Feedback Control of Linear SISO systems Process Dynamics and Control 1
Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals 2
Closed-Loop System 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
Feedback Control 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
Feedback Control 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
Block Diagram of Closed-Loop Process Closed-Loop Transfer Function Computer Actuator + - + + Process Sensor - Open-Loop Process Transfer Function - Controller Transfer Function - Sensor Transfer Function - Actuator Transfer Function 6
Closed-Loop Transfer Function 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-loop Transfer Functions 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-loop Transfer function 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-loop Transfer Functions Sensitivity functions: The sensitivity function: The complementary sensitivity function: The control sensitivity function: 10
Closed-loop Transfer Functions Overall transfer function for the output: SERVO RESPONSE REGULATORY 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-loop Transfer Functions Servo mechanism requires that: Regulatory response requires that: Since The two objectives are complementary 12
Closed-loop Transfer Functions Note that or requires that the controller is large This leads to large control sensitivity 13
PID Controller 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
PID Control PID Control Equation Proportional Action Derivative Action Integral Action Controller Bias PID Controller Parameters K c Proportional gain Integral Time Constant Derivative Time Constant Controller Bias 15
PID Control 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
PID Control 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
Proportional Feedback Form: Transfer function: or, Closed-loop form: 18
Proportional Feedback Example: Given first order process: for P-only feedback closed-loop dynamics: Closed-Loop Time Constant 19
Proportional Feedback 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
High-order process e.g. second order underdamped process Proportional Feedback increasing controller gain reduces offset, speeds response and increases oscillation 22
Proportional Feedback 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
Integral Control 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
PI Feedback Closed-loop response more complex expression degree of denominator is increased by one Assuming the closed-loop system is stable, we get 25
PI Feedback Example PI control of a first order process Closed-loop transfer function Note: offset is removed closed-loop is second order 26
PI Feedback 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
Effect of integral time constant on servo dynamics PI Feedback Small integral time constant induces oscillatory (underdamped) closed-loop response 28
PI Feedback Effect of controller gain on servo dynamics affects speed of response increasing gain eliminates offset quicker 29
Effect of integral action of regulatory response PI Feedback reducing integral time constant removes effect of disturbances makes behavior more oscillatory 30
PI Feedback 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
Derivative of error signal Used to compensate for trends in output measure of speed of error signal change provides predictive or anticipatory action Derivative 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
PID Feedback Transfer Function Closed-loop Transfer Function Slightly more complicated than PI form 33
PID Feedback 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
PD Feedback 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
PD Feedback Transfer Function Closed-loop Transfer Function Slightly more complicated than PI form 38
PD Feedback DC Motor example: In terms of angular velocity (velocity control) In terms of the angle (position control) 39
PD Feedback Closed-loop transfer function Simplifying Notice that Same effect as a PID controller. 40
Derivative Action 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 Polynomial Coefficients Routh Array 2. 36s 5 + 149. s 4! 0. 58s 3 + 121. s 2 + 0. 42s + 0. 78 = 0 a5 = 2. 36, a4 = 149., a3 =! 0. 58, a2 = 121., a1 = 0. 42, a0 = 0. 78 a5( 2. 36) a3(! 0. 58) a1( 0. 42) a4( 149. ) a2( 121. ) a0( 0. 78) b1 (! 2. 50) b2 (! 0. 82) b3( 0) c1( 0. 72) c2 ( 0. 78) d1( 189. ) 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 closedloop 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 Controllers P 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, For 2nd order open-loop process,, it leads to PI control, 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, G c 55
1. Process model factored into two parts IMC Controller Tuning where to 1. contains dead-time and RHP zeros, steady-state gain scaled 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
Example 1. IMC Tuning: a) Taylor Series: Filter Controller (PI) b) Padé approximation: Filter Controller (Commercial PID) 58
Example Servo Response 59
Example Regulatory response 60
IMC Tuning 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
Example Consider the process Consider the filter Let then solve for Yields a PI controller 62
Example Servo response 63
Pole placement 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
We say that and any common factors Pole placement are prime if they do not have Result: Assume that and are (co) prime. Let be an arbitraty polynomial of degree. Then there exist polynomials and of degree such that 65
Pole Placement 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
Pole Placement Performance objective: 3rd order polynomial Characteristic polynomial is given by Solving for and coefficients on both sides by equating polynomial Obtain a system of 4 equations in 4 unknowns 67
Pole Placement 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 1. Step in U is introduced 2. Observe behavior 3. Fit a first order plus dead time model Manual Control 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 K c T i T d P-only ( 1/ )(! /" )[1 + " / 3! ] PI K p ( 1/ )(! /" )[0.9 + " /12! ] K p "[30 + 3( " /!)] 9 + 20( " /! ) PID (1/ K p 3" + 16! )(! /" )[ ] 12! "[32 + 6( " /! )] 13 + 8( " /! ) 4" 11+ 2( " /! ) Example: PI: K c =10.27 τ I =18.54 PID: K c =15.64 τ I =19.75 τ d =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) penalizes large errors 3. Integral of time-weighted absolute error (ITAE) penalizes errors that persist ITAE is most conservative ITAE is preferred " ISE =! e ( t ) 2 dt 0 ITAE " =! t e ( t ) dt 0 77
ITAE Relations Choose K c, τ I and τ d 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 Y = A! = KK d c = " " " " = I " Setpoint Settings: B " d ( ) c ", ( ) Y = A! = KK = " " = A + B! " I " Example 79
ITAE Relations Example (contd) Setpoint Settings Kc Load Settings: Kc ( )! KKc = 0 965 9 0. 85. 30 = 2. 6852 = 2. 6852 K = 2. 6852 0. 3 = 8. 95 ( )! KKc = 1357 9 0. 947. 30 = 4. 2437 = 4. 2437 = 4. 2437 = 14. 15 K 0. 3 ( ) ( )!! = 0. 796 " 01465. 9 I 30 = 0. 7520!! I = 0. 7520 = 30 0. 7520 = 39. 89! d! = 0 308( 9 0. 929. 30) = 01006.! d = 01006.! = 3. 0194! "! = 0 842 9 0. 738. I 30 = 2. 0474!! I = = 30 2. 0474 2. 0474 = 14. 65! d! = 0 381( 9 0. 995. 30) = 01150.! d = 01150.! = 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, Ziegler-Nichols and Cohen-Coon tuning relations yield aggressive control with oscillatory response (requires detuning)! d! I = 0. 25 ITAE provides conservative performance (not aggressive)! " 83