Linear parameter-varying control: Difference between revisions

Revision as of 18:33, 21 April 2024

Revision as of 17:39, 18 June 2024

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===Parameter dependent systems===

In [[control engineering]], a [[state-space representation]] is a [[mathematical model]] of a physical system as a set of input, <math>u</math> output, <math>y</math> and [[State variable|state]] variables, <math>x</math> related by first-order [[Differential equation|differential]] equations. The dynamic evolution of a [[nonlinear]]{{disambiguation needed|date=April 2024}}, non-[[autonomous]] system is represented by

::<math>\dot{x} = f(x,u,t)</math>

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::<math>x(t_0)=x_0 ,u(t_0)=u_0</math>

The state variables describe the mathematical "state" of a [[dynamical system]] and in modeling large complex [[nonlinear]]{{disambiguation needed|date=April 2024}} systems if such state variables are chosen to be compact for the sake of practicality and simplicity, then parts of dynamic evolution of system are missing. The state space description will involve other variables called exogenous [[Variable (mathematics)|variables]] whose evolution is not understood or is too complicated to be modeled but affect the state variables evolution in a known manner and are measurable in real-time using [[sensors]].

When a large number of sensors are used, some of these sensors measure outputs in the system theoretic sense as known, [[wikt:explicit|explicit]] nonlinear functions of the modeled states and time, while other sensors are accurate estimates of the exogenous variables. Hence, the model will be a time varying, nonlinear system, with the future time variation unknown, but measured by the sensors in real-time.

In this case, if <math>w(t),w</math> denotes the exogenous variable [[Vector (mathematics and physics)|vector]], and <math>x(t)</math> denotes the modeled state, then the state equations are written as