Frequency theorem

A theorem that states conditions for the solvability of the Lur'e equations in control theory:

$$ \tag{1 } P ^ {*} H + H P + h h ^ {*} = G ,\ \ H q - h \kappa = g , $$

where $ P $, $ G = G ^ {*} $, $ q $, $ g $, $ \kappa $ are given $ n \times n $, $ n \times n $, $ n \times m $, $ n \times m $, and $ m \times m $ matrices respectively, and $ H = H ^ {*} $, $ h $ are the required $ n \times n $ and $ n \times m $ matrices. The Lur'e equations have two other equivalent forms: If $ \mathop{\rm det} \kappa \neq 0 $,

$$ \tag{2 } H Q _ {0} H + ( P _ {0} ^ {*} H + H P _ {0} ) + G _ {0} = 0 , $$

where $ Q _ {0} = Q _ {0} ^ {*} \geq 0 $, $ G _ {0} = G _ {0} ^ {*} $, and in the general case

$$ \tag{3 } 2 \mathop{\rm Re} x ^ {*} H ( P x + q \xi ) = {\mathcal G} ( x , \xi ) - | h ^ {*} x - \kappa \xi | ^ {2} \ \ ( \forall x , \xi ) , $$

where $ {\mathcal G} ( x , \xi ) $ is a given Hermitian form of two vectors $ x \in \mathbf C ^ {n} $, $ \xi \in \mathbf C ^ {m} $;

$$ {\mathcal G} ( x , \xi ) = x ^ {*} G x + 2 \mathop{\rm Re} ( x ^ {*} g \xi ) + \xi ^ {*} \Gamma \xi , $$

Moreover, $ \Gamma = \kappa ^ {*} \kappa \geq 0 $, $ G _ {0} = g \Gamma ^ {-1} g ^ {*} - G $, $ P _ {0} = P - g \Gamma g ^ {*} $, $ Q _ {0} = q \Gamma ^ {-1} q ^ {*} $.

Let the pair $ \{ P , q \} $ be controllable: $ \mathop{\rm rank} \| q , Pq \dots P ^ {n-1} q \| = n $. Then the Lur'e equations reduce to the case where

$$ P = \mathop{\rm diag} [ \lambda _ {1} \dots \lambda _ {h} ] ,\ \ \lambda _ {j} + \lambda _ {h} \neq 0 ,\ \lambda _ {j} \in \mathbf R . $$

If $ m = 1 $ and all the matrices are real, the Lur'e equations in scalar notation take the form

$$ \sum _ { k=1} ^ { n } q _ {k} \frac{h _ {j} h _ {k} }{\lambda _ {j} + \lambda _ {k} } - h _ {j} \sqrt \Gamma = \gamma _ {j} ,\ j = 1 \dots n ; $$

here $ h = [ h _ {1} \dots h _ {n} ] $ is the required vector.

The frequency theorem asserts that for the Lur'e equations to be solvable it is necessary and sufficient that

$$ {\mathcal G} [ ( i \omega I - P ) ^ {-1} q \xi , \xi ] \geq 0 $$

for all $ \xi \in \mathbf C ^ {m} $, $ \omega \in \mathbf R ^ {1} $, $ \mathop{\rm det} \| i \omega I - P \| \neq 0 $ ($ I $ is the identity matrix). The frequency theorem also formulates a procedure for determining the matrices $ H $ and $ h $ and asserts that if

$$ \mathop{\rm det} \Gamma \neq 0 ,\ \mathop{\rm det} \| i \omega I - P \| \neq 0 ,\ {\mathcal G} [ \| i \omega I - P \| ^ {-1} q \xi ,\ \xi ] > 0 $$

(for all $ \xi \neq 0 $, and all $ \omega $), then there exist (unique) matrices $ H $ and $ h $ such that (except for the case of equation (3)) the following is true: $ P + q \kappa ^ {-1} h ^ {*} $ is a Hurwitz matrix (see [3]).

The Lur'e equations in the form (2) are also sometimes called the matrix algebraic Riccati equation. The frequency theorem is used when solving problems on absolute stability [2], [4]–[6], control and adaptation (see, for example, [7]–[9]).

References

[1]	A.I. Lur'e, "Some non-linear problems of the theory of automatic control" , Moscow-Leningrad (1951) (In Russian)
[2]	V.M. Popov, "Hyperstability of control systems" , Springer (1973) (Translated from Rumanian)
[3]	V.A. Yakubovich, "A frequency theorem in control theory" Sib. Math. J. , 14 : 2 (1973) pp. 265–289 Sibirsk. Mat. Zh. , 14 : 2 (1973) pp. 384–420
[4]	A.K. Gelig, G.A. Leonov, V.A. Yakubovich, "Stability of non-linear systems with a unique equilibrium state" , Moscow (1978) (In Russian)
[5]	, Methods for studing non-linear systems of automatic control , Moscow (1975) (In Russian)
[6]	D.D. Siljak, "Nonlinear systems. Parameter analysis and design" , Wiley (1969)
[7]	V.N. Fomin, A.L. Fradkov, V.A. Yakubovich, "Adaptive control of dynamic objects" , Moscow (1981) (In Russian)
[8a]	J.C. Willems, "Almost invariant subspaces: an approach to high gain feedback design I. Almost controlled invariant subspaces" IEEE Trans. Autom. Control , 1 (1981) pp. 235–252
[8b]	J.C. Willems, "Almost invariant subspaces: an approach to high gain feedback design I. Almost conditionally invariant subspaces" IEEE Trans. Autom. Control , 5 (1982) pp. 1071–1084
[9]	W. Coppel, "Matrix quadratic equations" Bull. Austr. Math. Soc. , 10 (1974) pp. 377–401

Comments

The frequency theorem is better known as the Kalman–Yakubovich lemma or Kalman–Yacubovich lemma.

References

[a1]	R.E. Kalman, "Lyapunov functions for the problem of Lurie in automatic control" Proc. Nat. Acad. Soc. USA , 49 : 2 (1963) pp. 201–205
[a2]	B.D.O. Anderson, S. Vongpanitlerd, "Network analysis and synthesis: a modern systems theory approach" , Prentice-Hall (1973)