Course roadmap ME451: Control Systems Modeling Analysis Laplace transform Lecture 22 Frequency response Transfer function Models for systems • electrical • mechanical • electromechanical Block diagrams Linearization Dr. Jongeun Choi Department of Mechanical Engineering Michigan State University Design Time response • Transient • Steady state 1 Design specs Root locus Frequency response 3 Frequency domain 3 • Bode plot Stability • RouthRouth-Hurwitz 3 • Nyquist 1 PID & LeadLead-lag 2 Design examples (Matlab simulations &) laboratories 1 What is frequency response? 2 A simple example RC circuit System u(t) u(t) We would like to analyze a system property by applying a test sinusoidal input u(t) and observing a response y(t). Steady state response yss(t) (after transient dies out) of a system to sinusoidal inputs is called frequency response. R C y(t) y(t) Input a sinusoidal voltage u(t) What is the output voltage y(t)? 3 4 An example (cont’d) TF (R=C=1) An example (cont’d) 1 Derivation of y(t) r y 0.8 0.6 0.4 0.2 u(t)=sin(t) Inverse Laplace 0 Partial fraction expansion -0.2 -0.4 -0.6 0 as t goes to infinity. -0.8 -1 0 5 10 15 20 25 30 35 40 45 50 At steadysteady-state, u(t) u(t) and y(t) y(t) has same frequency, but different amplitude and phase! (Derivation for general G(s) G(s) is given at the end of lecture slide.) 5 6 Response to sinusoidal input Frequency response function How is the steady state output of a linear system when the input is sinusoidal? For a stable system G(s), G(jω) (ω is positive) is called frequency response function (FRF). FRF is a complex number, and thus, has an amplitude and a phase. First order example Im y(t) y(t) G(s) Steady state output Frequency is same as the input frequency Re Amplitude is that of input (A) multiplied by Phase shifts Gain 7 8 Another example of FRF First order example revisited Second order system FRF Im Two graphs representing FRF Bode diagram (Bode plot) (Today) Re Nyquist diagram (Nyquist (Nyquist plot) 9 Bode diagram (Bode plot) of G(jω) Bode diagram consists of gain plot & phase plot 10 Bode plot of a 1st order system TF Corner frequency 0 -10 -20 -30 -40 -50 -2 10 LogLog-scale -1 10 0 10 1 10 2 10 0 -20 -40 -60 -80 -100 -2 10 11 -1 10 0 10 1 10 2 10 12 Exercises of sketching Bode plot First order system Remarks on Bode diagram Bode diagram shows amplification and phase shift of a system output for sinusoidal inputs with various frequencies. It is very useful and important in analysis and design of control systems. The shape of Bode plot contains information of stability, time responses, and much more! It can also be used for system identification. (Given FRF experimental data, obtain a transfer function that matches the data.) 13 System identification Summary and exercises Sweep frequencies of sinusoidal signals and obtain FRF data (i.e., gain and phase). Select G(s) so that G(jω) fits the FRF data. Frequency response is a steady state response of systems to a sinusoidal input. For a linear system, sinusoidal input generates sinusoidal output with same frequency but different amplitude and phase. Bode plot is a graphical representation of frequency response function. (“bode.m”) Next, Bode diagram of simple transfer functions Exercise: Read Section 8. Agilent Technologies: FFT Dynamic Signal Analyzer Generate sin signals Collect FRF data Sweep frequencies Select G(s) G(s) 14 Unknown system 15 16 Derivation of frequency response Term having denominator of G(s) G(s) 0 as t goes to infinity. 17