Multi-Loop Control. Department of Chemical Engineering,

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1 Interaction ti Analysis and Multi-Loop Control Sachin C. Patawardhan Department of Chemical Engineering, I.I.T. Bombay

2 Outline Motivation Interactions in Multi-loop control Loop pairing using Relative Gain Array Analysis Singular Value Analysis Decoupling controller Conclusions ons 2

3 Motivation Industrial Processes: multivariable (multiple inputs influence same output) and exhibit strong interaction among the variables Conventional Control scheme: Multiple Single Input Single Output PID controllers used for controlling plant (Multi-Loop Control) Consequences: Loop Interactions Lack of coordination between different PID loops Neighboring PID loops can co-operate with each other or end up opposing / disturbing each other 3

4 Example: Shell Control Problem Controlled Outputs : (y) Top End Point (y2) Side Endpoint (y3) Bottom Reflux Temperature Manipulated Inputs : (u) Top Draw (u2) Side Draw (u3) Bottom Reflux Duty Unmeasured Disturbances: (d ) Upper reflux (d 2 ) Intermediate reflux 4

5 Which Scheme is Better? Control Scheme PID-: (y) Top End Point - (u) Top Draw PID-2: (y2) Side Endpoint - (u2) Side Draw PID-3: (y3) Bottom Reflux Temperature (u3) Bottom Reflux Duty Control Scheme 2 PID-: (y2) Side End Point - (u) Top Draw PID-2: (y) Top Endpoint - (u2) Side Draw PID-3: (y3) Bottom Reflux Temperature (u3) Bottom Reflux Duty How to examine above options systematically and reach a decision? 5

6 Tennessee Eastman Problem Primary controlled variables: Product concentration of G Product Flow rate 6

7 TE Problem: Inputs and Outputs 7

8 Loop Interactions Large loop interactions : can lead to poor quality of control due to lack of coordination among PID controllers Solution strategy: Choose controller pairing with minimal interactions De-tune the controllers to minimize loop interactions Design multi-variable controllers, which simultaneously change all inputs by considering errors in all the outputs 8

9 Quadruple Tank Setup Input Input 2 Output Output 2 9

10 Block Diagram Representation PID- u Level Control Valve Control Valve 2 PID-2 Level 2 u 2 0

11 Loop Interactions CV- PID- Level PID-2 Level 2 CV-2

12 Interaction Analysis Assume a loop pairing say y -u and perform the following experiments With all loops open, make a step change in u to u + Δu and measure the change in output Δy. We will term this as a direct effect. With all loops except the u -y loop closed, repeat the change in u. There will be change in y because of the direct effect but also there will be a retaliatory t effect because u 2 changes to keep y 2 constant. We will term this change as Δy +Δy r. 2

13 Interaction Analysis Ratio of these two terms can be defined as λ (for the y -u pairing) as λ = Δy /( Δy +Δy r. ) This is called relative gain Compute relative gain for each assumed inputoutput pairing Depending on the values of this index for various assumed loop pairings (step ), decision can be taken on the final loop pairing. i This decision making is based on steady state analysis only. 3

14 Relative Gain Array (RGA) RGA: A measure of loop interaction used for deciding loop pairing i of SISO controllers λ ij [ Δ / Δ ] y i u j = [ Δy ] i / Δu j open_loop [ Δ y / Δu ] i j all but y u loop closed y : Measured outputs u : Manipulated Inputs : Steady state gain / sensitivity between i'th op and j'th thi/p i j 4

15 Calculations of RGA Let us assume pairing y -u Steady state t model k ij Δy Δ y 2 = = k k Δu Δ u + k + k 2 Δu Δ u : Stesdy state gain / sensitivity between output i and input j Δy Δu = k Open Loop Step in Δu Δy Process Δu2 = 0 Δy 2 0 5

16 Calculations of RGA Now consider the situation where y 2 -u 2 loop is closed and perfectly controlled Step Change in Δ u Setpo int = 0 Process PID Δy Δy 2 = 0 Δy Δ y = k Δ u + k Δ u = 2 Δu 2 2 = ( k / k ) = kδu + k2δu2 = k Δu k Δu k 0 k 6

17 Calculations of RGA Other Δy Δu y 2 u 2 λ Loop Closed = = k k 2k k k 2 22 λ = λ = λ k2k k relative gains can be easily computed as follows 2 λ 22 2 = λ 22 RGA Matrix ( Λ) for 2 2system λ λ Λ = λ λ 2 RGA is independent of variable scaling 7

18 Example: Wood-Berry Column Pilot scale distillation column Distillate Composition Bottom Composition s 3s 2.8e 8.9e x D( s) = 6.7s + 2s + R( s) 7s s x B ( s ) 6.66 e 9.4 e S ( s ) 0.9s + 4.4s + Steady Stete Gain Matrix Δx D ΔR = 6 Δx B ΔS Reflux Rate Steam Flow-rate 8

19 Wood-Berry Column: RGA RGA( Λ) λ = / = Any loop pairing will result in loop interactions Pairing x x D B R B k c τ (min) I

20 Wood-Berry Column Multi-Loop PID Control Change in closed loop behavior due to loop interactions 20

21 RGA for MIMO Process u u u j y i = j K ij Steady When all but y u u j... = c... u n u j (steady state gain between input j and output i) state model : y i u j loop y =... ~ yi = K u j... 0 are closed = Ku u = K [ 0... y... 0 ] T i y and perfectly 0 i ~ yi y K ji = u c j u ~ Relative gain ( λ ) = K K ij c ij K ~... y = K~ ni ji i j c = K = ~ K ij ji [ K ] T ij ~ = Ky i u j controlled c i th column of gain Matrix inverse 2

22 RGA Calculations For general (n n)system Given Steady Stage Gain Matrix K RGA ( Λ) = K ( K ) where denotes Sh Schur T Product (element by element multiplication of two matrices) Properties. Summation of all elements in any row = 2. Summation of all elements in any column = 3. RGA is independent of scaling used for input of output variables 22

23 Analysis of RGA If λ = retaliatory action is not present. So assumed loop pairing is correct because there is no interaction from the other loop. If 0 < λ < retaliatory action is comparable to the direct action but is in the same direction. The assumed loop pairing may be chosen only if the index is closer to (say 0.8). If λ =0 Retaliatory action is much greater than the direct action. The assumed loop pairing is incorrect. The loop ppairing u -y 2 is preferable. 23

24 Analysis of RGA If λ > Retaliatory action is in opposite direction to the direct action but is smaller in magnitude than the direct. The assumed loop pairing may be chosen only if the index is close to. If λ < 0, Retaliatory effect is larger and opposite in direction to the main effect. Do not chose this loop pairing. 24

25 Suggested Loop Pairing Refinery Distillation Column RGA = y y y y u u2 u3 u Outputs Y : Top Composition Y 2 : Bottom Composition Y 3 : Side stream Composition Y 4 : Side Stream 2 Composition Inputs u : Top Draw u 2: Bottom Draw u 3 : Side Draw u 4 : Side Draw 2 25

26 RGA: Non-square Systems with 2 measurements and 3 inputs process Consider u = Δ Δ u u u y y Δ Δ Δ y u u Best Combination Δ Δ Δ Δ = y y RGA Δ Δ Δ Δ = Δ Δ y y RGA u u Δ Δ = Δ Δ y y RGA u u 26 Δ y y 2

27 Singular Value Analysis Powerful analytical tool for Selection of controlled, measured and manipulated variables Determination of best multi-loop configuration Evaluation of robustness (insensitivity to changes in plant behavior) of a control scheme ΔY = KΔU ( K: Steady State Gain Matrix) Singular Values ( σ, σ,... σ ) 2 +ve Square Root of Eigen-values of Roots of polynomial T det( si K K) = 0 Condition Number (CN) = K K = r 2 2 T KK max( σ ) min( σ ) i i 27

28 Singular Value Analysis Non-square systems (number of inputs not equal to number of outputs) SVA can be used to find a square subset with least difficulties i in control Larger condition number implies difficulties in controlling a system Among multiple possibilities, choose subset with minimum mum condition number Limitation: Singular values are dependent on scaling of input and output variables 28

29 Example: Furnace F air Furnace T HydroCarbon F O fuel 2, Conc. T O 4s e e ( s) = 4.22s s + 4s 4s ( s) 0.445e.96e s s + s F F ( s) ( ) HC air 2 fuel s λ = RGA( Λ) = Suggested Pairing THC Ffuel and O2 F air Singular Values σ = σ 2 = Condition Number = Large CN : Difficult to Control 29

30 4 Component Distillation Column 30

31 Neiderlinski Index Consider MIMO system whose inputs and outputs are paired as y u u,..., u,, y 2 2, yn n i.e. T.F. matrix G(s) is arranged such that transfer functions relating paired inputs and outputs are arranged along the diagonal. Further, let each element of G(s) be (a) rational and (b) open loop stable. Also, let n SISO feedback controllers with integral action be designed such that each SISO loop is stable when all the rest (n-) loops are open. Under these asumptions, the multi-loop control system will be unstable for all possible values of controller parameters if the Neiderlinski index defined as N det n G i = [ G(0) ] i = < 0 G ii (0) 3

32 Example: Furnace 4s 4s 3.235e e T HC( s) = 4.22s s + Fair ( s) 4s 4s O2 ( s ) 0.445e.96e Ffuel ( s) 4.434s s RGA( Λ) = Suggested Pairing : T HC F and O2 fuel F air Rearrange tarnsfer function matrix such that paired variables are on main diagonal T HC O 2( ( s ) s ) e = 4. 75s e 3. 98s + s s e 4. 22s + 4s e s + 4 s F F fuel air ( s ) ( s ) Ni = det =. > Process is integral controllable 32

33 Multi-loop PID Control After selection of loop pairings with minimum i interactions, ti one can design controllers for individual loops. Presence of interaction n and retaliatory t r effects from other loops may require that the controller be detuned for acceptable performance. BLT Detuning method Sequential Loop Tuning Independent Loop Tuning 33

34 Biggest Log Modulus Tuning SISO Loop Design Review Closed loop characteristic equation + g g ( s ) : Process Transfer Function ; p c ( s ) gp ( s ) = + G ( s ) = 0 g ( s ) : Controller transfer c function Nyquist plot: depicts real part of G(j ω) on x-axis and imaginary part of G(j ω) on y-axis as ω Nyquist Stability Criteria : A feedback control system will be unstable encircles point (-,0) on the - ve real axis as if the Nyquist plot of G(jω) ω. The number of encirclements correspond to the number of roots of the characteristic equation that lie in R.H.P. of s -plane, assuming that the process is open loop stable. 34

35 BLT Method A major of distance of G(jω) contour from point (-,0) is given as L ( s ) = 20 L c G ( s ) log + G ( s ) max Suggested design specification for log modulus: Lc 2dB Log modulus design : iteratively choose PI controller parameters (k, c τ I ) such that design specification is met. Multi -loop PI control of MIMO Processes Y(s) = - { [ I + G ( s ) G ( s ) ] G ( s ) G ( s } R (s) p c p c ) Closed Loop Characteristic Equation det [ I + G p ( s ) Gc ( s )] = 0 [ I + G ( jw ) G ( jw ] If Nyquist plot of det p ) c encircles the closed loop system is unstable. origin, 35

36 BLT Method then If we define define a new function f(s) as f(s) = - + det[ I + G ( s ) G ( s )] encirclement of (,0) by f(jω) would indicate instability. p c Defining a multi- variable closed loop log modulus L, = 20 c m f ( s ) log + f ( s ) Suggested design specification for log lodulus : L where n is dimension of MIMO system. max c,m ( 2n)dB. Calculate Tuning Procedure (Luyben, 986) Ziegler -Nichol's tuning for n individual PI controller s 2. Assume a factor F such that 2 F 5 3. De - tune PI controllers as follows k ZN ZN = k j / F and τ I = F i, j for j =,2,... n c, j τ j max 4. Iteratively choose F such that criteria L c, m ( 2n) db is satisfied 36

37 BLT: Examples 37

38 BLT: Examples 38

39 BLT: Examples 39

40 De-tuning of PID BLT De-tuning Method BLT : measure of k ZN kc = ; τ I = F τ ZN F ( k ZN, τ ZN ) : Ziegler Nichol' s Tuning F : Detuning Factor (2 F 5) Adjust F till Bigest Log Modulus (BLT) achieves a specified value how far system is from being unstable Sequential Tuning Method. Tune one of the controllers in selected pairing and close the loop 2. Tune next controller with the first loop closed and so on 40

41 Multi-Variable Decoupling Dynamic Decouplers 4

42 Decoupled Loops Loop interactions are eliminated Set point change in one loop does not affect the other loop 42

43 Design of Decouplers Choose T such 2 ( Gp 2 p 22 2 or T 2 that Choose T + G T ) U = 0 ( G + G T ) U = 0 ( s) = G G p2 p22 ( s) ( s) 2 such that ( p 2 p 2 22 or T ( s ) = 2 G G p2 p ( s ) ( s ) Difficulty: Model should be perfectly known Decoupling elements may be unrealizable if time-delays are present T Gain Decoupling k 2 2 = ; k 22 T 2 = k k 2 43

44 Example: Wood-Berry Column Pilot scale distillation column xd ( s ) xb ( s ) s 3s 2.8e 8.9e ( ) = s + s + R s 7s s ( ) e e S s 0.9s + 4.4s + PI Controller Pairing x R and x S D B T ( s ) 2 = G G p 2 p 22 ( s ) ( s ) Steady State Decoupler T2 ( s ) = 0.34 ; T2( s ) =.477 Dynamic Deoupler Design G T2( s ) = G 7s s 6.6e 9.4e = 0.9s + 4.4s + p 2 p ( s ) ( s ) 6.7 s + =.477 e 2.8s + 4.4s + = 0.34 e 0.9s + 2s 6s 44

45 Dynamic Decoupling: Difficulty Difficulty: De-coupler T. F. can be unrealizable xd ( s ) xb ( s ) = 3s s 8.9e 2.8e 2s + 6.7s + S ( s ) s 7s 9.4e 6.6e R ( s ) 4.4s + 0.9s + Let us assme PI Controller Pairing x S D and x R B ~ T ( s ) 2 Dynamic Deoupler Design for above pairing ~ s s Gp s 7 2( ) 9.4e 6.6e 0.9s + 6 = ~ = = 2.94 e G ( s ) 4.4s + 0.9s + 4.4s + p 22 ~ Gp2 ( s ) 2.8 s + 2 T2( s ) = = e G ( s ) 6.7s + p s s Decouplers are physically unrealizable as they include terms of type e ds with d > 0, which would require current decoupler output to depend on the future inputs 45

46 CSTR Example Consider non-isothermal CSTR dynamics dc feed flow rate A = f( CA, T, F, Fc, CA0, Tcin) dt dt = f2( CA, T, F, Fc, CA0, Tcin) dt States ( X ) coolant flow rate T [ C T ] MeasuredOutput ( Y ) [ T ] Manipulated Inputs( U ) [ F MeasuredDisturbances ( D ) A m F ] T c [ CA ] [ T ] cin Feed conc. UnmeasuredDisturbances (D u) 0 Cooling water Temp. 46

47 Multi-loop PID with and without Steady State Decoupling Conc.(mol/m3) Controlled Outputs Multi-loop PID Setpoint PID with s.s. s decoupling Time (min) CSTR Example PID Pairing C A -F c T - F 396 Temp.(K K) Time (min) k c = 6.34 τ k I, c,2 = 0.2 = τ I, =

48 Multi-loop PID with and without Steady State Decoupling Coole ent Flow (m3/m min) Manipulated Inputs Multi-loop PID PID with s.s. s decoupling Time (min) PID with Steady state decoupling PID Pairing C A m T m 2.5 Inflow (m3 3/min) Time (min) k c τ k τ I, c,2 I,2 = 0. = 0.2 = 0. =

49 Control of CSTR Control schemes Measure only temperature Estimate concentration and temperature using state observer Develop MIMO / Multi-loop control schemes to control estimated concentration and temperature Controllers developed Multi-loop PI Multi-loop PI with gain decoupling Multi-variable Controller (Linear Quadratic Optimal Control) Control problem: Compare performances for step change in conc. setpoint at k=50 followed by step disturbance at t = 50 49

50 CSTR: Multi-Loop PI Performance ) Co onc.(mol/m3 Temp.(K K) Controlled Outputs Time (min) Time (min) PID Pairing C A - F c T - F Linear Plant Simulation k = 6.34 c τ k I, c,2 = 0.2 = τ I, =

51 CSTR: Multi-Loop PI Performance (m3/min) Coolent Flow Inflow (m3/min) Inlet Conc. (mol/m3 3) Manipulated Inputs and Disturbance Time (min) Time (min) Time (min) Linear Plant Simulation 5

52 CSTR: PI with gain Decoupling Conc.(mol/m m3) Controlled Outputs Time (min) PID Pairing C A m T m 2 Linear Plant Simulation Temp p.(k) Time (min) k c = 0. τ k I, c,2 = 0.2 = 0. τ I, =

53 CSTR: PI with gain Decoupling m3/min) 30 Manipulated Inputs and Disturbance Coolent Flow ( Inflow (m3/min) Conc. (mol/m3 3) Time (min) Time (min) Linear Plant Simulation Inlet Time (min) 53

54 CSTR: PI with gain Decoupling 0.8 Controlled Outputs Conc.(mol/m3) Time (min) Non-Linear Plant Simulation i 420 Temp.(K K) Time (min) 54

55 CSTR: PI with gain Decoupling Coolent Flow (m3/min) Inflow (m3/min) Conc. (mol/m m3) Manipulated Inputs and Disturbance Time (min) Time (min) Non-Linear Plant Simulation i Inlet Time (min) 55

56 Multivariable State Feedback Controller Discrete time State Space Model x ( k + ) = Φx ( k ) + Γu ( k ) y( k ) = Cx( k ) State feedback multivariable control law ( ( k ) x( )) u( k ) = G xs k Step : Assume the states are measurable and design a stable control law / controller Step 2: Design a state estimator which constructs estimates of states t s by fusing measurements m with model predictions Step 3: Implement the controller using the estimated states Design Method: Linear Quadratic Optimal Control Advantage: Multi-variable systems can be controlled relatively easily 56

57 CSTR: LQOC Performance 0.45 Controlled Outputs Conc.(mol/m3) Temp.(K K) Time (min) Linear Plant Simulation (No Plant Model Mismatch Case) Time (min) 57

58 CSTR: LQOC Performance Coolent Flow (m3/min) Inlet Conc. (mol/m m3) Inflow (m3/min) Manipulated Inputs and Disturbance Time (min) Time (min) Time (min) Linear Plant Simulation (No Plant Model Mismatch Case) 58

59 CSTR: LQOC Performance 0.45 Controlled Outputs Co onc.(mol/m3 3) Temp.(K K) Time (min) Non-Linear Plant Simulation (Plant Model Mismatch Case) Time (min) 59

60 CSTR: LQOC Performance Co oolent Flow (m m3/min) Inlet Co onc. (mol/m3) Inflow (m m3/min) Manipulated Inputs and Disturbance Sampling Instant Sampling Instant Sampling Instant Non-Linear Plant Simulation (Plant Model Mismatch Case) 60

61 Conclusions Multi-variable processes are more difficult to control because of loop interactions RGA and SVD analysis provides a systematic approach to choose pairing of input-output t t variables for multi-loop l control When such simple measured do not work, it is advisable to use multi-variable controller (such as LQOC) 6

62 References Books Process Dynamics and Control, D. E. Seborg, T. F. Edgar, D. A. Mellichamp, Wiley, Process Dynamics, Modeling, and Control by B. A. Ogunnaike and W. H. Ray, Oxford University Press, 994. Control System Design, by Graham C. Goodwin, Stefan F. Graebe, Mario E. Salgado, Prentice Hall,

63 Internal Model Control Alternate parameterization of feed-back controller Controller Inputs Process Outputs Dynamic Model Setpoint Plant-model mismatch Internal Model Control Structure 63

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