augoffset

Map offset contribution to extra input channel

Since R2024a

Syntax

asys = augoffset(sys)

Description

For a state-space model with offsets, augoffset returns an offset-free state-space model with one extra input channel.

The function constructs a model such that the response of the original model to an input u(t) matches the response of the augmented model to [u(t) ; 1]. That is, the response of the augmented model is the same as the original model when you apply a unit step to the extra input channel and an input u(t) to the remaining input channels.

For a state-space model with offsets given by:

$\begin{array}{l} \dot{x} = δ_{0} + A (x - x_{0}) + B (u - u_{0}) \\ y = y_{0} + C (x - x_{0}) + D (u - u_{0}), \end{array}$

the transfer function of the augmented model is [H(s) h(s)], where H(s) and h(s) are the linear and affine parts of the original model, respectively. Therefore, the frequency response is the frequency-domain expression of the time-domain connection.

$\begin{matrix} Y (s) = H (s) U (s) + h (s) \frac{1}{s} \\ = [\begin{matrix} H (s) & h (s) \end{matrix}] [\begin{matrix} U (s) \\ \frac{1}{s} \end{matrix}] \end{matrix}$

Here,

$\begin{array}{l} H (s) = D + C (^{s I - A) - 1} B \\ h (s) = y_{0} + C (^{s I - A) - 1} (δ_{0} - A x_{0} - B u_{0}) \end{array}$

Note that H(s) is the usual transfer function without offsets.

You can use augoffset to turn offsets into an additional input and visualize the corresponding frequency response.

example

asys = augoffset(sys) takes a state-space model sys with offsets and returns an offset-free state-space model with one extra input channel.

Examples

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Convert State-Space Model with Offsets to Offset-Free Model

Open Live Script

This example shows how to obtain an offset-free state-space from a model with offsets using the augoffset function. The augoffset function constructs a model with an extra input channel and maps the contribution of the offsets in the original model to that extra input channel.

For this example, consider a linear time-varying model defined by the data function ltvssDataFcn.m.

Create an LTV model.

ltvSys = ltvss(@ltvssDataFcn)

Continuous-time state-space LTV model with 1 outputs, 1 inputs, and 1 states.

Sample the LTV dynamics at 5 seconds to obtain the local LTI model.

t = 5;
sys = psample(ltvSys,t);

The model sys contains an output offset.

sys.Offsets

ans = struct with fields:
    dx: 0
     x: 0
     u: 0
     y: -0.0132

Use augoffset to eliminate the offset.

asys = augoffset(sys);

Compare the model response to an input $\sin (t)$ . The response of the augmented model is the same as the original model when you apply a unit step to the extra input channel.

tsim = 0:0.01:5;
usim = sin(tsim);
[y1,~,x1] = lsim(sys,usim,tsim);
[y2,~,x2] = lsim(asys,[usim;ones(size(tsim))],tsim);
plot(tsim,y1,tsim,y2,'r--')
legend("Original model with offsets","Offset-free model")

View the data function code.

type ltvssDataFcn.m

function [A,B,C,D,E,dx0,x0,u0,y0,Delays] = ltvssDataFcn(t)
% SISO, first order
A = -(1+0.5*sin(t));
B = 1;
C = 1;
D = 0;
E = [];
dx0 = [];
x0 = [];
u0 = [];
y0 = 0.1*sin(5*t);
Delays = [];