232 Welcome back! Coming week labs: Lecture 9 Lab 16 System Identification (2 sessions) Today: Review of Lab 15 System identification (ala ME4232) Time domain Frequency domain 1
Future Labs To develop control systems 1. Determine a system model Lab 15 Simple open loop approach Lab 16 System identification (time domain, frequency domain) 2. Determine controller structure for desired properties Stability Performance (tracking); Disturbance rejection; Noise immunity; Robustness to model uncertainty Sophistication (analysis/design) better tradeoffs Lab 17 Proportional; Lab 18 Proportional-integral Lab 19 Feedforward; Lab 20: Internal model control; [ Lab 20 Adaptive control ] Lab 21: Force control (integrative lab) 233 2
Review of Lab 15 234 Servo-valve model for equal area actuator: Q L 1 ( xv, PL ) Cd wxv ( Ps sgn( xv ) PL ) Velocity = Q L /A act For small load pressure (Force/A act ), speed is proportional to xv For servo valve, xv is proportional xv 3
235 For unequal area actuator, formula has to be modified Question: for same valve opening, will extension or retraction be faster??? (what did you see from lab?) r = area ratio > 1 u = valve input For derivation see: Appendix in this paper: P.Y. Li, A New Passive Control for Hydraulic Human Power Amplifier, ASME IMECE2006-15056, Chicago, IL, November 2006. 4
Systems Analysis 236 5
Basic Systems Analysis Skills (What we reviewed last week) 237 Convert signals between Time domain and Laplace domain System description from differential equation to transfer function Find system response to inputs Block diagram simplification Pole locations and characteristic responses Frequency response 6
238 Transfer Functions Y(s) = G(s) U(s) Three uses of transfer functions: Prediction: Given input, find output (Everything!) Control: Given desired output, find input to achieve it (Labs 17-22) System Identification: Given input and output, find the system (Labs 15-16) 7
239 Determine G(s) by testing the system Probe with input U(s) Measure output Y(s) System Identification Often you have choice of what U(s) to use Time domain (step, impulse, ) Frequency domain (sine/cosine) 8
System Identification Approach 240 9
Repertoire of System Response 241 Time domain Frequency domain 10
242 Step response First Order System # of measurements = # of unknown parameters = 2 Not unique choice pick the ones that are most distinct Final value - Initial slope (other choices 1/e of final value.) Plot a line and use least squares to find the parameters! 11
Impulse Response 243 Ideal impulse impulse Approximate impulse How to preserve the shape of the impulse response? Signal to noise ratio becomes low 12
Frequency Response 244 Given a stable transfer function G(s) such that Y(s) = G(s) R(s) If the input is a sinusoid, r(t) = A sin ( t), then the output is given, after the transient has died down, by y(t) = B sin( t + ) with B = A G(j ) and = phase of G(j ) in radians This allows sinusoidal response to be easily characterized 13
Fourier series 245 Any periodic time function, r(t) of frequency can be represented by a sum of sinusoids of harmonics of r(t) = k = 01 M k sin(k t + k ) Thus the steady state response of system G(s) to a periodic input is: where y(t) = k=01 B k sin(k t + k ) B k = M k G(j k ) and k = k + Å G(j k ) 14
246 Fourier Transform / Frequency content An arbitrary signal (with finite power) can be considered to be a periodic signal with a very long period, i.e. T = 1/ -> 1 Then, -> 0, and the harmonics 0,, 2,, k,. Become a continuum.. For some R(j ) which turns out to be R(s = j ) where R(s) is the Laplace transform 15
247 Frequency Response (Bode Plot) Analytical method (learn this first) G(jw) and Phase (G(jw) ) Program this in Matlab Matlab tool - know what to expect first!!! (Garbage in/garbage out) Bode plot 20 log10( G(jw) ) vs log10(w) Freqresp Sys = tf(num,den); Bode(sys) [H, w] = freqresp(sys) 16
First Order System 248 Shape of the bode plot for first order system Approximate with straight lines for sketching What happens when w = pole? 17