Fuzzy model predictive control of flexible air-breathing hypersonic vehicles with unmodeled actuator sensor failures

Abstract

In this article, fuzzy model predictive control of flexible air-breathing hypersonic vehicles with unmodeled actuator sensor failures is proposed. The actuator sensor of flexible air-breathing hypersonic vehicle is assumed to suffer from some failures, which are difficult to be modeled. A control-oriented model of sensor failures is derived from the statistics law of input and output data of the sensor. A Takagi–Sugeno fuzzy model is utilized to approach the longitudinal dynamics model of flexible air-breathing hypersonic vehicles. Then, fuzzy model predictive control is proposed. A Lyapunov function approach is utilized and a set of linear matrix inequalities are given to solve the corresponding controller optimization problem. Simulation results are given to show the excellent reference altitude and velocity tracking performance of the proposed controller design method.

Introduction

Flexible air-breathing hypersonic vehicles (FAHVs) are a kind of novel vehicles which can fly at a speed with more than 5 Mach number. FAHVs can carry more payload than traditional flight vehicles since these can obtain oxygen directly from atmosphere. FAHVs present a more cost-efficient way to make access to space routine, or even make the space travel routine and intercontinental travel as easy as intercity travel. ¹ But because of the adopting of airframe integrated technology, the interactions between the elastic airframe, the propulsion system, and the structural dynamics are very strong, which make modeling and flight control of such vehicles a challenge problem. ²

The control of FAHVs has attracted much attention in recent years. In Parker et al., ³ a control-oriented model is developed using curve fits calculated directly from the forces and moments included in the truth model. Based on the control-oriented model, several studies on the flight control are conducted. In Xu et al., ⁴ and in Jankovsky et al., ⁵ output feedback control design methods for FAHVs are discussed. In Fiorentini et al., ⁶ a nonlinear robust adaptive control design method is presented for FAHVs. The non-minimum phase characteristics of FAHVs are discussed in Hu et al., ⁷ and then a stable inversion-based nonlinear controller is given. Adaptive sliding mode controller is also proposed to drive the nonlinear dynamics of FAHVs. ⁸ But the nonlinear dynamics and the flight environment of FAHVs are high complex, the aerodynamics parameters vary quickly, so the robust control of FAHVs is still a pendent problem.

Takagi–Sugeno (T-S) fuzzy control scheme is an efficient way for the control approach of nonlinear system, which could approximate any smooth nonlinear function to any specified accuracy within any compact set. ⁹ Through fuzzy If-Then rules, the nonlinear system can be represented by a set of linear models, and then the traditional linear controller design methods can be utilized. The fuzzy controller design for FAHVs has been discussed in literature. In Hu et al., ¹⁰ T-S fuzzy approach of FAHVs and fuzzy-guaranteed cost controller are proposed, and in Hu et al., ¹¹ fuzzy stable inversion-based tracking controller of FAHVs is presented. So in this article, T-S fuzzy approach is utilized for the controller design of FAHVs.

Similar to other airplanes and space vehicles, the actuator sensor of FAHVs is not completely reliable, which means that possible failures are not avoidable. The failures of actuator sensor are caused by electromagnetism disturbances, imperfect communication links, break or loss of effectiveness of the inner component and so on, so it is difficult to develop an accurate model for the actuator sensor failures. The actuator sensor failures may make the control system of FAHVs unsafe or unreliable, so the reliable control against possible actuator sensor failures is unavoidable, especially when the failures are hard to be modeled. The fuzzy-reliable control of FAHVs has been studied in Hu et al., ¹² but the failures considered in this article is loss of effectiveness, which is a simplified case of the failures. Actually, the accurate model of actuator failures is hard to be built, and what we can get is only the input and output data of actuator sensor. By the measurable data, the statistical characteristics of the failures can be derived. In this case, we can model the actuator sensor failures via a stochastic approach. Based on the stochastic model of the actuator sensor failures, we can design a fuzzy-reliable controller to stabilize the whole system. Motivated by the above discussions, fuzzy model predictive control (FMPC) of FAHVs with actuator sensor failures is studied in this article. Based on the T-S fuzzy modeling technology, a T-S fuzzy model of FAHVs is first constructed, and then actuator sensor failures are control-oriented modeled by stochastic approach. The control objective is proposed and a FMPC design method is developed by Lyapunov approach. The existence conditions of the corresponding controller optimization problem are formulated in the form of liner matrix inequalities. Finally, an illustrative example is provided to show the effectiveness and advantage of the proposed control design method.

The rest of this article is organized as follows. In section “Problem Formulation,” the dynamic model of FAHVs, the actuator sensor failures, and the control objective of the article are established. The design method of FMPC is presented in section “Fuzzy model predictive control design.” An example is given to illustrate the applicability of the proposed approach in section “Simulation Results.” Finally, we conclude the article in section “Conclusion.”

Notation

The notations used throughout the article are fairly standard. The superscript ‘T’ stands for matrix transposition; ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space; ${\mathbb{R}}^{n\times m}$ denotes the set of all $n\times m$ real matrices; $\mathrm{diag}\{\dots \}$ stands for a block-diagonal matrix; I and $0$ denote the identity matrix and zero matrix, respectively, with compatible dimensions. In addition, $\mathrm{E}\{x\}$ and $\mathrm{E}\{x\}$ mean expectation of x and expectation of x conditional on y, respectively. The notation |·| refers to the Euclidean vector norm. In symmetric block matrices or complex matrix expressions, we use an “*” to represent a term that is induced by symmetry. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.

Problem formulation

In this section, the nonlinear model and fuzzy model of FAHVs are established. The actuator sensor failures of the system are also control-oriented model. Then, the FMPC problem is organized.

Nonlinear model of FAHVs

The nonlinear model of FAHVs considered in this article is developed by Bolender and Doman.^13,14 The model is complex since the flexibility effects are included in the model. For describing conveniently, the sketch of the vehicle is given in Figure 1. Associated with Figure 1, the nonlinear dynamics of FAHVs can be described as follows

$$\{\begin{array}{lll}\stackrel{.}{h}\hfill & =\hfill & V\sin \left(\theta -\alpha \right)\hfill \\ \stackrel{.}{V}\hfill & =\hfill & \frac{1}{m}\left(T\cos \alpha -D\right)-g\sin \left(\theta -\alpha \right)\hfill \\ \stackrel{.}{\alpha }\hfill & =\hfill & \frac{1}{mV}\left(-T\sin \alpha -L\right)+Q+\frac{g}{V}\cos \left(\theta -\alpha \right)\hfill \\ \stackrel{.}{\theta }\hfill & =\hfill & Q\hfill \\ \stackrel{.}{Q}\hfill & =\hfill & \frac{M}{I_{yy}}\hfill \\ {\stackrel{··}{\eta }}_1\hfill & =\hfill & -2{\varsigma }_1{\omega }_1{\stackrel{.}{\eta }}_1-{\omega }_1^2{\eta }_1+N_1\hfill \\ {\stackrel{··}{\eta }}_2\hfill & =\hfill & -2{\varsigma }_2{\omega }_2{\stackrel{.}{\eta }}_2-{\omega }_2^2{\eta }_2+N_2\mathrm{,}\hfill \end{array}$$

(1)

In equation (1), h and V represent the flight altitude and velocity, respectively; $\alpha$ is the angle of attack of the vehicle; $\theta$ is the flight pitch angle with Q representing the pitch rate; ${\eta }_i$ denotes the $i\mathrm{th}$ generalized elastic coordinate. T, L, D, and $N_i$ are the thrust, lift, drag, and generalized elastic forces, respectively; M is the pitching moment. Polynomial expressions of the forces and moments $T,L,D,M,N_1$ , and $N_2$ are constructed by curve-fits approximations in Parker et al., ³ and the details are shown as follows

$$L\approx \frac{1}{2}\rho V^{2}S{C}_L(\alpha ,{\delta }_e)$$

$$D\approx \frac{1}{2}\rho V^{2}S{C}_D(\alpha ,{\delta }_e)$$

$$M\approx z_{T}T+\frac{1}{2}\rho V^{2}S\overline{c}(C_{M,\alpha }(\alpha )+C_{M,{\delta }_e}({\delta }_e))$$

$$T\approx C_T^{{\alpha }^3}{\alpha }^3+C_T^{{\alpha }^2}{\alpha }^2+C_T^{\alpha }\alpha +C_T^0$$

$$N_1\approx N_1^{{\alpha }^2}{\alpha }^2+N_1^{\alpha }\alpha +N_1^0$$

$$N_2\approx N_2^{{\alpha }^2}{\alpha }^2+N_2^{\alpha }\alpha +N_2^{\delta }{\delta }_e+N_2^0$$

with

$$\rho ={\rho }_0\exp \left(\frac{-(h-h_0)}{h_s}\right)$$

$$C_L=C_L^{\alpha }\alpha +C_L^{{\delta }_e}{\delta }_e+C_L^0$$

$$C_D=C_D^{{\alpha }^2}{\alpha }^2+C_D^{\alpha }\alpha +C_D^{{\delta }_e^2}{\delta }_e^2+C_D^{{\delta }_e}{\delta }_e+C_D^0$$

$$C_{M,\alpha }=C_{M,\alpha }^{{\alpha }^2}\alpha +C_{M,\alpha }^{\alpha }\delta +C_{M,\alpha }^0,C_{M,{\delta }_e}=c_e{\delta }_e$$

$$C_T^{{\alpha }^3}={\beta }_1(h,\overline{q})\Phi +{\beta }_2(h,\overline{q}),\overline{q}=\frac{1}{2}\rho V^2$$

$$C_T^{{\alpha }^2}={\beta }_3(h,\overline{q})\Phi +{\beta }_4(h,\overline{q})$$

$$C_T^{\alpha }={\beta }_5(h,\overline{q})\Phi +{\beta }_6(h,\overline{q})$$

$$C_T^0={\beta }_7(h,\overline{q})\Phi +{\beta }_8(h,\overline{q})$$

This nonlinear model is composed of five rigid body state variables $[h,V,\alpha ,\theta ,Q{]}^T$ and four flexible states $[{\eta }_1,{\stackrel{·}{\eta }}_1,{\eta }_{2,}{\stackrel{·}{\eta }}_2{]}^T$ , the control inputs are fuel-to-air ratio $\Phi$ and elevator deflection ${\delta }_e$ .

Figure 1.

Geometry of the flexible hypersonic vehicle model.

Fuzzy model of FAHVs

From section “Nonlinear model of FAHVs,” we can see that the nonlinear model of FAHVs is extremely complex, so it is difficult to design a nonlinear controller directly. Fuzzy model technology has been widely used for approaching and controller design of nonlinear system and is said to be greatly useful, so we utilize the well-known T-S fuzzy model technology to approach the nonlinear systems (1). By selecting some states or the combination of the states as premise variables, the nonlinear model (1) of FAHVs can then be described by a set of linear models combined by fuzzy membership functions. Then, the nonlinear model of FAHVs (1) can be represented by the following fuzzy model

If $z_1(t)$ is $M_1^i$ , $z_2(t)$ is $M_2^i$ $M_1^i$ is $M_p^i$ , Then

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}\stackrel{·}{x}(t)=A_{i}x(t)+B_{i}u(t)+\omega (t)\hfill \\ \multicolumn{1}{c}{y(t)=\mathrm{Cx}(t),}\end{array}}\\ \multicolumn{1}{c}{i=1,\dots ,l}\end{array}$$

(2)

where $M_1^i,\dots ,M_p^i$ are the fuzzy sets; $z(t)=\left[\begin{array}{c}{z}_1(t)z_2(t)\dots z_p(t)\hfill \end{array}\right]$ are the premise variables; $x(t)\in {\mathbb{R}}^n$ is the state vector; $x(t)=[h,V,\alpha ,\theta ,Q,{\eta }_1,{\stackrel{·}{\eta }}_1,{\eta }_{2,}{\stackrel{·}{\eta }}_2{]}^T$ ; $u(t)\in {\mathbb{R}}^m$ is the input vector; $u(t)=[\begin{array}{c}\Phi {\delta }_e\hfill \end{array}{]}^T$ , where $\Phi$ is fuel-to-air ratio and ${\delta }_e$ is the elevator deflection; $A_i,B_i,$ and C are known constant matrices; $\omega (t)$ is the fuzzy modeling error of FAHVs; l is the number of the fuzzy rules. Then, the T-S fuzzy model which represents the nonlinear hypersonic vehicle model (1) can be inferred as follows

$$\{\begin{array}{c}\stackrel{·}{x}(t)=\stackrel{l}{\sum _{i=1}}{\mu }_i(z)A_{i}x(t)+B_{i}u(t)+\omega (t)\hfill \\ \multicolumn{1}{c}{y(t)=\mathrm{Cx}(t)}\end{array}$$

(3)

where ${\mu }_i(z)=\frac{m_i(z)}{\sum _{i=1}^l{m}_i(z)},$ $m_i(z)=\underset{j=1}{\overset{q}{\Pi }}{M}_j^i(z)$ , with ${\mu }_i(z)\ge 0$ , and $\stackrel{l}{\sum _{i=1}}{\mu }_i(z)=1$ .

Actuator sensor failures

When the control signal has been computed according to the designed control strategy, this signal should be transferred from the computer to actuating mechanism. For FAHVs, the actuating mechanism is scramjet engine and aerodynamic elevator. The sensor which transferring this control signal is actuator sensor. In this article, we consider the failures of actuator sensor. The actuator sensor failures are hard to be accurately modeled, and what we can got is just the input and output data of the sensor. From the input–output data, the statistical characteristics of the sensor failures can be derived. According to the statistical characteristics, in this article, we model the actuator sensor failures via a stochastic approach, that is

$$u_F(t)=\pi u_c(t)$$

(4)

where $u_c(t)$ is the desired actuator input of the plant, and $u_F(t)$ is the actual input. $\{\pi \}$ is a stochastic process, which means the success rate of the data transmission from computer to controller. More specifically, when $\pi =0,$ the data transmission fails, the actuator receives no signal; when $\pi =1,$ the data transmission is successful, the actual actuator input is equal to the desired input. Based on the control-oriented model of actuator sensor failures, a natural assumption on $\{\pi \}$ can be made as

$$\mathrm{Prob}\{\pi =1\}=\mathrm{E}\left\{\pi \right\}=\overline{\pi }$$

$$\mathrm{Prob}\{\pi =0\}=1-\overline{\pi }$$

Remark 1

The actuator sensor failures considered in this article are more complex than original ones mentioned in literature. ¹² Since in the previous literature, the actuator sensor faults are considered to be loss of effectiveness, which are easily to be dealt with.

Control objective

For a flight vehicle, the main task is to track a given command without steady-error, which means

$$\underset{t\to \infty }{lim}e(t)=\underset{t\to \infty }{lim}(y(t)-y_{\mathrm{com}}(t))=0.$$

where $y_{\mathrm{com}}(t)=[V_{\mathrm{com}}(t),h_{\mathrm{com}}(t){]}^T$ is the velocity and altitude vector. For the built fuzzy system, an error integral action is introduced here to eliminate the steady-state tracking error. According to Hu et al., ¹⁰ define

$$d(t)={\int }_0^{t}e(\tau )d\tau ={\int }_0^t(y(\tau )-y_{\mathrm{com}}(\tau ))d\tau$$

then

$$\stackrel{·}{d}(t)=e(t)=y(t)-y_{\mathrm{com}}(t)$$

Combining the tracking error and the fuzzy model of FAHVs, an augmented fuzzy state-space system can be obtained

If $z_1(t)$ is $M_1^i$ , $z_2(t)$ is $M_2^i$ $\dots z_p(t)$ is $M_p^i$ , then

$$\{\begin{array}{c}\stackrel{·}{\zeta }(t)={\overline{A}}_i\zeta (t)+{\overline{B}}_{i}u(t)+G\overline{w}\left(t\right)\hfill \\ \multicolumn{1}{c}{y(t)=\overline{C}\zeta (t)}\end{array}$$

(5)

where

$$\zeta (t)=\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{d(t)}\end{array}\right],\overline{w}\left(t\right)=\left[\begin{array}{c}w\left(t\right)\hfill \\ \multicolumn{1}{c}{y_{\mathrm{com}}(t)}\end{array}\right],{\overline{A}}_i=\left[\begin{array}{cc}{A}_i\hfill & 0\\ \multicolumn{1}{c}{C}& 0\end{array}\right]$$

$${\overline{B}}_i=\left[\begin{array}{c}{B}_i\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]\overline{C}=\left[\begin{array}{cc}C\hfill & 0\hfill \end{array}\right],G=\left[\begin{array}{cc}I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -I\hfill \end{array}\right]$$

For description brevity, the above equation is written as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\zeta }(t)={\overline{A}}_z\zeta (t)+{\overline{B}}_{z}u(t)+G\overline{w}\left(t\right)}\\ \multicolumn{1}{c}{y(t)=\overline{C}\zeta (t)}\end{array}$$

(6)

where

$${\overline{A}}_z=\stackrel{l}{\sum _{i=1}}{\mu }_i(z){\overline{A}}_i,{\overline{B}}_z=\stackrel{l}{\sum _{i=1}}{\mu }_i(z){\overline{B}}_i$$

Based on the parallel distributed compensation (PDC) concept, when the actuator sensor failures do not exist, an ideal fuzzy state-feedback controller for (6) can be constructed as

$$u(t)=\stackrel{l}{\sum _{i=1}}{\mu }_i(z)K_i\zeta (t)=K_z\zeta (t)$$

(7)

where $K_z=\sum _{i=1}^l{\mu }_i(z)K_i$ . The augmented closed-loop system can be written as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\zeta }(t)=({\overline{A}}_z+{\overline{B}}_z{K}_z)\zeta (t)+G\overline{w}\left(t\right)}\\ \multicolumn{1}{c}{=\stackrel{l}{\sum _{i=1}}\stackrel{l}{\sum _{i=1}}{\mu }_i(z){\mu }_j(z)({\overline{A}}_i+{\overline{B}}_i{K}_j)\zeta (t)+G\overline{w}\left(t\right)}\end{array}$$

(8)

Unfortunately, in the existence of actuator sensor failures, the fuzzy controller (7) is not suitable. In this case, the closed-loop system is given by

$$\stackrel{·}{\zeta }(t)=\stackrel{l}{\sum _{i=1}}\stackrel{l}{\sum _{i=1}}{\mu }_i(z){\mu }_j(z)(A_{\overline{\pi }\mathrm{ij}}+\tilde{\pi }{B}_{\overline{\pi }\mathrm{ij}})\zeta (t)+G\overline{w}\left(t\right)$$

(9)

where $A_{\overline{\pi }\mathrm{ij}}={\overline{A}}_i+{\overline{B}}_i{K}_j$ , $B_{\overline{\pi }\mathrm{ij}}=B_i{K}_j$ , and $\tilde{\pi }=\pi -\overline{\pi }$ . Then, $\mathrm{E}\{\tilde{\pi }\}=0$ and $\mathrm{E}\{\tilde{\pi }\tilde{\pi }\}=\overline{\pi }(1-\overline{\pi }).$

Then, the command tracking of FAHVs can be transformed into the closed-loop stability of the augmented state-space system (9). Consider FAHVs with actuator sensor failures, the control objective is to find a state-feedback controller, such that

• The closed-loop system is robustly stable;

• The output of the system can track a command vector $y_{\mathrm{com}}(t)=[V_{\mathrm{com}}(t),h_{\mathrm{com}}(t){]}^T$ without steady-error;

• In the existence of unknown actuator sensor failures, the stability and the tracking performance of the system can be guaranteed.

FMPC design

In this section, stochastic control of FAHVs with actuator sensor failures will be investigated. Since the dynamic model of the actuator sensor faults is difficult to be built, we model the actuator sensor failures via a stochastic approach. This approach makes the control design of FAHVs a challenging problem. Model predictive control (MPC) is said to be a great way for control design of constraint system, no matter what kind of constraint. The stochastic model of the actuator sensor failures can also be regarded as a constraint, so in this article, MPC will be utilized here to solve the stochastic control design problem of FAHVs with actuator sensor faults.

For the application of MPC, the fuzzy closed-loop model (9) should be transformed into a discrete model, ¹⁵ which is listed as follows

$$\zeta (k+1)=\stackrel{l}{\sum _{i=1}}\stackrel{l}{\sum _{i=1}}{\mu }_i(k){\mu }_j(k)(A_{\overline{\pi }\mathrm{ij}}+\tilde{\pi }(k)B_{\overline{\pi }\mathrm{ij}})\zeta (k)+G\overline{w}\left(k\right)$$

(10)

where $\tilde{\pi }(k)=\pi (k)-\overline{\pi }$ . Then, $\mathrm{E}\{\tilde{\pi }(k)\}=0$ and $\mathrm{E}\{\tilde{\pi }(k)\tilde{\pi }(k)\}=\overline{\pi }(1-\overline{\pi }).$

Considering the control objective given in section “Control objective” and the design approach of MPC, the following performance functions are chosen

$$J=E\{\stackrel{N_f}{\sum _{s=1}}\zeta {(k+s|k)}^{T}Q\zeta (k+s|k)+u{(k+s|k)}^T\mathrm{Ru}(k+s|k)\}$$

(11)

where $Q>0$ and $R>0$ are weighting matrices; $\zeta (k+s|k)$ and $u(k+s|k)$ are state and control input of the closed-loop system at time $k+s$ predicted by signals at time k, respectively. $N_f$ is the running time of the system.

Remark 2

Since the fuzzy model (3) cannot exactly approach the nonlinear model of FAHVs (1), fuzzy modeling error is inevitable. In this case, the proposed FMPC strategy can only guarantee the robust stability of the closed-loop system. When the state of (9) only converges to a bounded region, a performance function $J=E\{\sum _{s=1}^{\infty }\zeta {(k+s|k)}^{T}Q\zeta (k+s|k)+u{(k+s|k)}^T\mathrm{Ru}(k+s|k)\}$ would be infinite. So in this article, we choose a performance index defined as (11), in which we use $N_f$ instead of $\infty$ , where $N_f$ is the running time of the system. Based on (11), we only need to analyze the minimization of J in finite time, which avoids the infinity of J.

So, the control objective of FMPC minimized the performance function (11), and stabilized the closed-loop system (9). In the existence of the stochastic actuator sensor failures model, the fuzzy model (9) should be stochastically stable, so we define a quadratic function $V(\zeta (k))=\zeta (k{)}^T{P}_i(k)\zeta (k)$ , where $P_i(k)$ is a common positive definite matrix at time k. Suppose $V(\zeta (k))$ satisfies the following constraint

$$\begin{array}{c}\multicolumn{1}{c}{E\{V(\zeta (k+s+1|k))-V(\zeta (k+s|k))\}}\\ \multicolumn{1}{c}{\le -E\{\zeta {(k+s|k)}^{T}Q\zeta (k+s|k)+u{(k+s|k)}^T\mathrm{Ru}(k+s|k)\}}\end{array}$$

(12)

Then, we can easily get that from $S=0$ to $N_f$

$$\begin{array}{c}\multicolumn{1}{c}{E\{V(\zeta (k+N_f|k))-V(\zeta (k|k))\}}\\ \multicolumn{1}{c}{\le -E\{\stackrel{N_f}{\sum _{i=1}}\zeta {(k+s|k)}^{T}Q\zeta (k+s|k)+u{(k+s|k)}^T\mathrm{Ru}(k+s|k)\}}\end{array}$$

Also, with $E\{\zeta (k+N_f|k)\}=0$ , hence, $E\{V(\zeta (k+N_f|k))\}=0$ . The performance (11)

$$J\le E\{V(\zeta (k|k)\}=\zeta (k{)}^T{P}_i(k)v(k)$$

Then, we can choose a variable $\chi$ to achieve the minimization of J, which is

$$\zeta (k{)}^T{P}_i(k)\zeta (k)\le \chi$$

A fuzzy predictive controller, which can stabilize the fuzzy system and minimize the performance function in (11), is designed and converted to a solution of an optimization problem.

Theorem 1

Suppose the success rate of the data transmission $\overline{\pi }$ is known, for the T-S fuzzy system (10), if there exist matrices $X_i(k)>0$ , $Y_i(K)$ , and $Z(k)$ , and a scalar $\delta >0$ , satisfying following convex minimization problem

$$\underset{\chi ,X_i(k),\mathrm{Yi}(K)}{\min }\chi$$

with the following inequalities

$$\left[\begin{array}{cc}-1\hfill & \zeta {(k|k)}^T\\ \multicolumn{1}{c}{\zeta (k|k)}& X_i(k)\end{array}\right]\le 0$$

(13)

$$\left[\begin{array}{cccccc}{X}_i(k)-Y_i(K)-Z^T(k)\hfill & *& *& *& *& *\\ \multicolumn{1}{c}{A_{l}Z(k)+\overline{\pi }{B}_l{Y}_i(k)}& -X_i(k)& *& *& *& *\\ \multicolumn{1}{c}{f{B}_l{Y}_i(k)}& 0& -X_i(k)& *& *& *\\ \multicolumn{1}{c}{\sqrt{\overline{\pi }}{Y}_i(k)}& 0& 0& -\chi R^{-1}& *& *\\ \multicolumn{1}{c}{\sqrt{Q}{Y}_i(k)}& 0& 0& 0& -\chi I& *\\ \multicolumn{1}{c}{G}& 0& 0& 0& 0& -\delta I\end{array}\right]<0$$

(14)

where $f=\sqrt{\overline{\pi }(1-\overline{\pi })}$ . Then, the fuzzy state-feedback controller can keep the system (10) asymptotically stable and the performance index (11) has an upper bound. The desired state-feedback control law is given by

$$K_i(k)=Y_i(k)Z^{-1}(k)$$

(15)

Proof

Suppose there exist $X_i(k)$ $>0$ and a nonsingular matrix $Z(k)$ satisfying (14). Since $X_i(k)$ $>0$ , we have

$$(X_i(k)-Z(k){)}^T{X}_i^{-1}(k)(X_i(k)-Z(k))\ge 0$$

which means

$$X_i(k)-Z(k)-Z^T(k)\ge -Z^T(k)X_i^{-1}(k)Z(k)$$

(16)

Then, (14) can be transformed into

$$\left[\begin{array}{cccccc}-Z^T(k)X_i^{-1}(k)Z(k)\hfill & *& *& *& *& *\\ \multicolumn{1}{c}{A_{l}Z(k)+\overline{\pi }{B}_l{Y}_i(k)}& -X_i(k)& *& *& *& *\\ \multicolumn{1}{c}{f{B}_l{Y}_i(k)}& 0& -X_i(k)& *& *& *\\ \multicolumn{1}{c}{\sqrt{\overline{\pi }}{Y}_i(k)}& 0& 0& -\chi R^{-1}& *& *\\ \multicolumn{1}{c}{\sqrt{Q}{Y}_i(k)}& 0& 0& 0& -\chi I& *\\ \multicolumn{1}{c}{G}& 0& 0& 0& 0& -\delta I\end{array}\right]<0$$

(17)

Perform a congruence transformation to (17) by diag $\{\begin{array}{c}{Z}^{-1}(k)IIIII\hfill \end{array}\}$ and simultaneous consideration of $K_i(k)=Y_i(k)Z^{-1}(k),$ we have

$$\left[\begin{array}{cccccc}-X_i^{-1}(k)\hfill & *& *& *& *& *\\ \multicolumn{1}{c}{A_l+\overline{\pi }{B}_l{K}_i(k)}& -X_i(k)& *& *& *& *\\ \multicolumn{1}{c}{f{B}_l{K}_i(k)}& 0& -X_i(k)& *& *& *\\ \multicolumn{1}{c}{\sqrt{\overline{\pi }}{K}_i(k)}& 0& 0& -\chi R^{-1}& *& *\\ \multicolumn{1}{c}{\sqrt{Q}{K}_i(k)}& 0& 0& 0& -\chi I& *\\ \multicolumn{1}{c}{G}& 0& 0& 0& 0& -\delta I\end{array}\right]<0$$

(18)

By Schur complement, (18) is equivalent to

$$\begin{array}{l}-X_i^{-1}(k)+A_l^T{X}_i^{-1}(k)A_l+f^2{B}_l^T{X}_i^{-1}(k)B_l\\ +\frac{Q}{\chi }+\frac{1}{\chi }\overline{\pi }{K}_i^T(k)R{K}_i(k)-{\delta }^{-1}G{G}^T<0\end{array}$$

(19)

By defining $X_i^{-1}(k)=\chi P_i^{-1}(k)$ , we have

$$\begin{array}{l}-\frac{P_i^{-1}(k)}{\chi }+A_l^T\frac{P_i^{-1}(k)}{\chi }{A}_l+f^2{B}_l^T\frac{P_i^{-1}(k)}{\chi }{B}_l\\ +\frac{Q}{\chi }+\frac{1}{\chi }\overline{\pi }{K}_i^T(k)R{K}_i(k)-{\delta }^{-1}G{G}^T<0\end{array}$$

So, we can get the following inequality

$$\begin{array}{l}-P_i^{-1}(k)+A_l^T{P}_i^{-1}(k)A_l+f^2{B}_l^T{P}_i^{-1}(k)B_l\\ +Q+\overline{\pi }{K}_i^T(k)R{K}_i(k)-\delta {\prime }^{-1}G{G}^T<0\end{array}$$

where $\delta \prime =\chi \delta$ . Along the closed-loop fuzzy system (10), we can obtain

$$E\{V(\zeta (k+s+1|k))-V(\zeta (k+s|k))\}$$

$$+E\{\zeta {(k+s|K)}^{T}Q\zeta (k+s|K)+u{(k+s|K)}^T\mathrm{Ru}(k+s|K)\}$$

$$\begin{array}{l}=\zeta (k+s|K)T\stackrel{l}{\sum _{i=1}}\stackrel{l}{\sum _{i=1}}{\mu }_i(k){\mu }_j(k)\\ \left(A_l^T{P}_i^{-1}(k)A_l+f^2{B}_l^T{P}_i^{-1}(k)B_l\right)\zeta (k+s|K)\\ -\zeta (k+s|K)T{P}_i^{-1}(k)\zeta (k+s|K)\\ +\zeta (k+s|K)T\left(Q+E\left(\overline{\pi }{K}_i^T(k)R{K}_i(k)\right)-{\delta }^{-1}G{G}^T\right)\zeta (k+s|K)\\ =\zeta (k+s|K)T\stackrel{l}{\sum _{i=1}}\stackrel{l}{\sum _{i=1}}{\mu }_i(k){\mu }_j(k)\\ (A_l^T{P}_i^{-1}(k)A_l+f^2{B}_l^T{P}_i^{-1}(k)B_l\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{-2P_i^{-1}(k)+2Q+2\overline{\pi }{K}_i^T(k)R{K}_i(k)}\\ \multicolumn{1}{c}{-{\delta }^{-1}G{G}^T}\end{array})\zeta (k+s|K)$$

So, we can seen from (12) that

$$\begin{array}{c}\multicolumn{1}{c}{E\{V(\zeta (k+s+1|k))-V(\zeta (k+s|k))\}}\\ \multicolumn{1}{c}{\le \zeta (k+s|K{)}^T(Q+\overline{\pi }{K}_i^T(k)R{K}_i(k))\zeta (k+s|K)}\end{array}$$

According to Lemma 1 proposed in Zhao et al., ¹⁶ we know that

$$\begin{array}{c}\multicolumn{1}{c}{E\{V(\zeta (k+s+1|k))-V(\zeta (k+s|k))\}}\\ \multicolumn{1}{c}{\le -{\lambda }_{\min }(Q+\overline{\pi }{K}_i^T(k)R{K}_i(k))\zeta (k+s|K{)}^T\mathrm{Ru}(k+s|K)}\end{array}$$

Summing up the inequality on both sides from $k=0,\dots ,d,$ we have

$$\begin{array}{c}\multicolumn{1}{c}{E\{V(\zeta (k+s+1|k))-V(\zeta (0))\}}\\ \multicolumn{1}{c}{\le -{\lambda }_{\min }^{-1}(Q+\overline{\pi }{K}_i^T(k)R{K}_i(k))}\\ \multicolumn{1}{c}{\mathrm{E}(\stackrel{d}{\sum _{k=0}}\zeta {(k+s|K)}^T\mathrm{Ru}(k+s|K))}\end{array}$$

where $\zeta (0)$ is the initial condition. So

$$\mathrm{E}\left(\stackrel{d}{\sum _{k=0}}{|\zeta (k+s|K)|}^2\right)\le -{\lambda }_{\min }\left(Q+\overline{\pi }{K}_i^T(k)R{K}_i(k)\right)V(\zeta (0))$$

Since ${\lambda }_{\min }^{-1}(Q+\overline{\pi }{K}_i^T(k)R{K}_i(k))>0$ , according to the Definition 1 of Zhao et al., ¹⁶ the closed-loop system is stochastically stable.

Multiplying a congruence transformation to (13) by diag $\{\begin{array}{c}I{X}_i^{-T}(k)\hfill \end{array}\}$ , we can get

$$\left[\begin{array}{c}-1\zeta {(k|k)}^T\frac{P_i(k)}{\chi }\hfill \\ \multicolumn{1}{c}{\frac{P_i(k)}{\chi }\zeta (k|k)-\frac{P_i^{-1}(k)}{\chi }}\end{array}\right]\le 0$$

By Schur complement, it is equivalent to

$$\zeta (k|k{)}^T{P}_i(k)\zeta (k|k)\le \chi$$

So, the performance function (11) is satisfied. The proof is completed. ▪

Simulation results

In this section, a numerical example is given to test the effectiveness and advantages of the FMPC method for FAHVs with actuator sensor failures. The parameter values of FAHVs model are similar to Hu et al. ¹⁰ When constituting fuzzy model, velocity V is selected as the premise variable, and is supposed to have three levels: a lower bound, an upper bound, and an equilibrium point, which is named as “small (S),”“big (B),” and “middle (M),” respectively. More specifically, $V_B=9000\mathrm{ft}/\mathrm{s}$ , $V_M=7702.0808\mathrm{ft}/\mathrm{s}$ , and $V_S=6400\mathrm{ft}/\mathrm{s}$ . Using the modeling method described in section “Problem formulation,” the fuzzy model of FAHVs can be constructed as follows:

• Rule 1. If V is small $(V_S)$ , then

$$\{\begin{array}{c}\stackrel{·}{x}(t)=A_{1}x(t)+B_{1}u(t)+\omega (t)\hfill \\ \multicolumn{1}{c}{y(t)=\mathrm{Cx}(t)}\end{array}$$

• Rule 2. If V is middle $(V_M)$ , then

$$\{\begin{array}{c}\stackrel{·}{x}(t)=A_{2}x(t)+B_{2}u(t)+\omega (t)\hfill \\ \multicolumn{1}{c}{y(t)=\mathrm{Cx}(t)}\end{array}$$

• Rule 3. If V is big $(V_B)$ , then

$$\{\begin{array}{c}\stackrel{·}{x}(t)=A_{3}x(t)+B_{3}u(t)+\omega (t)\hfill \\ \multicolumn{1}{c}{y(t)=\mathrm{Cx}(t)}\end{array}$$

where the matrices of the fuzzy model are

$$A_1=\left[\begin{array}{ccccccccc}0\hfill & 0& -6400& 6400& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{1.512\times {10}^{-3}}& -0.032\times {10}^{-2}& -0.3587& -31.5538& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{1.334\times {10}^{-7}}& -0.965\times {10}^{-6}& -0.0571& 0& 1& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 1& 0& 0& 0& 0\\ \multicolumn{1}{c}{-3.019\times {10}^{-6}}& 6.199\times {10}^{-5}& 1.911& 0& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 0& 0& 1& 0& 0\\ \multicolumn{1}{c}{0}& 0& 4648& 0& 0& -272.3& -0.66& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 0& 0& 0& 0& 1\\ \multicolumn{1}{c}{0}& 0& 2598& 0& 0& 0& 0& -400& -0.8\end{array}\right]$$

$$B_1={\left[\begin{array}{ccccccccc}0\hfill & 24.42& -1.01\times {10}^{-4}& 0& 0.1226& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& -12.86& -0.0093& 0& -1.029& 0& 0& 0& 1247\end{array}\right]}^T$$

$$A_2=\left[\begin{array}{ccccccccc}0\hfill & 0& -7702& 7702& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{0.2433\times {10}^{-3}}& -0.1349\times {10}^{-2}& 20.95& -31.92& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{1.769\times {10}^{-7}}& -1.027\times {10}^{-6}& -0.06961& 0& 1& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 1& 0& 0& 0& 0\\ \multicolumn{1}{c}{3.964\times {10}^{-7}}& -2.199\times {10}^{-6}& 2.946& 0& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 0& 0& 1& 0& 0\\ \multicolumn{1}{c}{0}& 0& 4648& 0& 0& -272.3& -0.66& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 0& 0& 0& 0& 1\\ \multicolumn{1}{c}{0}& 0& 2598& 0& 0& 0& 0& -400& -0.8\end{array}\right]$$

$$B_2={\left[\begin{array}{ccccccccc}0\hfill & 24.42& -8.384\times {10}^{-5}& 0& 0.1226& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& -37.2& -0.01122& 0& -1.491& 0& 0& 0& 1247\end{array}\right]}^T$$

$$A_3=\left[\begin{array}{ccccccccc}0\hfill & 0& -9000& 9000& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{3.357\times {10}^{-5}}& -4.536\times {10}^{-4}& -20.45& -31.55& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{1.027\times {10}^{-7}}& -5.87\times {10}^{-7}& -0.0803& 0& 1& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 1& 0& 0& 0& 0\\ \multicolumn{1}{c}{6.023\times {10}^{-6}}& 8.923\times {10}^{-5}& 3.836& 0& 0& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 0& 0& 1& 0& 0\\ \multicolumn{1}{c}{0}& 0& 4648& 0& 0& -272.3& -0.66& 0& 0\\ \multicolumn{1}{c}{0}& 0& 0& 0& 0& 0& 0& 0& 1\\ \multicolumn{1}{c}{0}& 0& 2598& 0& 0& 0& 0& -400& -0.8\end{array}\right]$$

$$B_3={\left[\begin{array}{ccccccccc}0\hfill & 24.42& -7.18\times {10}^{-5}& 0& 0.1226& 0& 0& 0& 0\\ \multicolumn{1}{c}{0}& -25.42& -0.0131& 0& -2.0357& 0& 0& 0& 1247\end{array}\right]}^T$$

The fuzzy modeling error $\omega (t)$ is assumed to be bounded. Here, we choose a random disturbance, described by

$$\omega (t)=d_{\max }N(•)$$

(20)

whose maximum absolute $d_{\max }$ is fixed to $0.1$ , and $N(•)$ denotes the normal distribution with mean = 0 and standard deviation = 1.

The fuzzy membership functions of V is defined as follows

$$\mathrm{if}V>V_M,\{\begin{array}{c}{\mu }_S(V)=0\hfill \\ \multicolumn{1}{c}{{\mu }_M(V)=1-{\mu }_B(V),}\\ \multicolumn{1}{c}{{\mu }_B(V)=\exp (-3.5\times {10}^{-12}{|V(t)-V_B|}^4)}\end{array}$$

$$\mathrm{if}V<V_M,\{\begin{array}{c}{\mu }_S(V)=\exp (-3.5\times {10}^{-12}{|V(t)-V_S|}^4)\hfill \\ \multicolumn{1}{c}{{\mu }_M(V)=1-{\mu }_B(V)}\\ \multicolumn{1}{c}{{\mu }_B(V)=0}\end{array}$$

The membership function of the fuzzy model is shown in Figure 2.

Figure 2.

Membership functions of V.

Considering the structural strength, the given command should have reasonable requirements for FAHVs, so the set point input reference commands are forced to pass through a pre-filter as

$$H(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$$

(21)

where $\zeta$ denotes the damping ratio, and ${\omega }_n$ stands for natural frequency. The signal out of the pre-filter is defined as reference command.

The success rate of the data transmission from the sensor to the controller $\overline{\pi }$ is known, and $\overline{\pi }=0.8$ . Letting $Q=1\times {10}^{-6}{I}_{11\times 11}$ , $R=1\times {10}^{-2}{I}_{2\times 2}$ , the running time $N_f$ is chosen as $N_f=100\mathrm{s}$ , then by Theorem 1, the controller gains can be obtained and the fuzzy-reliable state-feedback controller can be constructed.

In simulation, we consider a climbing maneuver using separate reference commands for altitude and velocity. The altitude reference $h_{\mathrm{com}}(t)$ is chosen as $1000\mathrm{ft}$ and the velocity reference $V_{\mathrm{com}}(t)$ is chosen as $100\mathrm{ft}/\mathrm{s}$ . $\zeta$ and ${\omega }_n$ are chosen as 0.9 and $0.1\mathrm{rad}/\mathrm{s}$ , respectively, which can make the climbing of velocity and altitude finish in about 50 s. The effectiveness and tracking performance is shown in Figure 3. From the above figure, we can see that the proposed fuzzy model predictive controller can achieve good performance for the tracking problem.

Figure 3.

Tracking performance.

The other important states and the inputs of FAHVs are shown in Figures 4 and 5, respectively. The angle of attack and flight path angle are shown in Figure 4. The inputs of the plant are shown in 5. The angle of attack, flight path angle, and inputs are all smooth and bounded.

Figure 4.

Angle of attack and flight path angle.

Figure 5.

Input of plant.

Conclusion

In this article, a FMPC strategy has been presented for FAHVs with actuator sensor failures. The fuzzy approach is utilized to model the nonlinear dynamics of FAHVs, and the actuator sensor failures are modeled by a stochastic approach. By defining an augmented system, the tracking problem of FAHVs has been transformed into a stable problem of the closed-loop system. Using Lyapunov method, sufficient conditions for such a controller design problem have been proposed in terms of linear matrix inequalities (LMIs). Simulation results have shown the effectiveness of the proposed strategy.