I have 9 coupled nonlinear ordinary differential equations. and i have to convert it in to state space form. here i have 9 state variable including theta.

Answer 1

[Torsten]

Write your system as

M(U,U')*U'' = f(U,U')

with a (9x9) matrix M.

Then you can use ode15s to solve the system

V'        = W
M(V,W)*W' = f(V,W)

or

[I , 0 ; 0 M] * [V' ; W'] = [W ; f(V,W)]

with V = U, W = U' and the mass matrix [I , 0 ; 0 M] (I = identity matrix, 0 = null matrix, both of size (9x9)).

[prabin kumar meher]

Thanks for the response. could you please elaborate more. because i have second order thate tearm in eq. 7.

[Torsten]

If you order your second-order derivatives as a vector [u1'' u2'' u3'' u4'' phi1'' phi2'' phi3'' phi4'' theta''].', your matrix M and vector f would be (if iI'm not mistaken):

M = [m11 m13 0 0 m12 m14 0 0 0;...
     m21 m23 0 0 m22 m24 0 0 0;...
     m31 m33 m35 0 m32 0 m36 0 0;...
     m41 0 m45 0 m42 m44 m46 0 0;...
     0 m53 m55 m57 0 m54 0 m58 0;...
     0 m63 m66 m67 0 m64 0 m68 0; ...    
     0 0 m75 m77 0 0 m76 m78 -me*sin(theta);...
     0 0 m85 m87 0 0 m86 m88 0;...
     0 0 0 -me*sin(theta) 0 0 0 0 me^2+I_p];
f = -[Z1; Z2; Z3; Z4; Z5; Z6; Z7-me*theta_dot^2*cos(theta); Z8; Z9-tau];