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SCIENCE CHINA Information Sciences, Volume 61, Issue 9: 092203(2018) https://doi.org/10.1007/s11432-017-9185-6

Achievable delay margin using LTI control for plants with unstable complex poles

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  • ReceivedApr 18, 2017
  • AcceptedJun 20, 2017
  • PublishedJan 4, 2018

Abstract

We consider the achievable delay margin of a real rational and strictly proper plant, with unstable complex poles, by a linear time-invariant (LTI) controller.The delay margin is defined as the largest time delay such that, for any delay less than this value, the closed-loop stability is maintained. Drawing upon a frequency domain method, particularly a bilinear transform technique, we provide an upper bound of the delay margin, which requires computing the maximum of a one-variable function. Finally, the effectiveness of the theoretical results is demonstrated through a numerical example.


Acknowledgment

This work was partially supported by Taishan Scholar Construction Engineering by Shandong Government and National Natural Science Foundation of China (Grant Nos. 61573220, 61573221, 61633014).


Supplement

Appendix

Proof of Lemma 1

We first write $P_{\alpha}(s)=P_0(s)B_{\alpha}(s)$ as the ratio of two polynomials, which have no common zeros in $\bar{\mathcal{C}}^+$ for small $\alpha\geq~0~$.

If i) holds, write

\begin{eqnarray}P_0(s)=g\cdot \frac{\bar{n}_0(s)}{[(s-a)^2 +b^2]\cdot\bar{d}_0(s)} \end{eqnarray}

with $\bar{n}_0(s)$ and $[(s-a)^2+b^2]\cdot\bar{d}_0(s)$ monic and coprime. Let

\begin{eqnarray}\left\{ \begin{aligned}n_{\alpha}(s):=&[(s+a)^2+b^2-\alpha]\cdot(1-c_{\alpha} s)\cdot\bar{n}_0(s), \\ d_{\alpha}(s):=&[(s+a)^2+b^2]\cdot[(s-a)^2+b^2-\alpha]\cdot(1+\bar{c}_{\alpha}s)\cdot\bar{d}_0(s). \end{aligned} \right. \end{eqnarray}

If ii) holds, write

\begin{eqnarray}P_0(s)=g\cdot \frac{\bar{n}_0(s)(z-s)}{[(s-a)^2 +b^2]\cdot\bar{d}_0(s)} \end{eqnarray}

with $\bar{n}_0(s)(z-s)$ and $[(s-a)^2+b^2]\cdot\bar{d}_0(s)$ monic and coprime. Let

\begin{eqnarray}\left\{ \begin{aligned}n_{\alpha}(s):=&[(s+a)^2+b^2-\alpha]\cdot(1-c_{\alpha} s)\cdot[z+s]\cdot\bar{n}_0(s), \\ d_{\alpha}(s):=&[(s+a)^2+b^2]\cdot[(s-a)^2+b^2-\alpha]\cdot(1+\bar{c}_{\alpha}s)\cdot\bar{d}_0(s). \end{aligned} \right. \end{eqnarray}

If iii) holds, write

\begin{eqnarray}P_0(s)=g\cdot \frac{\bar{n}_0(s)(z-s)}{\bar{d}_0(s)} \end{eqnarray}

with $\bar{n}_0(s)(z-s)$ and $\bar{d}_0(s)$ monic and coprime. Let

\begin{eqnarray}\left\{ \begin{aligned}n_{\alpha}(s):=&[a+b\text{j} -\alpha s]\cdot[a-b\text{j} -\alpha s]\cdot[z+s]\cdot\bar{n}_0(s), \\ d_{\alpha}(s):=&[a-b\text{j} +\alpha s]\cdot[a+b\text{j} +\alpha s]\cdot\bar{d}_0(s). \end{aligned} \right. \end{eqnarray}

Then, combing the above three cases, we can write

\begin{eqnarray}P_{\alpha}(s)=P_0(s)B_{\alpha}(s)=g\cdot \frac{n_{\alpha}(s)}{d_{\alpha}(s)}. \end{eqnarray}

It is straightforward to verify that $n_{\alpha}(s)$ and $d_{\alpha}(s)$ have no common zeros in $\bar{\mathcal{C}}^+$ for small $\alpha\geq0$. But this property would be lost at $\hat{\alpha}>0$ for there is an unstable pole-zero cancellation in $P_{\alpha}(s)$. Based on the continuity of the zeros of the characteristic polynomial as a function of the parameter $\alpha$, there exist an $\alpha^*\in~[0,~\hat{\alpha})$ so that the polynomial has a zero on the imaginary axis, i.e., $s=\text{j}\omega^*$, which means that $B_{\alpha^*}(\text{j}\omega^*)P_0(\omega^*)C(\omega^*)=-1$. Thus, the proof is completed.

Proof of Claim 1

Following the notation of $A=a^2+b^2$, and

\begin{eqnarray}Q_\alpha(s)=[a+b\text{j} -\alpha s]\cdot[a-b\text{j} -\alpha s]\cdot[z+s], \end{eqnarray}

at $s=\text{j}\omega$, we have

\begin{eqnarray}\begin{aligned}Q_\alpha(\text{j}\omega) =&[(\text{j}\omega-a)^2+b^2][(\text{j}\omega+a)^2+b^2-\alpha][1-\text{j}\omega c_\alpha] \tag{11} \\ =&[A-\omega^2-2a\omega \text{j} ][A-\omega^2+2a\omega \text{j} -\alpha][1-\text{j}\omega c_\alpha] \tag{12} \\ =&[(A-\omega^2)^2-\alpha(A-\omega^2)+4a^2\omega^2+2a\alpha\omega \text{j} ](1-\text{j} c_\alpha \omega) \tag{13} \\ =&[\omega^4+\omega^2(4a^2+\alpha-2A+2a\alpha)+A^2-\alpha A] \tag{14} \\ &-\text{j}[c_\alpha\omega^5+c_\alpha\omega^3(4a^2+\alpha-2A)+c_\alpha (A^2-\alpha A)-2a\alpha]. \end{aligned} \end{eqnarray}

Since

\begin{eqnarray}c_\alpha=\frac{2a\alpha}{A^2-\alpha A}, \end{eqnarray}

so $c_\alpha~(A^2-\alpha~A)-2a\alpha=0$, and note that $\Lambda(\alpha,\omega)={\omega}^2+4a^2+\alpha-2A$, then from the above equation, we have

\begin{eqnarray}\begin{aligned}Q_\alpha(\text{j}\omega)=&[\omega^4+\omega^2(4a^2+\alpha-2A+2a\alpha)+A^2-\alpha A] -\text{j}[c_\alpha\omega^5+c_\alpha\omega^3(4a^2+\alpha-2A)] \tag{15} \\ =& [ \Lambda(\alpha,\omega){\omega}^2 +2a\alpha c_{\alpha}{\omega}^2+A^2-\alpha A]-\text{j} c_{\alpha}\Lambda(\alpha,\omega){\omega}^3 \tag{16} \\ =&{\rm Re}Q_{\alpha}(\text{j}\omega)-\text{j}{\rm Im}Q_{\alpha}(\text{j}\omega), \end{aligned} \end{eqnarray}

as claimed.


References

[1] Krstic M. Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Boston: Birkhauser, 2009. Google Scholar

[2] Michiels W, Niculescu S I. Stability, Control and Computation of Time-delay Systems: An Eigenvalue Based Approach. 2nd ed. Philadelphia: SIAM Publications, 2014. Google Scholar

[3] Fridman E. Introdution to Time-delay Systems: Analysis and Control. London: Springer, 2014. Google Scholar

[4] Richard J P. Time-delay systems: an overview of some recent advances and open problems. Automatica, 2003, 39: 1667-1694 CrossRef Google Scholar

[5] Gu K, Niculescu S I. Survey on Recent Results in the Stability and Control of Time-Delay Systems. J Dyn Sys Meas Control, 2003, 125: 158-165 CrossRef Google Scholar

[6] Sipahi R, Niculescu S I, Abdallah C T, et al. Stability and stabilization of systems with time delay: limitations and opportunities. IEEE Contr Syst Mag, 2011, 31: 38--65. Google Scholar

[7] Davison D E, Miller D E. Determining the least upper bound on the achievable delay margin. In: Open Problems in Mathematical Systems and Control Theory. Blondel V D, Megretski A, eds. Princeton: Princeton University Press, 2004. 276--279. Google Scholar

[8] Middleton R H, Miller D E. On the Achievable Delay Margin Using LTI Control for Unstable Plants. IEEE Trans Automat Contr, 2007, 52: 1194-1207 CrossRef Google Scholar

[9] Ju P, Zhang H. Further Results on the Achievable Delay Margin Using LTI Control. IEEE Trans Automat Contr, 2016, 61: 3134-3139 CrossRef Google Scholar

[10] Qi T, Zhu J, Chen J. On delay radii and bounds of MIMO systems. Automatica, 2017, 77: 214-218 CrossRef Google Scholar

[11] Qi T, Zhu J, Chen J. Fundamental limits on uncertain delays: when is a delay system stabilizable by LTI controllers? IEEE Trans Automat Control, 2017, 62: 1314--1328. Google Scholar

[12] Michiels W, Engelborghs K, Vansevenant P. Continuous pole placement for delay equations. Automatica, 2002, 38: 747-761 CrossRef Google Scholar

[13] Zhu Y, Krstic M, Su H. Adaptive Output Feedback Control for Uncertain Linear Time-Delay Systems. IEEE Trans Automat Contr, 2017, 62: 545-560 CrossRef Google Scholar

[14] Glizer V Y. Controllability conditions of linear singularly perturbed systems with small state and input delays. Math Control Signals Syst, 2016, 28: 1-39 CrossRef Google Scholar

[15] Gaudette D L, Miller D E. Stabilizing a SISO LTI Plant With Gain and Delay Margins as Large as Desired. IEEE Trans Automat Contr, 2014, 59: 2324-2339 CrossRef Google Scholar

[16] Mazenc F, Malisoff M, Niculescu S I. Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach. SIAM J Control Optim, 2017, 55: 533-556 CrossRef Google Scholar

[17] Vidyasagar M. Control System Synthesis: A Factorization Approach. Cambridge: MIT Press, 1985. Google Scholar

[18] Rekasius Z V. A stability test for systems with delays. In: Proceedings of Joint Automatic Control Conference, San Francisco, 1980. Google Scholar

[19] Olgac N, Sipahi R. An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Trans Automat Contr, 2002, 47: 793-797 CrossRef Google Scholar

[20] Gu K Q, Kharitonov V L, Chen J. Stability of Time-delay Systems. Boston: Birkhäuser, 2003. Google Scholar

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