Observer-based backstepping dynamic surface control for stochastic nonlinear strict-feedback systems

Abstract

An observer-based dynamic surface control approach is proposed for a class of stochastic nonlinear strict-feedback systems in order to solve the problem of ‘explosion of complexity’ in the backstepping design; that is, the dynamic surface control approach is extended to the stochastic setting. The circle criterion is applied to designing a nonlinear observer, and so no linear growth condition is imposed on nonlinear functions depending on system states. It is proved that the closed-loop system is semi-globally uniformly ultimately bounded in fourth moment, and the ultimate boundedness can be tuned arbitrarily small. Two examples are given to demonstrate the effectiveness of the control scheme proposed in this paper.

Appendices

Appendix 1: The proof of Theorem 1

Proof: Define μ = x − v and $\Uplambda(x,\mu)=J(x)-J(x-\mu)$. From this together with $\Upphi(x)=HJ(x)$ by Assumption 2, it follows that

$$ \Upphi(x)-\Upphi(v)=H(J(x)-J(x-\mu))=H\Uplambda(x,\mu) $$

(44)

$$ \mu^{\rm T}=x^{\rm T}-(\hat{x}-KC\tilde{x})^{\rm T}=\tilde{x}^{\rm T}(I+KC)^{\rm T} $$

(45)

and then, by the Mean Value Theorem, we have $\Uplambda(x,\mu)=\int_0^1\frac{\partial J}{\partial s}|_{s=x-\tau\mu}\mu d\tau$ which, together with Assumption 2, implies that

$$ \mu^{\rm T}\Uplambda(x,\mu)=\frac{1}{2}\mu^{\rm T}\left(\int\limits_0^1\left[\frac{\partial J}{\partial s}+\left(\frac{\partial J}{\partial s}\right)^{\rm T}\right]_{s=x-\tau\mu}{\rm d}\tau\right)\mu\geq0 $$

(46)

and then, recalling (12) and (44), we have

$$ \tilde{x}^{\rm T}P(\Upphi(x)-\Upphi(v))=\tilde{x}^{\rm T}PH\Uplambda(x,\mu)=-\mu^{\rm T}\Uplambda(x,\mu)\leq0. $$

(47)

Along the trajectory of (14), one has

$$ \begin{aligned} {{\mathcal{L}}}V_0&=(\tilde{x}^{\rm T}P\tilde{x})\tilde{x}^{\rm T}[(A+LC)^{\rm T}P+P(A+LC)]\tilde{x}+2(\tilde{x}^{\rm T}P\tilde{x})\tilde{x}^{\rm T}P(\Upphi(x)-\Upphi(v))\\ &+2(\tilde{x}^{\rm T}P\tilde{x})\tilde{x}^{\rm T}P F+2{\rm Tr}\{G^{\rm T}(2P\tilde{x}\tilde{x}^{\rm T}P+\tilde{x}^{\rm T}P\tilde{x}P)G\}. \end{aligned} $$

(48)

where for convenience, F(y) and G(y) are denoted by F and G, respectively. Substituting (11) and (47) into (48) yields

$$ {{\mathcal{L}}}V_0\leq-(\tilde{x}^{\rm T}P\tilde{x})\tilde{x}^{\rm T}Q\tilde{x}+2(\tilde{x}^{\rm T}P\tilde{x})\tilde{x}^{\rm T}P F+2{\rm Tr}\{G^{\rm T}(2P\tilde{x}\tilde{x}^{\rm T}P+\tilde{x}^{\rm T}P\tilde{x}P)G\}. $$

(49)

Using Young’s inequality (see 4), together with (6) and (7), we have

$$ 2(\tilde{x}^{\rm T}P\tilde{x})\tilde{x}^{\rm T}P F\leq 2|P|^2|\tilde{x}|^3|F|\leq\frac{3}{2}\epsilon_1^{4/3}|P|^{8/3}|\tilde{x}|^4+\frac{1}{2\epsilon_1^4}y^4|\bar{F}(y)|^4 $$

(50)

$$ 2{\rm Tr}\{G^{\rm T}(2P\tilde{x}\tilde{x}^{\rm T}P+\tilde{x}^{\rm T}P\tilde{x}P)G\}\leq\frac{3n\sqrt{n}}{\epsilon_2^2}y^4|\bar{G}(y)|^4+3n\sqrt{n}\epsilon_2^2|P|^4|\tilde{x}|^4 $$

(51)

where $\epsilon_1$ and $\epsilon_2$ are the positive design constants. The detailed derivation of the inequality (51) is similar to [28, Eq. (A.7)]. Substituting (50) and (51) back into (49) yields (16).

Appendix 2: The proof of Theorem 2

Consider the following Lyapunov function

$$ V=\frac{1}{2}(\tilde{x}^{\rm T}P\tilde{x})^2+\frac{1}{4}\sum_{i=1}^nz_i^4+\frac{1}{4}\sum_{i=1}^{n-1}\varrho_{i+1}^4 $$

(52)

and then along the trajectory of (35), noting Theorem 1, we have

$$ \begin{aligned} {{\mathcal{L}}}V\leq&\left(-\lambda_{\rm min}(P)\lambda_{\rm min}(Q)+\frac{3}{2}\epsilon_1^{4/3}|P|^{8/3} +3n\sqrt{n}\epsilon_2^2|P|^4\right)|\tilde{x}|^4\\ &\quad+y^4\left(\frac{1}{2\epsilon_1^4}|\bar{F}(y)|^4 +\frac{3n\sqrt{n}}{\epsilon_2^2}|\bar{G}(y)|^4\right)-y^4\Upxi(y)+\frac{3}{2}y^2g_1^{\rm T}(y)g_1(y)\\ &\quad+a_{1,2}y^3\tilde{x}_2-\sum_{i=1}^{n}c_iz_i^4+\sum_{i=1}^{n-1}a_{i,i+1}z_i^3z_{i+1}+\sum_{i=1}^{n-1}a_{i,i+1}z_{i}^3\varrho_{i+1}\\ &\quad-\sum_{i=1}^{n-1}\frac{1}{\iota_{i+1}}\varrho_{i+1}^4+\sum_{i=1}^{n-1}\varrho_{i+1}^3B_{i+1}+\frac{3}{2}\sum_{i=1}^{n-1}\varrho_{i+1}^2{\rm Tr}\{C_{i+1}^{\rm T}C_{i+1}\}. \end{aligned} $$

(53)

Since for any M ₀ > 0, the sets $\Uppi_i:=\{\frac{1}{2}(\tilde{x}^{\rm T}P\tilde{x})^2+\frac{1}{4}\sum_{j=1}^iz_j^4+\frac{1}{4}\sum_{j=1}^{i-1}\varrho_{j+1}^4 \leq M_0 \}, i=1,\ldots, n$ are compact. Therefore, B _i+1 and Tr{C ^T _i+1 C _i+1} have their maximum on $\Uppi_i$, denoted by M _i+1 and N _i+1, respectively.

Using Young’s inequality (see Lemma 2), we have

$$ a_{1,2}y^3\tilde{x}_2\leq \frac{3}{4}(\epsilon_3a_{1,2})^{4/3}y^4+\frac{1}{4\epsilon_3^4}|\tilde{x}|^4 $$

(54)

$$ \sum_{i=1}^{n-1}a_{i,j}z_i^3z_{i+1}\leq \frac{3}{4}\sum_{i=1}^{n-1}(\delta_ia_{i,j})^{4/3}z_i^4+\frac{1}{4}\sum_{i=2}^n\frac{1}{\delta_{i-1}^4}z_i^4 $$

(55)

$$ \begin{aligned} &\sum_{i=1}^{n-1}a_{i,j}z_{i}^3\varrho_{i+1}\leq\frac{3}{4}\sum_{i=1}^{n-1}(\varsigma_ia_{i,j})^{4/3}z_i^4 +\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\varsigma_{i}^4}\varrho_{i+1}^4\\ &\sum_{i=1}^{n-1}\varrho_{i+1}^3B_{i+1}\leq\frac{3}{4}\sum_{i=1}^{n-1}(\nu_iM_{i+1})^{4/3}\varrho_{i+1}^4 +\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\nu_{i}^4} \end{aligned} $$

(56)

$$ \frac{3}{2}\sum_{i=1}^{n-1}\varrho_{i+1}^2{\rm Tr}\{C_{i+1}^{\rm T}C_{i+1}\}\leq\frac{3}{4}\sum_{i=1}^{n-1}(\xi_iN_{i+1})^{4/3}\varrho_{i+1}^4 +\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\xi_{i}^4}. $$

(57)

Substituting (54–57) and (21) into (53), we have

$$ \begin{aligned} {{\mathcal{L}}}V&\leq-\left(\lambda_{\rm min}(P)\lambda_{\rm min}(Q)-\frac{3}{2}\epsilon_1^{4/3}|P|^{8/3} -3n\sqrt{n}\epsilon_2^2|P|^4-\frac{1}{4\epsilon_3^4}\right) |\tilde{x}|^4\\ &\quad-\left(c_1-\frac{3}{4}(a_{1,2}\epsilon_3)^{4/3}-\frac{3}{4}(a_{1,2}\delta_1)^{4/3}-\frac{3}{4}(a_{1,2}\varsigma_1)^{4/3}\right)y^4\\ &\quad-\sum_{i=2}^{n-1}\left(c_i-\frac{3}{4}(a_{i,i+1}\delta_i)^{4/3}-\frac{1}{4\delta_{i-1}^{4}}-\frac{3}{4}(a_{i,i+1}\varsigma_i)^{4/3}\right)z_i^4 -\left(c_n-\frac{1}{4\delta_{n-1}^{4}}\right)z_n^4\\ &\quad-\sum_{i=1}^{n-1}\left(\frac{1}{\iota_{i+1}}-\frac{3}{4}\varsigma_i^{4/3} -\frac{3}{4}(\nu_iM_{i+1})^{4/3}-\frac{3}{4}(\xi_iN_{i+1})^{4/3}\right)\varrho_{i+1}^4\\ &\quad+\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\nu_{i}^4} +\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\xi_{i}^4}. \end{aligned} $$

(58)

Choose the design parameter $\epsilon_1, \epsilon_2,\epsilon_3, c_i, \delta_i, \varsigma_i, \nu_i, \xi_i$ such that

$$ \lambda_{\rm min}(P)\lambda_{\rm min}(Q)-\frac{3}{2}\epsilon_1^{4/3}|P|^{8/3} -3n\sqrt{n}\epsilon_2^2|P|^4-\frac{1}{4\epsilon_3^4}=\lambda^0>0 $$

(59)

$$ c_1-\frac{3}{4}(a_{1,2}\epsilon_3)^{4/3}-\frac{3}{4}(a_{1,2}\delta_1)^{4/3}-\frac{3}{4}(a_{1,2}\varsigma_1)^{4/3}=c_1^0>0 $$

(60)

$$ c_i-\frac{3}{4}(a_{i,i+1}\delta_i)^{4/3}-\frac{1}{4\delta_{i-1}^{4}}-\frac{3}{4}(a_{i,i+1}\varsigma_i)^{4/3}=c_i^0>0, \,i=2,\ldots,n-1 $$

(61)

$$ c_n-\frac{1}{4\delta_{n-1}^{4}}=c_n^0>0 $$

(62)

$$ \frac{1}{\iota_{i+1}}-\frac{3}{4}\varsigma_i^{4/3} -\frac{3}{4}(\nu_iM_{i+1})^{4/3}-\frac{3}{4}(\xi_iN_{i+1})^{4/3}=\iota_{i+1}^0>0, \,i=1,\ldots,n-1. $$

(63)

Substituting (59–63) into (58) yields

$$ \begin{aligned} {{\mathcal{L}}}V\leq&-\lambda^0|\tilde{x}|^4-\sum_{i=1}^{n}c_i^0z_i^4-\sum_{i=1}^{n-1}\iota_{i+1}^0\varrho_{i+1}^4+\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\nu_{i}^4} +\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\xi_{i}^4}\\ \leq&-\ell V+\hbar \end{aligned} $$

(64)

where

$$ \begin{aligned} \ell=&\hbox{min}\left\{2/\lambda_{\max}^2(P), 4c_1^0,\ldots,4c_n^0, 4\iota_{2}^0,\ldots, 4\iota_n^0 \right\},\\ \hbar=&\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\nu_{i}^4} +\frac{1}{4}\sum_{i=1}^{n-1}\frac{1}{\xi_{i}^4}. \end{aligned} $$

Based on (64), we easily obtain

$$ \frac{{\rm d}(EV)}{{\rm d}t}=E({{\mathcal{L}}}V)\leq-\ell E V+\hbar. $$

(65)

Let $\ell>\hbar/M$, then d(EV)/dt ≤ 0 on EV = M. Thus, V ≤ M is an invariant set, that is, if EV(0) ≤ M, then EV(t) ≤ M for all t > 0. Thus, (65) holds for all V(0) < M and all t > 0.

Based on Lemma 1, inequality (64) further implies that

$$ 0\leq EV(t)\leq V(0)e^{-\ell t}+\frac{\hbar}{\ell}, \forall t\geq0. $$

(66)

The above inequality means that EV(t) is eventually bounded by $\frac{\hbar}{\ell}$. Thus, recalling (52), all error signals of the closed-loop system, that is, $\tilde{x}, z_i, \varrho_{i+1}$ are SGUUB in the sense of fourth moment. Moreover, by adjusting the design parameters, we can increase the value of ℓ and reducing the value of $\hbar$. In other words, the boundedness of closed-loop error signals above can be made arbitrarily small. This completes the whole proof.