mss2mss

Affine state transformations of MTI models (currently only permutation, scaling and offset allowed)

Contents

Syntax

[sys_sc] = mss2mss(sys, T, c)

Description

[mtiSystemTransformed] = mss2mss(mtiSystem, transformation, offset)

Linear state transformation (and optionally input transformation) of an explicit MTI model in CPN1 format. Currently only permutation, scaling and offset is possible (no rotation, i.e. the transformation matrix must have exactly one non-zero element per row and column).

The tranformation is done with the linear state transformation

$$ \mathbf{x}=\mathbf{T}\tilde{\mathbf{x}}+\mathbf{c} $$

and

$ \tilde{\mathbf{x}}=\mathbf{T}^{-1}(\mathbf{x}-\mathbf{c}) $.

where T can have only 1 non-zero element per row and column.

The transformed MTI system in normalized representation is given by

$$\mathbf{\dot{\tilde{x}}}=\mathbf{\tilde{F}}_\phi\prod{( 1-|\mathbf{\tilde{F}}_U|+\mathbf{\tilde{F}}_U\mathbf{\tilde{x}})}$$

$$\mathbf{y}=\mathbf{\tilde{G}}_\phi\prod{( 1-|\mathbf{\tilde{G}}_U|+\mathbf{\tilde{G}}_U\mathbf{\tilde{x}})}$$

The output is a MTI model with the same input and output behavior as the original system, but with scaled states (and optionally scaled inputs) according to the transformation matrix and offset vector.

If the transformation matrix has dimensions n x n (with n being the number of states), then only states will be scaled. If the transformation matrix has dimensions (n + m) x (n + m) then the inputs will be scaled, too (where m is the number of inputs).

Combining this function with dataProcessing.scaleData allows for easy scaling of states (and optionally inputs) between lower and upper bounds.

Input arguments

mtiSystem - mss object (eMTI system)

transformation - transformation matrix (must be diagonal)

offset - vector of offsets

Output arguments

mtiSystemTransformed - mss object (transformed eMTI system)

Example state transformation

Build multilinear model (MTI), 2.order, 1 input

F.U = [[0.83, -0.1]; [0, 0.53]; [1, -0.1]];    % state transition matrix
F.phi = [[1.0080, 0]; [0, 1.50]];              % state weigting matrix
u = [zeros(20, 1); ones(20, 1); -ones(20, 1)]; % input signal
t = 1:1:60;                                    % time vector
Ts = 1;                                        % sampling time (discrete)
obj = mss(CPN1(F.U, F.phi), Ts);               % build mti model object

Simulate MTI Model

x0 = [0; 0];                                   % initial state
[y, ~, xsim] = msim(obj, u, t, x0);

Set limits for states (inputs not scaled in this example)

lbx = [0, 0];                                  % lower limit state
ubx = [1, 1];                                  % upper limit state

Linear state transformation of mti model

[xsc, ~, T, c] = dataProcessing.scaleData(xsim, u, lbx, ubx);
% scale data within limits and obtain transformation matrix and offset
msys = mss2mss(obj, T, c);                 % transform (scale) system

Simulation of transformed mti model

[ysc, ~, xsimsc] = msim(msys, u, t, xsc(1,:));
% simulate scaled mti model with scaled initial state

Plot results

figure()                                        % plot original mti model
sgtitle('original MTI model')
subplot(3, 1, 1)
plot(u)
xlim([min(t), max(t)])
xlabel('time')
legend('input')
subplot(3, 1, 2)
plot(xsim)
xlim([min(t), max(t)])
xlabel('time')
legend({'state 1', 'state 2'})
subplot(3, 1, 3)
plot(y)
xlim([min(t), max(t)])
xlabel('time')
legend('output')

figure()                                        % plot scaled mti model
sgtitle('scaled MTI model')
subplot(3,1,1)
plot(u)
xlim([min(t), max(t)])
xlabel('time')
legend('input')
subplot(3,1,2)
plot(xsimsc)
xlim([min(t), max(t)])
xlabel('time')
legend({'state 1','state 2'})
subplot(3,1,3)
plot(ysc)
xlim([min(t), max(t)])
xlabel('time')
legend('output')

Example transformation in state and input

Build multilinear model (MTI), 2.order, 1 input

F.U = [[0.83, -0.1]; [0, 0.53]; [1, -0.1]];    % state transition matrix
F.phi = [[1.0080, 0]; [0, 1.50]];              % state weigting matrix
u = [zeros(20, 1); ones(20, 1); -ones(20, 1)]; % input signal
t = 1:1:60;                                    % time vector
Ts = 1;                                        % sampling time (discrete)
obj = mss(CPN1(F.U, F.phi), Ts);               % build mti model object

Simulate MTI Model

x0 = [0; 0];                                   % initial state
[y, ~, xsim] = msim(obj, u, t, x0);

Set limits for states and inputs

lbx = [0, 0];                                  % lower limit state
ubx = [1, 1];                                  % upper limit state
lbu = 0;                                       % lower limit input
ubu = 1;                                       % upper limit input

Linear state transformation of mti model

[xsc, usc, T, c] = dataProcessing.scaleData(xsim, u, lbx, ubx, lbu, ubu);
% scale data within limits and obtain transformation matrix and offset
msys = mss2mss(obj, T, c);                 % transform (scale) system

Simulation of transformed mti model

[ysc, tOut, xsimsc] = msim(msys, usc, t, xsc(1,:));
% simulate scaled mti model with scaled initial state

plot results

figure()                                        % plot original mti model
sgtitle('original MTI model')
subplot(3, 1, 1)
plot(u)
xlim([min(t), max(t)])
xlabel('time')
legend('input')
subplot(3, 1, 2)
plot(xsim)
xlim([min(t), max(t)])
xlabel('time')
legend({'state 1','state 2'})
subplot(3, 1, 3)
plot(y)
xlim([min(t), max(t)])
xlabel('time')
legend('output')

figure()                                        % plot scaled mti model
sgtitle('scaled MTI model')
subplot(3, 1, 1)
plot(usc)
xlim([min(t), max(t)])
xlabel('time')
legend('input')
subplot(3, 1, 2)
plot(xsimsc)
xlim([min(t), max(t)])
xlabel('time')
legend({'state 1','state 2'})
subplot(3, 1, 3)
plot(ysc)
xlim([min(t), max(t)])
xlabel('time')
legend('output')

See also

mss CPN1


Published with MATLAB® R2024b