Decentralized control Sigurd Skogestad Department of Chemical - PowerPoint PPT Presentation

Decentralized control Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Tecnology (NTNU) Trondheim, Norway 1

Outline • Multivariable plants • RGA • Decentralized control • Pairing rules • Examples 2

MIMO (multivariable case) Distillation column “Increasing L from 1.0 to 1.1 changes y D from 0.95 to 0.97, and x B from 0.02 to 0.03” “Increasing V from 1.5 to 1.6 changes y D from 0.95 to 0.94, and x B from 0.02 to 0.01” Steady-State Gain Matrix ∆ Y ∆ L ⎛ ⎞ ⎛ ⎞ ( ) D ⎜ ⎟ = ⎜ ⎟ G 0 ⎜ ⎟ ⎜ ⎟ ∆ x ∆ V ⎝ ⎠ ⎝ ⎠ B 0.97 − 0.95 0.94 − 0.95 ⎡ ⎤ ( ) ⎢ ⎥ g g 0 0.2 − 0.1 ⎡ ⎤ ⎡ ⎤ 1.1 − 1.0 1.6 − 1.5 ( ) 11 12 G 0 = = = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ( ) g g 0 0.03 − 0.02 0.01 − 0.02 0.1 − 0.1 ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ 21 22 ⎢ ⎥ ⎣ 1.1 − 1.0 1.6 − 1.5 ⎦ ( ) ( ) Effect of input 1 ∆ L on output 2 ∆ x B Can also include dynamics : ⎡ ⎤ (Time constant 50 min for y D ) 0. 2 0.1 → ∆ ⎢ ⎥ y − D ⎢ ⎥ 1 + 50s 1 + 50s = G ( ) → ∆ s ⎢ ⎥ x 0.1 0.1 B ⎢ ⎥ − (time constant 40 min for x B ) + + ⎣ 1 40s 1 40s ⎦ 3

Analysis of Multivariable processes What is different with MIMO processes to SISO: The concept of “ directions ” (components in u and y have different magnitude ” Interaction between loops when single-loop control is used INTERACTIONS Process Model y 1 g 11 u 1 − " Open loop " g 12 ( ) ( ) ( ) ( ) = + y s g s u ( ) s g s u s 1 11 1 12 2 ( ) ( ) ( ) ( ) = + y s g s u ( ) s g s u s g 21 2 21 1 22 2 y 2 g 22 G u 2 4

Consider Effect of u 1 on y 1 1) “ Open-loop ” (C 2 = 0): y 1 = g 11 (s) · u 1 2) Closed-loop ” (close loop 2, C 2 ≠ 0) ⎛ ⎞ g g ⋅ C ⎟ ⎜ ( ) ⎟ y = g s − 12 21 2 ⎜ u ⎟ ⎜ 1 ⎜ 11 ⎟ 1 + ⋅ ⎝ 1 g C ⎠ 22 2 Change caused by “ interactions ” 5

Limiting Case C 2 →∞ (perfect control of y 2 ) ⎛ ⎞ g g ⎟ ⎜ ( ) ⎟ y = g s − 12 21 ⎜ u ⎟ ⎜ ⎟ 1 ⎜ 11 1 ⎝ g ⎠ 22 How much has “gain” from u 1 to y 1 changed by closing loop 2 with perfect control? ( ) y /µ ⋅ g 1 def 1 1 = = = = RGA Relative Gain OL 11 λ ( ) CL g g g g 11 y /µ − − g 12 21 1 12 21 1 1 11 g g g 22 11 22 The relative Gain Array (RGA) is the matrix Example from before formed by considering all the relative gains − ⎡ 0.2 0.1 ⎤ 1 G = , λ = = 2 ⎢ ⎥ 11 ⎡ ⎤ − 0.1 0.1 ( ) ( ) 0.1 0.1 y /u y /u ⎣ ⎦ 1 − ⎢ ⎥ 1 1 1 2 OL OL 0.2 0.1 ⎢ ⎥ ( ) ( ) ⎡ ⎤ y /u y /u � �� � � λ λ ⎢ ⎥ 1 1 1 2 ⎢ ⎥ 11 12 CL CL RGA = Λ = = ⎢ ⎥ 0.5 ⎢ ⎥ ( ) ( ) λ λ ⎢ y y /u ⎥ ⎣ ⎦ u 2 − 1 ⎡ ⎤ 21 22 2 1 2 2 ⎢ ⎥ OL OL RGA = ⎢ ⎥ ( ) ( ) ⎢ ⎥ y u y /u − 1 2 ⎣ ⎦ ⎣ ⎦ 2 1 2 2 CL CL 6

Property of RGA: Columns and rows always sum to 1 RGA independent of scaling (units) for u and y . Note: RGA as a function of frequency is the most important for control! 7

Use of RGA: (1) Interactions From derivation: Interactions are small if relative gains are close to 1 Choose pairings corresponding to RGA elements close to 1 Traditional: Consider Steady-state Better: Consider frequency corresponding to closed- loop time constant But: Avoid pairing on negative steady-state relative gain – otherwise you get instability if one of the loops become inactive (e.g. because of saturation) 8

Example: ⋅ ⋅ y =0.2 u -0.1 u − ⎡ ⎤ 0.2 0.1 1 1 2 ( ) = ⎢ G o ⎥ − y =0.1u -0.1u ⎣ 0.1 0.1 ⎦ 2 1 2 ⎡ ⎤ 2 -1 ⎢ ⎥ RGA = -1 ⎢ ⎥ 2 ⎣ ⎦ Onlyacceptablepairings : W ith integral action: ↔ u y ⇒ 1 1 Negative RGA individual ↔ y u 2 2 loop unstable + overall system unstable Not recommended : when loops saturate ↔ u y 1 2 ↔ u y 2 1 9

(2) Sensitivity measure But RGA is not only an interaction measure: Large RGA-elements signifies a process that is very sensitive to small changes (errors) and therefore fundamentally difficult to control Large (BAD!) example − ⎡ ⎤ ⎡ ⎤ 1 1 91 90 = G= RGA ⎢ ⎥ ⎢ ⎥ − ⎣ 0.9 0.91 ⎦ ⎣ 90 91 ⎦ 1 + = + 1.1% 90 1 = Relativechange - makesmatrixsingular! λ 21 ⎛ ⎞ 1 ⎟ ⎜ = + = ⎟ ˆ Then g g ⎜ 1 0.91 ⎜ ⎟ � ⎝ ⎠ 12 12 90 0.9 10

Singular Matrix: Cannot take inverse, that is, decoupler hopeless. Control difficult 11

Exercise. Blending process sugar u 1 =F 1 y 1 = F (given flowrate) water u 2 =F 2 y 2 = x (given sugar fraction) • Mass balances (no dynamics) – Total: F 1 + F 2 = F – Sugar: F 1 = x F (a) Linearize balances and introduce: u 1 =dF 1 , u 2 =dF 2 , y 1 =F 1 , y 2 =x, (b) Obtain gain matrix G (y = G u) (c) Nominal values are x=0.2 [kg/kg] and F=2 [kg/s]. Find G (d) Compute RGA and suggest pairings (e) Does the pairing choice agree with “common sense”? 12

Decentralized control 13

Two main steps • Choice of pairings (control configuration selection) • Design (tuning) of each controller 14

Design (tuning) of each controller k i (s) • Fully coordinated design – can give optimal – BUT: requires full model – not used in practice • Independent design – Base design on “paired element” – Can get failure tolerance – Not possible for interactive plants (which fail to satisfy our three pairing rules – see later) • Sequential design – Each design a SISO design – Can use “partial control theory” – Depends on inner loop being closed – Works on interactive plants where we may have time scale separation 15

Effective use of decentralized control requires some “natural” decomposition • Decomposition in space – Interactions are small – G close to diagonal – Independent design can be used • Decomposition in time – Different response times for the outputs – Sequential design can be used 16

Independent design: Pairing rules Pairing rule 1. RGA at crossover frequencies. Prefer pairings such that the rearranged system, with the selected pairings along the diagonal, has an RGA matrix close to identity at frequencies around the closed- loop bandwidth. Pairing rule 2. For a stable plant avoid pairings ij that correspond to negative steady-state RGA elements, ij (0) · 0 . Pairing rule 3. Prefer a pairing ij where g ij puts minimal restrictions on the achievable bandwidth. Specifically, its effective delay i j should be small. 17

Example 1: Diagonal plant • Simulations (and for tuning): Add delay 0.5 in each input • Simulations setpoint changes: r 1 =1 at t=0 and r 2 =1 at t=20 • Performance: Want |y 1 -r 1 | and |y 2 -r 2 | less than 1 • G (and RGA): Clear that diagonal pairings are preferred 18

Diagonal pairings Get two independent subsystems: 19

Diagonal pairings.... Simulation with delay included: 20

Off-diagonal pairings (!!?) Pair on two zero elements !! Loops do not work independently! But there is some effect when both loops are closed: 21

Off- diagonal pairings for diagonal plant • Example: Want to control temperature in two completely different rooms (which may even be located in different countries). BUT: – Room 1 is controlled using heat input in room 2 (?!) – Room 2 is controlled using heat input in room 1 (?!) TC ?? 2 1 TC 22

Off-diagonal pairings.... Controller design difficult. After some trial and error: – Performance quite poor, but it works because of the “hidden” feedback loop g 12 g 21 k 1 k 2 !! – No failure tolerance 23

Example 2: One-way interactive (triangular) plant • Simulations (and for tuning): Add delay 0.5 in each input • RGA: Seems that diagonal pairings are preferred • BUT: RGA is not able to detect the strong one-way interactions (g 12 =5) 24

Diagonal pairings One-way interactive: 25

Diagonal pairings.... Closed-loop response (delay neglected): With 1 = 2 the “interaction” term (from r 1 to y 2 ) is about 2.5 Need loop 1 to be “slow” to reduce interactions: Need 1 ≥ 5 2 26

Diagonal pairings..... 27

Off-diagonal pairings * =0) Pair on one zero element (g 12 =g 11 * * BUT pair on g 21 =g 22 =5: may use sequential design: Start by tuning k 2 28

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Comparison of diagonal and off-diagonal pairings – OK performance, – but no failure tolerance if loop 2 fails 30

Example 3: Two-way interactive plant - Already considered case g 12 =0 (RGA=I) g 12 =0.2: plant is singular (RGA= ∞ ) - - will consider diagonal parings for: (a) g 12 = 0.17, (b) g 12 = -0.2, (c) g 12 = -1 Controller: with 1 =5 and 2 =1 31

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Conclusions decentralized examples • Performance is OK with decentralized control (even with wrong pairings!) • However, controller design becomes difficult for interactive plants – and independent design may not be possible – and failure tolerance may not be guaranteed 35