Infinite Horizon Optimal Output Feedback Control for Linear Systems with State Equality Constraints

  • Jihyoung Cha
  • Seungeun Kang
  • Sangho KoEmail author
Original Paper


This note deals with the infinite horizon linear regulation problem using output feedback with state equality constraints. Similar to the corresponding state feedback problem of a previous work, an existence condition for the output feedback gain and, if it exists, all constrainable output feedback gains are determined. However, different from the fore-mentioned state feedback case, only the necessary conditions for the optimal output feedback gain which minimizes the given standard cost function are determined. The performance of the developed algorithm is demonstrated using numerical simulations for a simple model of divert control system and the lateral dynamics of an F-16 aircraft.


Linear quadratic control Regulation Output feedback control State equality constraints 


\( {\mathbf{A}} \)

Matrices (upper case boldface)

\( {\mathbf{A}}^{\text{T}} \)

Transpose matrix of \( {\mathbf{A}} \)

\( {\mathbf{A}}^{\dag } \)

Moore–Penrose inverse of \( {\mathbf{A}} \)

\( {\text{tr}}({\mathbf{A}}) \)

Trace of \( {\mathbf{A}} \)

\( {\mathbf{P}} > {\mathbf{0}} \)

Positive definite matrices

\( {\mathbf{P}} \ge {\mathbf{0}} \)

Positive semidefinite matrices

\( {\mathcal{A}} \)

Linear spaces (calligraphic uppercase)

\( {\mathcal{N}}({\mathbf{A}}) \)

Null-space of \( {\mathbf{A}} \)

\( {\mathbf{x}} \)

Column vectors (lower case boldface)

\( E\{ {\mathbf{x}}\} \)

Expectation of a random vector \( {\mathbf{x}} \)

\( y, \, Y \)

Scalars (lower or upper case)



This work was supported by 2018 Korea Aerospace University Faculty Research Grant.


  1. 1.
    Goodwin GC, Seron MM, Dona JA (2005) Constrained control and estimation: an optimization approach. Springer, LondonCrossRefzbMATHGoogle Scholar
  2. 2.
    Maciejowski JM (2002) Predictive control: with constraints. Pearson Education Ltd., LondonzbMATHGoogle Scholar
  3. 3.
    Ko S (2005) Performance limitations in linear estimation and control: Quantization and constraint effects, Ph.D. Thesis, University of California, San Diego, La Jolla, CAGoogle Scholar
  4. 4.
    Ko S, Bitmead RR (2007) Optimal control for linear systems with state equality constraints. Automatica 43(9):1573–1582. MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lewis FL, Vrabie DL, Syrmos VL (2012) Optimal control, 3rd edn. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  6. 6.
    Skelton RE, Iwasaki T, Grigoriadis K (1998) A unified algebraic approach to linear control design. Taylor & Francis, LondonGoogle Scholar
  7. 7.
    Stevens BL, Lewis FL (1992) Aircraft control and simulation. Wiley, HobokenGoogle Scholar
  8. 8.
    Athans M, Schweppe FC (1965) Gradient matrices and matrix calculations, No. TN-1965-53, Massachusetts Institute of Technology Lexington Lincoln LaboratoryGoogle Scholar
  9. 9.
    Napior J, Garmy V (2006) Controllable solid propulsion for launch vehicle and spacecraft application. In: Proceedings of the 57th international astronautical congress, Valencia, Spain.
  10. 10.
    Lee W, Eun Y, Bang H, Lee H (2013) Efficient thrust distribution with adaptive pressure control for multinozzle solid propulsion system. J Propuls Power 29(6):1410–1419. CrossRefGoogle Scholar
  11. 11.
    Nguyen LT, Ogburn ME, Gilbert WP, Kibler KS, Brown PW, Deal PL (1979) Simulator study of stall/post-stall characteristics of a fighter airplane with relaxed longitudinal static stability. [F-16], NASA-TP-1538Google Scholar
  12. 12.
    Kang S, Cha J, Ko S (2016) Linear quadratic regulation and tracking using output feedback with direct feedthrough. Int J Aeronaut Space Sci 17(4):593–603. CrossRefGoogle Scholar

Copyright information

© The Korean Society for Aeronautical & Space Sciences 2019

Authors and Affiliations

  1. 1.School of Aerospace and Mechanical EngineeringKorea Aerospace UniversityGoyangRepublic of Korea

Personalised recommendations