Stefan B. Williams May, 211 AMME35: System Dynamics & Control Assignment 4 Note: This assignment contributes 15% towards your final mark. This assignment is due at 4pm on Monday, May 3 th during Week 13 or via WebCT prior to that time. Submit your report to the assignment box on the 3 rd floor outside of the drawing office in the Mechanical Engineering Building. Late assignments will not be marked unless a Doctor s certificate or equivalent is provided. Plagiarism will be dealt with in accordance with the University of Sydney plagiarism policy. You must complete and submit the compliance statement available on-line. The objective of this assignment is to investigate frequency domain modelling of systems and to apply the control system design methods discussed in the lectures to realistic design problems. This assignment should take an average student 3-4 hours to complete. Total Marks: 1 The front page of your report should include: Your name and SID Your tutorial group number 1. Adding a controller modifies the frequency response of a system. One of the first steps in control system design is to determine what type of controller transfer function would be most suitable. In order to make this decision, the designer examines how the system Bode diagram and the gain and phase margins change for different types of controllers. Draw the Bode diagrams for the following types of controllers and comment on their influence on the system to be controlled. [5 marks] a. Proportional controller b. PI controller c. Lead controller d. Lag controller Note that lag and lead controllers have the form below. For a lag controller, τ 1 <τ 2 and K = 1, for a lead controller τ 1 >τ 2 and K = τ 2 /τ 1 G 1 1 ( s s ) K! + =! s 2 + 1 2. Find analytical expressions for the magnitude and phase response for each G(s) below. For each transfer function, sketch the Bode plot by hand (you may use Matlab to verify your plot but must clearly indicate the asymptotes used to guide you on the plots included in this handout these should be submitted with your assignment). Indicate the asymptotes and which poles and zeros they correspond to on your plot. Provide an estimate of the Gain and Phase Margins based on your plots. [5 marks]
5 a. G(s) = s s +1 Magnitude (db) 4 2-2 -4-6 -8-1 -12 ( )( s + 5) Bode Diagram -14 1-3 1-2 1-1 1 1 1 1 2 1 3 Frequency (rad/sec) -45 Phase (deg) -9-135 -18-225 -27 1-3 1-2 1-1 1 1 1 1 2 1 3 b. G(s) = Magnitude (db) 4 2-2 -4-6 -8-1 -12 ( s +1) s +.1 Frequency (rad/sec) ( )( s +1) ( s + 5) Bode Diagram -14 1-3 1-2 1-1 1 1 1 1 2 1 3 Frequency (rad/sec) -45 Phase (deg) -9-135 -18-225 -27 1-3 1-2 1-1 1 1 1 1 2 1 3 Frequency (rad/sec)
3. A CD drive requires a control system to position a read/write head over a disk. A physical representation of the system and a block diagram are shown below. [15 marks] a. The specifications for closed loop response are 1% overshoot and a settling time of.5 seconds. Translate these time-domain specifications into frequency domain specifications on Phase Margin, bandwidth and crossover frequency. b. Design a lead compensator to attain the frequency domain specifications from part (a). Use Matlab to produce a Bode plot showing the frequency response of the original system, the compensator, and the compensated system. Simulate the transient response of the system and discuss any discrepancies between the expected and actual transient response. c. Use Matlab to generate a Bode plot for the closed loop system and determine the -3 db bandwidth. d. Now add a lag compensator to reduce the steady state error to a ramp input by a factor of 1. Verify the resulting system performance.
4. A system exhibits the following closed loop step response with unity feedback and a gain of K=1. For gains lightly larger, the response becomes unstable. Such a response is characteristic of systems with lightly damped high frequency resonances. When implementing feedback control, designers must be wary of exciting the resonance. The transfer function for this system is 3 G(s) = (s +.3)(s 2 +.5s +1) Note the lightly damped poles at ωn = 1rad/s. These are responsible for the high-frequency oscillations present in the step response. Please answer the following questions. You are encouraged to use Matlab for generating plots. Matlab s sisotool should be particularly helpful for part (c). [1 marks] (a) Use Matlab to draw the Bode plot for this system and for the same system but without the resonant poles, 3 G o ( s) = ( s +.3) What are the Gain Margins for these two systems and what do they tell you about the stability of each system? (b) If the loop gain K were the only parameter you could adjust, what value of K would attain the highest possible crossover frequency for the system with the resonance without driving it unstable? What is the Phase Margin for this choice of K? (c) In part (b) your value for K should have resulted in a system with excessive Phase Margin and low bandwidth. By applying compensation to reduce the high frequency gain, an approach known as gain stabilisation, we should be able raise the loop gain K and attain a system with improved bandwidth. One approach is to add a lag compensator near the resonant peak. Note that usually we use lag compensation to increase low frequency gain, whereas here we are using it to reduce high frequency gain. We can get away with this because the system has large Phase Margin at crossover. Use Matlab s sisotool to design a lag compensator so that the final system has a crossover frequency of about ωc =1. 5 rad/s and a Phase Margin of about 6º. Add the Bode plot for your compensated
system to your plot from part (a) and compare the step response of your compensated system the one shown above for K = 1. By about how much has the system s closed loop bandwidth improved over your answer for part (b)? 5. The following questions explore the state space techniques for modelling that we considered during Week 1 and 11. [15 marks] a. Convert the following systems to their equivalent state space representation in phase-variable form. 2 i. G(s) = (s +1)(s 2 + 5s + 2) ii. G(s) = 5(s 2 + 2s + 2) (s +1)(s + 5)(s +15)(s +1) b. Are the following systems controllable and/or observable? i.!!x = # " 1.5 y =! 1 " $ % x $! &x + # % ".25 $ &u % ii. "!x = $ # 2!1 1 y = " 1 1 # % & x % " 'x + $ & # c. Given the following open-loop plant, 1 G(s) = (s +1)(s + 4)(s + 6) design a controller using pole placement with full state feedback to yield a 16% overshoot and a settling time of.75 seconds. Place the third pole 5 times as far from the imaginary axis as the dominant pole pair. Use the phase variables for state-variable feedback. Verify your design. Lab 6. You should complete Lab 3 during either Week 1 or 11 with your lab group (check the lab schedule to find out when you are expected to complete the lab). As a group, prepare a short report outlining your findings in the lab and answering the questions contained in the lab handout. This report can be submitted as a group, clearly indicating the names and SIDs of the members of the group. [2 marks] Group Based Question 7. We would like you to undertake a final group task and design a control system as a team. The team should comprise at most four or five students and may be the same groups you used for your labs. The scenario: a helicopter is flying over the ocean when suddenly the craft springs a massive fuel leak (5L/s). A small island is visible 21 m dead ahead but does not appear to have a suitable landing site on it. One of the pilots, who happens to be an experienced sky-diver figures that in order to parachute onto the island they will need to achieve an altitude of 5 m and jump clear of the craft 1 m shy of the island (i.e. 2 m dead ahead). They will need time to don their parachutes, so the craft should hover in this location. We would like you to design a control system for a simplified 2D 1!1 % 'u &
version of the helicopter, as shown in the figure below, that will allow the pilots to jump to safety before fuel runs out. The helicopter pose can be characterised by its 2D position (x, z) and pitch angle, θ, relative to the vertical axis. The aircraft is controlled by varying the thrust of the main rotor, T, and the rotor tilt angle, δ, relative to the vertical axis of the craft. The vertical displacement, h t, between the centre of gravity and the main rotor is 1m. The gravitational constant is 9.81m/s 2. Because of the leak, you have a limited fuel budget and the variable mass of the fuel will change the dynamics of the vehicle. Generating thrust consumes fuel at a rate of 1(L/hr)/kN. Specifications of the vehicle are included in the following table. You may ignore drag in the vertical and pitch axes but the vehicle will be subject to a drag force, F d, proportional to the square of the forward velocity. The vehicle should be limited to a maximum pitch of 3 o for pilot comfort. Begin by working out the dynamic models that govern the system and generating a Simulink model of the system. You may need to consider the control in stages to control the pitch, lateral and vertical motion. You may choose to undertake this design using the classical, LTI methods we have studied by treating pitch, lateral and vertical control separately or you may wish to consider exploring the State Space techniques more fully, which will allow you to account for the coupled dynamics. Submit a group report detailing your design and an analysis of its performance, clearly indicating the names and SIDs of the group members. [25 marks] Table 1 - Helicopter specifications Specification Value Mass (without fuel) 7,kg Moment of Inertia 5, kg m 2 Max. Ft propulsion 25kN Max. rotor deflection 5 o Initial fuel 15 L Fuel density.7 kg/l Fuel consumption 1 (L/hr)/kN Fuel leak 5 L/s Forward drag coefficient (quadratic) 4.5 N s 2 /m 2
Selected Equations Time Response 2 ω σ = ζω n n Gs () = C s 2 2 + 2ςωn s + ω 2 n ωd = ωn 1 ζ 1.8 4 tr t s ωn σ πζ ln( M ) p 2 ζ = 1 ζ M 2 2 p = e, ζ < 1 π + ln ( M ) p M p 5%, ζ =.7 16%, ζ =.5 2%, ζ =.45 Steady-State Error 1 e ( ) = 1 1 step e 1 + ramp ( ) = eparabola( ) = 2 lim Gs ( ) lim sg( s) lim sgs ( ) s s s Root Locus KG() s H() s = 1 KG() s H() s = (2k + 1)18 Frequency Response [ G jω ] [ G jω ] 1 tan G( jω) o σ = a (2k + 1) π θa = n m ( [ ]) ( [ ]) 2 2 M = G ( jω ) = G ( s ) = Re G ( jω ) + Im G ( jω ) s= jω Im ( ) φ = = Re ( ) GM = 2log K PM = tan 1 State Space!x = Ax + Bu y = Cx + Du 2ζ 2 4 2ζ + 1+ 4ζ Controllability R = B AB A 2 B! A n!1 B finite poles [ ] n m finite zeroes Observability " C % $ CA ' $ ' O = $ CA 2 ' $! ' $ ' # $ CA n!1 &'
Laplace Transform Tables