Accepted: July 14, 2022 | Published Online: July 16, 2022

Citation: Trung VA, Phong CT (2022) Synthesis of Optimal Control Laws for Air Missile using Gas Rudder. Int J Astronaut Aeronautical Eng 7:059

Copyright: © 2022 Trung VA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Abstract

Air missile using gas rudder to work quickly change the direction of the missile. However, to control the missile optimally, we need to synthesize the missile control laws. The article refers to the synthesis problem optimal control laws missile with used gas rudders for quickly moving in the direction of impact.

Keywords

Air missile, Gas rudder, Control laws, Optimal control laws

Introduction

To synthesize the optimal control law for aerospace missiles, we need to know the complete system of equations describing the air floor's motion as a control object in space. By ignoring the problems of durability, deformation and oscillation of the missile structure, only interested in the movement of the mass centre of the air floor and its rotation around the centre of mass, we can The problem limit in the motion of a solid body has 6 degrees of freedom (including three translational movements and three rotations). In addition, when ignoring the influence of moments generated by the dynamic systems and considering only the influence of aerodynamic forces and moments in combination with the above assumptions, we can show that most fully show the motion of missilion in the vertical plane by the following seven differential equations [1-3]:

$$\left\{\begin{array}{l}m\frac{dV}{dt}=P\cos \alpha -X-G\sin \theta \\ mV\frac{d\theta }{dt}=Psin\alpha +Y-G\cos \theta \\ J_z\frac{d{\omega }_z}{dt}={\displaystyle \sum M_z}\\ \frac{dy}{dt}=V\sin \theta \\ \frac{dx}{dt}=V\cos \theta \\ \frac{d\vartheta }{dt}={\omega }_z\\ \theta =\vartheta -\alpha \end{array}\right.\text{ (1)}$$

These forces are lift (Y), drag (X), and side force (Z).

Suppose we consider the variation in mass m of the air floor to be insignificant during the operation of the engine operation, the missilion has sufficient static stability (angle of attack is always very small and angle). The above equation is converted to:

$$\left\{\begin{array}{l}m\frac{dV}{dt}=P-X-G\sin \theta \\ mV\frac{d\theta }{dt}=P\alpha +Y-G\cos \theta \\ \frac{dy}{dt}=V\sin \theta \\ \frac{dx}{dt}=V\cos \theta \end{array}\right.\text{ (2)}$$

To complete the equation for the missilon input state as required, we need to add a control equation. According to [4], the lead methods aircraft using the dynamic throttle blade is implemented according to the proportional guide method. The method's essence is to ensure that the overload of the missilion is proportional to the angular speed of the line of sight. Therefore, the equation for the input status of an missilon is shown as follows [1]:

$$\left\{\begin{array}{l}\frac{dV}{dt}=\frac{P}{m}-\frac{X}{m}-g\sin \theta \\ \frac{d\theta }{dt}=\frac{P\alpha }{mV}+\frac{g}{V}(\frac{Y}{mg}-\cos \theta )\\ \frac{dy}{dt}=V\sin \theta \\ \frac{dx}{dt}=V\cos \theta \\ \frac{d{n}_y}{dt}=u\end{array}\right.$$

Instead, $X\text{ = }{C}_x\frac{\rho V^2}{2}S$ We have:

$$\left\{\begin{array}{l}\frac{dV}{dt}=\frac{P}{m}-C_x\frac{\rho V^2}{2}\frac{S}{m}-g\sin \theta \\ \frac{d\theta }{dt}=\frac{P\alpha }{mV}+\frac{g}{V}{n}_y-\frac{g}{V}\cos \theta \\ \frac{dy}{dt}=V\sin \theta \\ \frac{dx}{dt}=V\cos \theta \\ \frac{d{n}_y}{dt}=u\end{array}\right.\text{ (3)}$$

Inside: $\rho \text{ = }1.225{(1-\frac{H}{44300})}^{4.256}$ - air density at altitude H;

Thus, the problem is explicitly stated as follows:

Determine the optimal control law u (t) to take missilion with the mathematical model (3) from a first state point $\underset{¯}{x}(0)\text{ = }{\underset{¯}{x}}_0$ arbitrarily given to the end state point ${\underset{¯}{x}}_T$ given in the fastest period satisfying the function target [3,5]:

$$Q(\underset{¯}{x},\underset{¯}{u})\text{ = }{\displaystyle \underset{0}{\overset{T}{\int }}dt}\text{ (4)}$$

With the top conditions:

$$\begin{array}{l}V(0)=V_0,\theta (0)={\theta }_0,y(0)=y_0,x(0)=x_0,\\ n_y(0)=n_{y0},\left|n_y\right|\le n_{ygioihan}\end{array}$$

Principle of maximum Pontryagin

Consider an optimization problem for a semi-linear object with a form object model:

$$\left\{\begin{array}{l}\frac{d\underset{¯}{x}}{dt}=A\underset{¯}{x}+\underset{¯}{h}(\underset{¯}{u})\\ Q(\underset{¯}{x},\underset{¯}{u})={\displaystyle \underset{0}{\overset{T}{\int }}\left[{\underset{¯}{a}}^T\underset{¯}{x}+r(\underset{¯}{u})\right]}dt\to \min \end{array}\right.\text{ (5)}$$

With $\underset{¯}{x}(0)\text{ = }{\underset{¯}{x}}_0$ và $\underset{¯}{u}\in U$ is a closed subset of Rm has an early state score $\underset{¯}{x}(0)\text{ = }{\underset{¯}{x}}_0$ and a given T period. Moreover, the final state point is optional (or bound by the surface ST). Now we define the Hamilton function as a form:

$$H(\underset{¯}{x},\underset{¯}{u},\underset{¯}{p})\text{ = }{\underset{¯}{P}}^T\underset{¯}{f}\left[A\underset{¯}{x}+\underset{¯}{h}\underset{¯}{u}\right]-\left[{\underset{¯}{a}}^T\underset{¯}{x}+r(\underset{¯}{u})\right]\text{ (6)}$$

The formula calculates the co-state variables:

$$\frac{d\underset{¯}{x}}{dt}\text{ = }{\left(\frac{\partial H}{\partial \underset{¯}{p}}\right)}^T,\frac{d\underset{¯}{p}}{dt}\text{ = }-{\left(\frac{\partial H}{\partial \underset{¯}{x}}\right)}^T\text{ (7)}$$

Since U is a closed set, it is impossible to use the classical variational method or conditional $\frac{\partial H}{\partial \underset{¯}{u}}\text{ = }{\underset{¯}{0}}^T$ to determine the optimal control signal $\underset{¯}{u}(t)$. To solve this problem, we will call $\stackrel{\sim }{\underset{¯}{u}}(t)$ and $\underset{¯}{u}(t)$ are two certain control signals belonging to U, as $\stackrel{\sim }{\underset{¯}{x}}(t)$ and $\underset{¯}{x}(t)$ are two corresponding state orbits going from ${\underset{¯}{x}}_0$ brought by them. From problem results combined with if the hypothesis $\underset{¯}{u}(t)\in U$ is the optimal control signal of the problem (5) with the solution $\underset{¯}{p}(T)$ of equation (6) is satisfied, $\underset{¯}{p}(T)\text{ = }\underset{¯}{0}$ we must have:

$$H(\underset{¯}{x},\underset{¯}{p},\underset{¯}{u})\text{ = }\underset{\underset{¯}{\tilde{u}}\in U}{\max }H(\underset{¯}{x},\underset{¯}{p},\underset{¯}{\tilde{u}})\text{ (8)}$$

This property is precisely the content of the maximum principle that states for the semi-linear object optimization problem. Thus, we can find the optimal control signal $\underset{¯}{u}(t)\in U$ for the above problem, follow the steps below [6]:

Step 1: Establish a Hamilton function (6);

Step 2: Determine the relationship $\underset{¯}{u}(\underset{¯}{x},\underset{¯}{p})$ must have of optimal signal $\underset{¯}{u}(t)\in U$ from maximum principle (8);

Step 3: Replace the relationship found into Euler-Lagrange equation (7) and solve those equations with boundary conditions. $\underset{¯}{x}(0)\text{ = }{\underset{¯}{x}}_0,\underset{¯}{p}(T)\text{ = }\underset{¯}{0}$ to have $\underset{¯}{x}(t),\underset{¯}{p}(t)$;

Step 4: Replace $\underset{¯}{x}(t),\underset{¯}{p}(t)$ was found in step 3 into the relationship $\underset{¯}{u}(\underset{¯}{x},\underset{¯}{p})$ from step 2; or go to the model (5) of the object for an optimal solution $\underset{¯}{u}(t)$.

Synthesis of missile control laws according to the principle of maximum

We see that the input state equation (8) of the missile can be rewritten as a vector equation:

$$\frac{dx}{dt}\text{ = }f(t,x)+F(t,x)u\text{ (9)}$$

There are

$$x=\left[\begin{array}{l}V\\ \theta \\ y\\ x\\ ny\end{array}\right];f(t,x)=\left[\begin{array}{l}\frac{P}{m}-C_x\rho \frac{V^2}{2}\frac{S}{m}-g\sin \theta \\ \frac{P}{mV}\alpha +\frac{g}{V}ny-\frac{g}{V}\cos \theta \\ V\sin \theta \\ V\cos \theta \\ 0\end{array}\right];F(t,x)=\left[\begin{array}{l}0\\ 0\\ 0\\ 0\\ 1\end{array}\right].$$

To solve this problem, follow these steps in turn:

Step 1. Create a reduced Hamiltonian function

H(x,u,p) = pT(f(t,x) + F(t,x)u)

with $p^T\text{ = }\left(p_V,p_{\theta },p_y,p_x,p_{ny}\right)$:

We have

$$H(x,u,p)\text{ = }{p}_V\frac{P}{m}-{\text{p}}_V{C}_x\rho \frac{V^2}{2}\frac{S}{m}-{\text{p}}_{V}g\sin \theta +p_{\theta }\frac{P}{mV}\alpha +p_{\theta }\frac{g}{V}ny-p_{\theta }\frac{g}{V}\cos \theta +p_{y}V\sin \theta +p_{x}V\cos \theta +p_{ny}u$$

Step 2. Apply the maximum principle to find optimal control signal u(t) According to the principle of maximum

$H(x,u,p)\text{ = }\underset{\stackrel{\sim }{u}\in U}{\max H}(x,\stackrel{\sim }{u},p)$ ta có: $\frac{\partial H}{\partial u}\text{ = }{p}_{ny}$

So the optimal control signal has the form:

u(t) = sgn(pny) (10)

To determine the value of the optimal control signal u(t) more specifically, we need to calculate the solution pny(t).

Step 3. Prepare the Euler-Lagrange equation According to the formula

$$\frac{d\underset{¯}{p}}{dt}\text{ = }-\frac{\partial H}{\partial \underset{¯}{x}}$$

ta có:

$$\frac{d{p}_V}{dt}\text{ = }-\frac{\partial H}{\partial V}{\text{ = p}}_V{C}_x\rho V\frac{S}{m}+p_{\theta }\frac{P}{m{V}^2}\alpha -p_{\theta }\frac{g}{V^2}\cos \theta -p_y\sin \theta -p_x\cos \theta ;$$

$$\frac{d{p}_{\theta }}{dt}\text{ = }-\frac{\partial H}{\partial \theta }{\text{ = p}}_{V}gc\text{os}\theta -p_{\theta }\frac{g}{V}\sin \theta -p_{y}V\cos \theta +p_{x}V\sin \theta ;$$

$$\frac{d{p}_y}{dt}\text{ = }-\frac{\partial H}{\partial y}\text{ = }0;$$

$$\frac{d{p}_x}{dt}\text{ = }-\frac{\partial H}{\partial x}\text{ = }0;$$

$$\frac{d{p}_{ny}}{dt}\text{ = }-\frac{\partial H}{\partial ny}\text{ = }-\frac{g}{V}{p}_{\theta }.$$

Because u(t) = sgn(pny) so we are only interested in the solutions pny. Solving the system of Euler-Lagrange equations we obtain the form Pny:

$$P_{ny}\text{ = }{C}_5+C_{3}t+C_1{e}^{0.0097t}\cos (0.042t)-C_2{e}^{0.0097t}\sin (0.042t)\text{ (11)}$$

There are: C1, C2, C3, C5 are constants. These constants depend on the boundary conditions for the co-state vector $\underset{¯}{p}(t)$. Replace (11) with (10) we have:

$$u(t)\text{ = }sgn(C_5+C_{3}t+C_1{e}^{0.0097t}\cos (0.042t)-C_2{e}^{0.00097t}\sin (0.042t))\stackrel{}{\iff }$$

$$u(t)\text{ = }\left\{\begin{array}{l}1\stackrel{}{}\stackrel{}{}\stackrel{}{}khi\stackrel{}{}\stackrel{}{\begin{array}{l}{C}_5+C_{3}t+C_1{e}^{0.0097t}\cos (0.042t)\\ >C_2{e}^{0.00097t}\sin (0.042t)\end{array}}\\ -1\stackrel{}{}\stackrel{}{}khi\stackrel{}{}\stackrel{}{\begin{array}{l}{C}_5+C_{3}t+C_1{e}^{0.0097t}\cos (0.042t)\\ <C_2{e}^{0.00097t}\sin (0.042t)\end{array}}\end{array}\right.\text{ (12)}$$

This shows that the optimal control signal u(t) has the only value of either -1 or 1.

Thus, with the optimal control signal u(t) found, instead of equations (3), we will build the components' state orbits. $\theta (t)$, y(x), ny(t).

Simulation and evaluation of the results

To simulate and evaluate the results, we choose the first conditions [7]:

V(0) = 221 m/s; ; y(0) = 2000 m; x(0) = -1200m; ny(0) = 1.

The simulation results in Matlab are as follows:

Discussion

- For the orbital inclination angle $\theta$, when u = 1 the angle $\theta$ increases gradually $0÷0.45$ radians (The first of Figure 1). This is consistent with the air up arrow trajectory while in flight. Minimal change angle ensures no arrow has excellent stability. Similarly, when u = -1 (The first of Figure 2) is the case of air up name flying down.

- For the orbital variation of x, we see that the distance x always decreases to zero. This is in accordance with the problem requirement when the air up name goes from the first state point to the end state point after time t. As for the state trajectory of the altitude y when u = 1 (The second of Figure 1) corresponds to the case air floor ascends when u = -1 (The second of Figure 2) corresponds to the air up arrow is falling.

Thus, in this case, the optimal control signal of the air floor name according to the maximum principle Pontryagin is relatively consistent with reality.

Conclusion

By calculating and evaluating the results, we see that the results achieved are consistent with the control process's physical nature. It is possible to use the maximum method to synthesize the optimal control law for air train name using the dynamic throttle as the basis for researching, designing and manufacturing air floor names with different purposes in practice.

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