Anisotropy-Based Filtering for Linear Discrete Time-Varying Systems with Multiplicative Noises on a Finite Horizon

Abstract

We consider the anisotropy-based filtering problem for a linear discrete time-varying multiplicative noise system on a finite time horizon. A random disturbance with an inaccurately known distribution is fed to the system input. The uncertainty in the description of this distribution is characterized by a constraint on the anisotropic functional. The output of the original system is estimated using the selected filter of a certain type according to the principle of minimizing the performance indicator in the form of the anisotropic norm of the filtering error system. Based on the lemma on the boundedness of the anisotropic norm of the filtering error system, relations for the unknown filter matrices are derived. As special cases, we consider anisotropy-based filtering problems for a system with measurement dropouts as well as a similar problem for a system with a filter of a special type. A numerical example where the method is used to synthesize an anisotropy-based filter is given.

APPENDIX 1

Proof of Theorem 1. Let us reduce the expression (4.23) to the form of an inequality while considering the positive definiteness of the matrix $\mathcal {S}(k)$,

$$ \begin {aligned} \Theta (k)&\succ 0: \Theta (k)\succ \mathcal {S}(k),\quad \Theta ^{-1}(k)\prec \mathcal {S}^{-1}(k),\\ \Theta ^{-1}(k) &= \Psi (k) \prec I_{m_w} - \left (\thinspace q \sum \limits _{i=0}^{M}\thinspace \mathcal {D}_{i}^{\top }(k)\thinspace \mathcal {D}_{i}(k) + \sum \limits _{i=0}^{M}\thinspace \mathcal {B}_{i}^{\top }(k)\thinspace \mathcal {R}(k + 1)\thinspace \mathcal {B}_{i}(k) \right ),\\ \Psi ^{-1}(k)&\succ \mathcal {S}(k). \end {aligned}$$

(A.1.1)

Applying (A.1.1) to system (4.21)–(4.25), we obtain a system of inequalities of the form

$$ \begin {aligned} \widehat {\mathcal {R}}(k) & \succ \mathcal {R}(k), \quad \widehat {\mathcal {R}}(N+1)=0,\nonumber \\ \widehat {\mathcal {R}}(k) &\succ \mathcal {A}_0^{\top }(k)\widehat {\mathcal {R}}(k + 1)\mathcal {A}_0^{}(k)+\sum \limits _{i=1}^{M}a_i\mathcal {A}_i^{\top }(k)\widehat {\mathcal {R}}(k + 1) \mathcal {A}_i(k) + q\sum \limits _{i=0}^{M}c_i\mathcal {C}_{i}^{\top }(k) \mathcal {C}_{i}(k)\\ &\quad {} + \left ( \mathcal {B}_{0}^{\top } \widehat {\mathcal {R}}(k + 1)\mathcal {A}_0 + q\mathcal {D}_{0}^{\top }\mathcal {C}_{0} \right )^{ \top } \Psi ^{-1}(k) \left ( \mathcal {B}_{0}^{\top }\widehat {\mathcal {R}}(k + 1)\mathcal {A}_0 + q\mathcal {D}_{0}^{\top }\mathcal {C}_{0} \right ) , \end {aligned} $$

(A.1.2)

$$ \Psi (k) \prec I_{m_w} - \left ( q\sum \limits _{i=0}^{M}\mathcal {D}_{i}^{\top }(k)\mathcal {D}_{i}(k) + \sum \limits _{i=0}^{M} \mathcal {B}_{i}^{\top }(k)\widehat {\mathcal {R}}(k + 1)\mathcal {B}_{i}(k) \right ) ,\qquad \qquad \qquad \enspace \thinspace \thinspace $$

(A.1.3)

$$ \sum \limits _{k=0}^{N}\ln \det \Psi (k) \geqslant 2a + m_w^{}(N+1)\ln (1-q\gamma ^2).$$

(A.1.4)

Note that inequality (A.1.4) contains the product $q\gamma ^2$ of two variables of the system. To solve the system of inequalities, it is necessary to eliminate this nonlinear component by means of a transformation of the very expression or a change of variables. Consequently, we introduce the new variables

$$ \begin {gathered} \eta = q^{-1},\quad \overline {\Psi } = \eta \Psi ,\\ \mathcal {P}(k) = \eta \widehat {\mathcal {R}}(k),\quad \mathcal {P}(N+1) = \eta \widehat {\mathcal {R}}(N+1). \end {gathered} $$

(A.1.5)

For conciseness of notation of the calculations with linear matrix inequalities to follow, introduce the notation

$$ \overline {\mathcal {A}}(k) = \left ( \begin {array}{{ccc}} \mathcal {A}_1^{\top } & \ldots & \mathcal {A}_{M_1}^{\top } \end {array}\right ), \quad \overline {\mathcal {B}}(k) = \left ( \begin {array}{{ccc}} \mathcal {B}_1^{\top } & \ldots &\mathcal {B}_{M_2}^{\top } \end {array}\right ), $$

(A.1.6)

$$ \overline {\mathcal {C}}(k) = \left ( \begin {array}{{ccc}} \mathcal {C}_1^{\top } & \ldots & \mathcal {C}_{M_3}^{\top } \end {array}\right ), \quad \overline {\mathcal {D}}(k) = \left ( \begin {array}{{ccc}} \mathcal {D}_1^{\top } & \ldots & \mathcal {D}_{M_4}^{\top } \end {array}\right ). $$

(A.1.7)

To bring inequalities (A.1.2) and (A.1.3) to the form of a system of linear matrix inequalities (LMI), we need the Schur complement lemma. The statement of this lemma can be found in [20].

Let us apply the Schur complement lemma to the system of Riccati inequalities (A.1.2)–(A.1.3) with allowance for the change of variables (A.1.5) and notation (A.1.6)–(A.1.7),

$$ \left (\begin {array}{{cccccc}} \mathcal {P}(k) & \ast & \ast & \ast & \ast & \ast \\ 0 & \eta I_{m_w} & \ast & \ast & \ast & \ast \\ \mathcal {A}_0(k) & \mathcal {B}_{0}(k) & \mathcal {P}^{-1}(k+1) & \ast & \ast & \ast \\ \overline {\mathcal {A}}^{\top }(k) & 0 & 0 & \ddots & \ast & \ast \\ \mathcal {C}_0(k) & \mathcal {D}_{0}(k) & 0 & 0 & \mathcal {P}^{-1}(k+1) & \ast \\ \overline {\mathcal {C}}^{\top }(k) & 0 & 0 & 0 & 0 & I \\ \end {array}\right )\succ 0,$$

(A.1.8)

$$ \left (\begin {array}{{cccc}} \mathcal {P}(N) & \ast & \ast & \ast \\ 0 & \eta I_{m_w^{}}^{} & \ast & \ast \\ \mathcal {C}_0(N) & \mathcal {D}_0(N) & I_{p_z}^{} & \ast \\ \overline {\mathcal {C}}(N) & 0 & 0 & I\\ \end {array} \right )\succ 0,$$

(A.1.9)

$$ \left (\begin {array}{{ccccc}} \eta I_{m_w^{}}^{}-\overline {\Psi }(k) & \ast & \ast & \ast & \ast \\ \mathcal {B}_0^{\top }(k) & \mathcal {P}^{-1}(k+1) & \ast & \ast & \ast \\ \vdots & 0 & \ddots & \ast & \ast \\ \mathcal {B}_{M_2}^{\top }(k) & 0 & 0 & \mathcal {P}^{-1}(k+1) & \ast \\ \overline {\mathcal {D}}^{\top }(k) & 0 & \ldots & 0 & I \end {array}\right )\succ 0,$$

(A.1.10)

$$ \left (\begin {array}{{cc}} \eta I_{m_w^{}}^{}-\overline {\Psi }(N) & \ast \\ \mathcal {D}(N) & I_{p_z} \end {array}\right )> 0, $$

(A.1.11)

$$ \prod \limits _{k=0}^{N}\det \overline {\Psi }(k) \geqslant e^{2a}(\eta -\gamma ^2)^{m_w(N+1)},$$

(A.1.12)

$$ \mathcal {P}(k) \succ 0,\quad k=N-1,\ldots ,0\nonumber .$$

To eliminate the inverse matrices from inequalities (A.1.8) and (A.1.10), we perform congruent transformations of the inequalities using block-diagonal matrices of the form

$$ \begin {gathered} \mathrm {blockdiag}\left (I_{2n_x},I_{m_w},\mathcal {P}(k+1),\ldots ,\mathcal {P}(k+1),I_{p_z(M_3+1)}\right )\\[.5em] \text {and}\\[.5em] \mathrm {blockdiag}\left (I_{2n_x},\mathcal {P}(k+1),\ldots ,\mathcal {P}(k+1),I_{p_z(M_4+1)}\right ). \end {gathered} $$

To simplify the notation of the transformed LMI, we introduce the notation

$$ \overline {\mathcal {A}}(k)\mathcal {P}(k+1) = \left ( \begin {array}{{ccc}} \mathcal {A}_1^{\top }\mathcal {P}(k+1) & \ldots & \mathcal {A}_{M_1}^{\top }\mathcal {P}(k+1) \end {array}\right ), $$

(A.1.13)

$$ \overline {\mathcal {B}}(k)\mathcal {P}(k+1) = \left ( \begin {array}{{ccc}} \mathcal {B}_1^{\top }\mathcal {P}(k+1) & \ldots & \mathcal {B}_{M_2}^{\top }\mathcal {P}(k+1) \end {array}\right ). $$

(A.1.14)

Applying notation (A.1.13)–(A.1.14) and necessary congruent transformations to inequalities (A.1.8)–(A.1.12), we obtain the desired linear matrix inequalities (4.26)–(4.30). $\quad \blacksquare$

APPENDIX 2

Proof of Theorem 2. Let us write the condition for the boundedness of the anisotropic norm for the filtering error system with allowance for the above calculations,

$$ \begin {gathered} \left [ \begin {array}{{ccccc}} \mathcal {P}(k) & 0 & \mathcal {A}_0^{\top }\mathcal {P}(k+1) & \overline {\mathcal {A}}(k)\mathcal {P}(k+1) & \overline {\mathcal {C}}(k)\\ * & \eta I & \mathcal {B}_{0}^{\top }(k)\mathcal {P}(k+1) & 0 & 0\\ * & * & \mathcal {P}(k+1) & 0 & 0 \\ * & * & * & \mathrm {\Phi }_{M}\big (\mathcal {P}(k+1)\big ) & 0 \\ * & * & * & * & I \\ \end {array} \right ] \succ 0,\\ \left [ \begin {array}{{ccc}} \eta I - \overline {\Psi }(k) & \mathcal {B}_{0}^{\top }(k)\mathcal {P}(k+1) & \overline {\mathcal {B}}(k)\mathcal {P}(k+1) \\ * & \mathcal {P}(k+1) & 0 \\ * & * & \mathrm {\Phi }_{M}(\mathcal {P}(k+1))\\ \end {array} \right ]\succ 0,\\ \left [ \begin {array}{{ccc}} \mathcal {P}(N) & 0 & \overline {\mathcal {C}}(N)\\ * & \eta I & 0 \\ * & * & I \\ \end {array} \right ] \succ 0, \quad \left [ \begin {array}{{cc}} \eta I-\overline {\Psi }(N) & 0\\ 0 & I \end {array} \right ]\succ 0,\\ \prod \limits _{k=0}^{N}\det \overline {\Psi }(k) \geqslant e^{2a}(\eta -\gamma ^2)^{m_w(N+1)}. \end {gathered}$$

Considering (4.56) and (4.59), it is obvious that the inequalities contain nonlinear components in the form of the products of unknown matrices in the products $\mathcal {A}_i(k)\mathcal {P}(k+1)$ and $\mathcal {B}_{i}^{\top }(k)\mathcal {P}(k+1)$.

To eliminate nonlinear terms from the inequalities, we use the following change of variables in the inequalities:

$$ \mathcal {X}(k) = \left ( \begin {array}{{cc}} 0 & 0\\ W(k) & H(k) \end {array} \right ),\quad \mathcal {Y}(k) = \mathcal {X}(k)\mathcal {P}(k+1).$$

(A.2.1)

Applying the change of variables (A.2.1) to the system of inequalities (4.56), (4.59), we obtain

$$ \mathcal {A}(k)= \mathcal {A}^0_0(k)+\mathcal {X}(k)\mathcal {A}^1_0(k)+\sum \limits _{i=1}^{M} \lambda _{1i}(k)\left (\mathcal {A}_{i}^{0}(k)+\mathcal {X}(k)\mathcal {A}_{i}^{1}(k)\right ), $$

(A.2.2)

$$ \mathcal {B}(k)= \mathcal {B}^0_0(k)+\mathcal {X}(k)\mathcal {B}^1_0(k)+\sum \limits _{i=1}^{M} \lambda _{1i}(k)\left (\mathcal {B}_{i}^{0}(k)+\mathcal {X}(k)\mathcal {B}_{i}^{1}(k)\right ), $$

(A.2.3)

where

$$ \mathcal {A}^0_0(k) = \left (\begin {array}{{cc}} A_0(k)& 0\\ A_0(k) & 0 \end {array}\right ),\quad \mathcal {A}^1_0(k) = \left (\begin {array}{{cc}} -I & I\\ -C_{y_0}(k)& 0\end {array}\right ), $$

(A.2.4)

$$ \mathcal {A}^0_i(k) = \left (\begin {array}{{cc}}A_i(k) & 0\\ A_i(k) & 0\end {array}\right ),\quad \mathcal {A}^1_i(k) = \left (\begin {array}{{cc}} 0 & 0\\ -C_{y_i}(k)& 0 \end {array}\right ), $$

(A.2.5)

$$ \mathcal {B}^0_0(k) = \left (\begin {array}{c} B_0(k) \\ B_0(k)\end {array}\right ),\quad \mathcal {B}^1_0(k) = \left (\begin {array}{c}0\\ -D_{y_0}\end {array}\right ), $$

(A.2.6)

$$ \mathcal {B}^0_i(k) = \left (\begin {array}{c} B_i(k) \\ B_i(k)\end {array}\right ),\quad \mathcal {B}^1_i(k) = \left (\begin {array}{c} 0\\ -D_{y_i}\end {array}\right ). $$

(A.2.7)

Substituting (A.2.1)– (A.2.7) into the matrix inequalities, we eliminate the nonlinear components from the system of matrix inequalities. As a result, we obtain the desired LMI (4.60)–(4.64). $\quad \blacksquare$