Feedback Control of Linear SISO systems

Process Dynamics and Control

1 Open-LoopOpen-Loop ProcessProcess

 The study of dynamics was limited to open-loop systems  Observe process behavior as a result of specific input signals

2 Closed-LoopClosed-Loop SystemSystem

 In study and design of control systems, we are concerned with the dynamic behavior of a controlled or Closed-loop Systems

Feedback Control System

3 FeedbackFeedback ControlControl

 Control is meant to provide regulation of process outputs about a reference, , despite inherent disturbances

Controller System

Feedback Control System

 The deviation of the plant output, ,from its intended reference is used to make appropriate adjustments in the plant input,

4 FeedbackFeedback ControlControl

 Process is a combination of sensors and actuators

 Controller is a computer (or operator) that performs the required manipulations

Computer Actuator + + + - Process

Sensor

e.g. Classical one degree-of-freedom feedback control loop

5 Closed-LoopClosed-Loop TransferTransfer FunctionFunction

Block Diagram of Closed-Loop Process

Computer Actuator + + + - Process

Sensor

 - Open-Loop Process Transfer Function  - Controller Transfer Function  - Sensor Transfer Function  - Actuator Transfer Function

6 Closed-LoopClosed-Loop TransferTransfer FunctionFunction

 For analysis, we assume that the impact of actuator and sensor dynamics are negligible

 Closed-loop process reduces to the block diagram:

Feedback Control System

7 Closed-loopClosed-loop TransferTransfer FunctionsFunctions

 The closed-loop process has  Two inputs  The reference signal  The disturbance signal  Two outputs  The manipulated (control) variable signal  The output (controlled) variable signal

 We want to see how the inputs affect the outputs  Transfer functions relating , and ,

8 Closed-loopClosed-loop TransferTransfer functionfunction

 There are four basic transfer functions

 They arise from three so-called sensitivity functions

 Highlights the dilemma of control system design  Only one degree of freedom to shape the three sensitivity functions

9 Closed-loopClosed-loop TransferTransfer FunctionsFunctions

 Sensitivity functions:  The sensitivity function:

 The complementary sensitivity function:

 The control sensitivity function:

10 Closed-loopClosed-loop TransferTransfer FunctionsFunctions

 Overall transfer function for the output:

SERVO REGULATORY RESPONSE RESPONSE

 Servo response is the response of the output to setpoint change  Regulatory response is the response of the output to disturbance changes

11 Closed-loopClosed-loop TransferTransfer FunctionsFunctions

 Servo mechanism requires that:

 Regulatory response requires that:

 Since

 The two objectives are complementary

12 Closed-loopClosed-loop TransferTransfer FunctionsFunctions

 Note that

or

requires that the controller is large

This leads to large control sensitivity

13 PIDPID ControllerController

Most widespread choice for the controller is the PID controller

The acronym PID stands for: P - Proportional I - Integral D - Derivative

PID Controllers: greater than 90% of all control implementations dates back to the 1930s very well studied and understood optimal structure for first and second order processes (given some assumptions) always first choice when designing a control system

14 PIDPID ControlControl

PID Control Equation

Proportional Derivative Action Action

Integral Controller Action Bias PID Controller Parameters

Kc Proportional gain Integral Time Constant Derivative Time Constant Controller Bias

15 PIDPID ControlControl

PID Controller Transfer Function

or:

Note:

 numerator of PID transfer function cancels second order dynamics  denominator provides integration to remove possibility of steady-state errors

16 PIDPID ControlControl

Controller Transfer Function:

or,

Note:

Many variations of this controller exist Easily implemented in MATLAB/SIMULINK each mode (or action) of controller is better studied individually

17 ProportionalProportional FeedbackFeedback

Form:

Transfer function: or,

Closed-loop form:

18 ProportionalProportional FeedbackFeedback

Example: Given first order process:

for P-only feedback closed-loop dynamics:

Closed-Loop Time Constant

19 ProportionalProportional FeedbackFeedback

Final response:

Note:  for “zero offset response” we require

Tracking Error Disturbance rejection

 Possible to eliminate offset with P-only feedback (requires infinite controller gain)

 Need different control action to eliminate offset (integral)

20 Proportional Feedback

Servo dynamics of a first order process under proportional feedback

 increasing controller gain eliminates off-set 21 Proportional Feedback

High-order process e.g. second order underdamped process

 increasing controller gain reduces offset, speeds response and increases oscillation 22 ProportionalProportional FeedbackFeedback

Important points:  proportional feedback does not change the order of the system started with a first order process closed-loop process also first order order of characteristic polynomial is invariant under proportional feedback

 speed of response of closed-loop process is directly affected by controller gain increasing controller gain reduces the closed-loop time constant

 in general, proportional feedback reduces (does not eliminate) offset speeds up response for oscillatory processes, makes closed-loop process more oscillatory

23 IntegralIntegral ControlControl

Integrator is included to eliminate offset

 provides reset action  usually added to a proportional controller to produce a PI controller PID controller with derivative action turned off PI is the most widely used controller in industry optimal structure for first order processes

PI controller form

Transfer function model

24 PIPI FeedbackFeedback

Closed-loop response

 more complex expression  degree of denominator is increased by one

 Assuming the closed-loop system is stable, we get

25 PIPI FeedbackFeedback

Example PI control of a first order process

Closed-loop transfer function

Note: offset is removed closed-loop is second order 26 PIPI FeedbackFeedback

Example (contd) effect of integral time constant and controller gain on closed-loop dynamics

 (time constant) natural period of oscillation

damping coefficient

 integral time constant and controller gain can induce oscillation and change the period of oscillation

27 PI Feedback

Effect of integral time constant on servo dynamics

 Small integral time constant induces oscillatory (underdamped) 28 closed-loop response PI Feedback

Effect of controller gain on servo dynamics

 affects speed of response  increasing gain eliminates offset quicker 29 PI Feedback

Effect of integral action of regulatory response

 reducing integral time constant removes effect of disturbances  makes behavior more oscillatory 30 PIPI FeedbackFeedback

Important points:

 integral action increases order of the system in closed-loop

 PI controller has two tuning parameters that can independently affect speed of response final response (offset)

 integral action eliminates offset

 integral action should be small compared to proportional action tuned to slowly eliminate offset can increase or cause oscillation can be de-stabilizing

31 DerivativeDerivative ActionAction

Derivative of error signal  Used to compensate for trends in output measure of speed of error signal change provides predictive or anticipatory action  P and I modes only response to past and current errors  Derivative mode has the form

if error is increasing, decrease control action if error is decreasing, decrease control action

 Usually implemented in PID form

32 PIDPID FeedbackFeedback

Transfer Function

Closed-loop Transfer Function

 Slightly more complicated than PI form

33 PIDPID FeedbackFeedback

Example: PID Control of a first order process

Closed-loop transfer function

34 PID Feedback

Effect of derivative action on servo dynamics

 Increasing derivative action leads to a more sluggish servo response

35 PID Feedback

Effect of derivative action on regulatory response

 increasing derivative action reduces impact of disturbances on controlled variable  slows down servo response and affects oscillation of process 36 PDPD FeedbackFeedback

 PD Controller

 Proportional Derivative Control is common in mechanical systems  Arise in application for systems with an integrating behaviour

 Example : System in series with an integrator

37 PDPD FeedbackFeedback

Transfer Function

Closed-loop Transfer Function

 Slightly more complicated than PI form

38 PDPD FeedbackFeedback

 DC Motor example:  In terms of angular velocity (velocity control)

 In terms of the angle (position control)

39 PDPD FeedbackFeedback

 Closed-loop transfer function

 Simplifying

 Notice that

 Same effect as a PID controller.

40 DerivativeDerivative ActionAction

Important Points:

 Characteristic polynomial is similar to PI  derivative action does not increase the order of the system  adding derivative action affects the period of oscillation of the process good for disturbance rejection poor for tracking

 the PID controller has three tuning parameters and can independently affect, speed of response final response (offset) servo and regulatory response  derivative action should be small compared to integral action has a stabilizing influence difficult to use for noisy signals usually modified in practical implementation 41 Closed-loop Stability

Every control problem involves a consideration of closed-loop stability

General concepts:

Bounded Input Bounded Output (BIBO) Stability:

An (unconstrained) linear system is said to be stable if the output response is bounded for all bounded inputs. Otherwise it is unstable.

Comments: Stability is much easier to prove than instability This is just one type of stability

42 Closed-loop Stability

Closed-loop dynamics

Let

then,

The closed-loop transfer functions have a common denominator

called the characteristic polynomial

43 Closed-loop stability

General Stability criterion:

“ A closed-loop feedback control system is stable if and only if all roots of the characteristic polynomial are negative or have negative real parts. Otherwise, the system is unstable.”

 Unstable region is the right half plane of the complex plane.

 Valid for any linear systems.

44 Closed-loop Stability

Problem reduces to finding roots of a polynomial (for polynomial systems, without delay)

Easy (1990s) way : MATLAB function ROOTS (or POLE)

Traditional: 1. Routh array:

 Test for positivity of roots of a polynomial 2. Direct substitution

 Complex axis separates stable and unstable regions

 Find controller gain that yields purely complex roots 3. Root locus diagram

 Vary location of poles as controller gain is varied

 Of limited use

45 Closed-loop stability

Routh array for a polynomial equation is

where

Elements of left column must be positive to have roots with negative real parts 46 Example: Routh Array

Characteristic polynomial

2.36s5 +1.49s4 ! 0.58s3 +1.21s2 + 0.42s + 0.78 = 0

Polynomial Coefficients

a5 = 2.36, a4 = 1.49, a3 = !0.58, a2 = 1.21, a1 = 0.42, a0 = 0.78

Routh Array

a5(2.36) a3(!0.58) a1(0.42) a4(1.49) a2(1.21) a0(0.78) b1(!2.50) b2(!0.82) b3(0) c1(0.72) c2(0.78) d1(1.89) d2(0) e1(0.78)

 Closed-loop system is unstable

47 Direct Substitution

 Technique to find gain value that de-stabilizes the system.

 Observation: Process becomes unstable when poles appear on right half plane

Find value of that yields purely complex poles

 Strategy:  Start with characteristic polynomial

 Write characteristic equation:

 Substitute for complex pole

 Solve for and 48 Example: Direct Substitution

Characteristic equation

Substitution for

Real Part Complex Part

 System is unstable if

49 Root Locus Diagram

Old method that consists in plotting poles of characteristic polynomial as controller gain is changed e.g.

Characteristic polynomial 50 Stability and Performance

 Given plant model, we assume a stable closed-loop system can be designed

 Once stability is achieved - need to consider performance of closed- loop process - stability is not enough

 All poles of closed-loop transfer function have negative real parts - can we place these poles to get a “good” performance

S C Space of all P Controllers

S: Stabilizing Controllers for a given plant P: Controllers that meet performance 51 Controller Tuning

Can be achieved by  Direct synthesis : Specify servo transfer function required and calculate required controller - assume plant = model

 Internal Model Control: Morari et al. (86) Similar to direct synthesis except that plant and plant model are concerned

 Pole placement

 Tuning relations: Cohen-Coon - 1/4 decay ratio designs based on ISE, IAE and ITAE

 Frequency response techniques Bode criterion Nyquist criterion

 Field tuning and re-tuning

52 Direct Synthesis

From closed-loop transfer function

Isolate

For a desired trajectory and plant model , controller is given by

 not necessarily PID form  inverse of process model to yield pole-zero cancellation (often inexact because of process approximation)  used with care with unstable process or processes with RHP zeroes

53 Direct Synthesis

1. Perfect Control

 cannot be achieved, requires infinite gain

2. Closed-loop process with finite settling time

 For 1st order open-loop process, , it leads to PI control  For 2nd order open-loop process, , get PID control

3. Processes with delay

 requires  again, 1st order leads to PI control  2nd order leads to PID control 54 IMC Controller Tuning

Closed-loop transfer function

In terms of implemented controller, Gc

55 IMC Controller Tuning

1. Process model factored into two parts where contains dead-time and RHP zeros, steady-state gain scaled to 1.

2. Controller where is the IMC filter

 The constant is chosen such the IMC controller is proper  based on pole-zero cancellation

56 Example

PID Design using IMC and Direct synthesis for the process

Process parameters:

1. Direct Synthesis: (Taylor Series) (Padé)

 Servo Transfer function

57 ExampleExample

1. IMC Tuning:

a) Taylor Series: • Filter

• Controller (PI)

b) Padé approximation: • Filter

• Controller (Commercial PID)

58 ExampleExample

 Servo Response

59 ExampleExample

 Regulatory response

60 IMCIMC TuningTuning

 For unstable processes,

 Must modify IMC filter such that the value of at is 1

 Usual modification

 Strategy is to specify and solve for such that

61 ExampleExample

 Consider the process

 Consider the filter

 Let then solve for

 Yields a PI controller

62 ExampleExample

 Servo response

63 PolePole placementplacement

 Given a process model

a controller of the form,

and an arbitrary polynomial

 Under what condition does there exist a unique controller pair and such that

64 PolePole placementplacement

 We say that and are prime if they do not have any common factors

 Result:

Assume that and are (co) prime. Let be an arbitraty polynomial of degree . Then there exist polynomials and of degree such that

65 PolePole PlacementPlacement

 Example

 This is a second order system  The polynomials and are prime  The required degree of the characteristic polynomial is

 The degree of the controller polynomial and are

 Controller is given by

66 PolePole PlacementPlacement

 Performance objective: 3rd order polynomial

 Characteristic polynomial is given by

 Solving for and by equating polynomial coefficients on both sides

 Obtain a system of 4 equations in 4 unknowns

67 PolePole PlacementPlacement

 System of equations

 Solution is

 Corresponding controller is a PI controller

68 Tuning Relations

Process reaction curve method:  based on approximation of process using first order plus delay model

Manual Control

1. Step in U is introduced 2. Observe behavior 3. Fit a first order plus dead time model

69 Tuning Relations

Process response

1.2

1

0.8

0.6

0.4

0.2

0

-0.2 0 1 2 3 4 5 6 7 8 4. Obtain tuning from tuning correlations Ziegler-Nichols Cohen-Coon ISE, IAE or ITAE optimal tuning relations

70 Ziegler-Nichols Tunings

Controller P-only PI PID

- Note presence of inverse of process gain in controller gain - Introduction of integral action requires reduction in controller gain - Increase gain when derivation action is introduced

Example: PI: PID: 71 Example

Ziegler-Nichols Tunings: Servo response

72 Example

Regulatory Response

Z-N tuning  Oscillatory with considerable overshoot  Tends to be conservative 73 Cohen-Coon Tuning Relations

Designed to achieve 1/4 decay ratio  fast decrease in amplitude of oscillation

Controller Kc Ti Td P-only (1/ K p )(! /" )[1+" / 3! ] PI (1/ K p )(! /" )[0.9 +" /12! ] "[30 + 3(" /!)] 9 + 20(" /! )

PID 3" +16! "[32 + 6(" /! )] 4" (1/ K p )(! /" )[ ] 12! 13 + 8(" /! ) 11+ 2(" /! )

Example:

PI: Kc=10.27 τI=18.54

PID: Kc=15.64 τI=19.75 τd=3.10

74 Tuning relations

Cohen-Coon: Servo

 More aggressive/ Higher controller gains  Undesirable response for most cases 75 Tuning Relations

Cohen-Coon: Regulatory

 Highly oscillatory

 Very aggressive 76 Integral Error Relations

1. Integral of absolute error (IAE) " IAE = ! e(t) dt 0

2. Integral of squared error (ISE) " 2 ISE = ! e(t) dt 0 penalizes large errors 3. Integral of time-weighted absolute error (ITAE) " ITAE = ! t e(t) dt 0 penalizes errors that persist

 ITAE is most conservative  ITAE is preferred

77 ITAE Relations

Choose Kc, τI and τd that minimize the ITAE:

 For a first order plus dead time model, solve for:

! ITAE ! ITAE ! ITAE = 0, = 0, = 0 ! Kc !" I !"d  Design for Load and Setpoint changes yield different ITAE optimum

Type of Type of Mode A B Input Controller Load PI P 0.859 -0.977 I 0.674 -0.680 Load PID P 1.357 -0.947 I 0.842 -0.738 D 0.381 0.995 Set point PI P 0.586 -0.916 I 1.03 -0.165 Set point PID P 0.965 -0.85 I 0.796 -0.1465 D 0.308 0.929

78 ITAE Relations

From table, we get Load Settings: B "d Y = A ! = KKc = " = ( " ) " I " Setpoint Settings:

B "d Y = A ! = KKc = , " = A + B ! ( " ) " " I ( " )

Example

79 ITAE Relations

Example (contd) Setpoint Settings

! 9 = 0.796 " 0.1465( 30)= 0.7520 !0.85 ! I KK 0.965 9 2.6852 c = ( 30) = ! 30 ! I = 0.7520 = 0.7520 = 39.89 K 2.6852 2.6852 8.95 c = K = 0.3 = 0.929 !d 0.308 9 0.1006 ! = ( 30) = !d = 0.1006! = 3.0194 Load Settings:

"0.738 ! = 0.842 9 = 2.0474 ! I ( 30) 9 !0.947 KKc = 1.357( 30) = 4.2437 ! 30 ! I = 2.0474 = 2.0474 = 14.65 K 4.2437 4.2437 14.15 c = K = 0.3 = 0.995 !d 0.381 9 0.1150 ! = ( 30) = !d = 0.1150! = 3.4497

80 ITAE Relations

Servo Response

 design for load changes yields large overshoots for set-point changes 81 ITAE Relations

Regulatory response

82 Tuning Relations

 In all correlations, controller gain should be inversely proportional to process gain

 Controller gain is reduced when derivative action is introduced

 ! Controller gain is reduced as " increases

!  Integral time constant and derivative constant should increase as " increases  In general, !d = 0.25 ! I  Ziegler-Nichols and Cohen-Coon tuning relations yield aggressive control with oscillatory response (requires detuning)  ITAE provides conservative performance (not aggressive)

83