MIMO Systems: Decentralized Control

In class activities

Author

Ranjeet Utikar

Published

September 17, 2023

Modified

September 22, 2025

Activities

Explain what is controller pairings in a decentralized control strategy.
Describe the factors that affect the controller pairings.
Explain how decoupling control work and its advantages and limitations.
Consider the multivariable processes below and suggest suitable controller pairings based on steady-state RGA and dynamic RGA (DRGA) for each of the process.

2x2 MIMO

$\begin{matrix} (1) & G (s) = [\begin{array}{cc} \frac{5 e^{- 3 s}}{4 s + 1} & \frac{2.5 e^{- 5 s}}{15 s + 1} \\ \frac{- 4 e^{- 6 s}}{20 s + 1} & \frac{e^{- s}}{5 s + 1} \end{array}] \end{matrix}$

RGA is given by

$\begin{matrix} (2) & λ_{11} = \frac{k_{11}}{k_{11} - \frac{k_{12} k_{21}}{k_{22}}} = \frac{1}{1 - \frac{k_{12} k_{21}}{k_{11} k_{22}}} \end{matrix}$

Solution

The code for calculating RGA is given in Matlab file/ mlx file.

A simulink implementation using lti system block is in relative_gain_array_example_lti_system.slx.

2x2 MIMO

$\begin{matrix} (3) & G = [\begin{array}{cc} \frac{1.7 e^{- 2 s}}{10 s + 1} & \frac{1.1 e^{- 3 s}}{15 s + 1} \\ \frac{- 1.2 e^{- 2.5 s}}{11 s + 1} & \frac{1.4 e^{- s}}{8 s + 1} \end{array}] \end{matrix}$

Solution

The code for calculating RGA is given in Matlab file/ mlx file.

Steady-state gain matrix

$\begin{matrix} (4) & K = [\begin{array}{ccc} 0.7 & 0.3 & - 0.4 \\ 1.1 & - 0.6 & - 0.2 \\ 0.2 & 0.5 & - 0.9 \end{array}] \end{matrix}$

Solution

The code for calculating RGA is given in Matlab file/ mlx file.

3x3 MIMO

$\begin{matrix} (5) & G (s) = [\begin{array}{ccc} \frac{119 e^{- 5 s}}{21.7 s + 1} & \frac{153 e^{- 5 s}}{337 s + 1} & \frac{- 2.1 e^{- 5 s}}{10 s + 1} \\ \frac{37 e^{- 5 s}}{500 s + 1} & \frac{76.7 e^{- 5 s}}{28 s + 1} & \frac{- 5 e^{- 5 s}}{10 s + 1} \\ \frac{93 e^{- 5 s}}{500 s + 1} & \frac{- 66.7 e^{- 5 s}}{166 s + 1} & \frac{- 103.3 e^{- 5 s}}{23 s + 1} \end{array}] \end{matrix}$

Solution

The code for calculating RGA is given in Matlab file/ mlx file.

2x2 MIMO

$\begin{array}{ll} G_{11} (s) = \frac{1.0}{100 s + 1} & G_{12} (s) = \frac{0.3}{10 s + 1} \\ G_{21} (s) = \frac{- 0.4}{10 s + 1} & G_{22} (s) = \frac{2.0}{100 s + 1} \end{array}$

Solution

The code for calculating RGA is given in Matlab file/ mlx file.

Citation

BibTeX citation:

@online{utikar2023,
  author = {Utikar, Ranjeet},
  title = {MIMO {Systems:} {Decentralized} {Control}},
  date = {2023-09-17},
  url = {https://amc.smilelab.dev/content/notes/08-decentralized_control/in-class-activities.html},
  langid = {en}
}

For attribution, please cite this work as:

Utikar, Ranjeet. 2023. “MIMO Systems: Decentralized Control.” September 17, 2023. https://amc.smilelab.dev/content/notes/08-decentralized_control/in-class-activities.html.