Before starting with Linear Analysis

Linear Analysis is a powerful analysis tool to directly characterize the dynamic behavior of your system independently from its real inputs. It is completely different from the traditional time-domain analysis where dynamic behavior is deduced from the analysis of the system outputs as time-responses to given inputs in time-domain.

Let's consider a non-linear system in time-domain, with its input and outputs expressed as a function of time. A schematic representation is the following one:

Figure 1: Non linear system in Time-Domain - Input(s) and Output(s)

In order to represent such system using linear differential equations, it becomes necessary to linearize the system equations.

Consider first the simpler case of a non-linear function Y=f(x) as shown in figure Figure 2.

Figure 2: Linearization of a nonlinear function

Assume that it is necessary to operate in the vicinity of point a on the curve (operating point). For small perturbations X and Y around the operating point a, the slope at the operating point a is:

And the approximate linear relationship of the equivalent linear system becomes:

A quite similar process is applied in Simcenter Amesim to the complete model, differentiating all system functions regarding all variables to obtain the matrices representing the equivalent linear system.

Figure 3: Simplified view of the Linearization process in Simcenter Amesim around an operation point, corresponding to the specified LA Times t1 and t2

The linearized system could be represented either with its transfer function H(s) in the Laplace formalism, or with the usual [A, B, C, D] state-space representation:

Figure 4: Linearized system in Frequency-domain - Inputs(s) and Outputs(s)

This last [A, B, C, D] state-space representation is adopted in Simcenter Amesim, and it is stored in the *._jac0 jacobian file.

with:

• x: vector of the state variables (states of the system)

• u: vector of the control variables (input vector)

• y: vector of the observer variables (output vector)

While one could think that this [A, B, C, D] state-space representation is quite complex to understand, it is actually only the linearized system represented by linear matrices [A, B, C, D] that link its inputs U(s) and to its output Y(s).

Practically it is completely invisible in Simcenter Amesim for users who can just focus on the practical use of the available Linear Analysis tools, as described below.