Design of adaptive switching control for hypersonic aircraft

Abstract

This article proposes a novel adaptive switching control of hypersonic aircraft based on type-2 Takagi–Sugeno–Kang fuzzy sliding mode control and focuses on the problem of stability and smoothness in the switching process. This method uses full-state feedback to linearize the nonlinear model of hypersonic aircraft. Combining the interval type-2 Takagi–Sugeno–Kang fuzzy approach with sliding mode control keeps the adaptive switching process stable and smooth. For rapid stabilization of the system, the adaptive laws use a direct constructive Lyapunov analysis together with an established type-2 Takagi–Sugeno–Kang fuzzy logic system. Simulation results indicate that the proposed control scheme can maintain the stability and smoothness of switching process for the hypersonic aircraft.

Introduction

Hypersonic aircraft always switch flight control signals while flying in a large range of space and change their flight states continuously. It is, however, more difficult to steadily and smoothly control the switching processes for the aircraft, with their rapid time variation, strong nonlinearity, strong coupling, and model uncertainty.^1,2 Complex nonlinear systems cannot be well-controlled by means of traditional control methods, which will increase the difficulty of switching control. ³

The following are some methods that have been proposed to analyze and handle these problems. Stability of switched systems has been researched and analyzed in Lin and Antsaklis ⁴ and Zhang et al. ⁵ Zhang et al. ⁶ and Zhang ⁷ were concerned with the filter design problem for a class of switched system with average dwell time (ADT) switching. They were also concerned with $H_{\infty }$ filtering for a class of switched linear systems in discrete-time domain. ⁸ A logic-based supervisor has been designed to manage the switching moments between two main controllers. ⁹ Inverse dynamics has generated the trajectory for transition from vertical takeoff and landing (VTOL) to wing-borne flight. ¹⁰ A class of “policy holding” strategies have been proposed in the presence of measurement noise in Garone et al. ¹¹ $H_2$ and $H_{\infty }$ were usually used to handle control problems. ¹² A generalized $H_2$ state feedback and dynamic output-feedback controllers have been designed for a class of linear discrete-time hybrid systems with time-varying delays. ¹³ A switched linear parameter-varying approach has been designed for rigid longitudinal dynamics of a hypersonic aircraft under the ADT and mode-dependent average dwell time (MDADT) switching logics which can ensure the tracking of given commands from the closed-loop system in control of hypersonic aircraft. ¹⁴ An extended bump-less switching technique has been proposed to offer an optimal control for continuous-time systems of hypersonic aircraft via linear quadratic optimal control and internal model principle. ¹⁵ A switching fault-tolerant control approach has been applied to an air-breathing hypersonic vehicle subject to time-varying actuator and sensor faults. ¹⁶ The piece-wise linear Lyapunov functions have been used to investigate the stabilizing switching control laws for discrete- and continuous-time linear systems. ¹⁷ An adaptive switching learning proportional–derivative (PD) control has been proposed for trajectory tracking of robot manipulators in an iterative operation mode. ¹⁸ Mahmoud and Al-Rayyah ¹⁹ and Mahmoud ²⁰ established a new robust delay-dependent adaptive control design method for a class of nominally linear continuous-time systems with time-varying delays and unknown nonlinearities.

Interval type-2 Takagi–Sugeno–Kang (TSK) fuzzy,^21,22 as a novel kind of fuzzy, has been proved to be more suitable to deal with uncertainty problems^23,24 and has been widely used in control processes. Traditional type-1 TSK fuzzy logic system ²⁵ has been used to resolve the control issue.^26,27 It was often combined with some other methods to achieve adaptive and robust control of systems.^28,29 However, as the system becomes more complex, especially for nonlinear systems, the exact fuzzy sets and membership functions are no longer enough to accomplish the task. Thus, type-2 TSK fuzzy logic controller arises and has undergone rapid development because of its ability to deal with problems of uncertainty more effectively.

Recently, type-2 TSK fuzzy has also been studied for controllers of nonlinear systems. An interval type-2 TSK fuzzy has been proposed to confront uncertainties for the supervisory adaptive tracking control of robot manipulators in the robot dynamics. ³⁰ The structure of a type-2 TSK fuzzy neural system has been presented to handle uncertainties for identification and control of time-varying plants. ³¹

This article proposes an adaptive switching control of hypersonic aircraft based on type-2 TSK fuzzy sliding mode control. The proposed method is similar to the scheme for single-input and single-output (SISO) system in Hwang et al., ³² but the biggest difference between the proposed scheme and the reference is that the control plant in this article is a hypersonic aircraft, which has a multi-input and multi-output system of nonlinearity, variation, and instability with uncertain parameters.

The outline of this article is as follows: section “Background” presents the model structures of hypersonic aircraft and full-state feedback linearization as background. Section “Design of switching control” documents the design of the novel adaptive switching control in switching process. In section “Simulation of controller”, simulation studies are given to verify and analyze the effectiveness of the new switching controller. Finally, we present our conclusions in section “Conclusion”.

Background

The model of hypersonic aircraft

The hypersonic aircraft has features of rapid time variation, strong nonlinearity, strong coupling, and model uncertainty. The three views of X-24B configuration are shown in Figure 1.

Figure 1.

Three views of X-24B configuration.

Using the longitudinal force and moment equilibrium of hypersonic aircraft, the longitudinal model of hypersonic aircraft ³³ can be obtained as follows

$$\stackrel{·}{V}=\frac{T\cos \alpha -D}{m}-\frac{\mu \sin \gamma }{r^2}$$

(1)

$$\stackrel{·}{\gamma }=\frac{L+T\sin \alpha }{\mathrm{mV}}-\frac{(\mu -V^{2}r)\cos \gamma }{V{r}^2}$$

(2)

$$\stackrel{·}{q}=\frac{M_y}{I_y}$$

(3)

$$\stackrel{·}{\alpha }=q-\stackrel{·}{\gamma }$$

(4)

$$\stackrel{·}{h}=V\sin \gamma$$

(5)

$$\stackrel{·}{m}=-\frac{T}{I_{\mathrm{sp}}}$$

(6)

$$\stackrel{··}{\beta }=-2\xi \omega \stackrel{·}{\beta }-{\omega }^2\beta +{\omega }^2{\beta }_c$$

(7)

In this model, $V$ , $\gamma$ , $q$ , $\alpha$ , and $h$ are the velocity of hypersonic, flight path angle, pitch rate, angle of attack, and altitude, respectively; $\beta$ , $\omega$ , and $\xi$ are the throttle setting, natural frequency, and damping coefficient, respectively; $m$ , $\mu$ , $r$ , $M_y$ , and $I_y$ are the mass, gravitational constant, radial distance from Earth’s center, pitching moment, and moment of inertia, respectively. For the lift $L$ , drag $D$ , and thrust $T$ , there are

$$L=\frac{1}{2}\rho V^{2}s{C}_L$$

(8)

$$D=\frac{1}{2}\rho V^{2}s{C}_D$$

(9)

$$T=\frac{1}{2}\rho V^{2}s{C}_T$$

(10)

$$r=h+R_E$$

(11)

where $C_L$ , $C_D$ , and $C_T$ are the lift coefficient, drag coefficient, and thrust coefficient, respectively; $\rho$ , $s$ , and $R_e$ are the density of air, reference area, and radius of the earth, respectively.

Uncertain parameters as an additional variance to the nominal values are used for controller design

$$m=m_0(1+\Delta m)$$

(12)

$$I_y=I_{y0}(1+\Delta I_y)$$

(13)

$$s=s_0(1+\Delta s)$$

(14)

$$\overline{c}={\overline{c}}_0(1+\Delta c)$$

(15)

$$c_e=c_{e0}(1+\Delta c_e)$$

(16)

$$\rho ={\rho }_0(1+\Delta \rho )$$

(17)

Full-state feedback linearization

For a longitudinal nonlinear model of a hypersonic aircraft, the third derivative of $V$ and the fourth derivative of $h$ in equations (1)–(5) can be obtained as follows

$$\{\begin{array}{c}\hfill \stackrel{·}{V}=\frac{T\cos \alpha -D}{m}-\frac{\mu \sin \gamma }{r^2}=f_1(x)\hfill \\ \multicolumn{1}{c}{\stackrel{··}{V}=\frac{\partial f_1(x)}{\partial x}\stackrel{·}{x}={\omega }_1\stackrel{·}{x}}\\ \multicolumn{1}{c}{\stackrel{\dots }{V}={\stackrel{·}{x}}^T\frac{\partial {\omega }_1}{\partial x}\stackrel{·}{x}+{\omega }_1\stackrel{··}{x}={\omega }_1\stackrel{··}{x}+{\stackrel{·}{x}}^T{\omega }_2\stackrel{·}{x}}\end{array}$$

(18)

$$\{\begin{array}{c}\hfill \stackrel{·}{h}=V\sin \gamma \hfill \\ \multicolumn{1}{c}{\stackrel{··}{h}=\stackrel{·}{V}\sin \gamma +V\cos \gamma ·\stackrel{·}{\gamma }}\\ \multicolumn{1}{c}{\stackrel{\dots }{h}=\stackrel{··}{V}\sin \gamma +2\stackrel{·}{V}\stackrel{·}{\gamma }\cos \gamma -V{\stackrel{·}{\gamma }}^2\sin \gamma +V\stackrel{··}{\gamma }\cos \gamma }\\ \multicolumn{1}{c}{h^{(4)}=\stackrel{\dots }{V}\sin \gamma +3\stackrel{··}{V}\cos \gamma ·\stackrel{·}{\gamma }-3\stackrel{·}{V}\sin \gamma ·{\stackrel{·}{\gamma }}^2}\\ \multicolumn{1}{c}{+3\stackrel{·}{V}\cos \gamma ·\stackrel{··}{\gamma }-V\cos \gamma ·{\stackrel{·}{\gamma }}^3}\\ \multicolumn{1}{c}{-3V\sin \gamma ·\stackrel{·}{\gamma }·\stackrel{··}{\gamma }+V\cos \gamma ·\stackrel{\dots }{\gamma }}\end{array}$$

(19)

where the first, second, and third derivatives of $\gamma$ are obtained according to equation (2) as

$$\{\begin{array}{c}\hfill \stackrel{·}{\gamma }=\frac{L+T\sin \alpha }{\mathrm{mV}}-\frac{(\mu -V^{2}r)\cos \gamma }{V{r}^2}=f_2(x)\hfill \\ \multicolumn{1}{c}{\stackrel{··}{\gamma }=\frac{\partial f_2(x)}{\partial x}\stackrel{·}{x}={\pi }_1\stackrel{·}{x}}\\ \multicolumn{1}{c}{\stackrel{\dots }{\gamma }={\stackrel{·}{x}}^T\frac{\partial {\pi }_1}{\partial x}\stackrel{·}{x}+{\pi }_1\stackrel{··}{x}={\pi }_1\stackrel{··}{x}+{\stackrel{·}{x}}^T{\pi }_2\stackrel{·}{x}}\end{array}$$

(20)

In equations (18) and (20), $x=[\begin{array}{ccccc}\hfill V\hfill & \hfill \gamma \hfill & \hfill \alpha \hfill & \hfill \beta \hfill & \hfill h\hfill \end{array}{]}^T$ , ${\omega }_1=\partial f_1(x)/\partial x$ , ${\omega }_2=\partial {\omega }_1/\partial x$ , ${\pi }_1=\partial f_2(x)/\partial x$ , and ${\pi }_2=\partial {\pi }_1/\partial x$ . Therefore, the linearized model is obtained as follows

$$\left[\begin{array}{c}\hfill \stackrel{\dots }{V}\hfill \\ \multicolumn{1}{c}{h^{(4)}}\end{array}\right]=\left[\begin{array}{c}\hfill f_V\hfill \\ \multicolumn{1}{c}{f_h}\end{array}\right]+\left[\begin{array}{cc}\hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& \hfill b_{22}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\beta }_c\hfill \\ \multicolumn{1}{c}{{\delta }_e}\end{array}\right]$$

(21)

where

$$f_V=\frac{({\omega }_1{\stackrel{··}{x}}_0+\stackrel{·}{x}{\omega }_2\stackrel{·}{x})}{m}$$

(22)

$$\begin{array}{cc}\multicolumn{1}{c}{f_h=}& 3\stackrel{··}{V}\stackrel{·}{\gamma }\cos \gamma -3\stackrel{·}{V}{\stackrel{·}{\gamma }}^2\sin \gamma +3\stackrel{·}{V}\stackrel{··}{\gamma }\cos \gamma \hfill \\ \multicolumn{1}{c}{}& -3V\stackrel{·}{\gamma }\stackrel{··}{\gamma }\sin \gamma -V{\stackrel{·}{\gamma }}^3\cos \gamma \hfill \\ \multicolumn{1}{c}{}& +({\omega }_1{\stackrel{··}{x}}_0+{\stackrel{·}{x}}^T{\omega }_2\stackrel{·}{x})\sin \gamma /m\hfill \\ \multicolumn{1}{c}{}& +V\cos \gamma ({\pi }_1{\stackrel{··}{x}}_0+{\stackrel{·}{x}}^T{\pi }_2\stackrel{·}{x})\hfill \end{array}$$

(23)

$$b_{11}=\frac{\rho V^{2}s·C_{\beta }{\omega }^2}{2m}\cos \alpha$$

(24)

$$b_{12}=-\frac{\rho V^{2}s\overline{c}·c_e}{2m{I}_y}(T\sin \alpha +D_{\alpha })$$

(25)

$$b_{21}=\frac{\rho V^{2}s{C}_{\beta }{\omega }^2}{2m}\sin (\alpha +\gamma )$$

(26)

$$b_{22}=\frac{\rho V^{2}s\overline{c}·c_e}{2m{I}_y}[T\cos (\alpha +\gamma )+L_{\alpha }\cos \gamma -D_{\alpha }\sin \gamma ]$$

(27)

where $D_{\alpha }=\partial D/\partial \alpha ,L_{\alpha }=\partial L/\partial \alpha ,T_{\alpha }=\partial T/\partial \alpha ,\mathrm{and}{C}_{\beta }=\partial C_T/\partial \beta$ .

The nominal control law is as follows

$$U=\left[\begin{array}{c}\hfill {\beta }_c\hfill \\ \multicolumn{1}{c}{{\delta }_e}\end{array}\right]={\left[\begin{array}{cc}\hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& \hfill b_{22}\hfill \end{array}\right]}^{-1}·\left[\begin{array}{c}\hfill \stackrel{\dots }{V}-f_V\hfill \\ \multicolumn{1}{c}{\stackrel{⃜}{h}-f_h}\end{array}\right]$$

(28)

Design of switching control

We can get different control signals $U_1,U_2,\dots$ in different conditions. $U_1$ is the control signal under condition 1, $U_2$ is the control signal under condition 2, and so on. A hypersonic aircraft switches condition 1 to condition 2 at $t_1$ , and steady work in condition 2 at $t_2$ , while the switching of control signals takes the transition time. The control law in the whole process is

$$U=\{\begin{array}{cc}{U}_1& t\le t_1\\ U_t& t_1<t<t_2\\ U_2& t\ge t_2\end{array}$$

(29)

where $U_t$ is the switching control signal, which is in the form of

$$U_t=a*U_1+b*U_2$$

(30)

The system structure of switching control is shown in Figure 2.

Figure 2.

System structure of switching control.

As Figure 2 shows, for the block of switching control, the system structure of adaptive switching control based on type-2 TSK fuzzy sliding mode control is shown in Figure 3.

Figure 3.

System structure of adaptive switching control based on type-2 TSK fuzzy sliding mode control.

As Figure 3 shows, control signals $U_1,U_2$ are the inputs of type-2 TSK fuzzy controller, respectively; ${\hat{A}}_{{\beta }_c},{\hat{A}}_{{\delta }_e}$ , produced by adaptive laws, are used for type-2 TSK fuzzy controller, which made the system become more stable and adapt to the environment more quickly; the variable values ${\hat{E}}_{{\beta }_c},{\hat{E}}_{{\delta }_e}$ , estimated by the estimator, are used for the compensate controller, which compensate the difference between optimal control signal and type-2 TSK fuzzy sliding mode control signal.

Interval type-2 TSK fuzzy

The interval type-2 fuzzy, as a special case of type-2 fuzzy, is similar to type-1 fuzzy structure. The biggest difference between type-1 and type-2 fuzzy is the type reducer. The system structure of interval type-2 fuzzy is shown in Figure 4.

Figure 4.

Structure of an interval type-2 fuzzy logic system (IT2 FLS).

In Figure 4, for the group of input–output data, there are two inputs, $x_1\in X_1,x_2\in X_2$ , and one output, $y\in Y$ . In a type-2 TSK fuzzy logic (A2-C0) case, ³⁴ IF-THEN rules are also used to describe the relationship between input data and output data. In type-2 TSK fuzzy rules, the jth rule can be expressed as

$R^j$ If $x_1$ is ${{\tilde{F}}_1}^j$ and $x_2$ is ${{\tilde{F}}_2}^j$ and ⋯ and $x_n$ is ${{\tilde{F}}_n}^j$ , then $y^j=c_0^j+c_1^j{x}_1+c_2^j{x}_2+\cdots +c_n^j{x}_n$

where $j=1,\dots ,M$ , ${\tilde{F}}_i^j$ are type-2 antecedent fuzzy sets which have upper boundary and lower boundary, and $c_i^j(i=0,1,\dots ,n)$ are the consequent parameters.

We use Gaussian primary membership function with an uncertain mean as the input membership function

$${\mu }_i^j(x_i)=\exp [-\frac{1}{2}{\left(\frac{x_i-m_i^j}{{\sigma }_i^j}\right)}^2]m_i^j\in [m_{i1}^j,m_{i2}^j]$$

(31)

where $i=1,\dots ,n$ and $j=1,\dots ,M$ . The upper membership function ${\overline{\mu }}_i^j(x_i)$ and the lower membership function ${\underset{¯}{\mu }}_i^j(x_i)$ are as follows

$${\overline{\mu }}_i^j(x_i)=\{\begin{array}{c}\hfill N(m_{i1}^j,{\sigma }_i^j;x_i)x_i<m_{i1}^j\hfill \\ \multicolumn{1}{c}{1m_{i1}^j<x_i<m_{i2}^j}\\ \multicolumn{1}{c}{N(m_{i2}^j,{\sigma }_i^j;x_i)x_i>m_{i2}^j}\end{array}$$

(32)

$${\underset{¯}{\mu }}_i^j(x_i)=\{\begin{array}{c}\hfill N(m_{i2}^j,{\sigma }_i^j;x_i)x_i\le \frac{m_{i1}^j+m_{i2}^j}{2}\hfill \\ \multicolumn{1}{c}{N(m_{i1}^j,{\sigma }_i^j;x_i)x_i>\frac{m_{i1}^j+m_{i2}^j}{2}}\end{array}$$

(33)

$${\mu }_{{\tilde{F}}_i^j}(x_i)=[{\overline{\mu }}_{{\tilde{F}}_i^j}(x_i),{\underset{¯}{\mu }}_{{\tilde{F}}_i^j}(x_i)]i=1,\dots ,n$$

(34)

The secondary membership function ${\mu }_{{\tilde{F}}_i^j}(x_i)$ is expressed as

$${\mu }_{{\tilde{F}}_i^j}(x_i)=\underset{w^j\in [{\overline{\mu }}_{{\tilde{F}}_i^j}(x_i),{\underset{¯}{\mu }}_{{\tilde{F}}_i^j}(x_i)]}{\int }\frac{1}{w^j}$$

(35)

$$F^j=\underset{x\in X}{\cup }\left[\underset{i=1}{\overset{n}{\cap }}{\mu }_{{\tilde{F}}_i^j(x_i)}\right]$$

(36)

where ${\mu }_{{{\tilde{F}}_i}^j}$ are the interval sets. ( $j=1,\dots ,M$ ; $i=0,1,\dots ,n$ ). $F^j$ , which joins the results from meet operations, can be an interval type-1 set as

$$F^j=[{\underset{¯}{f}}^j,{\overline{f}}^j{]}^T$$

(37)

where

$$\{\begin{array}{c}\hfill {\underset{¯}{f}}^j={\underset{¯}{\mu }}_{{\tilde{F}}_1^j(x_1)}*\cdots *{\underset{¯}{\mu }}_{{\tilde{F}}_n^j(x_n)}\hfill \\ \multicolumn{1}{c}{{\overline{f}}^j={\overline{\mu }}_{{\tilde{F}}_1^j(x_1)}*\cdots *{\overline{\mu }}_{{\tilde{F}}_n^j(x_n)}}\end{array}$$

(38)

For the type reducer in Figure 4, we use the center-of-set type reducer ³⁵

$$y_{\cos }=[y_l,y_r]=\underset{f^1\in [{\underset{¯}{f}}^1,{\overline{f}}^1]}{\int }\cdots \underset{f^M\in [{\underset{¯}{f}}^M,{\overline{f}}^M]}{\int }\frac{1}{\frac{\sum _{j=1}^M{f}^j{y}^j}{\sum _{j=1}^M{f}^j}}$$

(39)

In the A2-C0 case, $y_l^j=y_r^j=y^j$ . Consequently, the de-fuzzified crisp output $y_{\mathrm{TSK},2}$ is

$$y_{\mathrm{TSK},2}(x)=\frac{\sum _{j=1}^M{f}^j(\sum _{i=1}^p{c}_i^j{x}_i+c_0^j)}{\sum _{j=1}^M{f}^j}$$

(40)

where $c_i^j$ , $i=0,1,\dots n$ are the consequent parameters.

Type-2 TSK fuzzy sliding mode control

In the switching process, we define the sliding mode variables as

$$\{\begin{array}{c}\hfill S_V={(d/\mathrm{dt}+{\lambda }_V)}^2{e}_V(t)=\stackrel{··}{V}-\stackrel{··}{V}d+2{\lambda }_V{\stackrel{·}{e}}_V+{\lambda }_V^2{e}_V\hfill \\ \multicolumn{1}{c}{S_h={(d/\mathrm{dt}+{\lambda }_h)}^3{e}_h(t)=\stackrel{\dots }{h}-{\stackrel{\dots }{h}}_d+3{\lambda }_h{\stackrel{··}{e}}_h+3{\lambda }_h^2{\stackrel{·}{e}}_h+{\lambda }_h^3{e}_h}\end{array}$$

(41)

Then, we can obtain the derivatives of $S_V$ and $S_h$ and equate them to 0 to define the sliding mode surfaces

$$\{\begin{array}{c}\hfill {\stackrel{·}{S}}_V=\stackrel{\dots }{V}-{\stackrel{\dots }{V}}_d+2{\lambda }_V{\stackrel{··}{e}}_V+{\lambda }_V^2{\stackrel{·}{e}}_V=0\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{S}}_h=\stackrel{....}{h}-{\stackrel{....}{h}}_d+3{\lambda }_h{\stackrel{\dots }{e}}_h+3{\lambda }_h^2{\stackrel{··}{e}}_h+{\lambda }_h^3{\stackrel{·}{e}}_h=0}\end{array}$$

(42)

We define $U^{*}=[{\beta }_c^{*},{\delta }_e^{*}{]}^T$ as the value of the control signal $U=[{\beta }_c,{\delta }_e{]}^T$ in equation (28) that achieves $\stackrel{·}{S}=0$ in equation (42)

$$\begin{array}{cc}\multicolumn{1}{c}{}& \left[\begin{array}{c}\hfill {\stackrel{\dots }{V}}_d-2{\lambda }_V{\stackrel{··}{e}}_V-{\lambda }_V^2{\stackrel{·}{e}}_V\hfill \\ \multicolumn{1}{c}{h_d^{(4)}-3{\lambda }_h{\stackrel{\dots }{e}}_h-3{\lambda }_h^2{\stackrel{··}{e}}_h-{\lambda }_h^3{\stackrel{·}{e}}_h}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& =\left[\begin{array}{cc}\hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& \hfill b_{22}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {{\beta }_c}^{*}\hfill \\ \multicolumn{1}{c}{{{\delta }_e}^{*}}\end{array}\right]+\left[\begin{array}{c}\hfill f_V\hfill \\ \multicolumn{1}{c}{f_h}\end{array}\right]\hfill \end{array}$$

(43)

The output of type-2 TSK fuzzy control is $U_f$ , which we propose to estimate the ideal control signal $U^{*}$ .

We use the Gaussian primary membership function with uncertain mean as the input membership function and the output membership function. We describe the ith rule of interval type-2 TSK fuzzy logic system as

Rule i: If $U_1$ is ${\tilde{F}}_1^i$ , and $U_2$ is ${\tilde{F}}_2^i$ , then $U_f$ is $c_1^i{U}_1+c_2^i{U}_2$

where $U_1$ is the control signal under condition 1 and $U_2$ is the control signal under condition 2; ${\tilde{F}}_1^i$ and ${\tilde{F}}_2^i$ are the label of the interval type-2 fuzzy set; $U_f$ stands for the output of type-2 fuzzy; and $c_1^i$ and $c_2^i$ are the consequent parameters.

Define $C^i=[c_1^i,c_2^i]$ , $U=[U_1,U_2]$ , then

$${\alpha }^i=c_1^i{U}_1+c_2^i{U}_2=C^i{U}^T$$

(44)

From equations (37)–(40), we establish the control output as

$$\begin{array}{cc}\multicolumn{1}{c}{}& U_f=\hfill \\ \multicolumn{1}{c}{}& \frac{1}{2}(\frac{\sum _{i=1}^L{{\overline{f}}_s}^i{\alpha }^i+\sum _{i=L+1}^M{\underset{¯}{f}}_s^i{\alpha }^i}{\sum _{i=1}^L{{\overline{f}}_s}^i+\sum _{i=L+1}^M{{\underset{¯}{f}}_s}^i}+\frac{\sum _{i=1}^R{{\underset{¯}{f}}_s}^j{\alpha }^i+\sum _{i=R+1}^M{{\overline{f}}_s}^i{\alpha }^i}{\sum _{i=1}^R{\underset{¯}{f}}_s^j+\sum _{i=R+1}^M{{\overline{f}}_s}^i}\hfill \end{array})$$

(45)

where ${\underset{¯}{f}}_s^i$ and ${\overline{f}}_s^i$ stand for the lower and upper membership functions of ${\tilde{F}}_s^i$ , respectively.

Consequently, the control output is

$$U_f=\frac{1}{2}(\sum _{i=1}^M{\alpha }^i{\xi }_l^i+\sum _{i=1}^M{\alpha }^i{\xi }_r^i)=\frac{1}{2}(A^T{\xi }_l+A^T{\xi }_r)=A^T\xi$$

(46)

where $\xi =[(1/2){\xi }_l^T,(1/2){\xi }_r^T{]}^T$ and

$${\xi }_l^i=\frac{h_l^i}{{\displaystyle \sum _{i=1}^L{\overline{f}}_s^i}+{\displaystyle \sum _{i=L+1}^M{\underset{¯}{f}}_s^i}}$$

(47)

in which

$$h_l^i=\{\begin{array}{c}\hfill {\overline{f}}_s^i,i=1,\dots ,L\hfill \\ \multicolumn{1}{c}{{\underset{¯}{f}}_s^i,i=L+1,\dots ,M}\end{array}$$

(48)

$${\xi }_r^i=\frac{h_r^i}{\sum _{i=1}^R{\overline{f}}_s^i+\sum _{i=R+1}^M{\underset{¯}{f}}_s^i}$$

(49)

$$h_r^i=\{\begin{array}{c}\hfill {\overline{f}}_s^i,i=1,\dots ,R\hfill \\ \multicolumn{1}{c}{{\underset{¯}{f}}_s^i,i=R+1,\dots ,M}\end{array}$$

(50)

For the control signal $U=[{\beta }_c,{\delta }_e{]}^T$ , interval type-2 TSK fuzzy is used in the channel of ${\beta }_c$ and ${\delta }_e$ , separately, where ${\beta }_{c1},{\beta }_{c2}$ and ${\delta }_{e1},{\delta }_{e2}$ are their respective inputs. $U_f=[{\beta }_{\mathrm{cf}},{\delta }_{\mathrm{ef}}{]}^T$ is the output of interval type-2 TSK fuzzy.

According to the universal approximation theorem, $U_f$ has an optimal resolution $U_f^{*}$ . Therefore

$$U^{*}=U_f^{*}+\epsilon =A^{*T}\xi +\epsilon$$

(51)

where $\epsilon$ is the approximation error satisfying $\left|{\epsilon }_i\right|<E_i$ for $i=1,2,\dots$ . $\hat{E}$ is the estimated bound value of the approximation error

$$\overline{E}=\hat{E}-E$$

(52)

Applying ${\hat{U}}_f$ to approximate $U^{*}$ , where $A^{*}$ is estimated by $\hat{A}$ , there is

$${\hat{U}}_f={\hat{A}}^T\xi$$

(53)

Therefore, the adaptive switching control signal of type-2 TSK fuzzy sliding mode control can be expressed as

$$U={\hat{U}}_f+U_n$$

(54)

where ${\hat{U}}_f=[{\hat{\beta }}_{\mathrm{cf}},{\hat{\delta }}_{\mathrm{ef}}{]}^T$ , as the main control signal to mimic the sliding mode control signal $U^{*}$ ; $U_n=[{\beta }_{\mathrm{cn}},{\delta }_{\mathrm{en}}{]}^T$ compensates the difference between optimal control signal and type-2 TSK fuzzy sliding mode control signal.

Stability analysis

According to equations (51) and (53), ${\overline{U}}_f$ is defined as

$${\overline{U}}_f={\hat{U}}_f-U^{*}={\hat{U}}_f-U_f^{*}-\epsilon ={\overline{A}}^T\xi -\epsilon$$

(55)

in which $\overline{A}=\hat{A}-A^{*}$ .

Therefore, equation (21) can be rewritten as

$$\left[\begin{array}{c}\hfill \stackrel{\dots }{V}\hfill \\ \multicolumn{1}{c}{h^{(4)}}\end{array}\right]=\left[\begin{array}{cc}\hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& \hfill b_{22}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\hat{\beta }}_{\mathrm{cf}}+{\beta }_{\mathrm{cn}}\hfill \\ \multicolumn{1}{c}{{\hat{\delta }}_{\mathrm{ef}}+{\delta }_{\mathrm{en}}}\end{array}\right]+\left[\begin{array}{c}\hfill f_V\hfill \\ \multicolumn{1}{c}{f_h}\end{array}\right]$$

(56)

From equations (42), (43) and (56), we obtain

$$\left[\begin{array}{c}\hfill {\stackrel{·}{S}}_V\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{S}}_h}\end{array}\right]=\left[\begin{array}{cc}\hfill b_{11}\hfill & \hfill b_{12}\hfill \\ \multicolumn{1}{c}{b_{21}}& \hfill b_{22}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\hat{\beta }}_{\mathrm{cf}}+{\beta }_{\mathrm{cn}}-{{\beta }_c}^{*}\hfill \\ \multicolumn{1}{c}{{\hat{\delta }}_{\mathrm{ef}}+{\delta }_{\mathrm{en}}-{{\delta }_e}^{*}}\end{array}\right]$$

(57)

In order to prove the whole system stability, the Lyapunov stability theory is adopted. To make the sliding mode variables $S$ , ${\overline{A}}_{{\beta }_c}$ , ${\overline{A}}_{{\delta }_e}$ , ${\overline{E}}_{{\beta }_c}$ , and ${\overline{E}}_{{\delta }_e}$ approach 0, we define the Lyapunov function as

$$\begin{array}{cc}\multicolumn{1}{c}{V=}& \frac{1}{2}{S}^{T}S+\frac{1}{2{\eta }_1}{\overline{A}}_{{\beta }_c}^T{\overline{A}}_{{\beta }_c}+\frac{1}{2{\eta }_2}{\overline{A}}_{{\delta }_e}^T{\overline{A}}_{{\delta }_e}\hfill \\ \multicolumn{1}{c}{}& +\frac{1}{2{\eta }_3}{\overline{E}}_{{\beta }_c}^T{\overline{E}}_{{\beta }_c}+\frac{1}{2{\eta }_4}{\overline{E}}_{{\delta }_e}^T{\overline{E}}_{{\delta }_e}\hfill \end{array}$$

(58)

where $S=[S_V,S_h{]}^T$ , $\overline{E}=[{\overline{E}}_{{\beta }_c},{\overline{E}}_{{\delta }_e}{]}^T$ , $\epsilon =[{\epsilon }_{{\beta }_c},{\epsilon }_{{\delta }_e}{]}^T$ , ${\eta }_1$ , ${\eta }_2$ , ${\eta }_3$ , and ${\eta }_4$ are all positive constants.

The derivative of equation (58) with respect to time is obtained as

$$\begin{array}{c}\stackrel{.}{V}=S^T\stackrel{.}{S}+\frac{1}{{\eta }_1}{\overline{A}}_{{\beta }_c}^T{\stackrel{.}{\overline{A}}}_{{\beta }_c}+\frac{1}{{\eta }_2}{\overline{A}}_{{\delta }_e}^T{\stackrel{.}{\overline{A}}}_{{\delta }_e}+\frac{1}{{\eta }_3}{\overline{E}}_{{\beta }_c}^T{\stackrel{.}{\overline{E}}}_{{\beta }_c}+\frac{1}{{\eta }_4}{\overline{E}}_{{\delta }_e}^T{\stackrel{.}{\overline{E}}}_{{\delta }_e}\\ =[S_V,S_h]\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]\left[\begin{array}{l}{\widehat{\beta }}_{cf}+{\beta }_{cn}-{{\beta }_c}^{*}\\ {\widehat{\delta }}_{ef}+{\delta }_{en}-{{\delta }_e}^{*}\end{array}\right]+\frac{1}{{\eta }_1}{\overline{A}}_{{\beta }_c}^T{\stackrel{.}{\overline{A}}}_{{\beta }_c}+\frac{1}{{\eta }_2}{\overline{A}}_{{\delta }_e}^T{\stackrel{.}{\overline{A}}}_{{\delta }_e}+\frac{1}{{\eta }_3}({\widehat{E}}_{{\beta }_c}-E_{{\beta }_c}){\stackrel{.}{\widehat{E}}}_{{\beta }_c}+\frac{1}{{\eta }_4}({\widehat{E}}_{{\delta }_e}-E_{{\delta }_e}){\stackrel{.}{\widehat{E}}}_{{\delta }_e}\\ ={\overline{A}}_{{\beta }_c}^T\left[(S_V{b}_{11}+S_h{b}_{21}){\xi }_{{\beta }_c}+\frac{1}{{\eta }_1}{\stackrel{.}{\overline{A}}}_{{\beta }_c}\right]+{\overline{A}}_{{\delta }_e}^T\left[(S_V{b}_{12}+S_h{b}_{22}){\xi }_{{\delta }_e}+\frac{1}{{\eta }_2}{\stackrel{.}{\overline{A}}}_{{\delta }_e}\right]+(S_V{b}_{11}+S_h{b}_{21})({\beta }_{cn}-{\epsilon }_{{\beta }_c})\\ +(S_V{b}_{12}+S_h{b}_{22})({\delta }_{en}-{\epsilon }_{{\delta }_e})+\frac{1}{{\eta }_3}({\widehat{E}}_{{\beta }_c}-E_{{\beta }_c}){\stackrel{.}{\widehat{E}}}_{{\beta }_c}+\frac{1}{{\eta }_4}({\widehat{E}}_{{\delta }_e}-E_{{\delta }_e}){\stackrel{.}{\widehat{E}}}_{{\delta }_e}\end{array}$$

(59)

To achieve $\stackrel{·}{V}\le 0$ , we design the adaptive laws as

$${\stackrel{.}{\overline{A}}}_{{\beta }_c}={\stackrel{.}{\widehat{A}}}_{{\beta }_c}=-{\eta }_1(S_V{b}_{11}+S_h{b}_{21}){\xi }_{{\beta }_c}$$

(60)

$${\stackrel{·}{C}}_{{\beta }_c}=(-{\eta }_1(S_V{b}_{11}+S_h{b}_{21}){\xi }_{{\beta }_c}){({\beta }_c{\beta }_c^T)}^{-1}{\beta }_c$$

(61)

$${\stackrel{.}{\overline{A}}}_{{\delta }_e}={\stackrel{.}{\widehat{A}}}_{{\delta }_e}=-{\eta }_2(S_V{b}_{12}+S_h{b}_{22}){\xi }_{{\delta }_e}$$

(62)

$${\stackrel{·}{C}}_{{\delta }_e}=(-{\eta }_2(S_V{b}_{12}+S_h{b}_{22}){\xi }_{{\delta }_e}){({\delta }_e{\delta }_e^T)}^{-1}{\delta }_e$$

(63)

$${\stackrel{.}{\widehat{E}}}_{{\beta }_c}={\eta }_3|S_V{b}_{11}+S_h{b}_{21}|$$

(64)

$${\stackrel{.}{\widehat{E}}}_{{\delta }_e}={\eta }_4|S_V{b}_{12}+S_h{b}_{22}|$$

(65)

$${\beta }_{\mathrm{cn}}=-{\hat{E}}_{{\beta }_c}\mathrm{sgn}(S_V{b}_{11}+S_h{b}_{21})$$

(66)

$${\delta }_{\mathrm{en}}=-{\hat{E}}_{{\delta }_e}\mathrm{sgn}(S_V{b}_{12}+S_h{b}_{22})$$

(67)

Then, $\stackrel{·}{V}$ in equation (59) can be transformed as

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{V}=}& -{\epsilon }_{{\beta }_c}(S_V{b}_{11}+S_h{b}_{21})-E_{{\beta }_c}|S_V{b}_{11}+S_h{b}_{21}|\hfill \\ \multicolumn{1}{c}{}& -{\epsilon }_{{\delta }_e}(S_V{b}_{12}+S_h{b}_{22})-E_{{\delta }_e}|S_V{b}_{12}+S_h{b}_{22}|\hfill \\ \multicolumn{1}{c}{}& \le \left|{\epsilon }_{{\beta }_c}\right||S_V{b}_{11}+S_h{b}_{21}|-E_{{\beta }_c}|S_V{b}_{11}+S_h{b}_{21}|\hfill \\ \multicolumn{1}{c}{}& +\left|{\epsilon }_{{\delta }_e}\right||S_V{b}_{12}+S_h{b}_{22}|-E_{{\delta }_e}|S_V{b}_{12}+S_h{b}_{22}|\hfill \\ \multicolumn{1}{c}{}& =(\left|{\epsilon }_{{\beta }_c}\right|-E_{{\beta }_c})|S_V{b}_{11}+S_h{b}_{21}|\hfill \\ \multicolumn{1}{c}{}& +(\left|{\epsilon }_{{\delta }_e}\right|-E_{{\delta }_e})|S_V{b}_{12}+S_h{b}_{22}|\hfill \\ \multicolumn{1}{c}{\le 0}\end{array}$$

(68)

which illustrates that the system is stable.

In order to overcome the chattering phenomenon, we replace the sign function with the saturation function

$$\mathrm{sat}\left(\frac{A}{k}\right)=\{\begin{array}{c}\hfill \frac{A}{k},\mathrm{if}\left|\frac{A}{k}\right|\le 1\hfill \\ \multicolumn{1}{c}{\mathrm{sign}\left(\frac{A}{k}\right),\mathrm{otherwise}}\end{array}$$

(69)

In this way, we can rewrite ${\beta }_{\mathrm{cn}}$ and ${\delta }_{\mathrm{en}}$ as

$${\beta }_{\mathrm{cn}}=-{\hat{E}}_{{\beta }_c}\mathrm{sat}\left[\frac{(S_V{b}_{11}+S_h{b}_{21})}{k_V}\right]$$

(70)

$${\delta }_{\mathrm{en}}=-{\hat{E}}_{{\delta }_e}\mathrm{sat}\left[\frac{(S_V{b}_{12}+S_h{b}_{22})}{k_h}\right]$$

(71)

Simulation of controller

Set up of experiments

Assume a hypersonic aircraft with the following model parameters and initial states: $m=136,820\mathrm{kg}$ , $s=334.73{\mathrm{m}}^2$ , $\overline{c}=24.38\mathrm{m}$ , $I_y=9,490,740\mathrm{kg}·{\mathrm{m}}^2$

$$C_L=0.6203\alpha$$

(72)

$$C_D=0.645{\alpha }^2+0.0043378\alpha +0.003772$$

(73)

$$C_M(\alpha )=-0.035{\alpha }^2+0.036617\alpha +5.3216\times {10}^{-6}$$

(74)

$$C_M(q)=\left[\frac{\overline{c}}{(2v)}\right]q(-6.796{\alpha }^2+0.3015\alpha -0.2289)$$

(75)

$$C_M({\delta }_e)=0.0292({\delta }_e-\alpha )$$

(76)

$$\rho =1.2266e^{-h/7315.2}$$

(77)

$$M_y=0.5\rho V^{2}s\overline{c}[C_M(\alpha )+C_M(q)+C_M(q)]$$

(78)

In the climbing mode, the thrust coefficient and the fuel ratio are computed as follows

$$C_T=\{\begin{array}{c}\hfill 0.1288\beta \mathrm{if}\beta <1\hfill \\ \multicolumn{1}{c}{0.112+0.0168\beta \mathrm{if}\beta \ge 1}\end{array}$$

(79)

$$I_{\mathrm{sp}}=\{\begin{array}{c}\hfill 4700-\frac{h}{100}\mathrm{Ma}<4\hfill \\ \multicolumn{1}{c}{5680-245\mathrm{Ma}-\frac{h}{100}\mathrm{Ma}\ge 4}\end{array}$$

(80)

In the cruising mode, the thrust coefficient is

$$C_T=\{\begin{array}{c}\hfill 0.02576\beta \mathrm{if}\beta <1\hfill \\ \multicolumn{1}{c}{0.0224+0.00336\beta \mathrm{if}\beta \ge 1}\end{array}$$

(81)

Initial states of the hypersonic aircraft flight are $V_0=4390\mathrm{m}/\mathrm{s}$ , $h_0=32,028\mathrm{m}$ , ${\gamma }_0=0^{\circ }$ , ${\alpha }_0=2.{745}^{\circ }$ , $q_0=0^{\circ }$ , and ${\beta }_{c0}=0.2133$ . Tables 1–4 show the initial membership function parameters of type-2 TSK fuzzy controller.

Table 1.

Membership function parameters of input ${\beta }_{c1}$ .

Negative			Zero			Positive
$m_{N1}$	$m_{N2}$	${\sigma }_N$	$m_{Z1}$	$m_{Z2}$	${\sigma }_Z$	$m_{P1}$	$m_{P2}$	${\sigma }_P$
−0.648	−0.552	0.163	−0.048	0.048	0.163	0.552	0.648	0.163

Table 2.

Membership function parameters of input ${\beta }_{c2}$ .

Negative			Zero			Positive
$m_{N1}$	$m_{N2}$	${\sigma }_N$	$m_{Z1}$	$m_{Z2}$	${\sigma }_Z$	$m_{P1}$	$m_{P2}$	${\sigma }_P$
−0.648	−0.552	0.163	−0.048	0.048	0.163	0.552	0.648	0.163

Table 3.

Membership function parameters of input ${\delta }_{e1}$ .

Negative			Zero			Positive
$m_{N1}$	$m_{N2}$	${\sigma }_N$	$m_{Z1}$	$m_{Z2}$	${\sigma }_Z$	$m_{P1}$	$m_{P2}$	${\sigma }_P$
−0.0206	−0.0194	0.002038	−0.0131	−0.0119	0.002038	−0.0056	−0.0044	0.002038

Table 4.

Membership function parameters of input ${\delta }_{e2}$ .

Negative			Zero			Positive
$m_{N1}$	$m_{N2}$	${\sigma }_N$	$m_{Z1}$	$m_{Z2}$	${\sigma }_Z$	$m_{P1}$	$m_{P2}$	${\sigma }_P$
−0.0206	−0.0194	0.002038	−0.0131	−0.0119	0.002038	−0.0056	−0.0044	0.002038

Simulation of response to switching from climbing mode to cruising mode

When the hypersonic aircrafts switch from climbing mode to cruising mode, they have to switch their control signals. In the climbing mode, the velocity command signal is a ramp signal, of which the slope is 10 m/s²; the altitude command signal is a ramp signal, of which the slope is 75 m/s. In the cruising mode, the velocity command signal is 4590 m/s, and the altitude command signal is 33,528 m. Before 20 s, the hypersonic aircraft is in climbing mode. After 25 s, the hypersonic aircraft is in cruising mode. The switching occurs during 20–25 s. The control effects of switching from climbing mode to cruising mode are detailed in Figure 5.

Figure 5.

Control effects of switching from climbing mode to cruising mode: (a) velocity, (b) altitude, (c) throttle setting, (d) elevator deflection, (e) angle of attack, and (f) flight path angle.

From the simulation results in Figure 5, we can draw the conclusion that the direct switch cannot keep the system stable, which illustrates the research on the control of switching from climbing mode to cruising mode is necessary. Because fuzzy control is suitable to resolve the uncertainty problems, so combined with sliding mode control, type-1 TSK fuzzy and type-2 fuzzy can keep the switching process stable. Compared with type-1 TSK fuzzy, type-2 fuzzy has more advantages that the uncertainties of its membership functions can increase the capability of dealing with uncertainty issues. Additionally, the adaptive laws tune the system adaptively to make the system stable faster. As a result, the switching process by using type-2 TSK fuzzy sliding mode control is smoother. Meanwhile, the track errors of switching and the elevator deflection are smaller, which verifies the switching performance of hypersonic aircraft is better. Type-2 TSK fuzzy sliding mode control proposed in this article can provide an effective adaptive switching control to realize the stable and smooth switching process.

Conclusion

This article presents a novel adaptive switching control of type-2 TSK fuzzy sliding mode control to maintain stability and smoothness in the switching process of control. The full-state feedback linearization is used to linearize the complex nonlinear model precisely. Considering the problems of parametric uncertainties, an interval type-2 TSK fuzzy logic control scheme is combined with sliding mode control to design a stable, smooth, and adaptive switching control for hypersonic aircraft. The fuzzy set structure of type-2 TSK fuzzy approach is found to be more suitable to resolve the aforementioned uncertainty problems. In order to formally establish stability, a constructive Lyapunov analysis approach is used in design of the control system. The effectiveness of the novel controller is illustrated by simulations of control effects of switching from climbing mode to cruising mode. The numerical simulation results indicate that the control scheme proposed in this article has stable and smoother switching performance.