A note on adaptive observer for the Lur’e differential inclusion system

Abstract

This paper considers the adaptive observer design problem for the Lur’e differential inclusion system with unknown parameters. Different from the current work, the proposed observer does not hold any set-valued mapping and the uncertain parameters can also be identified. Firstly, the adaptive observer is designed for the Lur’e differential inclusion system, and it is proved to be valid by Lyapunov stability theory. Then, based on the canonical form of the system matrix, the sufficient and necessary condition for one assumption is presented in both SISO and MIMO cases. Finally, numerical examples are simulated to show the effectiveness of the proposed method.

Appendix

In this section, we present one method which can be used to determine whether one signal is persistently exciting.

Definition 5

Function $u{:}[0,+\infty )\rightarrow R^{n}$ is called stationary signal if the following holds

$$\begin{aligned} \lim \limits _{T\rightarrow \infty }T^{-1}\int _{t}^{t+T}u(\tau )u^{T}(t+\tau )d\tau <\infty . \end{aligned}$$

The above limit is the function of t, and is called self-covariance matrix of u, denoted by $R_{u}(t)$. Furthermore, the Fourier transformation of $R_{u}(t)$ is defined by

$$\begin{aligned} S_{u}(\omega )=\int _{-\infty }^{\infty }R_{u}(\tau )\exp (-j\omega \tau )d\tau , \end{aligned}$$

which is spectral measure of u. A stationary signal is called k-th order multi-frequency if there exist at least k zeros of $S_{u}(\omega )$ for $\omega \in R$, and the nonzero real numbers are called subset of u.

Thus, we can obtain the following propositions. The proof is omitted.

Proposition 1

If $u=c~(c\ne 0)$, then the subset contains only one element, i.e. it is first-order multi-frequency.

Proposition 2

If $u=\sin \omega _{0}t~(\omega _{0}\ne 0)$, then the subset contains only two elements, i.e. it is second-order multi-frequency.

Proposition 3

If $u=\sum \limits _{i=1}^{l}A_{i}\sin \omega _{i}t(A_{i}\ne 0,\omega _{i}\ne 0, \omega _{i}\ne \omega _{j},i\ne j)$, then the subset contains 2l elements, i.e. it is 2l-th order multi-frequency.

The following lemma is important, and it can be referred to [29].

Lemma 3

[29] Let $y(s)=G(s)u(s)$, where $G(s)\in R^{n\times l}$ is transfer function, then there exist n different real numbers $\omega _{1},\omega _{2},\ldots , \omega _{n}$, such that $G(j\omega _{1}),G(j\omega _{2}),\cdots ,G(j\omega _{n})$ are linear independent, then y(t) is persistently exciting signal if and only if u is n-th order multi-frequency.

Thus, by Lemma 3, if f(y, u) in the system (3) is n-th order multi-frequency, then $\xi (t)$ is persistently exciting.