ME 375 System Modeling and Analysis

Size: px

Start display at page:

Download "ME 375 System Modeling and Analysis"

Philippa Stone
6 years ago
Views:

1 ME 375 System Modeling and Analysis G(s) H(s) Section 9 Block Diagrams and Feedback Control Spring 2009 School of Mechanical Engineering Douglas E. Adams Associate Professor

2 9.1 Key Points to Remember Block diagrams Each block is completely independent of the block before and after it (we assume there is NO LOADING) Solve for the highest order term and then add blocks and summing junctions until the equation is realized Graphical description of algebraic Laplace relationships Initial conditions are taken to be ZERO (particular solution) Feedback control If systems don t respond the way we like, and if we can t fi them, we measure the response and correct for any error Get there safely, fast, and do not oscillate/overshoot too much We use feedback control to change the poles and zeros We use feedback to stabilize, boost performance, reject disturbances, and reduce sensitivity

3 9.2 Two Simple Eamples First and second order systems First order systems (e.g. low pass filter, skydiver) G(s)

4 9.3 Two Simple Eamples First and second order systems Second systems (e.g. SDOF system, LCR oscillator) M G(s)

5 9.4 Inputoutput Block diagrams Decomposing G(s) Sometimes we want to decompose G(s) into its simplest components: M G(s)

6 9.5 Inputoutput Block diagrams Simplifying G(s) using Mason s Rule and B.D. algebra Sometimes we want to simplify combinations of blocks into a single G(s) block: G(s) = G(s) 1H(s)G(s) H(s) Mason s Rule

7 9.6 Inputoutput Block diagrams Simplifying G(s) using Mason s Rule and B.D. algebra We can choose which simplifications we want to make by combining certain blocks and pathways:

8 9.7 Block diagrams for a DC motor Coupling between electrical, magnetic, and mechanical domains How do we do this for coupled systems? (e.g., motor)

9 9.8 Eample Vehicle speed control system (demo) Input/ecitation Disturbance Assumptions Output/response Apply disturbance v(t) S.S. response Increasing grade S.S. error Transient response t Reduce response time Reduce S.S. error etc.

10 9.9 Why Use Feedback Control? Getting from point A to point B in the right way How do we get the system to respond the way we like? Too long Too much oscillation To here B A Take system from here Stability Speed Accuracy Just right Forward T.F. G(s) H(s) Unstable Closed Loop T.F. Feedback T.F. Openloop T.F.

11 9.10 A Practical Eample of Feedback Control Making the grade How hard should we study? Study time Performance Error Plant Graders OPENLOOP T.F. (w/o grades) Control law What happens if is large? CLOSEDLOOP T.F. (w/ grades) We get the grade we wanted and it doesn t matter how hard the eam is, in theory!

12 9.11 Feedback Control and Block Diagrams S.S. error, disturbance rejection, sensitivity, and performance In a servomechanism, we d like to specify a position and get there quickly and accurately: Steady state Disturbance Sensitivity Plant

13 9.12 Benefits of Feedback Control Stabilizes unstable systems (HIGHEST PRIORITY!) Enhances response characteristics Speed of response (bandwidth) Settling time Percent overshoot Reduces steadystate error Regulation (keep it in one place) Tracking (move it along a certain path) Reduces the effects of any disturbance Reduces sensitivity to system parameters In plant and control system

14 9.13 Openloop and Closedloop Changing system behavior without changing the system The system, or plant, may not respond the way we d like Poles we want Poles Rl Im How do we get to the poles?

15 9.14 Closedloop System with Sensors and Actuators Getting to the poles through feedback The key to the poles is in the internal feedback loops We get the new poles by measuring the velocity and position, and feeding that information back

16 9.15 Feedback Control Vehicular speed control with unity feedback OPENLOOP T.F.s CLOSEDLOOP T.F. (w/o disturbance) Sensitivity to M? S.S. error?

17 9.16 Feedback Control First order thermal environment control an electric thermos q i (t) R 1 OPENLOOP T.F.s Usually make these the same CLOSEDLOOP T.F. Sensitivity to T a? S.S. error?

18 9.17 Feedback Control Servomechanism speed control By measuring the error, we can compensate for it! SPEED CONTROL

19 9.18 Feedback Control Servomechanism speed control OPENLOOP T.F. CLOSEDLOOP T.F.

20 9.19 Feedback Control and the F.V. Theorem Servomechanism speed control for step inputs OPENLOOP T.F. Input speed CLOSEDLOOP T.F. For large K P and K o = K i

21 9.20 FluidLevel Control Block diagram generation w i C 1 C 2 R 1 R 2 Hoover Dam What if R j goes to zero or infinity?

22 9.21 Fluidlevel Control Block diagram reduction

23 9.22 Fluidlevel Control Block diagram (unity) feedback Error signal Actuation (control) signal Reference level CLOSEDLOOP T.F. Plant Unity feedback OPENLOOP T.F. Actual level

24 9.23 Steadystate Performance Servomechanism Speed control vs. Position Control TYPE 0 How can S.S. error go to zero? TYPE 1

25 9.24 System Type S.S. Error STEP TYPE 0 Nonzero S.S. error TYPE 1 Zero S.S. error RAMP TYPE 1 Nonzero S.S. error TYPE 2 Zero S.S. error Etc.

26 9.25 Steadystate Error Changing the system type using the control law q i (t) R 1 CLOSEDLOOP T.F.s How do we get zero S.S. error?

27 9.26 Transient Performance Getting there quickly and in the right way The plant responds according to the locations of its poles Poles we want Poles Im Rl y(t) 1 Time [s]

28 9.27 Transient performance Poles locations and transient response performance The plant may not respond the way we d like Im Rl

29 9.28 Feedback Control Design Vehicular speed control with unity feedback Stability Steadystate performance For M=1000 Kg, B=100 Ns/m, choose K such that S.S. error <1% What if there is a disturbance? Transient performance Choose K such that response time is less than 4 seconds

30 9.29 Feedback Control Design Servomechanism Proportional speed control OPENLOOP C.E. Openloop CLOSEDLOOP C.E. Closedloop What if we want to change the damping (simult.)?

31 9.30 Feedback Control Design Servomechanism Proportionalderivative speed control CLOSEDLOOP C.E. Closedloop C.L. w/ K P,K D Does this Make sense? What about the S.S. error? C.L. w/ K P O.L. Im Rl

32 9.31 Are We Following a Pattern? Motivating a need for root locus techniques CLOSEDLOOP C.E. Increasing K P Im Rl

33 9.32 Are We Following a Pattern? Motivating a need for root locus techniques CLOSEDLOOP C.E. C.L. w/ K P,K D O.L. Im Rl

34 9.33 Root Locus Plotting the poles of the closed loop system The characteristic equation changes when we close the loop: G c (s)g p (s)h(s) = "1 #G c (s)g p (s)h(s) = n180 o G c (s)g p (s)h(s) =1 CL poles Angle criterion X OL poles Magnitude criterion X Im Rl

35 9.34 Root Locus Rules for plotting root loci Get the characteristic equation, in the following form: Start at the OL poles, end at the OL zeros or infinity # of paths = # of OL poles Root loci are symmetric about the real ais and can t cross Root loci lie on the real ais to the left of an odd number of real poles and/or zeros Break away/in points are found from dk/ds=0 Im Asymptotes go out at the angle O X OL poles X Rl

36 9.35 Root Locus Two poles with unity feedback and proportional control Im symmetric Servoposition control System The system never becomes unstable even for infinite gain X X Rl To left of 1 pole

37 9.36 Root Locus Two poles with proportional control and sensor dynamics To left of 3 poles X OL poles pull the root loci to the right (Why? Bode?) (I.e. they destabilize the CL system) Servoposition control system with 1 st order sensor dynamics symmetric X X Im Rl To left of 1 pole

38 9.37 Root Locus Two poles with sensor dynamics and prop./deriv. control Im End at OL zero O X To left of 3 poles/zeros symmetric X X Rl To left of 1 pole OL zeros pull the root loci to the left (why? Bode?) (i.e. they stabilize the CL system)

39 9.38 Root Locus Two poles with sensor dynamics and prop./deriv. control Im End at OL zero X O To left of 3 poles/zeros symmetric X X Rl To left of 1 pole OL zeros pull the root loci to the left (more this timewhy?) (i.e. they stabilize the CL system)

40 9.39 PID Control Steadystate and transient performance using PID The plant may not respond the way we d like Speed, freq., SS acc. SS acc. ( phs.) Trans./damping ( phs.) Remember these?

41 9.40 Satellite Attitude Control A second order system with a double integrator EOM Im OLTF Plant poles X Rl Analysis: 1) What does the step response look like? 2) What is the SS error? 3) Is this acceptable? Control: 1) We want zero SS error in the step resp. 2) We want an overshoot of < 50% 3) We want a settling time of 10 sec

42 9.41 Satellite Attitude Control What about proportional control? CLTF Im CL poles X Rl 1) Root loci are symm. about the real ais 2) Loci approach OL zeros or Infinity at angles of This isn t good enough we get zero SS error (w/o dist.), but we can only choose the natural frequency we don t even have a damping ratio

43 9.42 Satellite Attitude Control What about proportionalderivative control? CLTF CL poles Im O X Rl Can you find this point? 1)! Root loci lie on real ais to left of odd # of OL poles/zeros 2) Root loci are symm. about the real ais 3) Loci approach OL zeros or Infinity at angles of What s net?

44 9.43 Satellite Attitude Control Designing the control law CLTF CL poles Im O X X Rl X What if we want to draw the loci for varying K P?

ANNA UNIVERSITY :: CHENNAI MODEL QUESTION PAPER(V-SEMESTER) B.E. ELECTRONICS AND COMMUNICATION ENGINEERING EC334 - CONTROL SYSTEMS

ANNA UNIVERSITY :: CHENNAI MODEL QUESTION PAPER(V-SEMESTER) B.E. ELECTRONICS AND COMMUNICATION ENGINEERING EC334 - CONTROL SYSTEMS ANNA UNIVERSITY :: CHENNAI - 600 025 MODEL QUESTION PAPER(V-SEMESTER) B.E. ELECTRONICS AND COMMUNICATION ENGINEERING EC334 - CONTROL SYSTEMS Time: 3hrs Max Marks: 100 Answer all Questions PART - A (10