University of Virginia PID Controllers Jack Stankovic University of Virginia Spring 2015

University of Virginia 2/44 Outline I Control: Integral Control PI Control: Proportional-Integral Control D Control: Derivative Control PD Control: Proportional-Derivative Control PID Control: Proportional-Integral-Derivative Control Summary

University of Virginia 3/44 P Control What is wrong with P control? –Non-zero steady-state error Why? –When (current, instantaneous) error becomes zero then there is no longer a control signal

University of Virginia 4/44 I Control What is I control? –The controller output is proportional to the integral of all past errors Integral Controller E(z) U(z)

University of Virginia 5/44 Integral Control KIKI G(z) R(z) E(z)T(z) U(z) D(z) + + N(z) Y(z) z/ z-1 Adds Pole

University of Virginia 6/44 PI Control

University of Virginia 7/44 D Control D Control: the control output is proportional to the rate of change of the error –D control is able to make an adjustment prior to the appearance of even larger errors. –D control is never used alone, because of its zero output when the error remains constant.

University of Virginia 8/44 PD Control

University of Virginia 9/44 PID Control

University of Virginia 10/44 I Control Mostly used with P control Here, by itself to demonstrate its main effects –Zero steady state error –Slow response

University of Virginia 11/44 Introducing I Control What is I control? –The controller output is proportional to the integral of all past errors Integral Controller E(z) U(z)

University of Virginia 12/44 I Control KIKI G(z) R(z) E(z)T(z) U(z) D(z) + + N(z) Y(z) z/ z-1

University of Virginia 13/44 Steady-State Error with I Control Start with Example 9.1: the IBM Lotus Domino Server Recall Here H(z) = 1

University of Virginia 14/44 Example (continued) Multiply top and bottom by (z-1) then set z = 1

University of Virginia 15/44 Why is the steady state error zero

University of Virginia 16/44 General Case for G(z) The steady-state error of a system with I control is 0, as long as the close-loop system is stable.

University of Virginia 17/44 The steady-state error due to disturbance

University of Virginia 18/44 Disturbance Rejection in the IBM Lotus Domino Server Example 9.3

University of Virginia 19/44 Transient Response with I Control I Control eliminates the steady-state error, but it slows the system down –The reason is the integrator adds an open-loop pole at 1, which generates a closed-loop pole that is usually close to 1. Example 9.2: Closed-loop poles of the IBM Lotus Domino Server

University of Virginia 20/44 Example 9.2 (continued) Observe the root locus –The largest closed-loop pole is always closer to the unit circle than the open-loop pole Here there is only P control Here there is I control

University of Virginia 21/44 PI Control Common controller Fast response by P control Accurate response by I control Good combination Another example of Pole Placement Design –Previously we did pole placement for P controller

University of Virginia 22/44 PI Control Design by Pole Placement Slightly different than for P control (see p 305 in text/handout) Design Goals: Assumption: G(z) is a first-order system –A higher-order system is approximated by a first- order system (chapter 3)

University of Virginia 23/44 PI Control Design by Pole Placement Example 9.5: Consider the IBM Lotus Domino server Determine transfer function of system and SASO requirements Stable – poles within unit circle Accuracy – zero steady state error Note: 2 poles (one from G(z) and one from I control)

University of Virginia 24/44 PI Control Design by Pole Placement Compute the desired closed loop poles Construct and expand the desired characteristic polynomial )

University of Virginia 25/44 Construct the modeled characteristic polynomial

University of Virginia 26/44 Example 9.5 (continued) Expand the modeled characteristic polynomial Set the desired equal to the modeled

University of Virginia 27/44 Example 9.5 (continued) Solve for

University of Virginia 28/44 Example 9.5 (continued) Then check Poles are ( ) and ( ) so the system is stable = 1 so there is no steady state error

University of Virginia 29/44 Example 9.5 (continued) P control leads to quicker response I control leads to 0 steady-state error

University of Virginia 30/44 PI Control Design Using Root Locus The new issue: –The root locus allows only one parameter to be varied –A PI controller has two parameters: The P control gain, and the I control gain Solution to this issue: –Determine possible locations of the PI controller’s zero, relative to other poles and zeros –For each relative location of the zero, draw the root locus –For the most promising relative locations, try a few possible exact locations –Simulate to verify the result

University of Virginia 31/44 Summary I control adjust the control input based on the sum of the control errors –Eliminate steady-state error –Increase the settling time D control adjust the control input based on the change in control error –Decrease settling time –Sensitive to noise P, I and D can be used in combination –PI control, PD control, PID control

University of Virginia 32/44 Summary (continued) Pole placement design –Find the values of control parameters based on a specification of desired closed-loop properties. Root locus design –Observe how closed-loop poles change as controller parameters are adjusted

University of Virginia 33/44 Relationship to WSN and RTS Consider PRR problem in WSN Consider Miss Ratio Problem in RTS

University of Virginia 34/44 Extra Slides

University of Virginia 35/44 Example 9.4 Moving-average filter plus I control –A moving average slows down the system responses –An I control also slows down the system response –So the combination leads to undesirable slow behavior Example: IBM Lotus Domino server + I control + Moving-average filer

University of Virginia 36/44 Moving-average filter vs. I controller An I controller works like a moving-average filter: –More response to sustained change in the output than a short transient disturbance An I controller drives the steady-state error to 0, but the moving-average filter does not.

University of Virginia 37/44 PI Control

University of Virginia 38/44 PI Control Design by Pole Placement Approaches: –Step 1: compute the desired closed-loop poles –Step 2,3,4: find the P control gain and I control gain –Step 5: Verify the result Check that the closed-loop poles lie within the unit circle Simulate transient response to assess if the design goals are met

University of Virginia 39/44 PI Control

University of Virginia 40/44 Steady-state Error with PI Control PI has a zero steady-state error, in response to a step change in the reference input –It also holds for the disturbance input

University of Virginia 41/44 PI Control Design Using Root Locus Example 9.6: PI control using root locus

University of Virginia 42/44 Example 9.6 (continue)

University of Virginia 43/44 Example 9.6 (continue) P control leads to quicker response I control leads to zero steady- state error

University of Virginia 44/44 CHR Controller Design Method

University of Virginia 45/44 CHR Controller Design Method

University of Virginia 46/44 CHR Controller Design Method Example 9.7:

University of Virginia 47/44 Example 9.7 (continue) No simulation is needed to verify it. Why? - Only one option in the table

University of Virginia 48/44 D Control

University of Virginia 49/44 D Control A Real CS Example: –An IBM Lotus Domino server is used for healthy consulting. (MaxUsr, RIS) –Bird flu happens in this area. More and more people request for the service. –More and more hardware is added to the server. –So the reference point keeps increasing. –To deal with the increasing reference point, do we have better choices than P/I control? –How about setting the control output proportional to the rate of error change?

University of Virginia 50/44 D Control D Control: the control output is proportional to the rate of change of the error –D control is able to make an adjustment prior to the appearance of even larger errors. –D control is never used alone, because of its zero output when the error remains constant. –The steady-state gain of a D control is 0.

University of Virginia 51/44 PD Control

University of Virginia 52/44 PD Control

University of Virginia 53/44 PD Control PD controllers are not appropriate for first-order systems because pole placement is quite limited PD controllers can be used to reduce the overshoot for a system that exhibits a significant amount of oscillation with P control Example: consider a second-order system

University of Virginia 54/44 Example (continue)

University of Virginia 55/44 Example (continue) It is real good It sounds good

University of Virginia 56/44 Example (continue)

University of Virginia 57/44 Example (continue)

University of Virginia 58/44 PID Control

University of Virginia 59/44 PID Control

University of Virginia 60/44 PID Control PI controllers are preferred over PID controller –D control is sensitive to the stochastic variations –A low-pass filter can be applied to smooth the system output. In that case, the D control only responds to large changes –But the filter slows down the system response PID Control Design by Pole placement –Compute the dominant poles based on the design goals –Compute the desired characteristic polynomial –Compute the modeled characteristic polynomial –Solve for the gains of the P, I and D control by coefficient matching –Verify the result

University of Virginia 61/44 PID control design by pole placement Example 9.8: –Consider the IBM Lotus Domino Server

University of Virginia 62/44 Example 9.8 (continue)

University of Virginia 63/44 Example 9.8 (continue)