ZigangPan/ NonlinearSystems

This is a paclet on nonlinear systems with the ultimate goal of robust MRAC for linear systems

Contributed by: Zigang Pan

This paclet introduces a ten-item list to represent a nonlinear time-invariant system in state space representation. With this special format, one can do all sorts of things for a nonlinear system, such as convert a linear system in special format (as described in ZigangPan`LinearSystems`) to a nonlinear system, linearize a nonlinear system around an operating point to obtain a LTI system, do system operations on the nonlinear system just as for a LTI system, block diagonal, concatenate, parallel, and feedback interconnect two nonlinear systems, etc. One can calculate the relative degree of the nonlinear system and dynamically extend a nonlinear system to one with vector relative degree.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["ZigangPan/NonlinearSystems"]

To load the code after installation, evaluate this code:
Needs["ZigangPan`NonlinearSystems`"]

Details

Version 1.1: Speed up in simulation of nonlinear systems using the RungeKutta45 method using the function simulationNLsystem.

Paclet Guide

This paclet implementation of nonlinear system is a bit versatile than the one offered in Mathematica.

NLsystemcheck

— check a given variable is in the special system representation (i.e., the ten-item list format).

NLsystemoperation

— Do various operations on a LTI system in the special ten-item format. Such as:
View -- view the system in a table like format
Organize -- sort the state, input, and output names and list the system in a more organized way
Organize and View -- do both of above
Add active inputs
Add active outputs
Drop active inputs
Drop active outputs
Set active inputs
Set active outputs
Add control inputs
Drop control inputs
Add disturbance inputs
Drop disturbance inputs
....
State transformation -- do state transformation using the pure transformation function provided. new states = transformation[old states]. The inverse of the transformation (a pure function) may be supplied as the third argument.
Drop states

NLsystemblockdiagonal

— build a bigger system that consists of two nonlinear systems in block diagonal composition.

NLsystemconcatenate

— concatenate two nonlinear systems in special format.

NLsystemparallel

— parallel interconnect two nonlinear systems in special format such that they share the common active inputs.

NLsystemfeedback

—

feedbackinterconnecttwononlinear

systems in special format.

simulationNLsystem

— simulate an nonlinear system in feedback with a nonlinear controller and a nonlinear disturbance system.

emptyNLsystem

— generate an empty nonlinear system in special format.

sivewavesystem

— generate a nonlinear system in special format that produces a stable sinusoidal output with specified amplitude and frequency. ▪

convert2NLsystem

— convert a LTI system in the special ten-item list to an equivalent nonlinear system in the special ten-item list (not the same ten-item list).

linearization

— linearize a nonlinear system around an operating point, and gives the linearization as an LTI system in special ten-item list.

NLcalculaterelativedegree

— calculate the relative degrees of a nonlinear system in the special format and determine whether it admits vector relative degree.

NLdynamicextension

— do dynamic extension on a nonlinear system in special format to reach (uniform) vector relative degree.

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NLsystemblockdiagonal

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NLcalculaterelativedegree

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NLdynamicextension

Examples

Initialization for Examples

In[1]:=

Basic Examples

In[2]:=

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Out[3]=

In[4]:=

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Out[7]=

In[8]:=

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In[9]:=

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