# Observability and Controllability of Second Order Linear Time Invariant Systems and Kalman Type Conditions

## TL;DR

This paper extends classical controllability and observability conditions to second order linear time-invariant systems without reducing them to first order, using Kalman-type criteria for both discrete and continuous cases.

## Contribution

It generalizes Kalman conditions directly for second order systems, deriving new algebraic criteria and transfer functions without system reduction.

## Key findings

- Controllability and observability matrices must have full rank for system properties.
- Derived transfer function for second order continuous-time systems.
- Numerical example confirms theoretical results.

## Abstract

In the present paper we consider controllability and observability of second order linear time invariant systems in matrix form. Without reducing into first order systems we show how the classical conditions for first order linear systems can be generalized to this case. In term of Kalman type criterions these concepts are investigated for second order discrete and continuous time linear systems. It should be pointed out that by repeated differentiation of state and output vector-functions we derive two different systems of linear algebraic equations. Then the initial values x_0, x_1 and input functions can be determined uniquely from these systems if and only if the observability and controllability matrices have full rank, respectively. Also the transfer function of the second order continuous-time linear state-space system is constructed. A numerical example is given to illustrate the feasibility and effectiveness of the theoretic results obtained.

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Source: https://tomesphere.com/paper/1906.06741