Lec 3. System Modeling

1. Lec 3. System Modeling • Transfer Function Model • Model of Mechanical Systems • Model of Electrical Systems • Model of Electromechanical Systems TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

2. Transfer Functions of LTI Systems Output of a linear time-invariant (LTI) system is given by where is the impulse response, output under input u(t)=(t) Take the Laplace transform: is called the transfer function

3. LTI Systems Given by Differential Equations Common LTI system models for practical systems: Assuming zero initial condition, transfer function model is (in general)

4. (Rational) Transfer Functions • n-th order system • Zeros: roots of , • Poles: roots of , Pole zero plot:

5. Standard Forms of Rational Transfer Functions Basic form: • Matlab commands: tf, tfdata Factored (or product) form: • Matlab commands: zpk, zpkdata Sum form (assume poles are distinct): If all poles are distinct p1,…,pn are the poles, r1,…,rn are the corresponding residues

6. Example

7. Example: Car Suspension Model (body) m2 shock absorber (wheel) m1 Input: road altitude r(t) Output: car body height y(t) road surface

8. Suspension System Model Differential equation model:

9. Translational Mechanical System Models • Identify all independent components of the system • For each component, do a force analysis (all forces acting on it) • Apply Newton’s Second Law to obtain an ODE, and take the Laplace transform of it • Combine the equations to eliminate internal variables • Write the transfer function from input to output

10. Rotational Mechanical Systems: Satellite gas jet Suppose that the antenna of the satellite needs to point to the earth Ignore the translational motions of the satellite Input: A force F generated by the release of reaction jet Output: orientation of the satellite given by the angle 

11. Satellite Model Torque: In general gas jet Newton’s Second Law:

12. Model of Electrical Systems Basic components resistor inductor capacitor

13. Impedance Basic components resistor inductor capacitor

14. Circuit Systems

15. Electromechanical System: DC Motor Armature resistance Torque T Basic motor properties: Torque proportional to current: Motor voltage proportional to shaft angular velocity: Friction B Input: voltage source e(t) Output: shaft angular position q(t) Newton law: Basic circuit properties (KVL):

16. Nonlinear System: Pendulum pendulum Input: external force F Output: angle  Dynamic equation from Newton’s law