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`|                     NOLIACPA : NOn LInear Analysis and Control PAckage                   |`

`|          Institut de Recherche en Communications et Cybernétique de Nantes          |`

`|                         Information : Alain.Glumineau@irccyn.ec-nantes.fr                      |`

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`------------------------------------------------------------------------------------`

`|                              Study of nonlinear observers                            |`

`|           Linearization by generalized Input-Output injection           |`

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`                                                                           `

`1   -> Computation of the observability indices                                                                                                       `

``

`A)  SINGLE OUTPUT SYSTEM:                                                                                                                                      `

`2    -> Linearization by generalized I/O injection Phi(u, du/dt,..., y,dy/dt,...)                                                   `

`3    -> Linearization by generalized I/O injection Phi(u, du/dt,..., y,dy/dt,...) and with output transformations        `

``

`B)  MULTIVARIABLE SYSTEM:                                                                                                                                        `

`4   -> Linearization by I/O injection Phi(u,y) without computation of I/O equations                                            `

`5   -> Linearization by generalized I/O injection Phi(y, du/dt,...)                                                                           `

`6   -> Linearization by generalized I/O injection with time derivatives of the outputs                                        `

`                          Phi(u, du/dt,..., y,dy/dt,...)         `

``

`Enter your choice :                                                                                          `

>    2

`**** RESULTS ****`

` Considered system :`

dx/dt = matrix([[-K1*x1(t)*x2(t)-(Ra+Rf)/K*x1(t)+u1(t)], [-K2*x2(t)-x3(t)+K1/J*K*x1(t)^2], [0]])

y = matrix([[x1(t)]])

`Particular point (xnull) where algebraic and numeric ranks will be compared:`

{x1(t) = 0, x2(t) = 0, x3(t) = 0}

__________________________________________________________________

`******  Computation of OBSERVABILITY INDICES   ******`

__________________________________________________________________

`The outputs are:`

y = matrix([[x1(t)]])

`After renumbering (if necessary) Observability indices are `

`for output  1: 3`

`Observability matrix :`

matrix([[1, 0, 0], [-(K1*x2*K+Ra+Rf)/K, -K1*x1, 0], [(K1^2*x2^2*K^2*J+2*K1*x2*K*J*Ra+2*K1*x2*K*J*Rf+K1*K^2*K2*x2*J+K1*K^2*x3*J-3*K1^2*x1^2*K^3+Ra^2*J+2*Ra*J*Rf+Rf^2*J)/K^2/J, K1/K*(2*K1*x1*x2*K+2*x1*Ra...
matrix([[1, 0, 0], [-(K1*x2*K+Ra+Rf)/K, -K1*x1, 0], [(K1^2*x2^2*K^2*J+2*K1*x2*K*J*Ra+2*K1*x2*K*J*Rf+K1*K^2*K2*x2*J+K1*K^2*x3*J-3*K1^2*x1^2*K^3+Ra^2*J+2*Ra*J*Rf+Rf^2*J)/K^2/J, K1/K*(2*K1*x1*x2*K+2*x1*Ra...
matrix([[1, 0, 0], [-(K1*x2*K+Ra+Rf)/K, -K1*x1, 0], [(K1^2*x2^2*K^2*J+2*K1*x2*K*J*Ra+2*K1*x2*K*J*Rf+K1*K^2*K2*x2*J+K1*K^2*x3*J-3*K1^2*x1^2*K^3+Ra^2*J+2*Ra*J*Rf+Rf^2*J)/K^2/J, K1/K*(2*K1*x1*x2*K+2*x1*Ra...
matrix([[1, 0, 0], [-(K1*x2*K+Ra+Rf)/K, -K1*x1, 0], [(K1^2*x2^2*K^2*J+2*K1*x2*K*J*Ra+2*K1*x2*K*J*Rf+K1*K^2*K2*x2*J+K1*K^2*x3*J-3*K1^2*x1^2*K^3+Ra^2*J+2*Ra*J*Rf+Rf^2*J)/K^2/J, K1/K*(2*K1*x1*x2*K+2*x1*Ra...

`Observability matrix at particular point :`

matrix([[1., 0., 0.], [-1.*(Ra+Rf)/K, 0., 0.], [(Ra^2+2.*Ra*Rf+Rf^2)/K^2, -1.*K1*u1(t), 0.]])

`WARNING : the formal rank is different `

`from the local rank computed with the initial conditions||`

`Two cases are possibles : 1) The local conditions are singular`

`or 2) There are simplification problems, and then we should`

`enter numerical values to solve them||`

`---------------------------------------------------`

`Linearization by generalized Input-Output injection   `

`with output and inputs time derivatives`

`---------------------------------------------------`

-diff(y(t),`$`(t,3))*` = `*(u1(t)*diff(y(t),`$`(t,2))*J*y(t)-2*J*u1(t)*diff(y(t),t)^2+2*K1^2*y(t)^4*K*diff(y(t),t)+u1(t)*K2*J*y(t)*diff(y(t),t)+2*J*diff(u1(t),t)*y(t)*diff(y(t),t)+K2*y(t)^2*diff(y(t),`...
-diff(y(t),`$`(t,3))*` = `*(u1(t)*diff(y(t),`$`(t,2))*J*y(t)-2*J*u1(t)*diff(y(t),t)^2+2*K1^2*y(t)^4*K*diff(y(t),t)+u1(t)*K2*J*y(t)*diff(y(t),t)+2*J*diff(u1(t),t)*y(t)*diff(y(t),t)+K2*y(t)^2*diff(y(t),`...
-diff(y(t),`$`(t,3))*` = `*(u1(t)*diff(y(t),`$`(t,2))*J*y(t)-2*J*u1(t)*diff(y(t),t)^2+2*K1^2*y(t)^4*K*diff(y(t),t)+u1(t)*K2*J*y(t)*diff(y(t),t)+2*J*diff(u1(t),t)*y(t)*diff(y(t),t)+K2*y(t)^2*diff(y(t),`...

`--`

`With s=0 et w=0`

`The NSC of linearization is not verified`

`--`

`With s=0 et w=1`

`The NSC of linearization is not verified`

`--`

`With s=0 et w=2`

`The NSC of linearization is not verified`

`--`

`With s=0 et w=3`

`The NSC of linearization is not verified`

`--`

`With s=1 et w=0`

`The NSC of linearization is not verified`

`--`

`With s=1 et w=1`

`The NSC of linearization is verified`

`--`

omega[2]*`=`

-1/y(t)*(d(diff(y(t),t))*u1(t)+d(diff(y(t),t))*K2*y(t)-3*d(diff(y(t),t))*diff(y(t),t)-d(diff(u1(t),t))*y(t))

omega[3]*`=`

-1/2*(-4*d(diff(y(t),t))*J*u1(t)*diff(y(t),t)+4*d(diff(y(t),t))*K1^2*y(t)^4*K+2*d(diff(y(t),t))*u1(t)*K2*J*y(t)+2*d(diff(y(t),t))*J*diff(u1(t),t)*y(t)-4*d(diff(y(t),t))*K2*y(t)*J*diff(y(t),t)+3*d(diff(...
-1/2*(-4*d(diff(y(t),t))*J*u1(t)*diff(y(t),t)+4*d(diff(y(t),t))*K1^2*y(t)^4*K+2*d(diff(y(t),t))*u1(t)*K2*J*y(t)+2*d(diff(y(t),t))*J*diff(u1(t),t)*y(t)-4*d(diff(y(t),t))*K2*y(t)*J*diff(y(t),t)+3*d(diff(...

phi[2]*` = `

-1/2*(-2*diff(u1(t),t)*y(t)+2*u1(t)*diff(y(t),t)+2*K2*y(t)*diff(y(t),t)-3*diff(y(t),t)^2)/y(t)

phi[3]*` = `

-1/2*(-2*J*u1(t)*diff(y(t),t)^2+4*K1^2*y(t)^4*K*diff(y(t),t)+2*u1(t)*K2*J*y(t)*diff(y(t),t)+2*J*diff(u1(t),t)*y(t)*diff(y(t),t)-2*K2*y(t)^2*diff(u1(t),t)*J-2*K2*y(t)*J*diff(y(t),t)^2+J*diff(y(t),t)^3)/...
-1/2*(-2*J*u1(t)*diff(y(t),t)^2+4*K1^2*y(t)^4*K*diff(y(t),t)+2*u1(t)*K2*J*y(t)*diff(y(t),t)+2*J*diff(u1(t),t)*y(t)*diff(y(t),t)-2*K2*y(t)^2*diff(u1(t),t)*J-2*K2*y(t)*J*diff(y(t),t)^2+J*diff(y(t),t)^3)/...

`--`

`With the local coordinates transformation :`

xi1*`=`*x1(t)

-xi2*`=`*(K1*x1(t)*x2(t)*K+x1(t)*Ra+x1(t)*Rf-u1(t)*K)/K

-1/2*xi3*`=`*(-2*K1*x1(t)^2*K^2*x3(t)*J+2*K1*x2(t)*K*J*x1(t)^2*Ra+2*K1*x2(t)*K*J*x1(t)^2*Rf+2*K1^2*x1(t)^4*K^3+Ra^2*J*x1(t)^2+Rf^2*J*x1(t)^2+J*u1(t)^2*K^2+K1^2*x2(t)^2*K^2*J*x1(t)^2+2*Ra*J*x1(t)^2*Rf-2...
-1/2*xi3*`=`*(-2*K1*x1(t)^2*K^2*x3(t)*J+2*K1*x2(t)*K*J*x1(t)^2*Ra+2*K1*x2(t)*K*J*x1(t)^2*Rf+2*K1^2*x1(t)^4*K^3+Ra^2*J*x1(t)^2+Rf^2*J*x1(t)^2+J*u1(t)^2*K^2+K1^2*x2(t)^2*K^2*J*x1(t)^2+2*Ra*J*x1(t)^2*Rf-2...
-1/2*xi3*`=`*(-2*K1*x1(t)^2*K^2*x3(t)*J+2*K1*x2(t)*K*J*x1(t)^2*Ra+2*K1*x2(t)*K*J*x1(t)^2*Rf+2*K1^2*x1(t)^4*K^3+Ra^2*J*x1(t)^2+Rf^2*J*x1(t)^2+J*u1(t)^2*K^2+K1^2*x2(t)^2*K^2*J*x1(t)^2+2*Ra*J*x1(t)^2*Rf-2...

`the new system equation is:`

Diff(xi1,t)*`=`

xi2

Diff(xi2,t)*`=`

xi3-1/2*(-2*diff(u1(t),t)*y(t)+2*u1(t)*diff(y(t),t)+2*K2*y(t)*diff(y(t),t)-3*diff(y(t),t)^2)/y(t)

Diff(xi3,t)*`=`

-1/2*(-2*J*u1(t)*diff(y(t),t)^2+4*K1^2*y(t)^4*K*diff(y(t),t)+2*u1(t)*K2*J*y(t)*diff(y(t),t)+2*J*diff(u1(t),t)*y(t)*diff(y(t),t)-2*K2*y(t)^2*diff(u1(t),t)*J-2*K2*y(t)*J*diff(y(t),t)^2+J*diff(y(t),t)^3)/...
-1/2*(-2*J*u1(t)*diff(y(t),t)^2+4*K1^2*y(t)^4*K*diff(y(t),t)+2*u1(t)*K2*J*y(t)*diff(y(t),t)+2*J*diff(u1(t),t)*y(t)*diff(y(t),t)-2*K2*y(t)^2*diff(u1(t),t)*J-2*K2*y(t)*J*diff(y(t),t)^2+J*diff(y(t),t)^3)/...

y*`=`*xi1

`this corresponds to the inverse transformation of coordinates:`

x1(t) = xi1

x2(t) = -(xi2*K+xi1*Ra+xi1*Rf-u1(t)*K)/K1/xi1/K

x3(t) = 1/2*(-2*J*K2*xi1*K*u1(t)+2*xi3*K*J*xi1+J*xi2^2*K+2*J*K2*xi1^2*Ra+2*J*K2*xi1^2*Rf+2*K1^2*xi1^4*K^2)/K1/xi1^2/K/J

`**** FIN ****`

`           `

`  End of  NOLIACPA   `

`  For a new computation do restart; and read noliacpa;    `

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>