Nonlinear leader-following MASs control: a data-driven adaptive sliding mode approach with prescribed performance

Abstract

This paper studies the prescribed performance tracking problem for nonlinear leader-following multiagent systems by developing a data-driven cooperative adaptive sliding mode controller. First, the synchronization measurement error is transformed by combining prescribed performance function with sliding mode surface. Then, the agent’s dynamics are described as a linearization model by the pseudo-partial derivative, and a data-based cooperative adaptive sliding mode control approach is designed to achieve the synchronization of all agents. The synchronization measurement error can converge to the predefined zone. Moreover, the proposed controller only depends on the input/output data of the agents. The effectiveness and advantages of the proposed controller are verified by numerical simulations.

Appendix

Proofs of Theorem 1

Proof

Substituting (43) into (41), we have

$$\begin{aligned} s_i(k+1) - s_i(k) = \frac{1}{2} \ln \bigg ( \frac{\varTheta _{i,1}(k)}{\varTheta _{i,2}(k)} \bigg ) \end{aligned}$$

(48)

where

$$\begin{aligned} \varTheta _{i,1}(k)= & {} \varPsi _{i,a}(k) + \varPsi _{i,b}(k) \\&\times \bigg [ N_{i,a}(k) -\varGamma _i \texttt {sign}(s_i(k)) \mathfrak {M}_i \bigg ] \\ \varTheta _{i,2}(k)= & {} \varPsi _{i,a}(k) - \varPsi _{i,c}(k) \\&\times \bigg [ N_{i,a}(k) - \varGamma _i \texttt {sign}(s_i(k)) \mathfrak {M}_i \bigg ] \end{aligned}$$

where $\mathfrak {M}_i = \frac{ \hat{\theta }_i(k)^2 }{\gamma _i + \hat{\theta }_i(k)^2 } < 1$.

If $s_i(k) < 0$, we have

$$\begin{aligned} \varTheta _{i,1}(k)= & {} \varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) +\varGamma _i \mathfrak {M}_i \bigg ] \\ \varTheta _{i,2}(k)= & {} \varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [ N_{i,a}(k) + \varGamma _i \mathfrak {M}_i \bigg ] \end{aligned}$$

Leveraging (44), we have

$$\begin{aligned}&\varPsi _{i,a}(k)< \varTheta _{i,1}(k)< \varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i \\&< \varPsi _{i,a}(k) + \varPsi _{i,a}(\infty ) \\&0 \le \varPsi _{i,a}(k) - \varPsi _{i,a}(\infty )< \varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i \\&< \varTheta _{i,2}(k) < \varPsi _{i,a}(k) \end{aligned}$$

Then,

$$\begin{aligned} 0< & {} s_i(k+1) - s_i(k)< \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \nonumber \\< & {} \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + \varPsi _{i,a}(\infty ) }{\varPsi _{i,a}(k) - \varPsi _{i,a}(\infty )} \bigg ) \end{aligned}$$

(49)

If $s_i(k) > 0$, we have

$$\begin{aligned}&\varPsi _{i,a}(k) - \varPsi _{i,a}(\infty )< \varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i \\&< \varTheta _{i,1}(k)< \varPsi _{i,a}(k) \\&0 \le \varPsi _{i,a}(k)< \varTheta _{i,2}(k)< \varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i \\&< \varPsi _{i,a}(k) + \varPsi _{i,a}(\infty ) \end{aligned}$$

Then,

$$\begin{aligned} 0> & {} s_i(k+1) - s_i(k)> \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \nonumber \\> & {} \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - \varPsi _{i,a}(\infty ) }{\varPsi _{i,a}(k) + \varPsi _{i,a}(\infty )} \bigg ) \end{aligned}$$

(50)

Using (49) and (50), we have

$$\begin{aligned}&\frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg )< s_i(k+1) - s_i(k) \nonumber \\&< \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \end{aligned}$$

(51)

Define $\varOmega _i$ as a bounded function

$$\begin{aligned} \varOmega _i = \Bigg \{ \varsigma _i(k)&:&\frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg )< s_i(k) \nonumber \\&< \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \Bigg \} \end{aligned}$$

(52)

Case A $s_i(k)$ out of the region $\varOmega _i$:

$$\begin{aligned} s_i(k) < \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \end{aligned}$$

or

$$\begin{aligned} s_i(k) > \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \end{aligned}$$

Using (49) and (50), step size $\triangle s_i(k) < \varOmega _i$ will make the sliding surface $s_i(k)$ reach and enter $\varOmega _i$.

Case B $s_i(k)$ gets into the region $\varOmega _i$:

$$\begin{aligned}&\frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg )< s_i(k) \nonumber \\&< \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \end{aligned}$$

(53)

In this case,

$$\begin{aligned}&s_i(k+1) - s_i(k)=\frac{1}{2} \nonumber \\&\quad \ln \left( \frac{\varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) -\varGamma _i \texttt {sign}(s_i(k)) \mathfrak {M}_i \bigg ] }{\varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [N_{i,a}(k) - \varGamma _i \texttt {sign}(s_i(k)) \mathfrak {M}_i \bigg ]} \right) \end{aligned}$$

(54)

1) If $0>s_i(k) > \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg )$, we have

$$\begin{aligned}&\frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) + \nonumber \\&\frac{1}{2} \ln \left( \frac{\varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) + \varGamma _i \mathfrak {M}_i \bigg ] }{\varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [ N_{i,a}(k) + \varGamma _i \mathfrak {M}_i \bigg ] }\right) \nonumber \\&< s_i(k+1) \nonumber \\&< \frac{1}{2} \ln \left( \frac{\varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) + \varGamma _i \mathfrak {M}_i \bigg ] }{\varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [ N_{i,a}(k) + \varGamma _i \mathfrak {M}_i \bigg ] } \right) \end{aligned}$$

(55)

Due to $\varGamma _i > |\mathfrak {N}_{i,a} |$, we have

$$\begin{aligned}&0< \frac{1}{2} \ln \left( \frac{\varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) + \varGamma _i \mathfrak {M}_i \bigg ] }{\varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [ N_{i,a}(k) + \varGamma _i \mathfrak {M}_i \bigg ] } \right) \nonumber \\&< \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) -2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \end{aligned}$$

(56)

2) If $0< s_i(k) < \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \bigg )$, we have

$$\begin{aligned}&\frac{1}{2} \ln \left( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \right) + \nonumber \\&\frac{1}{2} \ln \left( \frac{\varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) - \varGamma _i \mathfrak {M}_i \bigg ] }{\varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [ N_{i,a}(k) - \varGamma _i \mathfrak {M}_i \bigg ] } \right) \nonumber \\&> s_i(k+1) \nonumber \\&> \frac{1}{2} \ln \left( \frac{\varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) - \varGamma _i \mathfrak {M}_i \bigg ] }{\varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [ N_{i,a}(k) - \varGamma _i \mathfrak {M}_i \bigg ] } \right) \end{aligned}$$

(57)

Due to $\varGamma _i > | N_{i,a}(k) |$, we have

$$\begin{aligned}&\frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg )< \nonumber \\&\quad \frac{1}{2} \ln \left( \frac{\varPsi _{i,a}(k) + \varPsi _{i,b}(k) \bigg [ N_{i,a}(k) - \varGamma _i \mathfrak {M}_i \bigg ] }{\varPsi _{i,a}(k) - \varPsi _{i,c}(k) \bigg [ N_{i,a}(k) - \varGamma _i \mathfrak {M}_i \bigg ] } \right) < 0 \end{aligned}$$

(58)

Using (56) and (58), we have

$$\begin{aligned}&\frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) - 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) + 2 \varPsi _{i,c}(k) \varGamma _i} \bigg )< s_i(k+1) \nonumber \\&< \frac{1}{2} \ln \bigg ( \frac{\varPsi _{i,a}(k) + 2 \varPsi _{i,b}(k) \varGamma _i}{\varPsi _{i,a}(k) - 2 \varPsi _{i,c}(k) \varGamma _i} \bigg ) \end{aligned}$$

(59)

Thus, we conclude that the $s_i(k)$ is bounded. According to (32), $\varsigma _i(k)$ is bounded, which implies that $z_i(k)$ is bounded. According to Lemma 1, $e_i(k)$ is bounded. And $e_i(k)$ is constrained by the modified condition (11), which means the prescribed performance is guaranteed.

This completes the proof. $\square$