#  SCIENCE CHINA Information Sciences, Volume 64 , Issue 9 : 192201(2021) https://doi.org/10.1007/s11432-020-2979-0

## PID control of uncertain nonlinear stochastic systems with state observer • AcceptedMay 29, 2020
• PublishedAug 23, 2021
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### Abstract ### Acknowledgment

This work was supported by the Youth Scholars Fund of Beijing Technology and Business University (Grant No. PXM2018_014213_000033) and National Natural Science Foundation of China (Grant No. 61973329). The authors would like to thank Professor Lei GUO from Academy of Mathematics and Systems Science, Chinese Academy of Sciences, for valuable discussion on PID control of nonlinear stochastic systems.

### Supplement

Appendix

Definition A1 (17) . Let $(\Omega,~\mathcal{F},P)$ be a probability space. A filtration is a family of $\sigma$-algebra $\{\mathcal{F}_t\}_{t\geq~0}$ satisfying $\mathcal{F}_{t}\subset~\mathcal{F}_{s}\subset~\mathcal{F}$ for all $0\leq~t<s<\infty$. The filtration is considered to be right continuous if $\mathcal{F}_{t}=\bigcap_{s>t}\mathcal{F}_{s}$ for all $t\geq~0$. When the probability space is complete, the filtration is considered to satisfy the usual conditions if it is right continuous and $\mathcal{F}_{0}$ contains all $P$-null sets.

We consider the following stochastic system: \begin{equation*} \text{d}x(t)=f(x(t),t)\text{d}t+\sigma(x(t),t)\text{d}B(t), \eqno \text{(A1)}\end{equation*} where $x~\in~\mathbb{R}^{n}$ is the state of the system, $f~\in~\mathbb{R}^{n}$, $\sigma~\in~\mathbb{R}^{n}$, and $B(t)$ is (standard one-dimensional) Brownian motion. The $f$ term is referred to as a drift or vector field, and the noise term $\sigma~\text{d}B({t})$ is an uncertainty model. The uncertainty of this model could be caused by external random influences or by fluctuating coefficients and parameters in a mathematical model. The $\sigma$ function is referred to as a diffusion coefficient.

Itô's formula 17 . We denote $C^{2,1}(\mathbb{R}^n\times\mathbb{R}^{+},\mathbb{R}^{+})$ as the space of all nonnegative functions $V(x,t)$ defined on $\mathbb{R}^n\times\mathbb{R}^{+}$ that are continuously twice differentiable in $x$ and once in $t$. We define the differential operator ${L}$ associated with (A1) as follows: $${L}=\frac{\partial}{\partial t}+\sum\limits_{i = 1}^n {f_{i}(x,t)\frac{\partial}{\partial x_{i}}}+\frac{1}{2} \sum\limits_{i,j = 1}^n [\sigma(x,t)\sigma^{\text{T}}(x,t)]_{ij} \frac{\partial^{2}}{\partial x_{i}\partial x_{j}}.$$ If ${L}$ acts on function $V~\in~C^{2,1}(\mathbb{R}^n\times\mathbb{R}^{+},\mathbb{R}^{+})$, then $${L}V(x,t)=\frac{\partial V}{\partial t}(x,t)+f^{\text{T}}\triangledown V(x,t)+\frac{1}{2}\text{Tr}[\sigma\sigma ^{\text{T}}{H}(V)](x,t).$$ By Itô's formula, we can obtain $$\text{d}V(x(t),t)={L}V(x,t)\text{d}t+(\triangledown V(x(t),t))^{\text{T}}\sigma(x(t),t)\text{d}B(t),$$ where $\triangledown~V$ is the gradient of $V$, ${H}(V)=V_{x_{i}x_{j}}$ is the $n~\times~n$ Hessian matrix of $V$, and $\text{Tr}(A)$ denotes the trace of a matrix $A$.

Lemma A1 (Barbalat) . Assume that function $f:\mathbb{R}^{+}\rightarrow~\mathbb{R}$ is uniformly continuous and $\lim_{t~\to~\infty}~\int_0^t~f(\tau)~\text{d}\tau$ exists and is finite. Then $$\lim_{t \to \infty}f(t)=0.$$

See Lemma A.6 in 20 for a detailed discussion.

Theorem A1. Let $p\geq~2$ and $x_{0}\in~L^{p}(\Omega;\mathbb{R}^{d})$. Assume there exists a constant $\alpha>0$ such that, for all $(x,t)\in~\mathbb{R}^{d}\times~[t_{0},T]$: $$x^{\text{T}}f(x,t)+\frac{p-1}{2}|\sigma(x,t)|^{2}\leq \alpha(1+|x|^{2}).$$ Then, for the solution of (A1) on $t\in~[t_{0},~T]$, we obtain the following: $$\mathbb{E}|x(t)|^{p}\leq 2^{\frac{p-2}{2}}(1+\mathbb{E}|x_{0}|^{p}){\rm e}^{p\alpha (t-t_{0})}.$$

See Theorem 4.1 in 16 for a detailed discussion.

Lemma B1. Let the following linear growth condition hold for all $(x,t)\in\mathbb{R}^{d}\times[t_{0},\infty)$: \begin{equation*} \|f(x,t)\|\vee \|\sigma(x,t)\| \leq K\|x\|, \eqno \text{(B1)}\end{equation*} and let $x(t)$ be a solution to SDE (A1) on $[t_0,\infty)$. Let $h(t)=E\|x(t)\|^2$, and assume that $\sup_{t\geq~t_0}~h(t)<\infty$. Then, $h(t)$ is a uniformly continuous function of $t$ in $[t_0,\infty)$.

Proof. Let $C=\sup_{t\geq~t_0}~h(t)$, and assume that $t_0\le~t_1<t_2$. Then, we obtain the inequality $|h(t_2)-h(t_1)|\le~\mathbb{E}|\,\|x(t_2)\|^2-\|x(t_1)\|^2|$.

According to the Schwarz inequality, we obtain $$\mathbb{E}\big|\|x(t_2)\|^2-\|x(t_1)\|^2\big|\le \sqrt{\mathbb{E}(\|x(t_2)\|+\|x(t_1)\|)^2}\sqrt{\mathbb{E}(\|x(t_2)\|-\|x(t_1)\|)^2}.$$

Therefore, it is easy to see that \begin{equation*} \mathbb{E}(\|x(t_2)\|+\|x(t_1)\|)^2\le 2(\mathbb{E}\|x(t_2)\|^2+\mathbb{E}\|x(t_1)\|^2)\le 4\sup_{t\geq t_0} h(t)=4C, \eqno \text{(B2)}\end{equation*} where the RHS of (B2) is a constant that is independent of $t_1,t_2$.

In addition, we can obtain the following: \begin{align*}\mathbb{E}(\|x(t_2)\|-\|x(t_1)\|)^2&\le \mathbb{E}(\|x(t_2)-x(t_1)\|^2)=\mathbb{E}\left\|\int_{t_1}^{t_2} f(x(t),t)\text{d}t+\int_{t_1}^{t_2}\sigma(x(t),t)\text{d}B(t)\right\|^2 \\ &\le 2\left(\mathbb{E}\left\|\int_{t_1}^{t_2} f(x(t),t)\text{d}t\right\|^2+\mathbb{E}\left\|\int_{t_1}^{t_2}\sigma(x(t),t)\text{d}B(t)\right\|^2\right). \end{align*} From (B1), it is easy to obtain the following: \begin{align*}\mathbb{E}\left\|\int_{t_1}^{t_2} f(x(t),t)\text{d}t\right\|^2&\le \mathbb{E}\left[\int_{t_1}^{t_2} K\|x(t)\|\text{d}t\right]^2\le K^2(t_2-t_1)\mathbb{E}\left[\int_{t_1}^{t_2} \|x(t)\|^2\text{d}t\right] \\ &=K^2(t_2-t_1)\int_{t_1}^{t_2} h(t)\text{d}t\le K^2C(t_2-t_1)^2. \end{align*} From Itô's isometry, we obtain \begin{align*}\mathbb{E}\left\|\int_{t_1}^{t_2}\sigma(x(t),t)\text{d}B(t)\right\|^2 =\mathbb{E}\left[\int_{t_1}^{t_2}\|\sigma(x(t),t)\|^2\text{d}t\right]\le K^2\mathbb{E}\left[\int_{t_1}^{t_2}\|x(t)\|^2\text{d}t\right]=K^2\int_{t_1}^{t_2} h(t)\text{d}t\le K^2C(t_2-t_1). \end{align*} Therefore, we conclude that $$|h(t_2)-h(t_1)|\le \sqrt{4C}\sqrt{2K^2C[(t_2-t_1)^2+(t_2-t_1)]}<4KC\sqrt{(t_2-t_1)^2+(t_2-t_1)},$$ which implies $h(t)$ is uniformly continuous.

### References

 Åström K J, Hägglund T. PID Controllers: Theory, Design and Tuning. Instrument Society of America, 1995. Google Scholar

 Åström K J, Hägglund T. Advanced PID Control. Research Triangle Park: International Society of Automation, 2006. Google Scholar

 Ming-Tzu Ho , Chia-Yi Lin . PID controller design for robust performance. IEEE Trans Autom Control, 2003, 48: 1404-1409 CrossRef Google Scholar

 S?ylemez M T, Munro N, Baki H. Fast calculation of stabilizing PID controllers. Automatica, 2003, 39: 121-126 CrossRef Google Scholar

 Silva G J, Datta A, Bhattacharyya S P. PID Controllers for Time-Delay Systems. Boston: Birkhäuser, 2004. Google Scholar

 PID tuning using extremum seeking: online, model-free performance optimization. IEEE Control Syst, 2006, 26: 70-79 CrossRef Google Scholar

 Keel L H, Bhattacharyya S P. Controller Synthesis Free of Analytical Models: Three Term Controllers. IEEE Trans Autom Control, 2008, 53: 1353-1369 CrossRef Google Scholar

 Qiao D, Mu N, Liao X. Improved evolutionary algorithm and its application in PID controller optimization. Sci China Inf Sci, 2020, 63: 199205 CrossRef Google Scholar

 Guo L. Some perspectives on the development of control theory (in Chinese). J Syst Sci Math Sci, 2011, 31: 1014--1018. Google Scholar

 Guo L. Feedback and uncertainty: Some basic problems and results. Annu Rev Control, 2020, 49: 27-36 CrossRef Google Scholar

 Zhao C, Guo L. On the Capability of PID Control for Nonlinear Uncertain Systems. IFAC-PapersOnLine, 2017, 50: 1521-1526 CrossRef Google Scholar

 Zhao C, Guo L. PID controller design for second order nonlinear uncertain systems. Sci China Inf Sci, 2017, 60: 022201 CrossRef Google Scholar

 Cong X R, Guo L. PID control for a class of nonlinear uncertain stochastic systems. In: Proceedings of the 56th Annual Conference on Decision and Control, Melbourne, 2017. Google Scholar

 Koralov L B, Sinai Y G. Theory of Probability and Random Processes. Berlin: Springer, 2007. Google Scholar

 Khasminskii R. Stochastic Stability of Differential Equations. Berlin: Springer, 2012. Google Scholar

 Mao X R. Stochastic Differential Equations and Applications. Chichester: Horwood Publishing, 2008. Google Scholar

 Duan J Q. An Introduction to Stochastic Dynamics. Beijing: Science Press, 2015. Google Scholar

 Guo Y C. Stochastic Processes and Control Theory. Beijing: Tsinghua University Press, 2017. Google Scholar

 Zhang T, Deng F, Zhang W. Study on stability in probability of general discrete-time stochastic systems. Sci China Inf Sci, 2020, 63: 159205 CrossRef Google Scholar

 Reissig R, Sansone G, Conti R. Non-linear Differential Equations of Higher Order. Berlin: Springer, 1974. Google Scholar

 Zhong S, Huang Y, Guo L. A parameter formula connecting PID and ADRC. Sci China Inf Sci, 2020, 63: 192203 CrossRef Google Scholar