idNeuralStateSpace/linearize

Linearize a neural state-space model around an operating point

Since R2022b

Syntax

sys = linearize(nss,x0)

sys = linearize(nss,x0,u0)

sys = linearize(nss,t0,x0,u0)

Description

sys = linearize(nss,x0) linearizes the time-invariant autonomous (that is with no input) neural state-space model nss about the operating point specified by x0.

sys = linearize(nss,x0,u0) linearizes the time-invariant neural state-space model nss about the operating point specified by both x0 and u0.

sys = linearize(nss,t0,x0,u0) linearizes the time-varying neural state-space model nss about the operating point specified by both x0 and u0 at the time specified by t0.

Examples

Linearize Neural State-Space System

Use idNeuralStateSpace to create a time-invariant continuous-time neural state-space system with four states, three inputs and two outputs with direct feedthrough between input and output. The state and output networks are initialized randomly.

nss = idNeuralStateSpace(4, ...
    NumInputs=3, ...
    NumOutputs=4+2, ...
    HasFeedthrough=true);

Define a random operating point.

x0 = randn(4,1);
u0 = randn(3,1);

Linearize nss around the operating point.

sys = linearize(nss,x0,u0)

sys =
 
  A = 
             x1        x2        x3        x4
   x1   -0.3175    0.1922   -0.2388    0.1463
   x2   0.01367   0.01112    0.4915  0.001513
   x3  -0.07403   -0.1489    0.1822   -0.1584
   x4    0.1954   -0.1134    0.1546    0.1356
 
  B = 
             u1        u2        u3
   x1    0.1321    0.2498  -0.01543
   x2    0.1842   0.01501  -0.04696
   x3  -0.08879    0.1664   -0.0748
   x4   0.05851   0.02688  -0.04317
 
  C = 
             x1        x2        x3        x4
   y1         1         0         0         0
   y2         0         1         0         0
   y3         0         0         1         0
   y4         0         0         0         1
   y5    0.1542  -0.06943    0.2562   -0.3387
   y6   0.04741    0.2255  0.003164  -0.02298
 
  D = 
             u1        u2        u3
   y1         0         0         0
   y2         0         0         0
   y3         0         0         0
   y4         0         0         0
   y5   -0.3407  -0.04532    0.1005
   y6    0.1535    0.4711   0.08209
 
Continuous-time state-space model.

Since the first output elements are the four states of nss, the first four rows of C matrix form an identity matrix, and the first four rows of D are zero.

You can now apply linear control analysis and synthesis methods on the linear state-space system sys.

Input Arguments

nss — Neural state-space system

idNeuralStateSpace object

Neural state-space system, specified as an idNeuralStateSpace object.

Example: myNrlSS

t0 — Value of the time variable at the operating point

scalar

Value of the time variable at the operating point, specified as a scalar, for time-varying idNeuralStateSpace systems.

Example: 5.75

x0 — Value of the state at the operating point

column vector

Value of the state at the operating point, specified as a vector.

Example: [-0.2 0.3]'

u0 — Value of the input at the operating point

column vector

Value of the input at the operating point, specified as a vector, for non-autonomous idNeuralStateSpace systems.

Example: 2.4

Output Arguments

sys — Linearized model

ss (default) | idss

Linearized model, returned as an ss (Control System Toolbox) object, when Control System Toolbox™ is installed, or as an idss object otherwise. For more information on linearization, see Linearize Nonlinear Models (Simulink Control Design).

Note

The first output elements are all the states of nss. Therefore, their corresponding entries in the C matrix form an identity matrix, and their corresponding entries in the D matrix are zero.

Version History

Introduced in R2022b