PART-3
KNU/EECS/ELEC 835001
Decentralized control
for MIMO processes
Multivariable Control
Dr. Kalyana C. Veluvolu
Outline - Module 5.3
Decentralized control for MIMO processes
–
–
–
–
–
–
–
–
Multi-loop vs. Multivariable Control
Loop Decomposition
RGA detuning factor
Biggest Log Modulus Method
Design based on Loop Decomposition
IMC Design
Sequential Loop closing method
Simultaneous Equation Solving Method
Multivariable Control
Dr. Kalyana C. Veluvolu
2
Multi-loop Control Systems
Differences in SISO and interactive MIMO systems:
The manipulated variables that satisfy the desired controlled variables
must be determined simultaneously.
Differences between single-variable and multivariable behavior
increase as the transmission interaction increases.
Sensitivity of adjustments in manipulated variables to model changes
can be much greater in multi-loop than in single-loop systems.
Multivariable Control
Dr. Kalyana C. Veluvolu
3
Loop Decomposition
The decentralized system is diagonally paired, for any loop
ul ( s ) = g cl ( s )[rl ( s ) − yl ( s )]
Focusing on the loop
u j ( s ) − yi ( s )
only set-point change at u j ( s ) then
n
yi ( s ) = gij ( s )u j ( s ) −
∑
gil ( s ) gcl ( s ) yl ( s )
l =1,l ≠ j
y i ( s) = g ij ( s )u j ( s ) + aij ( s )u j ( s)
yk ( s ) = d kj ( s )u j ( s ),
Multivariable Control
∀k , k ≠ i
Dr. Kalyana C. Veluvolu
4
y1,sp
-
+
TITO Process
GC1
u1
G11(s)
G12(s)
-
GC2
u2
G22(s)
y2
++
+
y2,sp
Control Loop 2
c ( s) g12 ( s ) g 21 ( s )
a11 ( s ) = − 2
1 + c 2 ( s ) g 22 ( s )
d 21 ( s ) =
y1
G21(s)
c ( s) g12 ( s) g 21 ( s)
y1 ( s)
= g11 ( s) − 2
1 + c2 ( s) g 22 ( s)
u1 ( s)
y 2 ( s)
g 21 ( s)
= d 21 ( s) =
1 + c 2 ( s) g 22 ( s)
u1 ( s )
++
Control Loop 1
g 21 ( s )
1 + c2 ( s ) g 22 ( s )
Similarly
a22 ( s ) = −
d12 ( s) =
Multivariable Control
c1 ( s ) g12 ( s ) g 21 ( s )
1 + c1 ( s ) g11 ( s )
g12 ( s)
1 + c1 ( s) g11 ( s )
Dr. Kalyana C. Veluvolu
5
TITO Process Continued
The closed-loop transfer functions
y1 ( s ) g c1 ( s ) g11 ( s ) + g c1 ( s ) g c 2 ( s )[ g11 ( s ) g 22 ( s ) − g12 ( s ) g 21 ( s )]
=
r1 ( s )
gCL ( s )
y2 ( s ) g c 2 ( s ) g 22 ( s ) + g c1 ( s ) g c 2 ( s )[ g11 ( s ) g 22 ( s ) − g12 ( s ) g 21 ( s )]
=
r2 ( s )
gCL ( s )
y1 ( s ) g c 2 ( s ) g12 ( s )
=
r2 ( s )
gCL ( s )
y2 ( s ) g c1 ( s ) g 21 ( s )
=
r1 ( s )
gCL ( s )
with
gCL ( s ) = det [ I + G ( s )GC ( s ) ]

0 
 g ( s ) g12 ( s )   g c1 ( s )
= det  I +  11



g c1 ( s )  
 g 21 ( s ) g 22 ( s )   0

= 1 + g c1 ( s ) g11 ( s ) + g c 2 ( s ) g 22 ( s ) + g c1 ( s ) g c 2 ( s )[ g11 ( s ) g 22 ( s ) − g12 ( s) g 21 ( s )]
Multivariable Control
Dr. Kalyana C. Veluvolu
6
Design by Detuning Factor
Extend RGA with frequency dependant terms
λ ( s) =
– replacing steady-state gain with transfer function
1
G ( s )G21 ( s)
1 − 12
G11 ( s )G22 ( s)
GCL ( s ) = 1 + Gc1 ( s )G11 ( s ) + Gc 2 ( s )G22 ( s ) +
 G ( s )G21 ( s ) 
+Gc1 ( s )Gc 2 ( s )G11 ( s )G22 ( s )  1 − 12

 G11 ( s )G22 ( s ) 
For analysis of loop 1divided by 1 + Gc 2 ( s)G22 ( s)
results
Gc1 ( s )Gc 2 ( s )G11 ( s )G22 ( s )
λ ( s)
1 + Gc 2 ( s )G22 ( s )
1 + Gc 2 ( s )G22 ( s ) + Gc1 ( s )G11 ( s ) +
GCL ( s ) =
 1 + Gc 2 ( s )G22 ( s ) λ ( s ) 
= 1 + Gc1 ( s )G11 ( s ) 

 1 + Gc 2 ( s )G22 ( s ) 
Multivariable Control
Dr. Kalyana C. Veluvolu
7
Loop 1 much faster than loop 2
At the loop 1 critical frequency
Gc 2 ( jω )G22 ( jω ) → 0
as amplitude ratios decrease rapidly after comer frequency.
λ11
is not a strong function of frequency
1 + Gc 2 ( jω )G22 ( jω ) λ11
= 1.0
1 + Gc 2 ( jω )G22 ( jω )
tuned like a single-loop
controller without interaction.
GCL ( jω ) ≈ 1 + Gc1 ( jω )G11 ( jω )
Multivariable Control
Dr. Kalyana C. Veluvolu
8
Loop 1 Much Slower Than Loop 2
Gc 2 ( jω )G22 ( jω )
The fast loop,
very large at loop 1 critical frequency
big at a frequency much less than loop 2 critical frequency
Gc 2 ( jω )G22 ( jω ) >> 1.0
The characteristic equation in loop 1 can be simplified to
GCL ( jω ) ≈ 1 + Gc1 ( jω )G11 ( jω )
≈ 1 + Gc1 ( jω )
Gc 2 ( jω )G22 ( jω )
λ11 ( jω )Gc 2 ( jω )G22 ( jω )
G11 ( jω )
G ( jω )
≈ 1 + Gc1 ( jω ) 11
λ11 ( jω )
λ11
The gain of the process is changed. the controller gain has to be adjusted.
The phase lag unaffected, the integral time same.
It affect only the closed-loop process gain.
Multivariable Control
Dr. Kalyana C. Veluvolu
9
Same Dynamics for Loops 1 and 2
GCL ( s ) = 1 + Gc 2 ( s )G22 ( s ) + Gc1 ( s )G11 ( s ) + Gc1 ( s )Gc 2 ( s )G11 ( s )G22 ( s ) λ
= 1 + 2Λ ( s ) + Λ 2 ( s ) λ11
System stability
1 + 2Λ ( s) +
Λ 2 ( s)
λ
=0
To determine the closed-loop stability
1 + 2Λ( s) +
Multivariable Control
Λ 2 ( s)
λ



Λ ( s)
Λ ( s)
1
= 1 +
+


2
2
 λ + λ − λ  λ − λ − λ 
Dr. Kalyana C. Veluvolu
10
λ > 1.0
Case 1:
1 + 2Λ( s) +



Λ( s)
Λ( s)
= 1 +
1+



2
2
 λ + λ − λ  λ − λ − λ 
Λ 2 ( s)
λ
ωu1I = ωu1
Has real solution, ultimate frequency is unaffected by the interaction
the ultimate gain can be calculated from
ku1I g11 = λ − λ 2 − λ
ωu 1 I
k u1 I =
⇒
λ − λ2 − λ
ωu 1 I
g11
By the definition of gain margin
k u1 =
1
⇒
g11 ω
)
(
ku1I = λ − λ 2 − λ ku1
u1
Multivariable Control
Dr. Kalyana C. Veluvolu
11
Case 2: λ < 1.0
1 + 2Λ( s) +
Λ 2 ( s)
λ



Λ ( s)
Λ ( s)
= 1 +
1+



2
2
 λ + λ − λ  λ − λ − λ 
The factor with complex constant λ + λ 2 − λ
the gain margin calculated from
K u1I =
determine the stability limit
λ + λ2 − λ
ωu 1 I
g11
both phase and gain affected, for FOPDT process, can be approximated.
ku1 I
(
 tan −1 1 − 1

λ
= λ 1 −
π


Multivariable Control
)  k


(
u1
.  1.5 tan −1 1 − 1

λ
ωu1I =  1 −
π


Dr. Kalyana C. Veluvolu
)  ω


u1
12
Case 2: Correction factor
λ < 1.0
For 0.5 < λ < 1.0
ωu1I = ( 0.85λ + 0.1) ωu1

 ku1I = ( 0.62λ + 0.3) ku1
Check Table 3.2.1 of the
lecture notes for the
summary of the detuning
rules for all three cases
Multivariable Control
Dr. Kalyana C. Veluvolu
13
Example: WB Distillation Column
 12.8e − s

G ( s ) = 16.7 s−+7 s1
 6.6e
10.9 s + 1
− 18.9e −3 s 

21.0 s + 1 
− 19.4e −3 s 
14.4 s + 1 
 2.0 − 1.0
Λ=

− 1.0 2.0 
other loop open
Multivariable Control
NI > 0
other loop closed
Dr. Kalyana C. Veluvolu
14
Example: Continued
dynamic of the two loops are almost the same
λ>1=2
controller gains detuned by
(λ −
)
λ 2 − λ = (2 − 2) ≈ 0.59
Multivariable Control
Dr. Kalyana C. Veluvolu
15
BLM Method Review
Nyquist (Bode or Nichols) plot g (iω ) g c (iω ) made as ω from zero to infinity
Number of encirclements of (-1, 0) equal to number of closed-loop characteristic
equations roots lie in the right half of the s plane if the open-loop system is stable.
If (-1, 0) is encircled, closed-loop system is unstable.
closed-loop log modulus (Lcmax)
Lc = 20 log
gg c
1 + gg c
measure distance g (iω ) g c (iω ) from the (-1, 0) point
Farther away from (-1, 0), more stable
A commonly used specification is +2 dB for Lcmax
Multivariable Control
Dr. Kalyana C. Veluvolu
16
BLM Method for MIMO Processes
Design procedure
1. Calculate the Ziegler-Nichols setting for each individual loop. The ultimate
gain Ku and ultimate frequency ωu of each diagonal transfer function gii ( s )
calculated in SISO way.
g ci ( s ) = K ci (1 +
1
τ Ii s
K ZNi =
)
K ui
2.2
τ ZN =
i
2π
1.2ωui
2. Specify factor F, vary from 2 to 5, all feedback controllers gains (Kc) and
integral constant τ I calculated by correcting as
K ci =
Multivariable Control
K ZNi
τ Ii = Fτ ZN
F
i
Dr. Kalyana C. Veluvolu
17
BLM Method for MIMO Processes
3. Define a scalar function
W ( s ) = −1 + det( I + G ( s )Gc ( s ))
Multivariable closed-loop log modulus Lcm
Lcm = 20 log
W
1+ W
4. For MIMO processes, the best tuning criterion
( Lcm ) max = 2n
Multivariable Control
Dr. Kalyana C. Veluvolu
18
Example: WB Distillation Column
 12.8e − s

G (s ) = 16.7 s−+7 s1
 6.6e
10.9s + 1
− 18.9e −3 s 

21.0s + 1 
−3 s
− 19.4e 
14.4s + 1 
τ i ,ii
Controller
Kc,ii
Loop 1
0.375
8.29
Loop 2
-0.075
23.6
Multivariable Control
( Lcm ) max = 4
Dr. Kalyana C. Veluvolu
19
Structure Decomposition
The transfer function seen by controllers
g1 ( s ) = g11 ( s ) −
g12 ( s ) g21 ( s ) gc 2 ( s )
1 + g22 ( s ) gc 2 ( s )
g2 ( s ) = g22 ( s ) −
g12 ( s ) g21 ( s ) gc1 ( s )
1 + g11 ( s ) gc1 ( s )
The system can be separated into two SISO systems
The tuning of one controller depends on the other controller.
A set-point change on one loop seen as a disturbance by the other.
Multivariable Control
Dr. Kalyana C. Veluvolu
20
Approximation One
g1 ( s ) = g11 ( s ) −
g12 ( s ) g21 ( s ) g c 2 ( s )
1 + g22 ( s ) gc 2 ( s )
g2 ( s ) = g22 ( s ) −
g12 ( s ) g 21 ( s ) g c1 ( s )
1 + g11 ( s ) g c1 ( s )
The diagonal transfer functions contain no unstable zero
Both delay shorter than
L12 (s) + L21 (s)
frequencies lower than the cross-over frequency
g1 ( s ) = g11 ( s ) −
g12 ( s ) g 21 ( s )
g 22 ( s )
for
g 22 ( s ) gc 2 ( s ) > 1
g 2 ( s ) = g 22 ( s ) −
g12 ( s ) g 21 ( s )
g11 ( s )
for
g11 ( s ) gc1 ( s ) > 1
ω co
Argument 2: controllers designed so two loops have perfect control with
infinite bandwidth or infinite control gains
g c1 ( s ) g11 ( s )
= 1;
1 + g c1 ( s ) g11 ( s )
Multivariable Control
g c 2 ( s ) g 22 ( s )
=1
1 + g c 2 ( s ) g 22 ( s )
Dr. Kalyana C. Veluvolu
21
Approximation Two
•The transfer functions contain an unstable zero
•or a delay longer than
L12 ( s) + L21 ( s )
the following approximation can be used
Multivariable Control
g1 ( s ) = g11 ( s ) −
g12 ( s ) g 21 ( s )
K 22
g 2 ( s ) = g 22 ( s ) −
g12 ( s ) g21 ( s )
K11
Dr. Kalyana C. Veluvolu
22
Gain and phase margin design
G(s) =
Gc ( s ) =
b0
e − Ls
a2 s + a1s + 1
gc ( s) = K p +
2
K d s 2 + K p s + Ki
Ki
+ Kd s
s
b
As 2 + Bs + C
=k
⇒ Gc ( s )G ( s ) = k 0 e − sL
s
s
s
Phase cross Over frequency
Am Gc ( jωg )G ( jωg ) = 1
arg Gc ( jω g )G ( jω g )  = −π
Gain cross Over frequency
Φ m = π + arg Gc ( jω p )G ( jω p ) 
Gc ( jω p )G ( jω p ) = 1
Multivariable Control
Dr. Kalyana C. Veluvolu
23
Gain and phase margin design
Gc ( s )G ( s ) = k
k=
 

arg Gc ( jωg ) G ( jωg ) = −π


 Am × Gc ( jωg ) G ( jωg ) = 1

 G ( jω ) G ( jω ) = 1
p
p
 c

Φ m = π + arg Gc ( jω p ) G ( jω p )

b0 − sL
e
s
π
2 Am Lb0
Φm =
PID parameters
Multivariable Control
π
2
(1 −
1
)
Am
⇒ ωg L =
⇒ Am =
2
ωg
k
⇒ k = ωp
⇒ Φm =
K p 
K  = π
 i  2 A Lb
0
m
 K d 
Dr. Kalyana C. Veluvolu
π
π
2
− ωp L
 a1 
1
 
 a2 
24
Example: WB Distillation Column
 12.8e − s

1.Find the equivalent transfer function
G ( s ) = 16.7 s−+7 s1
 6.6e
10.9 s + 1
g12 ( s ) g 21 ( s ) 12.8e − s
6.43(14.4 s + 1)e −7 s
g1 ( s ) = g11 ( s ) −
=
−
g 22 ( s )
16.7 s + 1 (21.0s + 1)(10.9 s + 1)
− 18.9e −3s 

21.0s + 1 
− 19.4e −3s 
14.4 s + 1 
g12 ( s ) g 21 ( s ) −19.4e −3s
9.75(16.7 s + 1)e −9 s
g2 ( s ) = g 22 ( s ) −
=
+
g11 ( s )
14.4 s + 1 (21.0s + 1)(10.9 s + 1)
not FOPTD transfer functions, a model reduction is required. After model
reduction, Controller be designed
Gain and phase margin method: gain margin 3, phase margin π 3
Gc1 ( s) = 0.4660 +
0.1208
s
Multivariable Control
Gc2 ( s) = −0.2122 −
0.0343
s
Dr. Kalyana C. Veluvolu
25
Dr. Kalyana C. Veluvolu
26
Simulation
Detuning method
(dotted line)
better
performance on
load disturbance
rejection.
Independent
design method
(solid line) better
set-point tracking
Multivariable Control