As an extension of timeoptimal control, Bhat Bernstein proposed a saturated finitetime stabilizer for the double integrator [1]. Such a controller is continuous and consequently is free of jumping switch. In a recent work [2], a class of saturated finitetime stabilizers are presented for multiple integrators and a skillful analysis method was proposed for verifying the saturation reduction.
Constructing finitetime stabilizers for compensating input delays is a significant issue. In this line, profound results were achieved in [3] by means of timevarying distributed delay feedback; with the aid of the socalled Artstein's transformation [4], the mentioned problem has been reduced to solely assigning a finitetime stabilizer [5].
In addition, the stabilization design for compensating input saturation and input delays has long occupied the attention of the control community. In most cases, the kernel technical work is to verify the reduction of saturated terms by fully using suitable normal forms, NewtonLeibniz formula and Lyapunov functional (see for instance [6][9]).
However, there is currently no finitetime stabilizing design that compensates both input saturation and input delays. The algorithms in [6][9] do not take into account the finitetime stability and the classical timeoptimal control design; the works in [1], [2] do not consider delay compensation; and the works in [3], [5] do not deal with input saturation. Therefore, it will be an important theoretical work to design finitetime controllers that compensate both input saturation and input delay.
Moreover, exploration in this direction is of practical significance. As we know, the bounded bangbang control can bring about finitetime stability and the shortest transition time, but cannot allow a time delay. In fact, a delay in the input will make it difficult to achieve the prescribed performance and even give rise to serious consequences. For example, when the input is subject to a time delay, it may happen that a spacecraft has a downward velocity while it lands on the ground.
Due to the above observations, for a double integrator we would like to present a finitetime stabilizer that compensates both input saturation and input delay. The detailed technical works are stated as follows:
1) With the aid of the Artstein's transformation [4] and the reduction approach in [5], the problem under consideration is reduced to assigning a saturated finitetime stabilizer. This will largely simplify the design/analysis, for there is no need to deal with delayed terms when we focus on the equivalent system.
2) The saturated finitetime stabilizer is motivated by the method in [2]. As we know, it is hard to verify the saturation reduction in a cancellation way, for there is no suitable normal form due to fractional exponents. To this end, we will suggest a saturation reduction analysis method that slightly differs from the one in [2]. Specifically, we do not compute small time intervals for contradictions, but compute the derivatives of saturation functions in small domains and prove that the related derivatives are nonpositive. Once the saturated controller reduces to a linear one, we invoke the homogeneity based stability theory [1] to analyze the finitetime stability of the reduced system. These efforts allow us to obtain a saturated finitetime stabilizer that can get rid of
3) In addition, we will show that, in the case of the double integrator, the final control design has a simple form in the sense that it does not contain integral terms.
The rest of the note is organized as follows. In Section Ⅱ, we present the problem formulation and give mathematical preliminaries. In Section Ⅲ, we provide the control design and stability analysis. Simulations are also given in this section to illustrate the effectiveness of the design. In Section Ⅳ, we make comments on the work in this note, emphasizing the importance of suitable methods and pointing out a potential extension. Finally, concluding remarks are put into Section Ⅴ.
Ⅱ. PROBLEM FORMULATION AND PRELIMINARIES A. Problem FormulationThe bangbang timeoptimal controller is bounded and can guarantee the finitetime stability and the shortest convergence time. In the following, we will show that an input delay will lead to a serious consequence.
Consider the system
$ \begin{align}\label{eq1} \dot{y}_{1}=y_{2}, \dot{y}_{2}=u \end{align} $  (1) 
and find a suitable control
$ \begin{align}\label{eq2} u=\textrm{sgn}(y_{1}+2^{1}y_{2}y_{2}) \end{align} $  (2) 
and the shortest transition time
We now suppose that
$ \begin{equation} \begin{aligned}\label{eq3} &\dot{y}_{1}=y_{2}, \dot{y}_{2}=u(t0.2) \\ &y_{1}(t)=y_{2}(t)=0, t\in[0.2, 0] \\ &y_{1}(0)=y_{2}(0)=1. \end{aligned} \end{equation} $  (3) 
If one simply uses the following controller:
$ \begin{equation} \begin{aligned}\label{eq4} u(t0.2)=&\textrm{sgn}[y_{1}(t0.2)+ \\ &2^{1}y_{2}(t0.2)y_{2}(t0.2)] \end{aligned} \end{equation} $  (4) 
the histories of states and input will exhibit as follows.
From Fig. 1, it is observed that when
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Fig. 1 Histories of states and input of system (1), (4). 
We first recall the Artstein's transformation and its application in finitetime stabilization of systems subject to input delay (for details the reader is referred to [4], [5]).
Consider the system subject to an input delay
$ \begin{equation} \begin{aligned}\label{eq5} &\dot{x}(t)=Ax(t)+bu(th) \\ &A\in {\mathbb R}^{n\times n}, b\in {\mathbb R}^{n\times 1}, t\geq 0, h> 0. \end{aligned} \end{equation} $  (5) 
Making the Artstein's transformation
$ \begin{align}\label{eq6} y(t)=x(t)+\int^{t}_{th}e^{(tsh)A}bu(s)ds \end{align} $  (6) 
system (5) is transformed into
$ \begin{equation} \begin{aligned}\label{eq7} &\dot{y}(t)=Ay(t)+\bar{b}u(t) \\ &\bar{b}=e^{hA}b. \end{aligned} \end{equation} $  (7) 
Noting
$ \begin{align*} &\int^{t}_{th}e^{(tsh)A}bu(s)ds \\ &\overset{s=r+t}{=}\int^{0}_{h}e^{(t(r+t)h)A}bu(r+t)dr \\ &~~=\int^{0}_{h}e^{(rh)A}bu(r+t)dr \end{align*} $ 
the transformation (6) is the same as that in [5], where the Artstein's transformation is described by
$ \begin{align*} y(t)=x(t)+\int^{0}_{h}e^{(rh)A}bu(r+t)dr. \end{align*} $ 
From Theorem 7 of [5], we have
Lemma 1: If system (7) is finitetime stabilizable by a feedback control
$ \begin{align*} u(t)=k(t)f(y(t)) \end{align*} $ 
with
$ \begin{align}\label{eq8} f(y)\leq \alpha(\sqrt{y^{2}_{1}+\cdots+y^{2}_{n}}) \end{align} $  (8) 
then, system (5) is finitetime stabilizable by the feedback
$ \begin{align*} u(t)=k(t)f(x(t)+\int^{t}_{th}e^{(tsh)A}bu(s)ds). \end{align*} $ 
We then recall homogeneity concepts and a related lemma from [11]. Define the dilation
Definition 1: With respect to the dilation
Lemma 2: The origin is a finitetime stable equilibrium of
This criterion will facilitate the finitetime stability analysis.
Next, we introduce useful inequalities [12]:
1) For a constant
$ \begin{align*} ~~~~~~~~~~x^{p}y^{p}\leq2^{1p}xy^{p}, ~~0<p<1. \end{align*} $  (F1) 
2) Let
$ \begin{align*} ~~~~~~~~x^{c}y^{d}\leq\frac{c}{c+d}rx^{c+d}+\frac{d}{c+d}r^ {\frac{c}{d}}y^{c+d}. \end{align*} $  (F2) 
Finally, the saturation level function is defined as
$ \begin{align*} \textrm{sat}_{\varepsilon}(s)=\textrm{sign}(s)\textrm{min}\{s, \varepsilon\}, s\in {\mathbb R}, \varepsilon>0. \end{align*} $ 
Consider the double integrator that is subject to an input delay
$ \begin{equation} \begin{aligned}\label{eq9} \dot{X}(t)&=\left[ \begin{array}{cc} 0&1 \\ 0&0 \\ \end{array} \right]X(t)+\left[ \begin{array}{cc} 0 \\ 1 \\ \end{array} \right]u(th) \\ &\triangleq{}AX(t)+bu(th). \end{aligned} \end{equation} $  (9) 
By making an Artstein's transformation
$ \begin{align}\label{eq10} Y=X+\int^{t}_{th}e^{(tsh)A}bu(s)ds \end{align} $  (10) 
system (9) is described by
$ \begin{align}\label{eq11} \dot{Y}=AY+\bar{b}u \end{align} $  (11) 
where
$ \begin{align*} \bar{b}=e^{hA}b=\left[ \begin{array}{cc} 1 &h \\ 0&1 \\ \end{array} \right]\left[ \begin{array}{cc} 0 \\ 1 \\ \end{array} \right]=\left[ \begin{array}{cc} h \\ 1 \\ \end{array} \right]. \end{align*} $ 
Making another transformation
$ \begin{align}\label{eq12} z_{1}=Y_{1}+hY_{2}, z_{2}=Y_{2} \end{align} $  (12) 
we then have
$ \begin{align}\label{eq13} \dot{z}_{1}=z_{2}, \dot{z}_{2}=u. \end{align} $  (13) 
In this note, we will first present a saturated finitetime stabilizer for system (13), and then use Lemma 1 to show that, after substituting variables, this controller will be a globally finitetime stabilizer of system (9).
A. Saturated Finitetime Stabilizer of System (13)We will first focus on system (13), for which the following saturated controller is assigned:
$ \begin{equation} \begin{aligned}\label{eq14} u&=b_{2}\textrm{sat}^{1+2a}_{\varepsilon}(z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}}) \\ z_{2^{*}}&=b_{1}\textrm{sat}^{1+a}_{\varepsilon}(z_{1}) \end{aligned} \end{equation} $  (14) 
where
$ \begin{align}\label{eq15} a=2k_{1}/(2k_{2}+1), 1\leq k_{1}<k_{2}; \varepsilon > 0 \end{align} $  (15) 
and the control parameters
Remark 1: In [1], the saturated finitetime stabilizer for system (13) is described by
$ \begin{align*} u=&\textrm{sat}_{\varepsilon}(\textrm{sign}(z_{2})z_{2}^{\alpha}) \\ &\textrm{sat}_{\varepsilon}(\textrm{sign}(z_{1}+\frac{1}{2\alpha}\textrm{sign}(z_{2})z_{2}^{2\alpha}) \\ &\timesz_{1}+\frac{1}{2\alpha}\textrm{sign}(z_{2})z_{2}^{2\alpha}^{\frac{\alpha}{2\alpha}}) \end{align*} $ 
with
In the following, we show that the controller (14) is a finitetime stabilizer of system (13) under suitable parameter conditions.
Reduction Analysis of Saturated Terms:
Fact 1: The controller (14) of system (13) reduces to an unsaturated one in a finite time under the following parameter conditions:
$ \begin{align}\label{eq16} b_{2}\geq(1+a)b^{(1+a)^{1}}_{1}(1+b^{(1+a)^{1}}_{1})^{1+2a} \end{align} $  (16) 
$ \begin{align}\label{eq17} b_{1}>2^{a}. \end{align} $  (17) 
Proof: We prove this result in a bottomup recursive manner.
Step 1: Consider the
$ \begin{align}\label{eq18} \dot{z}_{2}=b_{2}\textrm{sat}^{1+2a}_{\varepsilon}(z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}})\leq b_{2}\varepsilon^{1+2a}. \end{align} $  (18) 
Noting
$ \begin{align*} \varepsilon&\leq z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}} \\ &\leq(z_{2}(0)b_{2}\varepsilon^{1+2a}t)^{(1+2a)^{1}}+b^{(1+a)^{1}}_{1}\varepsilon. \end{align*} $ 
As time goes to infinity, we obtain a contradiction
$ \begin{align*} \varepsilon\leq z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}} <0. \end{align*} $ 
So there exists a finite time
$ \begin{align}\label{eq19} z^{(1+a)^{1}}_{2}(t_{2})z^{(1+a)^{1}}_{2^{*}}(t_{2})=\varepsilon. \end{align} $  (19) 
We then calculate the time derivative of
$ \begin{equation} \begin{aligned}\label{eq20} \frac{d}{dt}z^{(1+a)^{1}}_{2}&=(1+a)^{1}z^{(1+a)^{1}1}_{2}\dot{z}_{2} \\ &\leq(1+a)^{1}b_{2}\varepsilon^{1+2a}z^{(1+a)^{1}1}_{2}. \end{aligned} \end{equation} $  (20) 
To compute
$ \begin{align*} \frac{d}{dt}z^{(1+a)^{1}}_{2^{*}}\leq b^{(1+a)^{1}}_{1}z_{2}. \end{align*} $ 
From (19), we have
$ \begin{align*} z_{2}=(1+b^{(1+a)^{1}}_{1})^{1+a}\varepsilon^{1+a}. \end{align*} $ 
For cancellation, we calculate
$ \begin{equation} \begin{aligned}\label{eq21} z_{2}&=z^{2(1+a)^{1}}_{2}z^{(1+a)^{1}1}_{2} \\ &\leq ((1+b^{(1+a)^{1}}_{1})^{1+a}\varepsilon^{1+a})^{2(1+a)^{1}}z^{(1+a)^{1}1}_{2} \\ &\leq (1+b^{(1+a)^{1}}_{1})^{1+2a}\varepsilon^{1+2a}z^{(1+a)^{1}1}_{2}. \end{aligned} \end{equation} $  (21) 
Then, the parameter condition (16) leads to
$ \begin{align*} (1+a)^{1}b_{2}\varepsilon^{1+2a}+b^{(1+a)^{1}}_{1}(1+b^{(1+a)^{1}}_{1})^{1+2a}\varepsilon^{1+2a}\leq0. \end{align*} $ 
Namely, we have
$ \begin{align}\label{eq22} \frac{d}{dt}(z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}})\leq 0. \end{align} $  (22) 
Combining (19) and (22), we have
$ \begin{align}\label{eq23} z^{(1+a)^{1}}_{2}(t)z^{(1+a)^{1}}_{2^{*}}(t)\leq \varepsilon, \forall t\geq t_{2}. \end{align} $  (23) 
Similarly, it can be shown that there exists a time
$ \begin{align}\label{eq24} z^{(1+a)^{1}}_{2}(t)z^{(1+a)^{1}}_{2^{*}}(t)\geq \varepsilon, \forall t\geq \overline{t_{1}}. \end{align} $  (24) 
Step 2: We claim that with
$ \begin{align*} \dot{z}_{1}=z_{2^{*}}+(z_{2}z_{2^{*}})\leq (b_{1}\varepsilon^{1+a}2^{a}\varepsilon^{1+a})<0. \end{align*} $ 
This implies that the claim is true.
Likewise, it can be shown that a finite time
Asymptotical Stability Analysis of the Reduced System:
After the time
$ \begin{equation} \begin{aligned}\label{eq25} u&=b_{2}(z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}})^{1+2a} \\ z_{2^{*}}&=b_{1}z^{1+a}_{1}. \end{aligned} \end{equation} $  (25) 
Following the backstepping method in [12], we can prove:
Fact 2: There are suitable
Proof: Fact 2 is proved in three steps.
Step 1: Define the function
$ \begin{align*} \dot{W}_{1}&=z^{1a}_{1}z_{2^{*}}+z^{1a}_{1}(z_{2}z_{2^{*}}) \\ &\leq b_{1}z^{1a}_{1}z^{1+a}_{1}+z^{1a}_{1}(z_{2}z_{2^{*}}). \end{align*} $ 
Letting
$ \begin{align*} \dot{W}_{1}&\leq b_{1}z^{2}_{1}+2^{a}z^{1a}_{1}z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}}^{1+a} \\ &\leq b_{1}z^{2}_{1}+2^{a}z^{1a}_{1}\xi^{1+a}_{2} \\ &\leq b_{1}z^{2}_{1}+2^{1a}(1a)l^{\frac{1+a}{1a}}z^{2}_{1}+2^{1a}(1+a)l\xi^{2}_{2} \end{align*} $ 
where
Step 2: For the whole system (13), (25), we define the function
$ \begin{align*} V=W_{1}+W_{2}, W_{2}=\int^{z_{2}}_{z_{2^{*}}}(s^{(1+a)^{1}}z^{(1+a)^{1}}_{2^{*}})^{12a}ds \end{align*} $ 
and have
$ \begin{align*} \dot{V}&\leq b_{1}z^{2}_{1}+2^{1a}(1a)l^{\frac{1+a}{1a}}z^{2}_{1}+2^{1a}(1+a)l\xi^{2}_{2}\\ &+\left(\frac{\partial W_{2}}{\partial z_{1}}\right)z_{2}+(z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}})^{12a}u. \end{align*} $ 
To calculate
$ \begin{align*} \frac{\partial W_{2}}{\partial z_{1}} =\, &(12a)\frac{\partial(z^{(1+a)^{1}}_{2^{*}})} {\partial z_{1}}\\ \times&\int^{z_{2}}_{z_{2^{*}}}(s^{(1+a)^{1}}z^{(1+a)^{1}}_{2^{*}})^{2a}ds \\ \leq& (12a)b^{(1+a)^{1}}_{1}z_{2}z_{2^{*}}z^{(1+a)^{1}}_{2}z^{(1+a)^{1}}_{2^{*}} ^{2a} \\ \leq &2^{a}(12a)b^{(1+a)^{1}}_{1}\xi_{2}^{1a}. \end{align*} $ 
Next, there holds
$ \begin{align*} z_{2}=z_{2}z_{2^{*}}+z_{2^{*}}\leq 2^{a}\xi_{2}^{1+a}+b_{1}z_{1}^{1+a}. \end{align*} $ 
We then have
$ \begin{align*} \left(\frac{\partial W_{2}}{\partial z_{1}}\right)z_{2}\,&\leq2^{a}(12a)b^{(1+a)^{1}}_{1}\\ &\times\xi_{2}^{1a}(2^{a}\xi_{2}^{1+a}+b_{1}z_{1}^{1+a}) \\ &\leq2^{2a}(12a)b^{(1+a)^{1}}_{1}\xi^{2}_{2} \\ &+2^{a}(12a)b^{(1+a)^{1}}_{1}b_{1}\xi_{2}^{1a}z_{1}^{1+a} \\ &\leq2^{2a}(12a)b^{(1+a)^{1}}_{1}\xi^{2}_{2} \\ &+2^{1a}(12a)(1a)b^{(1+a)^{1}}_{1}b_{1}m\xi^{2}_{2} \\ &+2^{1a}(12a)(1+a)b^{(1+a)^{1}}_{1}b_{1}m^{\frac{1a}{1+a}}z^{2}_{1}. \end{align*} $ 
where
$ \begin{align*} \dot{V}=\, &[b_{1}+2^{1a}(1a)l^{\frac{1+a}{1a}} \\ &+2^{1a}(12a)(1+a)b^{(1+a)^{1}}_{1}b_{1}m^{\frac{1a}{1+a}}]z^{2}_{1} \\ &+[b_{2}+2^{1a}(1+a)l+2^{2a}(12a)b^{(1+a)^{1}}_{1} \\ &+2^{1a}(12a)(1a)b^{(1+a)^{1}}_{1}b_{1}m]\xi^{2}_{2}. \end{align*} $ 
Direct computations show that, with
$ \begin{equation} \begin{aligned}\label{eq26} a=\frac{2}{9}, b_{1}=1.2, b_{2}=5.4 \\ l=1.8, m=1.3~~~~~~~~ \end{aligned} \end{equation} $  (26) 
there holds
Finitetime Stability Analysis:
Combining Facts 1 and 2, we know that the closedloop system (13), (14) is globally attractive and locally asymptotically stable, while also globally asymptotically stable at the origin in terms of [13].
Moreover, we can claim that the reduced system is finitetime stable. At first, we show that system (13), (25) is homogeneous of order
$ \begin{align}\label{eq27} F=(z_{2}, b_{2}(z^{(1+a)^{1}}_{2}+b^{(1+a)^{1}}_{1}z_{1})^{1+2a})^{{T}} \end{align} $  (27) 
is homogeneous of order
By
Thus, for system (13) we have actually constructed a globally finitetime stabilizing controller, since system (13), (14) is globally asymptotically stable at the origin and in a finite time the states enter a small domain
$ \begin{align*} \Omega=\{(z_{1}, z_{2}):z_{1}\leq \varepsilon, z^{(1+a)^{1}}_{2}+b^{(1+a)^{1}}_{1}z_{1}\leq\varepsilon\} \end{align*} $ 
in which system (13), (14) is finitetime stable.
So far we can have the following result:
Proposition 1: The controller (14) is a saturated finitetime stabilizer of system (13) if the parameter conditions (16), (17) and (26) are fulfilled.
Remark 2: The saturated finitetime stabilizer for system (13) is motivated by the method in [2]. A slight modification to the method in [2] is used: instead of calculating small time intervals for contradiction, we verify the saturation reduction in such a way that the derivatives of saturation functions are calculated in small domains and are proved to be nonpositive (see the Proof of Fact 1).
B. Saturated Finitetime Stabilizer of System (9)In this part, we give the entire expression of the saturated finitetime stabilizer for system (9).
We first give a candidate of controller (14) by computing parameters
$ \begin{align}\label{eq28} b_{1}=1.2, b_{2}=\textrm{max}\{1.6, 5.4\}=5.4. \end{align} $  (28) 
Now a saturated finitetime stabilizer of system (13) is described by
$ \begin{align}\label{eq29} u=5.4\textrm{sat}^{1+2a}_{\varepsilon}(z^{(1+a)^{1}}_{2}+1.2^{(1+a)^{1}}\textrm{sat}_{\varepsilon}(z_{1})). \end{align} $  (29) 
We then invoke Lemma 1 to show that, based on the controller (29), one can have a saturated finitetime stabilizer of (9).
Since
$ \begin{align*} u&=5.4\textrm{sat}^{1+2a}_{\varepsilon}(z^{(1+a)^{1}}_{2}+1.2^{(1+a)^{1}}\textrm{sat}_{\varepsilon}(z_{1})) \\ &\leq5.4(z^{(1+a)^{1}}_{2}+1.2^{(1+a)^{1}}z_{1})^{1+2a} \\ &\leq5.4(z^{(1+a)^{1}1}_{2}z_{2}+1.2^{(1+a)^{1}}z_{1})^{1+2a} \\ &\leq d(\sqrt{z^{2}_{2}+z^{2}_{1}})^{1+2a}. \end{align*} $ 
Namely, the condition (8) of Lemma 1 is fulfilled. Next, it has been proved that the controller (29) is a finitetime stabilizer of system (13). Thus, by Lemma 1, (12) and (29), we obtain a saturated finitetime stabilizer of system (9):
$ \begin{align} \label{eq30} u(t)=\, &5.4\textrm{sat}^{1+2a}_{\varepsilon}(Y^{(1+a)^{1}}_{2}(t) \nonumber\\ &+1.2^{(1+a)^{1}}\textrm{sat}_{\varepsilon}(Y_{1}(t)+hY_{2}(t))) \end{align} $  (30) 
where
Note that
At first, we have
$ \begin{align*} Y&=X+\int^{t}_{th}e^{(tsh)A}bu(s)ds \\ &=X+\int^{t}_{th}\left[ \begin{array}{cc} 1&tsh \\ 0&1 \\ \end{array} \right]\left[ \begin{array}{cc} 0 \\ 1 \\ \end{array} \right]u(s)ds \\ &=\left[ \begin{array}{cc} X_{1}+\int^{t}_{th}(tsh)u(s)ds \\ X_{2}+\int^{t}_{th}u(s)ds \\ \end{array} \right]. \end{align*} $ 
Furthermore, from equations in (9), we have
$ \begin{align*} &\int^{t}_{th}(tsh)u(s)ds \\ &=\int^{t+h}_{t}(tr)u(rh)dr \\ &=\int^{t+h}_{t}(tr)\dot{X}_{2}(r)dr \\ &=(tr)X_{2}(r)^{t+h}_{t}+\int^{t+h}_{t}X_{2}(r)dr \\ &=hX_{2}(t+h)+X_{1}(t+h)X_{1}(t). \end{align*} $ 
Also, we have
$ \begin{align*} \int^{t}_{th}u(s)ds=X_{2}(t+h)X_{2}(t). \end{align*} $ 
Thus, there hold
$ \begin{align} \label{eq31} &Y_{1}(t)=hX_{2}(t+h)+X_{1}(t+h) \nonumber\\ &Y_{2}(t)=X_{2}(t+h). \end{align} $  (31) 
Keeping in mind (30) and (31), we finally obtain the entire expression of the saturated finitetime stabilizer for system (9):
$ \begin{align} \label{eq32} u(t)=&5.4\textrm{sat}^{1+2a}_{\varepsilon}(Y^{(1+a)^{1}}_{2}(t) \nonumber\\ &+1.2^{(1+a)^{1}}\textrm{sat}_{\varepsilon}(Y_{1}(t)+hY_{2}(t))) \nonumber\\ =&5.4\textrm{sat}^{1+2a}_{\varepsilon}(X^{(1+a)^{1}}_{2}(t+h) \nonumber\\ &+1.2^{(1+a)^{1}}\textrm{sat}_{\varepsilon}(X_{1}(t+h))). \end{align} $  (32) 
Moreover, due to time delay, the actual control action will be
$ \begin{align*} u(th)=&5.4\textrm{sat}^{1+2a}_{\varepsilon}(X^{(1+a)^{1}}_{2}(t) \\ &+1.2^{(1+a)^{1}}\textrm{sat}_{\varepsilon}(X_{1}(t))). \end{align*} $ 
In the end, the result in this note is summed up as follows.
Theorem 1: For
Remark 3: In the case where the simple double integrator is considered, the final control design does not contain an integral term.
In the following, we use numerical simulations to show the effectiveness of the algorithm.
Take
$ \begin{align*} (X_{1}(t), X_{2}(t))=(0, 0), t\in[0.2, 0) \\ (X_{1}(0), X_{2}(0))=(1, 1).~~~~~~~~~~ \end{align*} $ 
Simulations in Fig. 2 show that the continuous controller (32) can guarantee that states converge to zero in finite time, although the input is subject to a time delay.
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Fig. 2 Histories of states and input of system (9), (32). 
Using the simulation example, we now explain that the causality requirement is met.
When
$ \begin{align*} u(00.2)&=5.4\textrm{sat}^{\frac{5}{9}}_{1}(X^{\frac{9}{7}}_{2}(0)+ 1.2^{\frac{9}{7}}\textrm{sat}_{1}(X_{1}(0))) \\ &=5.4. \end{align*} $ 
At this time instant,
When
$ \begin{align*} u(t0.2)=5.4\textrm{sat}^{\frac{5}{9}}_{1}(X^{\frac{9}{7}}_{2}(t)+1.2^{\frac{9}{7}}\textrm{sat}_{1}(X_{1}(t))). \end{align*} $ 
Clearly, in the time interval
Finally, it is noted that the controllers may be sensitive to input delay mismatches, since one needs to know the exact input delay when utilizing the suggested control design. Therefore, it would be a significant issue to address the robustness of such controllers.
Ⅳ. SOME DISCUSSIONSIn this section, we make some comments on the work in this note.
Firstly, we explain that searching for a suitable method is crucial to establish our algorithm.
To deal with the double integrator, we have made some attempts but ultimately failed. Then, we search for other solutions and find that the problem is largely simplified once we jointly use the results in [2], [4] and [5].
1) By making the Artstein's transformation [4] and using the result in [5], the concerned problem is reduced to how to obtain a saturated finitetime stabilizer;
2) By drawing inspiration from [2], a saturated finitetime stabilizer including no
Two failed attempts are listed as follows.
Scheme 1:
By introducing the transformation
$ \begin{align*} x_{1}=X_{1}, x_{2}=X_{2}+\int^{t}_{th}u(s)ds \end{align*} $ 
system (9) is described by
$ \begin{equation} \begin{aligned}\label{eq33} &\dot{x}_{1}=x_{2}+\psi, \dot{x}_{2}=u \\ &\psi=\int^{t}_{th}u(s)ds \end{aligned} \end{equation} $  (33) 
for which we assign the saturated control law:
$ \begin{align*} u&=b_{2}\textrm{sat}^{1+2a}_{\varepsilon}(x^{(1+a)^{1}}_{2}x^{(1+a)^{1}}_{2^{*}}) \\ x_{2^{*}}&=b_{1}\textrm{sat}^{1+a}_{\varepsilon}(x_{1}) \end{align*} $ 
where
Transferring delay
$ \begin{align*} u&=b_{2}(x^{(1+a)^{1}}_{2}+b^{(1+a)^{1}}_{1}x_{1})^{1+2a} \\ &\triangleq{}b_{2}(\xi_{2})^{1+2a} \end{align*} $ 
and
$ \begin{align*} \psi=\int^{t}_{th}u(s)ds=b_{2}\int^{t}_{th}\xi^{1+2a}_{2}(s)ds. \end{align*} $ 
Using the function
$ \begin{align*} W_{1}&\leq x^{1a}_{1}(b_{1}x^{1+a}_{1})+x^{1a}_{1}(x_{2}x_{2^{*}}) \\ &~~~+b_{2}x^{1a}_{1}\int^{t}_{th}\xi^{1+2a}_{2}(s)ds \\ &\leqb_{1}x^{2}_{1}+2^{a}x_{1}^{1a}\xi_{2}^{1+a} \\ &~~~+b_{2}x^{1a}_{1}\int^{t}_{th}\xi^{1+2a}_{2}(s)ds \end{align*} $ 
and here we need to specially focus on the term
Noting
$ \begin{align*} x^{1a}_{1}\int^{t}_{th}\xi^{1+2a}_{2}(s)ds&\leqx^{1a}_{1}\int^{t}_{th}\xi^{1+a}_{2}(s)ds \\ &\leq k_{1}hx^{2}_{1}+k_{2}\int^{t}_{th}\xi^{2}_{2}(s)ds, \\ &k_{1}, k_{2}>0. \end{align*} $ 
This estimate is useful for the stability analysis since we finally hope to obtain the dissipative inequality with the form
$ \begin{align*} \dot{V}\leqc_{1}z^{2}_{1}c_{2}\xi^{2}_{2}, c_{1}, c_{2}>0. \end{align*} $ 
But we have to let
$ \begin{align*} x^{1a}_{1}\int^{t}_{th}\xi^{1+2a}_{2}(s)ds&\leqx^{a}_{1}(x^{12a}_{1}\int^{t}_{th}\xi^{1+2a}_{2}(s)ds) \\ &\leq x^{a}_{1}(d_{1}hx^{2}_{1}+d_{2}\int^{t}_{th}\xi^{2}_{2}(s)ds), \\ &d_{1}, d_{2}>0. \end{align*} $ 
The term
Scheme 2:
Use the approach in [8] for system (9) and directly design
$ \begin{align*} u(th)&=b_{2}\textrm{sat}^{1+2a}_{\varepsilon}(X^{(1+a)^{1}}_{2}(th) \\ &~~~X^{(1+a)^{1}}_{2^{*}}(th)) \\ X_{2^{*}}(th)&=b_{1}\textrm{sat}^{1+a}_{\varepsilon}(X_{1}(th)) \end{align*} $ 
where
In this case, delayed terms need to be treated at each step of the saturation reduction analysis and some conservative estimates will be inevitable due to fractional exponents. At the same time, the delayed terms and the fractional exponents will also make it difficult to deal with the reduced system.
Secondly, we briefly discuss a potential extension of the suggested algorithm.
Initial investigation shows it is possible to deal with the
$ \begin{equation} \begin{aligned}\label{eq34} \dot{X}(t)&=\left[ \begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 0&0&0 \\ \end{array} \right]X(t)+\left[ \begin{array}{ccc} 0 \\ 0 \\ 1 \\ \end{array} \right]u(th) \\ &\triangleq{}AX(t)+bu(th). \end{aligned} \end{equation} $  (34) 
By making the Artstein's transformation
$ \begin{equation} \begin{aligned}\label{eq35} Y&=X+\int^{t}_{th}e^{(tsh)A}bu(s)ds \\ &=\left[ \begin{array}{cc} X_{1}+2^{1}\int^{t}_{th}(tsh)^{2}u(s)ds \\ X_{2}+\int^{t}_{th}(tsh)u(s)ds \\ X_{3}+\int^{t}_{th}u(s)ds \\ \end{array} \right] \end{aligned} \end{equation} $  (35) 
system (34) is described by
$ \begin{align*} &\dot{Y}_{1}=Y_{2}+2^{1}h^{2}u \\ &\dot{Y}_{2}=Y_{3}hu \\ &\dot{Y}_{3}=u. \end{align*} $ 
Making another transformation
$ \begin{align}\label{eq36} z_{1}=Y_{1}+hY_{2}+2^{1}h^{2}Y_{3}, z_{2}=Y_{2}+hY_{3}, z_{3}=Y_{3} \end{align} $  (36) 
we finally have
$ \begin{align}\label{eq37} \dot{z}_{1}=z_{2}, \dot{z}_{2}=z_{3}, \dot{z}_{3}=u. \end{align} $  (37) 
Then, it suffices to consider the saturated finitetime stabilization of the equivalent system.
As it can be imagined, we will face a parameter assignment problem in the case of the
In addition, at present, it is not clear whether the algorithm can be extended to some feedforward nonlinear systems. Apart from the problem of choosing control parameters, we have to ensure that the equivalent system has an uppertriangular structure. Moreover, once the equivalent system contains some delayed terms, we will bump into the same problem as in Scheme 1.
Ⅴ. CONCLUSIONFinitetime stabilization, compensation of input saturation, and compensation of input delay are important topics in the control community. In this note, through jointly using the existing approaches, we have for the first time provided a finitetime stabilizing design that compensates both input saturation and input delay. Hopefully, the analysis method in this note might motivate some new thinking.
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