A quadratic matrix inequality based PID controller design for LPV systems

Yan Wang, Rajesh Rajamani, Ali Zemouche

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


This paper develops a robust gain-scheduled proportional–integral–derivative (PID) controller design method for a linear-parameter-varying (LPV) system with parametric uncertainty. It is recognized in the literature that the robust fixed-order controller design can be formulated as a feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is an NP hard problem in general. Previous researchers have applied a linearization method, such as a variable change technique or a congruence transformation, to transform the BMI into a LMI. The applicability of the linearization method depends on the specific structure of the problem at hand and cannot be generalized. This paper instead formulates the gain-scheduled PID controller design as a feasibility problem of a quadratic matrix inequality (QMI) constraint, which covers the BMI constraint as a special case. An augmented sequential LMI optimization method is proposed to search for a feasible solution of the QMI constraint iteratively. As an illustrative application, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real-world output feedback control design system.

Original languageEnglish (US)
Pages (from-to)67-76
Number of pages10
JournalSystems and Control Letters
StatePublished - Apr 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier B.V.


  • Convex optimization
  • LPV system
  • Linear matrix inequality
  • PID controller
  • Quadratic matrix inequality
  • Robust control


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