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SCIENTIA SINICA Informationis, Volume 48 , Issue 7 : 947-962(2018) https://doi.org/10.1360/N112017-00282

Virtual equivalent system theory for adaptive control and simulation verification

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  • ReceivedDec 24, 2017
  • AcceptedFeb 14, 2018
  • PublishedJul 12, 2018

Abstract


Funded by

国家自然科学基金(61520106010,61741302)


Supplement

Appendix

引理及证明

以下引理在文献32里都有对应内容, 为方便中文阅读, 特意归纳整理, 并略有修正.

引理A1 确定性自校正控制的虚拟等价系统(图3)可以分解为3个子系统, 分别如图4$\sim$6所示, 且满足 $~y(k)=y_1(k)+y_2(k)+y_3(k)$, $~u(k)=u_1(k)+u_2(k)+u_3(k).$

proof 本引理是引理3的特殊情形, 在引理3中令$\omega(k)=0$, 即得本引理结论.

引理A2 给定 $\tilde{\phi}(k)=\tilde{\phi}_1(k)+\tilde{\phi}_2(k)$, $\|\tilde{\phi}_1(k)\|<\infty$, $\|\tilde{\phi}_2(k)\|=~o(\alpha+\|\tilde{\phi}(k)\|+\cdots+\|\tilde{\phi}(k-s)\|)$, 其中 $s$ 为有限正整数, $\alpha$ 为常数. 则必有 $\|\tilde{\phi}(k)\|<\infty$.

proof 首先, 我们知道 \begin{equation}\sum _{k=1}^{k=N}\|\tilde{\phi}(k)\|\le \sum _{k=1}^{k=N}\|\tilde{\phi}_1(k)\| + \sum _{k=1}^{k=N}\|\tilde{\phi}_2(k)\|. \tag{44}\end{equation} 下面采用反证法进行证明, 假设 $\|\tilde{\phi}(k)\|$ 无界. 那么必有无穷子列 $\|\tilde{\phi}(p_{k})\|\to~\infty$ 满足 \begin{equation}\sum _{k=1}^{k=N}\|\tilde{\phi}(p_k)\|\le \sum _{k=1}^{k=N}\|\tilde{\phi}_1(p_k)\| + \sum _{k=1}^{k=N}\|\tilde{\phi}_2(p_k)\|. \tag{45}\end{equation} 接下来分别考察(A2)式右面两项的性质. 首先考察第一项, 根据 Stolz 定理 以及 $\|\tilde{\phi}_1(p_k)\|<\infty$, 知道 \begin{equation}\frac{\sum _{k=1}^{k=N}\|\tilde{\phi}_1(p_k)\|}{\sum _{k=1}^{k=N}\|\tilde{\phi}(p_k)\|}\to\frac{\|\tilde{\phi}_1(p_k)\|}{\|\tilde{\phi}(p_k)\|}\to 0. \tag{46}\end{equation} 接下来考察第二项, 注意到 $~\|\tilde{\phi}(k)\|=0,~k\le~0$, 那么, \begin{equation}\sum _{k=1}^{k=N}\|\tilde{\phi}(k-i)\| \le \sum _{k=1}^{k=N}\|\tilde{\phi}(k)\|, i=1,\ldots,s. \tag{47}\end{equation} 因此, \begin{equation}\begin{split} \sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(k)\|+\cdots+\|\tilde{\phi}(k-s)\|) \le(s+1)\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(k)\|). \end{split} \tag{48}\end{equation} 上式结合本引理的给定条件导致 \begin{equation}\sum _{k=1}^{k=N}\|\tilde{\phi}_2(k)\|=o\left(\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(k)\|)\right). \tag{49}\end{equation} 进一步, 考虑到收敛序列及其子序列的极限相同, 我们有 \begin{equation}\sum _{k=1}^{k=N}\|\tilde{\phi}_2(p_k)\|=o\left(\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(p_k)\|)\right). \tag{50}\end{equation} 考虑到 (A2) 和(A3)式, 并利用夹逼原理有 \begin{equation}\frac{\sum _{k=1}^{k=N}\|\tilde{\phi}(p_k)\|}{\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(p_k)\|)}\to 0. \tag{51}\end{equation} 上式与如下事实相矛盾(根据 Stolz 定理): \begin{equation}\frac{\sum _{k=1}^{k=N}(\|\tilde{\phi}(p_k)\|)}{\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(p_k)\|)}\to 1. \tag{52}\end{equation} 因此原假设不成立. 从而证得结论, 即 $\|\tilde{\phi}(k)\|<\infty$.

引理A3 随机自校正控制的虚拟等价系统(图10)可以分解为3个子系统, 分别如11, 1213所示, 且有 \begin{equation}y(k)=y_1(k)+y_2(k)+y_3(k), u(k)=u_1(k)+u_2(k)+u_3(k).\end{equation}

proof 用数学归纳法给出证明. 首先, 我们知道, 对于 $k~\le~0~$ 有 \begin{equation}y(k)=y_1(k)+y_2(k)+y_3(k), u(k)=u_1(k)+u_2(k)+u_3(k). \tag{53}\end{equation} 接下来, 假设(A10)式对于$~k,k-1,\ldots,1.~$ 成立. 分别考虑图11$\sim$13, 有 \begin{align}& y(k+1)=\phi^{\rm T}(k-d+1)\hat\theta(t_k)+\omega(k+1)+e_i(k+1), \tag{54} \\ & y_1(k+1)=\phi^{\rm T}_1(k-d+1)\hat\theta(t_k)+\omega(k+1), \tag{55} \\ & y_2(k+1)=\phi^{\rm T}_2(k-d+1)\hat\theta(t_k)+e_i(k+1), \tag{56} \\ & y_3(k+1)=\phi^{\rm T}_3(k-d+1)\hat\theta(t_k). \tag{57} \end{align} 其中 \begin{eqnarray}\phi^{\rm T}_1(k-d+1)&=&[y_1(k),\ldots,y_1(k-n+1),u(k-d+1),\ldots,u_1(k-d-m+1)], \tag{58} \\ \phi^{\rm T}_2(k-d+1)&=&[y_2(k),\ldots,y_2(k-n+1),u_2(k-d+1),\ldots,u_2(k-d-m+1)], \tag{59} \\ \phi^{\rm T}_3(k-d+1)&=&[y_3(k),\ldots,y_3(k-n+1),u_3(k-d+1),\ldots,u_3(k-d-m+1)]. \tag{60} \end{eqnarray} 按照前面的假设, 即, (A10)式对于$~k,k-1,\ldots,1.~$ 成立, 显然有 \begin{equation}\phi^{\rm T}_1(k-d+1)+\phi^{\rm T}_2(k-d+1)+\phi^{\rm T}_3(k-d+1)=\phi^{\rm T}(k-d+1). \tag{61}\end{equation} 因此, 可以得到 \begin{equation}\begin{split} y_1(k+1)+y_2(k+1)+y_3(k+1)=\phi^{\rm T}(k-d+1)\hat\theta(t_k)+\omega(k+1)+e_i(k+1)=y(k+1). \end{split} \tag{62}\end{equation} 下面考察 $u_1(k+1)$, $u_2(k+1)$ 和 $u_3(k+1)$. 在图10中, 有 \begin{equation}u(k+1)=\phi^{\rm T}_{c}(k+1)\theta_c(t_{k+1})+\Delta u'(k+1). \tag{63}\end{equation} 类似地, 在图11$\sim$13中, 有 \begin{eqnarray}u_1(k+1)&=&\phi^{\rm T}_{c1}(k+1)\theta_c(t_{k+1}), \tag{64} \\ u_2(k+1)&=&\phi^{\rm T}_{c2}(k+1)\theta_c(t_{k+1}), \tag{65} \\ u_3(k+1)&=&\phi^{\rm T}_{c3}(k+1)\theta_c(t_{k+1})+\Delta u'(k+1), \tag{66} \end{eqnarray} 其中 \begin{eqnarray}\phi^{\rm T}_c(k+1)&=&[y(k+1),\ldots,y(k+1-s_1),u(k),\ldots,u(k+1-s_2),y_r(k+1),\ldots,y_r(k+1-s_3)], \tag{67} \\ \phi^{\rm T}_{c1}(k+1)&=&[y_1(k+1),\ldots,y_1(k+1-s_1),u_1(k),\ldots,u_1(k+1-s_2),y_r(k+1),\ldots,y_r(k+1-s_3)], \tag{68} \\ \phi^{\rm T}_{c2}(k+1)&=&[y_2(k+1),\ldots,y_2(k+1-s_1),u_2(k),\ldots,u_2(k+1-s_2),0,\ldots,0], \tag{69} \\ \phi^{\rm T}_{c3}(k+1)&=&[y_3(k+1),\ldots,y_3(k+1-s_1),u_3(k),\ldots,u_3(k+1-s_2),0,\ldots,0], \tag{70} \end{eqnarray} 其中, $s_1\geq~1,~s_2\geq~1,~s_3\geq~1$为正整数, 由不同的控制策略决定. 进一步基于 (A10)及(A19)式, 显然有 \begin{equation}\phi^{\rm T}_{c1}(k+1)+\phi^{\rm T}_{c2}(k+1)+\phi^{\rm T}_{c3}(k+1)=\phi^{\rm T}_c(k+1). \tag{71}\end{equation} 因此, \begin{equation}\begin{split} u_1(k+1)+u_2(k+1)+u_3(k+1) =\phi^{\rm T}_c(k+1)\theta_c(t_{k+1})+\Delta u'(k+1) =u(k+1). \end{split} \tag{72}\end{equation} 所以 (A10)式 对所有 $k$皆成立.

引理A4 若 $\widetilde{\phi}(k)=\widetilde{\phi}_1(k)+\widetilde{\phi}_2(k)$, $\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(k)\|^2~<\infty$, $\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_2(k)\|^2=~o(\frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}(k)\|^2)$, 则必有$\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}(k)\|^2~<\infty$.

proof 利用三角不等式以及显然的事实 $2ab\le~a^2+b^2$, 有 \begin{equation}\begin{split} \frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}(k)\|^2 =\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(k)+\widetilde{\phi}_2(k)\|^2 \le\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(k)\|^2+\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_2(k)\|^2. \end{split} \tag{73}\end{equation} 下面用反证法得出结论. 假设 $\frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}(k)\|^2$ 无界, 则一定存在无穷子列 $\frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}(p_k)\|^2\to\infty$, 满足 \begin{equation}\begin{split} \frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}(p_k)\|^2 =\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(p_k)+\widetilde{\phi}_2(p_k)\|^2 \le\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(p_k)\|^2+\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_2(p_k)\|^2. \end{split} \tag{74}\end{equation} 进一步考虑到收敛序列及其子序列的极限相同, 于是得到 \begin{equation}\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_2(p_k)\|^2= o\left(\frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}(p_k)\|^2\right). \tag{75}\end{equation} 上式结合(A31)式 以及事实 $\frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}_1(p_k)\|^2<\infty$, 利用夹逼定理, 得出明显的错误结论 $\frac{\sum^{n}_{k=1}\|\widetilde{\phi}(p_k)\|^2}{\sum^{n}_{k=1}\|\widetilde{\phi}(p_k)\|^2}\to~0$, 因此, 之前的假设不成立, 从而得到 $ \frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}(k)\|^2<\infty. $

引理A5 若 $\frac{1}{n}\sum^{n}_{k=1}[y_2(k)]^2=~o(1)$, $\frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]^2<\infty$, 则必有 $\frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)+y_2(k)]^2\to~\frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]^2.$

proof 显然 \begin{equation}\begin{split} \frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)+y_2(k)]^2 &=\frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]^2+\frac{1}{n}\sum^{n}_{k=1}[y_2(k)]^2 +\frac{2}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]y_2(k) \\ &\to \frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]^2+\frac{2}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]y_2(k). \end{split} \tag{76}\end{equation} 根据Cauchy不等式, 有 \begin{equation}\begin{split} 0\le\left\{\frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]y_2(k)\right\}^2\le\left\{\frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]^2\right\}.\left\{\frac{1}{n}\sum^{n}_{k=1}[y_2(k)]^2\right\} \to 0.\end{split} \tag{77}\end{equation} 于是, 运用夹逼定理得到 \begin{equation}\left\{\frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]y_2(k)\right \}^2\to 0, \frac{1}{n}\sum^{n}_{k=1}[y_1(k)-y_r(k)]y_2(k)\to 0. \tag{78}\end{equation}

代入 (A33)式, 即得所证结果.


References

[1] Kalman R E. Design of a self optimizing control system. Trans ASME, 1958, 80: 468--478. Google Scholar

[2] ?str?m K J, Wittenmark B. On self tuning regulators. Automatica, 1973, 9: 185-199 CrossRef Google Scholar

[3] Goodwin G C, Ramadge P J, Caines P E. Discrete time stochastic adaptive control. SIAM J Control Optim, 1981, 19: 829-853 CrossRef Google Scholar

[4] Guo L, Chen H F. The AAstrom-Wittenmark self-tuning regulator revisited and ELS-based adaptive trackers. IEEE Trans Automat Contr, 1991, 36: 802-812 CrossRef Google Scholar

[5] Guo L, Chen H F. Convergence and optimality of self-tuning regulators. Science China (Series A), 1991, 21: 905--913. Google Scholar

[6] Sin K S, Goodwin G C. Stochastic adaptive control using a modified least squares algorithm. Automatica, 1982, 18: 315-321 CrossRef Google Scholar

[7] Zhang You-Hong . Stochastic adaptive control and prediction based on a modified least squares--the general delay-colored noise case. IEEE Trans Automat Contr, 1982, 27: 1257-1260 CrossRef Google Scholar

[8] Anderson B D O, Johnstone R M G. Global adaptive pole positioning. IEEE Trans Automat Contr, 1985, 30: 11-22 CrossRef Google Scholar

[9] Elliott H, Cristi R, Das M. Global stability of adaptive pole placement algorithms. IEEE Trans Automat Contr, 1985, 30: 348-356 CrossRef Google Scholar

[10] Lozano R, Xiao-Hui Zhao R. Adaptive pole placement without excitation probing signals. IEEE Trans Automat Contr, 1994, 39: 47-58 CrossRef Google Scholar

[11] Goodwin G C, Sin K S. Adaptive Filtering Prediction and Control. Englewood: Prentice-hall, Inc., 1984. Google Scholar

[12] Chan C Y, Sirisena H R. Convergence of adaptive pole-zero placement controller for stable non-minimum phase systems. Int J Control, 1989, 50: 743-754 CrossRef Google Scholar

[13] Lai T, Wei C-Z. Extended least squares and their applications to adaptive control and prediction in linear systems. IEEE Trans Automat Contr, 1986, 31: 898-906 CrossRef Google Scholar

[14] Chen H F, Guo L. Asymptotically optimal adaptive control with consistent parameter estimates. SIAM J Control Optim, 1987, 25: 558-575 CrossRef Google Scholar

[15] Lei Guo . Self-convergence of weighted least-squares with applications to stochastic adaptive control. IEEE Trans Automat Contr, 1996, 41: 79-89 CrossRef Google Scholar

[16] Nassiri-Toussi K, Wei Ren K. Indirect adaptive pole-placement control of MIMO stochastic systems: self-tuning results. IEEE Trans Automat Contr, 1997, 42: 38-52 CrossRef Google Scholar

[17] Wittenmark B, Middleton R H, Goodwin G C. Adaptive decoupling of multivariable systems. Int J Control, 1987, 46: 1993-2009 CrossRef Google Scholar

[18] Chai T Y. The global convergence analysis of a multivariable decoupling self-tuning controller. Acta Autom Sin, 1989, 15: 432--436. Google Scholar

[19] Chai T Y. Direct adaptive decoupling control for general stochastic multivariable systems. Int J Control, 1990, 51: 885-909 CrossRef Google Scholar

[20] Chai T Y, Wang G. Globally convergent multivariable adaptive decoupling controller and its application to a binary distillation column. Int J Control, 1992, 55: 415-429 CrossRef Google Scholar

[21] Patete A, Furuta K, Tomizuka M. Stability of self-tuning control based on Lyapunov function. Int J Adapt Control Signal Process, 2008, 22: 795-810 CrossRef Google Scholar

[22] Katayama T, McKelvey T, Sano A. Trends in systems and signals. Annu Rev Control, 2006, 30: 5-17 CrossRef Google Scholar

[23] Li Q Q. Adaptive control. Comput Autom Meas Control, 1999, 7: 56--60. Google Scholar

[24] Li Q Q. Adaptive Control System Theory, Design and Application. Beijing: Science Press, 1990. Google Scholar

[25] Fekri S, Athans M, Pascoal A. Issues, progress and new results in robust adaptive control. Int J Adapt Control Signal Process, 2006, 20: 519-579 CrossRef Google Scholar

[26] Åström K J, Wittenmark B. Adaptive Control. Upper Saddle River: Addison-Wesley, 1995. Google Scholar

[27] Ioannou P A, Sun J. Robust Adaptive Control. Prentice-Hall: Englewood Cliffs, 1996. Google Scholar

[28] Kumar P R. Convergence of adaptive control schemes using least-squares parameter estimates. IEEE Trans Automat Contr, 1990, 35: 416-424 CrossRef Google Scholar

[29] van Schuppen J H. Tuning of Gaussian stochastic control systems. IEEE Trans Automat Contr, 1994, 39: 2178-2190 CrossRef Google Scholar

[30] Nassiri-Toussi K, Ren W. A unified analysis of stochastic adaptive control: asymptotic self-tuning. In: Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, 1995. 2932--2937. Google Scholar

[31] Morse A S. Towards a unified theory of parameter adaptive control. II. Certainty equivalence and implicit tuning. IEEE Trans Automat Contr, 1992, 37: 15-29 CrossRef Google Scholar

[32] Zhang W. On the stability and convergence of self-tuning control-virtual equivalent system approach. Int J Control, 2010, 83: 879-896 CrossRef Google Scholar

[33] Zhang W C. The convergence of parameter estimates is not necessary for a general self-tuning control system- stochastic plant. In: Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, 2009. Google Scholar

[34] Zhang W C, Li X L, Choi J Y. A unified analysis of switching multiple model adaptive control — virtual equivalent system approach. In: Proceedings of the 17th IFAC World Congress, Seoul, 2008. 41: 14403--14408. Google Scholar

[35] Zhang W C. Virtual equivalent system theory for self-tuning control. J Harbin Inst Tech, 2014, 46: 107--112. Google Scholar

[36] Zhang W C, Chu T G, Wang L. A new theoretical framework for self-tuning control. Int J Inf Tech, 2005, 11: 123--139. Google Scholar

[37] Liberzon D, Morse A S. Basic problems in stability and design of switched systems. IEEE Control Syst Mag, 1999, 19: 59-70 CrossRef Google Scholar

[38] Shorten R, Wirth F, Mason O. Stability Criteria for Switched and Hybrid Systems. SIAM Rev, 2007, 49: 545-592 CrossRef Google Scholar

[39] Desoer C A, Vidyasagar M. Feedback Systems: Input-Output Properties. New York: Academic Press, 1975. Google Scholar

[40] Chatterjee D, Liberzon D. On stability of stochastic switched systems. In: Proceedings of the 43rd IEEE Conference on Decision and Control, Nassau, 2004. 4: 4125--4127. Google Scholar

[41] Prandini M. Switching control of stochastic linear systems: stability and performance results. In: Proceedings of the 6th Congress of SIMAI, Cagliari, 2002. Google Scholar

[42] Prandini M, Campi M C. Logic-based switching for the stabilization of stochastic systems in presence of unmodeled dynamics. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, 2001. Google Scholar

[43] Guo L. A retrospect of the research on self-tuning regulators. All About Control Syst, 2014, 1: 50--61. Google Scholar

[44] Egardt B. Unification of some discrete-time adaptive control schemes. IEEE Trans Automat Contr, 1980, 25: 693-697 CrossRef Google Scholar

[45] Gawthrop P J. Some interpretations of the self-tuning controller. Proc IEEE, 1997, 124: 889--894. Google Scholar

[46] Ljung L, Landau I D. Model reference adaptive systems and self-tuning regulators — some connections. In: Proceedings of the 7th IFAC World Congress, Helsinki, 1978. 3: 1973--1980. Google Scholar

[47] Narendra K S, Valavani L S. Direct and indirect adaptive control. Automatica, 1979, 15: 663--664. Google Scholar

[48] Zhang W. Stable weighted multiple model adaptive control: discrete-time stochastic plant. Int J Adapt Control Signal Process, 2013, 27: 562-581 CrossRef Google Scholar

[49] Wei W, Zhang W C, Li D H, et al. On the stability of linear active disturbance rejection control: virtual equivalent system approach. In: Proceedings of the Chinese Intelligent Systems Conference, Yangzhou, 2015. 295--306. Google Scholar

[50] Li P W, Zhang W C. Towards a unified stability analysis of continuous-time T-S model based fuzzy control-virtual equivalent system approach. Int J Model Ident Control, 2018. Google Scholar