Keywords

1 Introduction

Switched systems [1, 2], consisting of a family of subsystems and a switching rule that orchestrates the switching between them, have been used to model many physical or man-made systems displaying switching features. The diverse switching signals differentiate switched systems from general time-varying systems, since the solutions of the switched systems are dependent on not only the system’s initial conditions but also the switching signals. This class of systems have numerous applications in the control of mechanical systems, the automotive industry, air traffic control, switching power converters and many other fields [2].

In switched systems, each subsystem is called a mode, and control problems are said to design a set of mode-dependent controllers or mode-independent controllers for the unforced system and find admissible switching signals such that the resulting systems is stable or satisfies certain performance criteria [2,3,4]. As we know, mode-dependent controller design is less conservative. However, for the control problem, it inevitably takes some time to identify the system modes and apply the matched controller. So, a very common assumption in the mode-dependent context, the controllers are switched synchronously with the switching of system modes, is quite unpractical. Therefore, the asynchronous phenomena between the system modes and the controller modes always exists. Recently, some efforts have been made to study asynchronous control problems [5,6,7,8,9]. In [5,6,7,8], each subsystem is stable. In [5], desirable controller is designed such that the energy function is decreasing in each switching interval (both mismatched period and matched period). This requirement is weakened in [6,7,8]. The energy function is not required decreasing in mismatched period any more. Most recently, in [9], the authors deal with asynchronous stabilization problem of switched system, which contains stable and unstable subsystems. However, the condition \(\inf _{t\ge t_0}[\frac{T^-(t)}{T^+(t)}]\ge -\frac{\beta }{\alpha }\) can not guarantee the condition \(-\gamma t=T^-(t)\alpha +T^+(t)\beta \) holds, which only can guarantee \(T^-(t)\alpha +T^+(t)\beta <0\) holds, where, \(T^-(t)\) and \(T^+(t)\) represent, respectively, the total active time of subsystems that are stable, not stable subsystems over (0, t); \(\alpha \), \(\beta \) and \(\gamma \) are constants. Therefore, a switching law is need.

In this paper, the problem of asynchronous control of switched nonlinear systems is studied. By using the piecewise Lyapunov function and average dwell time approach, the asynchronously switched stabilizing control problem for nonlinear systems is solved under the proposed switching law , which allows us to have stable and unstable subnonlinear system. Some examples are provided to show the effectiveness of the results.

2 Problem Descriptions and Preliminaries

Consider the following switched nonlinear system:

$$\begin{aligned} \dot{x}(t)=f_{\sigma (t)}(x(t),u(t)), \end{aligned}$$
(1)

where \(x(t)\in R^n\) is a state vector and \(u(t)\in R^m\) is a control input vector. \(f_{\sigma (t)}\) are a set of regularly nonlinear functions. \(\sigma (t):[0,\infty )\rightarrow \mathbb {S}\) is the switching signal, i.e., \(\sigma (t)=i_k\in \mathbb {S}\) for \(t\in [t_k,t_{k+1})\), where \(t_k\) is the kth switching time instant, \(\mathbb {S}=\{1,2,\ldots ,s\}\), \(s,k\in \mathbb {N}\). \(0=t_0<t_1< \ldots< t_k < \ldots \), \(\displaystyle {\lim _{k\rightarrow \infty }}t_k=\infty \), which can rule out Zeno behavior automatically.

In fact, for the control problem, it inevitably takes some time to identify the system modes and apply the matched controller. Therefore, the asynchronous phenomena between the system modes and the controller modes always exists. In this paper, we assume that the time lag of controllers modes to system modes is \(t_d>0\) \((t_d<t_{k+1}-t_k, k\in N)\). The state feedback control input can be written as \(u(t)=K_{\sigma (t-t_d)}x(t)\).

Before proceeding further, the following definitions are introduced.

Definition 2.1

[10] For a switching signal \(\sigma (t)\) and any \(t^{''}>t^{'}>t_0\), let \(N_{\sigma }(t^{'},t^{''})\) be the switching numbers of \(\sigma (t)\) over the interval \([t^{'},t^{''})\). If \(N_{\sigma }\!(t^{'},t^{''})\!\!\,\le \,\!\! N_0\!\!\,+\,\!\!\frac{t^{''}\,-\,t^{'}}{\tau _a}\) holds for \(N_0\ge 1\), \(\tau _a>0\), then \(N_0\) and \(\tau _a\) are called the chatter bound and the average dwell time, respectively.

Note that: When the active subsystems are changed at some time instant, a switching happens. Therefore, switching numbers mean the total numbers of switching.

Definition 2.2

[6] The equilibrium point of system (1) is globally uniformly exponentially stable under certain switching signals \(\sigma (t)\) if, for u(t), there exist constants \(K>0\) and \(\delta >0\) such that the solution of the system satisfies \(\Vert x(t)\Vert \le Ke^{-\delta (t-t_0)}\Vert x(t_0)\Vert \), \(\forall t\ge t_0\).

3 Main Results

In this section, we first proposed a switching law for the system (1). Under this switching law, the sufficient condition is given to guarantee the system (1) without control input is exponentially stable by using average dwell time. Second, the obtained result is extended to the system with control input.

3.1 Exponential Stability for the System (1) Without Control Input

Switching law 3.1

[11] Let \(0\!=\!t^0\!<\!t^1\!<\!t^2\!<\!\ldots (\lim _{j\rightarrow \infty }t^j\!=\!\infty )\) be a specified sequence of time instants satisfying \(\sup _j\{t^{j\!+\!1}\!-\!t^j\}\!=\!T\!<\!\infty \). Determine the switching signal \(\sigma (t)\) such that the inequality \(T^-\!(t^j,t^{j\!+\!1})\!/T^+\!(t^j,t^{j\!+\!1})\!\ge \!-\!(\beta +\alpha ^*)/(\alpha +\alpha ^*)\) holds on every time interval \([t^j,t^{j+1})(j=0,1,\ldots )\), where \(0<\alpha ^*<-\alpha \), \(\alpha \) and \(\beta \) are given constants, \(T^-\!(t^j,t^{j\!+\!1})\) and \(T^+\!(t^j,t^{j\!+\!1})\) denote the total active time of stable and unstable subsystems respectively over \((t^j,t^{j\!+\!1})\).

Based on the given switching law 3.1, the following theorem is presented to guarantee the system is exponentially stable.

Theorem 3.1

For the given scalars \(\alpha _{\sigma (t)}\) and \(\mu \ge 1\), the system (1) with \(u(t)=0\), under the switching law 3.1, is exponentially stable if there exist Lyapunov functions \(V_{\sigma (t)}(t):R^n\rightarrow R\), and two positive constants \(K_1\) and \(K_2\) such that \(\forall \sigma (t)=i\in \mathbb {S}\) the following inequalities hold

$$\begin{aligned}&K_1\Vert x(t)\Vert ^2\le V_i(t)\le K_2\Vert x(t)\Vert ^2, \end{aligned}$$
(2)
$$\begin{aligned}&\dot{V}_i(t)\le \alpha _iV_i(t), t\in [t_k,t_{k+1})\end{aligned}$$
(3)
$$\begin{aligned}&V_{\sigma (t_k)}(t_k)\le \mu V_{\sigma (t_k^-)}(t_k^-),\end{aligned}$$
(4)
$$\begin{aligned}&\tau _a> \frac{\ln \mu }{\alpha ^*}. \end{aligned}$$
(5)

Proof

When \(\forall t\in [t_k,t_{k+1}),\) for \(\sigma (t)=i\in \mathbb {S}\), \(k\in N\), it means the switched system is active within the ith subsystem. From (3) and (4), we have

$$\begin{aligned} V_{\sigma (t)}(t)&\le e^{\alpha _i(t-t_k)}V_{\sigma (t_k)}(t_k)\nonumber \\&\le \mu e^{\alpha _i(t-t_k)}V_{\sigma (t_k^-)}(t_k^-)\nonumber \\&\le \mu e^{\alpha _i(t-t_k)+\alpha _{\sigma (t_{k-1})}(t_k-t_{k-1})}V_{\sigma (t_{k-1})}(t_{k-1})\nonumber \\&\le \mu ^2e^{\alpha _i(t-t_k)+\alpha _{\sigma (t_{k-1})}(t_k-t_{k-1})}V_{\sigma (t_{k-1}^-)}(t_{k-1}^-)\nonumber \\&\le ...\nonumber \\&\le \mu ^{N_\sigma (t_0,t)}V_{\sigma (t_0)}(t_0)e^{\alpha T^-(t_0,t)+\beta T^+(t_0,t)}, \end{aligned}$$
(6)

where \(\alpha =\sup _{i\in \mathbb {S}}\{\alpha _i:\alpha _i<0\}\), \(\beta =\sup _{i\in \mathbb {S}}\{\alpha _i:\alpha _i\ge 0\}\), \(T^-\!(t_0,t)\) and \(T^+\!(t_0,t)\) denote the total active time of those subsystems that are stable, not stable subsystems over \((t_0,t)\), respectively.

Suppose \(0\!=\!t^0\!<\!t^1\!<\!t^2\!<\!\ldots (\lim _{j\rightarrow \infty }t^j\!=\!\infty )\) be a specified sequence of time instants satisfying Switching law 3.1. For any t, we have two cases:

  1. (1)

    For \(t_0\) and t satisfying \(t^{j-1}<t_0\le t^{j}<t^{j+1}<\ldots <t^k\le t\), one has

    $$\begin{aligned}&\frac{T^-\!(t^j,t^{j\!+\!1})}{T^+\!(t^j,t^{j\!+\!1})}\ge \!-\frac{\beta +\alpha ^*}{\alpha +\alpha ^*}\nonumber \\ \Rightarrow&T^-\!(t^j,t^{j\!+\!1})(-\alpha -\alpha ^*)\ge T^+\!(t^j,t^{j\!+\!1})(\beta +\alpha ^*)\nonumber \\ \Rightarrow&-\alpha ^*(T^-\!(t^j,t^{j\!+\!1})+T^+\!(t^j,t^{j\!+\!1}))\ge \alpha T^-\!(t^j,t^{j\!+\!1})+\beta T^+\!(t^j,t^{j\!+\!1})\nonumber \\ \Rightarrow&-\alpha ^*(t^{j\!+\!1}-t^j)\ge \alpha T^-\!(t^j,t^{j\!+\!1})+\beta T^+\!(t^j,t^{j\!+\!1}), \end{aligned}$$
    (7)
    $$\begin{aligned}&\frac{T^-\!(t^j,t^{j\!+\!1})}{T^+\!(t^j,t^{j\!+\!1})}\ge \!-\frac{\beta +\alpha ^*}{\alpha +\alpha ^*}\nonumber \\ \Rightarrow&T^-\!(t^j,t^{j\!+\!1})(-\alpha -\alpha ^*)\ge T^+\!(t^j,t^{j\!+\!1})(\beta +\alpha ^*)\nonumber \\ \Rightarrow&T^-\!(t^j,t^{j\!+\!1})(-\alpha -\alpha ^*)+T^+\!(t^j,t^{j\!+\!1})(-\alpha -\alpha ^*)\nonumber \\&\ge T^+\!(t^j,t^{j\!+\!1})(\beta +\alpha ^*)+T^+\!(t^j,t^{j\!+\!1})(-\alpha -\alpha ^*)\nonumber \\ \Rightarrow&T^+\!(t^j,t^{j\!+\!1})\le \frac{-\alpha -\alpha ^*}{\beta -\alpha }T, \end{aligned}$$
    (8)

    and whether or not the activated subsystems over the interval \([t_0,t^j]\) and \([t^k,t]\) are stable subsystems, we consider that the activated subsystems over the interval \([t_0,t^j]\) and \([t^k,t]\) are unstable subsystems. Then, one obtain

    $$\begin{aligned} e^{\alpha T^-\!(t_0,t)\!+\!\beta T^+\!(t_0,t)}\!&\le \!e^{\beta (t^j-t_0)+\sum _{q=j}^{k-1}[\beta T^+\!(t^q,t^{q+1})+\alpha T^-\!(t^q,t^{q+1})]+\beta (t-t^k)} \end{aligned}$$

    According to (7) and (8), one obtains

    $$\begin{aligned} e^{\alpha T^-\!(t_0,t)\!+\!\beta T^+\!(t_0,t)}&\le e^{\beta (t-t^k)-\alpha ^*\sum _{q=j}^{k-1}(t^{q+1}-t^q)+\beta (t^j-t_0)}\nonumber \\&=e^{\beta (t-t^k)-\alpha ^*(t^k-t^j)+\beta (t^j-t_0)}\nonumber \\&=e^{(\beta +\alpha ^*)(t-t^k+t^j-t_0)-\alpha ^*(t-t_0)}\nonumber \\&\le e^{(\beta +\alpha ^*)(T^+\!(t^k,t^{k+1})+T^+\!(t^{j-1},t^{j}))-\alpha ^*(t-t_0)}\nonumber \\&\le e^{\gamma -\alpha ^*(t-t_0)}, \end{aligned}$$
    (9)

    where \(\gamma =\frac{-2(\beta +\alpha ^*)(\alpha +\alpha ^*)}{\beta -\alpha }T\).

  2. (2)

    For \(t_0\) and t satisfying \(t^q\le t_0<t\le t^{q+1}\), one has

    $$\begin{aligned} e^{\alpha T^-\!(t_0,t)\!+\!\beta T^+\!(t_0,t)}&\le e^{\beta (t-t_0)}\nonumber \\&=e^{(\beta +\alpha ^*)(t-t_0)-\alpha ^*(t-t_0)}\nonumber \\&\le e^{\gamma -\alpha ^*(t-t_0)}, \end{aligned}$$
    (10)

    where \(\gamma \) has the same value as the one above.

Based on (6), (9) and (10) and average dwell time, for any t, the following inequality holds

$$\begin{aligned} V_{\sigma (t)}(t)&\le \mu ^{N_\sigma (t_0,t)}V_{\sigma (t_0)}(t_0)e^{\alpha T^-(t_0,t)+\beta T^+(t_0,t)}\nonumber \\&\le e^{N_0\ln \mu +\gamma }e^{(\frac{\ln \mu }{\tau _a}-\alpha ^*)(t-t_0)}V_{\sigma (t_0)}(t_0). \end{aligned}$$
(11)

From (2) and (5), we have

$$\begin{aligned} \Vert x(t)\Vert \le Ke^{-\kappa (t-t_0)}\Vert x(t_0)\Vert , \end{aligned}$$
(12)

where \(K=(\frac{K_2}{K_1}e^{N_0\ln \mu +\gamma })^{1/2}\) and \(\kappa =\frac{1}{2}(\alpha ^*-\frac{\ln \mu }{\tau _a})\).

Therefore, the system (1) without control input is exponentially stable.

3.2 Exponential Stability for the System (1) With Control Input

Switching law 3.2

Let \(0=t^0<t^1<t^2<\ldots (\lim _{j\rightarrow \infty }t^j=\infty )\) be a specified sequence of time instants satisfying \(\sup _j\{t^{j\!+\!1}\!-\!t^j\}\!=\!T\!<\!\infty \). Determine the switching signal \(\sigma (t)\) such that the inequality \((T^-\!(t^j,t^{j\!+\!1})-\!N_{\sigma }^-(t^j,t^{j\!+\!1})t_d)\!/(T^+\!(t^j,t^{j\!+\!1})+\!N_{\sigma }^-(t^j,t^{j\!+\!1})t_d)\!\ge \!-\!(\beta +\alpha ^*)/(\alpha +\alpha ^*)\) holds on every time interval \([t^j,t^j+1)(j=0,1,...)\), where \(0<\alpha ^*<-\alpha \), \(\alpha \) and \(\beta \) are given constants. \(T^-\!(t^j,t^{j\!+\!1})\), \(T^+\!(t^j,t^{j\!+\!1})\) and \(N_{\sigma }^-(t^j,t^{j\!+\!1})\) denote the total active time of those subsystems that are stable, not stable subsystems and the total switching numbers of stable subsystems over \((t^j,t^{j\!+\!1})\), respectively.

Based on the given switching law 3.2, the following theorem is presented to guarantee the system is exponentially stable.

Theorem 3.2

For the given scalars \(\alpha _{\sigma (t),\sigma (t-t_d)}\) and \(\mu \ge 1\), the system (1) with \(u(t)=K_{\sigma (t-t_d)}x(t)\), under the switching law 3.2, is exponentially stable if there exist Lyapunov functions \(V_{\sigma (t),\sigma (t-t_d)}(t):R^n\rightarrow R\), and two positive constants \(\hat{K}_1\) and \(\hat{K}_2\) such that \(\forall \sigma (t)=i\), \(\sigma (t-t_d)=p\), \(\forall i,p\in \mathbb {S}\) the following inequalities hold

$$\begin{aligned}&\hat{K}_1\Vert x(t)\Vert ^2\le V_{i,p}(t)\le \hat{K}_2\Vert x(t)\Vert ^2, \end{aligned}$$
(13)
$$\begin{aligned} \dot{V}_{i,p}(t)\le \left\{ \begin{aligned}&\alpha _{i,p}V_{i,p}(t), \quad t\in [t_k,t_{k}+t_d),i\ne p, \\&\alpha _{i,i}V_{i,i}(t), \quad t\in [t_k+t_d,t_{k+1}),i=p,\end{aligned} \right. \end{aligned}$$
(14)
$$\begin{aligned}&V_{\sigma (t_k),\sigma (t_k-t_d)}(t_k)\le \hat{\mu } V_{\sigma (t_k^-),\sigma (t_k^--t_d)}(t_k^-),\end{aligned}$$
(15)
$$\begin{aligned}&V_{\sigma (t_k+t_d),\sigma (t_k)}(t_k+t_d)\le \hat{\mu } V_{\sigma (t_k^-+t_d),\sigma (t_k^-)}(t_k^-+t_d),\end{aligned}$$
(16)
$$\begin{aligned}&\tau _a>\frac{2\ln \hat{\mu }}{\alpha ^*}. \end{aligned}$$
(17)

Proof

When \(\forall t\in [t_k+t_d,t_{k+1})\), \(\sigma (t)=i\in \mathbb {S}\); \(\forall t\in [t_k,t_k+t_d)\), \(\sigma (t-t_s)=p\in \mathbb {S}\), \(k\in N\). From (14), (15) and (16), we have

$$\begin{aligned} V_{\sigma (t),\sigma (t-t_d)}(t)&\le e^{\alpha _{i,i}(t-t_k-t_d)}V_{\sigma (t_k+t_d),\sigma (t_k)}(t_k+t_d)\nonumber \\&\le \hat{\mu } e^{\alpha _{i,i}(t-t_k-t_d)}V_{\sigma (t_k^-+t_d),\sigma (t_k^-)}(t_k^-+t_d)\nonumber \\&\le \hat{\mu } e^{\alpha _{i,i}(t-t_k-t_d)+\alpha _{i,p}t_d}V_{\sigma (t_k),\sigma (t_k-t_d)}(t_k)\nonumber \\&\le \hat{\mu } ^2 e^{\alpha _{i,i}(t-t_k-t_d)+\alpha _{i,p}t_d}V_{\sigma (t_k^-),\sigma (t_k^--t_d)}(t_k^-)\nonumber \\&\le ...\nonumber \\&\le \hat{\mu } ^{2N_{\sigma }(t_0,t)}e^{\alpha (T^-(t_0,t)-N_{\sigma }^-(t_0,t)t_d)+\beta (T^+(t_0,t)+N_{\sigma }^-(t_0,t)t_d)}\times \nonumber \\&V_{\sigma (t_0),\sigma (t_0-t_d)}V(t_0), \end{aligned}$$
(18)

where \(\alpha =\sup _{i\in \mathbb {S}}\{\alpha _{i,i},\alpha _{i,i}<0\}\), \(\beta =\sup _{i,p\in \mathbb {S}, i\ne p}\{\alpha _{i,p},\alpha _{i,i}>0\}\), \(T^-\!(t_0,t)\), \(T^+\!(t_0,t)\) and \(N_{\sigma }^-(t_0,t)\) denote the total active time of those subsystems that are stable, not stable subsystems and the total switching numbers of stable subsystems over \((t_0,t)\), respectively.

Combining Switching law 3.2 and following the similar proof procedure, we can conclude that the system (1) with \(u(t)=K_{\sigma (t-t_d)}x(t)\), under the switching law 3.2, is exponentially stable.

4 Numerical Examples

Example 1

Consider the following switched nonlinear system without control input

Switching Region 1: \(\sigma (t)=1\)

$$\begin{aligned}&\dot{x}_1(t)\!=\!0.2x_1(t)\!+\!0.1x_2(t)\!-\!0.15|sin(10x_2(t))|e^{-sin(10x_2(t))}x_2(t)\nonumber \\&\dot{x}_2(t)\!=\!0.7x_1(t)\!+\!0.02x_2(t) \end{aligned}$$
(19)

Switching Region 2: \(\sigma (t)=2\)

$$\begin{aligned}&\dot{x}_1(t)=-x_1(t)\nonumber \\&\dot{x}_2(t)=-0.2|cos(10x_1(t))|x_1(t)-x_2(t) \end{aligned}$$
(20)

Here, let \(\alpha _1=1.5\), \(\alpha _2=-0.6\), \(\alpha ^*=0.3\) and \(\mu =1.2\). According to Switching law 3.1, for \(t\in [0, 12]\), the switching signal \(\sigma (t)\) is given as follows:

$$\begin{aligned}&\sigma (t)=1:\quad t\in [0,0.3),\quad [2.9,3.4),\quad [6.8,7.1),\quad [8.9,9.4), \\&\sigma (t)=2:\quad t\in [0.3,2.9),\quad [3.4,6.8),\quad [7.1,8.9),\quad [9.4,12], \end{aligned}$$

where the specified sequence of time instants \(\{t_n\}_{n=0}^{4}\) is given as \(\{0,3,6,9,12\}\). Note that the condition \(1.5=\tau _a\ge \frac{\ln \mu }{\alpha ^*}=0.6077\) also holds. The simulation results are shown in Figs. 1 and 2, which well illustrate Theorem 3.1.

Fig. 1
figure 1

The switching signal \(\sigma (t)\)

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Fig. 2
figure 2

The state trajectories of the system (3)

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Example 2

Consider the following switched nonlinear system

Switching Region 1: \(\sigma (t)=1\)

$$\begin{aligned} \dot{x}_1(t)&=x_1(t)\!+\!0.1x_2(t)\!+\!0.15|sin(6x_1(t))|e^{-sin(6x_1(t))}x_2(t)\nonumber \\&\quad +(0.1+\frac{0.4}{e}|sin(6x_1(t))|e^{-sin(6x_1(t))})u_1(t)\nonumber \\ \dot{x}_2(t)&=0.7x_1(t)\!+\!0.2x_2(t), \end{aligned}$$
(21)

Switching Region 2: \(\sigma (t)=2\)

$$\begin{aligned}&\dot{x}_1(t)=0.6x_1(t)+0.3u_1(t)+0.7|cos(6x_2(t))|u_1(t)\nonumber \\&\dot{x}_2(t)=0.2|cos(10x_2(t))|x_1(t)-0.7x_2(t), \end{aligned}$$
(22)

Here, let \(\alpha _{11}=0.2\), \(\alpha _{12}=0.5\), \(\alpha _{21}=0.1\), \(\alpha _{22}=-0.3\), \(\alpha ^*=0.2\), \(\mu =1.1\) and \(t_d=0.2\). According to Switching law 3.2, for \(t\in [0,12]\), the switching signal \(\sigma (t)\) and \(\sigma (t-t_d)\) are given as follows:

$$\begin{aligned}&\sigma (t)=1:\quad t\in [0,0.3),\quad [2.9,3.4),\quad [6.8,7.1),\quad [9.0,9.4), \\&\sigma (t)=2:\quad t\in [0.3,2.9),\quad [3.4,6.8),\quad [7.1,9.0),\quad [9.4,12], \end{aligned}$$
$$\begin{aligned}&\sigma (t-t_d)=1:\quad t\in [0,0.5),\quad [3.1,3.6),\quad [7.0,7.3),\quad [9.2,9.6), \\&\sigma (t-t_d)=2:\quad t\in [0.5,3.1),\quad [3.6,7.0),\quad [7.3,9.2),\quad [9.6,12], \end{aligned}$$

where the specified sequence of time instants \(\{t_n\}_{n=0}^{4}\) is given as \(\{0,3,6,9,12\}\). Note that the condition \(1.5=\tau _a\ge \frac{2\ln \mu }{\alpha ^*}=0.9532\) also holds. The simulation results are shown in Figs. 3 and 4, which well illustrate Theorem 3.2.

Fig. 3
figure 3

The switching signal \(\sigma (t)\)

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Fig. 4
figure 4

The state trajectories of the system (3)

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5 Conclusion

In this paper, the piecewise Lyapunov function and average dwell time approach have been used to investigate the problem of asynchronous control of switched nonlinear systems. By using the proposed switching law, the asynchronously switched stabilizing control problem for nonlinear systems has been solved, which allows us to have stable and unstable subnonlinear system. Illustrative examples have been provided to show the effectiveness of the results.