# Least-Squares Parameter Estimation for State-Space Models with State Equality Constraints

## TL;DR

This paper develops a method for incorporating state equality constraints into least-squares estimation of state-space models, handling both known and uncertain constraints in time-invariant and time-varying systems.

## Contribution

It reformulates state equality constraints as matrix constraints and applies vectorized least squares, extending to uncertain constraints and dynamic systems.

## Key findings

- Effective incorporation of constraints improves estimation accuracy.
- Method handles both known and uncertain constraints.
- Applicable to time-invariant and time-varying systems.

## Abstract

If a dynamic system has active constraints on the state vector and they are known, then taking them into account during modeling is often advantageous. Unfortunately, in the constrained discrete-time state-space estimation, the state equality constraint is defined for a parameter matrix and not on a parameter vector as commonly found in regression problems. To address this problem, firstly, we show how to rewrite the state equality constraints as equality constraints on the state matrices to be estimated. Then, we vectorize the matricial least squares problem defined for modeling state-space systems such that any method from the equality-constrained least squares framework may be employed. Both time-invariant and time-varying cases are considered as well as the case where the state equality constraint is not exactly known.