Fall 2001 16.317-6 Typically assume that the system is operating about some nominal state solution a(t)(possibly requires a nominal input u(t)) Then write the actual state as a(t=a(t)+dx(t and the actual inputs as u(t)=u(t)+Su(t) The "8"is included to denote the fact that we expect the varia- tions about the nominal to be small Can then develop the linearized equations by using the Taylor series expansion of f(, about a (t) and a () Recall the vector equation i=f(a, u), each equation of which 正z=f(x,u) can be expanded as 0 dt )=f(x0+6x,b+) f(x0,)+ afi afi Su du where 0f「af;Of ax a and o means that we should evaluate the function at the nominal values of o and 2o The meaning of "small"deviations now clear-the variations in dc and Su must be small enough that we can ignore the higher order terms in the Taylor expansion of f(a, aFall 2001 16.31 7—6 • Typically assume that the system is operating about some nominal state solution x0(t) (possibly requires a nominal input u0(t)) — Then write the actual state as x(t) = x0(t) + δx(t) and the actual inputs as u(t) = u0(t) + δu(t) — The “δ” is included to denote the fact that we expect the varia￾tions about the nominal to be “small” • Can then develop the linearized equations by using the Taylor series expansion of f(·, ·) about x0(t) and u0(t). • Recall the vector equation x˙ = f(x, u), each equation of which x˙ i = fi(x, u) can be expanded as d dt(x0 i + δxi) = fi(x0 + δx, u0 + δu) ≈ fi(x0 , u0 ) + ∂fi ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0 δx + ∂fi ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0 δu where ∂fi ∂x =   ∂fi ∂x1 ··· ∂fi ∂xn   and ·|0 means that we should evaluate the function at the nominal values of x0 and u0. • The meaning of “small” deviations now clear — the variations in δx and δu must be small enough that we can ignore the higher order terms in the Taylor expansion of f(x, u)