linearize

Method of mss.

Linearize an explicit MTI model around an operating point (not necessarily an equilibrium).

Source: src/model/@mss/linearize.m

Syntax

lsys           = linearize(msys, x_op, u_op)
[A, B, C, D]   = linearize(msys, x_op, u_op)
lsys           = linearize(msys, x_op)          % only if nInput == 0

Description

linearize forms the first-order Taylor approximation of the explicit multilinear system

$\delta x = f(x,u), \qquad y = g(x,u)$

around $(x_\text{op}, u_\text{op})$ . The state evolution $\delta x$ is valid for both continuous-time $\dot x$ or discrete-time $x_{k+1}$ .

The result is a linear state-space model in the deviations $\Delta x = x - x_\text{op}$ , $\Delta u = u - u_\text{op}$ , and $\Delta y = y - y_\text{op}$ , as

$\delta\,\Delta x = A\,\Delta x + B\,\Delta u, \qquad \Delta y = C\,\Delta x + D\,\Delta u.$

The four matrices are taken straight off the model Jacobians as

$\begin{aligned} A = \frac{\partial f(x_\text{op}, u_\text{op})}{\partial x}, \quad B= \frac{\partial f (x_\text{op}, u_\text{op})}{\partial u} \\ C = \frac{\partial g(x_\text{op}, u_\text{op})}{\partial x}, \quad D= \frac{\partial g(x_\text{op}, u_\text{op})}{\partial u}, \end{aligned}$

see [1] or [2] for further details.

The operating point does not have to be an equilibrium — linearize never checks if $f(x_\text{op},u_\text{op}) = 0$ . Users can verify equilibrium conditions by calling functionValue and outputValue first.

Input arguments

Argument	Description
`msys`	The `mss` model.
`x_op`	State operating point (`msys.nState` elements).
`u_op`	Input operating point (`msys.nInput` elements); may be omitted when the model has no inputs.

Output arguments

Output	Description
`lsys`	(one output) `sparss` model with the sample time and signal names of `msys`.
`A, B, C, D`	(several outputs) Raw linearization matrices: `nState`×`nState`, `nState`×`nInput`, `nOutput`×`nState`, `nOutput`×`nInput`.

Example

sys = rmss(3,1,1,4);

lsys = linearize(sys, [1; 0; -1], 0.5);       % sparss model
[A,B,C,D] = linearize(sys, [1; 0; -1], 0.5);  % raw matrices
eig(A)              % local stability at the operating point

References

Kaufmann, Christoph, et al. Small-Signal Stability Analysis of Power Systems by Implicit Multilinear Models. IEEE access (2026).
Kaufmann, Christoph, et al. Efficient linearization of explicit multilinear systems using normalized decomposed tensors IFAC-PapersOnLine 56.2 (2023)