linearize

Linearize an explicit multilinear model around an operating point

Contents

Syntax

linearize(msys,x,u)

Description

linearize(msys,x,u) Linearizes an explicit MTI state-space model of type mss around an operating point of the states, given by the scalar or vector x, and the inputs, provided by the scalar or vector u. The linearization is done by computing the Jacobian, which was shown for explicit MTI models stored as a norm-1 CP-decomposed tensor in [1]. If no output equation of the mss object is specified, all states are outputs and feedthrough is set to zero.

Input Arguments

msys: multilinear MTI model, mss object

x : state operating point, vector

u : input operating point, vector

Output Arguments

linSs: linear LTI model, ss object

Example:

Second-order explicit MTI model with 2 states and 1 input (expanded and factored form)

$\dot{\mathbf{x}}=\left(\begin{array}{cc} x_2 + x_1x_2\\ 3u_1 + 2x_2 - 3u_1x_1 + 2x_1x_2 \end{array}\right) = \left(\begin{array}{cc} 2\cdot(0.5+0.5x_1)x_2 \\ 4\cdot(0.5+0.5x_1)x_2 + 6 \cdot(0.5-0.5x_1)u_1 \end{array}\right) $.

The factored explicit MTI model as CPN1 object thus has a structure matrix

    S = [0.5 -0.5; 1 0; 0 1];

and the parameter matrix

    phi = [2 0; 4 6];

Then we can create explicit MTI model as a CPN1 object and the mss-object

    tens = CPN1(S,phi);
    msys = mss(tens);

and assume the operating point for the states x and the input u

    x_op = [-1;1];
    u_op = 3;

Then we can linearize the explicit MTI model

    lsys = linearize(msys,x_op,u_op)
Warning: No Outputs defined, assuming states as outputs. 

lsys =
 
  A = 
       x1  x2
   x1   1   0
   x2  -7   0
 
  B = 
       u1
   x1   0
   x2   6
 
  C = 
       x1  x2
   y1   1   0
   y2   0   1
 
  D = 
       u1
   y1   0
   y2   0
 
Continuous-time state-space model.

You can find a full example here: open linearization example

References

[1] C. Kaufmann, D. Crespí, G. Lichtenberg, G. Pangalos, and C. Cateriano Yáñez, "Efficient Linearization of Explicit Multilinear Systems using Normalized Decomposed Tensors," IFAC-PapersOnLine, vol. 56, no. 2, pp. 7312–7317, Jan. 2023, doi: https://doi.org/10.1016/j.ifacol.2023.10.344.

See also

jacobian, cpn2Lin, cpn2LinSparse