1. Digital Control Systems Waseem Gulsher
BS (Evening)
2 Oct, 17
Mathematical Models
Lecture – 4
Digital Control Systems Waseem Gulsher

2. Satellite Model

3. Satellite Model
As the first example of the development of the mathematical model of a physical system, consider the attitude control system of a satellite.
Assume that the satellite is spherical and has the thrustor configuration shown in figure 1-5.

4. Satellite Model
Suppose that θ(t) is the yaw angle of satellite. In addition to the thrustors shown, thrustors will also control the pitch angle and the roll angle, giving complete three-axis control of the satellite.
Only consider the yaw-axis control systems, whose purpose is to control the angel θ(t).
For the satellite, the thrustors, when active apply a torque ĩ(t).
The torque of the two thrustors shown in figure (on previous slide) tends to reduce θ(t). The other two thrustors tends to increase θ(t).

5. Satellite Model
Since there is essentially no friction in the environment of the satellite, and assuming satellite rigid, we can write
Where J is the satellite’s moment of inertia about the yaw axis.

6. Satellite Model
Now derive the transfer function by taking Laplace transform of Eq. 1-1
(Initial condition are ignored when deriving transfer function). Eq. 1-2 can be expressed as
The ratio of the Laplace transforms of the output variable θ(t) to input variable ĩ(t) is called the plant transfer function, and is denoted as Gp(s).

7. Satellite Model
The model of the satellite may be a specified either by second-order differential equation of (1-1) or the second-order transfer function of (1-3).
The third model is the state-variable model.
Suppose that we define the variables x1(t) and x2(t) as
Where denotes the derivative of x1(t) with respect to time.

8. Satellite Model
Then from (1-4) and (1-5),

9. Satellite Model
Now equations (1-4) and (1-5) can be re-written in vector-matrix form
In this equation, x1(t) and x2(t) are called the state variables.
Hence, the satellite model may be specified in the form of equations (1-1), or (1-3) or (1-7).

10. Servomotor System Model

11. Servomotor System Model
In this section, the model of servo system (a position system) is derived.
An example of this type of system is an antenna tracking system.
In this system, an electric motor is utilized to rotate a radar antenna that tracks an aircraft automatically.
The error signal, which is proportional to the difference between pointing direction of the antenna and the line of sight of the aircraft, is amplified and drives the motor in the appropriate direction so as to reduces this error.

12. Servomotor System Model
A DC motor system is shown in figure 1-6.

13. Servomotor System Model
The motor is armature controlled with a constant field.
The armature resistance and inductance are Ra and La respectively.
Assume that inductance La can be ignored, which is the case for many servomotors.
The motor back emf em(t) is given by.

14. Servomotor System Model
Where θ(t) is the shaft position, ω(t) is the angular velocity, and Kb is the motor-dependant constant.
The total moment of inertia connected to the motor shaft is J, and B is the total viscous friction.
Letting ĩ(t) be the torque developed by the motor, we can write

15. Servomotor System Model
The developed torque for this motor is given by
Where i(t) is the armature current and KT is a parameter of motor.
The final equation required is the voltage equation for the armature current.

16. Servomotor System Model
These four equations may be solved for the output θ(t) as a function of the input e(t).
First from (1-11) and (1-8),

17. Servomotor System Model
Then from (1-9), (1-10) and (1-12),
This equation may written as
Which is the desired model. This model is second order; if armature resistance is not neglected, the model is third order.

18. Servomotor System Model
Then from (1-9), (1-10) and (1-12),

19. Thank You