【Advanced】MATLAB Aircraft Modeling and Simulation

# 1. Newton's Laws and Equations of Motion

Newton's laws are the foundation of classical mechanics and describe the motion of objects under the action of forces. For aircraft modeling, Newton's laws are particularly important as they provide the basic equations for describing the motion of aircraft.

**2.1.1 Newton's First Law**

Also known as the law of inertia, Newton's first law states: An object at rest will remain at rest, and an object in motion will remain in uniform motion in a straight line unless acted upon by an external force.

**2.1.2 Newton's Second Law**

Newton's second law states: The force acting on an object is equal to the product of the object's mass and its acceleration. The mathematical expression is:

F = m * a

Where:

* F is the resultant force acting on the object
* m is the mass of the object
* a is the acceleration of the object

# ***rcraft Dynamics Modeling

### 2.1 Newton's Laws and Equations of Motion

Newton's laws form the basis of classical mechanics, describing the rules of object motion. In aircraft dynamics modeling, Newton's laws are used to establish the equations of motion for aircraft.

**2.1.1 Newton's First Law**

Newton's first law states: If an object is not acted upon by any external forces, it will remain at rest or continue in uniform straight-line motion.

**2.1.2 Newton's Second Law**

Newton's second law states: The magnitude of the force exerted on an object is equal to the product of the object's mass and its acceleration, that is $F = ma$.

**2.1.3 Newton's Third Law**

Newton's third law states: For every action, there is an equal and opposite reaction.

### 2.2 Aerodynamics Modeling

Aerodynamics modeling is a crucial component of aircraft dynamics modeling. It describes the interaction between air and the aircraft, including lift, drag, and pitching moment.

**2.2.1 Lift, Drag, and Pitching Moment**

***Lift:** Lift is the upward force exerted by air on the aircraft, related to the shape of the wing and the angle of attack.
***Drag:** Drag is the force exerted by air parallel to the direction of motion, related to the shape and speed of the aircraft.
***Pitching Moment:** Pitching moment is the force exerted by air causing the aircraft to rotate around the lateral axis, related to the shape of the wing and the angle of attack.

**2.2.2 Calculation of Aerodynamic Coefficients**

Aerodynamic coefficients are dimensionless parameters describing aerodynamic characteristics. They can be obtained through wind tunnel tests or computational fluid dynamics (CFD).

### 2.3 Thrust Modeling

Thrust modeling is another significant aspect of aircraft dynamics modeling. It describes the thrust generated by engines and thrust vector control.

**2.3.1 Engine Thrust Model**

The engine thrust model describes the relationship between the thrust generated by the engine and engine speed, fuel flow rate, and other factors.

**2.3.2 Thrust Vector Control**

Thrust vector control is a technique for controlling the direction of engine thrust. It can be achieved by deflecting the nozzle or using other methods.

# ***rcraft Control System Design

### 3.1 Linear Control Theory

#### 3.1.1 State Space Representation

State space representation is a mathematical model for describing linear systems, consisting of state equations and output equations. State equations describe how the system's internal state changes over time, while output equations describe how the system's output is determined by its state and input.

The general form of state equations is:

ẋ = Ax + Bu

Where:

* x is the system state vector
* A is the state matrix
* B is the input matrix
* u is the input vector

The general form of output equations is:

y = Cx + Du

Where:

* y is the system output vector
* C is the output matrix
* D is the direct transmission matrix

#### 3.1.2 Transfer Function

Transfer function is a mathematical function describing the relationship between the input and output of a linear system. It can be obtained through state space representation or directly from input and output data. The general form of the transfer function is:

G(s) = C(sI - A)^{-1}B + D

Where:

* s is a complex variable
* I is the identity matrix

#### 3.1.3 Controller Design

The goal of linear controller design is to find a controller that meets specific performance requirements for the system, such as stability, response speed, ***mon linear controller design methods include:

***PID Control:** A PID controller is a classical proportional-integral-derivative controller that adjusts the proportional, integral, and derivative gains to control the system's output.
***State Feedback Control:** A state feedback controller designs control laws