Discussion of Three Examples to Recent Results of Finite- and Fixed-Time Convergent Algorithms

The recently published technical correspondence paper [1] addresses the results presented in [2, 3, 4]. Three examples questioning validity of the obtained results are provided.

This note discusses the examples given in [1]. It is shown that the first example ([1], Section 3) is irrelevant to the results of [2]. The second example ([1], Section 4) establishes the well-known fact about the result of [3] that a continuous differentiator can exactly differentiate a signal, only if its second derivative is equal zero. This note provides a method to extend the algorithms presented in [3] to the general case. It is shown that if the signal second derivative is not equal to zero, the differentiator can be modified by including discontinuous terms to achieve the goal. Finally, the third example ([1], Section 5) presents a special case of control gains and initial conditions, where the result of Theorem 1 in [4] on estimating the finite convergence time does not hold. However, this note demonstrates that the result of Theorem 1 in [4] still remains valid in the most practical case of selecting control gains. The result of Theorem 2 in [4] on estimating the fixed convergence time holds as well.

This note is organized as follows. Section 2-4 subsequently discuss the examples given in Sections 3, 4, and 5 of [1]. Section 6 summarizes the discussions.

Following the notation of Lemma 1 in [1], note that the paper [2] considers only systems with initial conditions in the form $[0,x_{20},x_{30},0]$. Therefore, Lemma 1 of [1] is applicable to the systems studied in [2], only if $a=b=0$, that is, $x(t_{0})=[0,0,0,0]$ is the origin. However, in this case, the system (2) of Section 3 in [1] has only the zero solution, $x(t)=0$, for $t\geq 0$, which is finite-time convergent to the origin. Remark 2 and Fig. 1 of Section 3 in [1] are not relevant to the result of [2], since $k_{3}^{2}=1<2$ and the conditions of Lemma 4 in [2] do not hold. Proposition 3 of Section 3 in [1] is not relevant to the result of [2] as well, since the paper [2] studies only attractivity (convergence) problems but not finite-time stability ones. The difference between finite-time stability and finite-time attractivity concepts can be consulted in Section 4 of [5].

This is a well-known fact that the finite- and fixed-time convergent differentiators proposed in Theorems 1 and 2 of [3] converge to the real system states exactly, only if the output $n$-th derivative is equal to zero, $y^{(n)}(t)=0$, for all $t\geq T$, where $T$ is a certain finite time and $n$ is the dimension of the differentiator. Note that in the series of papers mentioned in [1] the finite- or fixed-time convergent differentiators are used as parts of finite- or fixed-time convergent controllers, whose setpoints are represented by equilibria, that is, the condition $y^{(n)}(t)=0$ holds after a certain finite $T$.

Furthermore, those differentiators can be modified to achieve finite- or fixed-time convergence in the general situation by adding the term $-\lambda sign(z_{1}(t)-y(t))$ to the last differentiator equations, where $\lambda>\mid y^{(n)}(t)\mid$ is a uniform bound for the output $n$-th derivative. For example, in case of the fixed-time convergent differentiator proposed in Theorem 2 of [3], the corresponding equations take the form

$$\dot{z}_{1}(t)=z_{2}(t)-k_{1}\mid z_{1}(t)-y(t)\mid^{\alpha_{1}}sign(z_{1}(t)-y(t))$$

$$-\kappa_{1}\mid z_{1}(t)-y(t)\mid^{\beta_{1}}sign(z_{1}(t)-y(t)),$$

$(1)$

$$\vdots$$

$$\dot{z}_{i}(t)=z_{i+1}(t)-k_{i}\mid z_{1}(t)-y(t)\mid^{\alpha_{i}}sign(z_{1}(t)-y(t))$$

$$-\kappa_{i}\mid z_{1}(t)-y(t)\mid^{\beta_{i}}sign(z_{1}(t)-y(t)),$$

$$\quad i=1,\dots,n-1$$

$$\vdots$$

$$\dot{z}_{n}(t)=-k_{n}\mid z_{1}(t)-y(t)\mid^{\alpha_{n}}sign(z_{1}(t)-y(t))$$

$$-\kappa_{n}\mid z_{1}(t)-y(t)\mid^{\beta_{n}}sign(z_{1}(t)-y(t))$$

$$-\lambda sign(z_{1}(t)-y(t)),$$

$$k_{1},\dots,k_{n},\lambda>0,$$

where the gains $k_{1},\dots,k_{n}$ and $\kappa_{1},\dots,\kappa_{n}$ satisfy the conditions of Theorem 2 of [3] and $\lambda>\mid y^{(n)}(t)\mid$. This modification keeps the convergence fixed time estimates given in Theorem 2 of [3] and the convergence finite time estimates given in Theorem 1 of [3] in the most practical case of selecting the control gains, as noted in the next section.

The differentiator (1) is not smooth; however, a smooth differentiator for $(n-1)$-th derivative of the output can be constructed by increasing the dimension of the differentiator (1) by one, i.e., adding the equation for $z_{n+1}$ and moving the term $-\lambda sign(z_{1}(t)-y(t))$ to this equation, provided that the condition $\lambda>\mid y^{(n+1)}(t)\mid$ holds.

This note discussed the examples given in [1]. It has been shown that the results opposed in [1] remain valid in most practical cases or can be successfully modified or are irrelevant to the given examples.

The authors thank Prof. Y. Shtessel, with the University of Alabama in Huntsville, for his valuable and very useful discussions, comments, and suggestions.