Systems Analysis and Simulation I pp 89-96 | Cite as

# Bilinearization of Nonlinear Systems

Conference paper

## Abstract

In many control problems of practical interest, the cause-effect relation of a plant has to be modelled by a class of nonlinear differential equations. Because of the difficulty in dealing with general nonlinear equations, it is a normal way to linearize it regarding the operating steady state, using Taylor series expansion, and to deal with the resulting linear system. However, the linearized model is sometimes inadequate, and there is a growing need for a better approximation of the substantial nonlinear system when robust behaviour of nonlinear systems is required over the complete operating range, what means, that the nonlinear dynamics of the plant must be handled. One method to deal with systems of which the physical cause-effect relations may be modelled by ordinary differential equations of this type is “bilinearization”, with the goal to obtain approximations of the second order of the nonlinear equations, while linearization leads to first order approximations (Yousuf, 1984; Beater, 1987). The method of bilinearization will yield considerably better results for all those physical and technical systems, where the most relevant nonlinearity consists in a multiplicative connection of an energy and/or mass flow with the control variable. An essential advantage of this procedure is, that the system theory of bilinear systems where x∈R

$$\begin{array}{l}
\left. \begin{array}{l}
\dot x(t) = f(x(t)) + \mathop \Sigma \limits_{i = 1}^m {g_i}(x) {u_i}(t)\\
y(t) = g(x)
\end{array} \right\},\\
where x \in {R_n},u \in {R_m}and y \in {R_p}
\end{array}$$

(1.1)

$$\left. {\matrix{ {{\rm{\dot x}}\left( {\rm{t}} \right)} \hfill & = \hfill & {{\rm{Ax}}\left( {\rm{t}} \right) + \sum\limits_{{\rm{i = 1}}}^{\rm{m}} {{{\rm{N}}_{\rm{i}}}{\rm{x}}\left( {\rm{t}} \right){{\rm{u}}_{\rm{i}}}\left( {\rm{t}} \right) + {\rm{Bu}}\left( {\rm{t}} \right)} } \hfill \cr {{\rm{y}}\left( {\rm{t}} \right)} \hfill & = \hfill & {{\rm{Cx}}\left( {\rm{t}} \right)} \hfill \cr } } \right\},$$

(1.2)

_{n}, u∈R_{m}and y∈R_{p}and the system matrices A,B,C,N_{i}of appropriate order, is well established (Rugh, 1981; Schwarz, 1987). But on the other hand very few applications of this theoretic procedure for technical plants are known only.## Keywords

Step Response Nonlinear Control System Bilinear System Bilinear Model Hydraulic Drive
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- Beater, P. (1987). Zur Regelung nichtlinearer Systeme mit Hilfe bilinearer Modelle. Phd-Thesis University Duisburg (FRG). Düsseldorf: VDI-Verlag.Google Scholar
- Dorißen, H.-T. (1986). Zur mathematischen Realisierung bilinearer Systeme. Diploma-Thesis, MSRT, University Duisburg.Google Scholar
- Guo, L. (1987). Bilineare Modellbildung und Messung an dem sekundärgeregelten hydraulischen Rotationsantrieb (A4VSO40HS). Research-Report, MSRT, University Duisburg.Google Scholar
- Isidori, A. and Ruberti, A. (1973). Realization Theory of Bilinear Systems, in Mayne, D.Q. and Brockett, R. (editors). (1973). Geometric Methods in System Theory. Boston: Reidel.Google Scholar
- Isidori, A. (1985). Nonlinear Control Systems: An Introduction. Berlin: Springer.MATHCrossRefGoogle Scholar
- Marner, W. and Ulm, M. (1988). A Modern Concept for the Digital Control of Hydraulic Gears. 5th IAVD Congress on vehicle design components. 9-11 March 1989, Geneve (Switzerland).Google Scholar
- Schwarz, H. (1987). Stability of Discrete-Time Equivalent Homogeneous Bilinear Systems. Control Theory and Advanced Technology, 3,3, 263–269.MathSciNetGoogle Scholar
- Schwarz, H. and Dorissen, H. T. (1988). Systemidentification of Bilinear Systems via Realization Theory and its Application. (Contributed paper under review for publication).Google Scholar
- Yousuf, M. (1984). Regelung eines hydraulischen Antriebes mittels bilinearer Systemmodelle. Diploma-Thesis, MSRT, University Duisburg.Google Scholar

## Copyright information

© Akademie-Verlag Berlin 1988