Second order differential equation with large matrices

2022 12 4
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更新时间：2022 12 10
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Greetings,
I am tryig to solve this differential equation M * q'' + K * q= - K_d * q' - K_p * (q - q_d) where M is a 15x15 matrix, q is 15x1 vector, K is a 15x15 matrix, K_d and K_p are 15x15 known matrices and q_d is a 15x1 vector which is also known. q'' is a second time derivative. Which solver is the most applicable for this equation? and is there way to solve this in a matrix form and not expand the whole equation. Thanks in advance.

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Torsten
Torsten 2022-12-4
编辑：Torsten 2022-12-4
q_d must be 15x1, not 8x1.
Rewrite your system as a system of first-order differential equations and use ode45, e.g., to solve it.
Jan
Jan 2022-12-5
As Torsten has explained alreayd, rewrite to 2nd order ODE to a system of order 1 equations. You can use matrices to evaluate this system. This is one of the fundamental design ideas of Matlab. Simply write the system as matrix equation.
David Togonidze
David Togonidze 2022-12-10
One question guys. Let's say matrix M is not constant and is dependent on q, so at every time step M changes accordingly to the values of q calculated at previous time step (starting from initial condition q0). How can i incorporate that into the odesystem function for ode45 solver? Thank you in advance
Torsten
Torsten 2022-12-10
编辑：Torsten 2022-12-10
One question guys. Let's say matrix M is not constant and is dependent on q, so at every time step M changes accordingly to the values of q calculated at previous time step (starting from initial condition q0). How can i incorporate that into the odesystem function for ode45 solver? Thank you in advance
By dividing through M (assuming M is non-singular):
q'' = M \ (- K * q - K_d * q' - K_p * (q - q_d) )
If M is singular, define M as mass-matrix for the ODE solver in the options-structure and use ode15s or ode23t instead of ode45.

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Sam Chak
Sam Chak 2022-12-5
Ran in:

3 个投票

Ran in:
Hi @David Togonidze
SInce this is a linear system, if you have the Control System Toolbox, then you can manipulate the matrix differential equation and convert it to a State-Space Model, that is similar to the system of 1st-order differential equations as advised by @Torsten, and then run the simulation using the lsim() command.
The following example demonstrates a 2-DOF Mass-Spring-Damper System.
M = diag([1 1])
M = 2×2
1 0 0 1
Kd = 2*diag([1 1])
Kd = 2×2
2 0 0 2
Kp = 0.5*diag([1 1])
Kp = 2×2
0.5000 0 0 0.5000
K = 0.5*diag([1 1])
K = 2×2
0.5000 0 0 0.5000
Convert the system of ODEs to State-space Model:
where the state vector is defined as
% State matrix
A = [zeros(2) eye(2);
M\(-Kp-K) M\(-Kd)]
A = 4×4
0 0 1 0 0 0 0 1 -1 0 -2 0 0 -1 0 -2
% Input matrix
% B = [zeros(2); M\Kp] % based on your original Kp*qd
B = [zeros(2); M\(Kp+K)] % modified to track desired qd
B = 4×2
0 0 0 0 1 0 0 1
% Output matrix
C = [eye(2) zeros(2)]
C = 2×4
1 0 0 0 0 1 0 0
% Feedforward matrix
D = 0;
sys = ss(A, B, C, D)
sys = A = x1 x2 x3 x4 x1 0 0 1 0 x2 0 0 0 1 x3 -1 0 -2 0 x4 0 -1 0 -2 B = u1 u2 x1 0 0 x2 0 0 x3 1 0 x4 0 1 C = x1 x2 x3 x4 y1 1 0 0 0 y2 0 1 0 0 D = u1 u2 y1 0 0 y2 0 0 Continuous-time state-space model.
t = 0:0.02:20;
u1 = max(0,min(t-1,1)); % qd1
u2 = max(0,min(t-1,1)); % qd2
U = [u1; u2]; % qd array
lsim(sys, U, t)
% lsim(sys, U, t, x0) % Use this if there is a vector x0 of initial state values
grid on

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