How to linearizing a nonlinear system of differential equations?
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Linearization is the process of taking the gradient of a nonlinear function with respect to all variables and creating a linear representation at that point. It is required for certain types of analysis such as stability analysis, solution with a Laplace transform, and to put the model into linear state-space form. Consider a nonlinear differential equation model that is derived from balance equations with input u and output y.
dydt=f(y,u)dydt=f(y,u)
The right hand side of the equation is linearized by a Taylor series expansion, using only the first two terms.
dydt=f(y,u)≈f(¯y,¯u)+∂f∂y∣∣∣¯y,¯u(y−¯y)+∂f∂u∣∣∣¯y,¯u(u−¯u)dydt=f(y,u)≈f(y¯,u¯)+∂f∂y|y¯,u¯(y−y¯)+∂f∂u|y¯,u¯(u−u¯)
If the values of ¯uu¯ and ¯yy¯ are chosen at steady state conditions then f(¯y,¯u)=0f(y¯,u¯)=0 because the derivative term dydu=0dydu=0 at steady state. To simplify the final linearized expression, deviation variables are defined as y'=y−¯yy′=y-y¯ and u'=u−¯uu′=u-u¯. A deviation variable is a change from the nominal steady state conditions. The derivatives of the deviation variable is defined as dy'dt=dydtdy′dt=dydt because d¯ydt=0dy¯dt=0 in dy'dt=d(y−¯y)dt=dydt−d¯ydtdy′dt=d(y-y¯)dt=dydt-dy¯dt. If there are additional variables such as a disturbance variable dd then it is added as another term in deviation variable form d'=d−¯dd′=d-d¯.
dy′dt=αy′+βu′+γd′dy′dt=αy′+βu′+γd′
The values of the constants αα, ββ, and γγ are the partial derivatives of f(y,u,d)f(y,u,d) evaluated at steady state conditions.
α=∂f∂y∣∣∣¯y,¯u,¯dβ=∂f∂u∣∣∣¯y,¯u,¯dγ=∂f∂d∣∣∣¯y,¯u,¯d
dydt=f(y,u)dydt=f(y,u)
The right hand side of the equation is linearized by a Taylor series expansion, using only the first two terms.
dydt=f(y,u)≈f(¯y,¯u)+∂f∂y∣∣∣¯y,¯u(y−¯y)+∂f∂u∣∣∣¯y,¯u(u−¯u)dydt=f(y,u)≈f(y¯,u¯)+∂f∂y|y¯,u¯(y−y¯)+∂f∂u|y¯,u¯(u−u¯)
If the values of ¯uu¯ and ¯yy¯ are chosen at steady state conditions then f(¯y,¯u)=0f(y¯,u¯)=0 because the derivative term dydu=0dydu=0 at steady state. To simplify the final linearized expression, deviation variables are defined as y'=y−¯yy′=y-y¯ and u'=u−¯uu′=u-u¯. A deviation variable is a change from the nominal steady state conditions. The derivatives of the deviation variable is defined as dy'dt=dydtdy′dt=dydt because d¯ydt=0dy¯dt=0 in dy'dt=d(y−¯y)dt=dydt−d¯ydtdy′dt=d(y-y¯)dt=dydt-dy¯dt. If there are additional variables such as a disturbance variable dd then it is added as another term in deviation variable form d'=d−¯dd′=d-d¯.
dy′dt=αy′+βu′+γd′dy′dt=αy′+βu′+γd′
The values of the constants αα, ββ, and γγ are the partial derivatives of f(y,u,d)f(y,u,d) evaluated at steady state conditions.
α=∂f∂y∣∣∣¯y,¯u,¯dβ=∂f∂u∣∣∣¯y,¯u,¯dγ=∂f∂d∣∣∣¯y,¯u,¯d
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