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.

The constraint matrix A can be a double, ss, tf, zpk and frd object as well as an uncertain object, including umat, uss and ufrd. The result B will be of the same class.

Examples

Scalar Algebraic Constraint

Consider the constraint 4y + 7u = 0. Solving for y gives y = 1.75u. You form the equation using imp2exp:

A = [4 7]; 
Yidx = 1; 
Uidx = 2; 

and then

B = imp2exp(A,Yidx,Uidx) 
B = 
   -1.7500 

yields B equal to -1.75.

Matrix Algebraic Constraint

Consider two motor/generator constraints among 4 variables [V;I;T;W], namely [1 -1 0 -2e-3;0 -2e-3 1 0]*[V;I;T;W] = 0. You can find the 2-by-2 matrix B so that [V;T] = B*[W;I] using imp2exp.

A = [1 -1 0 -2e-3;0 -2e-3 1 0]; 
Yidx = [1 3]; 
Uidx = [4 2]; 
B = imp2exp(A,Yidx,Uidx) 
B = 
    0.0020    1.0000 
         0    0.0020 

You can find the 2-by-2 matrix C so that [I;W] = C*[T;V]

Yidx = [2 4]; 
Uidx = [3 1]; 
C = imp2exp(A,Yidx,Uidx) 
C = 
         500           0 
     -250000         500 

Uncertain Matrix Algebraic Constraint

Consider two uncertain motor/generator constraints among 4 variables [V;I;T;W], namely [1 -R 0 -K;0 -K 1 0]*[V;I;T;W] = 0. You can find the uncertain 2-by-2 matrix B so that [V;T] = B*[W;I].

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) 
UMAT: 2 Rows, 2 Columns 
  K: real, nominal = 0.002, variability = [-30  30]%, 2 occurrences 
  R: real, nominal = 1, variability = [-10  40]%, 1 occurrence     

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)

Algorithms

The number of rows of A must equal the length of yidx.