The article “Observer-Aided Output Feedback Synthesis as an Optimization Problem” by
B. T. Polyak and M. V. Khlebnikov was originally published electronically in Springer-Link
on 17 April 2022 in volume 83, issue 3, pages 303–324.
The formula for the gradient
\( \nabla _L f(K,L,\alpha ) \) in Lemma 3 is given inaccurately; it should read
$$
\frac 12\nabla _Lf(K,L,\alpha )=\rho _LL+ {M}^{\mathrm {T}}_2YP{N}^{\mathrm
{T}}_2-\frac {1}{\alpha }\begin {pmatrix}0 & I\end {pmatrix}Y\begin
{pmatrix}D\\D-LD_1\end {pmatrix}{D}^{\mathrm {T}}_1.
$$
(1)
Below find the full derivation of this result.
For differentiation of the function
\( f(K,L,\alpha ) \) with respect to
\( L \) under a constraint in the form of the Lyapunov equation
$$
\left (A_{K,L}+\frac {\alpha }{2}I\right )P+P {\left (A_{K,L}+\frac {\alpha
}{2}I\right )}^{\mathrm {T}} +\frac {1}{\alpha }\begin {pmatrix}D\\D-LD_1\end
{pmatrix} {\begin {pmatrix}D\\D-LD_1\end {pmatrix}}^{\mathrm {T}}=0,
$$
(2)
we give an increment of
\( \Delta L \) to the quantity
\( L \) and denote the corresponding increment in
\( P \) by
\( \Delta P \),
$$
\begin{aligned} &\left (\mathcal A+M_1KN_1+M_2(L+\Delta L)N_2+\frac {\alpha }{2}I\right
)(P+\Delta P) \\ &\qquad \qquad \qquad \qquad {}+(P+\Delta P) {\left (\mathcal
A+M_1KN_1+M_2(L+\Delta L)N_2+\frac {\alpha }{2}I\right )}^{\mathrm {T}} \\ &\qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad {}+\frac {1}{\alpha }\begin
{pmatrix}D\\D-(L+\Delta L)D_1\end {pmatrix} {\begin {pmatrix}D\\D-(L+\Delta
L)D_1\end {pmatrix}}^{\mathrm {T}}=0.\end{aligned}
$$
Leaving the notation
\( \Delta P \) for the principal part of the increment, we obtain
$$
\begin{aligned} &{}\left (A_{K,L}+M_2\Delta LN_2+\frac {\alpha }{2}I\right )P+P {\left
(A_{K,L}+M_2\Delta LN_2+\frac {\alpha }{2}I\right )}^{\mathrm {T}} \\ &\qquad
\qquad \qquad {}+\left (A_{K,L}+\frac {\alpha }{2}I\right )\Delta P+\Delta P{\left
(A_{K,L}+\frac {\alpha }{2}I\right )}^{\mathrm {T}} \\ &\qquad \qquad \qquad
\qquad \qquad {}+\frac {1}{\alpha }\left [\begin {pmatrix}D\\D-LD_1\end {pmatrix}
{\begin {pmatrix}D\\D-LD_1\end {pmatrix}}^{\mathrm {T}}- \begin {pmatrix}0\\
\Delta LD_1\end {pmatrix} {\begin {pmatrix}D\\D-LD_1\end {pmatrix}}^{\mathrm
{T}}\right . \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad {}\left .- \begin {pmatrix}D\\D-LD_1\end {pmatrix}
{\begin {pmatrix}0\\\Delta LD_1\end {pmatrix}}^{\mathrm {T}} \right
]=0.\end{aligned}
$$
After subtracting Eq. (2) from this
equation, we have
$$
\begin {aligned} &{}\left (A_{K,L}+\frac {\alpha }{2}I\right )\Delta P+\Delta P
{\left (A_{K,L}+\frac {\alpha }{2}I\right )}^{\mathrm {T}}+M_2\Delta LN_2P+P
{(M_2\Delta LN_2)}^{\mathrm {T}} \\ &\qquad \qquad \qquad \qquad \qquad {} -\frac
{1}{\alpha }\left [ \begin {pmatrix}0\\ \Delta LD_1\end {pmatrix} {\begin
{pmatrix}D\\D-LD_1\end {pmatrix}}^{\mathrm {T}}+ \begin {pmatrix}D\\D-LD_1\end
{pmatrix} {\begin {pmatrix}0\\\Delta LD_1\end {pmatrix}}^{\mathrm {T}}\right ]=0.
\end {aligned}
$$
(3)
Let us calculate the increment of the functional
\( f(K,L,\alpha ) \) with respect to
\( L \) by linearizing the relevant quantities,
$$
\begin{gathered} \Delta _Lf(K,L,\alpha )=\mathrm {tr}\, \mathcal C_2\Delta P {\mathcal
C}^{\mathrm {T}}_2+\rho _L\mathrm {tr}\, {L}^{\mathrm {T}}\Delta L+\rho _L\mathrm
{tr}\, {(\Delta L)}^{\mathrm {T}}L=\mathrm {tr}\,\Delta P {\mathcal C}^{\mathrm
{T}}_2\mathcal C_2+2\rho _L\mathrm {tr}\, {L}^{\mathrm {T}}\Delta L.\end{gathered}
$$
Consider the Lyapunov equation
$$
{\left (A_{K,L}+\frac {\alpha }{2}I\right )}^{\mathrm {T}}Y+Y\left (A_{K,L}+\frac
{\alpha }{2}I\right )+ {\mathcal C}^{\mathrm {T}}_2\mathcal C_2=0.
$$
(4)
From the dual equations (3)
and (4), we have
$$
\begin{aligned} \Delta _Lf(K,L,\alpha )&=2\mathrm {tr}\, Y\left [M_2\Delta LN_2P-\frac
{1}{\alpha } \begin {pmatrix}0\\ \Delta LD_1\end {pmatrix} {\begin
{pmatrix}D\\D-LD_1\end {pmatrix}}^{\mathrm {T}}\,\right ]+2\rho _L\mathrm {tr}\,
{L}^{\mathrm {T}}\Delta L \\ &=2\mathrm {tr}\left [N_2PYM_2\Delta L-\frac
{1}{\alpha }D_1 {\begin {pmatrix}D\\D-LD_1\end {pmatrix}}^{\mathrm {T}}Y\begin
{pmatrix}0\\ I\end {pmatrix}\Delta L\right ]+2\rho _L\mathrm {tr}\, {L}^{\mathrm
{T}}\Delta L \\ &=2\Bigl \langle \rho _L L+ {M}^{\mathrm {T}}_2YP {N}^{\mathrm
{T}}_2-\frac {1}{\alpha }\begin {pmatrix}0 & I\end {pmatrix}Y\begin
{pmatrix}D\\D-LD_1\end {pmatrix} {D}^{\mathrm {T}}_1,\Delta L\Bigr \rangle
,\end{aligned}
$$
whence formula (1) follows.
The original article can be found online at
https://doi.org/10.1134/S0005117922030018 .
