A new adaptive output-feedback controller against unknown control directions and intrinsic growth

ABSTRACT

This paper addresses global output-feedback stabilisation in the context of unknown control directions and intrinsic unmeasurable-states-dependent growth, and focuses on rectifying the substantial deficiencies in the related works and developing a new adaptive output-feedback controller essentially reducing conservativeness. First, a varying parameter which takes value 1 or $-1$, instead of in an unbounded countable set, is specialised to capture the unknown control direction and accordingly a supervisory mechanism is delicately constructed to decide when to switch the parameter from 1 to $-1$ or from $-1$ to 1. A dynamic high gain, rather than a switching one, is appointed to compensate the serious system uncertainties. Then, the wanted controller is constructed, for which the design functions and design parameters are recursively generated by backstepping method (instead of analytically defined owing to the unknown control coefficients). Based on the designations and supervisory mechanism, it turns out that no Zeno phenomenon occurs, and global boundedness and convergence (to zero) hold, both for the resulting closed-loop system. An example is given to show the effectiveness and the advantage of the proposed scheme.

KEYWORDS:

• Nonlinear systems
• unknown control directions
• unmeasurable states dependent growth
• dynamic adaptive compensation
• supervisory mechanism
• global output-feedback stabilisation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 * The system is called entire one since it covers all dynamics that will be involved in the controller design in Section 2 and the closed-loop system analysis in Section 4.

2 We first show that sequence $\left\{{\mu }_i\right\}$ is positive and strictly increasing if it indeed exists. Actually, from the left inequality of (8(8) $\begin{array}{rl}& 3{\mu }_{i-1}-2{\mu }_{i-2}<{\mu }_i<{\mu }_{i-1}+\frac{1-2{\mu }_{i-1}}{2^{n+1-i}+2^{n-i}-2},\\ & i=1,\dots ,n,\end{array}$(8) ), i.e.$3{\mu }_{i-1}-2{\mu }_{i-2}<{\mu }_i$, we gain $2({\mu }_{i-1}-{\mu }_{i-2})<{\mu }_i-{\mu }_{i-1}$, from which, ${\mu }_1-{\mu }_0>0$ and ${\mu }_0=0$, we can prove by induction that for each $i=1,\dots ,n$, there hold ${\mu }_i-{\mu }_{i-1}>0$, i.e. ${\mu }_i>{\mu }_{i-1}$, and the positiveness of ${\mu }_i$. We then show the existence of sequence $\left\{{\mu }_i\right\}$. It is easy to check that under constraint (8(8) $\begin{array}{rl}& 3{\mu }_{i-1}-2{\mu }_{i-2}<{\mu }_i<{\mu }_{i-1}+\frac{1-2{\mu }_{i-1}}{2^{n+1-i}+2^{n-i}-2},\\ & i=1,\dots ,n,\end{array}$(8) ), ${\mu }_1$ exists. Suppose for induction that there exist ${\mu }_i$'s up to $i\ge 2$ satisfying (8(8) $\begin{array}{rl}& 3{\mu }_{i-1}-2{\mu }_{i-2}<{\mu }_i<{\mu }_{i-1}+\frac{1-2{\mu }_{i-1}}{2^{n+1-i}+2^{n-i}-2},\\ & i=1,\dots ,n,\end{array}$(8) ). Then by the right inequality of (8(8) $\begin{array}{rl}& 3{\mu }_{i-1}-2{\mu }_{i-2}<{\mu }_i<{\mu }_{i-1}+\frac{1-2{\mu }_{i-1}}{2^{n+1-i}+2^{n-i}-2},\\ & i=1,\dots ,n,\end{array}$(8) ), i.e.${\mu }_i<{\mu }_{i-1}+\frac{1-2{\mu }_{i-1}}{2^{n+1-i}+2^{n-i}-2}$, there holds $1-{\mu }_i(2^{n+1-i}+2^{n-i}-2)>1-{\mu }_{i-1}(2^{n+1-i}+2^{n-i}-2)-(1-2{\mu }_{i-1})=-2{\mu }_{i-1}(2^{n-i}+2^{n-i-1}-2)$. Let's now compare the right and left terms of (8(8) $\begin{array}{rl}& 3{\mu }_{i-1}-2{\mu }_{i-2}<{\mu }_i<{\mu }_{i-1}+\frac{1-2{\mu }_{i-1}}{2^{n+1-i}+2^{n-i}-2},\\ & i=1,\dots ,n,\end{array}$(8) ) to confirm the existence of ${\mu }_{i+1}$ and in turn the existence of all ${\mu }_i$'s. One can see $\text{`Right'}-\text{`Left'}={\mu }_i+\frac{1-2{\mu }_i}{2^{n-i}+2^{n-i-1}-2}-(3{\mu }_i-2{\mu }_{i-1})=\frac{1-{\mu }_i(2^{n+1-i}+2^{n-i}-2)}{2^{n-i}+2^{n-i-1}-2}+2{\mu }_{i-1}>0$, which directly means the existence of ${\mu }_{i+1}$.

Additional information

Funding

This work was supported by the National Nature Science Foundation of China under [Grant Nos. 62033007, 61873146, and 61821004]; and by the Taishan Scholars Climbing Program of Shandong Province of China.