DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

EE 537 Homewors Friedland Text Updated: Wednesday November 8 Some homewor assignments refer to Friedland s text For full credit show all wor. Some problems require hand calculations. In those cases do not use MATLAB except to chec your answers. DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

EE 537 Homewor MATLAB Computer Simulation. Simulate the van der Pol oscillator y" + α ( y ) y' + y using MATLAB. Plot y(t) vs. t and also the phase plane plot y'(t) vs. y(t). Use y() y'().5. a. For α.. b. For α.8.. Do MATLAB simulation of the Lorenz Attractor chaotic system. Run for 5 sec. with all initial states equal to.3. Plot states versus time and also mae 3-D plot of x x x 3 using PLOT3(xxx3). x& σ( x x ) x& x& 3 rx bx use σ r 8 b 8/3. 3 x x x + x x 3 3.. A system has transfer function H ( s) s s + 4 + 4s + 3 a. Use MATLAB to mae a 3-D plot of the magnitude of H(s) b. Use MATLAB to mae a 3-D plot of the phase of H(s) c. Use MATLAB to draw magnitude and phase Bode plots

EE 537 Homewor State-Space Analysis. Show all wor do by hand. Mae plots with MATLAB.. x & x + u Ax + Bu y 8 6 [ 3 ] x Cx a. Find poles. b. Find Φ(s) and φ(t)e At. c. Find transfer function. d. Find impulse response. Use MATLAB to plot impulse response. e. Find step response. Use MATLAB to plot step response. f. Find output y(t) if u t) 3e 3t u ( t) x() [ ] T. ( a. Find state-space representation for the circuit in terms of L R R R3 C C. Now set L H C F CF and all resistors to ohm: b. Find transfer function from ABCD. Use MATLAB function sstf c. Find poles and zeros for these values of components. Use MATLAB not by hand. d. Simulate on MATLAB-- Plot the output y(t). Set input u(t) unit step and initial conditions equal to zero 3. x & x + u Ax + Bu y x Cx 8 6 4 a. Find poles and MV zeros b. Find transfer function c. Is system Minimal? d. Find minimal realization (ABCD)

EE 537 Homewor 3 The state-space models of four systems are below. For each system: a. Find poles and α ζ ω n POV for each complex pole pair. b. Find transfer function. c. Mae Bode plots of open-loop system. d. Plot root locus assuming P feedbac. e. Plot step response if the system is stable.. Inverted pendulum. [ ].. C B A 3. Ball balancer. [ ]. 9 C B A 4. Flexible-joint system. [ ] 3 3.... 3 C B A 5. Flexible-lin system.

[ ] 4 4.3 3... 4 C B A

EE 537 Homewor 4 Discrete Time SV Systems. Compound Interest. Consider the scalar DT SV system x ax + bu + Let initial condition be x and the input be the unit step u u ( )... The initial condition is called the initial balance and a constant input is nown as an annuity deposit. Find the state solution x. (Find analytic solution.). This is the compound interest formula in economics. 6. The DT SV is / x+ x 3/ + u y x [ ] a. Find poles and natural modes. b. Find resolvent Φ (z) and an analytic expression for the DT exponential c. Find transfer function d. Find pulse response (analytic expression) e. Find step response (analytic expression) f. Find analytic expression for the state x if x [ ] T u u ( ) 4 g. Find analytic expression for the output y in this case. h. Write the difference equation relating y to u. φ ( ) A and input is 7. For the system in problem a. Use iteration to compute x x x3 and y y y3 by hand. Do they agree with the analytic solution found in problem? b. Use the difference equation to compute y y y3 by hand. 8. a. Write a MATLAB program to compute output y if y.9y +. 995y u Use zero initial conditions and unit step input. Plot output for to 5. b. Find poles. Can you explain the behavior of y? Find natural frequency ω n.

EE 537 Homewor 5 Realization and Canonical Forms s + 3 H ( s) 3 s + 7s + 4s + 8. RCF a. Write SV equations for reachable canonical form b. Draw RCF bloc diagram. OCF a. Write SV equations for observable canonical form b. Draw OCF bloc diagram 3. Duality Show that RCF SV matrices are the same as OCF SV matrices if you reorder the states T T T bacwards and replace (ABC) by ( A C B ). 4. Parallel Canonical Form. a. Write SV equations for Parallel canonical form b. Draw Parallel Form bloc diagram 5. Gilbert s Method H ( s) s s s + + 4s + 3 s + + 7s + Find minimal SV realization. s s + 3s + s + s + 4

EE 537 Homewor 6 Ball Balancer Controller Design. The ball balancer is given by 9 A B C. [ ] x p p& θ θ &. We are going to do 3 feedbac control designs using MATLAB. with state [ ] T a. Use Acermann to find SVFB to place the poles at s ± j 5± j3. Plot time response over 5 seconds if x( ) [.. ] T. (MATLAB routine is called place or acer. Place is numerically more stable) b. Use LQR to design SVFB. Select R. Q diag{ 5 5}. Find closed-loop poles. Plot time response over 5 seconds if x( ) [.. ] T. b. Use LQR to design SVFB. Select R Q diag{ 5 5}. Find closed-loop poles. Plot time response over 5 seconds if x( ) [.. ] T.. A telescope pointing system is x& x + u a. Find poles. b. Use Acermann to mae settling time equal to ½ sec and POV equal to 4%. Do BY HAND. c. Verify BY HAND that the closed-loop poles are where desired. d. Use MATLAB to plot closed-loop step response. Chec settling time and POV. Chec closed-loop poles.

EE 537 Homewor 7 Design Project Ball Balancer LQR/LQE Regulator Design The inverted pendulum with both position and angle measurements is given by 9 A B C. [ ] x p p& θ θ &. with state [ ] T a. Use LQR to design a SVFB K. Select R. Q diag{ 5 5}. Find closed-loop poles. Use SVFB u -Kx and plot time response over 5 seconds if x( )... (You did this for hw 6.) [ ] T b. Use LQE to design an observer L. Select R o. Q o diag{ 5 5}. Find observer poles. c. Use observer to design a regulator u Kxˆ. Draw bloc diagram of the closed-loop system. Use MATLAB to plot system response with estimated state FB and observer. One way to do this is to write MATLAB M file with both the plant dynamics and the observer dynamics. Another is to mae a large augmented system with eight states. x( ).. and initial ~ x().5.5. Compare to results of part a. Plot time response over 5 seconds if initial state is [ ] T estimation error is [ ] T d. Use MATLAB to find the DOF regulator polynomials Z( s) T( s) S( s) in Z( s) U ( s) T( s) R( s) S( s) Y ( s) with r(t) the reference input. You can multiply polynomials in MATLAB using convolution operator conv( )

EE 537 Homewor 8 Digital PID Compensator Design The longitudinal dynamics of an aircraft have a short period mode whose dynamics are ω n q( s) δ ( s) ( s + α) + β with q(t) the pitch rate and δ(t) the elevator deflection. Select α. β 4. a. Use ODE3 in MATLAB to simulate the open-loop response. To do this write the aircraft dynamics in state-space form and write an M-file. (To write state equations you can first write down the differential equation describing the system.). Use ICs of q() q'(). The short period mode can be given desired handling qualities by using a PID controller d s i δ ( s) + p + e( s) s s + η with e(t) the tracing error. Select the derivative filtering pole at -η- and choose the PID gains for PID zeros at 3±3 j. b. Do a root locus versus the D gain. Using MATLAB is all right. c. Select a good value of d. Use MATLAB to simulate the closed-loop step response. Plot versus time the pitch rate q(t) the tracing error e(t) and the control input δ(t). Discretization: d. The aircraft mode time constant is sec. Select a sampling period of T msec (. sec) which is much faster than this. Convert the controller using the BLT to a digital PID tracing controller. Simulate the digital closed-loop system for the same time interval as you did the continuous system. Plot q e and δ. Show that the digital controller gives approximately the samples every T sec of the response with the continuous controller. (To simulate the digital controller you will need to convert both the PID controller and the aircraft dynamics to digital form using the BLT. Then write difference equations and write a Do Loop to plot the step response. Remember that t T.)

EE 537 Homewor 3. Polynomial Ran Tests A telescope pointing system can be written in state-space form as x& x + u y [ 3 ]x. a. Use polynomial ran test to select a B matrix to mae the stable mode unreachable. b. Use polynomial ran test to select a C matrix to mae the unstable mode unobservable.