the equation (16), we can see that the non-affine nonlinear model is simplified as a time-varying affine model with the global approximation by use of the dynamic approximation technique, meanwhile the tracking problem can be solved by use of the the affine nonlinear control approach.

Adaptive fuzzy controller design

Under normal circumstances, $f_{n d} (x, μ) and f_{d} (x, μ)$ are unknown or cannot be known precisely, so it is hard to design a satisfactory controller. In this article, we consider that $f_{n d} (x, μ) and f_{d} (x, μ)$ are replaced with fuzzy system ${\hat{f}}_{n d} (x, μ) and {\hat{f}}_{d} (x, μ)$ to realize the adaptive fuzzy tracking control.

At first, the fuzzy system ${\hat{f}}_{n d} (x, μ | θ_{n})$ is used to approximate $f_{n d} (x, μ)$ and can be built in the following two steps:

Step 1: For variables $x_{i} (i = 1, 2, \dots, n)$ and μ, p_i fuzzy sets are defined as $A_{i}^{l_{i}} (l_{i} = 1, 2, \dots, p_{i})$ and p_μ fuzzy sets are defined as $A_{μ}^{l_{μ}} (l_{μ} = 1, 2, \dots, p_{μ})$ , respectively.

Step 2: Fuzzy system ${\hat{f}}_{n d} (x, μ | θ_{n})$ are constructed using the following $\prod_{i = 1}^{n} p_{i} \cdot p_{μ}$ fuzzy rules:

R:IF x_{1} is A_{1}^{l_{1}} and \dots and x_{n} is A_{n}^{l_{n}} and μ is A_{μ}^{l_{μ}}, THEN {\hat{f}}_{n d} is E^{l_{1} \dots l_{n} \cdot l_{μ}}

By use of singleton fuzzifier, the product inference engine and center average defuzzifier, the output of fuzzy system is calculated as

\hat{f} (x, μ | θ_{n}) = \frac{\sum_{l_{1} = 1}^{p_{1}} \dots \sum_{l_{n} = 1}^{p_{n}} \sum_{l_{μ} = 1}^{p_{μ}} {\bar{y}}^{l_{1} \dots l_{n} \cdot l_{μ}} (\prod_{i = 1}^{n} μ_{A_{i}^{l_{i}}} (x_{i}) \cdot μ_{A_{μ}^{l_{μ}}} (μ))}{\sum_{l_{1} = 1}^{p_{1}} \dots \sum_{l_{n} = 1}^{p_{n}} \sum_{l_{μ} = 1}^{p_{μ}} (\prod_{i = 1}^{n} μ_{A_{i}^{l_{i}}} (x_{i}) \cdot μ_{A_{μ}^{l_{μ}}} (μ))}

where $μ_{A_{i}^{l_{i}}} (x_{i})$ is the membership function of x_i, $μ_{A_{μ}^{l_{μ}}} (μ)$ is the membership function of μ, and ${\bar{y}}^{l_{1} \dots l_{n} \cdot l_{μ}}$ is the free parameter which are stored in the set $θ_{n} \in R^{\prod_{i = 1}^{n} p_{i} \cdot p_{μ}}$ . By introducing the vector $ξ (x)$ , we can rewrite the equation (18) as

{\hat{f}}_{n d} (x, μ | θ_{n}) = θ_{n}^{T} ξ (x, μ)

where $ξ (x) is \prod_{i = 1}^{n} p_{i} \cdot p_{μ}$ dimensions vector and expressed as:

ξ (x, μ) = \frac{\prod_{i = 1}^{n} μ_{A_{i}^{l_{i}}} (x_{i}) \cdot μ_{A_{μ}^{l_{μ}}} (μ)}{\sum_{l_{1} = 1}^{p_{1}} \dots \sum_{l_{n} = 1}^{p_{n}} \sum_{l_{μ} = 1}^{p_{μ}} (\prod_{i = 1}^{n} μ_{A_{i}^{l_{i}}} (x_{i}) \cdot μ_{A_{μ}^{l_{μ}}} (μ))}

Similarly, fuzzy system ${\hat{f}}_{d} (x, μ | θ_{d})$ can approximate $f_{d} (x, μ)$ , which is expressed as

{\hat{f}}_{d} (x, μ | θ_{d}) = θ_{d}^{T} η (x, μ)

Then, the tracking error can be defined as

e = y - x = {(e, \dot{e}, \dots, e^{(n - 1)})}^{T}

where y is the reference trajectory.

For system (16), according to the fuzzy system (19) and (21), we can design the control law as

\begin{array}{l} u & = \frac{1}{{\hat{f}}_{d} (x, μ | θ_{d})} [- {\hat{f}}_{n d} (x, μ | θ_{n}) + {\dot{y}}_{n} + K^{T} e] \\ = \frac{1}{θ_{d}^{T} η (x, μ)} [- θ_{n}^{T} ξ (x, μ) + {\dot{y}}_{n} + K^{T} e] \end{array}

where $K = {(k_{n}, \dots, k_{1})}^{T}$ is root of $s^{n} + k_{1} s^{(n - 1)} + \dots + k^{n} = 0$ , which makes the characteristic equation stable, $ξ (x, μ) and η (x, μ)$ are the fuzzy vectors, and $θ_{d}^{T} and θ_{n}^{T}$ are adaptive parameter changed according to the adaptive law (24) and (25), which are designed as follows

{\dot{θ}}_{n} = γ_{1} Proj (θ_{n}, - e^{T} P b ξ (x, μ))

{\dot{θ}}_{d} = γ_{2} Proj (θ_{d}, - e^{T} P b η (x, μ) u)

where $Proj (\cdot, \cdot)$ is the projection operator (see the Appendix 1 for details) which bounds that $| θ_{n} | \leq θ and | θ_{d} | \leq Ω$ .

In Figure 2, the entire projection-based adaptive fuzzy tracking controller for the yaw channel of the UAVH is illustrated to obtain a clear idea.

Figure 2.

The flow chart of the projection-based adaptive fuzzy controller for the yaw channel of the UAVH. UAVH: unmanned aerial vehicle helicopter.

Stability analysis

The proof of stabilization of designed projection-based adaptive fuzzy controller is achieved by use of the Lyapunov function with the state tracking errors (equation (22)) and adaptive parameter estimation errors, which can be deduced as the following three steps:

Step 1: According to equations (23) and (22), state variable ${\dot{x}}_{n}$ can be computed as

\begin{array}{l} {\dot{x}}_{n} = {\hat{f}}_{d} (x, μ | θ_{d}) u + {\hat{f}}_{n d} (x, μ | θ_{n}) - K^{T} e - {\dot{e}}_{n} \end{array}

From equations (16) and (26), the closed loop dynamic equation of fuzzy control system can be written as

{\dot{e}}_{n} = - K^{T} e + [{\hat{f}}_{n d} (x, μ | θ_{n}) - f_{n d} (x, μ)] + [{\hat{f}}_{d} (x, μ | θ_{d}) - f_{d} (x, μ)] u

Then, for the need of the the Lyapunov stability theory, we can reformulate the dynamic equation (27) as the vector form

\dot{e} = Λ e + b {[{\hat{f}}_{n d} (x, μ | θ_{n}) - f_{n d} (x, μ)] + [{\hat{f}}_{d} (x, μ | θ_{d}) - f_{d} (x, μ)] u}

where Λ = [\begin{array}{l} 0 & 1 & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 0 & 1 \\ - k_{n} & - k_{n - 1} & - k_{n - 2} & - k_{n - 3} & \dots & - k_{2} & - k_{1} \end{array}], b = [\begin{array}{l} 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{array}]

Step 2: We define the minimum approximation error as

ω = {\hat{f}}_{n d} (x, μ | θ_{n}^{*}) - f_{n d} (x, μ) + [{\hat{f}}_{d} (x, μ | θ_{d}^{*}) - f_{d} (x, μ)] u

where $θ_{n}^{*} and θ_{d}^{*}$ are optimal parameters which are described as

θ_{n}^{*} = arg \min_{θ_{n} \in Ω_{n}} [sup_{x, μ \in R^{n}} | {\hat{f}}_{n d} (x, μ | θ_{n}) - f_{n d} (x, μ) |]

θ_{d}^{*} = arg \min_{θ_{d} \in Ω d} [sup_{x, μ \in R^{n}} | {\hat{f}}_{d} (x, μ | θ_{d}) - f_{d} (x, μ) |]

where $Ω_{n} and Ω_{d}$ are sets of $θ_{n} and θ_{d}$ . Then, from equations (28) and (29), we have

\dot{e} = Λ e + b {[{\hat{f}}_{n d} (x, μ | θ_{n}) - {\hat{f}}_{n d} (x, μ | θ_{n}^{*})] + [{\hat{f}}_{d} (x, μ | θ_{d}) - {\hat{f}}_{d} (x, μ | θ_{d}^{*})] u + ω}

Substituting equations (19) and (21) into equation (30), we can compute the closed loop dynamic equation as

\begin{array}{l} \dot{e} & = Λ e + b [{(θ_{n} - θ_{n}^{*})}^{T} ξ (x, μ) + {(θ_{d} - θ_{d}^{*})}^{T} η (x, μ) u + ω] \\ = Λ e + M \end{array}

The equation (31) clearly describes the relationship between the adaptive parameters $θ_{n} and θ_{d}$ and the tracking error e. The purpose of adaptive law is to provide a regulatory mechanism for $θ_{n} and θ_{d}$ to make the tracking error e and parameter error $θ_{n} - θ_{n}^{*} and θ_{d} - θ_{d}^{*}$ to achieve the minimum.

Step 3: Design the Lyapunov function as

V = \frac{1}{2} e^{T} P e + \frac{1}{2 γ_{1}} {(θ_{n} - θ_{n}^{*})}^{T} (θ_{n} - θ_{n}^{*}) + \frac{1}{2 γ_{2}} {(θ_{d} - θ_{d}^{*})}^{T} (θ_{d} - θ_{d}^{*})

where γ₁ and γ₂ are positive and P is a positive definite matrix which meets the following Lyapunov function:

Λ^{T} P + P Λ = - Q

where Q is an arbitrary $n \times n$ positive definite matrix.

According to the equations (31) and (32), the $\dot{V}$ along the dynamical tracking trajectory of error equations can be computed as

\begin{array}{l} \dot{V} & = \frac{1}{2} {\dot{e}}^{T} P e + \frac{1}{2} e^{T} P \dot{e} + \frac{1}{γ_{1}} {(θ_{n} - θ_{n}^{*})}^{T} {\dot{θ}}_{n} + \frac{1}{γ_{2}} {(θ_{d} - θ_{d}^{*})}^{T} {\dot{θ}}_{d} \\ = \frac{1}{2} (e^{T} Λ^{T} + M^{T}) P e + \frac{1}{2} e^{T} P (Λ e + M) + \frac{1}{γ_{1}} {(θ_{n} - θ_{n}^{*})}^{T} {\dot{θ}}_{n} + \frac{1}{γ_{2}} {(θ_{d} - θ_{d}^{*})}^{T} {\dot{θ}}_{d} \\ = \frac{1}{2} e^{T} (Λ^{T} P + P Λ) e + \frac{1}{2} (M^{T} P e + e^{T} P M) + \frac{1}{γ_{1}} {(θ_{n} - θ_{n}^{*})}^{T} {\dot{θ}}_{n} + \frac{1}{γ_{2}} {(θ_{d} - θ_{d}^{*})}^{T} {\dot{θ}}_{d} \\ = - \frac{1}{2} e^{T} Q e + e^{T} P M + \frac{1}{γ_{1}} {(θ_{n} - θ_{n}^{*})}^{T} {\dot{θ}}_{n} + \frac{1}{γ_{2}} {(θ_{d} - θ_{d}^{*})}^{T} {\dot{θ}}_{d} \\ = - \frac{1}{2} e^{T} Q e + e^{T} P b ω + {(θ_{n} - θ_{n}^{*})}^{T} e^{T} P b ξ (x, μ) + {(θ_{d} - θ_{d}^{*})}^{T} e^{T} P b η (x, μ) u + \frac{1}{γ_{1}} {(θ_{n} - θ_{n}^{*})}^{T} {\dot{θ}}_{n} + \frac{1}{γ_{2}} {(θ_{d} - θ_{d}^{*})}^{T} {\dot{θ}}_{d} \\ = - \frac{1}{2} e^{T} Q e + e^{T} P b ω + \frac{1}{γ_{1}} {(θ_{n} - θ_{n}^{*})}^{T} [{\dot{θ}}_{n} + γ_{1} e^{T} P b ξ (x, μ)] + \frac{1}{γ_{2}} {(θ_{d} - θ_{d}^{*})}^{T} [{\dot{θ}}_{d} + γ_{2} e^{T} P b η (x, μ) u] \end{array}

Substituting adaptive law (24) and (25) into (34) leads to:

\begin{array}{l} \dot{V} = & - \frac{1}{2} e^{T} Q e + e^{T} P b ω + {(θ_{n} - θ_{n}^{*})}^{T} [Proj (θ_{n}, - e^{T} P b ξ (x)) + e^{T} P b ξ (x, μ)] \\ + {(θ_{d} - θ_{d}^{*})}^{T} [Proj (θ_{d}, - e^{T} P b η (x) u) + e^{T} P b η (x, μ) u] \end{array}

Using the property 1A of the projection operator (see the appendix 1 for details), we can obtain that

{(θ_{n} - θ_{n}^{*})}^{T} [Proj (θ_{n}, - e^{T} P b ξ (x)) + e^{T} P b ξ (x, μ)] \leq 0

{(θ_{d} - θ_{d}^{*})}^{T} [Proj (θ_{d}, - e^{T} P b η (x) u) + e^{T} P b η (x, μ) u] \leq 0

Then, according to the above analysis, we can get

\begin{array}{l} \dot{V} = - \frac{1}{2} e^{T} Q e + e^{T} P b ω \end{array}

Since $- \frac{1}{2} e^{T} Q e \leq 0$ , and the fuzzy system is selected with very small minimum approximation error ω, we can obtain $\dot{V} \leq 0$ . Thus, the whole system is proved to be asymptotically stable.

Simulation results

In this part, simulations results are used to demonstrate the validity of the designed nonlinear projection-based adaptive fuzzy tracking controller for the yaw channel of an UAVH. First, the PM for simulation is achieved from the helicopter platform referenced by Zhao and Han.⁷

The parameters of nonlinear yaw channel model of an UAVH are summarized as following

\begin{array}{l} \dot{θ} = γ \\ \dot{γ} = a_{1} γ + a_{2} θ + a_{3} θ_{t r} + a_{4} θ_{t r}^{2} + a_{5} Ω θ_{t r} + d (t) \end{array}

with $a_{1} = - 1.38, a_{2} = - 3.33, a_{3} = 63.09, a_{4} = 11.65, a_{5} = - 0.14, and Ω = 1200$ . We can see from the equation (39) that $a_{3} θ_{t r} + a_{4} θ_{t r}^{2} + a_{5} Ω θ_{t r}$ is a nonlinear function about the controller input θ_tr.

Then, the initial conditions are $θ_{n} = 0.1 and θ_{d} = 0.1, θ (0) = 5, and r (0) = 0$ , and the filter parameter σ applied to approach u is selected as 50. The remaining parameters are selected as $Q = [\begin{matrix} 20 & 0 \\ 0 & 20 \end{matrix}], k_{1} = 50, k_{2} = 100, γ_{1} = 50, and γ_{2} = 0.5$ , which are chosen to achieve optimal control performance according to the adaptive law and the need of stability of control system.

Next, the control law of yaw dynamics of the UAVH is given by

\begin{array}{l} u = \frac{1}{θ_{d}^{T} η (x, μ)} [- θ_{n}^{T} ξ (x, μ) + {\dot{x}}_{2} + K^{T} e] \\ {\dot{θ}}_{n} = γ_{1} Proj (θ_{n}, - e^{T} P b ξ (x)) \\ {\dot{θ}}_{d} = γ_{2} Proj (θ_{d}, - e^{T} P b η (x) u) \end{array}

where, as for $ξ (x, μ) and η (x, μ)$ , five kinds of membership functions shown in Figure 3 are selected as following:

Figure 3.

The membership function of the fuzzy controller.

Finally, the following two cases are simulated to demonstrate the robustness of the designed controller for the yaw channel of the UAVH. In these simulations, the unknown uncertain disturbance is designed to change as a function of time, that is

d (t) = {\begin{array}{l} 0, {deg/s}^{2} & t \leq 10 \\ 5 sin (π t) + 4 \cos (0.5 π t) + 3 \cos (0.5 π t) sin (0.75 π t), {deg/s}^{2} & t > 10 \end{array}

Case 1: Proposed adaptive fuzzy control scheme (AFCS) is used in the step yaw signal tracking.

The tracking command of $θ_{c}$ is considered as

θ_{c} = {\begin{matrix} 10, & t \leq 20 \\ 15, & 20 < t \leq 40 \\ 25, & t > 40 \end{matrix}

Then, a filter which is $F_{c} = θ_{d} / θ_{c} = 0.8 / (s + 0.8)$ is applied for $θ_{c}$ to obtain the desired smooth trajectory $θ_{d} = y_{d} = 0.8 θ_{c} / (s + 0.8)$ . Meanwhile, the classical PID control method is used to compare with the AFCS to demonstrate the validity and robustness of the AFCS. The simulation results of tracking response, the related control effort u, and the filtering μ are illustrated in Figure 4. As can be seen in the Figure 4, the proposed control scheme for step yaw signal tracking control of the UAVH has satisfactory control performance (a minimum response time, without overshot and minimal tracking error) against the unknown time-varying disturbance. By comparing with PID control method, the proposed AFCS has more robust against unknown disturbances in yaw control, which further demonstrate the effectiveness of the designed controller. Moreover, Figure 5 shows the adaptive fuzzy system ${\hat{f}}_{n d} (x, μ | θ_{n}) and {\hat{f}}_{d} (x, μ | θ_{d})$ that are used to approximate f_nd and f_n are bounded, which also guarantees the robustness of the designed AFCS against the parameter uncertainties and unknown disturbances.

Figure 4.

System step response with the AFCS under the condition of the time-varying disturbance compared with PID controllers. AFCS: adaptive fuzzy control scheme; PID: proportional–integral–derivative.

Figure 5.

Fuzzy system ${\hat{f}}_{n d} (x, μ | θ_{n}) and {\hat{f}}_{d} (x, μ | θ_{d})$ in the case of the step response.

Case 2: Proposed AFCS is used in the sinusoidal yaw signal tracking.

The tracking command of $θ_{d}$ is considered as: $θ_{d} = 30 s i n (t)$ . Similarly, the classical PID control method is used to compare with the AFCS in the sinusoidal yaw signal tracking. The simulation results of tracking response, the related control effort u, and the filtering parameter μ are displayed in Figure 6. From the Figure 6, the proposed control scheme for sinusoidal yaw signal tracking control of the UAVH achieves good asymptotic tracking performance under the condition of the unknown time-varying disturbance. We can obviously draw that proposed AFCS comparing with PID control method has more robust against unknown disturbances in yaw control. Moreover, Figure 7 shows the adaptive fuzzy system ${\hat{f}}_{n d} (x, μ | θ_{n}) and {\hat{f}}_{d} (x, μ | θ_{d})$ . By comparing Figure 5 with Figure 7, we can find that the approximation value ${\hat{f}}_{n d} (x, μ | θ_{n}) and {\hat{f}}_{d} (x, μ | θ_{d})$ in the two case are similar and around −20 and 3, respectively.

Figure 6.

System sinusoidal response with the AFCS under the condition of the time-varying disturbance compared with PID controller. AFCS: adaptive fuzzy control scheme; PID: proportional–integral–derivative.

Figure 7.

Fuzzy system ${\hat{f}}_{n d} (x, μ | θ_{n}) and {\hat{f}}_{d} (x, μ | θ_{d})$ in the case of the sinusoidal response.

Conclusion

A systematic study is carried out for the yaw channel of the UAVH in this article. First, since the yaw channel of the UAVH is non-affine, a dynamic approximation method is developed to approach the non-affine system into an affine nonlinear form. Then, fuzzy control scheme is designed to handle the unknown uncertainties and disturbances in the yaw channel. By use of the projection-based adaptive law, the free parameters estimated function in the fuzzy control scheme can be bounded, and it can also guarantee the robustness of the controller against the uncertain disturbances. Remarkably, the proposed projection-based adaptive fuzzy controller is useful when the explicit model of the UAVH is hard to obtain. Finally, simulation results are illustrated to proved that proposed projection-based AFCS can maintain the progressive tracking of the output for the closed-loop control system of the UAVH and have satisfactory control performance in the case of the unknown uncertainties and time-varying disturbances. Nevertheless, there is a chattering phenomenon in the control u, which may be caused by the selection of fuzzy rule base. Therefore, the solution to solve the chattering phenomenon and supplementary of input and output constraints are our future works.

Footnotes

Appendix 1

The smoothened projection operator³⁴ is applied in this article to bound the parameter estimation equation. The projection operator is very helpful for robust adaptive controller that needs multiple derivative of adaptive law. Below, we describe the main definition and property from.³⁴

Definition 1. Parameter estimation vector q which belongs to a convex compact set is defined as following

where θ_max represents the norm bound of parameter vector θ. Then, for any obtained $x \in R^{n}$ , the normalized projection operator is defined as following

where ∇ represents the gradient vector of $f (\cdot)$ computed at θ and $f (\cdot) : R^{n} \to R$ represents the convex smooth function:

where ε represents an arbitrary positive.

The main property of the projection operator³⁴ is given as following

Property 1. If we choose the initial value $θ (0) \in Ω$ and make that the variable $θ (t)$ are changed according to the following equation:

Then

For any $t \dots t_{0}$ , the inequality can be established as

Acknowledgment

The authors sincerely thank the editor and all the anonymous reviewers for their valuable comments and suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Natural Science Foundation of China (61503156, 61403161, and 51405198) and the Fundamental Research Funds for the Central Universities (JUSRP11562 and NJ20150011).

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