Automated Nonlinear Control Structure Design by Domain of Attraction Maximization with Eigenvalue and Frequency Domain Specifications

  • Elias ReichensdörferEmail author
  • Dirk Odenthal
  • Dirk Wollherr
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 613)


This work proposes a new method for nonlinear control structure design for nonlinear dynamical systems by grammatical evolution. The optimization is based on a novel fitness function which implements three different design objectives. First, the closed loop eigenvalues of the linearized system dynamics are restricted to be located in a predefined region of the complex plane. Second, design specifications on the frequency magnitude of the sensitivity transfer functions of the linearized system dynamics are imposed. Third, the estimated domain of attraction of the nonlinear closed loop system dynamics is maximized by evaluating a quadratic Lyapunov function, obtained by the solution of the Lyapunov equation, with a Monte-Carlo sampling algorithm. Additionally, a general formula for the centroid of a design region for pole placement is derived analytically and embedded in the optimization framework. The generated controllers are evaluated on a common, nonlinear benchmark system. It is shown that the proposed method can efficiently generate control laws which meet the imposed design specifications on the closed loop system while maximizing the domain of attraction.


Nonlinear control structure design Lyapunov equation Grammatical evolution Robust control Domain of attraction maximization Frequency domain specifications Stability centroid 


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Elias Reichensdörfer
    • 1
    • 3
    Email author
  • Dirk Odenthal
    • 2
  • Dirk Wollherr
    • 3
  1. 1.BMW GroupMunichGermany
  2. 2.BMW M GmbHGarching near MunichGermany
  3. 3.Chair of Automatic Control EngineeringTechnical University of MunichMunichGermany

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