ssm-contraction-control

name: ssm-contraction-control description: "Controller design for Structured State-space Models (SSMs) using contraction theory with indirect data-driven output feedback. Use when: (1) designing controllers for nonlinear systems identified via SSM surrogate models, (2) implementing contraction-based stabilization with Linear Matrix Inequality (LMI) conditions, (3) establishing separation principle for observer-controller design, (4) applying scalable control design to time-series and dynamical systems. First controllability/observability analysis of SSMs with contraction theory framework."

SSM Controller Design via Contraction Theory

Indirect data-driven output feedback controller synthesis for nonlinear systems using Structured State-space Models (SSMs) as surrogate models.

Why SSMs?

Advantages over Transformers

Linear computational complexity: O(L) vs O(L²) for sequence length L
Long-term dependency capture: Efficient modeling of extended sequences
Scalable control design: LMI-based synthesis with tractable computation

State-space Structure

h_t = A h_{t-1} + B x_t
y_t = C h_t + D x_t

where:

h_t = hidden state (continuous-time dynamics)
x_t = input
y_t = output
A, B, C, D = learnable parameters

Core Contributions

1. Controllability Analysis

First SSM controllability characterization:

Linear-time-invariant (LTI) structure
Reachability condition: rank([B, AB, ..., A^(n-1)B]) = n
Connection to sequence learning capacity

2. Observability Analysis

First SSM observability characterization:

Output injectivity condition
rank([C^T, A^T C^T, ..., (A^T)^(n-1) C^T]) = n
Relation to state reconstruction

3. Separation Principle

Independent observer + controller design:

Observer: State estimation from outputs
Controller: State-feedback stabilization
Closed-loop stability preserved when designed independently

4. Contraction-Based Design

LMI conditions for exponential stability:

Incremental stability via contraction metric
Scalable synthesis: solve LMIs per subsystem
Robustness to bounded disturbances

Control Framework

Surrogate Model Learning

Collect data from nonlinear system
Train SSM as surrogate model
Extract learned parameters (A, B, C, D)
Apply control design to SSM

Contraction Theory Basics

Definition: System is contracting if all trajectories converge exponentially.

Metric condition:

M > 0,  A^T M A - M ≤ -Q (Q > 0)

Incremental stability:

|x(t) - y(t)| ≤ e^(-λt) |x(0) - y(0)|

LMI Controller Synthesis

Objective: Find K such that (A + BK) is contracting.

LMI formulation:

Find: M > 0, K, Y
Subject to:
  (A + BK)^T M (A + BK) - M < 0
  Y = K M

Scalable: Solve per subsystem, aggregate via contraction theory.

Observer Design

Luenberger observer:

ĥ_t = A ĥ_{t-1} + B x_t + L(y_t - C ĥ_t)

LMI condition:

(A - LC)^T M (A - LC) - M < 0

Implementation Workflow

Step 1: System Identification

Collect input-output trajectories
Train SSM via gradient descent
Validate model accuracy

Step 2: Controllability/Observability Check

Compute controllability matrix
Compute observability matrix
Verify rank conditions

Step 3: Controller Synthesis

Formulate contraction LMI
Solve for feedback gain K
Verify closed-loop contraction

Step 4: Observer Synthesis

Formulate observer LMI
Solve for observer gain L
Verify estimation contraction

Step 5: Output Feedback Control

Combine observer + controller
Apply separation principle
Verify exponential stability

Key Results

Theorem 1 (Controllability): SSM is controllable iff controllability matrix has full rank.

Theorem 2 (Observability): SSM is observable iff observability matrix has full rank.

Theorem 3 (Separation): Independent observer-controller design preserves exponential stability when both are contracting.

Theorem 4 (Scalability): Large-scale SSMs decompose into subsystems with local LMIs → tractable synthesis.

Advantages

Data-driven: No explicit physics model required
Scalable: Linear complexity for long sequences
Systematic: LMI-based with guaranteed stability
Robust: Contraction implies disturbance rejection

Applications

Time-series control: Regulate dynamical sequences
Robotics: Motion planning with learned dynamics
Process control: Chemical plant regulation
Economic systems: Market stabilization
Network control: Multi-agent coordination

References

arXiv: 2604.07069v1 - "Controller Design for Structured State-space Models via Contraction Theory"
Authors: Muhammad Zakwan, Vaibhav Gupta, Alireza Karimi, Efe C. Balta, Giancarlo Ferrari-Trecate
Published: 2026-04-08
PDF: papers/2026-04-09/ssm-contraction.pdf