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

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

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

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)

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.

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

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.

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

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

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

Results: RGA Gramian based methods ILQIA 1 235065 342149 263431 2 γ 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

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.

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)

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.

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

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

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.

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.

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.)

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

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.