Fall 2001 6.317-7 Since d ad=fi(z,u), we thus have that afi 6x+ afi x;)≈ dt 0 Combining for all n state equations, gives(note that we also set 0f1 0f1 0f2 af2 d 6x+ dt af A(tdc+b(tsu where afi af., afi. ax1 dx af? f A(t) 器 an afn ax and au ou afi af af af2 B(t) afn. afn afn.Fall 2001 16.31 7—7 • Since d dtx0 i = fi(x0, u0), we thus have that d dt(δxi) ≈ ∂fi ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0 δx + ∂fi ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0 δu • Combining for all n state equations, gives (note that we also set “≈” → “=”) that d dtδx =                         ∂f1 ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0 ∂f2 ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0 . . . ∂fn ∂x ¯ ¯ ¯ ¯ ¯ ¯ 0                         δx +                         ∂f1 ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0 ∂f2 ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0 . . . ∂fn ∂u ¯ ¯ ¯ ¯ ¯ ¯ 0                         δu = A(t)δx + B(t)δu where A(t) ≡                 ∂f1 ∂x1 ∂f1 ∂x2 ··· ∂f1 ∂xn ∂f2 ∂x1 ∂f2 ∂x2 ··· ∂f2 ∂xn . . . ∂fn ∂x1 ∂fn ∂x2 ··· ∂fn ∂xn                 0 and B(t) ≡                 ∂f1 ∂u1 ∂f1 ∂u2 ··· ∂f1 ∂um ∂f2 ∂u1 ∂f2 ∂u2 ··· ∂f2 ∂um . . . ∂fn ∂u1 ∂fn ∂u2 ··· ∂fn ∂um                 0