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# Special Topics ME 4803

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This 0 page Class Notes was uploaded by Chloe Reilly on Monday November 2, 2015. The Class Notes belongs to ME 4803 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/234247/me-4803-georgia-institute-of-technology-main-campus in Mechanical Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Supplement on the Power Spectral Density Its Meaning and Estimation ME 4803 Motion Control W Book 12303 Your lab instructions have asked you to examine the power spectral density of a transducer signal The power spectral density PSD is a way to evaluate the frequency content of a signal A rapidly varying signal has a high PSD at high frequencies Random white noise has equal content at all frequencies A signal with 60 Hz noise from the power lines would show higher power at 60 Hz and maybe its harmonics The concept of PSD calculation is represented in the diagram below A time varying signal Xt is first filtered by a bandpass filter centered at frequency f1 and of bandwidth Be Never mind that we can t really build a perfect bandpass filter as represented The filtered time signal is X1t which is then squared to get the signal power that is always positive This is an EE thing based on the power being the square of current That value is then averaged over time by integrating and dividing by the time If we wait long enough and the signal is stationary meaning its statistical nature doesn t change we get a good estimate CAf1B of GflB the true power in the signal in that frequency range We repeat that process as many times at as many frequencies as needed xl 1 X10 5 0 f1 frequency D A E X20 602 E 1 g f2 frequency Obviously this is a drag but in olden days it was really done that way with analog filters and analog computers sonny Today we use those newfangled digital computers as follows The true value of the PSD at a frequency f can be found as 60 2ggEIXfTIZ The steps in this equation are explained as follows The limit as t goes to in nity must be taken to account for variations in the random nature of the variable We will approximate in nity with a nite value depending on the signal and the lowest frequencies and the resolution in frequency we want E indicates the expected value often approximated by an average X f T is the Fourier transform calculated over a finite time T as given below XfT I xne zw dt It will be estimated at an integer number of frequency samples using the Fast Fourier Transform FFT algorithm calculated from samples of x You have dealt with this algorithm in your course on Computer Applications 0 Double vertical bars indicate the magnitude of the complex number If we use a single value of the Fourier transform as the estimate a real shortcut we get A 2 2 GM XfT Actually the magnitude is what is displayed on some oscilloscope calculations of the FFT This is the measurement you get in the lab if you use the FFT waveform math of the oscilloscope You can also compute the FFT using Matlab s function fft or to get the power spectral density directly with Matlab s function psd note the psd function has a lot of magic in the way it averages things and thus is not as good for leaming the process What does the FFT function give you As described in the Matlab help function for fft For length N input vector x the DFT is a length N vector X with elements N Xk sum xnexpj2pik lnlN l lt k lt N nl The inverse DFT computed by IFFT is given by N xn lN sum Xkexpj2piklnlN l lt n lt N kl The relationship between the DFT and the Fourier coef cients a and b in xn a0 sum akcos2piktnNdtbksin2piktnNdt kl is a0 XlN ak 2realXklN bk 2imagXklN where x is a length N discrete signal sampled at times t with spacing dt So the Digital Fourier Transform as implemented in an FFT gives you the amplitude of the sine and cosine terms of a Fourier series one period of which would create the same time domain function if there is no aliasing Aliasing must be avoided How good is your estimate of the PSD some extra info of relevance Measurements and calculations based on measurements are not perfect We can consider them to be an estimate which is equal to the true value plus error The error consists of a random component added to a bias A common way to view random error is by normalizing it relative to the true value so that the standard normalized random error is 8quot variancemean What kind of distribution does this random part come from The chi squared distribution describes a random variable which is the sum of the squares of n independent normal zero mean random variables z So 1 2 222 232 z is the distribution appropriate for estimates of CAX as described above but as discussed to this point we have only one calculation with a real and imaginary part so n2 If only a single reading of the FFT is used to estimate the PSD at a single frequency f you can t have much con dence in it due to random error The situation can be analyzed using the chisquared distribution and the normalized standard error will be found to be 8 o G f g G f n where n the number of degrees of freedom 2 for a single measurement and calculation of the waveform Hence the expected error is as large as the true value Not very good To increase n and do better we can 1 average several values of frequency together frequency smoothing thus reducing error but increasing the bandwidth ie decreasing the resolution in the frequency axis We can also 2 take multiple sets of data recalculating the FFT for each and average them with each other at each frequency value This is called ensemble averaging The effect on the error will be the same In fact if we have a total of T seconds of data we get the same error either way Since it is faster to calculate N FFTs of length TN averaging recalculating the FFT for separate segments of time is usually preferred Each average increases n by 2 so for an average of N values we get 8 r For qualitative indications of the frequency content of a signal this may not be necessary since one views the plot of the PSD and tends to average the high and low values appearing in a frequency range when there is a lot of uctuation Either form of smoothing results in a bandwidth or resolution on the frequency axis of Be The relation between Be the time over which data is collected T and the degrees of freedom of the chi squared distribution is A confidence interval of lOL results as follows axKw nZBT Irma2 Inna2 Bias Error Bendat and Pierson illustrate bias error for frequency smoothing but the effect is similar for ensemble averaging See the picture below If I want Gfg and get it by averaging across a range of frequencies from f0 32 to fly 32 it is clear that my average is biased low when the curvature of the function ie G is negative as shown For current purposes the bias will be treated non quantitatively with caution appropriate when Be is large relative to any peaks in the power spectral density Gfo EGfo An illustration of bias error due to frequency smoothing stirquothm of the Freauencv Function At this point we have considered a single signal and examined the content across a range of frequencies In system identification we want to know how the system is responding to inputs so we have two signals input and response Either may be corrupted by noise as shown in the diagram below Measurement noise 71 Input noise y Measurement x 1nput Signals in the system identification problem The frequency response function H 9 of the plant is what we want to estimate with a calculation of HO H f e l lf from measurements because it will help us create a linear mathematical model for the system since it gives us the magnitude and the angle 3 we need for the Bode plot We must carefully consider the effects of noise in most cases If noise were not an issue we could simply take the FFT of the output and divide by the FFT of the input then convert these compleX numbers to magnitude and phase Unfortunately life is not so simple due to both random and bias errors in the estimates Using the definitions from before that Ykf FFTykTSa Xkf FFTxkTSwe will further define the cross spectra and autospectra as 6 f 1imEXfTYfT 2 Gf11mElXfTIZ Taco Here the denotes complex conjugation Then we can get the frequency response function as G f H f W Gulf An alternative way to get at least the magnitude of the frequency response function is G f Hf w Gmf but this is known to produce poor results when the noise is significant How can we measure the effects of the noise The coherence function y f is a standard measure When the coherence function is l the effect of the noise has been eliminated whereas when coherence is 0 there is no effect of the signal Notice that the noise can be significant but its effect can be eliminated The coherence function is a real function defined as 2 72 f ny ml y 0ny f Lets simplify things slightly by assuming the input noise is zero but we have measurement noise Coherence is then the ratio of the response autospectra to the measurement autospectra y f z GWf W nyf so we can see that if there is no noise effect this will be one It is a measure ofthe amount of power that is linearly accounted for by the input 2 If yxy f 1s small at frequenc1es where H 9 1s small suspect that measurement n01se 1s overwhelming the response signal v If y f is small but H is not small suspect noise at the input which is driving the system but not accounted for by the input If the H 9 is peaked the coherence y f should be expected to be near 1 because of the strong signal otherwise expect that nonlinearities of the system are destroying the linearity of the relation between input and output Errors in the H 9 are complicated by the fact that there is a real and imaginary part that can also be expressed by a magnitude and angle If the uncertainty is depicted as a circle in the complex plane angle uncertainty will depend on the location of that circle Im Magnitude error Angle error Calculation of Estimates Calculation of the estimates are based on the DFT as implemented by calculating nd FFT s from time records T seconds long A 2 quotd 2 A 2 quotd 2 Gum ZIXkfT Gym ZIYfT ndT 161 quot4T k1 From the same experiment you must collect the time records of the measurement and calculate the cross spectrum A 2 quotd nyf ZXkfTYkfaT quot4T k1 The estimates of these two functions are then used in all other calculations as follows A 6 f H f AW Gm A Z 2 f Gm 7 y GMny f Collecting Data The time between samples must be fast enough to avoid aliasing The length of each data record must be long enough to get the resolution required This gives you the lowest frequency and the ability to measure peaks without bias LQR design 743 Symmetric Root Locus A most ellective and widely used technique of linear control systems design is the optimal linear quadratic regulator LQR The simpli ed version of the LQR problem is to nd the control such that the perfonnance index J pzzm u3tdz 790 O is minimized for the system it Fx Gu 791a z Hlx 791b where p in Eq 790 is a weighting factor of the designer s choice The control law that minimizes is given by linearstate feedback u 7 Kx 792 Here the optimal value of K is that which places the closedloop poles at the Symmetric root locus 74 Selection of Pole Locations for Good Design 511 stable roots those in the LHP of lhe rootlocus equation 1 VGA 9003 0 793 where G0 is the open loop transfer function from u to z z Ns 39 s 7F 1 7 G00 5 H101 G ms 94 Note that this is a root lows problem as discussed in Chapter 5 with respect to the parameter p which weights the cost tracking error 23 with respect to the control effort u2 in the performance index Eq 790 Note also that s and is affect Eq 793 in an identical manner therefore for any root s0 of Eq 793 there will also be a root at 30 We call the resulting root locus a symmetric root locus SRL since the locus in the LHP will have a mirror image in the right halfplane RHP that is there is symmetry with respect to the imaginary axis We may thus choose the optimal closedloop poles by rst selecting the matrix H1 which de nes the tracking error and which the designer wishes to keep small and then choosing p which balances the importance of this tracking error against the control effort Notice that the output we select as tracking error does not need to be the plant sensor output That is why we call the output in Eq 791 rather than 2 Selecting a set of stable poles from the solution of Eq 793 results in desired closedloop poles which we can then use in a poleplacement calcula tion such as Ackermann s formula Eq 770 to obtain K As with all root loci for real transfer functions Go the locus is also symmetric with respect to the real axis thus there is symmetry with respect to both the real and imaginary axes We can write the SRL equation in the standard rootlocus form Nf sNs I DSDS 0 795 obtain the locus poles and zeros by re ecting the openloop poles and zeros of the transfer function from u to 3 across the imaginary axis which doubles the number of poles and zeros and then sketch the locus Note that the locus could be either a 0 or 180 locus depending on the sign of Go sGDs in Eq 793 A quick way to determine which type of locus to use OS or 180 is to pick the one that has no part on the imaginary axis The realaxis rule of root locus plotting will reveal this right away For the controllability assumptions we have made here plus the assumption that all the system modes are present in the chosen output 3 the optimal closedloop system is guaranteed to be stable thus no part of the locus can be on the imaginary axis Examination of the SRL can provide us with pole locations that achieve varying balances by choosing different values of p between a fast response small values of z and a low control effort small values of u The following examples illustrate the use of the SRL 512 Chapter 7 Statespace Design 0 EXAMPLE 714 FIGURE 717 Symmetric root locus for a firstorder system A Firstorder Symmetric Root LOCMS Plot the SRL for the following rst order system with y Jquot 1j ll G 39 1 7 96 n A s u39 39 Solution The SRL equation Eq 793 for this example is l 1 7 97 0 p saJsa The SRL shown in Fig 717 is a 0quot locus The optimal stable pole can be determined explicitly in this case as 5 le p 798 Thus the closed loop root location that minimizes the performance index of Eq 790 lies on the real axis at the distance given by Eq 7 98 and is always to the left of the openloop root mm l 1 le O It is also possible to locate optimal pole locations for the design of an unstable system using the SRL and LQR method 0 EXAMPLE 7 15 SRL Design for an Unstable System Draw the SRL for the linearized equations of the simple inverted pendulum with 41 1 Take the output to be the sum of twice the position plus the velocity The equations of motion are H o 1 o mg 0 x71u Solution 799 FIGURE 718 Symmetric root locus for the inverted pendulum MATLAB lqr 74 Selection of Pole Locations for Good Design 513 llllll For the speci ed output of 2 xposition plus velocity we let the tracking error be 2 2 lx 7100 We then calculate from Eqs 799 and 7100 that 2 005 1101 s 7 15 The symmetric 0quot loci are shown in Fig 718 We would choose two stable roots for a given value of p and use them for poleplacement and control law design 9 As a nal example in this section we consider again the tape servomotor and introduce LQR design using the computer directly to solve for the optimal control law From Eqs 790 and 792 we know the information needed to nd the optimal control is given by the system matrices F and G and the output matrix H Most CACSD packages including MATLAB use a more general form of Eq 790 1 X J XTQX uTRudI 7102 0 Equation 7102 reduces to the simpler form of Eq 790 if we take Q pIITH1 and R l The direct solution for the optimal control gain is the MATLAB statement K IqrtF 6 Q R 7103

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