linearization of a non linear dinamic system

hi, I have to analyze the step response of a system of two differential equations:

vyp=(((Df*sin(Cf*atan(Bf*(((1-Ef)*(-delta+((vy+(a*r))/Vx)))+((Ef/Bf)*atan(Bf*(-delta+((vy+(a*r))/Vx))))))))/m)*cos(delta))+((Dr*sin(Cr*atan(Br*(((1-Er)*((vy-(b*r))/Vx))+((Er/Br)*atan(Br*((vy-(b*r))/Vx)))))))/m)-(Vx*r);

rp=(a/Iz)*(Df*sin(Cf*atan(Bf*(((1-Ef)*(-delta+((vy+(a*r))/Vx)))+((Ef/Bf)*atan(Bf*(-delta+((vy+(a*r))/Vx))))))))*cos(delta)-((b/Iz)*(Dr*sin(Cr*atan(Br*(((1-Er)*((vy-(b*r))/Vx))+((Er/Br)*atan(Br*((vy-(b*r))/Vx))))))))

the input of the system is delta and the output variables are vy and r. vyp=dvy/dt and rp=dr/dt. The other terms are constant. I think I have to linearizate this system but I have some problems about it. Any suggestions?

Answer 1

[Andreas Surya Sitorus]

I've some example for your case, by using JACOBIAN Matrix, may be it can help : % Variables declaration syms U1 U2 U3 U4 Omega Ixx Iyy Izz p q r phi theta psi syms dphi dtheta dpsi dp dq dr dW1 dW2 dW3 dW4 v1 v2 v3 v4 syms l b d m g Res J_TP K_M K_E W1 W2 W3 W4 dfxu x u

% Non Linear Diff.Equation dp=[((Iyy-Izz)*q*r/Ixx)-(J_TP*q*Omega/Ixx)+U2/Ixx] dq=[((Izz-Ixx)*p*r/Iyy)+(J_TP*p*Omega/Iyy)+U3/Iyy] dr=[((Ixx-Iyy)*p*q/Izz)+U4/Izz] dphi=[p+q*tan(theta)*sin(phi)+r*tan(theta)*cos(phi)] dtheta=[q*cos(phi)- r*sin(phi)] dpsi=[q*sec(theta)*sin(phi)+r*sec(theta)*cos(phi)]

% Non Linear State Space Equation dfxu=[dp;dq;dr;dphi;dtheta;dpsi;]

% State variable & Nonlinear Input Declaration x=[p; q; r; phi; theta; psi; ] u=[U2 U3 U4]

% Find matrix jacobian A & B A=jacobian(dfxu,x) B=jacobian(dfxu,u)

% substite parameter values & variable As=subs(A) Bs=subs(B)