Lemma on Boundedness of Anisotropic Norm for Systems with Multiplicative Noises under a Noncentered Disturbance

Abstract

We consider a linear discrete time-varying system with multiplicative noise acted upon by colored exogenous disturbance with nonzero first moment. Multiplicative noises are modeled in the form of linear combinations of deterministic matrices with pairwise independent random coefficients. For this system, we describe a method, based on a state-space realization, for calculating the anisotropic norm in terms of Riccati equations.

APPENDIX

Proof of Lemma 1. For the proof, consider the block matrix $\mathbf {E}[F_{0:N}^\mathrm {T} F_{0:N}] = \Lambda _{0:N}$, which will be an identity matrix of order $m(N+1)$ for an isometric system. The blocks of this matrix have the form

$$ \lambda _{i,j} = \sum \limits _{k=\max {\left \{i,j\right \}}}^{N} \mathbf {E}[f_{k,i}^\mathrm {T} f_{k,j}],$$

with the last block of the matrix being equal to

$$ {\lambda _{N,N} = \mathbf {E}[D_N^\mathrm {T} D_N] = \sum _{i=0}^M D_{i,N}^\mathrm {T} D_{i,N} = I_m}$$

in view of the mutual independence of the variables $\xi _{i,k}^D$ and $\xi _{j,k}^D$ and the block with number $N-1$ being, accordingly,

$$ \lambda _{N-1,N-1} = \sum \limits _{i=0}^M \left (D_{i,N-1}^\mathrm {T} D_{i,N-1} + B_{i,N-1}^\mathrm {T} C_{i,N}^\mathrm {T} C_{i,N} B_{i,N-1}\right ) = I_m, $$

where

$$ {\sum _{i=0}^M C_{i,N}^\mathrm {T} C_{i,N} = Q_{N}}; $$

obviously, a necessary condition for the last block $\lambda _{N,N}$ is $Q_{N+1} = 0$. Further, one can readily show by induction that relations (17) and (20) are satisfied.

Consider an element $\lambda _{i,j}$ with $i<j$,

$$ \lambda _{i,j} = \mathbf {E}\Big [B_{i-1}^\mathrm {T} T_{i-1,j-1}^\mathrm {T} \left (C_{j-1}^\mathrm {T} D_{j-1} + A_{j-1}^\mathrm {T} Q_j B_{j-1}\right )\Big ];$$

this relation can be reduced to the form

$$ \lambda _{i,j} = B_{0,i-1}^\mathrm {T} A_{0,i}^\mathrm {T}\cdots A_{0,j-2}^\mathrm {T} \left (C_{0,j-1}^\mathrm {T} D_{0,j-1} + A_{0,j-1}^\mathrm {T} Q_j B_{0,j-1}\right )$$

by virtue of the assumption on the joint independence of the variables $\xi _{i,k}^{\Omega }$, $\Omega =\left \{A,B,C,D\right \}$. The first $j-1$ entries in the $j$th column of the matrix $\Lambda _{0:N}$ have the form

$$ \begin {aligned} &\left [A_{0,j-2}A_{0,j-3}\cdots A_{0,1}B_{0,0}, \ldots , A_{0,j-4}B_{0,j-3}, B_{0,j-2}\right ]^\mathrm {T} \\ &\qquad \qquad \qquad \qquad {}\times \left (C_{0,j-1}^\mathrm {T} D_{0,j-1} + A_{0,j-1}^\mathrm {T} Q_{j} B_{0,j-1}\right ) = 0_{m(j-1)\times m}. \end {aligned}$$

Having multiplied the last equality by

$$ {\left [A_{0,j-2}A_{0,j-3}\cdots A_{0,1}B_{0,0}, \ldots , A_{0,j-4}B_{0,j-3}, B_{0,j-2}\right ]},$$

we obtain an expression similar to (18) with the notation

$$ {P_{j-1} = B_{0,j-2}B_{0,j-2}^\mathrm {T} + A_{0,j-2}P_{j-2}A_{0,j-2}^\mathrm {T}}, $$

coinciding with (19).

The proof of Lemma 1 is complete. $\quad \blacksquare$

Proof of Lemma 2. The proof is based on the invertibility of the matrix $S_k$ in view of its positive definiteness [27]. Further, one should apply Lemma 1 to the system $\left [\sqrt {q}F^\mathrm {T},G^{-\mathrm {T}}\right ]$ with allowance for notation (28) and (27); the parameter $q$ is determined in accordance with (23).

The proof of Lemma 2 is complete. $\quad \blacksquare$

Proof of Theorem 2. The proof of the theorem repeats the reasoning in [28] on the calculation of the anisotropic norm of a random matrix.

Consider the filter (24) that generates a disturbance on which the supremum of the gain (8) is attained. Since relation (23) holds, it suffices to relate the calculation of the special functions $\Phi (q)$ and $\Psi (q)$ to the covariance matrix of the disturbance $\Sigma _{0:N}(q)$. Consider separate blocks of the symmetric matrix $G_{0:N}G_{0:N}^\mathrm {T}$,

$$ g_{i,j} = \begin {cases} L_i\Delta _{i,j+1}B_{0,j}S_{j}^{1/2}, &\quad i>j,\\[.13em] S_j^{1/2}, &\quad i=j,\\[.13em] 0, &\quad i<j, \end {cases}$$

where $\Delta _{i,j} = (A_{0,i-1} + B_{0,i-1}L_{i-1})\Delta _{i-1,j}$ with the boundary condition $\Delta _{i,i} = I_n$. Proceeding from this representation, we obtain

$$ {\mathrm{tr}\, \Sigma _{0:N} = \mathrm{tr}\, \underset {0\leqslant i, 0\leqslant j}{\mathrm {block}}\left \{\sum _{k=0}^{\min \{i,j\}}g_{i,k}g_{j,k}^\mathrm {T}\right \}= \sum _{k=0}^{N}\mathrm{tr}\, (L_k\Upsilon _k L_k^\mathrm {T} + S_k)}.$$

The last equality implies formula (29). To prove (30), we use the invariance of the determinant of a matrix under transposition,

$$ {\det (G_{0:N}G_{0:N}^\mathrm {T}) = \det (G_{0:N}^\mathrm {T} G_{0:N}) = (\det G_{0:N})^2 = \prod \limits _{k=0}^N\det S_k}.$$

The proof of Theorem 2 is complete. $\quad \blacksquare$