Signal Processing and Control Tutorial Problems PDF

ZEIT3221 Signal Processing and Control Tutorial Problems 1 Discrete-time Signals using both the explicit form and the matrix form. 1.1. 2.4. Write out samples of the following discrete-time sign...

ZEIT3221 Signal Processing and Control Tutorial Problems 1 Discrete-time Signals using both the explicit form and the matrix form. 1.1. 2.4. Write out samples of the following discrete-time signals for Show that the DTFT X(ejω ) is periodic with period 2π. n = −2, −1, 0, 1, 2. (a) x[n] = ejπn/20 2.5. (b) x[n] = δ[n] Calculate the DTFT X(ejω ) of (c) x[n] = u[n] (d) x[n] = 0.8n u[n]  1, for − N ≤ n ≤ N (e) x[n] = cos(πn/10) x[n] = 0, otherwise 1.2. Suppose we have the finite-length signals x[n] = {3, 6, −2} and w[n] = {−1, 4, 2}. Determine 2.6. (a) the product signal y[n] = x[n]w[n] Show that X(ejω ) = X ∗ (e−jω ) if x[n] ∈ R. (b) the first-order difference signal y[n] = x[n] − x[n − 1] (c) the integrated signal y[n] = y[n − 1] + x[n] 2.7. (d) the energy of x[n] Calculate the DTFT of x[n − n0 ] given that X(ejω ) is the DTFT of x[n]. 1.3. 2.8. Suppose we have the finite-length signal x[n] = Calculate the DTFT of ejω0 n x[n] given that X(ejω ) is the {2, 6, −3, 1, −4}. Determine the signal arising from DTFT of x[n]. (a) a circular shift to the right by k = 3 (b) a circular shift to the left by k = 1 2.9. Show that X[k] = X ∗ [N − k] for k = 0, 1,... , N − 1 if x[n] ∈ 1.4. R. Consider the finite-length signal x[n] = {2, 1, 4}. (a) What is the corresponding finite-support signal x̄[n] for 2.10. n = 0 to n = 9? Panel (a) of the figure below shows 902 samples of a discrete- (b) What is the corresponding N -periodic signal x̃[n] for time signal x[n]. The sequence is the A3 note of an acoustic n = 0 to n = 9? guitar, sampled at Fs = 22.255 kHz. Panel (b) of the figure shows the magnitude of its DFT, |X[m]|, with frequency 1.5. range corresponding to 0 ≤ m ≤ 59 on a linear scale, and (a) Consider the finite-length signal x[n] = {3, 2, 1 − panel (c) shows the same information on a dB scale. j, −4, 7, −j9, 8, −j, 2}. Downsample the signal by N = 3 (a) Based on the plotted |X[m]|, what is the fundamental starting from the first element of the original signal. frequency (in Hz) of the guitar’s A3 note? (b) Consider the finite-length signal x[n] = {1 + j2, 0, 2}. (b) What is the frequency of the highest shown non-zero Upsample the signal by N = 4 starting from the first element spectral component of the guitar’s A3 note? of the original signal and ending with the last element of the original signal. 2 Fourier Analysis 2.1. List the basis vectors w(k) for N = 4. 2.2. Write out the matrices W and WH for N = 4. 2.3. (a) Evaluate the DFT of the sequence {2, −1, 3, 2} using both the explicit form and the matrix form of the DFT. (b) Subsequently calculate the inverse DFT of your results, Figure for 2.10. The A3 note of a guitar, represented by its and check that you get back to the original sequence, again time-domain representation and the magnitude of its DFT. 1 2.11. 3.9. Assume we have a time-domain signal x[n] of length Invert the lowpass frequency response N = 3800, on which we perform a radix-2 FFT.  (a) How many zeros must be appended to the end of x[n]? jω 1, |ω| ≤ ωc Hlp (e ) = (b) After the FFT is performed and assuming a sampling 0, ωc < |ω| ≤ π frequency of Fs = 10 kHz, what is the spacing in Hz between adjacent frequency-domain samples? to obtain the corresponding impulse response sin(ωc n) 2.12. hlp [n] = πn Suppose we want to perform Fourier analysis on a time- domain signal of length N = 32, 768. Estimate the ratio of the number of complex multiplications needed by a N -point DFT to the number of complex multiplications needed by a 3.10. N -point FFT (assuming a radix-2 implementation). Consider the code fragment shown in the figure. Suppose M = 3, N = 3, filter coefficients are b0 = 1, b1 = 2, b2 = 0.5, b3 = 1, a1 = 1, a2 = 0.5, the delay lines initially have values x = 1.7, x = 2.8, x = 1.9, y = 3.6, y = 3.1, 3 Discrete-time Filters y = 2.4, and the input value is 1.5. Determine the values in the delay lines and the variable y at each step of this code 3.1. fragment. (a) Is y[n] = 3x[n] + 5x[n − 1] linear and/or time-invariant? (b) Is y[n] = x[n]2 linear and/or time-invariant? (c) Is y[n] = nx[n] linear and/or time-invariant? 3.2. Show that the convolution operator is commutative. 3.3. If x[n] = ejω0 n and h[n] = an u[n], find y[n] = x[n] ∗ h[n]. 3.4. If x[n] = {1, 2, −1} and h[n] = {3, 2, 1}, find y[n]. 3.5. (a) Is y[n] = x[n − 1] causal/noncausal/anti-causal? (b) Is y[n] = x[n + 1] causal/noncausal/anti-causal? (c) Is y[n] = x[n − 1] + x[n + 1] causal/noncausal/anti-causal? 3.6. Calculate the output y[n] in terms of the input x[n] for a pure delay filter D having the impulse response h[n] = δ[n−1]. 3.7. (a) Calculate the frequency response of the moving average filter from its impulse response Figure for 3.10. C code fragment implementing an IIR  filter. 1/N, n = 0, 1,... , N − 1 h[n] = 0, otherwise 4 z-Transforms (b) Subsequently determine the magnitude response and the 4.1. phase response of the moving average filter. Calculate the z-transform of the discrete-time signal 3.8. x[n] = an u[n], a ∈ C Calculate the frequency response of the leaky integrator filter from its impulse response h[n] = (1 − λ)λn u[n] 4.2. Show that, if X(z) is the z-transform of x[n], then Z{x[n − N ]} = z −N X(z) 2  filter with cutoff frequency ωc = π/4. (b) Assuming a sampling frequency of Fs = 16 kHz, what 4.3. physical frequency does this cutoff frequency correspond to? Determine the transfer function corresponding to the CCDE 5.4. y[n] = x[n] + 3x[n − 1] + 2y[n − 1] + 2y[n − 2] − y[n − 3] Determine the FIR filter coefficients for a 5-tap lowpass filter with cutoff frequency ωc = π/4 and subject to a Blackman window. 4.4. 5.5. Determine the CCDE corresponding to the transfer function Suppose a continuous-time signal is sampled at Fs = 2 MHz, Y (z) 1 + 2z −1 + 3z −2 and the discrete-time signal is sent through a 70-tap linear- H(z) = = phase lowpass FIR filter with cutoff frequency of 600 kHz. X(z) 1 − z −1 + 0.5z −2 What would be the time delay, in seconds, between the filter’s input and output for a sinusoidal tone at 200 kHz. 4.5. 5.6. Determine whether or not the filter described by the following What is the transfer function of a resonator bandpass filter transfer function is stable centred at ω0 = 0.5π assuming λ = 0.9 and a gain of 1 in the Y (z) 1 + 2z −1 + 3z −2 centre of the passband? H(z) = = X(z) 1 − z −1 + 0.5z −2 5.7. Design a digital filter corresponding to the second-order Chebyshev analog lowpass filter with ripple 1 dB and a 1 dB 4.6. cutoff at 20 Hz. This has the transfer function Sketch (by hand) the pole-zero plot and the magnitude re- 17410.145 sponse of the filter having the following transfer function H(s) = 2 s + 137.94536s + 17410.145 Y (z) 1 + 2z −1 + 3z −2 Use the bilinear transform and assume a sampling frequency H(z) = = X(z) 1 − z −1 + 0.5z −2 of Fs = 100 Hz. 5.8. 4.7. Design an infinite impulse response filter, using the bilinear The impulse response of the moving average filter is given by transform method, corresponding to the analog Butterworth  lowpass filter with a 20 Hz cutoff frequency, unity DC gain, 1/N, 0 ≤ n < N and filter transfer function given by h[n] = 0, n < 0 and n > N − 1 15791.4 Hc (s) = 2. Calculate the transfer function of the moving average filter. s + 177.714s + 15791.4 Assume a sampling frequency of Fs = 200 Hz. 4.8. The impulse response of the leaky integrator filter is given by 5.9. Suppose there is a filter with a zero at zn = 0.8, poles at h[n] = (1 − λ)λn u[n] pn = 0.2 ± j0.3, and a DC gain of 1. (a) What is the corresponding transfer function? Calculate the transfer function of the leaky integrator. (b) Rewrite this transfer function using only real coefficients. (c) Write the CCDE corresponding to the transfer function. 5 Filter Design 5.1. 6 Dynamic Response (1+j)z+π (a) Is H(z) = z+2 a rational transfer function? 3z −1/2 +1 (b) Is H(z) = z−1 a rational transfer function? 5.2. Sketch the filter specifications for a bandpass filter from 0.3π to 0.4π, assuming a ±10% tolerance in both the passband and stopband, and transition region widths each of 10% of Figure for 6.1. Unity feedback system with proportional the total bandwidth π. control. 5.3. 6.1. (a) Determine the FIR filter coefficients for a 5-tap lowpass For the unity feedback system shown in the figure, specify 3 the gain K of the proportional controller so that the output y(t) has an overshoot of no more than 10% in response to a unit step. 6.2. Figure for 6.6. Unity feedback system with compensator. For the unity feedback system shown in the figure, specify the gain and pole location of the compensator so that the overall closed-loop response to a unit-step input has an Use Routh’s stability criterion to determine whether the overshoot of no more than 25% and a 1% settling time of no resulting closed-loop system will be stable. more than 0.1 s. 6.8. Find the range of K for which all the roots of the following polynomial are in the LHP: s5 + 5s4 + 10s3 + 10s2 + 5s + K = 0 7 Feedback Control Figure for 6.2. Unity feedback system with compensator. 7.1. A unity feedback control system has the open-loop transfer 6.3. function A certain servomechanism has dynamics dominated by a A G(s) = pair of complex poles and no finite zeros. The time-domain s(s + a) specifications on the rise time (tr ), percent overshoot (Mp ), Compute the sensitivities of the open-loop and closed-loop and settling time (ts ) are given by: tr ≤ 0.6 s, Mp ≤ 10%, transfer functions to changes in the parameter A. ts ≤ 3 s. Sketch the region in the s-plane where the poles could be placed so that the system will meet all three 7.2. specifications. A controller for a DC servo motor with transfer function 5 G(s) = s(s+10) has been designed with a unity feedback 6.4. A feedback system has the following response specifications: structure and has the transfer function Dc (s) = 6 (s+7)(s+9) s(s+12). Mp ≤ 16%, ts ≤ 6.9 s, tr ≤ 1.8 s. Sketch the region of (a) Find the system type for reference tracking and the acceptable closed-loop poles in the s-plane for the system, corresponding error constant for the system. assuming the transfer function can be approximated as (b) If a disturbance torque w adds to the control so that the simple second-order. input to the process is u + w, what is the system type and corresponding error constant with respect to disturbance 6.5. rejection? Approximate each of the following transfer functions with a second-order transfer function: 7.3. A compensated motor position control system is shown in (0.6s + 1)(0.35s + 1) G1 (s) = the figure. Assume the sensor dynamics are H(s) = 1. (0.38s + 1)(0.55s + 1)(s2 + 1.1s + 1) (a) Can the system track a step reference input r with zero (0.6s + 1)(0.35s + 1) steady-state error? If yes, give the value of the velocity error G2 (s) = (0.38s + 1)(0.55s + 1)(s2 + 0.2s + 1) constant. (0.6s + 1)(−0.35s + 1) (b) Can the system reject a step disturbance w with zero G3 (s) = 2 steady-state error? If yes, give the value of the velocity error (0.38s + 1)(0.55s + 1)(s + 1.1s + 1) constant. 6.6. Consider the system shown in the figure, where 1 K(s + z) G(s) = , Dc (s) = s(s + 20) s+p Find K, z, and p so the closed-loop system has a 10% overshoot to a step input and a settling time of 1.5 s (1% criterion). 6.7. Suppose that unity feedback is applied around a system with loop gain 0.2(s2 + 0.95s + 0.11) KG(s) = Figure for 7.3. Motor position feedback control system. s(s2 + 0.36s + 0.72) 4 7.4. from (a). Consider the system shown in the figure. (a) Compute the system transfer function from R(s) to E(s) 8.2. and determine the steady-state error (ess ) for a unit-step Sketch the asymptotes of the Bode plot magnitude and phase reference input signal, and a unit-ramp reference input for each of the following open-loop transfer functions. 5(s+12) signal. (a) L(s) = s(s+4)(s+9)(s+17) (b) Determine the locations of the closed-loop poles of the 2 s +5s+11 (b) L(s) = s(s 2 +5s+15) system. s+7.5 (c) Select the system parameters (k, kP , kI ) such that the (c) L(s) = s2 (s+75) closed-loop system has damping coefficient ζ = 0.707 and (s+50)(2s2 +5s+4) (d) L(s) = s2 (s+5)(s2 +60s+120) ωn = 1. What percent overshoot would you expect in y(t) for a unit-step reference input? 8.3. The Bode plot for 100[(s/10) + 1] G(s) = s[(s/1) − 1][(s/100) + 1] is shown in the figures. (a) Why does the phase start at −270◦ at low frequencies? (b) Is the closed-loop system for the Bode plot shown stable? (c) Will the system be stable if the gain is lowered by a Figure for 7.4. PI feedback control system. factor of 100? 7.5. For the system shown in the figure. (a) Design a proportional controller to stabilize the system. (b) Design a PD controller to stabilize the system. Figure for 7.5. Feedback control system with unstable plant. Figure for 8.3. Magnitude response. 7.6. Consider the feedback control system with the plant transfer 1 function G(s) = (s+0.1)(s+0.5). (a) Design a proportional controller so the closed-loop system has damping of ζ = 0.707. Under what conditions on kP is the closed-loop system stable? (b) Design a PI controller so that the closed-loop system has no overshoot. Under what conditions on (kP , kI ) is the closed-loop system stable? Figure for 8.3. Phase response. 8 Frequency-Response Methods 8.1. 8.4. (a) Calculate the magnitude and phase of Suppose that we have a unity feedback system with loop gain 1 25(s + 1) G(s) = G(s) = s+7 s(s + 2)(s2 + 2s + 16) by hand for ω = 0.1, 1, 7, 20, 100 s−1. Use Matlab’s margin to calculate the PM and GM for (b) Sketch the asymptotes for G(s) according to the Bode G(s) and, on the basis of the Bode plots, conclude which plot rules, and compare these with your computed results margin would provide more useful information to the control 5 designer for this system. 9.3. Consider the plant described by 8.5.     For the open-loop system 0 1 1 ẋ = x+ u 2 −9 8 K(s + 1) y = [2 4]x KG(s) = s2 (s + 10)2 (a) Find the transfer function using matrix algebra. determine the values for K that will yield PM ≥ 30◦ and the (b) Find the closed-loop characteristic equation if the maximum possible closed-loop bandwidth. feedback is: (i) u = −[K1 K2 ]x; (ii) u = −Ky. 8.6. 9.4. Consider the system with transfer function For the system 5 G(s) =  0 1   0  s(s + 1)(s/5 + 1) ẋ = x+ u −7.2 −9.3 1 Use Bode plot sketches to design a lead compensation Dc (s) y = [1 0]x with unity DC gain so that PM ≥ 40◦. design a state feedback controller that satisfies the following 8.7. specifications: (i) closed-loop poles having a damping coef- The open-loop transfer function of a unity feedback system is ficient ζ = 0.707; (ii) step-response settling time is under 0.5 s. K G(s) = 9.5. s(s/3 + 1)(s/40 + 1) For the system Design a lag compensator for G(s) using Bode plot sketches     0 1 0 so that the closed-loop system satisfies the following specifi- ẋ = x+ u 0 −7.5 1 cations: (i) the steady-state error to a unit ramp reference input is less than 0.01; (ii) PM ≥ 40◦. y = [1 0]x 8.8. design a state feedback controller so that the closed-loop step A DC motor with negligible armature inductance is to be used response has an overshoot of less than 18% and 1% settling in a position control system. Its open-loop transfer function time under 0.3 s. is given by 50 9.6. G(s) = Consider a system with the transfer function s(s/5 + 1) Design a compensator for the motor using Bode plot sketches s + 15 so that the closed-loop system satisfies the following speci- G(s) = 2 s − 15 fications: (i) the steady-state error to a unit ramp input is less than 1/200; (ii) the unit step response has an overshoot (a) Find (Ao , Bo , Co ) for this system in observer canonical of less than 20%; (iii) the bandwidth of the compensated form. system is no less than that of the uncompensated system. (b) Check if this system observable. (c) Is (Ao , Bo ) controllable? (d) Compute K so that the closed-loop poles are assigned to s = −15 ± j15. 9 State-Space Methods (e) Design an estimator with error poles at s = −15 ± j15. 9.1. Give the state description matrices in control canonical form 9.7. for the following transfer functions: A certain process has the transfer function (a) G(s) = s27s+1 +3s+2 (b) G(s) = s(s2s+7 4.5 +2s+2) G(s) = s(s − 4.5) 9.2. Given the system (a) Find Ao , Bo , Co for this system in observer canonical form. (b) If u = −Kx, compute K so that the closed-loop control     −3 2 0 ẋ = x+ u poles are located at s = −1.8 ± 2j. −5 −1 1 (c) Compute L so that the estimator-error poles are located with zero initial conditions, find the steady-state value of x at s = −15 ± 15j. for a step input u. (d) Give the transfer function of the resulting controller. 6 10 Discrete-time Control 10.1. A discrete-time controller is designed as a discrete equivalent of a continuous-time controller. This results in the controller transfer function U (z) 1 + 0.2z −1 D(z) = = E(z) (1 + 0.16z −1 )(1 − 0.25z −1 ) Determine the difference equation that implements this controller. 10.2. A unity feedback system has an open-loop transfer function given by 178 G(s) = s(s + 17) The following lag compensator added in series with the plant yields a phase margin of 50◦ and a crossover frequency of 30.6 rad/sec: 0.0475s + 1 Dc (s) = 4.141 0.02246s + 1 Using Tustin’s method, determine an equivalent digital realization of this compensator for a sample rate of 40 Hz. Express your result as both a normalized transfer function and as a difference equation. 10.3. The following transfer function is a lead network designed to add about 33.7◦ of phase at ω1 = 2.47 rad/sec: 0.7s + 1 H(s) = 0.2s + 1 Assuming a sampling period of T = 2/3 s, determine the poles and zeros of the resulting digital compensator. Subse- quently determine the realization of this digital compensator as a transfer function. 7