ROUTINES NOLIACPA

1 Outils

1.1 Extrank [Essen92]

Obtains the row rank of a matrix even if its entries are not rational

Restruct

2.1 Vectdegre   [Graciani94, pp.35 and 100], [Glumineau96a],[Conte93]

It calls recursively "degre" to compute the rel.deg. for each component. It gives them as a vector of integers.

It yields also the decoupling matrix

 

2.1.1 Degre [Graciani94, pp.34 et 98]

computes the relative degree of a function f(x)

it also gives the function f(x) under the affine representation a(x)+b(x)u

it uses the b(x) matrix to determine the relative degree

 

Separe   [Graciani94, pp.31 and 92]

Transform the system to analyze into an  affine form (if possible)

dy(ri)/dt(i) = f(x) +g(x)u

g

Dert

time derivative

1. Converga

See also: Converga.

2.2 Interact    [Graciani94, pp.35 and 100 ],[Di Benedetto94],[Glumineau92]

Interactor (or inversion) algorithm

2.2.1 Vectdegre

See also: Vectdegre.

2.2.2 Extrank

See also: Extrank.

2.2.3 Extgausselim   [Essen92]

Gauss Jordan elimination extended to handle  non-rational entries.

2.3 Converga

Transform the geometric syntax for non linear system into algebraic form (not used at this time)

2.4 Ordre [Graciani94, pp.35 and 102], [Glumineau89]

See also: Interact.

Computes the essential orders of a right-invertible system.

Resaccess [Aranda94]

Computes the H_k subspaces

Test the strong accessibility

3.1 Converga

3.2 Conshk [Graciani94, pp.36 and 94 ]

Computes the H_k and k* and H_inf

3.2.1 Deriverw [Graciani94, p.31 and 92 ]

See also: Deriverw.

Dert   [Graciani94, pp.30 and 92]

See also: Dert.

Converga

3.2.2 Extrank [Essen92]

See also: Extrank.

3.3 Cns   [Graciani94, p.36 and 95 ]

Checks the necessary and sufficient condition for integrability of Hk

3.3.1 Conshk [Graciani94, p.36 and 94 ]

See also: Conshk.

3.3.2 Integrab [Graciani94, p. 31 and 93]

Test the intergrability of a matrix mxn which represents the coefficients of a 1-form

Wedge product [Maple]

Simpform  [Maple, difform package]

3.4 Access [Graciani94, pp. 37 and 97 ]

Computes the Brunovsky's invariants and controllability indexes

3.4.1 Conshk [Graciani94, pp.36 and 94 ]

See also: Conshk.

3.5 Conswi [Graciani94, pp. 36 and 96 ]

Computes the differential forms wi of the linearising outputs

3.5.1 Extrank  [Essen92]

3.5.2 Kderiverw [Graciani94, pp. 31 and 93]

Computes the k-th time-derivative of a 1-form

Deriverw  [Graciani94, p. 31 and 93 ]

Computes the time-derivative of a one-form

3.5.3 Conshk [Graciani94, pp.36 and 94 ]

See also: Conshk.

3.6 Cs [Graciani94, pp. 37 and 98 ]

Sufficient condition for dynamic feedback input/state linearisation

(Lack of zero dynamics)

3.6.1 Conswi [Graciani94, pp.36 and 94 ]

3.6.2 Integrab [Graciani94, pp.35 and 100 ]

See also: Integrab.

Rescomp [Glumineau92]

4.1 Converga

4.2 Comp  [Graciani94, pp. 35 and 103 ]

Computes the dynamic decoupling compensator

It also gives the input/output linearisation

It does not compute de zero dynamics

 

4.2.1 Interact [Graciani94, p.35 and 100 ]

See also: Interact.

4.2.2 Ordre [Graciani94, p. 35 and 102 ],[Glumineau89]

See also: Ordre.

Gioi [Danjou95, pp.8, 19 and 61], [Plestan97],[Lopez99]

Linearization by generalized injection of input/output (and their time-derivatives) functions Phi(y,dy/dt,...,u,du/dt,...) for multivariable case

 

5.1 Ss2de2 [Danjou95, pp.19 and 61 ], [ Vignaud97 p.122]

Computation of Input/output differential equations of an obervable nonlinear system.

5.2 Intd [Danjou95, ]

Gioiam [Vignaud97 pp.78 and 112][Plestan97][lopez99]

Linearization by generalized injection of [input (and its time-derivatives)]/output functions Phi(y,u,du/dt,...) for multivariable case

 

6.1 Conv [Vignaud97 pp.68]

See also: Conv.

6.2 Ss2de2 [Danjou95, pp. 22 and 61], [Vignaud97 p.122]

See also: Ss2de.

6.3 Intd [Danjou95, pp. 20 and 61]

Integration of an one form with respect to several variables

6.4 Ioiwde [Vignaud97 pp.74 and 103]

Input/Output Injection Without Differential Equations.

System linearization via generalized input/output injection for a MIMO system.

Looks for a transformation to write a nonlinear system under the linear form via additive injection of input/ouput functions Phi(u,y) without computation of I/O equations.

 

6.5 Obser [Vignaud97 pp.69]

See also: Obser.

6.6 Intd [Danjou95, pp. 20 and 61]

See also: Intd.

6.7 Defform[Maple Help]

6.8 Simpform[Maple Help]

6.9 Liesymm [Maple Help]

6.9.1 Setup [Maple Help]

6.9.2 Getcoeff [Maple Help]

6.10 Extrank [Essen92]

6.11 Linsolve [Maple Help]

6.12 Ogioais (i.e. Sgioia) [Danjou95, pp.21 and 61 ]

The same as GIOIAS with a possible output transformation.

 

6.13 Gioias (i.e. Gioia2) [Danjou95, pp.18 and61  ]

See also: Gioias.

6.14 Obser [Vignaud97 pp.69]

Check the observability conditions and compute the observability indices

6.15 Gioias [Danjou95, pp. and  ]

Generalized Input/Output Injection Algorithm for Single output systems.

Looks for a transformation to write a nonlinear system under the linear form via generalized additive input/ouput injection Phi(u,du/dt,...,y,dy/dt,...).

6.16  Conv [Vignaud97 pp.68 and 98]

See also: Conv.

6.17  Ss2de   [Danjou95, pp.19 and 61  ]

Computes the input-output differential equation from the state space representation (state elimination problem)

If the system is  observable, there exists a solution. Whether the solution exists if it is not, is an open problem

 

HARD PROBLEM 

MUST BE CHECKED IF MAXIMA CAN DO IT!

6.18 Syslin2  [Danjou95, pp.20 and 61  ]

Computes the linearised system via the additive injection functions Phi_k

 6.19  Intd 

Integrates one 1-form

 7  Obser  [Vignaud97 pp .69]

Computes the observability indexes

Determines if the system is not completely observable

It reorders the outputs according to their observability index (decreasing)

 7.1  Extrank [Essen92]

See also: Extrank.

 7.2  Conv  [Vignaud97 pp. 68 and 98]

Converts a system from affine to general form

It is the inverse function of conversi

 


REFERENCES

[Aranda94]  Eduardo Aranda, Linéarisation par bouclage dynamique des systèmes non linéaires, Thèse de doctorat de l'Université de Nantes - École Centrale de Nantes, Mai 1994.

[Conte93] Conte G., Perdon A. M, Moog C. H., The differential  Field Associated to a General Analytic Nonlinear System, IEEE Trans. Automat. Contr., vol.38, pp 1120-1124, 1993.

[Danjou95] B. Danjou, J. Dulong, Observabilité des systèmes non linéaires et Calcul formel, Projet de Recherche option Automatique, Ecole Centrale de Nantes, Resp. A. Glumineau, 1995.

[De jager96] Bram de Jager, Symbolic analysis and design for nonlinear control systems, IFAC World Congress 30juin – 5 juillet, 1996.

[Desobry-01] Frédérc Desobry and Gwénaëlle Maillard, Développement en calcul formel d’une application pour l’observation des systèmes non linéaires,] Projet de spécialité –ECN- Encadrant : Ibrahim Souleiman, 2001.

[Di Benedetto94] M. Di Benedetto, A. Glumineau and C.H. Moog, The Nonlinear Interactor and its Application to Input-Output Decoupling, I.E.E.E. Trans. Aut. Contr., AC-39, n°6, pp. 1246-1250, 1994

[Essen92] Harm van Essen, Symbols speak louder than numbers, Faculteit Werktuigbouwkunde WFM 92.061, Technische Universiteit Eindhoven, Juin 1992.

[Glumineau89] A. Glumineau and C.H. Moog, The essential orders and the Nonlinear decoupling Problem, Int. J. Control, vol. 50, pp. 1825-1834, 1989.

[Glumineau92] Alain Glumineau, Solutions Algébriques pour l'Analyse et le Contrôle des Systèmes Non Linéaires, Thèse de Docteur Es Sciences, Université et Ecole Centrale de Nantes, Nantes, Novembre 1992.

[Glumineau96a]  A. Glumineau  and L. Graciani, Symbolic nonlinear analysis and control package, IFAC World Congress, 30juin – 5 juillet, 1996.

 [Glumineau96b] A. Glumineau, C.H.Moog and F.Plestan, New Algebro-geometric Conditions for the linearisation by Input-output injection, IEEE transaction on automatic control, Vol.41  No.4, avril 1996.

 

[Glumineau99] A. Glumineau and V. Lopez-Morales, Transformation to State Affine System and observer design, New Directions in Nonlinear observer design,  Lecture notes in Control and Information  Sciences, Springer, vol. 244, ISBN1-85233-134-8., pp. 59-69, 1999.

[Graciani94]  L. Graciani, "Commande Non linéaire et Calcul formel,  Rapport de DEA AIA, École Centrale de Nantes, Encadrant : Alain Glumineau, 6 septembre 1994.

[Isidori89]  A. Isidori,  Nonlinear Control Systems, 2 edition, Springer-Verlag, 1989

[Lopez99] V. Lopez-M., F. Plestan, A. Glumineau, Linearization by Completely Generalized Input Ouput Injection,  Kybernetika, vol.35, n° 6, pp. 793-802, 1999.

[Lopez01] V. Lopez-M., F. Plestan, A. Glumineau, An algorithm for the structural analysis of state space: synthesis of nonlinear observers., International Journal of Robust and Nonlinear Control, Vol. 11, Issue 12, pp. 1145-1160,  2001.

 

[Plestan97] F. Plestan et A. Glumineau, Linearization by generalized input-output injection, Systems Control Letters, vol. 31, pp. 115-128, 1997.

 

[Souleiman00] I. Souleiman, A. Glumineau, Equivalence and observers for state affine system, MTNS 2000, Mathematical Theory of Networks and systems, Perpignan, June 19-23, 2000.

 

[Souleiman01] I. Souleiman, A. Glumineau and G. Schreier, Direct Transformation of Nonlinear Systems into State Affine form. Nonlinear Control Systems, Proc. De  NOLCOS'01, 5th IFAC Symposium Nonlinear Control Systems, pp. 545-551, St. Petersburg, Russia,  4-6 Juillet  2001.

 

[Souleiman03] Souleiman, A. Glumineau and G. Schreier, Direct Transformation of Nonlinear Systems into State Affine Miso Form and Nonlinear Observers design, to appear in  I.E.E.E. Trans. Aut. Contr , 2003.

[Vignaud97]  Vincent Vignaud, Calcul Symbolique et Automatique Non Linéaire, Rapport de DEA AIA, École Centrale de Nantes, Encadrant : Alain Glumineau, 26 septembre 1997.