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.
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This work was supported by the National Natural Science Foundation of China (Nos. 61933006 and 61673183)
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
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
On the other hand, the uncorrelation of \(\varDelta _{i_1}\) and \(\varDelta _{i_2}\) for \({i_1}\ne {i_2}\) yields
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
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
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),
By the definition of \({\mathcal {H}}_2\) norm, it shows that
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
we have
As a result,
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
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
where \(\eta _{\varGamma } = \frac{\varGamma ^{-\frac{1}{2}}\eta }{\Vert \varGamma ^{-\frac{1}{2}}\eta \Vert _{2}}\). Thus, (21) directly yields that
and completes the proof.
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Li, J., Lu, J. & Su, W. Stabilizability of minimum-phase systems with colored multiplicative uncertainty. Control Theory Technol. 20, 382–391 (2022). https://doi.org/10.1007/s11768-022-00108-9
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DOI: https://doi.org/10.1007/s11768-022-00108-9