⚡ electrical engineering

Analysis of Non-Square Nonlinear MIMO Systems using Scaled Relative Graphs

This paper extends Scaled Relative Graph (SRG) analysis to non-square nonlinear MIMO systems by embedding operators into a common Hilbert space while restricting input dimensions to avoid conservatism, thereby enabling stability analysis and $L_2$ -gain bounds for a broad class of systems including those in Linear Fractional Representation (LFR) form.

Original authors: Julius P. J. Krebbekx, Roland Tóth, Amritam Das

Published 2026-04-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Julius P. J. Krebbekx, Roland Tóth, Amritam Das

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an engineer trying to keep a complex machine running smoothly. This machine has many inputs (like buttons you press) and many outputs (like lights that turn on or wheels that spin). In the world of control engineering, we call this a MIMO system (Multiple-Input, Multiple-Output).

For a long time, engineers had a fantastic toolkit to analyze these machines, but it only worked for "square" machines—where the number of buttons exactly matched the number of lights. If you had 3 buttons and 4 lights, the old tools broke down. They couldn't tell you if the machine would stay stable or if it would go haywire.

This paper introduces a new, super-powered tool called the Scaled Relative Graph (SRG) that finally works for these "non-square" machines. Here is how the authors solved the problem, explained with some everyday analogies.

1. The Problem: The "Square Peg in a Round Hole" Dilemma

Think of the old SRG method as a square cookie cutter. It works perfectly if your dough (the system) is also square. But what if your dough is a rectangle?

The Old Way: To use the square cutter, engineers used to just chop off the extra dough or stuff in extra "fake" dough to make it square.
- The Flaw: If you chop off the extra, you lose information. If you stuff in fake dough, you make the machine look "heavier" and more dangerous than it really is. This leads to conservatism: you might think a machine is unsafe and shut it down, even though it's actually perfectly fine.

2. The Solution: The "Magic Frame" (Embedding)

The authors came up with a clever trick. Instead of forcing the rectangular dough to become square, they built a special frame that holds the dough exactly as it is, but allows the square cookie cutter to work on it without distortion.

The Analogy: Imagine you have a tall, thin painting (a "tall" system) and a wide, short painting (a "wide" system). You want to hang them both in a gallery that only accepts square frames.
- The Naive Approach: You stretch the tall painting to fill a square frame (making it look weird) or you crop the wide painting (losing parts of the image).
- The Authors' Approach: They put the paintings in a custom frame that matches the gallery's square requirement, but they add a "black border" (zeros) to the empty space. Crucially, they tell the gallery inspector: "Ignore the black border. Only look at the actual painting."

This is the core of their math: Embedding. They mathematically add "dummy" inputs or outputs to make the system look square to the SRG tool, but they strictly restrict the analysis to the original inputs. This ensures they don't get fooled by the dummy data.

3. The Result: A Clearer Picture of Stability

By using this "Magic Frame," the authors can now:

See the Truth: They can calculate exactly how much "gain" (amplification) the system has. In the old method, the fake data would make the system look like it amplifies signals wildly, causing a false alarm. The new method ignores the fake data, giving a much tighter, more accurate safety margin.
Connect the Dots: They developed a set of rules (like a grammar for these graphs) that lets you combine different parts of a machine. If you have a linear part (like a spring) and a non-linear part (like a rubber band that gets stiff), you can now draw their SRGs and see if they will play nice together or crash.

4. Real-World Examples

The paper tests this on four different scenarios, like a mechanic testing a new engine:

The Lur'e System: A classic control setup. The authors showed that their method proves the system is stable even when the old "naive" method said it was unstable. It's like proving a bridge is safe when the old calculator said it would collapse.
Multiple Nonlinearities: A system with several "rubber bands" (non-linearities). Their method gave a much tighter safety bound than previous techniques, meaning engineers can push the machine harder without fear.
Mass-Spring-Damper: A system of weights and springs. They modeled a complex interaction where a spring pushes back (negative stiffness) for small movements. Their tool handled this tricky, non-square setup perfectly.
Comparison: They compared their results to other advanced methods (like IQC) and found their approach was just as good, but more general and easier to apply to non-square systems.

The Big Takeaway

Before this paper, if you had a control system where the number of inputs didn't match the number of outputs, you were flying blind or using tools that were too pessimistic.

This paper provides a universal adapter. It allows engineers to use powerful, visual, frequency-domain tools (like the SRG) on any system, regardless of its shape. It removes the "fake data" that used to scare engineers, allowing them to design systems that are both safer and more efficient.

In short: They figured out how to measure a rectangle using a square ruler without cutting off the corners or padding the edges, giving us a much more accurate way to keep our complex machines running safely.

1. Problem Statement

The paper addresses a significant gap in the analysis of Multiple-Input Multiple-Output (MIMO) nonlinear systems, specifically those that are non-square (i.e., the number of inputs $p$ differs from the number of outputs $q$ ).

Context: The Scaled Relative Graph (SRG) is a powerful graphical frequency-domain tool for analyzing nonlinear Single-Input Single-Output (SISO) systems, offering non-approximative stability conditions and $L_2$ -gain bounds. Recent efforts extended SRG to MIMO systems, but these were restricted to square systems ( $p=q$ ) due to the inherent Hilbert space structure of the SRG, which requires the input and output spaces to be identical.
The Challenge: When $p \neq q$ , the input and output Hilbert spaces differ. Naive approaches to handle this (e.g., "squaring up" the system by adding artificial zero inputs or outputs) introduce conservatism. This artificial expansion can destroy properties like passivity or fail to guarantee stability, leading to overly pessimistic or incorrect analysis results.
Objective: To develop a rigorous SRG framework that accommodates non-square operators without introducing conservatism, applicable to systems in Linear Fractional Representation (LFR) form.

2. Methodology

The authors propose a novel embedding strategy combined with restricted input spaces to generalize SRG analysis to non-square MIMO systems.

A. Operator Embedding with Restricted Input Spaces

The core innovation is embedding non-square operators into a common space of square operators acting on a larger Hilbert space ( $L_n^2$ where $n = \max\{p, q\}$ ), while strictly restricting the input space to the original dimension.

Wide Operators ( $p > q$ ): The output is embedded into $L_p^2$ by appending zero outputs. The SRG is computed on the original input space.
Tall Operators ( $p < q$ ): The input is embedded into $L_q^2$ by appending zero inputs. Crucially, the SRG is computed only on the subspace where these artificial inputs are zero. This prevents the "artificial" inputs from influencing the gain calculation, thereby avoiding the conservatism found in standard squaring-up methods.
Definition: The MIMO SRG is formally defined via these embeddings, ensuring that the radius of the SRG corresponds exactly to the incremental $L_2$ -gain of the original non-square operator.

B. Generalized Interconnection Rules

The paper generalizes the SRG calculus (sum, product, inverse) from square systems to these restricted-domain MIMO operators.

Proposition 1: Extends rules for scaling, addition, inversion, and composition to operators acting on subspaces.
Chord and Arc Properties: The authors introduce "improved" chord and arc completions (Lemma 9) to handle parallel and series interconnections more tightly, reducing conservatism when combining SRG bounds.

C. Stability Theorems for LFR Systems

The framework is applied to systems in Linear Fractional Representation (LFR) form, where a stable LTI block ( $G$ ) is feedback-connected to a static or dynamic nonlinear block ( $\Phi$ ).

Theorem 1 (Simple Feedback): Provides conditions for incremental stability and well-posedness of a feedback loop $[H_1, H_2]$ based on the distance between the SRG of $H_1$ and the scaled SRG of $H_2$ .
Theorem 2 (LFR Stability): A practical theorem for LFR systems. It guarantees well-posedness, causality, and incremental $L_2$ -gain bounds by analyzing the interconnection of the embedded LTI and nonlinear blocks. It utilizes homotopy arguments to ensure stability across the entire range of nonlinearity sectors.

D. Computational Formulas

The paper provides explicit algorithms to compute MIMO SRGs for:

Stable LTI Operators: An exact computation method using $H_\infty$ norms of shifted transfer matrices (Theorem 3). This generalizes previous square-only methods to tall/wide systems.
Static Nonlinearities: Bounds for diagonal static nonlinearities and specific non-square nonlinear blocks (Propositions 2 & 3).

3. Key Contributions

Non-Square SRG Definition: A rigorous definition of the SRG for non-square MIMO operators that avoids the conservatism of naive "squaring up" techniques by restricting the input space during embedding.
Generalized Calculus: Extension of SRG interconnection rules (sum, product, inverse) to operators with mismatched input/output dimensions and restricted domains.
Stability Theorems: New stability theorems (Theorems 1 & 2) that guarantee causality, well-posedness, and incremental $L_2$ -gain bounds for non-square nonlinear feedback systems.
Computational Tools: Explicit formulas for computing SRGs of non-square LTI systems and static nonlinearities, implemented in a publicly available Julia toolbox (SrgTools.jl).
LFR Framework: A comprehensive analysis framework for systems in LFR form, which is a standard representation in robust control and nonlinear system identification.

4. Results and Examples

The authors validate their theory through four examples, demonstrating reduced conservatism and improved performance compared to existing methods:

Non-Square Lur'e System:
- Comparison: Compared the proposed embedding method against a naive "squaring up" approach (adding non-zero artificial inputs/outputs).
- Result: The proposed method yielded a significantly tighter $L_2$ -gain bound (e.g., $\gamma \leq 0.4$ vs. $\gamma \leq 12.5$ in one case) and correctly identified stability where the naive method failed. It also preserved the passivity property of the system, which was lost in the naive approach.
SISO System with Multiple Nonlinearities:
- Comparison: Analyzed a system with two nonlinearities using the proposed MIMO SRG approach versus the SISO SRG approach from [15].
- Result: The MIMO approach provided a tighter incremental gain bound ($2.33$ vs. $4.81$) and rigorously proved causality and well-posedness, which were previously assumed in the SISO analysis.
MIMO Mass-Spring-Damper System:
- Application: A system with two masses, linear/damper elements, and multiple nonlinear springs.
- Result: Successfully analyzed the stability and gain bounds of the complex non-square interconnection, demonstrating the method's applicability to physical mechanical systems.
Comparison with IQC-Based Results:
- Comparison: Analyzed a MIMO feedback system studied in [7] using Integral Quadratic Constraints (IQC).
- Result: The SRG-based analysis produced results identical to the IQC method for the inner loop but provided a significantly tighter gain bound for the overall system ($1.79$ vs. $\approx 4.05$ ).

5. Significance

This work represents a major advancement in nonlinear control theory by bridging the gap between graphical frequency-domain analysis and multivariable non-square systems.

Theoretical Impact: It resolves the theoretical limitation of SRGs being restricted to square operators, providing a mathematically sound framework for general MIMO nonlinear systems.
Practical Impact: By eliminating the conservatism associated with artificial "squaring up," engineers can obtain tighter performance bounds and verify stability for a wider class of systems (e.g., under-actuated systems, systems with sensor/actuator mismatches) that were previously difficult to analyze using SRG methods.
Tooling: The provision of a software toolbox makes these advanced theoretical results accessible for practical control design and analysis.

In summary, the paper successfully extends the powerful Scaled Relative Graph framework to the general case of non-square MIMO nonlinear systems, offering a less conservative, rigorous, and computationally tractable method for stability and performance analysis.