Lec 3. System Modeling Transfer Function Model Model of Mechanical Systems Model of Electrical Systems Model of Electromechanical Systems Reading: 3.1-3.3, 3.6-3.8 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

Transfer Functions of General LTI Systems A linear time-invariant (LTI) system: Its impulse response h(t) is the output y(t) under input u(t)=(t) For arbitrary input u(t), the output is given by Take the Laplace transform: is called the transfer function

LTI Systems Given by Differential Equations Often times, the LTI systems modeling practical systems are Transfer function can be directly obtain by taking the Laplace transform (assuming zero initial condition) Transfer function H(s) is given by

(Rational) Transfer Functions Roots of B(s) are called the zeros of H(s): z1,…,zm Roots of A(s) are called the poles of H(s): p1,…,pn System is called an n-th order system Pole zero plot:

Standard Forms of Transfer Function Ratio of polynomial: Factored (or product) form: Sum form (assume poles are distinct): p1,…,pn are the poles, r1,…,rn are the corresponding residues If all poles are distinct

A Geometric Interpretation of Residues Distance to zeros Distance to poles (except itself) Pole zero plot: Remark: if a pole is very close to a zero, its residue will be small. (Approximate pole-zero cancellation)

Example

Example solve solve Zeros: Matlab code: Poles: Factored form: Sum form (by PFD):

Modified Example Zeros: Poles: (Compared with previous example, an extra zero very close to the pole p2=2 is introduced) Factored form: Sum form (by PFD): residue of the pole p2=2 is much smaller than in the previous case due to zero-pole cancellation.

Model of Mechanical Systems Car Suspension Model (body) m2 shock absorber (wheel) m1 Input: road altitude r(t) Output: car body height y(t) road surface

Force Analysis of Mass One Ignore the gravity forces (x,y are displacements from equilibrium positions) (body) m2 By Newton’s Second Law shock absorber (wheel) m1 (wheel) m1 road surface

Force Analysis of Mass Two Ignore the gravity forces (x,y are displacements from equilibrium positions) (body) m2 (body) m2 shock absorber (wheel) m1 By Newton’s Second Law road surface

Suspension System Model

Suspension System Model Differential equation model Taking Laplace transform: Compute X(s) from the second equation, plug in the first one: transfer function from input r(t) to output y(t)

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

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 

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

Model of Electrical Systems Basic components resistor inductor capacitor

Impedance Basic components resistor inductor capacitor

Circuit Systems +

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)

A Simple Nonlinear Control System pendulum Input: external force F Output: angle  Dynamic equation from Newton’s law A nonlinear differential equation! Linearization: approximate a nonlinear system by a linear one. When  is small, sin is approximately equal to . (see Section 3-10 of the textbook for more details)

A Simple Nonlinear Control System pendulum Input: external force F Output: angle  Dynamic equation from Newton’s law A nonlinear differential equation!