Transfer Function Matrix for Coupling Analysis

Hello! Welcome to your next lesson in our journey through multivariable control systems.

In the last lesson, we established the fundamental concepts of controllability and observability. We learned how to use the state-space matrices $A$ , $B$ , and $C$ to determine if it's theoretically possible to control our system's states and observe them through our chosen sensors. This is a crucial "go/no-go" check before we invest time in designing a controller.

Now that we know control is possible, we face a new challenge inherent to MIMO systems: coupling. In a system like your rotary inverted pendulum, the motor input that swings the arm will inevitably affect the pendulum's angle, and vice-versa. Our goal today is to understand and quantify this interaction.

This lesson directly addresses the learning outcome: Use the transfer function matrix to identify coupling between different input-output channels. We will introduce a powerful tool called the Relative Gain Array (RGA), which helps us make intelligent decisions about how to structure our controller in the face of these cross-couplings.

The Challenge of Input-Output Coupling

In a MIMO system, each input can influence multiple outputs. This is explicitly shown in the transfer function matrix, $G (s)$ , which we learned to derive in a previous module.

Transfer Function Matrix for MIMO Systems — This diagram shows the transfer function matrix $G(s)$ for a system with 'm' inputs and 'p' outputs. The off-diagonal elements, like $G_{12}(s)$, represent the coupling between channels—how input $U_2(s)$ affects output $Y_1(s)$.

This coupling complicates control. Imagine trying to design a simple controller where input $u_{1}$ is responsible for output $y_{1}$ , and $u_{2}$ for $y_{2}$ . When you use $u_{1}$ to adjust $y_{1}$ , the coupling term $G_{21} (s)$ causes an unwanted disturbance in $y_{2}$ , forcing your second controller to react. At best, this leads to sluggish performance; at worst, the two controllers can "fight" each other, leading to instability.

For your EE background, you can think of this as crosstalk between different channels in a communication system. Our goal is to figure out the best way to "wire" our controller to minimize the negative effects of this crosstalk.

The following video provides an excellent visual explanation of this problem.

Multivariable control configurations 2019-04-26

Watch this video segment from the 'Process dynamics and control' channel. It clearly illustrates how coupling creates unintentional interaction loops in a seemingly straightforward 'diagonal' control scheme.

Please watch from the beginning to 4:13. Focus on how a diagonal pairing (controller 1 manipulates input 1 to control output 1) still results in unintentional 'coupling loops' that affect the entire system's behavior.

Quantifying Coupling with the Relative Gain Array (RGA)

To make a smart decision about which input to pair with which output, we need a quantitative measure of the interactions. This is precisely what the Relative Gain Array (RGA) provides.

The RGA was developed by E.H. Bristol to solve this exact problem. It's an analytical tool that helps us choose the best pairings of manipulated variables (inputs, $u$ ) and controlled variables (outputs, $y$ ).

The core idea of a relative gain, $λ_{ij}$ , is to compare the effect of an input on an output under two conditions:

When all other control loops are "open" (i.e., other inputs are held constant).
When all other control loops are "closed" (i.e., other outputs are held constant by their own perfect controllers).

The relative gain is the ratio of these two effects:

λ_{ij} = \frac{Gain from u _{j} to y _{i} with other loops open}{Gain from u _{j} to y _{i} with other loops closed}

This comparison tells us how much the interaction from other control loops changes the fundamental gain between a given input-output pair.

To learn more about the fundamentals of RGA, please review the following resource.

MIMO control using RGA

Read the introduction and the 'What is RGA?' sections of this LibreTexts article to solidify your understanding of the purpose of RGA.

Read from the start down to the 'Understanding the Results of the RGA' subheading. This will introduce the concept and the matrix notation for RGA.

Calculating the RGA

While the conceptual definition is useful, it's not practical for calculations. A more direct method uses the steady-state gain matrix, $K_{p}$ . This matrix is simply the transfer function matrix $G (s)$ evaluated at $s = 0$ , which represents the DC gain of the system.

Given $y (s) = G (s) u (s)$ , the steady-state relationship is $y = K_{p} u$ , where $K_{p} = G (0)$ .

The RGA matrix, denoted by the capital Greek letter Lambda ( $Λ$ ), can be calculated directly from $K_{p}$ using the following formula:

Λ = K_{p} \otimes (K_{p}^{- 1})^{T}

Let's break this down:

$K_{p}$ is the $n \times n$ steady-state gain matrix.
$K_{p}^{- 1}$ is the standard matrix inverse of $K_{p}$ .
$(K_{p}^{- 1})^{T}$ is the transpose of the inverse.
$\otimes$ denotes the element-by-element product (also known as the Hadamard or Schur product), not standard matrix multiplication. This means you multiply the corresponding elements of the two matrices: $(Λ)_{ij} = (K_{p})_{ij} \times ((K_{p}^{- 1})^{T})_{ij}$ .

The resource you just read explains this calculation method.

MIMO control using RGA

Now, continue reading the 'MIMO control using RGA' article to see how to perform the calculation.

Read the section titled 'Method 2: Calculating RGA with Steady-State Gain Matrix'. Focus on the formula \lambda_{ij}=g_{ij} r_{ij} (where their 'g' is our 'K' and 'r' is our (K_p^{-1})^T) and the overall process.

A Shortcut for 2x2 Systems

For a 2x2 system, the RGA has a very useful property: the sum of the elements in any row or column is always 1. This means if you calculate just one element, say $λ_{11}$ , you can find all the others immediately.

If $Λ = [λ_{11} λ_{21} λ_{12} λ_{22}]$ , then:

$λ_{12} = 1 - λ_{11}$
$λ_{21} = 1 - λ_{11}$
$λ_{22} = λ_{11}$

So the RGA matrix becomes:

Λ = [λ_{11} 1 - λ_{11} 1 - λ_{11} λ_{11}]

And for a 2x2 system, $λ_{11}$ can be calculated with the formula:

λ_{11} = \frac{1}{1 - \frac{K _{12} K _{21}}{K _{11} K _{22}}}

Interpreting the RGA for Controller Pairing

Once you have the $Λ$ matrix, you can use it to decide on the best input-output pairings. The rules are straightforward:

Pair inputs $u_{j}$ and outputs $y_{i}$ corresponding to RGA elements $λ_{ij}$ that are positive and as close to 1 as possible.
A value of $λ_{ij} \approx 1$ means that the gain of the $y_{i} \leftarrow u_{j}$ loop is largely unaffected by other control loops. This is the ideal pairing.
A value of $λ_{ij} \approx 0$ indicates that $u_{j}$ has little to no effect on $y_{i}$ when other loops are open. This is a poor pairing choice.
A value of $λ_{ij} = 0.5$ indicates a high degree of interaction between the loops.
A negative value ( $λ_{ij} < 0$ ) is a strong warning. This implies that closing the other loops will cause the gain of the $y_{i} \leftarrow u_{j}$ loop to change sign. A controller designed for a positive gain would cause instability if the gain suddenly became negative. Avoid pairings with negative RGA values.

The following slides and video provide clear summaries and examples of these interpretation rules.

RELATIVE GAIN ARRAY (RGA; BRISTOL MATRIX)

These presentation slides offer a very concise summary of the RGA calculation for a 2x2 system and the rules for interpretation. They provide a clear, final recommendation.

Review slides 400-402. Slide 400 shows the derivation for the 2x2 case. Slides 401-402 explain how to interpret the results and provide the main recommendation for loop coupling.

Relative Gain Array RGA and Input Output Pairing

This video segment explains the properties of the RGA matrix and how to use its values to make pairing decisions.

Watch from 7:47 to 12:03. This section introduces the RGA matrix properties for a square system (rows/columns sum to 1) and walks through an example of interpreting the RGA values to select the best pairings.

Test your understanding!

A 2x2 MIMO system has the following steady-state gain matrix:

K_{p} = [21 - 1.8 - 1]

Calculate the RGA matrix $Λ$ .
Based on the RGA, what is the recommended controller pairing? (i.e., should you pair $u_{1} \to y_{1}, u_{2} \to y_{2}$ or $u_{1} \to y_{2}, u_{2} \to y_{1}$ )?

Show answer

First, let's find $λ_{11}$ .
$λ_{11} = \frac{1}{1 - \frac{K _{12} K _{21}}{K _{11} K _{22}}} = \frac{1}{1 - \frac{( - 1.8 ) ( 1 )}{( 2 ) ( - 1 )}} = \frac{1}{1 - \frac{- 1.8}{- 2}} = \frac{1}{1 - 0.9} = \frac{1}{0.1} = 10$
Since this is a 2x2 system, the full RGA matrix is:
$Λ = [10 - 9 - 9 10]$
The RGA recommends pairing based on the elements closest to 1. In this case, $λ_{11} = 10$ and $λ_{22} = 10$ are positive, while $λ_{12}$ and $λ_{21}$ are large and negative. The strong positive values on the diagonal suggest a diagonal pairing.

Therefore, the recommended pairing is $u_{1} \to y_{1}$ and $u_{2} \to y_{2}$ . However, the large value of $λ_{11} = 10$ indicates that the loops are still highly interactive. While this is the best pairing, we should expect significant coupling effects.

Conclusion

Today, we've tackled the critical issue of coupling in MIMO systems. You've learned how the off-diagonal terms in the transfer function matrix represent unwanted interactions and how these can complicate controller design.

Key Takeaways:

Coupling is the phenomenon where a single input affects multiple outputs, creating "crosstalk" between control loops.
The Relative Gain Array (RGA) is a powerful tool for quantifying steady-state coupling.
The RGA is calculated from the steady-state gain matrix $K_{p} = G (0)$ using the formula $Λ = K_{p} \otimes (K_{p}^{- 1})^{T}$ .
Interpretation is key: We should pair inputs and outputs that correspond to RGA elements that are positive and close to 1. We must avoid pairings with negative RGA values to prevent potential instability.

This analysis provides a systematic way to decide on a system's fundamental control structure before you begin designing the controllers themselves. For many systems, this means deciding on a "decentralized" or "diagonal" control scheme, where $u_{i}$ is tasked with controlling $y_{i}$ .

Preview of the Next Lesson:
Armed with this knowledge of system interactions, we are now ready to move into controller design. In the next module, we will begin with one of the most fundamental techniques in modern control: state feedback via pole placement. You will learn how to select desired closed-loop pole locations to meet performance specifications and calculate the feedback gain matrix $K$ to place the system's poles at those desired locations. The RGA analysis we did today helps justify the initial structure of that feedback matrix.