linearize

Linearize an explicit (mss) or hybrid implicit (dmss) multilinear model around an operating point.

Syntax
Description
Input Arguments
Output Arguments
Example (mss):
Example (dmss):
References
See also

Syntax

lsys = linearize(sys, op)                  % mss -> ss/sparss
dss  = linearize(sys, op)                  % dmss -> dss
[dss, dssSparse] = linearize(sys, op)      % dmss -> dense and sparse

Backward compatible signatures: linearize(msys, x, u) % mss (legacy) linearize(msys, xp, x, u, y, z) % dmss (legacy)

Description

linearize(sys, op) invokes the class-specific linearization implemented by both mss and dmss objects. It accepts a single operating point argument op as a struct for clarity and extensibility.

For mss objects:

op.x (n×1) – state operating point (required)
op.u (m×1) – input operating point (required if m>0)

For dmss objects:

op.xp (n×1) – state-derivative operating point (optional)
op.x (n×1) – state operating point (optional)
op.u (m×1) – input operating point (optional)
op.y (p×1) – algebraic variable operating point (optional)
op.z (q×1) – boolean variable operating point (optional)

Any omitted fields default to []. The class methods perform basic dimension checks.

Convenience forms are supported for parity with MATLAB’s style:

mss: numeric vector [x; u] or cell {x, u}
dmss: numeric vector [xp; x; u; y; z] or cell {xp, x, u, y, z}

Input Arguments

sys: multilinear MTI model (mss or dmss object)

op : struct, numeric vector, or cell containing operating point values (see above)

Output Arguments

lsys: linear LTI model, ss or sparss (for mss inputs)

dss, dssSparse: linear descriptor model(s) as dss and sparss (for dmss inputs)

Example (mss):

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 using an op struct

    op = struct('x', x_op, 'u', u_op);
    lsys = linearize(msys, 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.

Example (dmss):

For the following continuous-time implicit multilinear model with 2 states and 1 algebraic variable, a dmss object is created using the sym2dmss command:

    eqs = {'0==xp1-x1*x2*y1+4';
           '0==xp2-x2+2'
           '0==x2-y1';};
    dsys = sym2dmss(eqs, 0);
%
% The operating point point, where xp=0, is:
%
    xpop = [0; 0];
    xop  = [1; 2];
    yop  = 2;
%
% which is stored as a |op| struct
%
    op = struct('xp', xpop, 'x', xop,'y',yop);
%
% Then the |dmss| is linearized at the operating point as follows:
%
    dss = linearize(dsys, op);
%
% The result |dss| is a linear descriptor state-space model, which is a
% |ss| object.

Reduced by 2 column(s) and 0 equation(s) with trivial Reduction due to duplications.

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.

[2] C. Kaufmann, G. Pangalos, G. Lichtenberg, O. Gomis-Bellmunt, "Small-Signal Stability Analysis of Power Systems by Implicit Multilinear Models", submitted to IEEE Trans. Circuits Syst. I Regul. Pap., 2025.

You can find a full example here: open linearization example