MIMO optimal control design: The interplay between the h2 and the ℓ1 norms

Murti V. Salapaka, Mohammed Dahleh, Petros G. Voulgaris

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In this paper we consider controller design methods which can address directly the interplay between the H2 and ℓ1 performance of the closed loop. The development is devoted to multi-input/multi-output (MIMO) systems. Two relevant multiobjective performance problems are considered each being of interest in its own right. In the first, termed as the combination problem, a weighted sum of the ℓ1 norms and the square of the H2 norms of a given set of input-output transfer functions constituting the closed loop is minimized. It is shown that, in the one-block case, the solution can be obtained via a finite-dimensional quadratic optimization problem which has an a priori known dimension. In the four-block case, a method of computing approximate solutions within any a priori given tolerance is provided. In the second, termed as the mixed problem, the H2 performance of the closed loop is minimized subject to an ℓ1 constraint. It is shown that approximating solutions within any a priori given tolerance can be obtained via the solution to a related combination problem.

Original languageEnglish (US)
Pages (from-to)1374-1388
Number of pages15
JournalIEEE Transactions on Automatic Control
Issue number10
StatePublished - 1998

Bibliographical note

Funding Information:
Manuscript received June 30, 1995; revised December 2, 1997. Recommended by Associate Editor, J. Shamma. This work was supported by the National Science Foundation under Grants ECS-9733802, ECS-9632820, and ECS-9308481, by AFOSR under Grant F49620-97-1-0168, and by ONR under Grants N00014-95-1-0948 and N0014-97-1-0153. M. V. Salapaka is with the Electrical and Computer Engineering Department, Iowa State University, Ames, IA 50011 USA. M. Dahleh is with the Mechanical Engineering Department, University of California, Santa Barbara, CA 93106 USA. P. G. Voulgaris is with the Coordinated Science Laboratory, University of Illinois at Urbana Champaign, Urbana, IL 61801 USA. Publisher Item Identifier S 0018-9286(98)07730-7.


  • Duality
  • Mixed objectives
  • Robust control
  • ℓ optimization


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