Fall 2001 16.317-8 Similarly, if the nonlinear measurement equation is y=g(a, u),can show that, if y(t)=yo+8y,then ag1 ax lo 0 ag2 y 00u gp C(t)8.c+D(t)s Typically think of these nominal conditions u ,u as"set points or"operating points" for the nonlinear system. The equations O C=A(t)dx+b(to dy=c(tSx +d(tdu then give us a linearized model of the system dynamic behavior about these operating/set points Note that if ao, u are constants, then the partial fractions in the expressions for A-D are all constant -> LTI linearized model One particularly important set of operating points are the equilib- rium points of the system. Defined as the states control input combinations for which provides n algebraic equations to find the equilibrium pointsFall 2001 16.31 7—8 • Similarly, if the nonlinear measurement equation is y = g(x, u), can show that, if y(t) = y0 + δy, then δy =                         ∂g1 ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0 ∂g2 ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0 . . . ∂gp ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0                         δx +                         ∂g1 ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0 ∂g2 ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0 . . . ∂gp ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0                         δu = C(t)δx + D(t)δu • Typically think of these nominal conditions x0 , u0 as “set points” or “operating points” for the nonlinear system. The equations d dtδx = A(t)δx + B(t)δu δy = C(t)δx + D(t)δu then give us a linearized model of the system dynamic behavior about these operating/set points. • Note that if x0 , u0 are constants, then the partial fractions in the expressions for A—D are all constant → LTI linearized model. • One particularly important set of operating points are the equilib￾rium points of the system. Defined as the states & control input combinations for which x˙ = f(x0 , u0 ) ≡ 0 provides n algebraic equations to find the equilibrium points