Process control – FY · 2018-12-03 · Process control – FY Multivariable control Several...
Transcript of Process control – FY · 2018-12-03 · Process control – FY Multivariable control Several...
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Process control – FY
Multivariable control
◮ Several inputs and outputs
◮ Stability and interaction
◮ Connect inputs and outputs (RGA)
◮ Eliminate interaction (Feedback)
◮ Model predictive control (MPC)
Reading: Systems Engineering and Process Control: Y.1–Y.4
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Example 1: Shower flow and temperature
◮ Control signals: Cold water flow, Hot water flow
◮ Measurements: Total water flow, temperature
◮ Both inputs affect both outputs
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Example 2: Tank level and temperature
◮ Level control affects temperature
◮ Temperature control does not affect level
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Example 3: Evaporator level and temperature
◮ Level control affects temperature
◮ Temperature control affects level
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Example 4: Pressure and flow in transportation pipe
Mass tower TankPIC
PT
LIC
FIC
FT
LT
Pipe pressure to be kept constant, while desired flow decided by tank level
controller
◮ Pressure control affects flow
◮ Flow control affects pressure
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Multivariable system descriptions
State-space form:
dx
dt= Ax+ Bu
y = Cx+ Du
where u and y are vectors
Transfer function matrix, e.g.,:
Y (s) =
[
G11(s) G12(s)G21(s) G22(s)
]
U(s) = G(s)U(s)
Connection: G(s) = C(sI − A)−1B+ D
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Multivariable system descriptions
Example: The shower
[
yflow
ytemp
]
=
[
1 1
−e−s e−s
] [
ucold
uhot
]
Example: Transportation pipe
[
ypres
yflow
]
=
0.018e−2s
0.42s+ 1
−0.02e−1.4s
0.9s+ 1
4.1e−2.64s
3s+ 1
12e−4s
3s+ 1
[
upump
uvalve
]
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Multivariable system descriptions
Step response, transportation pipeT
o: O
ut(1
)
-0.03
-0.02
-0.01
0
0.01
0.02
0.03From: In(1)
0 5 10 15 20
To:
Out
(2)
-2
0
2
4
6
8
10
12
14
From: In(2)
0 5 10 15 20
Step Response
Time (seconds)
Am
plitu
de
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Multivariable system descriptions
Block diagrams, e.g.,:
u1
u2
y1
y2
G11
G12
G21
G22 Σ
Σ
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Block diagram computations for multivariable systems
Serial connection:
U(s)G1(s) G2(s)
Y (s)Y (s) = G2(s)G1(s)U(s)
Feedback:
U(s)G1(s)
−G2(s)
Y (s)Σ
Y (s) = G1(s)(
U(s) − G2(s)Y (s))
(
I + G1(s)G2(s))
Y (s) = G1(s)U(s)
Y (s) =(
I +G1(s)G2(s))−1
G1(s)U(s)
(Transfer function matrices, so order of multiplication important!)
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Stability for multivariable systems
Each feedback loop by itself might be stable when no other loops closed, but
full system might become unstable when more (stable) loops are closed
Example:
Σ
Σ
Σ
Σ
u1
u2
y1
y2
G11
G12
G21
G22
r1
r2
−1
−1
Gc1
Gc2
Process
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Stability for multivariable systems
Only first loop closed:
Σr1
−1
Gc1
u1
u2
y1
y2
Output dependence on reference r1:
Y1 =G11Gc1
1+ G11Gc1
R1
Y2 = G21U1 =G21Gc1
1+ G11Gc1
R1
Stability decided by characteristic equation
1+ G11Gc1 = 0
(Due to symmetry, similar connection for the other loop)
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Stability for multivariable systems
Both loops closed at the same time:
Σ
Σr1
r2
−1
−1
Gc1
Gc2
u1
u2
y1
y2
Output dependence on references r1 and r2:
Y1 =G11Gc1 + Gc1Gc2(G11G22 −G12G21)
AR1 +
G12Gc2
AR2
Y2 =G21Gc1
AR1 +
G22Gc2 + Gc1Gc2(G11G22 −G12G21)
AR2
Stability decided by characteristic equation
A(s) = (1+G11Gc1)(1+G22Gc2) − G12G21Gc1Gc2 = 0
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Multivariable stability – General case
Multivariable process Gp(s) and multivariable controller Gc(s):
rGc(s) Gp(s)
−1
yΣ
Y (s) = (I + Gp(s)Gc(s))−1
Gp(s)Gc(s)R(s)
Characteristic equation:
A(s) = det (I + Gp(s)Gc(s)) = 0
Closed loop asymptotically stableZ[ all roots in left half plane
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Multivariable stability – Example
Gp(s) =
10.1s+1
5s+1
10.5s+1
20.4s+1
, Gr(s) =
K1 0
0 K2
Each individually closed loop is stable for all K1, K2 > 0
det (I + Gp(s)Gr(s)) = det
K1
0.1s+1+ 1 5K2
s+1
K1
0.5s+12K2
0.4s+1+ 1
=
a(b)
b(s)
where
a(s) = 0.02s4 + 0.1(3.1+ 2K1 + K2)s3
+ (1.29+ 1.1K1 + 1.3K2 + 0.8K1 K2)s2
+ (2+ 1.9K1 + 3.2K2 + 0.5K1 K2)s
+ (1+ K1 + 2K2 − 3K1 K2) = 0
Last coefficient negative if 3K1 K2 > 1+ K1 + K2 [ unstable
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Relative Gain Array (RGA)
◮ A way to deduce coupling in “quadratic systems”
(as many inputs as outputs)
◮ Often based on static gain K = G(0)
◮ Normalization to avoid scaling problems
◮ Guidance in connection of inputs and outputs
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Relative gain
1. Open loop static gain when ∆uk = 0, k ,= j
ki j = Gi j(0) =∆yi
∆u j
2. Closed loop static gain when ∆yk = 0, k ,= i (assume “perfect” control
of the other outputs)
li j =∆yi
∆u j
3. Relative gain
λi j =ki j
li j
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RGA for 2$ 2-system
u1
u2
y1
y2
k11
k12
k21
k22 Σ
Σ
Open loop gain from u1 to y1:
y1 = k11u1
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RGA for 2$ 2-system
u1
u2
y1
y2
k11
k12
k21
k22 Σ
Σ
Closed loop gain from u1 to y1 when all outputs controlled to zero except y1:
y2 = k21u1 + k22u2 = 0 (perfect control)
u2 = −k21
k22
u1
y1 =
(
k11 −k12k21
k22
)
︸ ︷︷ ︸
=l11
u1
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RGA for 2$ 2-system
Relative gain:
λ11 =k11
l11
=k11
k11 −k12k21
k22
Note that l11 =det Kk22
Can show that matrix with elements 1/li j is given by
1
det K
[
k22 −k21
−k12 k11
]
=(
K−1)T
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RGA for general quadratic system
RGA-matrix Λ with elements λi j = ki j1li j
, can be computed as
Λ = K .∗(
K−1)T
Note! ”.∗” is element-wise multiplication
For Λ, we have
◮ Row and column sum equal
n∑
i=1
λi j =
n∑
j=1
λi j = 1
◮ System easy to control if Λ close to unit matrix (perhaps after
permutations)
◮ Negative elements imply difficult connectivity
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Connecting inputs and outputs
Rule of thumb: Inputs and outputs should be connected such that relative
gain is positive and as close to one as possible.
Example 1:
Λ Connectivity Interaction[
1 0
0 1
]
u1 − y1
u2 − y2None
[
0 1
1 0
]
u1 − y2
u2 − y1None
[
0.85 0.15
0.15 0.85
]
u1 − y1
u2 − y2Weak
[
2 −1
−1 2
]
u1 − y1
u2 − y2Difficult
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Connecting inputs and outputs
Example 2: 3$ 3-system
G(s) =
2s+3
1s+3
1s+1
1s+1
1s+2
1s+1
1s+1
1s+1
2s+2
RGA-computation:
G(0) =
23
13
1
1 12
1
1 1 1
, Λ =
−2 0 3
4 −1 −2
−1 2 0
Difficult multivariable interaction
Connectivity suggestion: u1 − y2, u2 − y3, u3 − y1
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Process
Gc2
Gc3
Gc1
u1
u2
u3
y1
y2
y3
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Decoupling
Idea: Introduce new inputs m and decoupling filter D(s) to make system
easier to control
m1
m2
u1
u2
y1
y2
Σ Σ
ΣΣ
D
Gc1
Gc2
G11
G12
G21
G22
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Decoupling
Outputs as a function of new inputs:
Y (s) = G(s)U(s) = G(s)D(s)M(s) = T(s)M(s)
where T(s) chosen with desirable properties
◮ Decoupling filter: D(s) = G(s)−1T(s)
◮ Choose T(s) diagonal
◮ 2$ 2-case:
T(s) =
[
T11 0
0 T22
]
, D(s) =1
det G
[
G22T11 −G12T22
−G21T11 G11T22
]
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Designing the decoupling
One choice of decoupling is the following:
D(s) =
1 −G12
G11
−G21
G22
1
(other choices of D not covered in course)
This choice gives the following decoupled system:
T(s) =
G11 − G12
G21
G22
0
0 G22 −G21
G12
G11
This can be controlled using two ordinary controllers
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Interpretation as two feedforwards:
m1
m2
u1
u2
y1
y2
Σ Σ
ΣΣ1
1Gc1
Gc2
G11
G12
G21
G22
−G21
G22
−G12
G11
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Implementing the decoupling
Decoupling elements −G12
G11and−G21
G22cannot be implemented if they contain
◮ pure derivatives
◮ negative time delays
Solution options:
◮ low pass filter
◮ use static gain only
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Example: Transportation pipe
Process:
G(s) =
0.018e−2s
0.42s+ 1
−0.02e−1.4s
0.9s+ 1
4.1e−2.64s
3s+ 1
12e−4s
3s+ 1
RGA:
Λ =
[
0.73 0.27
0.27 0.73
]
Conclusion: Control pressure with pump and flow with valve, some interaction
between loops
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Simulation with decoupling (solid) and without (dashed):
0 100 200 30055
60
65
70
75
80
85
0 100 200 30015
20
25
30
35
0 100 200 30030
35
40
45
50
0 100 200 30050
55
60
65
70
Flow Pressure
Valve Pump
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Model predictive control (MPC)
Advanced method for control of multivariable processes
Basics:
◮ Need multivariable process model – linear or nonlinear
◮ Model predicts future behavior of system from now to time T (horizon)
◮ Add model, performance objective, and input and state constraints to an
optimization problem
◮ Online solution to optimization problem in every sample
◮ Gives optimal trajectories for inputs
◮ Use first value of input trajectory only
◮ Repeat optimization in next sample (receding horizon)
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Optimization over receding horizon
k+H
Out
puts
Inpu
ts
y
y
u
u
k+Hk
k
t
t
1
2
2
1
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Optimization over receding horizon
k+H+1
Out
puts
Inpu
ts
y
y
u
u
t
t
1
2
2
1
k+1
k+1 k+H+1
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Optimization problem example
Linear model:
dx
dt= Ax+ Bu
y = Cx+ Du
Quadratic objective function:
J =
k+H∑
t=k+1
(
w1(r1 − y1)2 + w2(r2 − y2)
2)
+
k+H−1∑
t=k
(
ρ1∆u21 + ρ2∆u2
2
)
Optimization problem: Minimize J w.r.t. u1 and u2 while taking constraints into
account:
uimin≤ ui ≤ uimax
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Multivariable control – Summary
◮ Unidirected influence from control loop to another does not affect stability
and the effect can be eliminated by feedforward
◮ Multivariable interaction can affect both stability and performance to the
worse. Interaction can be reduced with decoupling filter
◮ Analysis tool RGA describes the difficulty of the interaction and what
outputs that should be fed back to what inputs
◮ Difficult interaction or high performance demands might require a
multivariable controller, e.g., MPC