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Robust Optimal Control Design Using a Differential Game Approach for Open-Loop Linear Quadratic Descriptor Systems

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Abstract

This paper studies the robust optimal control problem for descriptor systems. We applied differential game theory to solve the disturbance attenuation problem. The robust control problem was converted into a reduced ordinary zero-sum game. Within a linear quadratic setting, we solved the problem for finite and infinite planning horizons.

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Notes

  1. We would like to thank the referee pointing this out to us.

  2. The system (1) has at most index one, if and only if \(rank\left( {\left[ {\begin{array}{*{20}c} E &{} {A_EW} \\ \end{array}} \right] } \right) = n + r\) (see [11]).

  3. We write \(X>0 (X\ge 0)\) if \(X\) is positive (semi) definite.

  4. \({\mathbb {C}}^ - = \left\{ {\lambda \in {\mathbb {C}} \left| {{\mathop {\mathrm{Re}}\nolimits } \left( \lambda \right) < 0} \right. } \right\} {\mathbb {C}}_0^ + = \left\{ {\lambda \in {\mathbb {C}}\left| {{\mathop {\mathrm{Re}}\nolimits } \left( \lambda \right) \ge 0} \right. } \right\} .\)

  5. \(\lim _{t_f \rightarrow \infty } J_i \left( {t_f ,x_{1_0 } ,u} \right) = - \infty \left( \infty \right) \) if \(\forall r \in {\mathbb {R}},\exists T_f \in {\mathbb {R}} \text {, such that } t_f \ge T_f\) implies \(J_i \left( {t_f ,x_{1_0 } ,u} \right) \le r\left( { \ge r} \right) \).

  6. Matrix \(A\) is called stable if the real parts of all its eigenvalues are negative.

  7. Such a solution is called an LRS solution.

  8. Note from (2) that the assumption that \(A\) is stable is equivalent to the assumption that all finite eigenvalues of \(A_E\) are stable.

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Acknowledgments

The first author is grateful for the support of the Directorate General High Education of Ministry of Education and Culture of Indonesia through a scholarship.

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Authors and Affiliations

  1. UIN Sunan Kalijaga, Yogyakarta, Indonesia

    Muhammad Wakhid Musthofa

  2. Mathematics Department, Universitas Gadjah Mada, Yogyakarta, Indonesia

    Salmah & Ari Suparwanto

  3. Tilburg University, Tilburg, The Netherlands

    Jacob Engwerda

Authors
  1. Muhammad Wakhid Musthofa
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  2. Salmah
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  3. Jacob Engwerda
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  4. Ari Suparwanto
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Corresponding author

Correspondence to Muhammad Wakhid Musthofa.

Additional information

Communicated by Dean A. Carlson.

Appendix: State Transformation and Shorthand Notation (7, 8)

Appendix: State Transformation and Shorthand Notation (7, 8)

With \(X\) and \(Y\) defined as in (2), \(X =: \left[ {\begin{array}{*{20}c} {X_1 } &{} {X_2 } \\ \end{array}} \right] \) and \(Y^T =: \left[ {\begin{array}{*{20}c} {Y_1^T }&{Y_2^T } \end{array}} \right] \) are nonsingular matrices. With the state transformation \(\left[ {\begin{array}{*{20}c} {x_1 \left( t \right) } \\ {x_2 \left( t \right) } \\ \end{array}} \right] : = X^{ - 1} x\left( t \right) \), the corresponding state and control matrices for the reduced dynamical system are \(A:=J;\ B_1:=Y_1B_u\) and \(B_2:=Y_1B_w\).

The matrices used in Eqs. (7) and (8) are

\(C_1 X =: \bar{C}_1 =: \left[ {\begin{array}{*{20}c} {\bar{C}_{11} }&{\bar{C}_{12} } \end{array}} \right] , C_2 X =: \bar{C}_2 =: \left[ {\begin{array}{*{20}c} {\bar{C}_{21} }&{\bar{C}_{22} } \end{array}} \right] , Q: = X_1^T \bar{Q}X_1 ,\ V: = - X_1^T \bar{Q}X_2 Y_2 B_u ,\ N: = B_u^T Y_2^T X_2^T \bar{Q}X_2 Y_2 B_w ,\ W: = - X_1^T \bar{Q}X_2 Y_2 B_w ,\ R_{\bar{1} \bar{1}} : = B_u^T Y_2^T X_2^T \bar{Q} X_2 Y_2 B_u + \bar{R}_1\), and \(R_{\overline{22}\gamma } : = B_w^T Y_2^T X_2^T \bar{Q}X_2 Y_2 B_w - \gamma \bar{R}_2\).

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Musthofa, M.W., Salmah, Engwerda, J. et al. Robust Optimal Control Design Using a Differential Game Approach for Open-Loop Linear Quadratic Descriptor Systems. J Optim Theory Appl 168, 1046–1064 (2016). https://doi.org/10.1007/s10957-015-0750-8

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