Decentralized backstepping output-feedback control for stochastic interconnected systems with time-varying delays using neural networks

Abstract

This paper addresses the decentralized adaptive output-feedback control problem for a class of interconnected stochastic strict-feedback uncertain systems described by It$\hat{\hbox{o}}$ differential equation using neural networks. Compared with the existing literature, this paper removes the commonly used assumption that the interconnections are bounded by known functions multiplying unknown parameters, and all unknown interconnections are lumped in a suitable function which is compensated by only a neural network in each subsystem. So, the controller is simpler even than that for the strict-feedback systems described by the ordinary differential equation. Moreover, the circle criterion is applied to designing nonlinear observers for the estimates of system states. A simulation example is used to illustrate the effectiveness of control scheme proposed in this paper.

Appendices

Appendix A: The proof of Theorem 1

Proof: Define μ _i  = x _i  − v _i and $\Uplambda_i(x_i,\mu_i)=J_i(x_i)-J_i(x_i-\mu_i)$. From this together with $\Upphi_i(x_i)=H_iJ_i(x_i)$ by Assumption 3, it follows that

$$ \Upphi_i(x_i)-\Upphi_i(v_i)=H_i(J_i(x_i)-J_i(x_i-\mu_i))=H_i\Uplambda_i(x_i,\mu_i) $$

(47)

$$ \mu_i^{\rm T}=x_i^{\rm T}-(\hat{x}_i-K_iC_i\tilde{x}_i)^{\rm T}=\tilde{x}_i^{\rm T}(I+K_iC_i)^{\rm T} $$

(48)

and then, by the Mean Value Theorem, we have $\Uplambda_i(x_i,\mu_i)=\int\nolimits_0^1\frac{\partial J_i(s)}{\partial s}|_{s=x_i-\tau_i\mu_i}\mu_i {\rm d}\tau_i$ which, together with Assumption 3, implies that

$$ \mu_i^{\rm T}\Uplambda_i(x_i,\mu_i)=\frac{1}{2}\mu_i^{\rm T}\left(\int_0^1\left[\frac{\partial J_i(s)}{\partial s}+\left(\frac{\partial J_i(s)}{\partial s}\right)^{\rm T}\right]_{s=x_i-\tau\mu_i}{\rm d}\tau\right)\mu_i\geq0. $$

(49)

Then, we have

$$ \tilde{x}_i^{\rm T}P_i(\Upphi_i(x_i)-\Upphi_i(v_i))=\tilde{x}_i^{\rm T}P_iH_i\Uplambda_i(x_i,\mu_i)=-\mu_i^{\rm T}\Uplambda_i(x_i,\mu_i)\leq 0. $$

(50)

Along the trajectory of (25), one has

$$ \begin{aligned} {{\mathcal{L}}}V_{i,0}= \,& b_i(\tilde{x}_i^{\rm T}P_i\tilde{x}_i)\tilde{x}_i^{\rm T}[(A_i+L_iC_i)^{\rm T}P_i+P_i(A_i+L_iC_i)]\tilde{x}_i\\ &\quad +2b_i(\tilde{x}_i^{\rm T}P_i\tilde{x}_i)\tilde{x}_i^{\rm T}P_i(\Upphi_i(x_i)-\Upphi_i(v_i))+2b_i(\tilde{x}_i^{\rm T}P_i\tilde{x}_i)\tilde{x}_i^{\rm T}P_i F_i\\ &\quad +2b_i{\rm Tr}\{G_i^{\rm T}(2P_i\tilde{x}_i\tilde{x}_i^{\rm T}P_i+\tilde{x}_i^{\rm T}P_i\tilde{x}_iP_i)G_i\}. \end{aligned} $$

(51)

Substituting (22) and (50) into (51) yields

$$ \begin{aligned} {{\mathcal{L}}}V_{i,0}&\leq-b_i(\tilde{x}_i^{\rm T}P_i\tilde{x}_i)\tilde{x}_i^{\rm T}Q_i\tilde{x}_i+2b_i(\tilde{x}_i^{\rm T}P_i\tilde{x}_i)\tilde{x}_i^{\rm T}P_i F_i\\ &\quad +2b_i{\rm Tr}\{G_i^{\rm T}(2P_i\tilde{x}_i\tilde{x}_i^{\rm T}P_i+\tilde{x}_i^{\rm T}P_i\tilde{x}P_i)G_i\}. \end{aligned} $$

(52)

Using Young’s inequality(see Lemma 2), together with (17)–(18), we have

$$ \begin{aligned} &2b_i(\tilde{x}_i^{\rm T}P_i\tilde{x}_i)\tilde{x}_i^{\rm T}P_i F_i\\ &\quad \leq 2b_i|P_i|^2|\tilde{x}_i|^3|F_i|\\ &\quad \leq 2b_i\sum_{j=1}^{n_i}|P_i|^2|\tilde{x}_i|^3|f_{i,j}|\\ &\quad \leq\sum_{j=1}^{n_i}\left[ \frac{3b_i}{2}\epsilon_{i,1}^{4/3}|P_i|^{8/3}|\tilde{x}_i|^4+\frac{b_i} {2\epsilon_{i,1}^4}|f_{i,j}|^4\right]\\ &\quad \leq \sum_{j=1}^{n_i}\vphantom{\sum_{k=1}^Ny}\left[\frac{3b_i} {2}\epsilon_{i,1}^{4/3}|P_i|^{8/3}|\tilde{x}_i|^4\right.\\ &\quad \left.+\frac{b_i(2N)^3} {2\epsilon_{i,1}^4}\left(\sum_{k=1}^Ny_k^4(t-{\rm d}(t))\bar{\varphi}^4_{i,j,k,d}(y_k(t-{\rm d}(t))) +\sum_{k=1}^Ny_k^4\bar{\varphi}^4_{i,j,k}(y_k)\right)\right] \end{aligned} $$

(53)

and

$$ \begin{aligned} 2b_i{\rm Tr} &\{G_i^{\rm T}(2P_i\tilde{x}_i\tilde{x}_i^{\rm T}P_i+ \tilde{x}_i^{\rm T}P_i\tilde{x}_iP_i)G_i\}\\ &\quad \leq6b_in_i\sqrt{n_i}|P_i|^2|\tilde{x}_i|^2|G_i|^2\\ &\quad \leq\sum_{j=1}^{n_i}6b_in_i^2\sqrt{n_i}|P_i|^2|\tilde{x}_i|^2|g_{i,j}|^2\\ &\quad \leq\sum_{j=1}^{n_i}\left[\frac{3b_in_i^2\sqrt{n_i}}{\epsilon_{i,2}^2}|g_{i,j}|^4+3b_in_i^2\sqrt{n_i}\epsilon_{i,2}^2|P_i|^4|\tilde{x}_i|^4\right]\\ &\quad \leq\sum_{j=1}^{n_i}\left[\frac{3b_in_i^2\sqrt{n_i}} {\epsilon_{i,2}^2}(2N)^3\left( \sum_{k=1}^Ny_k^4(t-{\rm d}(t))\bar{\psi}^4_{i,j,k,d}(y_k(t-{\rm d}(t)))+ \sum_{k=1}^Ny_k^4\bar{\psi}^4_{i,j,k}(y_k)\right)\right.\\ &\quad \left.+3b_in_i^2\sqrt{n_i}\epsilon_{i,2}^2|P_i|^4|\tilde{x}_i|^4\vphantom{\sum_{k=1}^Ny}\right] \end{aligned} $$

(54)

where $\epsilon_{i,1}$ and $\epsilon_{i,2}$ are the positive design constants. The detailed derivation of the inequality (54) refers to the eq. (A.7) in [29]. Substituting (53)–(54) back into (52) yields (27).

Appendix B: The proof of Theorem 2

Consider the following Lyapunov function for the system (35)

$$ V_{i,1}=\frac{1}{4}y_i^4+\frac{1} {4}\sum_{j=2}^{n_i}z_{i,j}^4+\frac{1}{2}\tilde{W}_i^{\rm T}\Upgamma_i^{-1}\tilde{W}_i+\frac{1} {2}\gamma_i^{-1}\tilde{\theta}_i^{2} $$

(55)

where $\tilde{W}_i=W_i-\hat{W}_i$ and $\tilde{\theta}_i=\theta_i-\hat{\theta}_i$ denote the estimates of W _i and θ _i , respectively. Along the solutions of (34) and (35), we have

$$ \begin{aligned} {{\mathcal{L}}}V_{i,1} = & -c_{i,1}y_i^4-y_i^4\left(\hat{W}_i^{\rm T}S_i(y_i) +\psi_i(y_i)\hat{\theta}_i\right) +\underbrace{y_i^3(a_{i,1,2}z_2 +a_{i,1,2} \tilde{x}_{i,2}+f_{i,1}(t,y,y(t-{\rm d}(t))))}\limits_{\rm I}\\ & +\underbrace{\frac{3}{2}y_i^2g_{i,1} (t,y,y(t-{\rm d}(t)))g_{i,1}^{\rm T}(t,y,y(t-{\rm d}(t)))} \limits_{\rm II}\\ & +\sum_{j=2}^{n_i} \left[-c_{i,j}z_{i,j}^4-\Upxi_{i,j}z_{i,j}^4-\frac{1}{4} \lambda_{i,j} \left(\frac{\partial^2\alpha_{i,j-1}} {\partial y_i^2}\right)^{2}z_{i,j}^6\right]\\ & -\sum_{j=2}^{n_i-1}\frac{3}{4}(\delta_{i,j}a_{i,,j,j+1})^{4/3}z_{i,j}^4 +\underbrace{\sum_{j=2}^{n_i-1}a_{i,j,j+1}z_{i,j+1}z_{i,j}^3} \limits_{\rm III}\\ &\underbrace{-\sum_{j=2}^{n_i}z_{i,j}^3 \frac{\partial\alpha_{i,j-1}}{\partial y_i}(a_{i,1,2} \tilde{x}_{i,2}+f_{i,1}(t,y,y(t-{\rm d}(t))))}\limits_{\rm IV}\\ &\underbrace{-\frac{1}{2}\sum_{j=2}^{n_i}z_{i,j}^3 \left( \frac{\partial^2\alpha_{i,j-1}}{\partial y_i^2}\right) g_{i,1}(t,y,y(t-{\rm d}(t)))g_{i,1}^{\rm T}(t,y,y(t-{\rm d}(t)))} \limits_{\rm V}\\ & +\underbrace{\frac{3}{2} \sum_{j=2}^{n_i}z_{i,j}^2\left( \frac{\partial\alpha_{i,j-1}} {\partial y_i}\right)^2g_{i,1}(t,y,y(t-{\rm d}(t)))g_{i,1}^{\rm T} (t,y,y(t-{\rm d}(t)))}\limits_{\rm VI}\\ & -\tilde{W}_i^{\rm T}S_i(y_i)y_i^4-\tilde{\theta}_i\psi_i(y_i)y_i^4. \end{aligned} $$

(56)

Using Young’s inequality (see Lemma 2) and noting (11)–(12), all underlined terms in (56) satisfy the following inequalities

$$ \begin{aligned} {\rm I}:\,&y_i^3(a_{i,1,2}z_{i,2}+a_{i,1,2}\tilde{x}_{i,2}+f_{i,1})\leq \frac{3}{4}(\delta_{i,1}a_{i,1,2})^{4/3}y_i^4+\frac{3}{4}(\epsilon_{i,3}a_{i,1,2})^{4/3}y_i^4 \\ &\quad +\frac{3}{4}(\epsilon_{i,4}a_{i,1,2})^{4/3}y_i^4+ \frac{1}{4\delta_{i,1}^4}z_{i,2}^4+\frac{1}{4\epsilon_{i,3}^4}|\tilde{x}_i|^4\\ &\quad +\frac{1}{4\epsilon_{i,4}^4}(2N)^3\left[\sum_{k=1}^Ny_k^4(t-{\rm d}(t))\bar{\varphi}^4_{i,1,k,d}(y_k(t-{\rm d}(t))) +\sum_{k=1}^Ny_k^4\bar{\varphi}^4_{i,1,k}(y_k)\right] \end{aligned} $$

(57)

$$ \begin{aligned} {\rm II}:&\frac{3}{2}y_i^2g_{i,1}(t,y,y(t-{\rm d}(t)))g_{i,1}^{\rm T}(t,y,y(t-{\rm d}(t)))\leq \frac{3\epsilon_{i,5}}{4}y_i^4\\ &\quad + \frac{3}{4\epsilon_{i,5}}(2N)^3\left[\sum_{k=1}^Ny_k^4(t-{\rm d}(t))\bar{\psi}^4_{i,1,k,d} (y_k(t-{\rm d}(t))) +\sum_{k=1}^Ny_k^4\bar{\psi}^4_{i,1,k}(y_k)\right] \end{aligned} $$

(58)

$$ \begin{aligned} {\rm III}:&\sum_{j=2}^{n_i-1}a_{i,j,j+1}z_{i,j+1}z_{i,j}^3\leq \frac{3}{4}\sum_{j=2}^{n_i-1}(\delta_{i,j}a_{i,j,j+1})^{4/3}z_{i,j}^4 + \frac{1}{4}\sum_{j=3}^{n_i}\frac{1}{\delta_{i,j-1}^4}z_{i,j}^4 \end{aligned} $$

(59)

$$ \begin{aligned} {\rm IV}:&-\sum_{j=2}^{n_i}z_{i,j}^3 \frac{\partial\alpha_{i,j-1}}{\partial y_i}(a_{i,1,2}\tilde{x}_{i,2}+f_{i,1}(t,y,y(t-{\rm d}(t))))\\ &\quad \leq \frac{3}{2}\sum_{j=2}^{n_i}(\eta_{i,j}a_{i,1,2})^{4/3}\left( \frac{\partial\alpha_{i,j-1}}{\partial y_i}\right)^{4/3}z_{i,j}^4+\frac{1}{4}\sum_{j=2}^{n_i}\frac{1}{\eta_{i,j}^4}|\tilde{x}_i|^4\\ &\quad +\frac{1}{4}\sum_{j=2}^{n_i} \frac{1}{\eta_{i,j}^4}(2N)^3\left[\sum_{k=1}^Ny_k^4(t-{\rm d}(t))\bar{\varphi}^4_{i,1,k,d}(y_k(t-{\rm d}(t))) +\sum_{k=1}^Ny_k^4\bar{\varphi}^4_{i,1,k}(y_k)\right] \end{aligned} $$

(60)

$$ \begin{aligned} {\rm V}:&-\frac{1}{2}\sum_{j=2}^{n_i}z_{i,j}^3\left( \frac{\partial^2\alpha_{i,j-1}}{\partial y_i^2}\right)g_{i,1}(t,y,y(t-{\rm d}(t)))g_{i,1}^{\rm T}(t,y,y(t-{\rm d}(t)))\\ &\quad \leq \frac{1}{4}\sum_{j=2}^{n_i}\lambda_{i,j}\left( \frac{\partial^2\alpha_{i,j-1}}{\partial y_i^2}\right)^2z_{i,j}^6\\ &\quad + \frac{1}{4}\sum_{j=2}^{n_i} \frac{1}{\lambda_{i,j}}(2N)^3\left[\sum_{k=1}^Ny_k^4(t-{\rm d}(t))\bar{\psi}^4_{i,1,k,d} (y_k(t-{\rm d}(t))) +\sum_{k=1}^Ny_k^4\bar{\psi}^4_{i,1,k}(y_k)\right] \end{aligned} $$

(61)

$$ \begin{aligned} {\rm VI}:& \frac{3}{2}\sum_{j=2}^{n_i}z_{i,j}^2\left( \frac{\partial\alpha_{i,j-1}}{\partial y_i}\right)^2g_{i,1}(t,y,y(t-{\rm d}(t)))g_{i,1}^{\rm T}(t,y,y(t-{\rm d}(t)))\leq\frac{3}{4}\sum_{j=2}^{n_i}\xi_i\left(\frac{\partial\alpha_{i,j-1}} {\partial y_i}\right)^4z_{i,j}^4\\ &\quad +\frac{3}{4}\sum_{j=2}^{n_i} \frac{1}{\xi_{i,j}}(2N)^3\left[\sum_{k=1}^Ny_k^4(t-{\rm d}(t))\bar{\psi}^4_{i,1,k,d}(y_k(t-{\rm d}(t))) +\sum_{k=1}^Ny_k^4\bar{\psi}^4_{i,1,k}(y_k)\right] \end{aligned} $$

(62)

where $\epsilon_{i,3}>0, \epsilon_{i,4}>0$ and $\epsilon_{i,5}>0$. Substituting (57)–(62) back into (56) leads to

$$ \begin{aligned} {{\mathcal{L}}}V_{i,1}\leq&\left(\frac{1}{4}\sum_{j=2}^{n_i}\frac{1} {\eta_{i,j}^4}+\frac{1}{4\epsilon_{i,3}^4}\right)|\tilde{x}_i|^4 -\sum_{j=1}^{n_i}c_{i,j}z_{i,j}^4-y_i^4\left(W_i^{\rm T}S_i(y_i)+\psi_i(y_i)\theta_i\right)+y_i^4\beta_i\\ &\quad +\sum_{k=1}^Ny_k^4\Uppsi_{i,2,k}(y_k)+\sum_{k=1}^Ny_k^4(t-{\rm d}(t))\Uppsi_{i,2,k,d}(y_k(t-{\rm d}(t))) \end{aligned} $$

(63)

where $\beta_i=\frac{3}{4} (\delta_{i,1}a_{i,1,2})^{4/3}+\frac{3} {4}(\epsilon_{i,3}a_{i,1,2})^{4/3}+\frac{3} {4}(\epsilon_{i,4}a_{i,1,2})^{4/3} +\frac{3}{4}\epsilon_{i,5}$, and

$$ \begin{aligned} \Uppsi_{i,2,k,d}(y_k(t-{\rm d}(t)))&=\frac{(2N)^3}{4\epsilon_{i,4}^4}\bar{\varphi}^4_{i,1,k,d}(y_k(t-{\rm d}(t))) + \frac{3(2N)^3}{4\epsilon_{i,5}}\bar{\psi}^{4}_{i,1,k,d}(y_k(t-{\rm d}(t))) \\ &\quad +\frac{(2N)^3}{4}\sum_{j=2}^{n_i} \frac{1}{\eta_{i,j}^4}\bar{\varphi}^4_{i,1,k,d}(y_k(t-{\rm d}(t)))\\ &\quad +\frac{(2N)^3}{4}\sum_{j=2}^{n_i}\frac{1}{\lambda_{i,j}}\bar{\psi}^4_{i,1,k,d}(y_k(t-{\rm d}(t)))\\ &\quad +\frac{3(2N)^3}{4}\sum_{j=2}^{n_i}\frac{1}{\xi_{i,j}} \bar{\psi}^4_{i,1,k,d}(y_k(t-{\rm d}(t))).\\ \Uppsi_{i,2,k}(y)&=\frac{(2N)^3}{4\epsilon_{i,4}^4}\bar{\varphi}^4_{i,1,k}(y_k) + \frac{3(2N)^3}{4\epsilon_{i,5}}\bar{\psi}^{4}_{i,1,k}(y_k)+ \frac{(2N)^3}{4}\sum_{j=2}^{n_i} \frac{1}{\eta_{i,j}^4}\bar{\varphi}^4_{i,1,k}(y_k)\\ &\quad +\frac{(2N)^3}{4}\sum_{j=2}^{n_i} \frac{1}{\lambda_{i,j}}\bar{\psi}^4_{i,1,k}(y_k)+ \frac{3(2N)^3}{4}\sum_{j=2}^{n_i}\frac{1}{\xi_{i,j}}\bar{\psi}^4_{i,1,k}(y_k). \\ \end{aligned} $$

Now we consider the following positive definite function for the whole closed-loop large-scale system

$$ V=\sum_{i=1}^NV_{i,0}+\sum_{i=1}^NV_{i,1}+\frac{1} {1-\zeta}\sum_{i=1}^N\sum_{k=1}^N\int\limits_{t-{\rm d}(t)}^{t}y_k^4(\tau) \Uppsi_{i,k,d}(y_k(\tau)){\rm d}\tau $$

(64)

where the positive function $\Uppsi_{i,k,d}(y_k(\tau))=\Uppsi_{i,1,k,d}(y(\tau))+\Uppsi_{i,2,k,d}(y_k(\tau))$. From (27) and (63), we have

$$ \begin{aligned} {{\mathcal{L}}}V_i&\leq \sum_{i=1}^N\left[\vphantom{\underline {\beta_i+\sum_{k=1}^N}}\left(-b_i\lambda_{\rm min}(P_i)\lambda_{\rm min}(Q_i)+\frac{3b_in_i} {2}\epsilon_{i,1}^{4/3}|P_i|^{8/3} +3b_in_i^3\sqrt{n_i}\epsilon_{i,2}^2|P_i|^4+\frac{1} {4}\sum_{j=2}^{n_i}\frac{1}{\eta_{i,j}^4}\right.\right.\\ &\quad \left.+\frac{1}{4\epsilon_{i,3}^4}\right)|\tilde{x}_i|^4-\sum_{j=1}^{n_i} c_{i,j}z_{i,j}^4-y_i^4\left(W_i^{\rm T}S_i(y_i)+\psi_i(y_i)\theta_i\right)\\ &\quad \left.+y_i^4\left(\underline {\beta_i+\sum_{k=1}^N\psi_{k,1,i}(y_k) +\sum_{k=1}^N\psi_{k,2,i}(y_k)+\frac{1}{1-\zeta}\sum_{k=1}^N\Uppsi_{k,i,d}(y_i)}\right)\right]. \end{aligned} $$

(65)

The underlined function in (65) is unknown and can be approximated by a NN as following

$$ \beta_i+\sum_{k=1}^N\psi_{k,1,i,d}(y_k) +\sum_{k=1}^N\psi_{k,2,i,d}(y_k)+\frac{1} {1-\zeta}\sum_{k=1}^N\Uppsi_{k,i,d}(y_i)=W_i^{\rm T}S_i(y_i)+\sigma_i(y_i) $$

(66)

with the approximation error |σ _i (y _i )| ≤ ψ _i (y _i )θ _i . Substituting (59) into (58) results in

$$ \begin{aligned} {{\mathcal{L}}}V&\leq \sum_{i=1}^N\left[\vphantom{\frac{1}{4\epsilon_{i,3}^4}}\left(-b_i\lambda_{\rm min}(P_i) \lambda_{\rm min}(Q_i)+\frac{3b_in_i}{2}\epsilon_{i,1}^{4/3}|P_i|^{8/3} +3b_in_i^3\sqrt{n_i}\epsilon_{i,2}^2|P_i|^4\right.\right.\\ &\quad \left.\left.+\frac{1}{4}\sum_{j=2}^{n_i}\frac{1} {\eta_{i,j}^4}+\frac{1}{4\epsilon_{i,3}^4}\right)|\tilde{x}_i|^4 \right] -\sum_{i=1}^N\sum_{j=1}^{n_i}c_{i,j}z_{i,j}^4. \end{aligned} $$

(67)

Substituting (36) into (67) yields

$$ {{\mathcal{L}}}V_i\leq -\sum_{i=1}^N\nu_i|\tilde{x}_i|^4-\sum_{i=1}^N \sum_{j=1}^{n_i}c_{i,j}z_{i,j}^4. $$

(68)

Similar to the derivation of Theorem 3 in [34], from (68) together with Lemma 1, the conclusion is obvious.