What calculation is done to find the gains (k) of the zpk function?

State space matrix:

A = [ -0.669951646484827 1.28424762631945e-09 0.011224152177422 -0.137266467613442 0 8.3380759857086e-07;

1 0 0 0 0 0;

-15.5796093146461 -9.77564302835082 -0.0172201697024891 0.0437085492873758 0 -9.43217074987179e-05;

195.456682375095 -0.779222433532193 -0.0745486085894737 -0.730188697017893 0 -0.00127843959431113;

0 -2.89356048753657e-05 0.996838168810935 0.0794585752910095 0 0;

0 -197.035459421598 -0.0794585753231511 0.996838168812305 0 0];

B = [ -36.5034154566994;

0;

3.76413630113738;

-25.9051875864034;

0;

0];

C =[ 1 0 0 0 0 0;

195.456682375095 -0.779222433532193 -0.0745486085894737 -0.730188697017893 0 -0.00127843959431113];

D = [ 0;

-25.9051875864034];

[z,p,k] = ss2zp(A,B,C,D)

z =

Column 1

-0.62671948453317 + 0i

-0.0217530033855665 + 0i

-0.000365798857384335 + 0i

-5.92556062974436e-09 + 0i

5.92549129068837e-09 + 0i

Inf + 0i

Column 2

0 + 0i

-276.100495151319 + 0i

-0.0085919008871075 + 0.0722792555820489i

-0.0085919008871075 - 0.0722792555820489i

-0.00147218583594221 + 0i

-2.50275860753844e-15 + 0i

p =

0 + 0i

-0.696733151596068 + 5.19575335398669i

-0.696733151596068 - 5.19575335398669i

-0.0116313338403144 + 0.0872215466769184i

-0.0116313338403144 - 0.0872215466769184i

-0.000631542332441605 + 0i

k =

-36.5034154566994

-25.9051875864034

How is the calculation done to find the earnings: -36.5034154566994 and -25.9051875864034?

Answer 1

[Star Strider]

I am not certain that it is documented anywhere (I cannot find it), however it appears to be the coefficient of the highest-order component of the numerator of the transfer function representation.

Example —

A = [ -0.669951646484827      1.28424762631945e-09         0.011224152177422        -0.137266467613442         0       8.3380759857086e-07;
                         1                                          0                                              0                                   0                        0                         0;
         -15.5796093146461        -9.77564302835082          -0.0172201697024891        0.0437085492873758          0     -9.43217074987179e-05;
          195.456682375095        -0.779222433532193       -0.0745486085894737        -0.730188697017893            0      -0.00127843959431113;
                         0                    -2.89356048753657e-05        0.996838168810935        0.0794585752910095          0                         0;
                         0                       -197.035459421598          -0.0794585753231511         0.996838168812305           0                         0];
B = [    -36.5034154566994;
                         0;
          3.76413630113738;
         -25.9051875864034;
                         0;
                         0];
C =[                1                                             0                         0                                                   0                               0                         0;
        195.456682375095        -0.779222433532193       -0.0745486085894737       -0.730188697017893                0      -0.00127843959431113];
D = [ 0;
      -25.9051875864034];
[z,p,k] = ss2zp(A,B,C,D)

z = 
   1.0e+02 *

  -0.0063 + 0.0000i   0.0000 + 0.0000i
  -0.0002 + 0.0000i  -2.7610 + 0.0000i
  -0.0000 + 0.0000i  -0.0001 + 0.0007i
  -0.0000 + 0.0000i  -0.0001 - 0.0007i
   0.0000 + 0.0000i  -0.0000 + 0.0000i
      Inf + 0.0000i  -0.0000 + 0.0000i

p = 
   0.0000 + 0.0000i
  -0.6967 + 5.1958i
  -0.6967 - 5.1958i
  -0.0116 + 0.0872i
  -0.0116 - 0.0872i
  -0.0006 + 0.0000i

k = 2×1

-36.5034 -25.9052

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format shortG
[n,d] = ss2tf(A,B,C,D)

n = 2×7

1.0e+00 * 0 -36.503 -23.685 -0.50631 -0.00018204 -7.2022e-18 9.5167e-21 -25.905 -7152.9 -133.57 -38.076 -0.055788 -4.8735e-16 0

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

d = 1×7

1 1.4174 27.522 0.66746 0.2132 0.00013438 0

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.

[Rafaella Barrêto Campos]

But through which calculation is matrix B found?

[Star Strider]

The ‘B’ matrix (and the others) are provided as arguments to the ss2zp function.

We have no idea how you calculated them.

[Rafaella Barrêto Campos]

Regardless of values, what is the formula for calculating matrix B?

[Star Strider]

It is however you calculated it!

We have no idea what you are doing or how you designed that system.

Answer 2

[Sam Chak]

Hi @Rafaella Barrêto Campos

Short Answer:

Due to your specific state-space configuration, k(1) is derived from the first element of the input matrix , while k(2) is derived from the second element of the direct matrix .

format long g

A = [ -0.669951646484827      1.28424762631945e-09         0.011224152177422        -0.137266467613442         0       8.3380759857086e-07;
                         1                                          0                                              0                                   0                        0                         0;
         -15.5796093146461        -9.77564302835082          -0.0172201697024891        0.0437085492873758          0     -9.43217074987179e-05;
          195.456682375095        -0.779222433532193       -0.0745486085894737        -0.730188697017893            0      -0.00127843959431113;
                         0                    -2.89356048753657e-05        0.996838168810935        0.0794585752910095          0                         0;
                         0                       -197.035459421598          -0.0794585753231511         0.996838168812305           0                         0];
B = [    -36.5034154566994;
                         0;
          3.76413630113738;
         -25.9051875864034;
                         0;
                         0];
C = [                1                                             0                         0                                                   0                               0                         0;
        195.456682375095        -0.779222433532193       -0.0745486085894737       -0.730188697017893                0      -0.00127843959431113];
D = [ 0;
      -25.9051875864034];
sys = ss(A, B, C, D);
k1  = B(1)

k1 = 
         -36.5034154566994

k2  = D(2)

k2 = 
         -25.9051875864034

Long Answer:

MATLAB does not show the steps to obtain the solution; it typically provides only the numerical solution in most cases.

The state-space representation must be converted to transfer matrix form using the following formula, which can be found in any Modern Control Theory textbook:

.

In this equation, the transfer matrix relates the number of outputs to the number of inputs . The output matrix indicates that there are two outputs, while the input matrix indicates that there is only one input. Consequently, the transfer matrix will consist of two transfer functions.

The gain vector k obtained from the zpk() command is an array of the coefficients of the highest degree of the numerator in each transfer function.

G   = tf(sys)

G =
 
  From input to output...
         -36.5 s^4 - 23.68 s^3 - 0.5063 s^2 - 0.000182 s + 3.613e-20
   1:  ---------------------------------------------------------------
       s^5 + 1.417 s^4 + 27.52 s^3 + 0.6675 s^2 + 0.2132 s + 0.0001344
 
       -25.91 s^5 - 7153 s^4 - 133.6 s^3 - 38.08 s^2 - 0.05579 s - 2.377e-18
   2:  ---------------------------------------------------------------------
          s^5 + 1.417 s^4 + 27.52 s^3 + 0.6675 s^2 + 0.2132 s + 0.0001344
 
Continuous-time transfer function.

[num, den] = tfdata(G, 'v')

num = 2x1 cell array
    {[            0 -36.5034154566995 -23.6848135464053 -0.506311190296269 -0.000182040605087803 3.61253687043047e-20]}
    {[-25.9051875864034 -7152.91840645244 -133.573644890081 -38.0755416738836 -0.0557875976251336 -2.376582946259e-18]}

den = 2x1 cell array
    {[1 1.41736051320521 27.5223433539825 0.667458527478527 0.213195050723291 0.000134382418944796]}
    {[1 1.41736051320521 27.5223433539825 0.667458527478527 0.213195050723291 0.000134382418944796]}

%% Extract the data from the cell array
k1  = num{1}(2)

k1 = 
         -36.5034154566995

k2  = num{2}(1)

k2 = 
         -25.9051875864034

[Rafaella Barrêto Campos]

But through which calculation is matrix B found?

[Sam Chak]

Huh? 😵‍

[Rafaella Barrêto Campos]

That's right. But through which calculation is matrix B found?

[Sam Chak]

Isn't the input matrix given in the first place?

B = [-36.5034154566994;
                     0;
      3.76413630113738;
     -25.9051875864034;
                     0;
                     0]

B = 6×1

-36.5034 0 3.7641 -25.9052 0 0

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[Rafaella Barrêto Campos]

Surely there is a specific formula to calculate B, can you tell me?

[Rafaella Barrêto Campos]

X_trim =

0.0156566822191464

0.0795424262493709

196.412468864823

15.6561280499959

0

-3048

U_trim =

-0.091567106993385

[A,B,C,D]=linmod(modelName,X_trim,U_trim)

A =

Columns 1 through 2

-0.669951646484827 1.28424762631945e-09

1 0

-15.5796093146461 -9.77564302835082

195.456682375095 -0.779222433532193

0 -2.89356048753657e-05

0 -197.035459421598

Columns 3 through 4

0.011224152177422 -0.137266467613442

0 0

-0.0172201697024891 0.0437085492873758

-0.0745486085894737 -0.730188697017893

0.996838168810935 0.0794585752910095

-0.0794585753231511 0.996838168812305

Columns 5 through 6

0 8.3380759857086e-07

0 0

0 -9.43217074987179e-05

0 -0.00127843959431113

0 0

0 0

B =

-36.5034154566994

0

3.76413630113738

-25.9051875864034

0

0

C =

Columns 1 through 2

1 0

195.456682375095 -0.779222433532193

Columns 3 through 4

0 0

-0.0745486085894737 -0.730188697017893

Columns 5 through 6

0 0

0 -0.00127843959431113

D =

0

-25.9051875864034

How is matrix B calculated?

[Sam Chak]

Hi @Rafaella Barrêto Campos

Your new query regarding the computation of matrix appears to be unrelated to your original question about how to obtain the gain k, which has been adequately explained above by @Star Strider and me.

Since the technical explanations have been provided, please accept the answer that you find most satisfactory and vote for other helpful responses. You may then continue the discussion in your thread titled "function zpk and linmod," as the "short" answer can be found there.

Accepting answers to questions that have been adequately addressed, rather than leaving their status 'open,' will increase the likelihood of unanswered question receiving favorable responses from other users or experts.