Stabilizability of minimum-phase systems with colored multiplicative uncertainty

Abstract

This work addresses the mean-square stability and stabilizability problem for minimum-phase multi-input and multi-output (MIMO) plant with a novel colored multiplicative feedback uncertainty. The proposed uncertainty is generalization of the i.i.d. multiplicative noise and assumed to be a stochastic system with random finite impulse response (FIR), which has advantage on modeling a class of network phenomena such as random transmission delays. A concept of coefficient of frequency variation is developed to characterize the proposed uncertainty. Then, the mean-square stability for the system is derived, which is a generalization of the well-known mean-square small gain theorem. Based on this, the mean-square stabilizability condition is established, which reveals the inherent connection between the stabilizability and the plant’s unstable poles and the coefficient of frequency variation of the uncertainty. The result is verified by a numerical example on the stabilizability of a networked system with random transmission delay as well as analog erasure channel.

Appendices

Appendix A Proof of Lemma 1

It follows from Assumption 1 that $\tilde{\varvec{\omega }}(k, l) \equiv 0$ for $k<l$ and $k> l+{\bar{\tau }}$. Therefore, $r_{l}(\lambda )=0$ for $\lambda > {\bar{\tau }}$. For the case $0 \le \lambda \le {{\bar{\tau }}}$, it holds that

$$\begin{aligned} r_{l}(\lambda )&= \textstyle \sum \limits _{j=0}^{{\bar{\tau }}-\lambda } {\mathrm {E}}\{\tilde{\varvec{\omega }}({l+j+\lambda ,l}) \tilde{\varvec{\omega }}^*({l+j,l})\}\\&= \textstyle \sum \limits _{j=0}^{{\bar{\tau }}-\lambda }\varvec{\beta }_{(j+\lambda )j}, \end{aligned}$$

which is independent of l. On the other hand, it is easy to see that $r_{l}(\lambda ) = r_{l}(-\lambda )$ for $\lambda < 0$ by Change of Variables.

Appendix B Proof of Theorem 1

To see the boundedness of the variances of u(k) and $\varepsilon (k)$, it suffices to show $\Vert u_i\Vert _{\varvec{v}}$ and $\Vert \varepsilon _i\Vert _{\varvec{v}}$ are finite for $i=1,\ldots ,m$, where $u_i$ and $\varepsilon _i$ are the ith element of u and $\varepsilon$, respectively, and $\Vert u_i\Vert _{\varvec{v}}^2 = \lim _{k\rightarrow \infty }\mathrm {E}\{u_i^2(k)\}$.

Let $G_i$ and $g_i$ be the i-th row of G(z) and g(k), respectively. Since the input sequence $\left\{ v(k)\right\}$ is a white noise vector process, it is easy to get

$$\begin{aligned}&\mathrm {E}\{ {u_i^2(k)}\} \nonumber \\&\quad = \mathrm {E}\Big \{ \textstyle \sum \limits _{{l_1} = 0}^{\infty } {{g_i}( {{l_1}} )( {v( {{k - l_1}} ) + d( {{k - l_1}} )} )} \nonumber \\&\qquad \times {{\Big [ {\textstyle \sum \limits _{{l_2} = 0}^{\infty } {{g_i}( {{l_2}} )( {v( {{k - l_2}} ) + d( {{k - l_2}} )} )} } \Big ]}^*} \Big \} \nonumber \\&\quad ={\textstyle \sum \limits _{{l_1} = 0}^{\infty } {{g_i}( {{l_1}} )\varSigma _v g_i^*( {{l_1}} )} } \nonumber \\&\qquad +\textstyle \sum \limits _{{l_1},l_2 = 0}^{\infty } {{g_i}( {{l_1}} )\mathrm {E}\{ {d( {{k - l_1}} )d^*{{( {{k - l_2}} )}}} \}g_i^*( {{l_2}} )}. \end{aligned}$$

(27)

On the other hand, the uncorrelation of $\varDelta _{i_1}$ and $\varDelta _{i_2}$ for ${i_1}\ne {i_2}$ yields

$$\begin{aligned}&\mathrm {E}\{d(k_1)d^*(k_2)\} \nonumber \\&\quad =\mathrm {diag}\{\mathrm {E}\{d_{1}(k_1)d_{1}^*(k_2)\},\cdots ,\mathrm {E}\{d_m(k_1)d_{m}^*(k_2)\} \}, \end{aligned}$$

(28)

where $d_i$ is the ith element of d. Then each summand in the second term on the right-hand side of the equality of (27) can be written as

$$\begin{aligned}&{{g_i}( {{l_1}} )\mathrm {E}\{ {d( {{k - l_1}} )d^*{{( {{k - l_2}} )}}} \}g_i^*( {{l_2}} )} \nonumber \\&\quad =\textstyle \sum \limits _{i_1=1}^{m}{{g_{i i_1}}( {{l_1}} )g_{i i_1}( {{l_2}} )\mathrm {E}\{ {d_{i_1}( {{k - l_1}} )d_{i_1}{{( {{k - l_2}} )}}} \}}, \end{aligned}$$

(29)

where $g_{i_1 i_2}(k)$ is referred to as the impulse response of the transfer function $G_{{i_1 i_2}} := [G(z)]_{i_1 i_2}$. From (8), it holds that

$$\begin{aligned}&\mathrm {E}\{d_{i_1}(k-l_1)d_{i_2}(k-l_2)\} \nonumber \\&\quad = \textstyle \sum \limits _{ j_1=0 }^{{\bar{\tau }}_{i_1}} \textstyle \sum \limits _{ j_2=0 }^{{\bar{\tau }}_{i_1}}\delta ((k-l_1- j_1)-(k - l_2 -j_2)) \beta _{i_1,j_1j_2} \nonumber \\&\qquad \times \mathrm {E}\{u_{i_1}(k-l_1-j_1)u_{i_1}(k-l_2-j_2)\} \nonumber \\&\quad = \textstyle \sum \limits _{j=0}^{{\bar{\tau }}_{i_1}} \beta _{i_1,j(l_1+j-l_2)} \mathrm {E}\{u_{i_1}^2(k-l_1-j)\}. \end{aligned}$$

(30)

Thus, $\Vert d_{i}\Vert _{\varvec{v}}^2 = \textstyle \sum \limits _{j=0}^{{\bar{\tau }}_{i}} \beta _{i,jj} \Vert u_{i}\Vert _{\varvec{v}}^2$, and, from (27),

$$\begin{aligned} \Vert u_i\Vert _{\varvec{v}}^2 =&\lim _{k \rightarrow \infty } \textstyle \sum \limits _{l_1,l_2=0}^{\infty }\textstyle \sum \limits _{i_1=1}^{m} g_{i i_1}(l_1)g_{i i_1}(l_2) \\&\times \textstyle \sum \limits _{j=0}^{{\bar{\tau }}_{i_1}} \beta _{i_1,j(l_1-l_2+j)} \mathrm {E}\{u_{i_1}^2(k-l_1-j)\}\\ =&\!\!\textstyle \sum \limits _{i_1=1}^{m}\!\!\Big [ \textstyle \sum \limits _{l_1=0}^{\infty }\textstyle \sum \limits _{l_2=0}^{\infty } g_{i i_1}(l_1)g_{i i_1}(l_2)\textstyle \sum \limits _{j=0}^{{\bar{\tau }}_{i_1}} \beta _{i_1,j(l_1\!-\!l_2\!+\!j)} \Big ]\Vert u_{i_1}\Vert _{\varvec{v}}^2\\ =&\!\!\textstyle \sum \limits _{i_1=1}^{m}\!\!\Big [ \textstyle \sum \limits _{l=0}^{\infty }\textstyle \sum \limits _{\lambda =-\infty }^{\infty } g_{i i_1}(l)g_{i i_1}(l-\lambda ) r_{i_1}(\lambda ) \Big ] \Vert u_{i_1}\Vert _{\varvec{v}}^2. \end{aligned}$$

By the definition of ${\mathcal {H}}_2$ norm, it shows that

$$\begin{aligned}&\textstyle \sum \limits _{l=0}^{\infty }\textstyle \sum \limits _{\lambda =-\infty }^{\infty } g_{i i_1}(l)g_{i i_1}(l-\lambda ) r_{i_1}(\lambda )\\&\quad = \Vert G_{i i_1}(z)\varPhi _{i_1}(z)\Vert _2^2, \end{aligned}$$

provided that $\varPhi _{i_1}^\sim (z)\varPhi _{i_1}(z) = S_{i_1}(z)$, where $\varPhi _i$ and $S_i$ are the ith diagonal element of $\varPhi$ and S, respectively. By the fact that

$$\begin{aligned} {\textstyle \sum \limits _{{l} = 0}^{\infty } {{g_i}( {{l}} )\varSigma _v g_i^*( {{l}} )} } = \textstyle \sum \limits _{i_1=1}^{m}\Vert G_{i i_1}(z)\Vert _2^2 \sigma _{v_{i_1}}^2, \end{aligned}$$

we have

$$\begin{aligned} \Vert u_i\Vert _{\varvec{v}}^2 = \textstyle \sum \limits _{i_1=1}^{m}\Vert G_{i i_1}\Vert _2^2 \sigma _{v_{i_1}}^2 + \textstyle \sum \limits _{i_1=1}^{m}\Vert G_{i i_1}\varPhi _{i_1}\Vert _2^2 \Vert u_{i_1}\Vert _{\varvec{v}}^2. \end{aligned}$$

As a result,

$$\begin{aligned} \begin{bmatrix} \Vert u_1\Vert _{\varvec{v}}^2\\ \vdots \\ \Vert u_m\Vert _{\varvec{v}}^2 \end{bmatrix} =&\begin{bmatrix} {\left\| {{G_{11}}} \right\| _2^2}&{} \cdots &{}{\left\| {{G_{1m}}} \right\| _2^2}\\ \vdots &{}{}&{} \vdots \\ {\left\| {{G_{m1}}} \right\| _2^2}&{} \cdots &{}{\left\| {{G_{mm}}} \right\| _2^2} \end{bmatrix}\begin{bmatrix} \sigma _{v_1}^2 \\ \vdots \\ \sigma _{v_m}^2 \end{bmatrix} \nonumber \\&+\begin{bmatrix} {\left\| {{G_{11}}{\varPhi _1}} \right\| _2^2}&{} \cdots &{}{\left\| {{G_{1m}}{\varPhi _m}} \right\| _2^2}\\ \vdots &{}{}&{} \vdots \\ {\left\| {{G_{m1}}{\varPhi _1}} \right\| _2^2}&{} \cdots &{}{\left\| {{G_{mm}}{\varPhi _m}} \right\| _2^2} \end{bmatrix} \begin{bmatrix} \Vert u_1\Vert _{\varvec{v}}^2\\ \vdots \\ \Vert u_m\Vert _{\varvec{v}}^2 \end{bmatrix}. \end{aligned}$$

(31)

Then a necessary and sufficient condition for the existence of finite and unique $\Vert u_i\Vert _{\varvec{v}}^2$ is that the condition (13) holds, noting that $G_{i_1 i_2}\varPhi _{i_2} = T_{i_1 i_2}W_{i_2}$. Meanwhile, with the fact that

$$\begin{aligned} \Vert \varepsilon _i\Vert _{\varvec{v}}^2 = \sigma _{v_i}^2 + \Vert d_i\Vert _{\varvec{v}}^2, \end{aligned}$$

the proof is established.

Appendix C Proof of Corollary 2

It follows from [18] that the inner part of $\varGamma ^{\frac{1}{2}}M_\mathrm{\mathrm{in}}\varGamma ^{-\frac{1}{2}}$ can be constructed as

$$\begin{aligned} M_{\varGamma \mathrm{in}}=\left[ \begin{array}{l|l}{ \dfrac{1}{\lambda ^*}}&{}{{ \dfrac{\sqrt{|\lambda |^2-1}}{\lambda ^*}}\eta _{\varGamma }^*}\\ \hline {{ \dfrac{\sqrt{|\lambda |^2-1}}{\lambda ^*}}\eta _{\varGamma }}&{}{I-\left( 1+ \dfrac{1}{\lambda ^*}\right) \eta _{\varGamma }\eta _{\varGamma }^*},\end{array}\right] \end{aligned}$$

where $\eta _{\varGamma } = \frac{\varGamma ^{-\frac{1}{2}}\eta }{\Vert \varGamma ^{-\frac{1}{2}}\eta \Vert _{2}}$. Thus, (21) directly yields that

$$\begin{aligned} \pi _i = |W_i(\lambda )|^2 (\lambda ^2 - 1) \end{aligned}$$

and completes the proof.