MESB374 Chapter8 System Modeling and Analysis Time domain Analysis Transfer Function Analysis

Transfer Function Analysis Dynamic Response of Linear Time-Invariant (LTI) Systems –Free (or Natural) Responses –Forced Responses Transfer Function for Forced Response Analysis –Poles –Zeros General Form of Free Response –Effect of Pole Locations –Effect of Initial Conditions (ICs) Obtain I/O Model based on Transfer Function Concept

Dynamic Responses of LTI Systems Ex: Let’s look at a stable first order system: –Solve for the output: –Take LT of the I/O model and remember to keep tracks of the ICs: –Rearrange terms s.t. the output Y(s) terms are on one side and the input U(s) and IC terms are on the other: Free Response Forced Response Time constant

Free & Forced Responses Free Response ( u(t) = 0 & nonzero ICs ) –The response of a system to zero input and nonzero initial conditions. –Can be obtained by Let u(t) = 0 and use LT and ILT to solve for the free response. Forced Response ( zero ICs & nonzero u(t) ) –The response of a system to nonzero input and zero initial conditions. –Can be obtained by Assume zero ICs and use LT and ILT to solve for the forced response (replace differentiation with s in the I/O ODE model).

In Class Exercise Find the free and forced responses of the car suspension system without tire model: –Solve for the output: –Rearrange terms s.t. the output Y(s) terms are on one side and the input U(s) and IC terms are on the other: Free Response Forced Response –Take LT of the I/O model and remember to keep tracks of the ICs:

Given a general n-th order system model: The forced response (zero ICs) of the system due to input u(t) is: –Taking the LT of the ODE : Forced Response & Transfer Function Forced Response Transfer Function Inputs Transfer Function Inputs = =

Transfer Function Given a general nth order system: The transfer function of the system is: –The transfer function can be interpreted as: Differential Equation u(t) Input y(t) Output Time Domain G(s)G(s) U(s) Input Y(s) Output s - Domain Static gain

Transfer Function Matrix For Multiple-Input-Multiple-Output (MIMO) System with m inputs and p outputs: Inputs Outputs

Poles and Zeros Poles The roots of the denominator of the TF, i.e. the roots of the characteristic equation. Given a transfer function (TF) of a system: n poles of TF Zeros The roots of the numerator of the TF. m zeros of TF

(2) For car suspension system: Find TF and poles/zeros of the system. Examples ( 1) Recall the first order system: Find TF and poles/zeros of the system. Pole: Zero: No Zero Pole: Zero:

System Connections Cascaded System Input Output Parallel System Input Output + + Feedback System Input Output + -

Given a general nth order system model: The free response (zero input) of the system due to ICs is: –Taking the LT of the model with zero input (i.e., ) General Form of Free Response A Polynominal of s that depends on ICs Free Response (Natural Response) = Same Denominator as TF G(s)

Free Response (Examples) Ex: Find the free response of the car suspension system without tire model (slinker toy): Ex: Perform partial fraction expansion (PFE) of the above free response when: (what does this set of ICs means physically)? Q: Is the solution consistent with your physical intuition? Decaying rate: damping, mass Frequency: damping, spring, mass phase: initial conditions

The free response of a system can be represented by: Free Response and Pole Locations Real Img. exponential decrease constant exponential increase decaying oscillation Oscillation with constant magnitude increasing oscillation t

Complete Response Q:Which part of the system affects both the free and forced response ? Q:When will free response converges to zero for all non-zero I.C.s ? Complete Response U(s) Input Y(s) Output Denominator D(s) All the poles have negative real parts.

Obtaining I/O Model Using TF Concept (Laplace Transformation Method) Noting the one-one correspondence between the transfer function and the I/O model of a system, one idea to obtain I/O model is to: –Use LT to transform all time-domain differential equations into s-domain algebraic equations assuming zero ICs (why?) –Solve for output in terms of inputs in s-domain to obtain TFs (algebraic manipulations) –Write down the I/O model based on the TFs obtained

Step 1: LT of differential equations assuming zero ICs Example – Car Suspension System Step 2: Solve for output using algebraic elimination method 1.# of unknown variables = # equations ? 2.Eliminate intermediate variables one by one. To eliminate one intermediate variable, solve for the variable from one of the equations and substitute it into ALL the rest of equations; make sure that the variable is completely eliminated from the remaining equations x p

Example (Cont.) Step 3: write down I/O model from TFs from first equation Substitute it into the second equation