Bilinearization of Nonlinear Systems

  • H. Schwarz
  • H. T. Dorissen
  • L. Guo
Part of the Advances in Simulation book series (ADVS.SIMULATION, volume 1)


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
$$\begin{array}{l} \left. \begin{array}{l} \dot x(t) = f(x(t)) + \mathop \Sigma \limits_{i = 1}^m {g_i}(x) {u_i}(t)\\ y(t) = g(x) \end{array} \right\},\\ where x \in {R_n},u \in {R_m}and y \in {R_p} \end{array}$$
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
$$\left. {\matrix{ {{\rm{\dot x}}\left( {\rm{t}} \right)} \hfill & = \hfill & {{\rm{Ax}}\left( {\rm{t}} \right) + \sum\limits_{{\rm{i = 1}}}^{\rm{m}} {{{\rm{N}}_{\rm{i}}}{\rm{x}}\left( {\rm{t}} \right){{\rm{u}}_{\rm{i}}}\left( {\rm{t}} \right) + {\rm{Bu}}\left( {\rm{t}} \right)} } \hfill \cr {{\rm{y}}\left( {\rm{t}} \right)} \hfill & = \hfill & {{\rm{Cx}}\left( {\rm{t}} \right)} \hfill \cr } } \right\},$$
where x∈Rn, u∈Rm and y∈Rp and the system matrices A,B,C,Ni 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.


Step Response Nonlinear Control System Bilinear System Bilinear Model Hydraulic Drive 
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Copyright information

© Akademie-Verlag Berlin 1988

Authors and Affiliations

  • H. Schwarz
  • H. T. Dorissen
  • L. Guo
    • 1
    • 2
  1. 1.Department of Measurement and ControlFaculty of Mechanical Engineering, University of DuisburgDuisburg 1Germany
  2. 2.VW-foundation, Hannover, and the Deutsche ForschungsgemeinschaftBonnGermany

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