(1) Identify individual entries in a Control Loop Diagram. (2) Sketch Bode Plots by hand (when we could have used a computer

Transcription

1 Last day: (1) Identify individual entries in a Control Loop Diagram (2) Sketch Bode Plots by hand (when we could have used a computer program to generate sketches). How might this be useful? Can more clearly see how individualterms affect system performance and system output. Easier to understand why different changes in the control loop might be used compensate for specific shortcomings and specific terms in the system. 1

2 Stability Defining stability Interpreting stability from the magnitude and phase plots 2

3 Stability - definition A stable system is a system which generates a bounded output when given a bounded input. E.g. a sine wave input results in a sine wave output that doesn t grow unbounded until the system finally breaks. Resonance is still acceptable (the output can become very large, as long as it does not become unbounded) In analyzing systems, we are MOST concerned about stability in Feedback Loops. Let's have a closer look at how a Feedback Loop might become unstable by looking at a Feedback Loop Diagram. 3

4 A closer look at Feedback Loop diagram for simplification: Motor and amplifier behaviour Error X(s) Y(s) G(s) + F(s) Sensor behaviour Calculate the output Value using Algebra: Y = G * Error = G * (X-FY) = GX - GFY Y * (1+GF)=G X This is the transfer function for a Feedback Loop! Y G H X 1 GF Where: G= forward transfer function GF = loop transfer function Y/X= G/(1+GF) 4

5 Aside general rules for Simplifying System Block Diagrams You can often derive the overall transfer function of a system using just the System Block Diagrams The final result may have just a single block containing all the original control, process and feedback blocks Simplify the diagram by combining functions together at the different nodes. 5

6 Aside: Rules for simplifying block diagrams: This is a feedback loop simplification. We just solved for this relationship on Slide 4! Image from: 6

7 This is the ultimate simplification of a feedback system, into a single step Motor and amplifier behaviour Error X(s) Y(s) G(s) + F(s) Sensor behaviour X(s) G (s) 1 G( s) F (s) Y(s) 7

8 For feedback loops, putting the feedback loop into this form is useful to identify something important about stability of the system. X(s) G (s) 1 G (s) F (s) Y(s) Major Question: overall transfer function Y G H X 1 GF What happens when GF gets to be a value of -1? H gets bigger and bigger, becoming an unstable system. GF can become -1 depending on the gain and phase of the system. This is the whole point of stability analysis, to determine the gain and phase when the system becomes unstable.. 8

9 Stability analysis usually assumes that the feedback element F(s) =1 (unity gain), in order to simplify the analysis. Most of the time, stability analysis in control theory assumes that the feedback gain is a con value of 1. This simplifies analysis to just looking at the true open-loop gain G(s): X(s) G(s) Y(s) X(s) + - G(s) 1 G (s) Y(s) 1 overall transfer function Y G H X 1 G Therefore, H becomes unstable if G ever becomes a value of -1 G is called the open-loop gain since G is the gain in the system if the feedback loop is removed. 9

10 Stability comes from avoiding G(s)=-1 G(s) = -1 when magnitude plot = 1 = 0 db, and phase plot* = -180 degrees A stable system is one where the open-loop gain is less than 1 when the open-loop phase angle is -180 degrees An unstable system is one where the open-loop gain is greater than 1 at the same time that the open-loop phase is -180 degrees (since this guarantees that at some point, the value G(s) would be equal to -1**) Aside: ( * the word phase is from the fact that s, H(s), and G(s) are all really complex number (with real and imaginary components), which can be represented with a magnitude and phase value ) ( **actually, this is only the real part of G(s) which equals -1, since G(s) is a complex # with real and imaginary parts, but really that doesn t matter in this analysis, it still guarantees that the system will become unstable at some point ) 10

11 For clarification: Why analyze the open-loop response instead of the closed-loop response directly? H(s) Why not analyze this entire expression X(s) G(s) 1 G (s) Y(s) H(s) instead of just analyzing G(s)? Because G(s) remains bounded and stable even as the entire expression H(s) becomes unstable. That is, the open-loop system can remain stable, but as soon as it is used in a feedback loop there is a chance that the entire system will inherently become unstable! We can use Bode plots and straight-line asymptotes to examine G(s) even when the system becomes unstable, but we don t have any tools to accurately analyze H(s) as the entire expression becomes unbounded and unstable.

12 You check stability by looking at the gain and phas margin of the open-loop system Magnitude plot Gain Margin - the amount of gain that you could add to the system before reaching a gain of 1 = 0 db. Measured at the phase 0) crossing frequency (phase = +/-180 0db Phase Margin - the amount of phase you could add/subtract to the system before 0. Measured at the gain reaching +/-180 Phase plot wgain_cross w phase_cross 0 crossing frequency (gain = 1 = 0dB) wgain_cross= gain crossing (frequency at which gain = 0dB, and where the phase margin is measured) wphase_cross= phase crossing (frequency at which -180 phase = +/- 180deg, and where gain margin is measured) 12 Plot from lecture notes, 8B, pg. 12

13 What can you do with the gain and phase margin? Tells you how much proportional gain (k) you can add to the system (to make the system operate faster) without becoming unstable. Tells you how you might select and design a controller to compensate for aspects of the system (e.g. PID) to increase stability or improve performance 13

14 Changing the Performance by using Controllers/Compensators Controllers alter the Bode Plots of the overall system PID controls revisited Improving performance with Controllers / Compensators 14

15 Some Terminology Controllers /Compensators An operation introduced into a control loop to make it operate in a desired way (rise time, fall time, stability, etc). Bandwidth The range of frequencies over which a system operates. Sometimes defined as the range of frequencies over which the magnitudes exceeds some arbitrary threshold (e.g. gain of 1, magnitude of -40dB, magnitude of 0.05, etc). In control theory, often defined as a drop of -3dB drop from the maximum value (a drop of 1/sqrt(2)). 15

16 Bode Plots bandwidth measurements FC= corner frequency, F H= high cutoff frequency, F L= low cutoff frequency BandpassFilter (bandwidth = F H FL) Bode plot of pole (aka lowpassfilter ) (bandwidth = F C 0 = FC) Aside: Why is -3 db drop used? -3dB = 20 log( ) = 20 log (1 /sqrt(2)) A -3 db drop is equal to a drop of 1/sqrt(2) in the magnitude of the signal. This means thesquareof the amplitude drops by a factor of 2 at the -3dB point. This is BandpassFilter with narrow band (bandwidth = F H FL) also called the half-power point ( power is usually the magnitude squared)... Don t worry, not mentioned again in this course) Images from: 16

17 Controllers/compensators change the Bode Plot (and therefore the performance) of the system (Compensated System ) = (Controller )* (Uncompensated) The bode plot for the final Compensated System is the same as putting the Uncompensated and Controller bode plots in cascade series (and then combining multiple bode plots as we have done in previous examples). Controller 10 db

18 PID Controllers Revisited In the in-class demo (arm moving back and KI C (s) K P KDs s forth when connected to a potentiometer), turning one of the three tuning knobs changed the constant values Kp, Ki, and Kd (i.e. We were changing the constants for proportional, derivative and integral gain in the transfer function) 18

19 PID Controllers Proportional Gain increases speed of system Derivative Gain- Acts like the damping /friction term ; Used to get reduce overshoot. Integral Gain- reduces steady-state error (error value accumulates over time, changes inputs) KI KDs2 KPs KI C (s) K P KDs s s k ( s z1 )( s z 2 ) s Looks like this has 2 zeros (z1 and z2), proportional gain, and one integrator (s in denominator) Can use this controller to generate phase compensation as well as a higher operating frequency bandwidth 19

20 PID Controller - Bode plots KI KDs2 KPs KI C ( s) K P KDs s s k ( s z1 )( s z 2 ) s Amplitude boost at low frequency Higher gain at high reduces steady-state error frequencies might lead to problems. Phase margin increase at high frequencies, higher bandwidth 20 Image from:

21 Use PID or other types of controllers to get a more desirable output 1. How to get faster response? Increase proportional gain Increase the system bandwidth (frequency response) 2. How to reduce overshoot? Increase damping (e.g. Increase derivative gain) 3. How to eliminate steady-state error Increase integral gain 21

22 1. How to get faster response?in general: wider bandwidth = faster response higher gain = faster response System T4 has a higher bandwidth than System T3 on the magnitude Bode Plot (frequency domain) frequency Therefore, System T4 has a faster rise time than System T3 (time domain). May or may not result in higher overshoot, oscillations. Figure: Response of two second-order systems time (fromdorf,beiser) 22

23 1. How to get faster response?in general: wider bandwidth = faster response higher gain = faster response Time Response to a step-input of a system with increasing proportional gain Kp. Note that as the gain increases: rise time is faster overshoot and oscillations are higher steady-state error is reduced (gets closer to desired final value of 1) All of these were seen in the PID demo done in-class as well tion/ An increase in the proportional gain shifts the (fromdorf,beiser) Bode Plot vertically upwards (this may also 23 increase system bandwidth as well)

24 2. How to reduce overshoot Increase damping / increase derivative gain Time Response to a step-input of a system with increasing derivative gain KD. Note that as KDincreases, system overshoot and oscillation decreases and Increasing K D disappears; higher KDvalues can severely reduce response time (a very slow-reacting system). 24 Image from:

25 3. How to reduce steady-state error Increase integral gain If KIis chosen correctly, the steady-state error reduces to zero in a reasonable time. If KItoo small, results in slow asymptotic approach to zero (may take a long time) If KIis too large, oscillatory response, or even instability may result. 25 Example from: