1. Fredrik Bengtsson, Torsten Wik & Elin Svensson
Resolving issues of scaling for gramian based input-output pairing methods
Fredrik Bengtsson, Torsten Wik & Elin Svensson

2. Chalmers University of Technology
Input output pairing
Matching manipulated variables with controlled variables for decoupled control.
11/7/2018
Chalmers University of Technology

3. Chalmers University of Technology
Gramian based pairing methods
Here: PM, HIIA and Σ2
Based on
controllability and observability gramians
information about entire frequency range
For design of decentralized and sparse control configurations
Affected by input and output scaling
11/7/2018
Chalmers University of Technology

4. Gramian based pairing methods
Given a transfer function matrix:
each measure generates an interaction matrix (IM), with
using the Hilbert-Schmidt norm (PM), Hankel norm (HIIA) and 2-norm (Σ2)

5. Gramian based pairing methods
Decentralized pairing: Choose the pairing that yields the largest sum of elements from the IM.
Significant interactions not included in the decentralized pairing can be included in a sparse control scheme via feedforward or by using MIMO controllers on part of the system.

6. Scaling
Interaction matrix for gramian based measures depend on input and output scaling.
Generally inputs and outputs scaled from 0 to 1

7. A Heat exchanger network example
Controlled temperatures: T1-T4
Manipulated variables: Bypasses on heat exchangers U1-U4.
T5 is assumed to be controlled further down stream.

8. Chalmers University of Technology
Determining the controller configuration for the Heat Exchanger Network
Heat exchanger model: Series of mixing tanks
Heat is transferred from a mixing tank on the hot side to a corresponding tank on the cold side.
System linearized using Matlab/Simulink.
11/7/2018
Chalmers University of Technology

9. Finding Control configurations
Control configurations found using
PM, HIIA and Σ2 (gramian based measures)
RGA and ILQIA (for comparison)
RGA PM
HIIA Σ2
ILQIA T1 U3 U1 T2 U4 T3 U2 T4

10. Chalmers University of Technology
Comparing the control configurations
For each control structure:
PI controllers designed for each pairing using Lambda method (λ = γT, γ from 1 to 15)
Test with
reference steps on each output
disturbances on each of the inputs temperature and flow rate
Cost defined as: sum of squares of the outputs’ deviation from the reference.
11/7/2018
Chalmers University of Technology

11. Results: RGA Gramian based methods ILQIA 1 235065 342149 263431 2γ
RGA
Gramian based methods
ILQIA 1
235065
342149
263431 2
136606
210002
204696 3
95950
126502
156832
3,5
61002
83096
120940 4
28570
63220
101968
4,5
16528
41376
85213 5
7595
16443
69522
5,5
1788
4771
53765 6
1966
3533
35830
6,5
2154
3282
21232 7
2344
3235
9054
7,5
2537
3274
1570 8
2732
3358
990 10
3526
3897
1064 15
5568
5706
1687

12. Examining the interaction matrices
Examining the second column :
U2 seems best suited for controlling T4
Elements in this column are very small => little emphasis on U2 relative to U1, U3 and U4
However:
Gramian based measures all suggest using U2 to control T3
Non gramian based measures both used U2 to control T4, and yielded better results.

13. Conclusions Small amount of interaction in second column
=> gramian based measures place little emphasis on U2
Scaling issue (despite conventional scaling of system inputs and outputs)

14. Proposed Modifications
Scaling previously suggested for Σ2 method:
Each element in the IM is divided by the sum of all the elements in either its column or row.
Ensures that either each input or each output is given equal weight.

15. Scaling using the Sinkhorn Knopp algorithm
Alternating between normalizing the rows and normalizing the columns of the IM
=> Converge to a matrix where both the column and row sum is one.
Ensures that all inputs and outputs are given equal weight
Results independent of original scaling

16. Testing the new scaling methods
RGA
PM column Scaling
HIIA column scaling Σ2
column scaling PM
row Scaling
HIIA row scaling
row scaling
SK-scaling
HIIA SK-scaling
ILQIA T1 U3 U1 U2 T2 U4 T3 T4
Column and SK scaling all yield improved results, while no improvement is visible with
row scaling

17. Comparison of scaling methods
MIMO system generator: large number of 5by5 MIMO systems.
Different scaling methods were applied to the generated models and compared to standard scaling.

18. The evaluation
Lambda controllers with varying values of λ were implemented for each of the suggested systems.
The entire feedback system was then tested both in reference step and in load disturbance
A cost was defined as the squared deviation from the reference
For each configuration the cost was calculated for values of λ ranging from 0.1T to 10T and the lowest cost was saved.

19. The evaluation Each IM is given a score: where S = score of the IM
c = the IMs cost
cmin = is the lowest cost of all IMs for the system.
(Unstable closed loop systems were given a score of zero.)

20. Results for 150 systems Load Disturbance Reference Step No scaling
Column scaling
Row scaling
Row/column scaling
Sinkhorn-Knopp scaling PM
0.38
0.54
0.59
0.64 Σ2
0.37
0.53
0.57
HIIA
0.52
0.61
0.60
0.63
0.66
Load Disturbance
No scaling
Column scaling
Row scaling
Row/column scaling
Sinkhorn-Knopp scaling PM
0.42
0.61
0.62
0.66
0.71 Σ2
0.64
0.67
0.75
HIIA
0.57
0.70
0.74

21. Conclusion
All the scaling schemes yielded statistically significant improvements, both for reference steps and load disturbances.
Scaling using Sinkhorn Knopp algorithm yielded the largest improvement.
Systems with other properties were tested with similar results.
Sparse controllers were also tested with similar results.