Kalman–Yakubovich–Popov lemma
The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number [math]\displaystyle{ \gamma \gt 0 }[/math], two n-vectors B, C and an n x n Hurwitz matrix A, if the pair [math]\displaystyle{ (A,B) }[/math] is completely controllable, then a symmetric matrix P and a vector Q satisfying
- [math]\displaystyle{ A^T P + P A = -Q Q^T }[/math]
- [math]\displaystyle{ P B-C = \sqrt{\gamma}Q }[/math]
exist if and only if
- [math]\displaystyle{ \gamma+2 Re[C^T (j\omega I-A)^{-1}B]\ge 0 }[/math]
Moreover, the set [math]\displaystyle{ \{x: x^T P x = 0\} }[/math] is the unobservable subspace for the pair [math]\displaystyle{ (C,A) }[/math].
The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain.
The Kalman–Popov–Yakubovich lemma which was first formulated and proved in 1962 by Vladimir Andreevich Yakubovich[1] where it was stated that for the strict frequency inequality. The case of nonstrict frequency inequality was published in 1963 by Rudolf E. Kálmán.[2] In that paper the relation to solvability of the Lur’e equations was also established. Both papers considered scalar-input systems. The constraint on the control dimensionality was removed in 1964 by Gantmakher and Yakubovich[3] and independently by Vasile Mihai Popov.[4] Extensive reviews of the topic can be found in [5] and in Chapter 3 of.[6]
Multivariable Kalman–Yakubovich–Popov lemma
Given [math]\displaystyle{ A \in \R^{n \times n}, B \in \R^{n \times m}, M = M^T \in \R^{(n+m) \times (n+m)} }[/math] with [math]\displaystyle{ \det(j\omega I - A) \ne 0 }[/math] for all [math]\displaystyle{ \omega \in \R }[/math] and [math]\displaystyle{ (A, B) }[/math] controllable, the following are equivalent:
- for all [math]\displaystyle{ \omega \in \R \cup \{\infty\} }[/math]
- [math]\displaystyle{ \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right]^* M \left[\begin{matrix} (j\omega I - A)^{-1}B \\ I \end{matrix}\right] \le 0 }[/math]
- there exists a matrix [math]\displaystyle{ P \in \R^{n \times n} }[/math] such that [math]\displaystyle{ P = P^T }[/math] and
- [math]\displaystyle{ M + \left[\begin{matrix} A^T P + PA & PB \\ B^T P & 0 \end{matrix}\right] \le 0. }[/math]
The corresponding equivalence for strict inequalities holds even if [math]\displaystyle{ (A, B) }[/math] is not controllable. [7]
References
- ↑ Yakubovich, Vladimir Andreevich (1962). "The Solution of Certain Matrix Inequalities in Automatic Control Theory". Dokl. Akad. Nauk SSSR 143 (6): 1304–1307.
- ↑ Kalman, Rudolf E. (1963). "Lyapunov functions for the problem of Lur'e in automatic control". Proceedings of the National Academy of Sciences 49 (2): 201–205. doi:10.1073/pnas.49.2.201. PMID 16591048. PMC 299777. Bibcode: 1963PNAS...49..201K. http://www.pnas.org/content/49/2/201.full.pdf.
- ↑ Gantmakher, F.R. and Yakubovich, V.A. (1964). Absolute Stability of the Nonlinear Controllable Systems, Proc. II All-Union Conf. Theoretical Applied Mechanics. Moscow: Nauka.
- ↑ Popov, Vasile M. (1964). "Hyperstability and Optimality of Automatic Systems with Several Control Functions". Rev. Roumaine Sci. Tech. 9 (4): 629–890.
- ↑ Gusev S. V. and Likhtarnikov A. L. (2006). "Kalman-Popov-Yakubovich lemma and the S-procedure: A historical essay". Automation and Remote Control 67 (11): 1768–1810. doi:10.1134/s000511790611004x.
- ↑ Brogliato, B. and Lozano, R. and Maschke, B. and Egeland, O. (2020). Dissipative Systems Analysis and Control (3rd ed.). Switzerland AG: Springer Nature.
- ↑ Anders Rantzer (1996). "On the Kalman–Yakubovich–Popov lemma". Systems & Control Letters 28 (1): 7–10. doi:10.1016/0167-6911(95)00063-1.
 |  |
