imp2exp

Convert implicit linear relationship to explicit input-output relation

Syntax

B = imp2exp(A,yidx,uidx)

Description

B = imp2exp(A,yidx,uidx) transforms a linear constraint between variables y and u of the form A(:,[yidx;uidx])*[y;u] = 0 into an explicit input/output relationship y = B*u. The vectors yidx and uidx refer to the columns (inputs) of A as referenced by the explicit relationship for B.

Examples

Uncertain Matrix Algebraic Constraint

Consider two uncertain motor/generator constraints among 4 variables $$\left[\begin{array}{c}V\\ I\\ T\\ W\end{array}\right]$$, namely $$\left[\begin{array}{cccc}1& -R& 0& -K\\ 0& -K& 1& 0\end{array}\right]*\left[\begin{array}{c}V\\ I\\ T\\ W\end{array}\right]=0$$. Find the uncertain 2-by-2 matrix B so that $$\left[\begin{array}{c}V\\ T\end{array}\right]=B*\left[\begin{array}{c}W\\ I\end{array}\right]$$.

R = ureal('R',1,'Percentage',[-10 40]); 
K = ureal('K',2e-3,'Percentage',[-30 30]); 
A = [1 -R 0 -K;0 -K 1 0]; 
Yidx = [1 3]; 
Uidx = [4 2]; 
B = imp2exp(A,Yidx,Uidx) 

B =

  Uncertain matrix with 2 rows and 2 columns.
  The uncertainty consists of the following blocks:
    K: Uncertain real, nominal = 0.002, variability = [-30,30]%, 2 occurrences
    R: Uncertain real, nominal = 1, variability = [-10,40]%, 1 occurrences

Type "B.NominalValue" to see the nominal value, "get(B)" to see all properties, and "B.Uncertainty" to interact with the uncertain elements.

Scalar Dynamic System Constraint

Consider a standard single-loop feedback connection of controller C and an uncertain plant P, described by the equations e = r-y; u = Ce; f = d+u; y = Pf.

P = tf([1],[1 0]); 
C = tf([2*.707*1 1^2],[1 0]); 
A = [1 -1 0 0 0 -1;0 -C 1 0 0 0;0 0 -1 -1 1 0;0 0 0 0 -P 1]; 
OutputIndex = [6;3;2;5];  % [y;u;e;f] 
InputIndex = [1;4];       % [r;d] 
Sys = imp2exp(A,OutputIndex,InputIndex); 
Sys.InputName = {'r';'d'}; 
Sys.OutputName = {'y';'u';'e';'f'};

pole(Sys)

ans = 4×1 complex

  -0.7070 + 0.7072i
  -0.7070 - 0.7072i
  -0.7070 + 0.7072i
  -0.7070 - 0.7072i

stepplot(Sys)

Input Arguments

A — Implicit system

static or dynamic input/output model

Implicit system, specified as a static, or dynamic input/output model. A can be:

• A numeric LTI model such as tf, zpk, ss, or frd object

• An uncertain model such as umat, uss, or ufrd object

The number of rows in model A must equal the length of the index vector yidx.

yidx — Indices to partition implicit model channels into output signals

vector

Indices to partition the model channel into output signals, specified as a vector. The index vectors yidx and uidx specify how to partition the input channels of A into y and u signals, respectively.

If [yidx,uidx] does not include all inputs of A, imp2exp excludes the missing y channels from the output model B. In such cases, imp2exp retains only a subset B(I,:) of the outputs/rows of B and does not affect how the explicit model B is computed.

uidx — Indices to partition implicit model channels into input signals

vector

Indices to partition the model channel into input signals, specified as a vector. The index vectors yidx and uidx specify how to partition the input channels of A into y and u signals, respectively.

Output Arguments

B — Explicit system

static or dynamic input/output model

Explicit system, returned as a static, or dynamic input/output model. The output explicit model B is of the same subclass as model A. For instance, if A is specified as an ss model, then imp2exp also returns B as an ss model.