Design of Fast Response Compound Control System for Hypersonic Skid-to-Turn Missile

Abstract

A skid-to-turn (STT) missile is an axisymmetric structure missile, and its control system consists of a pitch channel and a yaw channel with an axisymmetric structure. To achieve the fast response of the STT missile system, a compound control method of aerodynamic force and lateral thrust based on regional pole assignment (RPA) is proposed. In the aerodynamic control system, the linear quadratic regulator (LQR) is used to design the controller, and the sliding mode control method is used to design the controller of the lateral thrust system. The regional pole assignment is introduced into the aerodynamic system to improve the compound control system response speed. The problems of regional pole assignment and system stability are solved by a linear matrix inequality (LMI). Considering that the missile flies at different altitudes, the missile system is controlled by gain scheduling. Compared to previous designs of time-varying compound control systems for STT missiles or hypersonic vehicles, in order to meet the practical requirement of a fast response for the vehicle, this time-varying compound control strategy can achieve faster tracking response and attitude control for the STT missile. Finally, through simulations of the pitch channel and yaw channel control systems of the STT missile, the effectiveness of the designed compound control system in achieving a fast response is verified.

1. Introduction

Hypersonic missiles are more than 5 times the speed of sound flight of missiles, have the advantages of fast response speeds and penetration abilities, and are important combat aircraft in the field of aviation [1]. Typical missiles are the Russian “Zircon” missile, the United States “Dark Eagle” missile, and the Chinese “DF-17” missile [2,3,4]. It can be divided into skid-to-turn (STT) missiles and bank-to-turn (BTT) missiles according to different turning modes in the course of flight [5,6]. The BTT missile is a non-axisymmetric aircraft. However, an STT missile is an axisymmetric aircraft with a fast response speed and a relatively simple system structure. The pitch channel, yaw channel, and roll channel of the STT missile can be designed separately [7,8,9]. With the rapid development of aircraft in the field of aviation, it is particularly important to design a fast response control system for the STT missile to further improve the response speed.

In the study of hypersonic missile control system design, many researchers only consider the control scheme of the aerodynamic system and do not consider the constraint when the aerodynamic rudder of the vehicle reaches saturation [10,11,12,13,14]. When the aerodynamic rudder is saturated, the control force and torque cannot be generated continuously, which makes the system unstable and then malfunction [15,16]. To make the missile control system stable, it is very important to add the lateral thrust provided by the engine to the system so that the missile control system can continue to work stably. There are many studies on compound control methods. Reference [17] applies neural networks and sliding mode control to robotic systems. In [18], neural network and proportional integral derivative (PID) controls are used to design the controller of the electromagnetic linear simulator. References [19,20] apply neural networks, adaptive control, and integral sliding modes to inertially stabilized platform systems. However, the research object of the above compound control method is not a hypersonic vehicle. To design the compound control system of a hypersonic missile, a compound control method based on linear quadratic regulator (LQR) and sliding mode control is proposed in this paper. At the same time, in order to achieve the fast response of the system, the regional pole assignment (RPA) is applied to the aerodynamic system to improve the response speed of the system.

The control methods used to improve the system response speed include the PID control method, adaptive control method, pulse control method, pole assignment method, etc. [21,22,23,24,25]. Among them, the PID control method is suitable for the controller design of the time-invariant system [26,27]. Although the adaptive control method is suitable for variable parameter systems, the algorithm complexity is high and the operation time is long [28,29]. Pulse control sends pulse signals to make the machine work, and the response speed is fast, but pulse control is widely used in such fields as computer numerical control machine tools and robots [30,31]. There are few studies on the fast response control method of the hypersonic vehicle. Reference [32] adopts the pole assignment control method to improve the response speed of the vehicle, but the control method is suitable for time-invariant systems. However, the regional pole assignment control method is used in this paper to improve the response speed of hypersonic missile variable parameter systems.

The gain scheduling control method has a wide range of applications, such as power plant control, automotive control, process control, and aircraft flight control [33,34,35,36]. Gain scheduling control can dynamically adjust the controller parameters according to the current state of the system, which is suitable for nonlinear systems or systems whose parameters change with the operating points, so that the system can maintain good performance at different operating points [37]. Moreover, by adjusting controller parameters at different operating points, gain scheduling control can design a controller for a nonlinear system by using linear control theory [38]. Reference [39] realizes the target trajectory tracking of discrete systems through gain scheduling. Reference [40] applies gain scheduling to the robust control system of a spacecraft rendezvous. In this paper, gain scheduling is used to control hypersonic missiles at different flight altitudes.

Below is a comparison table of common control methods in the linear time-varying composite control of hypersonic vehicles, covering LQR (linear quadratic regulator), sliding mode control (SMC), adaptive control, robust control, and model predictive control (MPC), including their features, advantages, disadvantages, and applicable scenarios. A comparison of common control methods for linear time-varying systems is shown in

Table 1. Comparison of common control methods for linear time-varying systems.

Control Method	LQR (Linear Quadratic Regulator)	Sliding Mode Control (SMC)	Adaptive Control	Robust Control	Model Predictive Control (MPC)
Core Idea	Designs an optimal control law by minimizing a quadratic performance index (state error and control input).	Designs a sliding surface to drive the system state to the surface in finite time, with strong robustness to uncertainties and disturbances.	Estimates system parameters online and adjusts the control law to adapt to parameter variations and uncertainties.	Designs controllers by optimizing disturbance rejection performance (e.g., H∞ control) to enhance system robustness.	Predicts future system states based on the model and achieves control objectives by optimizing future control sequences.
Advantages	- Optimal performance under nominal conditions. - Clear design, easy to implement. - Low computational cost, suitable for real-time control.	- Strong robustness to uncertainties and disturbances. - Finite-time convergence. - Suitable for nonlinear systems.	- Capable of online estimation and compensation for parameter uncertainties. - Suitable for systems with fast parameter variations.	- Strong robustness to unmodeled dynamics and external disturbances. - Standardized design, easy to analyze.	- Capable of handling multivariable constraints. - Suitable for complex dynamic systems. - Excellent tracking performance.
Disadvantages	- Weak robustness to uncertainties and disturbances. - Requires an accurate system model.	- May introduce high-frequency chattering. - Complex design, requiring careful design of the sliding surface and control law.	- Slow convergence. - Weak robustness to unmodeled dynamics and external disturbances. - Complex design.	- High computational cost. - Limited ability to handle parameter uncertainties.	- High computational complexity, poor real-time performance. - Requires an accurate system model.
Applicable Scenarios	- High-performance control of nominal systems. - Scenarios requiring high real-time performance.	- Systems with strong uncertainties and disturbances. - Nonlinear system control.	- Systems with fast parameter variations and significant uncertainties. - Scenarios requiring online adjustment of control laws.	- Systems with unmodeled dynamics and external disturbances. - Scenarios requiring high robustness.	- Multivariable constrained systems. - Complex dynamic systems. - Scenarios requiring high tracking performance.
Combination with Other Methods	Often combined with sliding mode control or robust control to enhance robustness.	Often combined with LQR to balance optimal performance and robustness.	Often combined with robust control to enhance robustness to unmodeled dynamics and disturbances.	Often combined with LQR to balance optimal performance and robustness.	Often combined with robust control to enhance robustness to uncertainties and disturbances.

.

In practical applications, multiple control methods are often combined to balance performance, robustness, and real-time requirements, e.g., LQR and sliding mode control, adaptive control, and robust control.

In the linear time-varying composite control of aircraft, the choice of optimal control over adaptive control is primarily based on the unique characteristics of the aircraft, control requirements, and a comparison of the advantages and disadvantages of the two methods. Below is a detailed analysis:

(1)

Control Requirements of Aircraft

High Performance: Aircraft demand extremely high control precision and dynamic performance, especially in tasks such as altitude control and trajectory tracking.

Strong Robustness: During flight, aircraft face complex aerodynamic environments and parameter uncertainties.

Real-Time Performance: Control algorithms must complete calculations and output control commands within a very short time to meet the fast dynamic response requirements of aircraft.

Reliability: The control system of aircraft must be highly reliable to avoid flight accidents caused by control algorithm failures.

(2)

Advantages of Optimal Control

Optimal Performance: By minimizing performance indices (e.g., energy consumption and state error), optimal control provides the best system performance under nominal conditions.

Clear Design: The design of optimal control is based on explicit mathematical models and performance indices, making the design process standardized and easy to implement.

Robustness: Although optimal control itself has weak robustness to uncertainties, it can be combined with robust control methods (e.g., sliding mode control) to form a composite control strategy, enhancing system robustness.

Real-Time Performance: Optimal control calculations are typically based on offline-designed control laws, with low online computational requirements, making them suitable for real-time applications in aircraft.

(3)

Limitations of Adaptive Control

While adaptive control has advantages in handling parameter uncertainties, it has the following limitations in aircraft control:

Slow Convergence: Adaptive control requires online parameter estimation, which can be slow and may not meet the fast dynamic response requirements of aircraft.

High Complexity: The design and implementation of adaptive control are complex, especially in multivariable, strongly coupled aircraft systems.

Reliability Issues: Adaptive control relies on online parameter estimation. If estimation errors or failures occur, the control system may become unstable.

(4)

Compound Control Strategies in Aircraft

In practical applications, aircraft control typically adopts compound control strategies, combining optimal control with other methods (e.g., robust control and sliding mode control) to compensate for the limitations of a single method. Such compound strategies retain the performance advantages of optimal control while enhancing system robustness and adaptability through the introduction of other methods. Although adaptive control has advantages in handling parameter uncertainties, its slow convergence, design complexity, and reliability issues limit its application in aircraft.

The contributions of this paper are summarized as follows: (1) The research object of this paper is a hypersonic STT missile, whose pitch channel and yaw channel are mutually axisymmetric. The compound control models of these two channels are given, and the compound control systems of aerodynamic force and lateral thrust are designed based on LQR and sliding mode control theory. (2) In order to improve the response speed of the system, the regional pole assignment (RPA) method is proposed to design the controller in the aerodynamic control system. The problems of RPA and system stability are solved by linear matrix inequality (LMI). (3) The control of STT missiles at different flight altitudes is realized by gain scheduling.

The structure of this paper is organized as follows: The mathematical model of the compound control system of the pitch channel and yaw channel of STT missiles is given in Section 2. In Section 3, the compound controllers of two channels are designed. To improve the response speed of the system, the regional pole assignment is proposed to design the controller in the aerodynamic control system. Section 4 gives the comparison simulation and the result analysis. In Section 5, the conclusion is given.

2. STT Missile Model

The hypersonic STT missile system can be decoupled into a pitch channel, a yaw channel, and a roll channel. The pitch channel and yaw channel are axisymmetric to each other. The hypersonic STT missile system is controlled by the aerodynamic force provided by the rudder and the lateral thrust provided by the engine. The compound control model of the pitch channel is established, and the yaw channel control model is given according to the axisymmetric structure.

2.1. Pitch Channel Model

According to the flight dynamics characteristics of STT missiles, a variable parameter system model is established, and the model is approximately linearized. The altitude dynamic equation of the STT missile is expressed as follows:

$$\dot{\alpha }={\omega }_z-\alpha Y^{\alpha }(t)/\left(mV\right)-{\delta }_z{Y}^{{\delta }_z}/\left(mV\right)-k_y{F}_{Ty}$$

(1)

$${\dot{\omega }}_z=M_z^{{\omega }_z}{\omega }_z/J_z+\alpha M_z^{\alpha }(t)/J_z+M_z^{{\delta }_z}{\delta }_z/J_z-l_z{F}_{Ty}$$

(2)

$$n_y={\displaystyle \frac{V}{g}}(\alpha Y^{\alpha }(t)/\left(mV\right)+{\delta }_z{Y}^{{\delta }_z}/\left(mV\right)+k_y{F}_{Ty})$$

(3)

By differentiating Equation (3), we obtain

$${\dot{n}}_y={\displaystyle \frac{V}{g}}(\dot{\alpha }{Y}^{\alpha }(t)/\left(mV\right)+{\dot{\delta }}_z{Y}^{{\delta }_z}/\left(mV\right)+k_y{\dot{F}}_{Ty})$$

(4)

The dynamic process of the actuator is described as

$${\dot{\delta }}_z=-{\tau }_1{\delta }_z+{\tau }_1{\delta }_{zc}$$

(5)

$${\dot{F}}_{Ty}=-{\tau }_2{F}_{Ty}+{\tau }_2{F}_{Tyc}$$

(6)

Equation (1), Equation (5), and Equation (6) are substituted into Equation (4) and sorted as

$${\dot{n}}_y=-{\displaystyle \frac{Y^{\alpha }(t)}{mV}}{n}_y+{\displaystyle \frac{Y^{\alpha }(t)}{mg}}{\omega }_z-{\displaystyle \frac{Y^{{\delta }_z}}{mg}}{\tau }_1{\delta }_z-{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{F}_{Ty}+{\displaystyle \frac{Y^{{\delta }_z}}{mg}}{\tau }_1{\delta }_{zc}+{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{F}_{Tyc}$$

(7)

By substituting Equation (3) into Equation (2), we obtain

$$\begin{array}{c}{\dot{\omega }}_z={\omega }_z{M}_z^{{\omega }_z}/J_z+M_z^{\alpha }(t)\left({\displaystyle \frac{g}{Y^{\alpha }(t)/m}}{n}_y-{\displaystyle \frac{Y^{{\delta }_z}}{Y^{\alpha }(t)}}{\delta }_z-{\displaystyle \frac{k_y}{Y^{\alpha }(t)/\left(mV\right)}}{F}_{Ty}\right)/J_z+{\delta }_z{M}_z^{{\delta }_z}/J_z-l_z{F}_{Ty}\\ ={\displaystyle \frac{M_z^{{\omega }_z}}{J_z}}{\omega }_z+{\displaystyle \frac{M_z^{\alpha }(t)mg}{Y^{\alpha }(t)J_z}}{n}_y-\left({\displaystyle \frac{M_z^{\alpha }(t)Y^{{\delta }_z}}{Y^{\alpha }(t)J_z}}-{\displaystyle \frac{M_z^{{\delta }_z}}{J_z}}\right){\delta }_z-\left({\displaystyle \frac{M_z^{\alpha }(t)k_{y}mV}{Y^{\alpha }(t)J_z}}+l_z\right)F_{Ty}\end{array}$$

(8)

Therefore, the pitch channel motion equation is described as follows:

$$\{\begin{array}{l}{\dot{n}}_y=-{\displaystyle \frac{Y^{\alpha }(t)}{mV}}{n}_y+{\displaystyle \frac{Y^{\alpha }(t)}{mg}}{\omega }_z-{\displaystyle \frac{Y^{{\delta }_z}}{mg}}{\tau }_1{\delta }_z-{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{F}_{Ty}+{\displaystyle \frac{Y^{{\delta }_z}}{mg}}{\tau }_1{\delta }_{zc}+{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{F}_{Tyc}\\ {\dot{\omega }}_z={\displaystyle \frac{M_z^{{\omega }_z}}{J_z}}{\omega }_z+{\displaystyle \frac{M_z^{\alpha }(t)mg}{Y^{\alpha }(t)J_z}}{n}_y-\left({\displaystyle \frac{M_z^{\alpha }(t)Y^{{\delta }_z}}{Y^{\alpha }(t)J_z}}-{\displaystyle \frac{M_z^{{\delta }_z}}{J_z}}\right){\delta }_z-\left({\displaystyle \frac{M_z^{\alpha }(t)k_{y}mV}{Y^{\alpha }(t)J_z}}+l_z\right)F_{Ty}\\ {\dot{\delta }}_z=-{\tau }_1{\delta }_z+{\tau }_1{\delta }_{zc}\\ {\dot{F}}_{Ty}=-{\tau }_2{F}_{Ty}+{\tau }_2{F}_{Tyc}\end{array}$$

(9)

where $n_y$ is the output acceleration of the pitch channel; $g$ is the gravitational acceleration; ${\omega }_z$ is the angular velocity of the pitch channel; ${\delta }_z$ and ${\delta }_{zc}$ represent the elevator deflection angle and its command in the pitch channel, respectively; ${\tau }_1$ and ${\tau }_2$ represent time constants of aerodynamic force and lateral thrust in the pitch channel, respectively; $m$ and $V$ represent the mass and flight speed of STT missiles, respectively; $k_y=1/(mV)$; $l_z=-l/J_z$; $l$ is the distance from the center of force to the center of mass; $J_z$ is the component of the moment of inertia on the Z axis; $F_{Ty}$ and $F_{Tyc}$ represent the thrust and thrust command of the engine in the pitch channel, respectively; $M_z^{{\omega }_z}$, $M_z^{{\delta }_z}$, and $M_z^{\alpha }(t)$ represent the partial derivatives of the pitching moment $M_z$ to ${\omega }_z$, ${\delta }_z$, and $\alpha$, respectively; and $Y^{{\delta }_z}$ and $Y^{\alpha }(t)$ represent the partial derivatives of lift $Y$ to ${\delta }_z$ and $\alpha$, respectively. The aerodynamic coefficient in the pitch channel is denoted as

$$\{\begin{array}{l}{a}_1=-M_z^{{\omega }_z}/J_z\hfill \\ a_2(t)=-M_z^{\alpha }(t)/J_z\hfill \\ a_3=-M_z^{{\delta }_z}/J_z\hfill \\ a_4(t)=Y^{\alpha }(t)/(mV)\hfill \\ a_5=Y^{{\delta }_z}/(mV)\hfill \end{array}$$

(10)

Then, the system model of the channel is written as follows:

$$\{\begin{array}{l}{\dot{n}}_y=-a_4(t)n_y+{\displaystyle \frac{V}{g}}{a}_4(t){\omega }_z-{\displaystyle \frac{V}{g}}{a}_5{\tau }_1{\delta }_z-{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{F}_{Ty}+{\displaystyle \frac{V}{g}}{a}_5{\tau }_1{\delta }_{zc}+{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{F}_{Tyc}\\ {\dot{\omega }}_z=-a_1{\omega }_z-{\displaystyle \frac{a_2(t)g}{a_4(t)V}}{n}_y+\left({\displaystyle \frac{a_2(t)a_5}{a_4(t)}}-a_3\right){\delta }_z+\left({\displaystyle \frac{a_2(t)k_y}{a_4(t)}}-l_z\right)F_{Ty}\\ {\dot{\delta }}_z=-{\tau }_1{\delta }_z+{\tau }_1{\delta }_{zc}\\ {\dot{F}}_{Ty}=-{\tau }_2{F}_{Ty}+{\tau }_2{F}_{Tyc}\end{array}$$

(11)

The tracking error of the output acceleration in the pitch channel is defined as

$$e_y=n_{yc}-n_y$$

(12)

where $n_{yc}$ is the pitch acceleration tracking command. To enhance the tracking accuracy of pitch acceleration $n_y$, the tracking error integral term is chosen. Equations (11) and (12) are described as the state-space form. The state vectors are $x_1={\displaystyle {\int }_0^t{e}_{y}dt}$, $x_2=e_y$, $x_3={\omega }_z$, $x_4={\delta }_z$, and $x_5=F_{Ty}$. The control vectors are $u_1={\delta }_{zc}$ and $u_2=F_{Tyc}$, and the equation of state is denoted as

$$\{\begin{array}{l}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=-a_4(t)x_2-{\displaystyle \frac{V}{g}}{a}_4(t)x_3+{\displaystyle \frac{V}{g}}{a}_5{\tau }_1{x}_4+{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{x}_5-{\displaystyle \frac{V}{g}}{a}_5{\tau }_1{u}_1-{\displaystyle \frac{V}{g}}{k}_y{\tau }_2{u}_2\hfill \\ {\dot{x}}_3={\displaystyle \frac{a_2(t)g}{a_4(t)V}}{x}_2-a_1{x}_3+\left({\displaystyle \frac{a_2(t)a_5}{a_4(t)}}-a_3\right)x_4+\left({\displaystyle \frac{a_2(t)k_y}{a_4(t)}}-l_z\right)x_5\hfill \\ {\dot{x}}_4=-{\tau }_1{x}_4+{\tau }_1{u}_1\hfill \\ {\dot{x}}_5=-{\tau }_2{x}_5+{\tau }_2{u}_2\hfill \end{array}$$

(13)

2.2. Yaw Channel Model

According to the axial symmetrical structure of the STT missile, the altitude dynamic equation of the STT missile is expressed as follows:

$$\dot{\beta }={\omega }_y+\beta Z^{\beta }(t)/\left(mV\right)+{\delta }_y{Z}^{{\delta }_y}/\left(mV\right)-k_z{F}_{Tz}$$

(14)

$${\dot{\omega }}_y={\omega }_y{M}_y^{{\omega }_y}/J_y+\beta M_y^{\beta }(t)/J_y+{\delta }_y{M}_y^{{\delta }_y}/J_y-l_y{F}_{Tz}$$

(15)

$$n_z={\displaystyle \frac{V}{g}}(\beta Z^{\beta }(t)/\left(mV\right)+{\delta }_y{Z}^{{\delta }_y}/\left(mV\right)-k_z{F}_{Tz})$$

(16)

By differentiating Equation (16), we obtain

$${\dot{n}}_z={\displaystyle \frac{V}{g}}(\dot{\beta }{Z}^{\beta }(t)/\left(mV\right)+{\dot{\delta }}_y{Z}^{{\delta }_y}/\left(mV\right)-k_z{\dot{F}}_{Tz})$$

(17)

The dynamic process of the actuator is described as

$${\dot{\delta }}_y=-{\tau }_1{\delta }_y+{\tau }_1{\delta }_{yc}$$

(18)

$${\dot{F}}_{Tz}=-{\tau }_2{F}_{Tz}+{\tau }_2{F}_{Tzc}$$

(19)

Equation (14), Equation (18), and Equation (19) are substituted into Equation (17) and sorted as

$${\dot{n}}_z={\displaystyle \frac{Z^{\beta }(t)}{mV}}{n}_z+{\displaystyle \frac{Z^{\beta }(t)}{mg}}{\omega }_y-{\displaystyle \frac{Z^{{\delta }_y}}{mg}}{\tau }_1{\delta }_y+{\displaystyle \frac{V}{g}}{k}_z{\tau }_2{F}_{Tz}+{\displaystyle \frac{Z^{{\delta }_y}}{mg}}{\tau }_1{\delta }_{yc}-{\displaystyle \frac{V}{g}}{k}_z{\tau }_2{F}_{Tzc}$$

(20)

By substituting Equation (16) into Equation (15), we obtain

$${\dot{\omega }}_y={\displaystyle \frac{M_y^{{\omega }_y}}{J_y}}{\omega }_y+{\displaystyle \frac{M_y^{\beta }(t)mg}{Z^{\beta }(t)J_y}}{n}_z+\left({\displaystyle \frac{M_y^{{\delta }_y}}{J_y}}-{\displaystyle \frac{M_y^{\beta }(t)Z^{{\delta }_y}}{Z^{\beta }(t)J_y}}\right){\delta }_y+\left({\displaystyle \frac{M_y^{\beta }(t)k_{z}mV}{Z^{\beta }(t)J_y}}-l_y\right)F_{Tz}$$

(21)

The dynamics model of the yaw channel is written as follows:

$$\{\begin{array}{l}{\dot{n}}_z={\displaystyle \frac{Z^{\beta }(t)}{mV}}{n}_z+{\displaystyle \frac{Z^{\beta }(t)}{mg}}{\omega }_y-{\displaystyle \frac{Z^{{\delta }_y}}{mg}}{\tau }_1{\delta }_y+{\displaystyle \frac{V}{g}}{k}_z{\tau }_2{F}_{Tz}+{\displaystyle \frac{Z^{{\delta }_y}}{mg}}{\tau }_1{\delta }_{yc}-{\displaystyle \frac{V}{g}}{k}_z{\tau }_2{F}_{Tzc}\\ {\dot{\omega }}_y={\displaystyle \frac{M_y^{{\omega }_y}}{J_y}}{\omega }_y+{\displaystyle \frac{M_y^{\beta }(t)mg}{Z^{\beta }(t)J_y}}{n}_z+\left({\displaystyle \frac{M_y^{{\delta }_y}}{J_y}}-{\displaystyle \frac{M_y^{\beta }(t)Z^{{\delta }_y}}{Z^{\beta }(t)J_y}}\right){\delta }_y+\left({\displaystyle \frac{M_y^{\beta }(t)k_{z}mV}{Z^{\beta }(t)J_y}}-l_y\right)F_{Tz}\\ {\dot{\delta }}_y=-{\tau }_1{\delta }_y+{\tau }_1{\delta }_{yc}\\ {\dot{F}}_{Tz}=-{\tau }_2{F}_{Tz}+{\tau }_2{F}_{Tzc}\end{array}$$

(22)

where $n_z$ is the output acceleration of the yaw channel; ${\omega }_y$ is the angular velocity of the yaw channel; ${\delta }_y$ and ${\delta }_{yc}$ represent the rudder deflection angle and its command in the yaw channel, respectively; ${\tau }_1$ and ${\tau }_2$ represent time constants of aerodynamic force and lateral thrust in the yaw channel, respectively, for the axisymmetric missile; ${\tau }_1$ and ${\tau }_2$ in the pitch and yaw channels are equal; $k_z=-1/(mV)$; $l_y=l/J_y$; $J_y$ is the component of the moment of inertia on the Y axis; $F_{Tz}$ and $F_{Tzc}$ represent the thrust and thrust command of the engine in the yaw channel, respectively; $M_y^{{\omega }_y}$, $M_y^{{\delta }_y}$, and $M_y^{\beta }(t)$ represent the partial derivatives of the yawing moment $M_y$ to ${\omega }_y$, ${\delta }_y$, and $\beta$, respectively; and $Z^{{\delta }_y}$ and $Z^{\beta }(t)$ represent the partial derivatives of lateral force $Z$ to ${\delta }_y$ and $\beta$. The aerodynamic coefficient in the yaw channel is denoted as

$$\{\begin{array}{l}{b}_1=-M_y^{{\omega }_y}/J_y\hfill \\ b_2(t)=-M_y^{\beta }(t)/J_y\hfill \\ b_3=-M_y^{{\delta }_y}/J_y\hfill \\ b_4(t)=-Z^{\beta }(t)/\left(mV\right)\hfill \\ b_5=-Z^{{\delta }_y}/\left(mV\right)\hfill \end{array}$$

(23)

Then, the system model of the channel is written as follows:

$$\{\begin{array}{l}{\dot{n}}_z=-b_4(t)n_z-{\displaystyle \frac{V}{g}}{b}_4(t){\omega }_y+{\displaystyle \frac{V}{g}}{b}_5{\tau }_1{\delta }_y-{\displaystyle \frac{V}{g}}{b}_5{\tau }_1{\delta }_{yc}+{\displaystyle \frac{V}{g}}{\tau }_2{F}_{Tz}{k}_z-{\displaystyle \frac{V}{g}}{k}_z{\tau }_2{F}_{Tzc}\\ {\dot{\omega }}_y=-b_1{\omega }_y+{\displaystyle \frac{b_2(t)g}{b_4(t)V}}{n}_z+\left({\displaystyle \frac{b_2(t)b_5}{b_4(t)}}-b_3\right){\delta }_y+\left({\displaystyle \frac{b_2(t)k_z}{b_4(t)}}-l_y\right)F_{Tz}\\ {\dot{\delta }}_y=-{\tau }_1{\delta }_y+{\tau }_1{\delta }_{yc}\\ {\dot{F}}_{Tz}=-{\tau }_2{F}_{Tz}+{\tau }_2{F}_{Tzc}\end{array}$$

(24)

The tracking error of the output acceleration in the yaw channel is defined as

$$e_z=n_{zc}-n_z$$

(25)

where $n_{zc}$ is the yaw acceleration tracking command. Equations (24) and (25) are described as the state-space form. The state vectors are ${\tilde{x}}_1={\displaystyle {\int }_0^t{e}_{z}dt}$, ${\tilde{x}}_2=e_z$, ${\tilde{x}}_3={\omega }_y$, ${\tilde{x}}_4={\delta }_y$, and ${\tilde{x}}_5=F_{Tz}$. The control vectors are ${\tilde{u}}_1={\delta }_{yc}$ and ${\tilde{u}}_2=F_{Tzc}$, and the state equation is denoted as

$$\{\begin{array}{l}{\dot{\tilde{x}}}_1={\tilde{x}}_2\hfill \\ {\dot{\tilde{x}}}_2=-b_4(t){\tilde{x}}_2+{\displaystyle \frac{V}{g}}{b}_4(t){\tilde{x}}_3-{\displaystyle \frac{V}{g}}{b}_5{\tau }_1{\tilde{x}}_4-{\displaystyle \frac{V}{g}}{k}_z{\tau }_2{\tilde{x}}_5+{\displaystyle \frac{V}{g}}{b}_5{\tau }_1{\tilde{u}}_1+{\displaystyle \frac{V}{g}}{k}_z{\tau }_2{\tilde{u}}_2\hfill \\ {\dot{\tilde{x}}}_3=-{\displaystyle \frac{b_2(t)g}{b_4(t)V}}{\tilde{x}}_2-b_1{\tilde{x}}_3+\left({\displaystyle \frac{b_2(t)b_5}{b_4(t)}}-b_3\right){\tilde{x}}_4+\left({\displaystyle \frac{b_2(t)k_z}{b_4(t)}}-l_y\right){\tilde{x}}_5\hfill \\ {\dot{\tilde{x}}}_4=-{\tau }_1{\tilde{x}}_4+{\tau }_1{\tilde{u}}_1\hfill \\ {\dot{\tilde{x}}}_5=-{\tau }_2{\tilde{x}}_5+{\tau }_2{\tilde{u}}_2\hfill \end{array}$$

(26)

3. Controller Design

The design of the STT missile compound controller includes the design of an aerodynamic system controller and the design of a lateral thrust system controller. In the aerodynamic system, LQR is used to design the controller. To make the system respond quickly, the regional pole assignment (RPA) method is proposed to design the controller. The problems of RPA and system stability are solved by a linear matrix inequality (LMI). When the aerodynamic rudder reaches saturation and cannot continuously produce control force, the lateral thrust generated by the engine is added to the system. In the lateral thrust system, the sliding mode method is used to design the controller. In the end, the gain scheduling method is used to control the STT missile at different flight altitudes. The control structure diagram is shown in Figure 1.

3.1. Controller Design of Aerodynamic System

3.1.1. Aerodynamic System Controller in Pitch Channel

When studying the aerodynamic system, the influence of the engine’s lateral thrust on the system is not considered; that is, $x_5=0$ and $u_2=0$. Then, the state equation of the aerodynamic system (13) is written as follows:

$${\dot{X}}_1=A_{1\mathrm{P}}(t)X_1+B_{1\mathrm{P}}{u}_1$$

(27)

where $X_1={\left[\begin{array}{cccc}{x}_1& x_2& x_3& x_4\end{array}\right]}^T$ is the state vector, $u_1={\delta }_{zc}$ is the control vector, and

$A_{1\mathrm{P}}(t)=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -a_4(t)& -{\displaystyle \frac{V}{g}}{a}_4(t)& {\displaystyle \frac{V}{g}}{a}_5{\tau }_1\\ 0& {\displaystyle \frac{a_2(t)g}{a_4(t)V}}& -a_1& {\displaystyle \frac{a_2(t)a_5}{a_4(t)}}-a_3\\ 0& 0& 0& -{\tau }_1\end{array}\right]$, $B_{1\mathrm{P}}=\left[\begin{array}{c}0\\ -{\displaystyle \frac{V}{g}}{a}_5{\tau }_1\\ 0\\ {\tau }_1\end{array}\right]$, $A_{1\mathrm{P}}(t)$, and $B_{1\mathrm{P}}$ represent the coefficient matrix of the pitch channel. Because $\mathrm{rank}({\mathit{A}}_{1\mathrm{P}},{\mathit{B}}_{1\mathrm{P}})=4$, system (27) is completely controllable. Considering the control method of the linear variable parameter system, LQR is used here to design the rudder deflection feedback control law, which is expressed as

$$u_1=K(t)X_1=K_1(t)x_1+K_2(t)x_2+K_3(t)x_3+K_4(t)x_4$$

(28)

where $K(t)=\left[\begin{array}{cccc}{K}_1(t)& K_2(t)& K_3(t)& K_4(t)\end{array}\right]$. System (27) meets the following quadratic performance indicator:

$${\widehat{J}}_{\min }={\displaystyle \frac{1}{2}}{\displaystyle {\int }_{t_0}^{t_f}[x^{\mathrm{T}}(t)Q(t)x(t)+}{u}^T(t)R(t)u(t)]dt$$

(29)

where $Q(t)\ge 0$, $R(t)>0$, $x(t)$ is the state vector, and $u(t)$ is the control vector. By substituting Equation (28) into Equation (27), the closed-loop system is as follows:

$${\dot{X}}_1=(A_{1\mathrm{P}}(t)+B_{1\mathrm{P}}K(t))X_1={\overline{A}}_{1\mathrm{P}}(t)X_1$$

(30)

where ${\overline{A}}_{1\mathrm{P}}(t)=\left[\begin{array}{cccc}0& 1& 0& 0\\ {\overline{a}}_{21}(t)& {\overline{a}}_{22}(t)& {\overline{a}}_{23}(t)& {\overline{a}}_{24}(t)\\ 0& {\overline{a}}_{32}(t)& -a_1& {\overline{a}}_{34}(t)\\ {\overline{a}}_{41}(t)& {\overline{a}}_{42}(t)& {\overline{a}}_{43}(t)& {\overline{a}}_{44}(t)\end{array}\right]$, with ${\overline{a}}_{21}(t)=-V{a}_5{\tau }_1{K}_1(t)/g$, ${\overline{a}}_{22}(t)=-a_4(t)-V{a}_5{\tau }_1{K}_2(t)/g$, ${\overline{a}}_{23}(t)=-V{a}_4(t)/g-V{a}_5{\tau }_1{K}_3(t)/g$,${\overline{a}}_{24}(t)=V{a}_5{\tau }_1/g-V{a}_5{\tau }_1{K}_4(t)/g$, ${\overline{a}}_{32}(t)=a_2(t)g/(a_4(t)V)$, ${\overline{a}}_{34}(t)=a_2(t)a_5/a_4(t)-a_3$, ${\overline{a}}_{41}(t)={\tau }_1{K}_1(t)$, ${\overline{a}}_{42}(t)={\tau }_1{K}_2(t)$, ${\overline{a}}_{43}(t)={\tau }_1{K}_3(t)$, ${\overline{a}}_{44}(t)=-{\tau }_1+{\tau }_1{K}_4(t)$.

To improve the response speed of the STT missile system, the regional pole assignment method is proposed to design the controller of the aerodynamic system. Regional pole assignment includes assigning the poles of a linear system to a certain region $D$, which is in the left half plane of the complex plane, and the stability of the system is called $D$-stability [41]. The controller of the regional pole assignment is solved by transforming the solution problem into a linear matrix inequality (LMI). As shown in Figure 2, the radius is $r$, and the circular region $D(d,r)$ with the center of the circle at $(-(d+r),0)$ is the LMI region [42].

Region $D(d,r)$ is represented as

$$D(d,r)=\{s\in C|(s+r+d)(\overline{s}+r+d)-r^2<0\}$$

(31)

where $s$ is a complex variable. The characteristic function of LMI region $D(d,r)$ is expressed as

$$f_{D(d,r)}(s)=\left[\begin{array}{cc}-r& (r+d)+s\\ (r+d)+\overline{s}& -r\end{array}\right]=\left[\begin{array}{cc}-r& (r+d)\\ (r+d)& -r\end{array}\right]+s\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]+\overline{s}{\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]}^T$$

(32)

For a given matrix $H$, the matrix $H$ has $D$-stability when all its eigenvalues are in the LMI region. And there is a symmetric positive definite matrix $N$ such that

$$\left[\begin{array}{cc}-rN& (r+d)N+HN\\ (r+d)N+HN& -rN\end{array}\right]<0$$

(33)

The farther the LMI region is from the circle point, that is, the farther the real part of the closed-loop system pole is from the virtual axis, the faster the system responds.

The state feedback controller for the regional pole assignment of system (27) is denoted as

$$u_{1\mathrm{P}}=K_{\mathrm{P}}(t)X_1$$

(34)

By substituting controller (34) into system (27), the closed-loop system is

$${\dot{X}}_1=(A_{1\mathrm{P}}(t)+B_{1\mathrm{P}}{K}_{\mathrm{P}}(t))X_1$$

(35)

According to the regional pole assignment method, the closed-loop pole of system (35) is assigned to LMI region $D_{\mathrm{P}}(d_{\mathrm{P}},r_{\mathrm{P}})$. If there exists a positive definite symmetric matrix $N_{\mathrm{P}}$, the following inequality is satisfied:

$$\left[\begin{array}{cc}-r_{\mathrm{P}}{N}_{\mathrm{P}}& (r_{\mathrm{P}}+d_{\mathrm{P}})N_{\mathrm{P}}+(A_{1\mathrm{P}}(t)+B_{1\mathrm{P}}{K}_{\mathrm{P}}(t))N_{\mathrm{P}}\\ (r_{\mathrm{P}}+d_{\mathrm{P}})N_{\mathrm{P}}+(A_{1\mathrm{P}}(t)+B_{1\mathrm{P}}{K}_{\mathrm{P}}(t))N_{\mathrm{P}}& -r_{\mathrm{P}}{N}_{\mathrm{P}}\end{array}\right]<0$$

(36)

Then, the system is in the circle $D_{\mathrm{P}}(d_{\mathrm{P}},r_{\mathrm{P}})$ and asymptotically stable. Let $W_{\mathrm{P}}=K_{\mathrm{P}}(t)N_{\mathrm{P}}$; substituting this into inequality (36), we obtain

$$\left[\begin{array}{cc}-r_{\mathrm{P}}{N}_{\mathrm{P}}& (r_{\mathrm{P}}+d_{\mathrm{P}})N_{\mathrm{P}}+A_{1\mathrm{P}}(t)N_{\mathrm{P}}+B_{1\mathrm{P}}{W}_{\mathrm{P}}\\ (r_{\mathrm{P}}+d_{\mathrm{P}})N_{\mathrm{P}}+A_{1\mathrm{P}}(t)N_{\mathrm{P}}+B_{1\mathrm{P}}{W}_{\mathrm{P}}& -r_{\mathrm{P}}{N}_{\mathrm{P}}\end{array}\right]<0$$

(37)

The matrices $N_{\mathrm{P}}$ and $W_{\mathrm{P}}$ are solved by LMI, and then the controller gain $K_{\mathrm{P}}$ is obtained:

$$K_{\mathrm{P}}(t)=W_{\mathrm{P}}{N}_{\mathrm{P}}^{-1}$$

(38)

3.1.2. Aerodynamic System Controller in Yaw Channel

In the aerodynamic system, the influence of the engine’s lateral thrust is not considered; that is, ${\tilde{x}}_5=0$ and ${\tilde{u}}_2=0$. Then, the state equation of aerodynamic system (26) is written as

$${\dot{\tilde{X}}}_1=A_{1\mathrm{Y}}(t){\tilde{X}}_1+B_{1\mathrm{Y}}{\tilde{u}}_1$$

(39)

where ${\tilde{X}}_1={\left[\begin{array}{cccc}{\tilde{x}}_1& {\tilde{x}}_2& {\tilde{x}}_3& {\tilde{x}}_4\end{array}\right]}^T$ is the state vector, ${\tilde{u}}_1={\delta }_{\mathrm{y}c}$ is the control vector, and

$A_{1\mathrm{Y}}(t)=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -b_4(t)& {\displaystyle \frac{V}{g}}{b}_4(t)& -{\displaystyle \frac{V}{g}}{b}_5{\tau }_1\\ 0& -{\displaystyle \frac{b_2(t)g}{b_4(t)V}}& -b_1& {\displaystyle \frac{b_2(t)b_5}{b_4(t)}}-b_3\\ 0& 0& 0& -{\tau }_1\end{array}\right]$, $B_{1\mathrm{Y}}=\left[\begin{array}{c}0\\ {\displaystyle \frac{V}{g}}{b}_5{\tau }_1\\ 0\\ {\tau }_1\end{array}\right]$, $A_{1\mathrm{Y}}(t)$, and $B_{1\mathrm{Y}}$ represent the coefficient matrix of the yaw channel. Because $\mathrm{rank}({\mathit{A}}_{1\mathrm{Y}},{\mathit{B}}_{1\mathrm{Y}})=4$, system (39) is completely controllable. The LQR is chosen here to design the rudder deflection feedback control law, which is expressed as

$${\tilde{u}}_1=\tilde{K}(t){\tilde{X}}_1={\tilde{K}}_1(t){\tilde{x}}_1+{\tilde{K}}_2(t){\tilde{x}}_2+{\tilde{K}}_3(t){\tilde{x}}_3+{\tilde{K}}_4(t){\tilde{x}}_4$$

(40)

where $\tilde{K}(t)=\left[\begin{array}{cccc}{\tilde{K}}_1(t)& {\tilde{K}}_2(t)& {\tilde{K}}_3(t)& {\tilde{K}}_4(t)\end{array}\right]$. System (39) meets the quadratic performance indicator (29). By substituting Equation (40) into Equation (39), the closed-loop system is

$${\dot{\tilde{X}}}_1=(A_{1\mathrm{Y}}(t)+B_{1\mathrm{Y}}\tilde{K}(t)){\tilde{X}}_1={\overline{A}}_{1\mathrm{Y}}(t){\tilde{X}}_1$$

(41)

where ${\overline{A}}_{1\mathrm{Y}}(t)=\left[\begin{array}{cccc}0& 1& 0& 0\\ {\overline{b}}_{21}(t)& {\overline{b}}_{22}(t)& {\overline{b}}_{23}(t)& {\overline{b}}_{24}(t)\\ 0& {\overline{b}}_{32}(t)& -b_1& {\overline{b}}_{34}(t)\\ {\overline{b}}_{41}(t)& {\overline{b}}_{42}(t)& {\overline{b}}_{43}(t)& {\overline{b}}_{44}(t)\end{array}\right]$, with ${\overline{b}}_{21}(t)=V{b}_5{\tau }_1{\tilde{K}}_1(t)/g$, ${\overline{b}}_{22}(t)=-b_4(t)+V{b}_5{\tau }_1{\tilde{K}}_2(t)/g$, ${\overline{b}}_{23}(t)=V{b}_4(t)/g+V{b}_5{\tau }_1{\tilde{K}}_3(t)/g$, ${\overline{b}}_{24}(t)=-V{b}_5{\tau }_1/g+V{b}_5{\tau }_1{\tilde{K}}_4(t)/g$, ${\overline{b}}_{32}(t)=-b_2(t)g/(b_4(t)V)$, ${\overline{b}}_{34}(t)=b_2(t)b_5/b_4(t)-b_3$, ${\overline{b}}_{41}(t)={\tau }_1{\tilde{K}}_1(t)$, ${\overline{b}}_{42}(t)={\tau }_1{\tilde{K}}_2(t)$, ${\overline{b}}_{43}(t)={\tau }_1{\tilde{K}}_3(t)$, ${\overline{b}}_{44}(t)=-{\tau }_1+{\tau }_1{\tilde{K}}_4(t)$.

To improve the response speed of the STT missile system, a regional pole assignment method is used to design the controller of the aerodynamic system. The state feedback controller for the regional pole assignment of system (39) is denoted as

$$u_{1\mathrm{Y}}=K_{\mathrm{Y}}(t){\tilde{X}}_1$$

(42)

By substituting controller (42) into system (39), the closed-loop system is

$${\dot{\tilde{X}}}_1=(A_{1\mathrm{Y}}(t)+B_{1\mathrm{Y}}{K}_{\mathrm{Y}}(t)){\tilde{X}}_1$$

(43)

According to the regional pole assignment method, the closed-loop pole of system (43) is assigned to LMI region $D_{\mathrm{Y}}(d_{\mathrm{Y}},r_{\mathrm{Y}})$. If there exists a positive definite symmetric matrix $N_{\mathrm{Y}}$, the following inequality is satisfied:

$$\left[\begin{array}{cc}-r_{\mathrm{Y}}{N}_{\mathrm{Y}}& (r_{\mathrm{Y}}+d_{\mathrm{Y}})N_{\mathrm{Y}}+(A_{1\mathrm{Y}}(t)+B_{1\mathrm{Y}}{K}_{\mathrm{Y}}(t))N_{\mathrm{Y}}\\ (r_{\mathrm{Y}}+d_{\mathrm{Y}})N_{\mathrm{Y}}+(A_{1\mathrm{Y}}(t)+B_{1\mathrm{Y}}{K}_{\mathrm{Y}}(t))N_{\mathrm{Y}}& -r_{\mathrm{Y}}{N}_{\mathrm{Y}}\end{array}\right]<0$$

(44)

Then, the system is in the circle $D_{\mathrm{Y}}(d_{\mathrm{Y}},r_{\mathrm{Y}})$ and asymptotically stable. Let $W_{\mathrm{Y}}=K_{\mathrm{Y}}(t)N_{\mathrm{Y}}$; substituting this into inequality (44), we obtain

$$\left[\begin{array}{cc}-r_{\mathrm{Y}}{N}_{\mathrm{Y}}& (r_{\mathrm{Y}}+d_{\mathrm{Y}})N_{\mathrm{Y}}+A_{1\mathrm{Y}}(t)N_{\mathrm{Y}}+B_{1\mathrm{Y}}{W}_{\mathrm{Y}}\\ (r_{\mathrm{Y}}+d_{\mathrm{Y}})N_{\mathrm{Y}}+A_{1\mathrm{Y}}(t)N_{\mathrm{Y}}+B_{1\mathrm{Y}}{W}_{\mathrm{Y}}& -r_{\mathrm{Y}}{N}_{\mathrm{Y}}\end{array}\right]<0$$

(45)

The matrices $N_{\mathrm{Y}}$ and $W_{\mathrm{Y}}$ are solved by LMI, and then the controller gain $K_{\mathrm{Y}}$ is obtained:

$$K_{\mathrm{Y}}(t)=W_{\mathrm{Y}}{N}_{\mathrm{Y}}^{-1}$$

(46)

3.2. Controller Design of Lateral Thrust System

For the compound control hypersonic STT missile, when the aerodynamic rudder is saturated, the control force cannot be generated continuously. In this case, the lateral thrust generated by the engine is added to the system.

3.2.1. Lateral Thrust System Controller in Pitch Channel

The system model with lateral thrust is described as

$${\dot{X}}_2=A_{2\mathrm{P}}(t)X_2+B_{2\mathrm{P}}{u}_2$$

(47)

where the state vector of the thrust system is $X_2={\left[\begin{array}{cc}{X}_1^{\mathrm{T}}& x_5\end{array}\right]}^{\mathrm{T}}$, and $u_2=F_{Tyc}$ is the control vector.

$A_{2\mathrm{P}}(t)=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ {\overline{a}}_{21}(t)& {\overline{a}}_{22}(t)& {\overline{a}}_{23}(t)& {\overline{a}}_{24}(t)& {\overline{a}}_{25}(t)\\ 0& {\overline{a}}_{32}(t)& -a_1& {\overline{a}}_{34}(t)& {\overline{a}}_{35}(t)\\ {\overline{a}}_{41}(t)& {\overline{a}}_{42}(t)& {\overline{a}}_{43}(t)& {\overline{a}}_{44}(t)& 0\\ 0& 0& 0& 0& -{\tau }_2\end{array}\right]$, with ${\overline{a}}_{25}(t)={\displaystyle \frac{V{k}_y{\tau }_2}{g}}$, ${\overline{a}}_{35}(t)={\displaystyle \frac{a_2(t)k_y}{a_4(t)}}-l_z$, $B_{2\mathrm{P}}={\left[\begin{array}{ccccc}0& -V{k}_y{\tau }_2/g& 0& 0& {\tau }_2\end{array}\right]}^{\mathrm{T}}$. Because $\mathrm{rank}({\mathit{A}}_{2\mathrm{P}},{\mathit{B}}_{2\mathrm{P}})=5$, system (47) is completely controllable. The sliding mode control method is chosen because the lateral thrust control system is controlled by an on–off mode. To solve the sliding mode controller, a sliding mode surface needs to be designed. First, system (47) is converted to the following standard form:

$${\dot{\overline{X}}}_{\mathrm{P}}=A_{3\mathrm{P}}(t){\overline{X}}_{\mathrm{P}}+B_{3\mathrm{P}}{u}_2$$

(48)

where ${\mathit{A}}_{3\mathrm{P}}(t)=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ -{\widehat{a}}_0(t)& -{\widehat{a}}_1(t)& -{\widehat{a}}_2(t)& -{\widehat{a}}_3(t)& -{\widehat{a}}_4(t)\end{array}\right]$, ${\mathit{B}}_{3\mathrm{P}}={\left[\begin{array}{ccccc}0& 0& 0& 0& 1\end{array}\right]}^{\mathrm{T}}$, ${\widehat{a}}_0(t)$, ${\widehat{a}}_1(t)$, ${\widehat{a}}_2(t)$, ${\widehat{a}}_3(t)$, and ${\widehat{a}}_4(t)$ are the variables of the standard form matrix; ${\overline{X}}_{\mathrm{p}}={\left[\begin{array}{ccccc}{\overline{x}}_{1\mathrm{P}}& {\overline{x}}_{2\mathrm{P}}& {\overline{x}}_{3\mathrm{P}}& {\overline{x}}_{4\mathrm{P}}& {\overline{x}}_{5\mathrm{P}}\end{array}\right]}^{\mathrm{T}}$ is the state vector of the standard equation; and ${\overline{X}}_{\mathrm{P}}={P_{\mathrm{P}}}^{-1}(t)X_2$, $P_{\mathrm{P}}(t)$ is called the transformation matrix, written as $P_{\mathrm{P}}(t)={\left[\begin{array}{ccccc}{P}_{1\mathrm{P}}& P_{2\mathrm{P}}& P_{3\mathrm{P}}& P_{4\mathrm{P}}& P_{5\mathrm{P}}\end{array}\right]}^{\mathrm{T}}$.

The sliding mode surface is defined as $S({\overline{X}}_{\mathrm{P}})$ and satisfies

$$S({\overline{X}}_{\mathrm{P}})=P_{1\mathrm{P}}{\overline{x}}_{1\mathrm{P}}+P_{2\mathrm{P}}{\overline{x}}_{2\mathrm{P}}+P_{3\mathrm{P}}{\overline{x}}_{3\mathrm{P}}+P_{4\mathrm{P}}{\overline{x}}_{4\mathrm{P}}+{\overline{x}}_{5\mathrm{P}}=0$$

(49)

During the selection of $P_{1\mathrm{P}}$ to $P_{4\mathrm{P}}$, it is ensured that the system state converges to zero and meets the following sliding mode arrival conditions:

$$S({\overline{\mathit{X}}}_{\mathrm{P}})\dot{S}({\overline{\mathit{X}}}_{\mathrm{P}})<0$$

(50)

The sliding mode controller is defined as

$$u_2=\{\begin{array}{ll}{F}_{\mathrm{P}},& S({\overline{X}}_{\mathrm{P}})>0\\ -F_{\mathrm{P}},& S({\overline{X}}_{\mathrm{P}})<0\end{array}$$

(51)

where $F_{\mathrm{P}}$ is the lateral thrust produced by the pitch channel engine. To mitigate the jump in the thrust amplitude of the engine and save fuel consumption, the lateral thrust system is designed to output thrust with different amplitude sizes for the STT missile. When the system state is far away from the sliding surface, the engine with larger thrust will work. When the system state is close to the sliding surface, the engine with smaller thrust works. In this case, the variable structure controller of the system is

$$u_2=\{\begin{array}{ll}{F}_{\mathrm{P}1},& S({\overline{X}}_{\mathrm{P}})\ge {\epsilon }_1\\ F_{\mathrm{P}2},& {\epsilon }_2\le S({\overline{X}}_{\mathrm{P}})<{\epsilon }_1\\ 0,& -{\epsilon }_2\le S({\overline{X}}_{\mathrm{P}})<{\epsilon }_2\\ -F_{\mathrm{P}2},& -{\epsilon }_1\le S({\overline{X}}_{\mathrm{P}})<-{\epsilon }_2\\ -F_{\mathrm{P}1},& S({\overline{X}}_{\mathrm{P}})<-{\epsilon }_1\end{array}$$

(52)

where $F_{\mathrm{P}1}$ is the larger value of the engine’s lateral thrust, $F_{\mathrm{P}2}$ is the smaller value of the engine’s lateral thrust, and ${\epsilon }_1$ and ${\epsilon }_2$ are the boundary constants of the sliding mode surface.

When an STT missile flies at different altitudes, gain scheduling is used to control the missile system of the different flight altitudes. The STT missile flight control system can be regarded as a fixed system with multiple steady-state operating points, each of which has a fixed stability control gain. Different flight altitudes are regarded as different stable operating points, and the missile controller at each altitude is designed. Therefore, the gain scheduling design method is appropriate. When limited to space, the controller of a steady-state operating point is designed every 2 km. The LQR controller, pole assignment controller, and sliding mode controller of the pitch channel are denoted, respectively, at different heights as

$$U_{\mathrm{P}}(h)=\{\begin{array}{l}{U}_{\mathrm{P}1}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),25\mathrm{km}\le h\le 26\mathrm{km}\\ U_{\mathrm{P}2}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),27\mathrm{km}\le h\le 28\mathrm{km}\\ U_{\mathrm{P}3}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),29\mathrm{km}\le h\le 30\mathrm{km}\\ U_{\mathrm{P}4}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),31\mathrm{km}\le h\le 32\mathrm{km}\\ U_{\mathrm{P}5}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),33\mathrm{km}\le h\le 34\mathrm{km}\\ U_{\mathrm{P}6}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),35\mathrm{km}\le h\le 36\mathrm{km}\\ U_{\mathrm{P}7}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),37\mathrm{km}\le h\le 38\mathrm{km}\\ U_{\mathrm{P}8}(u_1(h),u_{1\mathrm{P}}(h),u_2(h)),39\mathrm{km}\le h<40\mathrm{km}\end{array}$$

(53)

3.2.2. Lateral Thrust System Controller in Yaw Channel

The system model of the yaw channel with lateral thrust is described as

$${\dot{\tilde{X}}}_2=A_{2\mathrm{Y}}(t){\tilde{X}}_2+B_{2\mathrm{Y}}{\tilde{u}}_2$$

(54)

where the state vector of the thrust system is ${\tilde{X}}_2={\left[\begin{array}{cc}{\tilde{X}}_1^{\mathrm{T}}& {\tilde{x}}_5\end{array}\right]}^{\mathrm{T}}$, and ${\tilde{u}}_2=F_{Tzc}$ is the control vector.

$A_{2\mathrm{Y}}(t)=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ {\overline{b}}_{21}(t)& {\overline{b}}_{22}(t)& {\overline{b}}_{23}(t)& {\overline{b}}_{24}(t)& {\overline{b}}_{25}(t)\\ 0& {\overline{b}}_{32}(t)& -b_1& {\overline{b}}_{34}(t)& {\overline{b}}_{35}(t)\\ {\overline{b}}_{41}(t)& {\overline{b}}_{42}(t)& {\overline{b}}_{43}(t)& {\overline{b}}_{44}(t)& 0\\ 0& 0& 0& 0& -{\tau }_2\end{array}\right]$, with ${\overline{b}}_{25}(t)=-{\displaystyle \frac{V{k}_z{\tau }_2}{g}}$, ${\overline{b}}_{35}(t)={\displaystyle \frac{b_2(t)k_z}{b_4(t)}}-l_y$, $B_{2\mathrm{Y}}={\left[\begin{array}{ccccc}0& V{k}_z{\tau }_2/g& 0& 0& {\tau }_2\end{array}\right]}^{\mathrm{T}}$. Because $\mathrm{rank}({\mathit{A}}_{2\mathrm{Y}},{\mathit{B}}_{2\mathrm{Y}})=5$, system (54) is completely controllable. To solve the sliding mode controller, a sliding mode surface needs to be designed. First, system (54) is converted to the following standard form:

$${\dot{\overline{X}}}_{\mathrm{Y}}=A_{3\mathrm{Y}}(t){\overline{X}}_{\mathrm{Y}}+B_{3\mathrm{Y}}{\tilde{u}}_2$$

(55)

where ${\mathit{A}}_{3\mathrm{Y}}(t)=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ -{\widehat{b}}_0(t)& -{\widehat{b}}_1(t)& -{\widehat{b}}_2(t)& -{\widehat{b}}_3(t)& -{\widehat{b}}_4(t)\end{array}\right]$, ${\mathit{B}}_{3\mathrm{Y}}={\left[\begin{array}{ccccc}0& 0& 0& 0& 1\end{array}\right]}^{\mathrm{T}}$, ${\widehat{b}}_0(t)$, ${\widehat{b}}_1(t)$, ${\widehat{b}}_2(t)$, ${\widehat{b}}_3(t)$, and ${\widehat{b}}_4(t)$ are the variables of the standard form matrix; ${\overline{X}}_{\mathrm{Y}}={\left[\begin{array}{ccccc}{\overline{x}}_{1\mathrm{Y}}& {\overline{x}}_{2\mathrm{Y}}& {\overline{x}}_{3\mathrm{Y}}& {\overline{x}}_{4\mathrm{Y}}& {\overline{x}}_{5\mathrm{Y}}\end{array}\right]}^{\mathrm{T}}$ is the state vector of the standard equation; and ${\overline{X}}_{\mathrm{Y}}={P_{\mathrm{Y}}}^{-1}(t){\tilde{X}}_2$, $P_{\mathrm{Y}}(t)$ is called the transformation matrix, written as $P_{\mathrm{Y}}(t)={\left[\begin{array}{ccccc}{P}_{1\mathrm{Y}}& P_{2\mathrm{Y}}& P_{3\mathrm{Y}}& P_{4\mathrm{Y}}& P_{5\mathrm{Y}}\end{array}\right]}^{\mathrm{T}}$.

The sliding mode surface is defined as $S({\overline{X}}_{\mathrm{Y}})$ and satisfies

$$S({\overline{X}}_{\mathrm{Y}})=P_{1\mathrm{Y}}{\overline{x}}_{1\mathrm{Y}}+P_{2\mathrm{Y}}{\overline{x}}_{2\mathrm{Y}}+P_{3\mathrm{Y}}{\overline{x}}_{3\mathrm{Y}}+P_{4\mathrm{Y}}{\overline{x}}_{4\mathrm{Y}}+{\overline{x}}_{5\mathrm{Y}}=0$$

(56)

During the selection of $P_{1\mathrm{Y}}$ to $P_{4\mathrm{Y}}$, it is ensured that the system state converges to zero and meets the following sliding mode arrival conditions:

$$S({\overline{\mathit{X}}}_{\mathrm{Y}})\dot{S}({\overline{\mathit{X}}}_{\mathrm{Y}})<0$$

(57)

The sliding mode controller is defined as

$${\tilde{u}}_2=\{\begin{array}{ll}{F}_{\mathrm{Y}},& S({\overline{X}}_{\mathrm{Y}})>0\\ -F_{\mathrm{Y}},& S({\overline{X}}_{\mathrm{Y}})<0\end{array}$$

(58)

where $F_{\mathrm{Y}}$ is the lateral thrust of the yaw channel engine. To mitigate the jump in the thrust amplitude of the engine and save fuel consumption, the lateral thrust system is designed to output thrust with different amplitude sizes for the STT missile. In this case, the variable structure controller of the system is

$${\tilde{u}}_2=\{\begin{array}{ll}{F}_{\mathrm{Y}1}& S({\overline{X}}_{\mathrm{Y}})\ge {\epsilon }_1\\ F_{\mathrm{Y}2}& {\epsilon }_2\le S({\overline{X}}_{\mathrm{Y}})<{\epsilon }_1\\ 0& -{\epsilon }_2\le S({\overline{X}}_{\mathrm{Y}})<{\epsilon }_2\\ -F_{\mathrm{Y}2}& -{\epsilon }_1\le S({\overline{X}}_{\mathrm{Y}})<-{\epsilon }_2\\ -F_{\mathrm{Y}1}& S({\overline{X}}_{\mathrm{Y}})<-{\epsilon }_1\end{array}$$

(59)

where $F_{\mathrm{Y}1}$ is the larger value of the engine’s lateral thrust, and $F_{\mathrm{Y}2}$ is the smaller value of the engine’s lateral thrust.

When an STT missile flies at different altitudes, gain scheduling is used to control the missile system at different flight altitudes. When limited to space, the controller of a steady-state operating point is designed every 2 km. The LQR controller, pole assignment controller, and sliding mode controller of the yaw channel are denoted, respectively, at different heights as

$$U_{\mathrm{Y}}(h)=\{\begin{array}{l}{U}_{\mathrm{Y}1}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),25\mathrm{km}\le h\le 26\mathrm{km}\\ U_{\mathrm{Y}2}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),27\mathrm{km}\le h\le 28\mathrm{km}\\ U_{\mathrm{Y}3}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),29\mathrm{km}\le h\le 30\mathrm{km}\\ U_{\mathrm{Y}4}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),31\mathrm{km}\le h\le 32\mathrm{km}\\ U_{\mathrm{Y}5}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),33\mathrm{km}\le h\le 34\mathrm{km}\\ U_{\mathrm{Y}6}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),35\mathrm{km}\le h\le 36\mathrm{km}\\ U_{\mathrm{Y}7}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),37\mathrm{km}\le h\le 38\mathrm{km}\\ U_{\mathrm{Y}8}({\tilde{u}}_1(h),u_{1\mathrm{Y}}(h),{\tilde{u}}_2(h)),39\mathrm{km}\le h<40\mathrm{km}\end{array}$$

(60)

4. Simulation

In this section, the gain scheduling steady-state controller based on a regional pole assignment is given. Then, the fast response of the pitch channel and yaw channel of STT missiles at different flight altitudes is simulated. By comparing the response speed of the compound control system when the aerodynamic system is LQR control or regional pole assignment, the importance of designing regional pole assignment is highlighted.

4.1. Gain Scheduling Controller

The parameters of the STT missile control system are shown in

Table 2. Parameters of STT missile.

Parameter	Value	Unit
$m$	500	kg
$V$	1963.4	m/s
$g$	9.81	m/s2
${\tau }_1$	0.0025	s
${\tau }_2$	0.001	s
$J_y$	300	Nms2
$J_z$	300	Nms2
$l$	1.36	m
$F_{\mathrm{P}1}$	6000	N
$F_{\mathrm{P}2}$	400	N
$F_{\mathrm{Y}1}$	6000	N
$F_{\mathrm{Y}2}$	400	N

.

In the STT missile control system, the aerodynamic parameters are $a_1=0.0381$, $a_3=14.1014$, $a_5=0.0062$, $b_1=0.0381$, $b_3=14.1014$, and $b_5=0.0062$. Throughout the change in actual aerodynamic parameters, $a_2(t)$ and $a_4(t)$ are approximately proportional, and $b_2(t)$ and $b_4(t)$ are approximately proportional. Herein, the proportional relations are referred to as $a_4(t)=k_a{a}_2(t)$ and $b_4(t)=k_b{b}_2(t)$ separately. The proportional coefficients are $k_a=-0.042$ and $k_b=-0.042$.

According to the design requirements, such as the response speed and oscillation frequency of the system, the center and radius of the circular region for pole placement are determined. Since the system is required to have a response time of less than 0.3 s and a maximum oscillation frequency of less than 300 rad/s, considering the positions of the dominant poles and other poles, the pole region is set as $D(0.1,300)$. The poles of steady-state operating points at different flight altitudes of STT missiles and corresponding controllers are given below. When limited to space, the poles and controllers of a steady-state operating point are shown every 2 km; the pole region $D(0.1,300)$ is denoted as $D_1(0.1,300)$ to $D_8(0.1,300)$ at different flight altitudes.

Steady-state operating point 1 (25–26 km):

The pole of the region $D_1(0.1,300)$ is [−580, −25, −9.5, −0.1], and the controller at the pole is [−0.0884, 0.8773, −1.6138, 2.934].

Steady-state operating point 2 (27–28 km):

The pole of the region $D_2(0.1,300)$ is [−570, −20, −9, −0.1], and the controller at the pole is [−0.0573, 1.147, −2.19, 2.823].

Steady-state operating point 3 (29–30 km):

The pole of the region $D_3(0.1,300)$ is [−590, −25, −10, −0.5], and the controller at the pole is [−0.0784, 1.5681, −2.9972, 2.8231].

Steady-state operating point 4 (31–32 km):

The pole of the region $D_4(0.1,300)$ is [−585, −20, −8.5, −0.3], and the controller at the pole is [−0.2137, 2.1205, −4.073, 2.8054].

Steady-state operating point 5 (33–34 km):

The pole of the region $D_5(0.1,300)$ is [−575, −26, −9.2, −0.4], and the controller at the pole is [−0.2925, 2.9034, −5.5815, 2.8055].

Steady-state operating point 6 (35–36 km):

The pole of the region $D_6(0.1,300)$ is [−586, −28, −10, −0.2], and the controller at the pole is [−0.4009, 3.9783, −7.6543, 2.8055].

Steady-state operating point 7 (37–38 km):

The pole of the region $D_7(0.1,300)$ is [−592, −25, −8.7, −0.1], and the controller at the pole is [−0.4362, 4.3292, −8.877, 2.3389].

Steady-state operating point 8 (39–40 km):

The pole of the region $D_8(0.1,300)$ is [−585, −26, −9.3, −0.3], and the controller at the pole is [−0.3682, 7.3652, −14.137, 2.8233].

Because the pitch channel and yaw channel are axisymmetric to each other, the controllers obtained by pole assignment are equal in these two channels. The pole assignment controller is substituted into system (35) and system (43), respectively, and combined with sliding mode control, the dual channel compound control design is realized.

4.2. Simulation Analysis

In this subsection, simulations of the STT missile at 25 km, 30 km, and 35 km are given, respectively. At each altitude, the pitch channel and yaw channel, which are axisymmetric, are simulated. Each channel is compared when the aerodynamic system of the compound control system is LQR and regional pole assignment (RPA), respectively.

(1)

Simulation at 25 km

The acceleration command for tracking at 25 km is $n_{yc}=n_{zc}=15g$. The simulations of the pitch channel and yaw channel are shown in Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7.

Figure 3 shows the acceleration tracking response of the STT missile at 25 km. It can be seen from Figure 3a,b that the response speed of the compound control system based on regional pole assignment (RPA) is significantly faster than that of the compound control system based on LQR. Although the RPA response curve has a brief overshoot, it returns to a stable state instantaneously. From the comparison of (a) and (b) in Figure 4, Figure 5, and Figure 7, the simulation curves are equal in value but opposite in direction, reflecting the mutual axisymmetric structure of the pitch channel and yaw channel. Figure 4 is the angular rate response curve. The compound control system based on LQR converges to the zero axis slowly, while the response curve of the compound control system based on RPA has a reverse rate, but the system quickly returns to a stable state. It meets the design requirements of a fast response system. Figure 5 is the response curve of the rudder deflection angle. Figure 6 shows the thrust response curve of the engine. When the rudder deflection angle in Figure 5 reaches 30 degrees of saturation, the engine starts working at the corresponding time in Figure 6. The sliding mode surface in Figure 7 has a brief fluctuation and then quickly returns to a stable state.

(2)

Simulation at 30 km

The acceleration command for tracking at 30 km is $n_{yc}=n_{zc}=10g$. The simulations of the pitch channel and yaw channel are shown in Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

Figure 8 shows the acceleration tracking response of the STT missile at 30 km. It can be seen from Figure 8a,b that the response speed of the compound control system based on RPA is significantly faster than that of the compound control system based on LQR. Although the RPA response curve has a brief overshoot, it returns to a stable state instantaneously. It meets the need for a rapid response system. From the comparison of (a) and (b) in Figure 9, Figure 10, and Figure 12, the simulation curves are equal in value but opposite in direction, reflecting the mutual axisymmetric structure of the pitch channel and yaw channel. Figure 9 is the angular rate response curve. The response curve of the compound control system based on RPA quickly returns to a stable state. Figure 10 is the response curve of the rudder deflection angle. Figure 11 shows the thrust response curve of the engine. When the rudder deflection angle in Figure 10 reaches a saturated state, the engine starts working at the corresponding time in Figure 11. The sliding mode surface in Figure 12 has a short vibration at the initial moment and then quickly converges to a stable state.

(3)

Simulation at 35 km

The acceleration command for tracking at 35 km is $n_{yc}=n_{zc}=6g$. The simulations of the pitch channel and yaw channel are shown in Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17.

Figure 13 shows the acceleration tracking response of the STT missile at 35 km. It can be seen from Figure 13a,b that the response speed of the compound control system based on RPA is significantly faster than that of the compound control system based on LQR. From the comparison of (a) and (b) in Figure 14, Figure 15 and Figure 17, the simulation curves are equal in value but opposite in direction, reflecting the mutual axisymmetric structure of the pitch channel and yaw channel. Figure 15 is the response curve of the rudder deflection angle. Figure 16 shows the thrust response curve of the engine. When the rudder deflection angle in Figure 15 reaches a saturated state, the engine starts working at the corresponding time in Figure 16. The sliding mode surface in Figure 17 has a short vibration at the initial moment and then quickly converges to a stable state.

In the simulation diagram, the curve from simulation case (1) at 25 km is smooth; in simulation case (2), there is weak oscillation in the curve at 30 km; and simulation case (3) has oscillations in the curve at 35 km. With the increase in flight height, the oscillation gradually increases. This is because the increase in flight height decreases the air density, and the aerodynamic coefficient of the aircraft changes, resulting in a very short period of small amplitude oscillation in the system. From the point of view of requiring a fast response of the system, weak oscillation has little effect on the overall performance of the system, so it is beneficial that the controller designed by regional pole assignment can achieve a fast response of the system under the condition of small oscillation.

5. Conclusions

An aerodynamic and lateral thrust compound control method based on regional pole assignment is proposed to improve the response speed of the STT missile system. In the aerodynamic control system, the linear quadratic controller is used to design the controller, and the sliding mode control method is used to design the controller of the lateral thrust system. Regional pole assignment is introduced into the aerodynamic system to improve the response speed of the system. The problems of regional pole assignment and system stability are solved by linear matrix inequalities. At different flight altitudes, gain scheduling is used to control the missile system. Compared to previous designs of time-varying compound control systems for STT missiles or hypersonic vehicles, in order to meet the practical requirement of a fast response for the vehicle, this time-varying compound control strategy can achieve faster tracking responses and altitude control for the STT missile. Aiming at the axial symmetrical structure of the STT missile, the control system of the pitch channel and yaw channel is simulated. The response time of the designed compound control system is 0.2 s faster than that of the previous compound control scheme, which verifies the effectiveness of the fast response method of the designed compound control system.