Improved fast global sliding mode control based on the exponential reaching law

Abstract

In order to improve the control performance of the global sliding mode control method, a fast global sliding mode control method is proposed to accelerate the response of the system by changing the exponential decay function in the sliding surface as an exponential bilateral decay function which can make the dynamic sliding mode surface evolve into the linear sliding mode surface in the finite time. The exponential reaching law is used to design the control law, and Lyapunov stability theory is used to prove the stability of the system. The control method proposed in this article can be applied to control the uncertain nonlinear system. Simulation results show that the method has faster response rate than the conventional global sliding mode control method. The method can be used to control the quad-rotor unmanned helicopter, and its good practicability can be verified.

Keywords

Global sliding mode control Lyapunov stability theory uncertain nonlinear system quad-rotor unmanned helicopter exponential reaching law

Introduction

Control problem of nonlinear system is a hot issue which has been received more attention.^1,2 Some conventional control methods are hard to control the nonlinear system with uncertainties.^3–5 Uncertainties in the control system can be caused by the structured uncertainties and external disturbances.⁶ Therefore, some improved advanced control methods are proposed to resolve these isuues.^7–10 Among them, the sliding mode control (SMC) is widely used in the nonlinear system because of its simple physical implementation and good robustness to the uncertainties of the internal parameters as well as the external disturbances.¹¹ The dynamic response process of the conventional SMC is composed of two parts: the sliding mode and the reaching mode. However, the robustness to the uncertainties of both internal parameters and external disturbance exists only in the stage of the sliding mode. That is, the whole control process is not global robustness.

The global sliding mode controller can cancel the reaching mode, reduce the chatting, and overcome the disturbance and the time delay.^12,13 Therefore, a global sliding mode control (GSMC) is proposed and successfully applied in the brushless direct current (DC) motor servo system in Lu and Chen.¹⁴ Furthermore, for the second-order time-varying system, an improved GSMC which can drive the system states along the minimum time trajectory within the input torque limit is proposed in Choi et al.¹⁵ A type of integral sliding surface is also introduced to design a global robust controller in Yang et al.¹⁶ Furthermore, discrete GSMC is also applied in many fields, such as switching DC–DC converter and missile electromechanical actuator system.^17–19 Above all, GSMC can cancel the reaching mode by designing a nonlinear dynamic sliding mode surface. In this way, the system can enter into the sliding mode at the beginning. Based on the advantages of the GSMC, it has a good application prospect in many fields such as the flight simulators, rotating antenna, motor servo systems, as well as the missile electromechanical actuator.^20–23

Generally, the sliding mode surface equation of the GSMC is a dynamic sliding mode surface which is composed of two parts: the linear sliding mode surface and the nonlinear function. The nonlinear function is generally designed as the exponential decay function. Theoretically, when the time t is infinite, that is, when the nonlinear function decays to zero, the dynamic sliding mode surface can evolve into the linear sliding mode surface. Although the evolution speed of the sliding mode surface can be changed by adjusting the exponential decay parameters, the trajectory of the system cannot be guaranteed to reach the linear sliding mode in the finite time. For this, a fast global sliding mode control (FGSMC) is proposed in this article. The method can ensure that the sliding mode surface equation can evolve into the linear sliding mode surface in the finite time. In this way, the response rate of the system can be further improved.

In section “Improved FGSMC,” an improved FGSMC method is given, and the stability of the system is analyzed. In section “Simulation analysis,” the simulation analysis of the proposed method is given. In section “Application instance,” an application instance on the quad-rotor unmanned helicopter controlled by the proposed method is given. A brief conclusion is given in section “Conclusion.”

Improved FGSMC

System model

A nth-order uncertain nonlinear system is described as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} \\ \dots \\ {\overset{\cdot}{x}}_{n} = F (x_{1}, x_{2}, \dots, x_{n}, t) + b \cdot u (t) + Δ f (x_{1}, x_{2}, \dots, x_{n}, t) + d (t) \end{matrix}

(1)

where [x₁, x₂, …, x_n]^T is the state vector of the system, F(x₁, x₂, …x_n, t) is a nonlinear known function, b is a known constant, b ≠ 0, u is the control law to be designed, Δf(x₁, x₂, …, x_n, t) represents the model uncertainty, and d(t) is the external interference. The upper and lower boundaries of the sum of the uncertainty are D_max and D_min, that is

D_{min} \leq Δ f (x_{1}, x_{2}, \dots, x_{n}, t) + d (t) \leq D_{max}

(2)

The errors are defined as

{\begin{matrix} e_{1} = x_{d} - x_{1} \\ e_{2} = {\overset{\cdot}{x}}_{d} - x_{2} \\ \dots \\ e_{n} = x_{d}^{(n - 1)} - x_{n} \end{matrix}

(3)

where x_d is the desired signal. A suitable u(t) should be designed to make the system track the desired signal.

Improved dynamic sliding mode surface

For the system above, a global sliding mode surface is usually designed as

s = c_{1} \cdot e_{1} + c_{2} \cdot e_{2} + \dots + c_{n} \cdot e_{n} - H (t)

(4)

where the coefficient c_i satisfies the Hurwitz conditions, and the function H(t) satisfies the following three conditions:¹⁴

$H (0) = c_{1} \cdot e_{1} (0) + c_{2} \cdot e_{2} (0) + \dots + c_{n} \cdot e_{n} (0)$ ;

When t → ∞, H(t) → 0;

H(t) has the first-order derivative.

Therefore, H(t) is usually designed as a monotonic exponential decay function as follows

H (t) = H (0) \cdot e^{- kt}

(5)

Thus, the global sliding mode surface can be described as

s = c_{1} \cdot e_{1} + c_{2} \cdot e_{2} + \dots + c_{n} \cdot e_{n} - H (0) \cdot e^{- kt}

(6)

From equation (6), s(0) = 0. That is, any initial state of the system can remain on the sliding mode surface. Therefore, the reaching mode in the SMC is canceled. However, when H(t) ≠ 0, the sliding mode surface designed by equation (6) cannot directly go through the origin. Although the evolution speed of the sliding mode surface can be accelerated by adjusting the parameter k, the sliding mode surface cannot go through the origin in the finite time.

In order to accelerate evolution of the dynamic sliding mode surface, an improved dynamic sliding mode surface can be designed by changing the form of the function H(t). Suppose that the dynamic sliding mode surface evolves into the linear sliding mode surface at the finite given time t_z, then the function H(t) can be designed as

H (t) = {\begin{matrix} H (0) [\frac{(1 - C) e^{- α_{2} \cdot t_{z}} + C e^{- α \cdot t_{z}}}{e^{- α_{2} \cdot t_{z}} - e^{- α_{1} \cdot t_{z}}} e^{- α_{1} t} + \frac{- C e^{- α \cdot t_{z}} - (1 - C) e^{- α_{1} \cdot t_{z}}}{e^{- α_{2} \cdot t_{z}} - e^{- α_{1} \cdot t_{z}}} e^{- α_{2} t} + C e^{- α t}], t \leq t_{z} \\ 0, t > t_{z} \end{matrix}

(7)

It is easy to be proved that the function H(t) above satisfies the first condition in Lu and Chen.¹⁴ That is, when t = 0, H(t) = H(0). For the second condition in Lu and Chen,¹⁴ when t ≥ t_z, H(0) = 0. Therefore, the function H(t) in equation (7) can have faster decay rate than the function in equation (5).

Furthermore, in order to make H(t) have the first-order derivative at t = t_z, the left derivative must be equal to the right derivative at t = t_z, that is

H_{+}' (t_{z}) = H'_{-} (t_{z}) = 0

(8)

According to equation (7), the left derivative of H(t) can be calculated as

\begin{matrix} {H'}_{-} (t_{z}) \\ = \frac{[(1 - C) (α_{2} - α_{1}) e^{- (α_{1} + α_{2}) t_{z}} + C (α - α_{1}) e^{- (α + α_{1}) t_{z}} + C (α_{2} - α) e^{- (α + α_{2}) t_{z}}]}{H (0) (e^{- α_{2} \cdot t_{z}} - e^{- α_{1} \cdot t_{z}})} \end{matrix}

(9)

Thus, the constraint condition of the parameters can be described as

\begin{array}{l} (1 - C) (α_{2} - α_{1}) e^{- (α_{1} + α_{2}) t_{z}} \\ + C (α - α_{1}) e^{- (α + α_{1}) t_{z}} + C (α_{2} - α) e^{- (α + α_{2}) t_{z}} = 0, \\ α_{1} \neq α_{2} \end{array}

(10)

According to equation (10), when the parameters α₁, α₂, α, and C are selected, the evolution time t_z can be determined by the numerical method. Thus, the function H(t) also satisfies the third condition in Lu and Chen.¹⁴

For instance, set C = 0.5, α₁ = 150, α₂ = 100, and α = 10. From equation (7), H(t)/H(0) is shown in Figure 1. Thus, H(t) has a fast convergent rate. Therefore, the improved dynamic sliding mode surface can also rapidly evolved into the linear sliding mode surface at the time t_z, which can accelerate the dynamic response of the system.

Figure 1.

H(t)/H(0).

The function curve $H'_{-} (t_{z}) / H (0)$ can be obtained by numerical solution in Figure 2.

Figure 2.

$H'_{-} (t_{z}) / H (0)$ .

The value which makes the function $H'_{-} (t_{z})$ be zero at first time is determined as the best t_z. From Figure 2, the best t_z is about 0.011 s.

FGSMC law

The reaching law is designed as

\overset{\cdot}{s} = - ε \cdot sgn (s) - q \cdot s

(11)

where ε > 0, which indicates that the rate of the system reaching the switching surface s = 0, and the exponential reaching term, −qs, can guarantee that the system state can tend to the sliding mode with a large rate when s is bigger.

According to equations (4) and (11)

\begin{matrix} - ε \cdot sgn (s) - q \cdot s = c_{1} \cdot {\overset{\cdot}{e}}_{1} + c_{2} \cdot {\overset{\cdot}{e}}_{2} + \dots + c_{n} \cdot {\overset{\cdot}{e}}_{n} - \overset{\cdot}{H} \\ = c_{1} \cdot {\overset{\cdot}{x}}_{d} + c_{2} \cdot {\overset{\cdot\cdot}{x}}_{d} + \dots + c_{n} \cdot x_{d}^{(n)} \\ - c_{1} \cdot x_{2} - c_{2} \cdot x_{3} - \dots c_{n - 1} \cdot x_{n} \\ - c_{n} \cdot (F (x_{1}, x_{2}, \dots, x_{n}, t) + b \cdot u (t) \\ + Δ f (x_{1}, x_{2}, \dots, x_{n}, t) + d (t)) - \overset{\cdot}{H} \end{matrix}

(12)

Accordingly, the control law u(t) can be designed as

\begin{matrix} u (t) = \frac{1}{c_{n} \cdot b} {ε \cdot sgn (s) + q \cdot s + c_{1} \cdot {\overset{\cdot}{x}}_{d} + c_{2} \cdot {\overset{\cdot\cdot}{x}}_{d} + \\ \dots + c_{n} \cdot x_{d}^{(n)} - c_{1} \cdot x_{2} - c_{2} \cdot x_{3} - \dots - c_{n - 1} \cdot x_{n} - \\ c_{n} \cdot F (x_{1}, x_{2}, \dots, x_{n}, t) - c_{n} \cdot [Δ f (x_{1}, x_{2}, \dots, x_{n}, t) + d (t)] - \overset{\cdot}{H}} \end{matrix}

(13)

Because the term [Δf(x₁, x₂, …, x_n, t) + d(t)] is uncertainty, in order to ensure the reliability of the control law, the control law is designed as

\begin{matrix} u (t) = \frac{1}{c_{n} \cdot b} {ε \cdot sgn (s) + q \cdot s + c_{1} \cdot {\overset{\cdot}{x}}_{d} + c_{2} \cdot {\overset{\cdot\cdot}{x}}_{d} + \\ \dots + c_{n} \cdot x_{d}^{(n)} - c_{1} \cdot x_{2} - c_{2} \cdot x_{3} - \dots c_{n - 1} \cdot x_{n} - \\ c_{n} \cdot F (x_{1}, x_{2}, \dots, x_{n}, t) - c_{n} \cdot D_{c} - \overset{\cdot}{H}} \end{matrix}

(14)

where D_c is the boundary of the uncertainty

D_{c} = D_{2} - D_{1} \cdot sgn (s)

(15)

D_{1} = \frac{D_{max} - D_{min}}{2}

(16)

D_{2} = \frac{D_{max} + D_{min}}{2}

(17)

Analyzing the stability

Choose the Lyapunov function as

V = \frac{1}{2} s^{2}

(18)

The derivative of the Lyapunov function can be calculated as

\begin{array}{l} \dot{V} = s \cdot \dot{s} = s \cdot [- ε \cdot sgn (s) - q \cdot s \\ + D_{c} - Δ f (x_{1}, x_{2}, \dots, x_{n}, t) - d (t)] \end{array}

(19)

when s > 0

\dot{V} = s \cdot [- ε - q \cdot s + D_{\min} - Δ f (x_{1}, x_{2}, \dots, x_{n}, t) - d (t)] < 0

(20)

when s < 0

\overset{\cdot}{V} = s \cdot [ε - q \cdot s + D_{max} - Δ f (x_{1}, x_{2}, \dots, x_{n}, t) - d (t)] < 0

(21)

Therefore, the stability of the system can be proved.

Thus, it can be seen that the improved FGSMC can not only keep the merits of the original GSMC that the reaching mode can be canceled but also have a faster dynamic response.

Simulation analysis

Some simulation experiments are performed by GSMC and FGSMC to verify the feasibility of the method proposed in this article. A two-order uncertain nonlinear system is given as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - 25 \cdot {\overset{\cdot}{x}}_{1} + b \cdot u (t) + d (t) \end{matrix}

(22)

where b = 133, d(t) = 10 sin(πt), the desired signal is set as x_d = sin7t, and the initial state is set as [x₁(0), x₂(0)]^T = [−0.15, 0.15]^T.

In the simulation experiments, the parameters of the sliding mode surface are set as c₁ = 15 and c₂ = 1. The parameters in the reaching law are set as ε = 0.5 and q = 10. The parameters in H(t) are set as k = 150, α₁ = 150, α₂ = 100, and α = 10. The value t_z can be resolved as t_z = 0.011 s by the numerical method.

GSMC and FGSMC are separately used to control the system in equation (22), and their position and speed tracking results are shown in Figures 3 and 4.

Figure 3.

Tracking result obtained by GSMC: (a) position tracking and (b) speed tracking.

Figure 4.

Tracking result obtained by FGSMC: (a) position tracking and (b) speed tracking.

The comparison results of the two methods above are shown in Figure 5.

Figure 5.

The comparison results of the two methods: (a) position tracking, (b) speed tracking, and (c) the control input.

In Figures 3 –5, the solid line represents the given desired signal, and the dashed line and the dot dashed line represent the tracking results obtained by the GSMC and the FGSMC, respectively.

According to the tracking results above, it can be seen clearly that the convergence time of the improved FGSMC method proposed in this article can be reduced. That is, compared with the GSMC method, the dynamic sliding mode surface of fast global sliding model control method can quickly evolve into the linear sliding mode surface in the finite time. Therefore, the improved FGSMC method has a faster response rate.

Application instance

A quad-rotor unmanned helicopter is a high-order, nonlinear, strong-coupling complex system. A simplified structure model of the quad-rotor unmanned helicopter is described in Figure 6.

Figure 6.

Simplified quad-rotor unmanned helicopter model.

According to Zheng and Xiong,²² the simplified dynamic model of the quad-rotor unmanned helicopter can be described as

{\begin{matrix} \overset{\cdot\cdot}{x} = (\cos θ \cdot \sin φ \cdot \cos ψ + \sin θ \cdot \sin ψ) \cdot u_{1} - K_{1} \cdot \overset{\cdot}{x} / m \\ \overset{\cdot\cdot}{y} = (\cos θ \cdot \sin φ \cdot \sin ψ - \sin θ \cdot \cos ψ) \cdot u_{1} - K_{2} \cdot \overset{\cdot}{y} / m \\ \overset{\cdot\cdot}{z} = (\cos θ \cdot \cos φ) \cdot u_{1} - g - K_{3} \cdot \overset{\cdot}{z} / m \\ \overset{\cdot\cdot}{φ} = \frac{u_{3} \cdot l}{I_{y}} - \frac{K_{5} \cdot l}{I_{y}} \overset{\cdot}{φ} \\ \overset{\cdot\cdot}{θ} = \frac{u_{2} \cdot l}{I_{x}} - \frac{K_{4} \cdot l}{I_{x}} \overset{\cdot}{θ} \\ \overset{\cdot\cdot}{ψ} = \frac{u_{4}}{I_{z}} - \frac{K_{6}}{I_{z}} \cdot \overset{\cdot}{ψ} \end{matrix}

(23)

where (x, y, z) denotes the position of the center of the gravity of the quad-rotor unmanned helicopter. θ, φ, and ψ are the roll angle, the pitch angle, and the yaw angle, respectively. u_i represents the control law; l is the distance between the center of the rotor and the geometric center of the aircraft; I_x, I_y, and I_z are the inertias; m is the weight of the craft; g is the acceleration of gravity; and K_i is the resistance coefficient. The terms in equation (23) with the resistance coefficients are the frictional drag items which are uncertain. Therefore, there are some uncertainties in the model in equation (23), and FGSMC can be applied to control the system above.

The parameters of the quad-rotor unmanned helicopter are m = 1.4 kg, g = 9.8 kg/m², l = 0.2 m, I_x = 0.03 kg m²,I_y = 0.03 kg m²,I_z = 0.04 kg m²,x_d = 2,y_d = 1, z_d = 3, ψ_d = 0.01, K₁ = K₂ = K₃ = 0.01 N s/m, K₄ = K₅ = K₆ = 0.012 N s/m, and the boundary of the uncertainty D = 0.1. The parameters of the sliding mode surface are set as c₁ = 1, c₂ = 1, ε = 0.5, and q = 1. The parameters in H(t) are set as k = 150, α₁ = 150, α₂ = 140, and α = 10.

The above methods of GSMC and FGSMC are used to control the states of the quad-rotor unmanned helicopter and their simulation results are shown in Figure 7.

Figure 7.

Comparison of two methods: (a) the control result of the position x, (b) the control result of the position y, (c) the control result of the position z, (d) the control result of the roll angle, (e) the control result of the pitch angle, and (f) the control result of the yaw angle.

The control inputs of GSMC and FGSMC are shown in Figures 8 and 9, respectively.

Figure 8.

The control inputs of GSMC: (a) the control input u₁, (b) the control input u₂, (c) the control input u₃, and (d) the control input u₄.

Figure 9.

The control inputs of FGSMC: (a) the control input u₁, (b) the control input u₂, (c) the control input u₃, and (d) the control input u₄.

The dashed line and the dot dashed line represent the control results of GSMC and FGSMC, respectively. From Figures7 to 9, it can be clearly seen that FGSMC has a faster response rate than GSMC because the dynamic sliding mode surface can quickly evolve into the linear sliding mode surface in the finite time.

Conclusion

In this article, a new FGSMC method is proposed. The exponential decay function in the sliding surface is changed as an exponential bilateral decay function which can make the dynamic sliding mode surface evolve into the linear sliding mode surface in the finite time. The method can be applied to control the uncertain nonlinear system, for instance, the quad-rotor unmanned helicopter. Simulation results show that the FGSMC method has faster response rate than the conventional GSMC method.

Footnotes

Academic Editor: Nasim Ullah

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 was supported by the Tianjin Research Program of Application Foundation and Advanced Technology (14JCYBJC18900) and the National Natural Science Foundation of China (61203302).

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