Equivalence of robust stabilization and robust performance via feedback

  • Joseph A. Ball
  • Quanlei Fang
  • Gilbert J. Groenewald
  • Sanne ter Horst
Original Article


One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified L 2-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to zero, it is clear that any criterion for robust performance also produces a criterion for robust stability. Counter-intuitively, as a consequence of the so-called Main Loop Theorem, application of a result on robust stability to a feedback configuration with an artificial full-block uncertainty operator added in feedback connection between the error and disturbance signals produces a result on robust performance. The main result here is that this performance-to-stabilization reduction principle must be handled with care for the case of dynamic feedback compensation: casual application of this principle leads to the solution of a physically uninteresting problem, where the controller is assumed to have access to the states in the artificially-added feedback loop. Application of the principle using a known more refined dynamic-control robust stability criterion, where the user is allowed to specify controller partial-state dimensions, leads to correct robust-performance results. These latter results involve rank conditions in addition to linear matrix inequality conditions.


Multidimensional linear systems Output feedback Robust stabilization Robust performance Linear fractional transformations Linear matrix inequalities 


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Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  • Joseph A. Ball
    • 1
  • Quanlei Fang
    • 2
  • Gilbert J. Groenewald
    • 3
  • Sanne ter Horst
    • 1
  1. 1.Department of MathematicsVirginia TechBlacksburgUSA
  2. 2.Department of MathematicsState University of New York at BuffaloBuffaloUSA
  3. 3.Department of MathematicsNorth West UniversityPotchefstroomSouth Africa

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