getx0

Map initial conditions from a mechss object to a sparss object

Syntax

x0 = getx0(sys,q0,dq0)

x0 = getx0(sys,q1,q2)

Description

x0 = getx0(sys,q0,dq0) computes a matching initial condition x0 for an equivalent sparse first-order model using the initial conditions for displacement q0 and velocity dq0 from the continuous-time sparse second-order model sys. You can omit the second and third input arguments when q0 and dq0 are zero. The output, $$x0=\left[\begin{array}{c}q0\\ dq0\end{array}\right]=\left[\begin{array}{c}q\left(0\right)\\ \frac{dq}{dt}\left(0\right)\end{array}\right]$$ when the [sys.M;sys.G] matrices has no zero columns.

In general, the states of the sparss model is $$x=\left[\begin{array}{c}q\\ dq\left(jnz\right)\end{array}\right]$$ where jnz are the indices of the nonzero columns of [sys.M;sys.G]. The resulting number of states n_x is $$n_x=n_q+\text{numel}\left(jnz\right)\le 2n_q$$ where n_q is the number of nodes in the mechss object sys.

x0 = getx0(sys,q1,q2) computes a matching initial condition using initial values q1 = q[k] and q2 = q[k+1] for discrete-time systems. The second and third input arguments can be omitted when q0 and dq0 are zero. The output, $$x0=\left[\begin{array}{c}q1\\ q2\end{array}\right]=\left[\begin{array}{c}q\left[k\right]\\ q\left[k+1\right]\end{array}\right]$$ when the M and G matrices of the mechss model sys has no zero columns.

Examples

Map Initial Conditions from Continuous-Time Sparse Second-Order Model to Equivalent First-Order Model

For this example, consider a continuous-time sparse second-order model with 108 nodes. The row vectors q0 and dq0 are the initial conditions for displacement and velocity, respectively. The vectors q0 and dq0 have the same size as the number of nodes in sys.

Load the data in sparseSOC.mat and find the initial conditions for an equivalent sparse first-order model using getx0.

load('sparseSOC.mat','sys','q0','dq0');
x0 = getx0(sys,q0,dq0);
size(x0)

ans = 1×2

   216     1

In this case, vector x0 contains 216 states, since $$x0=\left[\begin{array}{c}q0\\ dq0\end{array}\right]$$.

Map Initial Conditions from Discrete-Time Sparse Second-Order Model to Equivalent First-Order Model

For this example, consider a discrete-time sparse second-order model with 7102 nodes. The row vectors q1 and q2 are the initial conditions for k and k+1 time steps, respectively. The vectors q1 and q2 have the same size as the number of nodes in sys.

Load the data in sparseSOD.mat and find the initial conditions for an equivalent discrete-time sparse first-order model using getx0.

load('sparseSOD.mat','sys','q1','q2');
x0 = getx0(sys,q1,q2);
size(x0)

ans = 1×2

       14204           1

In this case, vector x0 contains 14204 states since $$x0=\left[\begin{array}{c}q1\\ q2\end{array}\right]$$.

Input Arguments

sys — Continuous-time or discrete-time sparse second-order model

mechss model object

Continuous-time or discrete-time sparse second-order model to be converted to a sparse first-order model, specified as a mechss model object.

For more information, see the mechss reference page.

q0 — Initial values of the displacement vector

row vector of length Nq

Initial values of the displacement vector, specified as a row vector of length Nq, where Nq is the number of states in the continuous-time second-order sparse model sys.

dq0 — Initial values of the velocity vector

row vector of length Nq

Initial values of the velocity vector, specified as a row vector of length Nq, where Nq is the number of states in the continuous-time second-order sparse model sys.

q1 — Initial values of state vector at k

row vector of length Nq

Initial values of state vector at k, specified as a row vector of length Nq, where Nq is the number of states in the continuous-time second-order sparse model sys. In other words, q1 = q[k].

q2 — Initial values of state vector at k+1

row vector of length Nq

Initial values of state vector at k, specified as a row vector of length Nq, where Nq is the number of states in the continuous-time second-order sparse model sys. In other words, q2 = q[k+1].

Output Arguments

x0 — Mapped initial conditions for the equivalent sparse first-order model

array of doubles

Mapped initial conditions for the equivalent sparse first-order model, returned as an array of doubles. When the mass matrix M and the velocity-to-output matrix G of the mechss object contains no zero columns, then:

• $$x0=\left[\begin{array}{c}q0\\ dq0\end{array}\right]=\left[\begin{array}{c}q\left(0\right)\\ \frac{dq}{dt}\left(0\right)\end{array}\right]$$ for continuous-time systems.

• $$x0=\left[\begin{array}{c}q1\\ q2\end{array}\right]=\left[\begin{array}{c}q\left[k\right]\\ q\left[k+1\right]\end{array}\right]$$ for discrete-time systems.

Version History

Introduced in R2020b