Fixed-time stabilization of high-order integrator systems with mismatched disturbances

Abstract

The fixed-time stabilization of high-order integrator systems with both matched and mismatched disturbances is investigated. A continuous non-switching control law is designed based on the bi-limit homogeneous technique for arbitrary-order integrator systems. Combining with fixed-time disturbance observer, the proposed continuous control law for the system with matched and mismatched disturbances guarantees that the convergence time is uniformly bounded with respect to any initial states. Finally, the numerical results are provided to verify the efficiency of the developed method.

Appendix

For brevity, define $A^n_1(s)$ and $A^n_2(s)$ as follows

$$\begin{aligned} A^n_1(s)=s^n+k_ns^{n-1}+\cdots +k_2s+k_1 \end{aligned}$$

(26)

and

$$\begin{aligned} A^n_2(s)=s^n+k'_ns^{n-1}+\cdots +k'_2s+k'_1 \end{aligned}$$

(27)

with

$$\begin{aligned} k'_i=3k_i,\quad i=1,2,\ldots ,n \end{aligned}$$

(28)

For $n\le 4$, a direct application of Routh criterion [36] shows that $A^n_2(s)$ is Hurwitz if the parameters $k_i$ are selected such that the polynomial $A^n_1(s)$ is Hurwitz. For example, $A^4_1(s)$ with $n=4$ in (26) is Hurwitz if and only if the condition

$$\begin{aligned} k_2k_3k_4>k^2_2+k^2_4k_1,\quad k_i>0, \quad i=1,\ldots ,4 \end{aligned}$$

(29)

holds. Obviously, the condition $k'_2k'_3k'_4>k'^2_2+k'^2_4k'_1$ with $k'_i>0,i=1,\ldots ,4$ holds when $k'_i$ are defined by (28) with the selection of $k_i$ in (29). It follows that $A^4_2(s)$ is Hurwitz if $A^4_1(s)$ is Hurwitz. Similarly, it can be easily found that the conclusion holds for $n=1,2,3$.

For $n>4$, the polynomial $A^n_2(s)$ may not be Hurwitz even if $A^n_1(s)$ is Hurwitz. In this case, the following lemma can be applied to select $k_i,i=1,\ldots ,n$ such that $A^n_1(s)$ and $A^n_2(s)$ are both Hurwitz.

Lemma 5

[37] The polynomial $a_ns^n+\cdots +a_1x+a_0$ is Hurwitz if the condition $a_ia_{i+1}\ge 3a_{i-1}a_{i+2},i=1,\ldots ,n-2$ with $a_j>0,j=0,\ldots ,n$ holds.

Lemma 5 was first proposed in [38] using Chinese, and its English description was provided in [37]. It follows from Lemma 5 that the positive real numbers $k_j,j=1,\ldots ,n$ can be chosen as

$$\begin{aligned} k_ik_{i+1}\ge 3k_{i-1}k_{i+2},\quad i=2,\ldots ,n-1 \end{aligned}$$

(30)

with the definition $k_{n+1}=1$ such that $A^n_1(s)$ is Hurwitz. An equivalent expression of (30) is

$$\begin{aligned} (3k_i)(3k_{i+1})\ge 3(3k_{i-1})(3k_{i+2}),\quad i=2,\ldots ,n-1 \end{aligned}$$

(31)

Taking into account (28), (31) means the following inequality

$$\begin{aligned} k'_ik'_{i+1}\ge 3k'_{i-1}k'_{i+2},\quad i=2,\ldots ,n-1 \end{aligned}$$

(32)

holds with the definition $k'_{n+1}=1$. Although $k'_{n+1}=k_{n+1}=1$ does not satisfy the relationship (28), it can be easily verified that (32) can be derived from (31) when $i=n-1$. It follows from Lemma 5 that the polynomial $A^n_2(s)$ is Hurwitz with the positive real numbers $k'_i$ satisfying (28) and (30). Therefore, the positive real numbers $k_i$ can be selected in terms of (30) to ensure that $A^n_1(s)$ and $A^n_2(s)$ are both Hurwitz.