1 Introduction

Reusable reentry vehicles are capable of carrying out tasks by flexible maneuvering on orbit [1]. The reusable reentry vehicles can be launched repeatedly, and have a lot of advantages such as cost effectiveness, speed and so on [2,3,4]. Because of these advantages, the vehicles are of high importance in civil and military aspects. Nevertheless, during the reentry process, the vehicle exhibits complicated coupling effects, strong nonlinearities, uncertainties and external perturbations which bring challenges to the development of the vehicle’s attitude control [5,6,7]. Furthermore, several stringent constraints, such as dynamic pressure, heat flux, and structured loads add more challenges to the control design [8,9,10].

Over the past years, various control methods have been suggested for the vehicle’s flight control. Earlier flight control problems were based on the linearized model of the aerocrafts at a particular operating point. In [11,12,13,14], a LQR has been designed to stabilized the reentry vehicle. However, this method is unsuitable for the multivariable and highly nonlinear model of the vehicle. A more effective control can be achieved by linearizing the model at different operating regions based on gain-scheduling [15,16,17,18]. Although gain-scheduling approach can solve some vehicle control problems, robustness and global stability of the system can not be guaranteed, particularly in the event of sudden parametric variations. In order to improve the nonlinear performance of the gain scheduling method, a trajectory linearization control was proposed in [19,20,21,22] and dynamic inversion control method was utilized in [23,24,25,26]. Nonetheless, the downside of these approaches is poor robustness against parametric uncertainties and modeling errors if built poorly.

A good robust control performance of the reentry vehicle has been achieved using the robust backstepping method [27, 28]. A command filtered backstepping control of a hypersonic vehicle with dynamic perturbations has been studied in [29]. A robust attitude control of reusable reentry vehicle has been realized using the sliding mode control [30]. A fractional order sliding mode controller optimized with pigeon-inspired optimization has been developed for attitude control of the reentry vehicle [31]. In [32], a chattering-free sliding mode control of the reentry vehicle has been presented. A high-order sliding mode control was proposed for an uncertain space vehicle in the reentry period [33]. In [34], a robust model predictive control of the reentry vehicle has been investigated.

When the space vehicle is subjected to time-varying disturbances, an adaptive backstepping controller was utilized in [35]. In [36], an adaptive super-twisting sliding mode control was proposed for the vehicle in the reentry period. In [37], an adaptive fast terminal sliding mode control with disturbance observer has been designed for the reentry vehicle. In [38], a sliding mode based disturbance observer was designed for attitude control of a reentry vehicle with constrained inputs.

It is worth noting that the aforementioned control techniques are in continuous time domain. Recently, the applications of computer controlled systems are increasing due to the emergence of cheap micro-controllers with very fast sampling rates [39]. Dynamic systems such as reusable reentry vehicle requires controllers with very fast sampling rates. Therefore, discrete time control of reusable reentry vehicle is an important area to explore. In [40], attitude control of reentry hypersonic vehicle was achieved using a discrete time backstepping technique. In [41], a discrete time sliding control of reusable space vehicle vehicle was studied.

Motivated by the aforesaid discussion, this paper investigates the attitude optimal control of the reusable reentry vehicle subjected to external disturbances and parametric variations. Contrary to [1, 2, 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], the nonlinear kinematic model of the vehicle is transformed into a discrete time form using the Euler method. Unlike [11,12,13,14,15,16,17,18] where the nonlinear kinematic model of the vehicle was linearized near a particular operating point, we linearized the model using global feedback linearization. During the reentry phase, exact linearization is not possible due to parametric variations, coupled dynamics and disturbances. We modelled these effects as constant disturbances and a robust LMI based LQRPI + D is designed to tackle the problem. Unlike the robust control methods [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41], a convex optimization is formulated to compute the optimal controller gains. Comparing with the conventional LMI based LQR, the innovation points of this article are as follows:

  1. 1.

    We proposed a robust LQRPI + D based on LMI to stabilized the reentry vehicle. This controller combines the advantages of convex optimization, disturbance suppression and quick convergence.

  2. 2.

    The impact of the disturbances is penalized by including a new cost term to the cost function.

  3. 3.

    In addition, a cost term representing the integral of the deviation of the output from the equilibrium is added to the objective function to minimize the deviation of the output from the equilibrium.

This paper is arranged as follows: The kinematic model of the reentry vehicle is provided in Sect. 2. In Sect. 3, the LQRP, LRQP + D and LQRPI + D are designed. Numerical simulations and discussions are given in Sect. 4. The research is concluded in Sect. 5.

Fig. 1
figure 1

The complete model of the vehicle [42]

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2 Mathematical modelling

The complete model of the reentry vehicle is shown in Fig. 1. The attitude kinematic model of the reentry vehicle consists of three degree of freedom translational kinematic subsystem and three degree of freedom rotational kinematic subsystem. The nonlinear model is obtained under the following assumptions [34, 35]

  1. 1.

    The vehicle is unpowered rigid body.

  2. 2.

    Rotational effect of the earth is ignored.

  3. 3.

    Slow movement of the vehicle is neglected.

Then, the nonlinear dynamic model of the reentry vehicle can be described as [34, 35]

$$\begin{aligned} {\dot{\alpha }}&= \zeta _y - \zeta _x~ cos\alpha ~ tan \beta -\zeta _z~ sin \alpha ~ tan \beta \nonumber \\&\quad + \frac{1}{m \pi } (Y + mg~ cos \delta ~ cos \mu ). \end{aligned}$$
(1)
$$\begin{aligned} {\dot{\beta }}&= \zeta _x ~sin \alpha -\zeta _z ~cos \alpha + \frac{1}{m \pi } (Y + mg ~cos \delta ~sin \mu ). \end{aligned}$$
(2)
$$\begin{aligned} {\dot{\mu }}&= - \zeta _x~ cos \alpha ~ cos \beta -\zeta _z~ sin \alpha ~ cos \beta + \zeta _y sin \beta \nonumber \\&\quad -\Big [ \frac{1}{m \pi } (tan \beta + tan \delta ~ sin \mu ) \nonumber \\&\quad - \frac{g~cos \delta ~ cos \mu ~ tan \beta }{\pi } + \frac{Y~tan \delta ~ cos \mu }{m \pi }\big ]. \end{aligned}$$
(3)
$$\begin{aligned} {\dot{\zeta }}_x&= \frac{J_{zz} u_x}{J_{xx} J_{zz} -J_{xz}^2 } + \frac{J_{xz} u_z}{J_{xx} J_{zz} -J_{xz}^2 } \nonumber \\&\quad + \frac{(J_{xx} - J_{yy} + J_{zz}) J_{xz}}{J_{xx} J_{zz} -J_{xz}^2 } \zeta _x \zeta _y \nonumber \\&\quad + \frac{(J_{yy} - J_{zz})J_{zz} -J_{xz}^2 }{J_{xx} J_{zz} -J_{xz}^2 } \zeta _y \zeta _z. \end{aligned}$$
(4)
$$\begin{aligned} {\dot{\zeta }}_y&= \frac{ u_y}{J_{yy} } + \frac{J_{xz} u_z}{J_{yy} } (\zeta _z^2 - \zeta _x^2) + \frac{J_{zz} - J_{xx}}{J_{xx} J_{zz} -J_{xz}^2 } \zeta _x \zeta _z. \end{aligned}$$
(5)
$$\begin{aligned} {\dot{\zeta }}_z&= \frac{J_{zz} u_x}{J_{xx} J_{zz} -J_{xz}^2 } + \frac{J_{xz} u_z}{J_{xx} J_{zz} -J_{xz}^2 } \nonumber \\&\quad + \frac{(J_{xx} - J_{yy})J_{xx} + J_{xz}^2}{J_{xx} J_{zz} -J_{xz}^2 } \zeta _x \zeta _y \nonumber \\&\quad + \frac{(J_{yy} - J_{xx} -J_{zz}) J_{xz} }{J_{xx} J_{zz} -J_{xz}^2 } \zeta _y \zeta _z. \end{aligned}$$
(6)

where \(\alpha \), \(\beta \), and \(\mu \) stand for the attack angle, sideslip angle and bank angle respectively, g, L, m, \(\pi \), Y, and \(\delta \) denote the gravitational acceleration, aerodynamic lift force, side force, reentry velocity, flight path angle, and mass respectively; \( J_{xx},~J_{yy},~J_{zz},~ \& ~J_{xz}\) denote the moments of inertia. \(u = [u_x~u_y~u_z]^T\) represent the control torque. The kinematic equations can be simplified as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{\varOmega }}&{} = f(\varOmega ) + g(\zeta ) \zeta . \\ J{\dot{\zeta }}&{}= -\zeta ^{\times } J \zeta + u. \end{array}\right. } \end{aligned}$$
(7)

where \(\zeta = [\zeta _x~\zeta _y~\zeta _z]^T\) denote the angular velocity vector and \(\varOmega = [\alpha ~\beta ~\mu ]^T\) is the attitude angle vector, \(\zeta ^{\times } \in \mathfrak {R}^{3 \times 3}\) is the skew-symmetric operator.

$$\begin{aligned} f(\varOmega )&=\begin{bmatrix} (-L+ mg~cos \delta ~ cos \mu )/ (mV~cos \beta ) \\ (Y+mg~cos \delta ~sin \mu )/(m \pi ) \\ (tan \beta ~+tan \delta ~ sin \mu )L /(m \pi ) -g~cos \delta ~cos\mu ~tan \beta /\pi \\ +Y~tan \delta ~cos \mu /(m \pi )~~~~~ ~~~~~ ~~~~~~ ~~ ~~ ~~~~\end{bmatrix}. \\ g(\varOmega )&= \begin{bmatrix} -cos \alpha ~ tan \beta &{} 1 &{} -sin \alpha ~tan \beta \\ sin \alpha &{} 0 &{} -cos \alpha \\ -cos \alpha ~ cos \beta &{} sin \beta &{} -sin \alpha ~ cos \beta \end{bmatrix}. \\ J&= \begin{bmatrix} J_{xx} &{} 0 &{} -J_{xz} \\ 0 &{} J_{yy} &{} 0 \\ -J_{xz} &{} 0 &{} J_{zz} \end{bmatrix}: \zeta ^{\times } =\begin{bmatrix} 0 &{} -\zeta _z &{} \zeta _y \\ \zeta _z &{} 0 &{} -\zeta _x \\ -\zeta _y &{} \zeta _x &{} 0 \end{bmatrix}. \end{aligned}$$

2.1 Discrete time model

The nonlinear model (7) can be transformed into a discrete-time model using the Euler method [42]. If the sampling interval is chosen as \(T_s\), an approximate discrete time model can be derived as

$$\begin{aligned} {\left\{ \begin{array}{ll} \varOmega (k+1) = \varOmega (k) + T_sf(\varOmega (k)) + T_sg(\zeta (k)) \zeta (k). \\ \zeta (k+1) = \zeta (k)-J^{-1}\zeta ^{\times }(k) J \zeta (k) + T_s J^{-1} u(k). \end{array}\right. } \end{aligned}$$
(8)

2.2 Feedback linearization

Selecting the attitude angle vector, \(\varOmega \), as the output, and u as the control input vector, the relative degree of the system (7) is (2,2,2). By using the approach presented in [43], the following equation can be obtained

$$\begin{aligned} \varOmega (k+2)&= \varOmega (k+1) + T_sf(\varOmega (k+1)) \nonumber \\&\quad + T_sg(\zeta (k+1)) \zeta (k+1). \nonumber \\&= \varOmega (k+1) + T_sf(\varOmega (k+1)) \nonumber \\&\quad + T_sg(\zeta (k+1)) [\zeta (k)-J^{-1}\zeta ^{\times }(k) J \zeta (k) ] \nonumber \\&\quad + T_s^2g(\zeta (k+1)) J^{-1} u(k). \end{aligned}$$
(9)

Then, (9) can be rewritten as

$$\begin{aligned} \varOmega (k+2)&= F(\varOmega (k),\zeta (k)) + G(\zeta (k)) u(k). \end{aligned}$$
(10)

where

$$\begin{aligned} F(\varOmega (k),\zeta (k))&= \varOmega (k+1) + T_sf(\varOmega (k+1)) \\&\quad + T_s g(\zeta (k+1)) [\zeta (k)-J^{-1}\zeta ^{\times }(k) J \zeta (k) ]. \\ G(\zeta (k))&= T_s^2g(\zeta (k+1)) J^{-1}. \end{aligned}$$

From (10), the following discrete-time strict feedback system is derived

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbf {X}}_1(k+1) = {\mathbf {X}}_2(k).\\ {\mathbf {X}}_2(k+1) = F(\varOmega (k),\zeta (k)) + G(\zeta (k)) u(k). \\ \end{array}\right. } \end{aligned}$$
(11)

where \( {\mathbf {X}}_1(k) = \varOmega (k)\), \({\mathbf {X}}_2(k)=\varOmega (k+1)\). Equation (11) can be completely linearized using the following feedback control law

$$\begin{aligned} u(k) = G(\zeta (k))^{-1}[-F(\varOmega (k),\zeta (k)) + v(k)]. \end{aligned}$$
(12)

where \(v(k) = [v_x~v_y~v_z]^T\) is the vector of the new inputs to stabilize the linearized system. Using (12), the exact linearized system is as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbf {X}}_1(k+1) &{}= {\mathbf {X}}_2(k). \\ {\mathbf {X}}_2(k+1) &{}=v(k) + \varDelta . \end{array}\right. } \end{aligned}$$
(13)

Remark 1

In real situation, exact feedback linearization is not possible due to the parametric uncertainties and external disturbances affecting the vehicle during the reentry phase. As such, \(\varDelta \) is added to (13) to account for the bounded disturbances.

Equation (13) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbf {X}}(k+1) = A_0 {\mathbf {X}}(k) + B_0 v + \varGamma \varDelta .\\ y(k) = C_0 {\mathbf {X}}(k). \end{array}\right. } \end{aligned}$$
(14)
Fig. 2
figure 2

The block diagram of the optimal control of the reentry vehicle

Full size image

where \({\mathbf {X}}(k) = [{\mathbf {X}}_1(k)^T~{\mathbf {X}}_2(k)^T]^T \in \mathfrak {R}^{1 \times 6}\), \(A_0 \in \mathfrak {R}^{6 \times 6}\), \(B_0 \in \mathfrak {R}^{6 \times 3}\), \(\varGamma \in \mathfrak {R}^{6 \times 3}\) and \(C_0 \in \mathfrak {R}^{3 \times 6}\) are given by

$$\begin{aligned} A_0&= \begin{bmatrix} 0 &{} I \\ 0 &{} 0 \end{bmatrix};~~ B_0 = \begin{bmatrix} 0 \\ I \end{bmatrix}; ~~ \varGamma = \begin{bmatrix} 0 \\ I \end{bmatrix}; ~~ C_0 = \begin{bmatrix} I&0 \end{bmatrix}; \\ I&= diag \{1,1,1 \} . \end{aligned}$$

Remark 2

Using Matlab, rank([B AB])=6 which is the same as the order of the system. Therefore, the linearized system is controllable.

3 Control design

In this section, LQRP, LQRP + D and LQRIP + D are formulated in terms LMI for the system (14). The block diagram of the closed loop system is shown in Fig. 2.

3.1 LQRP design

In this section, a LMI based LQRP is designed for (14) with \(\varDelta =0\). Consider the following cost function

$$\begin{aligned} J = \sum _{k=0}^{\infty } [y_k^T Q y_k + v_k^T R v_k]. \end{aligned}$$
(15)

where \(Q>0\) and \(R>0\) represent the output error and the control weighting matrices respectively. We go a head to establish the linear optimal control law \(u(k)=L{\mathbf {X}}_k\) that stabilizes (14) while minimizing (15). Assuming that the Lyapunov function \(V_k\) has the following properties:

$$\begin{aligned} V_k = {\mathbf {X}}_k^T M {\mathbf {X}}_k, ~~M>0. \end{aligned}$$
(16)

There exists \(\vartheta _+ >0\) such that \({\mathbf {X}}_0^T M {\mathbf {X}}_0 \le \vartheta _+ \).

$$\begin{aligned} V_k \le -[{\mathbf {X}}_k^T C^T Q C {\mathbf {X}}_k + v_k^T R v_k]. \end{aligned}$$
(17)

Then, (14) under \(v_k\) is asymptotically stable and \(J_{\infty } \le V({\mathbf {X}}_0)\). Using \(v_k=L{\mathbf {X}}_k\), one has

$$\begin{aligned} {\mathbf {X}}_k^T[M(A+BL) + (A+BL)^TM^T]{\mathbf {X}}_k \nonumber \\ \le -{\mathbf {X}}_k^T[C^TQC + L^T R L]{\mathbf {X}}_k. \end{aligned}$$
(18)

If there exists L and M such that \({\mathbf {X}}_k \ne 0\), one gets

$$\begin{aligned} M(A+BL) + (A+BL)^TM^T + C^TQC + L^T R L \le 0. \end{aligned}$$
(19)

The upper bound on the cost \({\mathbf {X}}_0^TM{\mathbf {X}}_0\) can be minimized for a given \(\vartheta _+\) as

$$\begin{aligned} \min _{\vartheta _+, K,L} \vartheta _+ ~~ subject ~to ~(19) \end{aligned}$$

Equation (19) can be expressed as

$$\begin{aligned} \varUpsilon&= \begin{bmatrix} \psi &{} C^TQ &{} L^TR \\ * &{} -Q * 0\\ * &{} * &{} -R \end{bmatrix} \le 0. \end{aligned}$$
(20)

where \(\psi = M(A+BL) + (A+BL)^TM^T\). Pre- and post-multiplying (20) by \(diag \{ Y,I,I \}\) and considering \(Y=M^{-1}\) and \(S=LY\), the convex optimization of (19) is expressed as

$$\begin{aligned}&\begin{bmatrix} ({A} {Y}+ {B} {S})+( {A} {Y}+ {B} {S})^T &{} {Y} {Q} &{} {Y} {L}^T {R} \\ * &{} - {Q} &{} 0 \\ * &{} * &{}- {R} \end{bmatrix} \le 0, \nonumber \\&\begin{bmatrix} \vartheta _+ &{} \eta _0^T \\ * &{} Y \end{bmatrix} \ge 0. \end{aligned}$$
(21)

The bound of the Lyapunov function is thus

$$\begin{aligned} \begin{bmatrix} \vartheta _+ &{} {\mathbf {X}}_0^T \\ * &{} M^{-1} \end{bmatrix} \ge 0 \Leftrightarrow \begin{bmatrix} \vartheta _+ &{} \eta _0^T \\ * &{} Y \end{bmatrix} \ge 0. \end{aligned}$$
(22)

After attaining a feasible solution, one has \( {L} = {S} {Y}^{-1}\), \( {M} = {Y}^{-1}\)

3.2 LQRP + D

The LQRP designed in Sect. 3.1 does not consider the effects of external disturbances and unmodeled dynamics. In order to account for these problems, a new cost term is added to the cost function (15). The modified cost function is given by

$$\begin{aligned} J = \sum _{k=0}^{\infty } [y_k^T \tilde{Q} y_k + \varOmega v_k^T v_k - \varLambda ^2 \varDelta ^T \varDelta ]. \end{aligned}$$
(23)

Letting \({\tilde{R}} = diag[\varOmega ~~-\varLambda ^2]\) and \(U_k = [v_k^T~~\varDelta ^T]^T\), we get

$$\begin{aligned} J = \sum _{k=0}^{\infty }[y_k^T {\tilde{Q}} y_k + U_k^T {\tilde{R}} U_k]. \end{aligned}$$
(24)

We define the following Lyapunov function to ensure closed loop stability

$$\begin{aligned} {\tilde{V}}_k = {\mathbf {X}}_k^T P {\mathbf {X}}_k, ~~{\tilde{V}}_k>0. \end{aligned}$$
(25)

There exists \(\vartheta >0\) such that \({\mathbf {X}}_0^T P {\mathbf {X}}_0 \le \vartheta _+ \). Then, it follows that

$$\begin{aligned} {\tilde{V}}_{k+1} \le -[{\mathbf {X}}_k^T {\tilde{Q}} {\mathbf {X}}_k + U_k^T {{\tilde{R}}} U_k]. \end{aligned}$$
(26)

Equation (14) under \(U_k\) is asymptotically stable and \(J_{\infty } \le {\tilde{V}}_k({\mathbf {X}}_0)\). Substituting \(U_k =H {\mathbf {X}}_k\) into (26) yields

$$\begin{aligned} {\mathbf {X}}_k^T[P(A+GH) + (A+GH)^T P^T]{\mathbf {X}}_k \nonumber \\ \le -{\mathbf {X}}_k^T[{\tilde{Q}} + H^T {{\tilde{R}}} H]{\mathbf {X}}_k. \end{aligned}$$
(27)

where \(G = [B~~ \varGamma ]\). If there exist H and P with \({\mathbf {X}}_k \ne 0\), (27) becomes

$$\begin{aligned} P(A+GH) + (A+GH)^T P^T +{\tilde{Q}} + H^T {{\tilde{R}}} H \le 0. \end{aligned}$$
(28)

The minimization problem can be written as

$$\begin{aligned} \min _{\vartheta _+, P,H} \vartheta _+ ~~subject ~to ~(28) \end{aligned}$$

Using the same approach as the foregoing section, we have

$$\begin{aligned}&\begin{bmatrix} ({A} {Z}+ {B} {T})+( {A} {Z}+ {B} {T})^T &{} {Z} C^T \tilde{Q} &{} {Z} {F}^T {{\tilde{R}}} \\ * &{} - {\tilde{Q}} &{} 0 \\ * &{} * &{}- {{\tilde{R}}} \end{bmatrix} \le 0. \nonumber \\&\begin{bmatrix} \vartheta _+ &{} \eta _0^T \\ * &{} Z \end{bmatrix} \ge 0. \end{aligned}$$
(29)

On reaching feasible solution, one gets \( {H} = {T} {Z}^{-1}\), \( {P} = {Z}^{-1}\).

3.3 LQRPI + D

In this section, the LQRP + D formulation is modified by including an extra cost term denoting the integral of the deviation of the output from its initial state, \(\theta = \int _0^t y(\tau ) d \tau \) (\({\dot{\theta }} = Cx\)). The discrete time equivalent of the integral of the output can be expressed as

$$\begin{aligned} \varepsilon _{k+1} = C {\mathbf {X}}_k. \end{aligned}$$
(30)

where \(\varepsilon _{k+1}=\theta _{k+1} - \theta _{k} \). The new cost function is given by

$$\begin{aligned} J = \sum _{k=0}^{\infty } [y_k^T y_k + \varepsilon _k^T \varepsilon _k + \sigma v_k^T v_k - \varLambda ^2 \varDelta ^T \varDelta ]. \end{aligned}$$
(31)

Introducing \(U_k = [v_k^T~~\varDelta ^T]^T\), \(\chi =[{\mathbf {X}}_k^T~~\varepsilon _k^T]^T\), we obtain

$$\begin{aligned} {\dot{\chi }}&= \begin{bmatrix} A &{} 0 \\ C &{} 0 \end{bmatrix} \chi +\begin{bmatrix} B \\ 0 \end{bmatrix} v_k +\begin{bmatrix} \varGamma \\ 0 \end{bmatrix} \varDelta , \nonumber \\&= {\hat{A}} \chi + {\hat{B}} v_k + {\hat{\varGamma }} \varDelta . \end{aligned}$$
(32)

Equation (31) can be rewritten as

$$\begin{aligned} J = \sum _{k=0}^{\infty } [\chi _k^T {\hat{Q}} \chi _k + U_k^T {\hat{R}} U_k ]. \end{aligned}$$
(33)

where

$$\begin{aligned} {\hat{Q}}&= \begin{bmatrix} C^T C &{} 0 \\ 0 &{} \vartheta I \end{bmatrix};~~ {\hat{R}} =\begin{bmatrix} \sigma I &{} 0\\ 0 &{} \varLambda ^2 I \end{bmatrix} . \end{aligned}$$
Fig. 3
figure 3

Attack angle response

Full size image

We consider the following Lyapunov function

$$\begin{aligned} {\hat{V}}_k = {\mathbf {X}}_k^T {\hat{P}} {\mathbf {X}}_k, ~~{\hat{V}}_k>0. \end{aligned}$$
(34)

There exists \(\vartheta >0\) such that \(\chi _0^T {\hat{P}} \chi _0 \le \vartheta _+ \). Then,

$$\begin{aligned} {\hat{V}}_{k+1} \le -[\chi _k^T {\hat{Q}} \chi _k + U_k^T {\hat{R}} U_k]. \end{aligned}$$
(35)

Using \(U_k = {\hat{H}} {\mathbf {X}}_k\), \(\forall \chi \ne 0\), we have

$$\begin{aligned} {\hat{P}}({\hat{A}}+{\hat{G}} {\hat{H}}) + ({\hat{A}}+ {\hat{G}} {\hat{H}})^T {\hat{P}}^T +{\hat{Q}} +{\hat{H}}^T {\hat{R}} {\hat{H}} \le 0. \end{aligned}$$
(36)

where \({\hat{G}} = [{\hat{B}}~~ {\hat{\varGamma }}]\). The minimization problem is thus

$$\begin{aligned} \min _{\vartheta _+, {\hat{P}},{\hat{H}}} \vartheta _+ ~~subject ~to ~(36) \end{aligned}$$

Based on convex analysis, problem (36) is converted to

$$\begin{aligned}&\begin{bmatrix} ({\hat{A}} {\hat{Z}}+ {\hat{B}} {\hat{T}})+( {\hat{A}} {Z}+ {\hat{B}} {\hat{T}})^T &{} {\hat{Z}} C^T {\hat{Q}} &{} {\hat{Z}} {\hat{F}}^T {\hat{R}} \\ * &{} - {\hat{Q}} &{} 0 \\ * &{} * &{}- {\hat{R}} \end{bmatrix} \le 0. \nonumber \\&\begin{bmatrix} \vartheta _+ &{} \eta _0^T \\ * &{} {\hat{Z}} \end{bmatrix} \ge 0. \end{aligned}$$
(37)

A feasible solution gives \( {\hat{H}} = {\hat{T}} {\hat{Z}}^{-1}\), \( {\hat{P}} = {\hat{Z}}^{-1}\).

Fig. 4
figure 4

Side slip angle response

Full size image
Fig. 5
figure 5

Bank angle response

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4 Simulation results

Numerical simulations are presented in this section to illustrate the efficacy of the proposed scheme. The simulations are carried out using the LMI MATLAB toolbox. Moreover, a comparison is made with an existing discrete time backstepping [40], discrete time sliding mode control [41] and LMI based \(H_{\infty }\) control [44]. The parameters of the reentry vehicle are adopted from [34]. The external disturbances and parametric uncertainties are modelled as constants \(\varDelta = [0.1~ 0.4 ~ 0.3]^T\). The weight matrices are chosen as \(Q=C^TC\) and \(R=0.01 \).

The simulation results are presented in Figs. 3, 4 and 5. The response of the angle of attack is depicted in Fig. 3. Due to the presence of the disturbances, the LQRP struggle to stabilized the attack angle. The LQRP + D, \(H_{\infty }\), sliding mode control and backstepping are able to suppress the disturbances and improve the performance of the LQRP. The LQRPI + D has the least overshoot and converge to zero at a quicker rate. In Fig. 4, the bank angle response is plotted. It can be seen that the LQRPI + D provides superior performance compared to the LQRP, LQRP + D, \(H_{\infty }\), sliding mode control and backstepping with respect to overshoot and settling time. The sideslip angle response is shown in Fig. 5. The sideslip angle is stabilized to zero at a faster rate under the action of LQRPI + D compared to the LQRP, LQRP + D, \(H_{\infty }\), sliding mode control and backstepping. The superior performance of the proposed LMI based LQRPI + D is due to its optimal gains, disturbance rejection capability and integral action.

5 Conclusions

This paper presents LQRPI + D based on LMI optimization to stabilized the attitude angle of the reentry vehicle, by considering the external disturbances and parametric variations. The stability of the proposed controller is proved by Lyapunov function. Simulation results illustrate that the LQRPI + D can mitigate the effects of the external perturbations and enhance the vehicle robust control action. In addition, the proposed LQRPI + D outperforms LQRP, LQRP + D and \(H_{\infty }\).