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Lec 3. System Modeling Transfer Function Model
Model of Mechanical Systems Model of Electrical Systems Model of Electromechanical Systems Reading: , TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA
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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
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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
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(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:
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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
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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)
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Example
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Example solve solve Zeros: Matlab code: Poles: Factored form:
Sum form (by PFD):
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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.
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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
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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
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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
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Suspension System Model
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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)
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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
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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
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Satellite Model Torque: In general gas jet Newton’s Second Law:
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Model of Electrical Systems
Basic components resistor inductor capacitor
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Impedance Basic components resistor inductor capacitor
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Circuit Systems +
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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)
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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)
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A Simple Nonlinear Control System
pendulum Input: external force F Output: angle Dynamic equation from Newton’s law A nonlinear differential equation!
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