Up to this point, almost all design examples were effectively single-input–single-output (SISO): one control signal, one measured output, and classical tools such as Bode plots, Nyquist plots, and root locus were applied to scalar loop transfer functions.

Many physical systems, however, have multiple actuators and multiple sensor outputs. Let \( \mathbf{u}(t)\in\mathbb{R}^m \) be the vector of inputs and \( \mathbf{y}(t)\in\mathbb{R}^p \) the vector of outputs. For a linear time-invariant (LTI) plant, taking Laplace transforms yields

\[ \mathbf{y}(s) = \mathbf{G}(s)\,\mathbf{u}(s),\qquad \mathbf{G}(s) = \begin{bmatrix} G_{11}(s) & \cdots & G_{1m}(s) \\ \vdots & \ddots & \vdots \\ G_{p1}(s) & \cdots & G_{pm}(s) \end{bmatrix}. \]

The off-diagonal entries \( G_{ij}(s) \) with \( i\neq j \) represent coupling: input \( u_j \) directly influences output \( y_i \). When off-diagonal dynamics have magnitude comparable to or larger than the diagonal terms, the system is said to be strongly coupled.

Classical “loop-by-loop” design treats each pair \( (u_i,y_i) \) independently, assuming all other loops are either open or act as disturbances. In strongly coupled plants, this approximation is poor: changing the controller on one channel can profoundly alter the dynamics seen on another.

flowchart TD
  R["Reference vector r (r1,...,rp)"] --> C["Controller matrix C(s)"]
  C --> U["Actuator signals u (u1,...,um)"]
  U --> G["Plant matrix G(s) with strong off-diagonal terms"]
  G --> Y["Outputs y (y1,...,yp)"]
  Y --> E["Error signals e = r - y"]
  E --> C
  G --- COUP["Coupling: each ui affects multiple yi"]
        

This lesson focuses on why linear SISO tools are insufficient in such cases, not on providing a complete multivariable design methodology (which is the subject of modern control and robust control courses).

Consider a unity-feedback configuration where \( \mathbf{r} \) is a vector of set points, \( \mathbf{y} \) the measured output, and the controller is a matrix of transfer functions \( \mathbf{C}(s) \in \mathbb{C}^{m\times p} \). For simplicity we assume square systems \( m=p \) and noiseless measurements:

\[ \mathbf{u}(s) = \mathbf{C}(s)\big(\mathbf{r}(s) - \mathbf{y}(s)\big),\qquad \mathbf{y}(s) = \mathbf{G}(s)\,\mathbf{u}(s). \]

Eliminating \( \mathbf{u}(s) \) gives

\[ \mathbf{y}(s) = \mathbf{G}(s)\mathbf{C}(s)\big(\mathbf{r}(s)-\mathbf{y}(s)\big) = \mathbf{L}(s)\big(\mathbf{r}(s)-\mathbf{y}(s)\big), \quad \mathbf{L}(s) \triangleq \mathbf{G}(s)\mathbf{C}(s). \]

Rearranging:

\[ \big(\mathbf{I} + \mathbf{L}(s)\big)\mathbf{y}(s) = \mathbf{L}(s)\mathbf{r}(s), \qquad \mathbf{y}(s) = \big(\mathbf{I}+\mathbf{L}(s)\big)^{-1}\mathbf{L}(s)\mathbf{r}(s). \]

By analogy with the SISO case, we define the multivariable sensitivity and complementary sensitivity matrices

\[ \mathbf{S}(s) \triangleq \big(\mathbf{I}+\mathbf{L}(s)\big)^{-1},\qquad \mathbf{T}(s) \triangleq \mathbf{L}(s)\big(\mathbf{I}+\mathbf{L}(s)\big)^{-1}. \]

Then

\[ \mathbf{y}(s) = \mathbf{T}(s)\,\mathbf{r}(s),\qquad \mathbf{e}(s) = \mathbf{r}(s) - \mathbf{y}(s) = \mathbf{S}(s)\,\mathbf{r}(s). \]

In the scalar case, \( S(s) \) and \( T(s) \) are just complex numbers for each frequency. In the MIMO case they are matrices. Their entries tell how each component of \( \mathbf{r} \) affects each component of \( \mathbf{e} \) and \( \mathbf{y} \), while their matrix norms capture worst-case directions in which disturbances or references have the largest effect.

A key limitation of classical SISO reasoning is that it usually examines only the diagonal entries of \( \mathbf{S}(s) \) and \( \mathbf{T}(s) \). In strongly coupled plants, large off-diagonal entries can dominate the behavior of the closed loop even if all diagonal SISO loops look “well designed.”

The simplest multi-loop structure is a decentralized controller, where each loop has its own SISO controller and there is no dynamic cross-coupling inside the controller:

\[ \mathbf{C}(s) = \begin{bmatrix} C_{11}(s) & 0 \\ 0 & C_{22}(s) \end{bmatrix}. \]

Consider the symmetric, coupled 2×2 plant

\[ \mathbf{G}(s) = \frac{1}{s+1} \begin{bmatrix} 1 & \alpha \\ \alpha & 1 \end{bmatrix}, \qquad 0\leq\alpha\leq 1, \]

where \( \alpha \) measures coupling strength. Let the decentralized controller be purely proportional:

\[ \mathbf{C}(s) = k \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},\qquad k>0. \]

The loop transfer matrix is

\[ \mathbf{L}(s) = \mathbf{G}(s)\mathbf{C}(s) = \frac{k}{s+1} \begin{bmatrix} 1 & \alpha \\ \alpha & 1 \end{bmatrix}. \]

The sensitivity matrix is

\[ \mathbf{S}(s) = \big(\mathbf{I}+\mathbf{L}(s)\big)^{-1} = \frac{1}{\Delta(s)} \begin{bmatrix} 1 + \frac{k}{s+1} & -\frac{k\alpha}{s+1} \\ -\frac{k\alpha}{s+1} & 1 + \frac{k}{s+1} \end{bmatrix}, \]

with

\[ \Delta(s) = \left(1+\frac{k}{s+1}\right)^2 -\left(\frac{k\alpha}{s+1}\right)^2 = 1 + \frac{2k}{s+1} + \frac{k^2(1-\alpha^2)}{(s+1)^2}. \]

Even though the controller is diagonal, \( \mathbf{S}(s) \) is not diagonal as soon as \( \alpha\neq 0 \). A step change in the first reference component \( r_1 \) will, through the off-diagonal element \( S_{21}(s) \), induce a nonzero error component at the second loop. This is the core of interaction: each loop “feels” changes in the other loop.

As \( \alpha \) approaches 1 (strong coupling) and \( k \) increases, the determinant \( \Delta(s) \) can become very small near certain frequencies, making both diagonal and off-diagonal entries of \( \mathbf{S}(s) \) large. This implies:

• large amplification of reference changes and disturbances,
• high sensitivity of the closed loop to small modeling errors,
• loss of robustness, even though each SISO loop looks acceptable when analyzed in isolation.

flowchart TD
  Dstart["Design each loop separately"] --> L1["Tune C11(s) from y1/u1 as if SISO"]
  Dstart --> L2["Tune C22(s) from y2/u2 as if SISO"]
  L1 --> CL["Close both loops simultaneously"]
  L2 --> CL
  CL --> INT["Strong coupling: off-diagonal Gij(s) large"]
  INT --> OBS1["Unexpected overshoot / \noscillation in y2 from step in r1"]
  INT --> OBS2["Controller changes in \none loop degrade other loop"]
  OBS1 --> LIM["Conclusion: decentralized SISO design \nis limited for strongly coupled MIMO"]
        

For a given frequency \( \omega \), the complex matrix \( \mathbf{G}(\mathrm{j}\omega) \) maps input sinusoidal vectors \( \mathbf{u}(\mathrm{j}\omega) \) to outputs \( \mathbf{y}(\mathrm{j}\omega) = \mathbf{G}(\mathrm{j}\omega)\mathbf{u}(\mathrm{j}\omega) \). The amplification of a unit-norm input direction \( \mathbf{u} \) is

\[ a(\omega,\mathbf{u}) \triangleq \frac{\big\|\mathbf{G}(\mathrm{j}\omega)\mathbf{u}\big\|_2} {\|\mathbf{u}\|_2},\qquad \mathbf{u}\neq \mathbf{0}. \]

The induced 2-norm of \( \mathbf{G}(\mathrm{j}\omega) \) is defined as

\[ \big\|\mathbf{G}(\mathrm{j}\omega)\big\|_2 \triangleq \sup_{\mathbf{u}\neq \mathbf{0}} \frac{\big\|\mathbf{G}(\mathrm{j}\omega)\mathbf{u}\big\|_2} {\|\mathbf{u}\|_2}. \]

A fundamental result from linear algebra is that this supremum equals the largest singular value of \( \mathbf{G}(\mathrm{j}\omega) \).

\[ \big\|\mathbf{G}(\mathrm{j}\omega)\big\|_2 = \sigma_{\max}\!\big(\mathbf{G}(\mathrm{j}\omega)\big), \]

where \( \sigma_{\max} \) is the largest eigenvalue of \( \big(\mathbf{G}(\mathrm{j}\omega)\big)^{\ast}\mathbf{G}(\mathrm{j}\omega) \) under a square root.

Sketch of proof.

• For fixed \( \omega \), define the Hermitian, positive semidefinite matrix \( \mathbf{H} = \mathbf{G}^{\ast}\mathbf{G} \).
• For a unit vector \( \mathbf{u} \) one has \( \big\|\mathbf{G}\mathbf{u}\big\|_2^2 = \mathbf{u}^{\ast}\mathbf{H}\mathbf{u} \). The quantity \( \mathbf{u}^{\ast}\mathbf{H}\mathbf{u} \) is the Rayleigh quotient of \( \mathbf{H} \) at \( \mathbf{u} \).
• Linear algebra shows that \( \lambda_{\min}(\mathbf{H}) \leq \mathbf{u}^{\ast}\mathbf{H}\mathbf{u} \leq \lambda_{\max}(\mathbf{H}) \), and the maximum is achieved for an eigenvector corresponding to \( \lambda_{\max}(\mathbf{H}) \).
• Therefore \( \big\|\mathbf{G}\mathbf{u}\big\|_2^2 \leq \lambda_{\max}(\mathbf{H}) \), and the supremum of the ratio is \( \sqrt{\lambda_{\max}(\mathbf{H})} = \sigma_{\max}(\mathbf{G}) \).

In SISO control, we examine the magnitude \( |G(\mathrm{j}\omega)| \). In MIMO, \( \sigma_{\max}\big(\mathbf{G}(\mathrm{j}\omega)\big) \) plays an analogous role: it is the largest possible amplification over all input directions. Classical gain and phase margins computed per diagonal entry fail to capture dangerous directions in which the plant has very large gain.

This is one of the main motivations for multivariable robustness notions such as singular-value plots of \( \mathbf{L}(\mathrm{j}\omega) \), \( \mathbf{S}(\mathrm{j}\omega) \), and \( \mathbf{T}(\mathrm{j}\omega) \), studied in more advanced courses.

A practical design question in MIMO control is: which input should be paired with which output? For a 2×2 plant, should we design controllers between \( (u_1,y_1) \) and \( (u_2,y_2) \), or between \( (u_1,y_2) \) and \( (u_2,y_1) \)? The relative gain array (RGA) provides a steady-state measure of loop interaction.

Let \( \mathbf{G}(0) \) be the static (DC) gain matrix of the plant, assumed square and nonsingular. The RGA \( \boldsymbol{\Lambda} \) is defined elementwise as

\[ \boldsymbol{\Lambda} \triangleq \mathbf{G}(0)\circ\big(\mathbf{G}(0)^{-T}\big), \qquad \Lambda_{ij} = G_{ij}(0)\,\big(\mathbf{G}(0)^{-T}\big)_{ij}, \]

where \( \circ \) denotes elementwise (Hadamard) product. For a 2×2 matrix \( \mathbf{G}(0) = \begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix} \) with nonzero determinant, one can show

\[ \boldsymbol{\Lambda} = \frac{1}{g_{11}g_{22}-g_{12}g_{21}} \begin{bmatrix} g_{11}g_{22} & -g_{12}g_{21} \\ -g_{21}g_{12} & g_{22}g_{11} \end{bmatrix}. \]

A useful property is that each row and each column of \( \boldsymbol{\Lambda} \) sums to 1. For example,

\[ \Lambda_{11} + \Lambda_{12} = \frac{g_{11}g_{22}-g_{12}g_{21}}{g_{11}g_{22}-g_{12}g_{21}} = 1. \]

For pairing decisions, the following rules of thumb are widely used:

• Pair \( y_i \) with the input \( u_j \) for which \( \Lambda_{ij} \) is positive and closest to 1.
• Values of \( \Lambda_{ij} \) much larger than 1 in magnitude indicate strong interaction and poor pairing (large effective loop gain at high decentralization).
• Negative \( \Lambda_{ij} \) indicate sign changes of effective gain when closing other loops, which often leads to severe robustness problems.

The RGA thus quantifies limitations of simple decentralized linear control: if no pairing yields \( \Lambda_{ij} \) values near 1, then any purely diagonal controller will struggle, and true multivariable design (or explicit decoupling networks) become necessary.

We now simulate the 2×2 coupled plant from Section 3 using Python and the python-control library. We compare weak coupling (\( \alpha = 0.2 \)) and strong coupling (\( \alpha = 0.9 \)) under identical decentralized proportional controllers.

import numpy as np
import matplotlib.pyplot as plt
import control as ctl

# 2x2 coupled plant: G(s) = 1/(s+1) * [[1, alpha], [alpha, 1]]
s = ctl.TransferFunction.s

def make_plant(alpha):
    g11 = 1/(s + 1)
    g22 = 1/(s + 1)
    g12 = alpha/(s + 1)
    g21 = alpha/(s + 1)
    G = ctl.TransferFunction([[g11, g12],
                              [g21, g22]])
    return G

def closed_loop(alpha, k):
    G = make_plant(alpha)
    # Decentralized proportional controller: C(s) = k * I
    C = ctl.TransferFunction([[k, 0],
                              [0, k]])
    L = G * C
    # Closed-loop from r to y: T(s) = L (I + L)^(-1)
    I2 = ctl.tf2ss(ctl.TransferFunction([[1, 0],
                                         [0, 1]]))
    T = ctl.feedback(L, I2)  # feedback(L, I) = L (I+L)^(-1)
    return T

k = 2.0
alphas = [0.2, 0.9]
t = np.linspace(0, 10, 1000)

# Step in r1, r2 = 0
r1 = np.ones_like(t)
r2 = np.zeros_like(t)
R = np.vstack((r1, r2))

plt.figure()
for alpha in alphas:
    T = closed_loop(alpha, k)
    tout, y, _ = ctl.forced_response(T, T=t, U=R)
    plt.plot(tout, y[0, :], label=f"y1, alpha={alpha}")
    plt.plot(tout, y[1, :], "--", label=f"y2, alpha={alpha}")

plt.xlabel("Time (s)")
plt.ylabel("Outputs y1, y2")
plt.title("Step in r1 with decentralized control for different coupling alpha")
plt.legend()
plt.grid(True)
plt.show()
      

For small \( \alpha \), the cross-effect on \( y_2 \) is limited and the loop behaves roughly as expected from SISO design. For large \( \alpha \), the step in \( r_1 \) produces substantial excursion in \( y_2 \), illustrating the interaction problem and the limits of fully decentralized linear control for strongly coupled plants.

We implement a simple forward-Euler integration of the same plant in state-space form. With internal state \( \mathbf{x}(t)\in\mathbb{R}^2 \) we choose

\[ \dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t),\qquad \mathbf{y}(t) = \mathbf{C}\mathbf{x}(t), \]

with

\[ \mathbf{A} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix},\quad \mathbf{B} = \begin{bmatrix} 1 & \alpha \\ \alpha & 1 \end{bmatrix},\quad \mathbf{C} = \mathbf{I}_2. \]

This state-space model has the same transfer matrix as the plant in Section 3. The C++ code below simulates the closed loop with a proportional decentralized controller.

#include <iostream>
#include <cmath>

int main() {
    double alpha = 0.9;   // coupling strength
    double k = 2.0;       // proportional gain
    double T = 10.0;      // simulation horizon (s)
    double dt = 1e-3;     // time step
    int steps = static_cast<int>(T / dt);

    // State x = [x1; x2], outputs y = x (C = I)
    double x1 = 0.0, x2 = 0.0;

    // Reference: step in r1, r2 = 0
    double r1 = 1.0, r2 = 0.0;

    for (int i = 0; i < steps; ++i) {
        double t = i * dt;

        // Outputs
        double y1 = x1;
        double y2 = x2;

        // Decentralized proportional control u = K (r - y)
        double e1 = r1 - y1;
        double e2 = r2 - y2;
        double u1 = k * e1;
        double u2 = k * e2;

        // State derivatives: xdot = A x + B u
        double x1dot = -x1 + 1.0 * u1 + alpha * u2;
        double x2dot = -x2 + alpha * u1 + 1.0 * u2;

        // Forward Euler step
        x1 += dt * x1dot;
        x2 += dt * x2dot;

        // Print a coarse sample for plotting later
        if (i % 100 == 0) {
            std::cout << t << " " << y1 << " " << y2 << "\n";
        }
    }

    return 0;
}
      

By post-processing the printed data (for example in Python, MATLAB, or GNUplot), one can compare \( y_1(t) \) and \( y_2(t) \) for different \( \alpha \). As \( \alpha \) increases, \( y_2(t) \) deviates more from zero, emphasizing the strong coupling in the plant and the limitations of independent loop tuning.

A similar simulation can be written in Java using plain arrays and a simple integration loop. The structure mirrors the C++ example.

public class CoupledPlantSimulation {
    public static void main(String[] args) {
        double alpha = 0.9;
        double k = 2.0;
        double T = 10.0;
        double dt = 1e-3;
        int steps = (int)(T / dt);

        double x1 = 0.0, x2 = 0.0;
        double r1 = 1.0, r2 = 0.0;

        for (int i = 0; i < steps; i++) {
            double t = i * dt;

            double y1 = x1;
            double y2 = x2;

            double e1 = r1 - y1;
            double e2 = r2 - y2;

            double u1 = k * e1;
            double u2 = k * e2;

            double x1dot = -x1 + 1.0 * u1 + alpha * u2;
            double x2dot = -x2 + alpha * u1 + 1.0 * u2;

            x1 += dt * x1dot;
            x2 += dt * x2dot;

            if (i % 100 == 0) {
                System.out.println(t + " " + y1 + " " + y2);
            }
        }
    }
}
      

Libraries such as EJML or Apache Commons Math can be introduced in further courses to handle larger MIMO systems with more systematic matrix operations, but the essential limitation illustrated here is independent of the numerical tool: strong coupling reduces the effectiveness of purely diagonal linear controllers.

Using MATLAB’s Control System Toolbox, we can work directly with MIMO transfer-function matrices and visualize step responses for different coupling strengths.

% Parameters
alpha_values = [0.2, 0.9];
k = 2;

s = tf('s');

for idx = 1:numel(alpha_values)
    alpha = alpha_values(idx);

    % 2x2 coupled plant: G(s) = 1/(s+1)*[1 alpha; alpha 1]
    G11 = 1/(s + 1);
    G22 = 1/(s + 1);
    G12 = alpha/(s + 1);
    G21 = alpha/(s + 1);
    G = [G11 G12; G21 G22];

    % Decentralized proportional controller
    C = k * eye(2);

    % Loop transfer and closed-loop from r to y
    L = G * C;
    I2 = eye(2);
    T = feedback(L, I2);  % T(s) = L(I+L)^(-1)

    % Step in r1 only
    t = 0:0.01:10;
    r = [ones(size(t)); zeros(size(t))];
    [y, t_out] = lsim(T, r', t);

    figure;
    plot(t_out, y(:,1), 'b', 'LineWidth', 1.5); hold on;
    plot(t_out, y(:,2), 'r--', 'LineWidth', 1.5);
    grid on;
    xlabel('Time (s)');
    ylabel('Outputs');
    title(sprintf('Step in r1, alpha = %.2f', alpha));
    legend('y1', 'y2');
end
      

In Simulink, the same system can be built using two State-Space or Transfer Fcn blocks representing the plant matrix, along with vector signals for \( \mathbf{r} \) and \( \mathbf{y} \), and a diagonal gain block implementing the controller. The observed cross-coupling in the step responses clearly demonstrates the limitations of classical SISO-based design workflows.

Wolfram Mathematica provides high-level support for multivariable state-space models. The following code constructs the 2×2 coupled plant and simulates a step in the first reference.

alpha = 0.9;
k = 2.0;

(* State-space matrices *)
a = { {-1, 0}, {0, -1} };
b = { {1, alpha}, {alpha, 1} };
c = IdentityMatrix[2];
d = { {0, 0}, {0, 0} };

plant = StateSpaceModel[{a, b, c, d}];

(* Decentralized proportional controller: C(s) = k I *)
kMat = k IdentityMatrix[2];
controller = StateSpaceModel[{ConstantArray[{0, 0}, {2, 2}],
                              ConstantArray[{0, 0}, {2, 2}],
                              kMat, ConstantArray[{0, 0}, {2, 2}]}];

(* Series interconnection and unity feedback *)
loop = SystemsModelSeries[plant, controller];
closed = SystemsModelFeedback[loop];

tmax = 10;
resp = OutputResponse[closed, {UnitStep[t], 0}, {t, 0, tmax}];

Plot[Evaluate[{resp[[1]], resp[[2]]}], {t, 0, tmax},
     PlotLegends -> {"y1(t)", "y2(t)"},
     AxesLabel -> {"t", "y"},
     PlotRange -> All,
     GridLines -> Automatic]
      

By varying alpha and k, one can explore how strong coupling amplifies the response on the uncommanded output, again emphasizing the limitations of simple decentralized linear controllers in MIMO settings.

Problem 1 (Deriving MIMO Sensitivity and Complementary Sensitivity):
Consider a square MIMO LTI plant \( \mathbf{G}(s) \) with controller \( \mathbf{C}(s) \) in unity feedback. Starting from \( \mathbf{u} = \mathbf{C}(\mathbf{r}-\mathbf{y}) \) and \( \mathbf{y} = \mathbf{G}\mathbf{u} \), derive the expressions for the multivariable sensitivity and complementary sensitivity matrices \( \mathbf{S}(s) \) and \( \mathbf{T}(s) \).

Solution:

From the equations

\[ \mathbf{u}(s) = \mathbf{C}(s)\big(\mathbf{r}(s)-\mathbf{y}(s)\big),\qquad \mathbf{y}(s) = \mathbf{G}(s)\,\mathbf{u}(s), \]

substitute \( \mathbf{u}(s) \) into the second equation:

\[ \mathbf{y}(s) = \mathbf{G}(s)\mathbf{C}(s)\big(\mathbf{r}(s)-\mathbf{y}(s)\big) = \mathbf{L}(s)\mathbf{r}(s) - \mathbf{L}(s)\mathbf{y}(s), \]

where \( \mathbf{L}(s) \triangleq \mathbf{G}(s)\mathbf{C}(s) \). Collecting terms with \( \mathbf{y}(s) \) on the left-hand side:

\[ \big(\mathbf{I}+\mathbf{L}(s)\big)\mathbf{y}(s) = \mathbf{L}(s)\mathbf{r}(s). \]

Assuming \( \mathbf{I}+\mathbf{L}(s) \) is invertible at the frequencies of interest,

\[ \mathbf{y}(s) = \big(\mathbf{I}+\mathbf{L}(s)\big)^{-1}\mathbf{L}(s)\mathbf{r}(s). \]

The error is \( \mathbf{e}(s) = \mathbf{r}(s)-\mathbf{y}(s) \), so

\[ \mathbf{e}(s) = \Big(\mathbf{I} - \big(\mathbf{I}+\mathbf{L}(s)\big)^{-1}\mathbf{L}(s)\Big)\mathbf{r}(s) = \big(\mathbf{I}+\mathbf{L}(s)\big)^{-1}\mathbf{r}(s). \]

This suggests the definitions

\[ \mathbf{S}(s) \triangleq \big(\mathbf{I}+\mathbf{L}(s)\big)^{-1},\qquad \mathbf{T}(s) \triangleq \mathbf{L}(s)\big(\mathbf{I}+\mathbf{L}(s)\big)^{-1}, \]

with \( \mathbf{e}(s) = \mathbf{S}(s)\mathbf{r}(s) \) and \( \mathbf{y}(s) = \mathbf{T}(s)\mathbf{r}(s) \).

Problem 2 (Off-Diagonal Sensitivity in the 2×2 Example):
For the 2×2 plant and decentralized proportional controller of Section 3, show that the off-diagonal sensitivity element \( S_{21}(s) \) is given by

\[ S_{21}(s) = -\frac{\dfrac{k\alpha}{s+1}}{ 1 + \dfrac{2k}{s+1} + \dfrac{k^2(1-\alpha^2)}{(s+1)^2} }. \]

Solution:
From Section 3,

\[ \mathbf{S}(s) = \frac{1}{\Delta(s)} \begin{bmatrix} 1 + \dfrac{k}{s+1} & -\dfrac{k\alpha}{s+1} \\ -\dfrac{k\alpha}{s+1} & 1 + \dfrac{k}{s+1} \end{bmatrix}, \quad \Delta(s) = 1 + \frac{2k}{s+1} + \frac{k^2(1-\alpha^2)}{(s+1)^2}. \]

The element in row 2, column 1 is \( S_{21}(s) = -\dfrac{k\alpha}{s+1}/\Delta(s) \), which matches the stated expression. Unless \( \alpha = 0 \), this term is nonzero, confirming that reference changes in the first loop induce error in the second loop.

Problem 3 (Induced Norm and Singular Values):
Let \( \mathbf{M}\in\mathbb{C}^{p\times m} \) and define the induced 2-norm \( \|\mathbf{M}\|_2 = \sup_{\mathbf{u}\neq 0} \|\mathbf{M}\mathbf{u}\|_2/\|\mathbf{u}\|_2 \). Show that \( \|\mathbf{M}\|_2^2 = \lambda_{\max}\big(\mathbf{M}^{\ast}\mathbf{M}\big) \), where \( \lambda_{\max} \) denotes the largest eigenvalue of \( \mathbf{M}^{\ast}\mathbf{M} \).

Solution:

For any nonzero \( \mathbf{u} \),

\[ \frac{\|\mathbf{M}\mathbf{u}\|_2^2}{\|\mathbf{u}\|_2^2} = \frac{\mathbf{u}^{\ast}\mathbf{M}^{\ast}\mathbf{M}\mathbf{u}} {\mathbf{u}^{\ast}\mathbf{u}} = R_{\mathbf{M}^{\ast}\mathbf{M}}(\mathbf{u}), \]

the Rayleigh quotient of \( \mathbf{M}^{\ast}\mathbf{M} \) at \( \mathbf{u} \). Spectral theory for Hermitian matrices states \( \lambda_{\min} \leq R_{\mathbf{M}^{\ast}\mathbf{M}}(\mathbf{u}) \leq \lambda_{\max} \), and the maximum value \( \lambda_{\max} \) is achieved for eigenvectors associated with \( \lambda_{\max} \).

Therefore \( \sup_{\mathbf{u}\neq 0} \|\mathbf{M}\mathbf{u}\|_2^2/\|\mathbf{u}\|_2^2 = \lambda_{\max}(\mathbf{M}^{\ast}\mathbf{M}) \), and taking square roots yields \( \|\mathbf{M}\|_2 = \sqrt{\lambda_{\max}(\mathbf{M}^{\ast}\mathbf{M})} \).

Problem 4 (RGA of a Simple 2×2 Plant):
Let \( \mathbf{G}(0) = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \). Compute the RGA \( \boldsymbol{\Lambda} \) and comment on the preferred pairing.

Solution:

First compute \( \det\mathbf{G}(0) = 2\cdot 2 - 1\cdot 1 = 3 \). The inverse is

\[ \mathbf{G}(0)^{-1} = \frac{1}{3} \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}, \qquad \mathbf{G}(0)^{-T} = \mathbf{G}(0)^{-1}, \]

because the matrix is symmetric. Then the RGA is

\[ \boldsymbol{\Lambda} = \mathbf{G}(0)\circ \mathbf{G}(0)^{-T} = \begin{bmatrix} 2\cdot \tfrac{2}{3} & 1\cdot \left(-\tfrac{1}{3}\right) \\ 1\cdot \left(-\tfrac{1}{3}\right) & 2\cdot \tfrac{2}{3} \end{bmatrix} = \begin{bmatrix} \tfrac{4}{3} & -\tfrac{1}{3} \\ -\tfrac{1}{3} & \tfrac{4}{3} \end{bmatrix}. \]

Since \( \Lambda_{11} = \Lambda_{22} = 4/3 \gt 1 \) and \( \Lambda_{12} = \Lambda_{21} = -1/3 \lt 0 \), all potential pairings are problematic: pairing along the diagonal yields effective static gains larger than expected from SISO design, and off-diagonal pairings yield sign changes. This suggests that a purely diagonal controller is not well suited and that multivariable or decoupling strategies should be considered.

Problem 5 (Effect of Coupling on Closed-Loop Poles):
For the coupled 2×2 plant and decentralized proportional controller of Section 3, derive the characteristic polynomial of the closed loop and show that it is second order, with coefficients depending on \( k \) and \( \alpha \).

Solution:

The closed-loop characteristic equation is given by \( \det\big(\mathbf{I}+\mathbf{L}(s)\big) = 0 \), with

\[ \mathbf{L}(s) = \frac{k}{s+1} \begin{bmatrix} 1 & \alpha \\ \alpha & 1 \end{bmatrix}. \]

Thus

\[ \mathbf{I}+\mathbf{L}(s) = \begin{bmatrix} 1 + \dfrac{k}{s+1} & \dfrac{k\alpha}{s+1} \\ \dfrac{k\alpha}{s+1} & 1 + \dfrac{k}{s+1} \end{bmatrix}, \]

and

\[ \det\big(\mathbf{I}+\mathbf{L}(s)\big) = \left(1+\frac{k}{s+1}\right)^2 - \left(\frac{k\alpha}{s+1}\right)^2 = 1 + \frac{2k}{s+1} + \frac{k^2(1-\alpha^2)}{(s+1)^2}. \]

Multiplying by \( (s+1)^2 \) yields the polynomial

\[ (s+1)^2 + 2k(s+1) + k^2(1-\alpha^2) = s^2 + 2(1+k)s + \big(1 + 2k + k^2(1-\alpha^2)\big). \]

Therefore, the closed-loop poles are the roots of this second-order polynomial. As \( \alpha \) approaches 1 and \( k \) grows, the constant term becomes small, yielding lightly damped modes (poles closer to the imaginary axis), which in turn increases overshoot and sensitivity. This illustrates how strong coupling constrains achievable closed-loop performance even when the plant and controller are both linear.

In this lesson we examined the limitations of classical linear control in the presence of strongly coupled multi-input–multi-output plants. We extended the SISO notions of loop transfer, sensitivity, and complementary sensitivity to matrices, and saw that even purely diagonal controllers produce non-diagonal sensitivity matrices whenever the plant is coupled.

Using linear-algebra tools such as matrix norms, singular values, and the relative gain array, we showed how interaction and directionality of gain make SISO-based robustness interpretations unreliable for MIMO systems. Numerical simulations in Python, C++, Java, MATLAB/Simulink, and Wolfram Mathematica demonstrated that increasing coupling leads to larger cross-effects and reduced robustness, even though each loop may appear acceptable when analyzed alone.

These observations motivate the development of modern multivariable control methodologies (state-space design, optimal control, robust control), which explicitly handle coupling, directionality, and uncertainty using matrix-valued tools. Those advanced topics build naturally on the mathematical structures introduced in this lesson.