PROCESS CONTROL DESIGN PASI 2005

Transcription

1 PROCESS CONTROL DESIGN PASI 25 Pan American Advanced Studies Institute Program on Process Systems Engineering August 16-25, 25, Iguazu Falls Thomas E. Marlin Department of Chemical Engineering And McMaster Advanced Control Consortium McMASTER U N I V E R S I T Y 128 Main Street West, Hamilton, ON, Canada L8S 4L7 Copyright 25 by Thomas Marlin

2 Design is a challenging task. We must use all of our technical and problem-solving skills. Where do we start? When are we finished? v8 F2 F1 T1 T3 T4 v3 T5 F3 T6 F4 F5 P1 v1 v2 L1 v5 v7 v6 T7 L2 T2 T8 Hot Oil T9 Hot Oil F6

3 You will be the leaders in the PSE community; practitioners, educators, and researchers. You need to know The process needs, independent of PSE technology F2 F1 T1 T3 v1 v2 T2 Hot Oil T4 v3 T5 L1 T9 F3 v5 T6 F4 v7 T8 Hot Oil v6 F6 T7 F5 v8 P1 L2 The current best practices, strengths and gaps Principles informing research and practice Key breakthroughs in associated technologies

4 Course Goals Be able to define the control objectives for this goal-driven engineering task. Evaluate unique features of multivariable dynamic systems resulting from interaction Design simple control strategies using process insights and performance metrics Design challenging problems using a systematic, optimization method Become enthusiastic and investigate further

5 Course Resources Lecture Notes Annotated Reading List Solutions to Workshops (38 problems, 73 Pages) WEB sites Undergraduate Control: Graduate Control Design: This course will be a success if you study, apply, and improve good control design techniques

6 Theory & Practice Course Content Defining the Control Design Problem & Workshop Single-Loop Control Concepts & Workshop Multivariable Principles - Interaction & Workshop - Controllability & Workshop - Integrity & Workshop - Directionality & Workshop Slides provided but not covered because of limited me. Short-cut Design Procedure & Workshop Optimization-Based Control Design This is a small subset of the topics in the graduate course noted on the previous slide.

7 Course Emphasis So many great topics! A Foss: Key Challenge is which variables to measure, which inputs to manipulate, and what links should be made between these two sets CONTROL STRUCTURE! C. Nett: Objective minimize control system complexity subject to achievement of accuracy specifications SIMPLICITY! PERFORMANCE!

8 CONTROL DESIGN: DEFINING THE PROBLEM Control Design is a Goal-Driven Problem v8 F2 F1 T1 T3 T4 v3 T5 F3 T6 F4 F5 P1 v1 v2 L1 v5 v7 v6 T7 L2 T2 T8 Hot Oil T9 Hot Oil F6 The problem is defined based on knowledge of safety, process technology, sales, market demands, legal requirements, etc. Process systems technology is applied to achieve these goals.

9 CONTROL DESIGN: DEFINING THE PROBLEM Outline of the Topic. Defining the control design problem - Categories Measures of control performance - Controlled variable - Manipulated variable Benefits of reduced variation - Class exercise Workshop

10 CONTROL DESIGN: DEFINING THE PROBLEM The Control Design Form The form provides a useful check list for items that should be considered. It also provides a concise yet complete presentation of the important decisions that must be reviewed by all stake holders. The objectives and performance descriptions must be stated in process terms, not limited by anticipated control performance. See Marlin (2), Chapter 24 CONTROL DESIGN FORM TITLE: ORGANIZATION: PROCESS UNIT: DESIGNER: DRAWING ORIGINAL DATE: PROCESS FLOW: REVISION No. DATE: PIPING AND INSTR.: PAGE No. OF CONTROL OBJECTIVES: 1) SAFETY OF PERSONNEL 2) ENVIRONMENTAL PROTECTION 3) EQUIPMENT PROTECTION 4) SMOOTH, EASY OPERATION 5) PRODUCT QUALITY 6) EFFICIENCY AND OPTIMIZATION 7) MONITORING AND DIAGNOSIS [Define the control performance required by specifying standard deviations, approach to constraint, response to major upset, or other relevant, quantitative performance measure for each objective.] MEASUREMENTS: VAR. SENSOR RANGE ACCURACY&DYNAMICS RELIABILITY PRINCIPLE LOCATION REPRODUCIBILITY MANIPULATED VARIABLES: VAR. AUTO/ LOCAL/ CONTIN. RANGE PRECISION DYNAMICS RELIABIL. MANUAL REMOTE DISCRETE CONSTRAINTS: VARIABLE LIMIT VALUES MEASURED/ PENALTY FOR VIOLATION INFERRED DISTURBANCES: SOURCE MAGNITUDE FREQUENCY MEASURED? DYNAMIC RESPONSES: (ALL INPUTS, MANIPULATED AND DISTURBANCE) INPUT OUTPUT GAIN DYNAMIC MODEL ADDITIONAL CONSIDERATIONS: CURRENT OPERATION: CONTROL DESIGN: (INCLUDE DRAWING) BENEFITS: COSTS:

11 CONTROL DESIGN: DEFINING THE PROBLEM Let s design controls for this process How do we start? What are the common steps? When are we finished? T6 P1 Vapor Product, mostly C 1 and C 2 (L. Key) Feed Methane Ethane (LK) Propane Butane Pentane T1 F1 T2 T4 T5 T3 L1 P 1 kpa T 298 K F2 Process fluid F3 Steam A1 L. Key Liquid product

12 CONTROL DESIGN: DEFINING THE PROBLEM Control Design Form Objectives 1. Safety 2. Environmental protection 3. Equipment protection 4. Smooth operation 5. Product quality WORKSHOP: Complete a Control Design Form Vapor product 6. Profit Measurements 7. Monitoring and diagnosis T6 P1 Manipulated variables T4 Constraints T1 T2 T5 Disturbances F1 T5 L1 Dynamic responses Additional considerations F2 T3 F3 Process fluid Steam A1 L. Key Liquid product

13 CONTROL DESIGN: DEFINING THE PROBLEM Entries must be specific and measurable to guide design CONTROL OBJECTIVES: 1) SAFETY OF PERSONNEL a) the maximum pressure of 12 kpa must not be exceeded under any (conceivable) circumstances 2) ENVIRONMENTAL PROTECTION a) material must not be vented to the atmosphere under any circumstances 3) EQUIPMENT PROTECTION a) the flow through the pump should always be greater than or equal to a minimum 4) SMOOTH, EASY OPERATION a) the feed flow should have small variability 5) PRODUCT QUALITY a) the steady-state value of the ethane in the liquid product should maintained at its target of 1 mole% for operating condition changes of +2 to -25% feed flow, 5 mole% changes in the ethane and propane in the feed, and -1 to +5 C in the feed temperature. b) the ethane in the liquid product should not deviate more than ±1 mole % from its set point during transient responses for the following disturbances i) the feed temperature experiences a step from to 3 C ii) the feed composition experiences steps of +5 mole% ethane and -5 mole% of propane iii) the feed flow set point changes 5% in a step 6) EFFICIENCY AND OPTIMIZATION a) the heat transferred should be maximized from the process integration exchanger before using the more expensive steam utility exchanger 7) MONITORING AND DIAGNOSIS a) sensors and displays needed to monitor the normal and upset conditions of the unit must be provided to the plant operator b) sensors and calculated variables required to monitor the product quality and thermal efficiency of the unit should be provided for longer term monitoring

14 CONTROL DESIGN: DEFINING THE PROBLEM Entries must be specific and measurable to guide design CONSTRAINTS: VARIABLE LIMIT VALUES MEASURED/ HARD/ PENALTY FOR VIOLATION INFERRED SOFT drum pressure 12 kpa, high P1, measured hard personnel injury drum level 15%, low L1, measured hard pump damage Ethane in F5 ± 1 mole%, A1, measured & soft reduced selectivity in product (max deviation) T6, inferred downstream reactor DISTURBANCES: SOURCE MAGNITUDE DYNAMICS feed temperature (T1) -1 to 55 C infrequent step changes of 2 C magnitude feed rate (F1) 7 to 18 set point changes of 5% at one time feed composition ±5 mole% feed ethane frequent step changes (every 1-3 hr) For an excellent problem definition, see the Tennessee Eastman design challenge problem. Downs, J. and E. Vogel (1993) A Plant-wide Industrial Process Control Problem, Comp. Chem. Engr., 17,

15 CONTROL DESIGN: DEFINING THE PROBLEM What measures of control performance would we use? 1.5 Controlled Variable Time Manipulated Variable Time

16 CONTROL DESIGN: DEFINING THE PROBLEM = IAE = SP(t)-CV(t) dt A B.5 B/A = Decay ratio Return to set point, zero offset Rise time Time 1.5 C/D = Maximum overshoot of manipulated variable 1 C.5 D Time

17 CONTROL DESIGN: DEFINING THE PROBLEM = IAE = SP(t)-CV(t) dt.8.6 Maximum CV deviation from set point Time Time

18 CONTROL DESIGN: DEFINING THE PROBLEM An important, but often overlooked, factor All objectives and CVs are not of equal importance! CV Priority Ranking A product A recycle

19 CONTROL DESIGN: DEFINING THE PROBLEM An important, but often overlooked, factor Our primary goal is to maintain the CV near the set point. Besides not wearing out the valve, why do we have goals for the MV? Steam flow Manipulated Variable Time AC

20 CONTROL DESIGN: DEFINING THE PROBLEM Our primary goal is to maintain the CV near the set point. Besides not wearing out the valve, why do we have goals for the MV? PI 1 AT 1 PI 4 Fuel flow FT 1 TI 1 PI 5 TI 2 5 TI 2 15 PT 1 TI 3 TI TI 6 TI 7 TC TI 1 Manipulated Variable FT 2 TI 8 FI 3 TI 11 Time PI 2 PI 3 PI 6 Fuel For more on Life Extending Control, see Li, Chen, and Marquez (23)

21 CONTROL DESIGN: DEFINING THE PROBLEM What measures of control performance would we use? Controlled Variable Time Manipulated Variable Time

22 CONTROL DESIGN: DEFINING THE PROBLEM Often, the process is subject to many large and small disturbances and sensor noise. The performance measure characterizes the variability. 2 Controlled Variable 1-1 Variance or standard deviation of CV Time 2 Manipulated Variable 1-1 Variance or standard deviation of MV Time

23 CONTROL DESIGN: DEFINING THE PROBLEM Process performance = efficiency, yield, production rate, etc. It measures performance for a control objective. Calculate the process performance using the CV distribution, not the average value of the key variable!

24 Class Exercise: Benefits for reduced variability for chemical reactor Goal: Maximize conversion of feed ethane but do not exceed 864C Which operation, A or B, is better and explain why. A B

25 Class Exercise: Benefits for reduced variability for boiler A Goal: Maximize efficiency and prevent fuel-rich flue gas Which operation, A or B, is better and explain why. B

26 DEFINING THE PROBLEM: Workshop 1 Complete a control design form for a typical 2-product distillation tower. Make reasonable assumptions and note questions you would ask to verify your assumptions. Note that the figure is not complete; you are allowed to make changes to sensors and final elements.

27 DEFINING THE PROBLEM: Workshop 2 Complete a control design form for a typical fired heater. Make reasonable assumptions and note questions you would ask to verify your assumptions. Note that the figure is not complete; you are allowed to make changes to sensors and final elements.

28 DEFINING THE PROBLEM: Workshop 3 Typically, we have only a steady-state flowsheet (if that) when designing a plant. Discuss the information in the Control Design Form that can be determined at this stage of the design. Discuss the information in the Control Design Form that is not known at this stage of the design.

29 DEFINING THE PROBLEM: Workshop 4 Two process examples show the benefit of reduced variability, the fired heater reactor and the boiler. Discuss the difference between the two examples. Can you think of another example that shows the principle of each? Squeeze down the variability.4 frequency of occurrence deviation from mean

30 DEFINING THE PROBLEM: Workshop 5 Discuss an important assumption that is made on the procedure proposed for calculating the average process performance. (Hint: consider dynamics) How would you evaluate the assumption?

31 DEFINING THE PROBLEM: Workshop 6 The following performance vs. process variable correlations are provided. All applications require the same average value for the process variable (see arrow). What is the best distribute for each case? (Sketch histogram as your answer.) Distribution A Distribution B Distribution C Process performance Process performance Process performance Process variable Process variable Process variable

32 Principles of Single-Loop Feedback Control Multiloop control contains many single-loop systems. Conclusion: We need to understand single-loop principles. Vapor product, C2- Lesson Outline? TC6 PC1? Ten observations on what affects single-loop feedback T5 T3 LC1? Workshop F3 Steam A1 L. Key Liquid product, C3+

33 Principles of Single-Loop Feedback Control Performance Observations #. Very obvious, but not so obvious for multivariable systems. Process gain must not be zero (K p ). Gain should not be too small (range) or too large (sensitivity). Which valves can be used to control each measured variable? Would the answer change if many singleloops were implemented at the same time? F1 T A Product concentration L F2

34 Principles of Single-Loop Feedback Control Performance Observation #1. Feedback dead time limits best possible performance Controlled Variable S-LOOP plots deviation variables (IAE = 9.791) Time 1.5 θ p, feedback dead time Discuss why the red area defines deviation from set point that cannot be reduced by any feedback. Manipulated Variable 1.5 T in v1 TC v Time

35 Principles of Single-Loop Feedback Control Performance Observation #1. Feedback dead time limits best possible performance Controlled Variable S-LOOP plots deviation variables (IAE = ) Time θ p, feedback dead time Discuss why the red area defines deviation from set point that cannot be reduced by any feedback. Manipulated Variable T in v1 TC v Time

36 Principles of Single-Loop Feedback Control Performance Observation #2. Large disturbance time constants slow disturbances and improve performance..8 T in T in v1 TC v2 Controlled Variable Time.8 Please discuss v1 TC Controlled Variable Smaller maximum deviation v Time

37 Principles of Single-Loop Feedback Control Performance Observation #3. Feedback must change the MV aggressively to improve performance. Controlled Variable Manipulated Variable Kc = 1.3, TI = 7, Td = 1.5 S-LOOP plots deviation variables (IAE = ) Time Time Controlled Variable Manipulated Variable S-LOOP plots deviation variables (IAE = ) Time 5-5 Kc =.6, TI = 1, Td = Time Please discuss

38 Principles of Single-Loop Feedback Control Performance Observation #4. Sensor and final element dynamics are in feedback loop, slow responses degrade performance. 8 S-LOOP plots deviation variables (IAE = ) 8 S-LOOP plots deviation variables (IAE = ) Controlled Variable Manipulated Variable Time IAE = 88 θ 5, τ = 5, τ =, τ = p = p sensor valve Time Controlled Variable Manipulated Variable Time IAE = 113 θ 5, τ = 5, τ = 1, τ = 1 p = p sensor valve Time retuned

39 Principles of Single-Loop Feedback Control Performance Observation #5. Inverse response (RHP zero) degrades feedback control performance. Process reaction curve for the effect of solvent flow rate on the reactor effluent concentration. Two, isothermal CSTRs with reaction A B and F S >> F A reactant F A C A solvent F S C AS = C A1 V1 C A2 V2

40 Principles of Single-Loop Feedback Control Performance Observation #5. Inverse response degrades feedback control performance. PARALLEL STRUCTURES X(s) G 1 (s) Y(s) G 2 (s) Inverse response occurs when parallel paths have different signs for their steady-state gains and the path with the smaller magnitude gain is faster.

41 Principles of Single-Loop Feedback Control Performance Observation #6. Disturbance frequencies around and higher than the critical frequency cannot be controlled. Slow A B C medium fast T in v1 v2 TC Behavior of the tank temperature for three cases?

42 Principles of Single-Loop Feedback Control Performance Observation #6: Frequency Response Region 1 1 I: Control is needed, and it is effective 1 Region II: Control is needed, but it is not effective B Region III: Control is not needed, and it is not effective This is CV / D, small is good. Amplitude Ratio, CV / D A Recall, this is the disturbance Frequency, frequency, w (rad/time) low frequency = long period C

43 Performance Observation #6. Disturbance frequencies around and higher than the critical frequency cannot be controlled. Let s apply frequency response concepts to a practical example. Can we reduce this open-loop variation? T in v1 A v2 Feedback dynamics are: A(s) v(s) 2s 1. e = 2s + 1 We note that the variation has many frequencies, some much slower than the feedback dynamics.

44 Performance Observation #6. Disturbance frequencies around and higher than the critical frequency cannot be controlled. Yes, we can we reduce the variation substantially because of the dominant low frequency of the disturbance effects. T in v1 AC v2 Feedback dynamics are: A(s) v(s) = 1. e 2s 2s + 1 Low frequencies reduced a lot. Higher frequencies remain!

45 Principles of Single-Loop Feedback Control Performance Observation #7. Long controller execution periods degrade feedback control performance..8 S-LOOP plots deviation variables (IAE = ) T in v1 TC v2 Controlled Variable Time -.2 Manipulated Variable Time

46 Principles of Single-Loop Feedback Control Performance Observation #8. Use process understanding to provide a CV-MV pairing with good steady-state and dynamic behavior. Which valve Should be adjusted for good control performance? feed v1 AC A CSTR with A B v2 cooling

47 Principles of Single-Loop Feedback Control Performance Observation #8. Use process understanding to provide a CV-MV paring with good steady-state and dynamic behavior. v1 Component material balance v1 CSTR with A B v2 Energy balance Component material balance AC A v1 gives faster feedback dynamics v2 cooling

48 Principles of Single-Loop Feedback Control Performance Observation #9. Some designs require a loop to adjust two valves to achieve the desired range and precision. Two heating valves are available for manipulation. T1 T2 Split range T5 TC PC-1 Vapor product Adjust only one at a time. FC-1 T3 LC-1 F2 Process fluid F3 Steam L. Key AC-1 Liquid product

49 Principles of Single-Loop Feedback Control Performance Observation #1. Some designs require several loops to adjust the same manipulated variable to ensure that the highest priority objective is achieved. Control the effluent concentration of A but do not exceed a maximum reactor temperature. Only the cooling medium may be adjusted. signal select

50 Single-loop Control, Workshop #1 What if the % ethane is sometimes 2% and other times 5%? TC6 PC1 Feed Methane Ethane (LK) Propane Butane Pentane T5 T3 LC1 Vapor product, C2- F3 Steam A1 L. Key Liquid product, C3+

51 Single-loop Control, Workshop #2 The consumers vary and we must satisfy them by purchasing fuel gas. Therefore, we want to control the pressure in the gas distribution network. Design a control system. By the way, fuel A is less expensive.

52 Single-loop Control, Workshop #3 Design a controller that will control the level in the bottom of the distillation tower and send as much flow as possible to Stream A

53 Single-loop Control, Workshop #4 Stream A (cold) 2 1 Stream A (cold) Stream B (hot) TC TC Freedom to adjust flows Stream A Stream B 1. Constant Adjustable 2. Adjustable Constant 3. Constant Constant 3 Stream A (cold) TC Stream B (hot) Stream B (hot) You can add valve(s) and piping.

54 Single-loop Control, Workshop #5 Class exercise: Distillation overhead system. Design a pressure controller. (Think about affecting U, A and T) PC CW NC No vapor product You can add valve(s) and piping.

55 Single-loop Control, Workshop #6 Class exercise: Distillation overhead system. Design a pressure controller. (Think about affecting U, A and T) PC Refrigerant LC NC You can add valve(s) and piping.

56 Single-loop Control, Workshop #7 The following control system has a very large gain near ph = 7. For a strong acid/base, performance is likely to be poor. How can we improve the situation? F1 Strong base ph A1 C L Strong acid

57 Principles of Multivariable Dynamic Processes and MultiLoop Control Basically, multiloop designs are simply many single-loop (PID) controllers. Then, what is new? ** INTERACTION ** When we adjust one valve, how many measurements change and what are responses? Step valve v8 F2 F1 T1 T3 T4 v3 T5 F3 T6 F4 F5 P1 v1 v2 L1 v5 v7 v6 T7 L2 T2 T8 Hot Oil T9 Hot Oil F6

58 Principles of Multivariable Dynamic Processes and MultiLoop Control ** INTERACTION ** LESSON OUTLINE Interaction A brief definition Controllability Can desired performance be achieved? Integrity What happens to the system when a controller stops functioning? Directionality A key factor in control performance Class Exercises and Workshops throughout

59 INTERACTION: The difference in multivariable control Definition: A multivariable process has interaction when input (manipulated) variables affect more than one output (controlled) variable Step change to reflux with constant reboiler XD (mol frac) Time (min) x 1 4 XB (mol frac) Time (min) x R (mol/min) V (mol/min) Time (min) Time (min)

60 Multivariable Interaction, Workshop #1-3 m 3 /h -6 m 3 /h F A, x A =1 F S, x AS = F M, x AM Total flow, FM Composition (fraction A), x AM The ranges of the two mixing flows are given in the figure. Sketch the feasible steadystate operating window in the Figure. Note: You may assume no disturbances for this exercise.

61 Controllability: Is the desired performance achievable? Manipulated variables Disturbances T1 F1 T4 T2 T5 T5 T6 P1 L1 Vapor product Control Design Form 1. Safety 2. Environmental protection 3. Equipment protection 4. Smooth operation 5. Product quality 6. Profit 7. Monitoring and diagnosis Controlled F2 T3 F3 variables Process fluid Can we control the CVs with the MVs? If the required performance is not achievable, fix the process; don t design controllers!

62 Controllability: Is the desired performance achievable? CONTROLLABILITY -A characteristic of the process that determines whether a specified dynamic behavior can be achieved with a defined set of controlled and manipulated variables and a defined scenario. Various specifications for dynamic behavior are possible; we will review a few commonly used. We seek a fundamental property of the process independent of a specific control design or structure.

63 Controllability: Is the desired performance achievable? Interaction influences Controllability. Is this behavior possible? Goals: Maintain cold effluent T 1 and Maintain hot effluent at T 2 final steady-states = set points Freedom to adjust flows Stream A Stream B Adjustable Adjustable Stream A (cold) Stream B (hot) T2 T1

64 Controllability: Is the desired performance achievable? Interaction influences Controllability. Quick check for each loop T2 Can we control T2 with v2? YES Stream A (cold) v1 T1 Can we control T1 with v1? YES v2 Stream B (hot) Since each individual loop is OK, both loops are OK?

65 Controllability: Is the desired performance achievable? Interaction influences Controllability. Energy balance on each stream Q Q Hot Cold = = F Hot F C Cold p C H p ( T C Hout ( T Cout T Hin T Cin ) ) Stream A (cold) v1 T2 T1 v2 Stream B (hot)

66 Controllability: Is the desired performance achievable? A few typical controllability performance specifications Steady-state: Achieve desired steady-state when disturbances occur For processes that operate at steady-state, but no information about transition. Point-wise state (output): Move to specified initial values of states (or CVs) to final values in finite time Often in textbooks. Perhaps, useful for batch end-point control. Functional: Strictly follow any defined trajectory for CVs Useful for transition control between steady states and for batch processes. However, likely too restrictive (strictly, any).

67 Controllability: Is the desired performance achievable? Steady-state Controllable: A mathematical test. The system will be deemed controllable if the steady-state I/O gain matrix can be inverted, i.e., Det [ G() ] [G()] -1 exists This is only applicable to open-loop stable plants. It is a point-wise test that gives no (definitive) information about other conditions. No information about the transient behavior or the changes to MV s to achieve the desired CV s (set points).

68 Controllability: Is the desired performance achievable? Pointwise State Controllable: A process example for p.s. controllability. T 1 = x 1 T 2 = x 2 T 3 = x 3 T 4 = x 4 From Skogestad & Postlethwaite, 1996 u =T One MV and four CVs. The control performance can be achieved! Nothing is specified for MV at any time or for CV between t and t 1 or after the time t 1.

69 Controllability: Is the desired performance achievable? Functional Controllable* - A system is controllable if it is possible to adjust the manipulated variables MV(t) so that the system will follow a (smooth) defined path from CV(t ) to CV(t 1 ) in a finite time. A system G(s) is (output) functionally controllable when dimensions of CV and MV are the same (say n) The rank of G(jω) = n Stated differently, G -1 (jω ) exists for all ω Stated again, σ min (G(jω)) > [minimum singular value] Unfortunately, dead times and RHP zeros prevent the controller from implementing the inverse for most process. Also, no specification on the MV s.

70 Controllability: Is the desired performance achievable? Class Exercise: Let s define controllability with a test that is computable. MV 1 G 11 (s) + + CV 1 G 21 (s) G d1 (s) D G 12 (s) G d2 (s) MV 2 G 22 (s) + + CV 2

71 Controllability: Is the desired performance achievable? Controllability Test: Solve as open-loop optimization problem, which can be an LP or convex QP. If all violation slacks (s 2n ) on the performance specifications are zero, i.e. if f =, the system is controllable!, f u n o o n n n n n n n n n n n n n n SP n n n n SP n n n n n n n n n n n d y x u given s s s s s s u u u u u u u y s s y y s s y Cx y Dd Bu Ax x t s s s 2 2 max 1 1 max 1 1 max 1 min max min ,,, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (.. ) ( min = + + = + = Note that in the formulation, slacks s 1n define allowable deviation from desired output, and s 2n are violations for excessive deviation.

72 Controllability: Is the desired performance achievable? Example of controllability test for 2x2 Fluidized Catalytic Cracking Reactor Product riser Must be tightly controlled Large fluidized vessel regenerator Fcat T Tris Tubular reactor With short space time T Trgn Feed Keep in safe range Fair

73 Controllability: Is the desired performance achievable? Example of controllability test for 2x2 Fluidized Catalytic Cracking Reactor FCC Case 1 Tris SP = +1 F at t= With no bounds on the speed of adjustment, the set points can be tracked exactly. Is the system controllable? Trgn (F) Tris (F) time (sec) time (sec) Fair (lb air/lb feed) Fcat (lb cat/lb feed) time (sec) time (sec)

74 Controllability: Is the desired performance achievable? Example of controllability test for 2x2 Fluidized Catalytic Cracking Reactor FCC Case 2 Tris SP = +1 F at t= With bounds on the speed of adjustment, Fair.1 (lb air/lb feed)/2sec Fcat.1 (lb cat/lb feed)/2sec Is the system controllable? Trgn (F) Tris (F) time (sec) time (sec) Fair (lb air/lb feed) Fcat (lb cat/lb feed) time (sec) time (sec)

75 Controllability: Is the desired performance achievable? Evaluation of the proposed approach GOOD ASPECTS Can define relevant timedomain performance specifications - on CVs (or states) -on MVs Great flexibility on form of specifications Easily computed No limitation on disturbance Includes most other definitions/tests as special cases SHORTCOMINGS Linear Model No robustness measure No guarantee that any controller will achieve the performance Finite horizon True for all controllability tests

76 Controllability: Is the desired performance achievable? Determining controllability from process sight Lack of controllability when 1. One CV cannot be effected by any valve 2. One MV has no effect on CVs These are generally easy to determine. This requires care and process insight to determine. 3. Lack of independent effects. Look for contractions

77 CONTROLLABILITY : Workshop 1 We need to control the mixing tank effluent temperature and concentration. F 1 T 1 C A1 F 2 T 2 C A2 You have been asked to evaluate the steady-state controllability of the process in the figure. Discuss good and poor aspects and decide whether you would recommend the design. T A Controlled variables are the temperature and concentration in the tank effluent.

78 CONTROLLABILITY : Workshop 2 The sketch describes a simplified boiler for the production of steam. The boiler has two fuels that can be manipulated independently. We want to control the steam temperature and pressure. Analyze the controllability of this system and determine the loop pairing.

79 CONTROLLABILITY : Workshop 3 The sketch describes a simplified flash drum. A design is proposed to control the temperature and pressure of the vapor section. Analyze the controllability of this system and determine if the loop pairing is correct. Hot streams Feed Methane Ethane (LK) Propane Butane Pentane vapor liquid

80 CONTROLLABILITY : Workshop 4 A non-isothermal CSTR Does interaction exist? Are the CVs (concentrations) independently controllable? A B + 2C -r A = k e -E/RT C A G 11 (s) + + v1 G 21 (s) G d1 (s) C B A A G 12 (s) G d2 (s) C C G 22 (s) + + v2

81 Integrity: Is the design robust to changes in controller status? FC AC LC What is the effect of turning off AC (placing in manual)? TC We will first introduce the Relative Gain; then, we will apply it to the Integrity Question

82 RELATIVE GAIN: Definition The relative gain between MV j and CV i is λ ij. It is defined in the following equation. MV 1 (s) G 11 (s) G 21 (s) G 12 (s) + + G d1 (s) G d2 (s) CV 1 (s) D(s) Explain in words. MV 2 (s) G 22 (s) + + CV 2 (s) λ ij = CV MV i CV MV j i j MV CV k k = constant = constant = CV MV CV MV i i j j other loops other loops open closed What have we assumed about the other controllers?

83 RELATIVE GAIN: Properties 1. The RGA can be calculated from openloop gains (only). λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed [ CV ] = [ K][ MV ] 1 [ MV ] = [ K] [ CV ] k ij ki ij CV = i MV = MV i j CV The relative gain array is the element-by-element product of K with K -1. ( = product of ij elements) j MV i CV j Open-loop Closed-loop Λ ( ) T K ( k )( ki ) 1 λ = K = ij ij ji

84 RELATIVE GAIN: Properties 2. In some cases, the RGA is very sensitive to small errors in the gains, K ij. λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed λ 11 = 1 1 K K K K When this term is nearly equal 1. When is this equation very sensitive to errors in the individual gains?

85 RELATIVE GAIN: Properties 2. In some cases, the RGA is very sensitive to small errors in the gains, K ij. λ ij = CV MV i CV MV j i j MV = constant k CV = constant k We must perform a thorough study to ensure that numerical derivatives are sufficiently accurate! Change in F D used in finite difference for derivative λ 11 for a positive change in F D λ 11 for a negative change in F D Average λ 11 for positive and negative changes in F D 2% % % % Results for a distillation tower, from McAvoy, 1983 The δx must be sufficiently small (be careful about roundoff).

86 RELATIVE GAIN: Properties 2. In some cases, the RGA is very sensitive to small errors in the gains, K ij. λ ij = CV MV i CV MV j i j MV = constant k CV = constant k We must perform a thorough study to ensure that numerical derivatives are sufficiently accurate! Convergence tolerance of equations (some of all errors squared) λ 11 for a positive change in F D λ 11 for a negative change in F D Average λ 11 for positive and negative changes in F D Results for a distillation tower, from McAvoy, 1983 The convergence tolerance must be sufficiently small. Average gains from +/-

87 RELATIVE GAIN: Properties 3. We can evaluate the RGA of a system with integrating processes, such as levels. λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed m 1 Redefine the output as the derivative of the level; then, calculate as normal. ρ = density L A D = density solvent m 2 dl / D dt (1 m 1 D / ρ D / ρ) (1 m 2 D / ρ) D / ρ

88 RELATIVE GAIN: Properties Additional Properties, stated but not proved here 4. Rows and column of RGA sum to Elements in RGA are independent of variable scalings 6. Permutations of variables results in same permutation in RGA 7. RGA is independent of a specific I/O pairing it need be evaluated only once

89 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed λ ij < In this case, the steady-state gains have different signs depending on the status (auto/manual) of the other loops. A Solvent v A v S A1 C A CSTR with A B Discuss interaction What sign is process gain of A2 loop with A1 C A A2 (a) in automatic (b) in manual.

90 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed λ ij < In this case, the steady-state gains have different signs depending on the status (auto/manual) of other loops! We can achieve stable multiloop feedback by using the sign of the controller gain that stabilizes the multiloop system. Discuss what happens when the other interacting loop is placed in manual!

91 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed λ ij < the steady-state gains have different signs For λ ij <, one of three BAD situations occurs 1. Multiloop is unstable with all in automatic. 2. Single-loop ij is unstable when others are in manual. 3. Multiloop is unstable when loop ij is manual and other loops are in automatic

92 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed λ ij = In this case, the steady-state gain is zero when all other loops are open, in manual. LC FC Could this control system work? What would happen if one controller were in manual?

93 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed <λ ij <1 In this case, the multiloop (ML) steady-state gain is larger than the single-loop (SL) gain. What would be the effect on tuning of opening/closing the other loop? Discuss the case of a 2x2 system paired on λ ij =.1

94 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed λ ij = 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. What is generally true when λ ij = 1? Does λ ij = 1 indicate no interaction? MV 1 (s) G 11 (s) G 21 (s) G 12 (s) + + G d1 (s) G d2 (s) CV 1 (s) D(s) MV 2 (s) G 22 (s) + + CV 2 (s)

95 RELATIVE GAIN: Interpretations λ ij = 1 In this case, the steady-state gains are identical in both the ML and the SL conditions. λ ij = CVi MV j CVi MV j MV = constant k CV = constant k = CVi MV CVi MV j j other loops open other loops closed Diagonal gain matrix Lower diagonal gain matrix K K = k 11 k k =.... k n1 k k Both give an RGA that is diagonal! 1 1 RGA = 1 = I 1 1

96 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed 1<λ ij In this case, the steady-state multiloop (ML) gain is smaller than the single-loop (SL) gain. If a ML process has a smaller process gain, why not just increase the associated controller gain by λ ij? What would be the effect on tuning of opening/closing the other loop? Discuss a the case of a 2x2 system paired on λ ij = 1.

97 RELATIVE GAIN: Interpretations MV j CV i λ ij = CV MV i CV MV j i j MV = constant k CV = constant k = CV MV CV MV i i j j other loops other loops open closed λ ij = In this case, the gain in the ML situation is zero. We conclude that ML control is not possible. Have we seen this result before?

98 INTEGRITY Integral Stabilizability - Can be stabilized with feedback using same sign of controller gains as single-loop Increasingly restrictive Integral Controllability - Can be stabilized with feedback using same sign of controller gains as single-loop and all controllers can be detuned equally Integral Controllability with Integrity - Can be stabilized with feedback using same sign of controller gains as single-loop and some controllers can be placed off, on manual while retaining stability (with retuning) Decentralized Integral Controllability - Can be stabilized with feedback using same sign of controller gains as single-loop and any controller(s) can be detuned any amount

99 INTEGRITY INTEGRITY is strongly desired for a control design. SOME ASSUMPTIONS FOR RESULTS PRESENTED Limited to stable plants; if open-loop unstable plants, extensions to analysis are available All controllers have integral modes. They provide zero steady-state offset for asymptotically constant ( step-like ) inputs All simple loops ; variable structure (split range and signal select) are not considered unless explicitly noted. See references (Campo and Morari, Skogestad and Postlethwaite, etc.) for limitations on the transfer functions.

100 INTEGRITY Integral Stabilizability - Can be stabilized with feedback using same sign of controller gains as single-loop Integral Controllability - Can be stabilized with feedback using same sign of controller gains as single-loop and all controllers can be detuned equally Integral Controllability with Integrity - Can be stabilized with feedback using same sign of controller gains as single-loop and some controllers can be off ( manual ) while retaining stability (with retuning) Decentralized Integral Controllability - Can be stabilized with single-loop feedback using same sign of controller gains as any individual controller(s) can be detuned any amount Niederlinski Index Not very important Relative Gain Array* Important, no short-cut test * All possible sub-systems with controller(s) off

101 Integrity Workshop 1 The process in the figure is a simplified head box for a paper making process. The control objectives are to control the pressure at the bottom of the head box (P1) tightly and to control the slurry level (L) within a range. The manipulated variables are the slurry flow rate in (F lin ) and the air vent valve opening. 1. Determine the integrity of the two possible pairings based process insight. 2. Recommend which pairing should be used. 3. Discuss the integrity of the resulting system. Air Pulp and water slurry Paper mat on wire mesh

102 Integrity Workshop 2 The following transfer function matrix and RGA are given for a binary distillation tower. Discuss the integrity for the two loop pairings. AC XD( s) XB( s) = 3s.747e 12s e 11.75s + 1 s.667e 15s e 1.2s + 1 2s 2s F F R V ( s) ( s) FR FV AC XD XB

103 Integrity Workshop 3 The following transfer function matrix and RGA are given for a binary distillation tower. Discuss the integrity for the two loop pairings. XF AC XD F D XD( s) XB( s) =.747e 12s e 11.75s + 1 3s 3.3s.8e 5s + 1.8e 3s + 1 2s 2s F F D V ( s) ( s) F V XD FD.39 FV.61 AC XB XB.61.39

104 Integrity Workshop 4 We will consider a hypothetical 4 input, 4 output process. How many possible combinations are possible for the square mutliloop system? For the system with the RGA below, how many loop pairings have good integrity? mv1 mv2 mv3 mv4 CV1 1 CV CV CV 4 1

105 Integrity Workshop 5 The table presents RGA(1,1) for the same 2x2 process with different level controllers (considered part of the process ) and different operation conditions. What do you conclude about the effects of regulatory level controls and operating conditions on the RGA? Small distillation column XF F R F V AC XD AC XB Rel. vol = 1.2, R = 1.2 R min XD, XB Feed RGA RGA Comp..998, , , , , , , , , XF F V AC XD F D AC XB From McAvoy, 1983

106 Directionality: Its effect on Control Performance Let s gain insight about control performance and learn one short-cut metric Do we need more than AC Controllability Integrity (RGA) Before we design controls? XF AC XD F D F V AC AC XB λ(1,1) = 6.1 λ(1,1) =.39

107 Important Observation: Case 1 Design A FR XD FRB XB Design B FD XD FRB XB RGA = 6.9 AC RGA =.39 AC AC AC.988 IAE = ISE = IAE = ISE = IAE =.5956 ISE = IAE =.4577 ISE = e-5 XD, light key XB, light key XD, light key XB, light key SAM = SSM = SAM = SSM = SAM =.133 SSM = SAM = SSM =.1748 Reflux flow Time Reboiled vapor Time Reflux flow Time Reboiled vapor Time

108 Important Observation: Case 2 Design A FR XD Design B FD XD FRB XB FRB XB RGA = 6.9 AC RGA =.39 AC AC AC XD, light key IAE = ISE = XB, light key IAE = ISE =.3839 XD, light key IAE = ISE =.786 XB, light key IAE = ISE = Reflux flow SAM = SSM =.2517 Reboiled vapor SAM = SSM = Reflux flow SAM =.5154 SSM = Reboiled vapor SAM = SSM = Time Time Time Time

109 Directionality: Its effect on Control Performance Conclusion from the two observations (and much more evidence) The best performing loop pairing is not always the pairing with Relative gains elements nearest 1. The least interaction This should not be surprising; we have not established a direct connection between RGA and performance. So, what is going on?

110 Directionality: Its effect on Control Performance Strong directionality is a result of Interaction The best multivariable control design depends on the feedback and input! A key factor is the relationship between the feedback and disturbance directions. Disturbances in this direction are difficult to correct. Disturbances in this direction are easily corrected.

111 Directionality: Its effect on Control Performance Strong directionality is a result of Interaction Distillation with energy balance A Disturbance direction easily controlled A Disturbance direction not easily controlled

112 KEY MATHEMATICAL INSIGHT For Short-cut Metric Definition of the Laplace transform and take lim as s E( s) Directionality: Its effect on Control Performance = st E( t) e dt lim E( s) = E( t) dt s Measure of control performance, Large value = BAD By the way, this is an application of the method for evaluating the moment of a dynamic variable. n th moment : t n Y ( t) dt = ( 1) n d ds n n Y ( s) s=

113 Directionality: Its effect on Control Performance KEY MATHEMATICAL INSIGHT APPLIED FIRST TO A SINGLE-LOOP SYSTEM PI Controller 1/s D(s) G d (s) SP(s) E(s) + MV(s) + G C (s) G v (s) G P (s) - + CV m (s) G S (s) Apply this concept to a single-loop control system CV(s) Please verify E( s) Gd ( s) = D( s) 1+ Gp( s) Gc( s) E SL ( t) dt = K K D p T K I c

114 Directionality: Its effect on Control Performance Apply the approach just introduced to a 2x2 multiloop system Identify the key aspects - group terms! SP 1 (s) SP 2 (s) G c1 (s) MV 1 (s) MV 2 (s) G c2 (s) G 11 (s) G 21 (s) G 12 (s) G 22 (s) + + G d1 (s) G d2 (s) + + CV 1 (s) D(s) CV 2 (s) E iml ( t) dt = RDG f E ( t) dt ij tune isl

115 Directionality: Its effect on Control Performance E iml ( t) dt = RDG f E ( t) dt ij tune isl Relative Disturbance Gain dimensionless only s-s gains can be +/- and > or < 1. different for each disturbance Usually the dominant term for interaction Tune Factor Change in tuning for multi-loop Single-loop performance (dead times, large disturbances, etc. are bad)

116 Directionality: Its effect on Control Performance 1 K 1 K E iml Relative disturbance gain 1 K K d 2 K K d1 22 K11K22 ( t) dt = RDG f E ( t) dt 12 ij K K What unique information is here? What is this term? c c tune T T I I SL ML isl K K Typical range D p T K I c

117 Directionality: Its effect on Control Performance E iml ( t ) dt / E ( t ) dt isl = The dominant factor! RDG ij f tune The change in performance due to interaction in multiloop system RDG include feedback interaction (RGA) and disturbance direction

118 Directionality: Its effect on Control Performance E iml ( t) dt = RDG f E ( t) dt RDG is easily calculated ij tune 2x2 Multiloop General Square Multiloop isl 1 K 1 K 1 K K d 2 K K d1 22 K11K22 12 Multiloop : Single loop : RDG ij ( MV ) = D j ML [ MV ] [ MV ] ( MV ) j D ML SL SL = K = ( K 1 p P K d diagonal ) 1 K d

119 Directionality: Its effect on Control Performance Relative Disturbance Gain (RDG) Gives effects of Interaction on Performance E iml ( t Advantages ) dt Steady-state information Includes feedback and disturbance directions Direct measure of CV performance / E ( t ) dt isl = RDG ij f tune Shortcomings Allows +/- cancellation - large is bad performance - small might be good performance Represents only CV performance Measures only one aspect of CV behavior

120 Directionality: Its effect on Control Performance Some important results. You can prove them yourself. For a single set point change, RDG = RGA For a disturbance with same effect as an MV, the RDG = to 2. (depending on the output variable) For one-way interaction, RDG = 1 Decouple only for unfavorable directionality, i.e., large RDG Large RGA indicates poor performance for SP For these common disturbances, interaction is favorable and performance similar to SL! Performance similar to SL! Decoupling can make performance worse!

121 Process Example: FOSS packed bed chemical reactor O2 H2 TC FC A T C F Q T Q 1. Calculate the RGA and tuning 2. Select pairings and predict performance A. Temperature set point change (single SP) B. Quench pressure change (disturbance same as MV) = ) ( ) ( ) ( ) (.... s T s F s e s e s e s e s C s T Q Q s s s s A B Directionality: Its effect on Control Performance

122 Directionality: Its effect on Control Performance 1. Calculate the RGA and tuning O2 H2 For T - FQ and C - TQ pairing RGA = 13.7, Detune factor = 2. Kc T = -.115, TI T = 1.37 Kc C = -.61, TI C = Select pairings and predict performance A. For set point change, RDG = RGA = large!! Predict poor performance! B. For disturbance, RDG small (between and 2)! Predict good performance, near single-loop TC T Q F Q FC T A C

123 Directionality: Its effect on Control Performance EMLdt RGA ) fdetune ESLdt ( EMLdt ( RGA )( Kij / Kii ) fdetune ESLdt 1 IAE = ISE = IAE = ISE = 1.27 Set point change Looks like poor tuning! Discuss the performance Time Time

124 Directionality: Its effect on Control Performance Disturbance in quench pressure, which is through MV dynamics. Discuss the performance 1-1 IAE = ISE = E dt f ESLdt ML detune E ML dt = Time IAE = ISE = Time How can this good performance occur with high interaction?

125 ) (... ) ( ) ( ) ( ) (. s X s e s e s F s F s e s e s e s e s XB s XD F s s V R s s s s = AC AC Process Example: Binary distillation shown in figure. 1. Calculate the RGA, RDG and tuning 2. Predict performance A. XD set point change B. Feed composition disturbance XD XB F R F V XF Directionality: Its effect on Control Performance

126 Directionality: Its effect on Control Performance Distillation tower (R,V) with both controllers in automatic for XD set point change For input = SP XD, RDG = RGA = 6 RDG large indicates much worse than singleloop Transient confirms the short-cut prediction XD, light key Reflux flow IAE = ISE = XD.23 XB SAM = SSM = Time XB, light key Reboiled vapor IAE = ISE = SAM = SSM = Time Looks like poor tuning!

127 Directionality: Its effect on Control Performance Distillation for XD set point change Let s explain why the interaction is unfavorable From SP X D From FV X D F R AC F V Note that the interaction from the bottom loop tends to counteract the action of the XD controller! AC From F R X B Unfavorable interaction, poor performance