Abstract
Algebraic Riccati Equation (ARE) solutions play an important role in many optimal/suboptimal control problems. However, a key assumption in formulation and solution of the ARE is a certain ‘regularity condition’ on the feedthrough term D of the system. For example, formulation of the ARE requires nonsingularity of \(D+D^T\) in positive real dissipative systems and, in the case of bounded real dissipative systems, one requires nonsingularity of \(I-D^T D\). Note that for lossless systems \(D+D^T=0\), while for all-pass systems \(I-D^TD=0\); this rules out the formulation of the ARE. Noting that the ARE solutions are also extremal “storage functions” for dissipative systems, one can speak of storage function for the lossless case too. This contributed chapter formulates new properties of the ARE solution; we then show that this property is satisfied by the storage function for the lossless case too. The formulation of this property is via the set of trajectories of minimal dissipation. We show that the states in a first-order representation of this set satisfy static relations that are closely linked to ARE solutions; this property holds for the lossless case too. Using this property, we propose an algorithm to compute the storage function for the lossless case.
With best wishes to Harry L. Trentelman on the occasion of his 60th birthday.
The last author adds: “to my Ph.D. supervisor, who always has been very friendly, and imparted a sense of discipline and rigour in teaching and research during the course of my Ph.D. Through your actions, I learnt the meaning of ‘zero tolerance’ to careless and hasty work. Thanks very much for all the skills I imbibed from you: both technical and non-technical.”
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- 1.
Lossless systems, with u input and y output, are conservative with respect to the “positive real supply rate” \(u^Ty\) and have \(D+D^T=0\). Similarly, all-pass systems are conservative with respect to the “bounded real supply rate” \(u^Tu-y^Ty\). For all-pass systems \(I-D^TD=0\). Hence, all arguments about ARE solutions and storage functions made for lossless systems are applicable to all-pass systems as well.
- 2.
For trajectories \((w_1,x_1)\) and \((w_2,x_2)\), their concatenation at \({t_0}\), denoted by \((w_1,x_1) {\wedge _{t_0}} (w_2,x_2)\), is defined as
$$ \begin{aligned} (w_1,x_1) \wedge _{t_0} (w_2,x_2)(t):= {\left\{ \begin{array}{ll} (w_1,x_1)(t) \quad \text {for} \quad t<t_0 \\ (w_2,x_2)(t)\quad \text {for} \quad t \geqslant t_0. \end{array}\right. } \end{aligned} $$.
- 3.
As in [5], for a square and nonsingular polynomial matrix R(s), we call the values of \(\lambda \in \mathbb {C} \) at which rank of \(R(\lambda )\) drops the eigenvalues of the polynomial matrix R(s) and we call the vectors in the nullspace of \(R(\lambda )\) the eigenvectors of R(s) corresponding to \(\lambda \).
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Acknowledgments
This work was supported in part by SERB-DST, IRCC (IIT Bombay) and BRNS, India.
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Bhawal, C., Kumar, S., Pal, D., Belur, M.N. (2015). New Properties of ARE Solutions for Strictly Dissipative and Lossless Systems. In: Belur, M., Camlibel, M., Rapisarda, P., Scherpen, J. (eds) Mathematical Control Theory II. Lecture Notes in Control and Information Sciences, vol 462. Springer, Cham. https://doi.org/10.1007/978-3-319-21003-2_5
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