ME375 Handouts - Fall 2002 MESB374 Chapter8 System Modeling and Analysis Time domain Analysis Transfer Function Analysis

Transfer Function Analysis ME375 Handouts - Fall 2002 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 ME375 Handouts - Fall 2002 Dynamic Responses of LTI Systems Ex: Let’s look at a stable first order system: Time constant 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: Solve for the output: Forced Response Free Response

Free & Forced Responses ME375 Handouts - Fall 2002 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. Assume zero ICs and use LT and ILT to solve for the forced response (replace differentiation with s in the I/O ODE model).

ME375 Handouts - Fall 2002 In Class Exercise Find the free and forced responses of the car suspension system without tire model: 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: Solve for the output: Forced Response Free Response

Forced Response & Transfer Function ME375 Handouts - Fall 2002 Forced Response & Transfer Function 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 = Transfer Function Inputs = Inputs

Transfer Function Given a general nth order system: ME375 Handouts - Fall 2002 Transfer Function Given a general nth order system: The transfer function of the system is: The transfer function can be interpreted as: Static gain u(t) Input y(t) Output U(s) Input Y(s) Output Differential Equation G(s) Time Domain s - Domain

Transfer Function Matrix ME375 Handouts - Fall 2002 Transfer Function Matrix For Multiple-Input-Multiple-Output (MIMO) System with m inputs and p outputs: Inputs Outputs

Poles and Zeros Poles Zeros ME375 Handouts - Fall 2002 Poles and Zeros Given a transfer function (TF) of a system: Poles The roots of the denominator of the TF, i.e. the roots of the characteristic equation. Zeros The roots of the numerator of the TF. m zeros of TF n poles of TF

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

System Connections + + - Cascaded System Input Output Parallel System ME375 Handouts - Fall 2002 Cascaded System Input Output Parallel System Input Output + Feedback System + - Input Output

General Form of Free Response ME375 Handouts - Fall 2002 General Form of Free Response 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., ) Free Response (Natural Response) A Polynominal of s that depends on ICs = Same Denominator as TF G(s)

Free Response (Examples) ME375 Handouts - Fall 2002 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)? phase: initial conditions Decaying rate: damping, mass Frequency: damping, spring, mass Q: Is the solution consistent with your physical intuition?

Free Response and Pole Locations ME375 Handouts - Fall 2002 Free Response and Pole Locations The free response of a system can be represented by: exponential decrease Real Img. constant exponential increase decaying oscillation Oscillation with constant magnitude increasing oscillation t

Complete Response Y(s) U(s) Output Input Complete Response ME375 Handouts - Fall 2002 Complete Response U(s) Input Y(s) Output 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 ? 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

Example – Car Suspension System Step 1: LT of differential equations assuming zero ICs x p Step 2: Solve for output using algebraic elimination method # 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

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