Finite-time Stabilization of the Double Integrator Subject to Input Saturation and Input Delay
  IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(5): 1017-1024   PDF    
Finite-time Stabilization of the Double Integrator Subject to Input Saturation and Input Delay
Huawen Ye, Meng Li, Chunhua Yang, Weihua Gui     
School of Information Science and Engineering, Central South University, Changsha 410083, China
Abstract: The time-optimal control design of the double integrator is extended to the finite-time stabilization design that compensates both input saturation and input delay. With the aid of the Artstein's transformation, the problem is reduced to assigning a saturated finite-time stabilizer.
Key words: Finite-time stabilization     input delay     saturated design    
Ⅰ. INTRODUCTION

As an extension of time-optimal control, Bhat Bernstein proposed a saturated finite-time 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 finite-time stabilizers are presented for multiple integrators and a skillful analysis method was proposed for verifying the saturation reduction.

Constructing finite-time stabilizers for compensating input delays is a significant issue. In this line, profound results were achieved in [3] by means of time-varying distributed delay feedback; with the aid of the so-called Artstein's transformation [4], the mentioned problem has been reduced to solely assigning a finite-time 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, Newton-Leibniz formula and Lyapunov functional (see for instance [6]-[9]).

However, there is currently no finite-time stabilizing design that compensates both input saturation and input delays. The algorithms in [6]-[9] do not take into account the finite-time stability and the classical time-optimal 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 finite-time controllers that compensate both input saturation and input delay.

Moreover, exploration in this direction is of practical significance. As we know, the bounded bang-bang control can bring about finite-time 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 finite-time 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 finite-time 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 finite-time 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 non-positive. Once the saturated controller reduces to a linear one, we invoke the homogeneity based stability theory [1] to analyze the finite-time stability of the reduced system. These efforts allow us to obtain a saturated finite-time stabilizer that can get rid of $\textrm{abs}(\cdot)$ and $\textrm{sign}(\cdot)$ functions.

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 Formulation

The bang-bang time-optimal controller is bounded and can guarantee the finite-time 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 $u(|u|\leq1)$ to minimize $J=\int^{t_{f}}_{0}dt=t_{f}$. Invoking [10], one knows that under the initial condition $y_{1}(0)=y_{2}(0)=1$, the bang-bang control law is described by

$ \begin{align}\label{eq2} u=-\textrm{sgn}(y_{1}+2^{-1}y_{2}|y_{2}|) \end{align} $ (2)

and the shortest transition time $t_{f}=1+\sqrt{6}$ can be calculated directly.

We now suppose that $u$ is subject to an input delay, namely, we will focus on a system like

$ \begin{equation} \begin{aligned}\label{eq3} &\dot{y}_{1}=y_{2}, \dot{y}_{2}=u(t-0.2) \\ &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(t-0.2)=&-\textrm{sgn}[y_{1}(t-0.2)+ \\ &2^{-1}y_{2}(t-0.2)|y_{2}(t-0.2)|] \end{aligned} \end{equation} $ (4)

the histories of states and input will exhibit as follows.

From Fig. 1, it is observed that when $y_{1}$ reaches zero, $y_{2}$ takes a negative value. If system (1) represents the motion of a landing spacecraft, simulations in Fig. 1 show that the spacecraft bumps into the ground. Namely, some serious consequences will be brought about, once the time-optimal controller is subject to a time delay. Therefore, it would be theoretically and practically significant to develop a finite-time stabilization algorithm for compensating both input saturation and input delay; which is exactly what this note addresses.

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Fig. 1 Histories of states and input of system (1), (4).
B. Preliminaries

We first recall the Artstein's transformation and its application in finite-time stabilization of systems subject to input delay (for details the reader is referred to [4], [5]).

Consider the system subject to an input delay $h$

$ \begin{equation} \begin{aligned}\label{eq5} &\dot{x}(t)=Ax(t)+bu(t-h) \\ &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}_{t-h}e^{(t-s-h)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}_{t-h}e^{(t-s-h)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^{(-r-h)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^{(-r-h)A}bu(r+t)dr. \end{align*} $

From Theorem 7 of [5], we have

Lemma 1: If system (7) is finite-time stabilizable by a feedback control

$ \begin{align*} u(t)=k(t)f(y(t)) \end{align*} $

with $k(t)$ bounded and $f:{\mathbb R}^{n}\rightarrow {\mathbb R}$ continuous such that $f(0)=0$ and there is a strictly increasing function $\alpha$ such that

$ \begin{align}\label{eq8} |f(y)|\leq \alpha(\sqrt{y^{2}_{1}+\cdots+y^{2}_{n}}) \end{align} $ (8)

then, system (5) is finite-time stabilizable by the feedback

$ \begin{align*} u(t)=k(t)f(x(t)+\int^{t}_{t-h}e^{(t-s-h)A}bu(s)ds). \end{align*} $

We then recall homogeneity concepts and a related lemma from [11]. Define the dilation $\Delta_{k}(x_{1}, \ldots, x_{2})=(k^{r_{1}}x_{1}, \ldots, k^{r_{n}}x_{n}), k>0$, where $x_{1}, \ldots, x_{n}$ are suitable coordinates on $R^{n}$ and $r_{1}, \ldots, r_{n}$ are positive real numbers.

Definition 1: With respect to the dilation $\Delta_{k}(x_{1}, \ldots, x_{2})$, a vector field $F$ is homogeneous of degree $m$ if and only if the $i$th component $F_{i}$ is homogeneous of degree $r_{i}+m$.

Lemma 2: The origin is a finite-time stable equilibrium of $F$ if and only if the origin is an asymptotically stable equilibrium of $F$ and $m<0$.

This criterion will facilitate the finite-time stability analysis.

Next, we introduce useful inequalities [12]:

1) For a constant $p$ (a ratio of positive odd integers) and for any $x, y\in R$, the following inequalities hold:

$ \begin{align*} ~~~~~~~~~~|x^{p}-y^{p}|\leq2^{1-p}|x-y|^{p}, ~~0<p<1. \end{align*} $ (F1)

2) Let $c$ and $d$ be positive constants. Given any positive number $r>0$, the following inequality holds:

$ \begin{align*} ~~~~~~~~|x|^{c}|y|^{d}\leq\frac{c}{c+d}r|x|^{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*} $
Ⅲ. CONTROL DESIGN AND STABILITY ANALYSIS

Consider the double integrator that is subject to an input delay $h$:

$ \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(t-h) \\ &\triangleq{}AX(t)+bu(t-h). \end{aligned} \end{equation} $ (9)

By making an Artstein's transformation

$ \begin{align}\label{eq10} Y=X+\int^{t}_{t-h}e^{(t-s-h)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 finite-time stabilizer for system (13), and then use Lemma 1 to show that, after substituting variables, this controller will be a globally finite-time stabilizer of system (9).

A. Saturated Finite-time 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 $b_{i}\, (i=1, 2)$ will be determined later.

Remark 1: In [1], the saturated finite-time 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}) \\ &\times|z_{1}+\frac{1}{2-\alpha}\textrm{sign}(z_{2})|z_{2}|^{2-\alpha}|^{\frac{\alpha}{2-\alpha}}) \end{align*} $

with $\alpha\in(0, 1), \varepsilon>0$. Getting rid of functions $\textrm{sign}(\cdot)$ and $|\cdot|$, here we will achieve a new saturated finite-time stabilizer.

In the following, we show that the controller (14) is a finite-time 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 bottom-up recursive manner.

Step 1: Consider the $z_{2}$ subsystem and suppose that $z^{(1+a)^{-1}}_{2}-z^{(1+a)^{-1}}_{2^{*}}\geq \varepsilon$ holds for all $t\geq0$. There holds

$ \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 $|z^{(1+a)^{-1}}_{2^{*}}|\leq b^{(1+a)^{-1}}_{1}\varepsilon$, we obtain

$ \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 $t_{2}$ such that

$ \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 $z^{(1+a)^{-1}}_{2}-z^{(1+a)^{-1}}_{2^{*}}$ at the time instant $t_{2}$. By (15), we have $z^{(1+a)^{-1}-1}_{2}\geq0$; together with (19), we obtain

$ \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 $\frac{d}{dt}z^{(1+a)^{-1}}_{2^{*}}$, we only need to consider the case of $|z_{1}|<\varepsilon$, since $\frac{d}{dt}z^{(1+a)^{-1}}_{2^{*}}=0$ holds for all $|z_{1}|\geq\varepsilon$. For $|z_{1}|<\varepsilon$, $z^{(1+a)^{-1}}_{2^{*}}$ is described by $z^{(1+a)^{-1}}_{2^{*}}=-b^{(1+a)^{-1}}_{1}z_{1}$. We thus have

$ \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 $|z_{2}|$ in the following way:

$ \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 $\overline{t_{1}}$ such that

$ \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 $b_{1}>2^{-a}$, there exists a finite time $t_{1}\, (\geq T_{2}=\max\{t_{2}, \overline{t_{1}}\})$ such that $z_{1}(t)\leq\varepsilon$ holds for all $t\geq t_{2}$. In fact, using (23), (24) and (F1), we first have $|z_{2}-z_{2^{*}}|\leq2^{-a}\varepsilon^{1+a}$; next, for $z_{1}\geq\varepsilon$, we have $z_{2^{*}}=-b_{1}\varepsilon^{1+a}$. Hence, by (17), for $z_{1}\geq\varepsilon$ there holds

$ \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 $\overline{t_{1}}\, (\geq T_{2})$ exists such that $z_{1}(t)\geq\varepsilon$ holds for all $t\geq \overline{t_{1}}$.

Asymptotical Stability Analysis of the Reduced System:

After the time $T_{1}=\max\{t_{1}, \overline{t_{1}}\}$, the controller (14) is reduced to

$ \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 $b_{1}, b_{2}$ such that system (13) with the controller (25) is finite-time stable.

Proof: Fact 2 is proved in three steps.

Step 1: Define the function $W_{1}=(2-a)^{-1}z^{2-a}_{1}$ for the subsystem $\dot{z}_{1}=z_{2}$. Keeping (25) in mind, we obtain

$ \begin{align*} \dot{W}_{1}&=z^{1-a}_{1}z_{2^{*}}+z^{1-a}_{1}(z_{2}-z_{2^{*}}) \\ &\leq -b_{1}z^{1-a}_{1}z^{1+a}_{1}+z^{1-a}_{1}(z_{2}-z_{2^{*}}). \end{align*} $

Letting $\xi_{2}=z^{(1+a)^{-1}}_{2}-z^{(1+a)^{-1}}_{2^{*}}$ and using $(\textrm{F1})$ and $(\textrm{F2})$, we have

$ \begin{align*} \dot{W}_{1}&\leq -b_{1}z^{2}_{1}+2^{-a}z^{1-a}_{1}|z^{(1+a)^{-1}}_{2}-z^{(1+a)^{-1}}_{2^{*}}|^{1+a} \\ &\leq -b_{1}z^{2}_{1}+2^{-a}z^{1-a}_{1}\xi^{1+a}_{2} \\ &\leq -b_{1}z^{2}_{1}+2^{-1-a}(1-a)l^{-\frac{1+a}{1-a}}z^{2}_{1}+2^{-1-a}(1+a)l\xi^{2}_{2} \end{align*} $

where $l>0$ is tunable.

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^{*}})^{1-2a}ds \end{align*} $

and have

$ \begin{align*} \dot{V}&\leq -b_{1}z^{2}_{1}+2^{-1-a}(1-a)l^{-\frac{1+a}{1-a}}z^{2}_{1}+2^{-1-a}(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^{*}})^{1-2a}u. \end{align*} $

To calculate $(\partial W_{2}/\partial z_{1})z_{2}$, we obtain

$ \begin{align*} \frac{\partial W_{2}}{\partial z_{1}} =\, &(1-2a)\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& (1-2a)b^{(1+a)^{-1}}_{1}|z_{2}-z_{2^{*}}||z^{(1+a)^{-1}}_{2}-z^{(1+a)^{-1}}_{2^{*}} |^{-2a} \\ \leq &2^{-a}(1-2a)b^{(1+a)^{-1}}_{1}|\xi_{2}|^{1-a}. \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}(1-2a)b^{(1+a)^{-1}}_{1}\\ &\times|\xi_{2}|^{1-a}(2^{-a}|\xi_{2}|^{1+a}+b_{1}|z_{1}|^{1+a}) \\ &\leq2^{-2a}(1-2a)b^{(1+a)^{-1}}_{1}\xi^{2}_{2} \\ &+2^{-a}(1-2a)b^{(1+a)^{-1}}_{1}b_{1}|\xi_{2}|^{1-a}|z_{1}|^{1+a} \\ &\leq2^{-2a}(1-2a)b^{(1+a)^{-1}}_{1}\xi^{2}_{2} \\ &+2^{-1-a}(1-2a)(1-a)b^{(1+a)^{-1}}_{1}b_{1}m\xi^{2}_{2} \\ &+2^{-1-a}(1-2a)(1+a)b^{(1+a)^{-1}}_{1}b_{1}m^{-\frac{1-a}{1+a}}z^{2}_{1}. \end{align*} $

where $m$ is a tunable positive constant. Noting $u=-b_{2}\xi^{1+2a}_{2}$, we finally have

$ \begin{align*} \dot{V}=\, &[-b_{1}+2^{-1-a}(1-a)l^{-\frac{1+a}{1-a}} \\ &+2^{-1-a}(1-2a)(1+a)b^{(1+a)^{-1}}_{1}b_{1}m^{-\frac{1-a}{1+a}}]z^{2}_{1} \\ &+[-b_{2}+2^{-1-a}(1+a)l+2^{-2a}(1-2a)b^{(1+a)^{-1}}_{1} \\ &+2^{-1-a}(1-2a)(1-a)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 $\dot{V}\leq-0.0514z^{2}_{1}-0.0679\xi^{2}_{2}$, which implies that system (13), (25) is asymptotically stable.

Finite-time Stability Analysis:

Combining Facts 1 and 2, we know that the closed-loop 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 finite-time stable. At first, we show that system (13), (25) is homogeneous of order $a$ with respect to the dilation $\triangle_{\lambda}(z)=(\lambda^{1}z_{1}, \lambda^{1+a}z_{2})$. Since $z_{2}$ and $u=-b_{2}(z^{(1+a)^{-1}}_{2}+b^{(1+a)^{-1}}_{1}z_{1})^{1+2a}$ are homogeneous of degree $1+a$ and $1+2a$, respectively, it follows from Definition 1 that the vector field

$ \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 $a$.

By $a<0$ and Lemma 2, the claim is true.

Thus, for system (13) we have actually constructed a globally finite-time 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 finite-time stable.

So far we can have the following result:

Proposition 1: The controller (14) is a saturated finite-time stabilizer of system (13) if the parameter conditions (16), (17) and (26) are fulfilled.

Remark 2: The saturated finite-time 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 non-positive (see the Proof of Fact 1).

B. Saturated Finite-time Stabilizer of System (9)

In this part, we give the entire expression of the saturated finite-time stabilizer for system (9).

We first give a candidate of controller (14) by computing parameters $a, b_{1}$ and $b_{2}$. Let $a=-2/9$; set $b_{1}=1.2, b_{2}=1.6$ according to (17) and (16) of Fact 1; assign $b_{1}=1.2, b_{2}=5.4$ according to (26) of Fact 2. So we finally set

$ \begin{align}\label{eq28} b_{1}=1.2, b_{2}=\textrm{max}\{1.6, 5.4\}=5.4. \end{align} $ (28)

Now a saturated finite-time 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 finite-time stabilizer of (9).

Since $z_{1}$ and $z_{2}$ are proved to be bounded and there holds $(1+a)^{-1}>1$, there is a certain $d>0$ such that

$ \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 finite-time stabilizer of system (13). Thus, by Lemma 1, (12) and (29), we obtain a saturated finite-time 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 $Y_{1}, Y_{2}$ are given in (10).

Note that $Y_{1}, Y_{2}$ contain integral terms and involve the history of the control signals. Interestingly, by further using equations in (9), the expression of the final controller will not include integral terms.

At first, we have

$ \begin{align*} Y&=X+\int^{t}_{t-h}e^{(t-s-h)A}bu(s)ds \\ &=X+\int^{t}_{t-h}\left[ \begin{array}{cc} 1&t-s-h \\ 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}_{t-h}(t-s-h)u(s)ds \\ X_{2}+\int^{t}_{t-h}u(s)ds \\ \end{array} \right]. \end{align*} $

Furthermore, from equations in (9), we have

$ \begin{align*} &\int^{t}_{t-h}(t-s-h)u(s)ds \\ &=\int^{t+h}_{t}(t-r)u(r-h)dr \\ &=\int^{t+h}_{t}(t-r)\dot{X}_{2}(r)dr \\ &=(t-r)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}_{t-h}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 finite-time 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(t-h)=&-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 $a=-2/9$ and any $h, \varepsilon>0$, the controller (32) globally finite-time stabilizes system (9) if the parameter conditions (16), (17) and (26) are fulfilled.

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 $a=-2/9, \varepsilon=1, h=0.2$. Run simulations for system (9), (32) under the initial conditions

$ \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.

Download:
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 $t=0$, the control action is taken as

$ \begin{align*} u(0-0.2)&=-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, $u(0-0.2)$ has an explicit value that is determined from $(X_{1}(0), X_{2}(0))=(1, 1)$.

When $0<t\leq0.2$, the control action is taken as

$ \begin{align*} u(t-0.2)=-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 $(0, 0.2]$, $u(t-0.2)$ can always be determined from the input value at the previous instant. Therefore, it is safe to say that the causality requirement is met.

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 DISCUSSIONS

In 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 finite-time stabilizer;

2) By drawing inspiration from [2], a saturated finite-time stabilizer including no $\textrm{abs}(\cdot)$ and $\textrm{sign}(\cdot)$ functions can be explicitly constructed.

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}_{t-h}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}_{t-h}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 $0<\varepsilon<1, a=-2k_{1}/(2k_{2}+1), 1\leq k_{1}<k_{2}$, and multiplying coefficients $b_{i}\, (i=1, 2)$ are to be determined.

Transferring delay $h$ into the perturbed term $\psi$, we hope that the problem is simplified and only the saturated finite-time stabilization problem needs to be considered. But we find that the finite-time stability of the reduced system is hard to prove. In fact, due to the perturbed term $\psi$ and particularly the negative parameter $a$, it is hard to choose suitable $b_{i}\, (i=1, 2)$ to ensure the finite-time stability of the reduced system. A more detailed explanation is given as follows. Suppose that the saturation reduction is already verified and there hold

$ \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}_{t-h}u(s)ds=b_{2}\int^{t}_{t-h}\xi^{1+2a}_{2}(s)ds. \end{align*} $

Using the function $W_{1}=(2-a)^{-1}x^{2-a}_{1}$ for the $x_{1}$ subsystem of (33), and $x_{2^{*}}=-b_{1}x^{1+a}_{1}$, we obtain

$ \begin{align*} W_{1}&\leq x^{1-a}_{1}(-b_{1}x^{1+a}_{1})+x^{1-a}_{1}(x_{2}-x_{2^{*}}) \\ &~~~+b_{2}|x^{1-a}_{1}|\int^{t}_{t-h}|\xi^{1+2a}_{2}(s)|ds \\ &\leq-b_{1}x^{2}_{1}+2^{-a}|x_{1}|^{1-a}|\xi_{2}|^{1+a} \\ &~~~+b_{2}|x^{1-a}_{1}|\int^{t}_{t-h}|\xi^{1+2a}_{2}(s)|ds \end{align*} $

and here we need to specially focus on the term $|x^{1-a}_{1}|\int^{t}_{t-h}|\xi^{1+2a}_{2}(s)|ds$.

Noting $|\xi_{2}|\leq\varepsilon<1$, if $a>0$ we will have

$ \begin{align*} |x^{1-a}_{1}|\int^{t}_{t-h}|\xi^{1+2a}_{2}(s)|ds&\leq|x^{1-a}_{1}|\int^{t}_{t-h}|\xi^{1+a}_{2}(s)|ds \\ &\leq k_{1}hx^{2}_{1}+k_{2}\int^{t}_{t-h}\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}\leq-c_{1}z^{2}_{1}-c_{2}\xi^{2}_{2}, c_{1}, c_{2}>0. \end{align*} $

But we have to let $a<0$ for finite-time stability. Thus, if we make a computation in the following way

$ \begin{align*} |x^{1-a}_{1}|\int^{t}_{t-h}|\xi^{1+2a}_{2}(s)|ds&\leq|x^{a}_{1}|(|x^{1-2a}_{1}|\int^{t}_{t-h}|\xi^{1+2a}_{2}(s)|ds) \\ &\leq |x^{a}_{1}|(d_{1}hx^{2}_{1}+d_{2}\int^{t}_{t-h}\xi^{2}_{2}(s)ds), \\ &d_{1}, d_{2}>0. \end{align*} $

The term $|x^{a}|(|x^{a}|\geq\varepsilon^{a}>1)$ will become an obstacle to take suitable $b_{i}\, (i=1, 2)$. Besides, other computational manners will give rise to the same problem.

Scheme 2:

Use the approach in [8] for system (9) and directly design

$ \begin{align*} u(t-h)&=-b_{2}\textrm{sat}^{1+2a}_{\varepsilon}(X^{(1+a)^{-1}}_{2}(t-h) \\ &~~~-X^{(1+a)^{-1}}_{2^{*}}(t-h)) \\ X_{2^{*}}(t-h)&=-b_{1}\textrm{sat}^{1+a}_{\varepsilon}(X_{1}(t-h)) \end{align*} $

where $0<\varepsilon<1, a=-2k_{1}/(2k_{2}+1), 1\leq k_{1}<k_{2}$, the control parameters $b_{i}\, (i=1, 2)$ are to be determined.

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 $n\text{-}th$ order integrator, since we can still reduce the problem into how to assign saturated finite-time stabilizers. As an example, let us consider the triple integrator that is subject to an input delay $h$:

$ \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(t-h) \\ &\triangleq{}AX(t)+bu(t-h). \end{aligned} \end{equation} $ (34)

By making the Artstein's transformation

$ \begin{equation} \begin{aligned}\label{eq35} Y&=X+\int^{t}_{t-h}e^{(t-s-h)A}bu(s)ds \\ &=\left[ \begin{array}{cc} X_{1}+2^{-1}\int^{t}_{t-h}(t-s-h)^{2}u(s)ds \\ X_{2}+\int^{t}_{t-h}(t-s-h)u(s)ds \\ X_{3}+\int^{t}_{t-h}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 finite-time stabilization of the equivalent system.

As it can be imagined, we will face a parameter assignment problem in the case of the $n$th order integrator. In fact, we now have to use the contradiction method in [2] to do the saturation reduction analysis. As a consequence, some multiplying coefficients will become quite large as the system dimension increases. Likewise, the multiplying coefficients needed in treating the reduced system will also be too large.

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 upper-triangular structure. Moreover, once the equivalent system contains some delayed terms, we will bump into the same problem as in Scheme 1.

Ⅴ. CONCLUSION

Finite-time 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 finite-time 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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