---_._-_ _._._-... _ (56-r1S:_~0.~~\<;).e~~~b::;.~ 11<3<3 G

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1 ~-~-- ---_._-_ _._._-... _ _ , _ :( ~~,A ~~AT1~ H.Qr~!j.'-~ f'-\ ;. A-P- S':1S'"[E-r.-t.>_ L~'lt1I\. (56-r1S:_~0.~~\<;).e~~~b::;.~ 11<3<3 G Consider now the general nonlinear dynamic control system in matrix form represented by d dtx(t) = F(x(t), u(t)) 0.66) where x(t), u(t), and F are, respectively, the n-dimensional state space vector, the r-dimensional control vector, and the n-dimensional vector function. Assume that the nominal (operating) system trajectory Xn (t) is known and that the nominal control that keeps the system on the nominal trajectory is given by un(t). Using the same logic as for the scalar case, we can assume that the actual system dynamics in the immediate proximity of the system nominal trajectories can be approximated by the first terms of the Taylor series. That is, starting with x(t) = xn(t) +.6.x(t), u(t) = un(t) +.6.u(t) (1.67) and (1.68) we expand equation (1.66) as follows d d dtxn + dt.6.x = F(xn +.6.x, Un +.6.u) = F(xn' Un) + (~F)..6.x + (~F)..6.u + high-order terms ux I Xn{t) uu "",,,(t) unit) I un(') (1.69) High-order terms contain at least quadratic quantities of.6.x and.6.u. Since.6.x and.6.u are small their squares are even smaller, and hence the high-order terms can be neglected. Using (1.67) and neglecting high-order terms, an approximation is obtained d (OF) (OF) dt.6.x(t)= -;r.6.x(t) + ~.6.u(t) (1.70) ox IXn(t) uu Ix"{') u.. (t) unit)

2 @ INTRODUCTION 27 Partial derivatives in (1.70) represent the Jacobian matrices given by (OF) ox IXn(t) un(t) _Anxn_ - - 8F, ~ 8F, 8x, 8X2 8xn li 8F2 8x, ~ 8Fi 8xj 8Fn 8Fn 8Fn 8x, 8x, 8xn I Xn(t) Un (t) (l.71a) 8F, 8F, &F, &u, e,;:; 8ur &F2 li &u, BUr BFi (~~) IXn(t) Bu] un(t) =B nxt = (1.71b) BFn BFn BFn I Bu, Xn(t) 8U2 BUr Un (t) Note that the Jacobian matrices have to be evaluated at the nominal points, i.e. at xn(t) and un(t). With this notation, the linearized system (1.70) has the form d dt 6.x(t) = A6.x(t) +B6.u(t), 6.x(to) = x(to) - xn(to) (1.72) The output of a nonlinear system, in general, satisfies a nonlinear algebraic equation, that is y(t) = 9(x(t), u(t)) (1. 73) This equation can be also linearized by expanding its right-hand side into a Taylor series about nominal points Xn ( t) and Un(t). This leads to + high-order term,? (1.74) Note that Yn cancels term 9(xn,yn). By neglecting high-order terms in (1.74), the linearized part of the output equation is given by 6.y(t) = Cax(t) + nau(t) (1.75)

3 28 INTRODUCTION where the Jacobian matrices C and D satisfy ~ ~ f!flj.. ax, ax2 ax" ag2 ag2 ax, ax" cpxn _ ag, (09) axj - ox Ix,,(t) (1.76a) u,,(t) ag p ag p ag p I x,,(t) ax, ax2 ax" u,,(t) (09) ~ ag~ au, ~ au, DPxr _ ~ - OU auj IX"(') un(t).q... OUr ag2 OUr!2.r.!2.r.!2.r. I x,,(t) au, au2 OUr u" (t) (1.76b) Example 1.2: Let a nonlinear system be represented by dxi. at = Xl SlllX2 + X2 U dx2 2 dt =:: Xle- x2 + u y = 2XIX2 + x~ Assume that the values for the system nominal trajectories and control are known and given by Xl n, X2n, and Un. The linearized state space equation of the above nonlinear system is obtained as [ L\~l(t)] = [ sinx 2n XlncoSX2n+un] [L\XI(t)] + [X2n]L\u(t) L\x2(t) e-x2n -xlne-x2n L\X2(t) 2un L\y(t) = [2X2n 2Xln + 2X2n 1[~:~~~~] + O~u(t) Having obtained the solution of this linearized system under the given control input 6.u( t), the corresponding approximation'of the nonlinear system trajectories is <>

4 ( INTRODUCTION 29 Example 1.3: Consider the mathematical model of a single-link robotic manipulator with a flexible joint (Spong and Vidyasagar, 1989) IBI + mgl sinbi +k(bi - ( 2 ) = 0 JB2 - k(bi - ( 2 ) = U where BI, ()2 are angular positions, I, J are moments of inertia, m and l are, respectively, the link's mass and length, and k is the link's spring constant. Introducing the change of variables as the manipulator's state space nonlinear model equivalent to (1.66) is given by Xl = Xz. mgl. k ( ) X2 = ---smxi - - Xl - X3 I I X3 = X4. k ( ) 1 X4 = J Xl - X3 + JU Take the nominal points as (Xl n, Xzn, X3n, X4n, un), then the matrices A and B defined in (1.71) are given by _ k+m~l o A= [ I k J COSX' n 1 o o o In Spong (1995) the following numerical values are used for system parameters: mgl = 5, I = J = 1, k = Assuming that the output variable is equal to the link's angular position, that is the matrices C and D, defined in (1.76), are given by C = [ ], D = 0

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