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

• AcceptedFeb 14, 2018
• PublishedJul 12, 2018
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### Abstract

A general and unified analysis of a self-tuning control (STC) system composed of any possible control strategy, any possible parameter estimation algorithm, and any linear plant is presented. Based on virtual equivalent system (VES) theory, several stability and convergence criteria were developed without the convergence requirement of parameter estimation. These criteria could be used to guide control engineering practice. Considering that model reference adaptive control (MRAC) can be regarded as a special case of STC, VES theory is also suitable for the stability and convergence analysis of MRAC. Finally, simulation results verified the effectiveness of VES theory.

### Supplement

Appendix

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.

proof 首先, 我们知道 $$\sum _{k=1}^{k=N}\|\tilde{\phi}(k)\|\le \sum _{k=1}^{k=N}\|\tilde{\phi}_1(k)\| + \sum _{k=1}^{k=N}\|\tilde{\phi}_2(k)\|. \tag{44}$$ 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 $$\sum _{k=1}^{k=N}\|\tilde{\phi}(p_k)\|\le \sum _{k=1}^{k=N}\|\tilde{\phi}_1(p_k)\| + \sum _{k=1}^{k=N}\|\tilde{\phi}_2(p_k)\|. \tag{45}$$ 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 $$\frac{\sum _{k=1}^{k=N}\|\tilde{\phi}_1(p_k)\|}{\sum _{k=1}^{k=N}\|\tilde{\phi}(p_k)\|}\to\frac{\|\tilde{\phi}_1(p_k)\|}{\|\tilde{\phi}(p_k)\|}\to 0. \tag{46}$$ 接下来考察第二项, 注意到 $~\|\tilde{\phi}(k)\|=0,~k\le~0$, 那么, $$\sum _{k=1}^{k=N}\|\tilde{\phi}(k-i)\| \le \sum _{k=1}^{k=N}\|\tilde{\phi}(k)\|, i=1,\ldots,s. \tag{47}$$ 因此, $$\begin{split} \sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(k)\|+\cdots+\|\tilde{\phi}(k-s)\|) \le(s+1)\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(k)\|). \end{split} \tag{48}$$ 上式结合本引理的给定条件导致 $$\sum _{k=1}^{k=N}\|\tilde{\phi}_2(k)\|=o\left(\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(k)\|)\right). \tag{49}$$ 进一步, 考虑到收敛序列及其子序列的极限相同, 我们有 $$\sum _{k=1}^{k=N}\|\tilde{\phi}_2(p_k)\|=o\left(\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(p_k)\|)\right). \tag{50}$$ 考虑到 (A2) 和(A3)式, 并利用夹逼原理有 $$\frac{\sum _{k=1}^{k=N}\|\tilde{\phi}(p_k)\|}{\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(p_k)\|)}\to 0. \tag{51}$$ 上式与如下事实相矛盾(根据 Stolz 定理): $$\frac{\sum _{k=1}^{k=N}(\|\tilde{\phi}(p_k)\|)}{\sum _{k=1}^{k=N}(\alpha+\|\tilde{\phi}(p_k)\|)}\to 1. \tag{52}$$ 因此原假设不成立. 从而证得结论, 即 $\|\tilde{\phi}(k)\|<\infty$.

proof 用数学归纳法给出证明. 首先, 我们知道, 对于 $k~\le~0~$ 有 $$y(k)=y_1(k)+y_2(k)+y_3(k), u(k)=u_1(k)+u_2(k)+u_3(k). \tag{53}$$ 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 $$\phi^{\rm T}_1(k-d+1)+\phi^{\rm T}_2(k-d+1)+\phi^{\rm T}_3(k-d+1)=\phi^{\rm T}(k-d+1). \tag{61}$$ 因此, 可以得到 $$\begin{split} y_1(k+1)+y_2(k+1)+y_3(k+1)=\phi^{\rm T}(k-d+1)\hat\theta(t_k)+\omega(k+1)+e_i(k+1)=y(k+1). \end{split} \tag{62}$$ 下面考察 $u_1(k+1)$, $u_2(k+1)$ 和 $u_3(k+1)$. 在图10中, 有 $$u(k+1)=\phi^{\rm T}_{c}(k+1)\theta_c(t_{k+1})+\Delta u'(k+1). \tag{63}$$ 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 $$\phi^{\rm T}_{c1}(k+1)+\phi^{\rm T}_{c2}(k+1)+\phi^{\rm T}_{c3}(k+1)=\phi^{\rm T}_c(k+1). \tag{71}$$ 因此, $$\begin{split} u_1(k+1)+u_2(k+1)+u_3(k+1) =\phi^{\rm T}_c(k+1)\theta_c(t_{k+1})+\Delta u'(k+1) =u(k+1). \end{split} \tag{72}$$ 所以 (A10)式 对所有 $k$皆成立.

proof 利用三角不等式以及显然的事实 $2ab\le~a^2+b^2$, 有 $$\begin{split} \frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}(k)\|^2 =\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(k)+\widetilde{\phi}_2(k)\|^2 \le\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(k)\|^2+\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_2(k)\|^2. \end{split} \tag{73}$$ 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 $$\begin{split} \frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}(p_k)\|^2 =\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(p_k)+\widetilde{\phi}_2(p_k)\|^2 \le\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_1(p_k)\|^2+\frac{2}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_2(p_k)\|^2. \end{split} \tag{74}$$ 进一步考虑到收敛序列及其子序列的极限相同, 于是得到 $$\frac{1}{n}\sum_{k=1}^{n}\|\widetilde{\phi}_2(p_k)\|^2= o\left(\frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}(p_k)\|^2\right). \tag{75}$$ 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 $\frac{1}{n}\sum^{n}_{k=1}\|\widetilde{\phi}(k)\|^2<\infty.$

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

### 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

[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

• Figure 1

Self-tuning control system

• Figure 2

Deterministic VES I

• Figure 3

Deterministic VES II

• Figure 4

Subsystem 1 of deterministic VES II

• Figure 5

Subsystem 2 of deterministic VES II

• Figure 6

Subsystem 3 of deterministic VES II

• Figure 7

VES for one-step-ahead STC (self-tuning control)

• Figure 8

Stochastic self-tuning control system

• Figure 9

Stochastic VES I

• Figure 10

Stochastic VES II

• Figure 11

Subsystem 1 of stochastic VES II

• Figure 12

Subsystem 2 of stochastic VES II

• Figure 13

Subsystem 3 of stochastic VES II

• Figure 14

VES for minimum variance self-tuning control

• Figure 15

Equivalence of output signals of VES before and after decomposition

• Figure 16

Equivalence of control signals of VES before and after decomposition

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