Skip to main content

Exponential stability of nonlinear systems with impulsive effects and disturbance input

Abstract

In this paper, exponential stability of nonlinear systems with impulse time window, disturbance input and bounded gain error is investigated. By means of the above result and the construction of a linear stabilizing feedback controller, another criterion of exponential stability is established. A numerical example is given to demonstrate the effectiveness of the theoretical results.

Introduction

Customarily, \(R_{+}\) denotes the set of positive real numbers. \(R^{n}\) is an n-dimensional real Euclidean space with the norm \(\Vert \cdot \Vert \). \(R^{m \times n}\) refers to the set of all \(m \times n\)-dimensional real matrices. \(\lambda _{M} ( A ) \), \(\lambda _{m} ( A ) \), \(A^{T} \), and \(A^{-1}\) are the maximum, the minimum eigenvalue, the transpose, and the inverse of matrix A, respectively. I represents the identity matrix with proper dimension. The positive definite matrix A is represented by \(A > 0\). Define \(f ( {x ( {b^{-} } )} ) = \lim_{t \to b^{-} } f ( {x ( t )} )\).

Over the past two decades, nonlinear systems have been paid considerable attention because many systems in many practical applications can be modeled by nonlinear systems, for instance, robotics, information science, artificial intelligence, automatic control systems, and so forth [8, 14, 15, 17, 24]. Due to impulsive effects, the stability of systems will become oscillations and instability. Therefore, it is significant to discuss stability of nonlinear systems with impulsive effects [9, 10, 19, 21, 22]. In recent years, many sufficient criteria on the asymptotic stability for impulsive control of nonlinear systems have been published under some conditions [1, 16]. We consider not only the asymptotic satiability of the nonlinear impulsive control systems but other aspects in the design of nonlinear impulsive control systems. In particular, it is often desirable that nonlinear impulsive control systems converge fast enough in order to reach fast response. Obviously, exponential stability is a fast convergence rate to the equilibrium point [7, 11, 13].

Many scholars just assume that impulses occur at fixed-time points [12, 18, 20]. However, in many practical applications, impulses occur stochastically. Therefore, it is necessary to study a more practical impulsive scheme which concerns the above case. In what follows, we will discuss the following nonlinear impulsive control systems with impulse time window, disturbance input and bounded gain error:

$$ \textstyle\begin{cases} \dot{x}(t) = Ax ( t ) + Bw ( t ) +Cu ( t )+ f ( {x ( t )} ), &kT\leq t< kT+\tau_{k},\\ x(t)=x ( {t^{-} } ) + Qx ( {t^{-} } ) + \phi ( {x ( {t^{-} } )} ), &t=kT+\tau_{k},\\ \dot{x}(t) = Ax ( t ) + Bw ( t ) +Cu ( t )+ f ( {x ( t )} ), &kT+\tau_{k}< t< (k+1)T , \end{cases} $$
(1.1)

where \(x ( t )\in R^{n}\) is the state variable, \(w ( t ) \in R^{r} \) denotes the disturbance input, \(u ( t ) \in R^{p} \) is the control input, \(\phi ( {x ( {t} )} )\) is the gain error, \(f: R^{n}\rightarrow R^{n}\) and \(\phi : R^{n}\rightarrow R^{n}\) are said to be continuous nonlinear functions satisfying \(f(0)=0\) and \(\phi(0)=0\), respectively, \(T>0\) represents the control period, \(\tau_{k} \in ( {kT, ( {k + 1} )T} ) \) is unknown. \(A\in R^{n \times n}\), \(B \in R^{n \times r}\), \(C \in R^{n \times p}\), and \(Q\in R^{n \times n} \) are constant matrices. In general, let

$$\begin{aligned} &\bigl\Vert {f \bigl( {x ( t )} \bigr)} \bigr\Vert \le l \bigl\Vert {x ( t )} \bigr\Vert ,\\ & \bigl\Vert {w ( t )} \bigr\Vert \le l_{1} \bigl\Vert {x ( t )} \bigr\Vert ,\\ & \bigl\Vert {\phi \bigl( {x ( t )} \bigr)} \bigr\Vert \le l_{2} \bigl\Vert {x ( t )} \bigr\Vert , \end{aligned}$$

where \(l, l_{1}\), and \(l_{2}\) are nonnegative constants. In system (1.1), the impulse is stochastic in an impulse time window, which is wider than an impulse occurring at fixed-time points. For more information on an impulse time window, the reader is referred to [35, 23].

In order to obtain exponential stability, a linear feedback controller \(u ( t ) = Gx ( t )\) is considered, where \(G\in R^{r \times n}\) is a constant matrix. We rewrite system (1.1) as follows:

$$ \textstyle\begin{cases} \dot{x}(t) = (A+CG)x ( t ) + Bw ( t ) + f ( {x ( t )} ), &kT\leq t< kT+\tau_{k},\\ x(t)=x ( {t^{-} } ) + Qx ( {t^{-} } ) + \phi ( {x ( {t^{-} } )} ), &t=kT+\tau_{k},\\ \dot{x}(t) =(A+CG)x ( t )+ Bw ( t ) + f ( {x ( t )} ), &kT+\tau_{k}< t< (k+1)T . \end{cases} $$
(1.2)

The main purpose of this paper is to investigate the exponential stability of system (1.1). By employing the obtained result, system (1.2) is exponentially stable via constructing a linear feedback gain matrix G. A numerical example is given to demonstrate the effectiveness of the theoretical results.

Main results

We need the following definitions and lemmas which play a major role in the proof of the theorems.

Definition 2.1

([11])

The function \(V:[t_{0} - \alpha ,\infty ) \times R^{n} \to R_{+} \) belongs to class \(v_{0}\) if

  1. (1)

    V is continuous on each of the sets \([\tau _{k - 1} ,\tau _{k} ) \times R^{n} \) and \(\lim_{(t,y) \to (\tau _{k}^{-} ,x)} V(t,y) = V(\tau _{k}^{-} ,x)\) exists;

  2. (2)

    \(V(t,x)\) is locally Lipschitzian in \(x \in R^{n} \) and \(V(t,0) \equiv 0\).

Definition 2.2

([11])

For \(V \in v_{0} \), the right and upper Dini’s derivative of V is defined as

$$D^{+} V\bigl(t,x(t)\bigr) = \lim_{h \to 0^{+} } \sup { {1 \over h}}\bigl[V\bigl(t + h,x(t) + hf\bigl(t,x(t)\bigr)\bigr) - V \bigl(t,x(t)\bigr)\bigr]. $$

Lemma 2.1

([6])

Let \(x,y \in R^{n}\) and \(\eta >0 \), then

$$2x^{T} y \le \eta x^{T} x+\eta ^{ - 1}y^{T} y. $$

Lemma 2.2

([2])

The following linear matrix inequality (LMI)

$$ \begin{bmatrix} Q & S \\ {S^{T} } & G \end{bmatrix} < 0, $$

where \(Q^{T}=Q\), \(G^{T}=G\), is equivalent to

$$G < 0,\quad Q - SG^{ - 1} S^{T} < 0. $$

Lemma 2.3

([6])

Let \(x \in R^{n}\) and \(A \in R^{n \times n} \) be a symmetric matrix, then

$$\lambda _{m} ( A )x^{T} x \le x^{T} Ax \le \lambda _{M} ( A )x^{T} x. $$

Theorem 2.1

Let the assumptions about \(w ( t )\), \(f ( {x ( t )} )\), \(\phi ( {x ( t )} ) \) be satisfied and \(u ( t )=0 \). If there exist positive numbers \(\varepsilon, \eta\) and \(0 < P \in R^{n \times n}\) satisfying conditions as follows:

$$\begin{aligned} & ( 1 )\quad \begin{bmatrix} {A^{T} P + PA + ( {l^{2}+ \eta l_{1}^{2} } )I} & P \\ P & { ( { - I - \eta ^{ - 1} BB^{T} } )^{ - 1} } \end{bmatrix} < 0, \\ & ( 2 )\quad \ln \gamma + T ( {h + \varepsilon } )\le 0, \end{aligned}$$

where \(\beta = \lambda _{M} ( {P^{ -1} ( {I + Q} )^{T} P ( {I + Q} ) } )\), \(\beta _{1} = \lambda _{M} ( P )\), \(\beta _{2} = \lambda _{m} ( P )\), \(h = \lambda _{M} ( P^{ - 1} ( PA + A^{T} P + \eta ^{ - 1} PBB^{T} P + P^{2} + ( l^{2}+ \eta l_{1}^{2} )I ) )\), \(\gamma = ( {\sqrt {\beta } +\sqrt { \frac{{\beta _{2} }}{{\beta _{3} }}}}l_{2} )^{2} \). Then system (1.1) is exponentially stable at origin.

Proof

Define

$$V \bigl( {x ( t )} \bigr) = x^{T} ( t )Px ( t ). $$

Let \(t \in [ {kT,kT + \tau _{k} } )\), we have

$$\begin{aligned} D^{+} \bigl( {V \bigl( {x ( t )} \bigr)} \bigr) =& 2x^{T} ( t )P \bigl( {Ax ( t ) + Bw ( t ) + f \bigl( {x ( t )} \bigr)} \bigr) \\ =& x^{T} ( t ) \bigl(PA+A^{T}P\bigr)x ( t ) + 2x^{T} ( t )P \bigl( {Bw ( t ) + f \bigl( {x ( t )} \bigr)} \bigr) . \end{aligned}$$
(2.1)

By Lemma 2.1, it is clear that

$$ 2x^{T} ( t )PBw ( t ) \le \eta ^{ - 1} x^{T} ( t )PBB^{T} Px ( t ) + \eta w^{T} ( t )w ( t ) $$
(2.2)

and

$$ 2x^{T} ( t )Pf \bigl( {x ( t )} \bigr) \le x^{T} ( t )P^{2} x ( t ) + f^{T} \bigl( {x ( t )} \bigr)f \bigl( {x ( t )} \bigr). $$
(2.3)

From the assumptions about \(f ( {x ( t )} )\), \(w ( t )\), substituting (2.2) and (2.3) into (2.1) yields

$$\begin{aligned} D^{+} \bigl( {V \bigl( {x ( t )} \bigr)} \bigr) \le& x^{T} ( t ) \bigl(PA+A^{T}P\bigr)x ( t ) + \eta ^{ - 1} x^{T} ( t )PBB^{T} Px ( t ) \\ &{}+ \eta w^{T} ( t )w ( t ) + x^{T} ( t )P^{2} x ( t ) + f^{T} \bigl( {x ( t )} \bigr)f \bigl( {x ( t )} \bigr) \\ \le & x^{T} ( t ) \bigl( {PA + A^{T} P + \eta ^{ - 1} PBB^{T} P + P^{2} + \bigl( {l^{2} + \eta l_{1}^{2} } \bigr)I} \bigr)x ( t ). \end{aligned}$$
(2.4)

By Lemma 2.2, condition (1) and inequality (2.4), we have

$$D^{+} \bigl( {V \bigl( {x ( t )} \bigr)} \bigr) \le hV \bigl( {x ( t )} \bigr), $$

which yields that

$$ V \bigl( {x ( t )} \bigr) \le V \bigl( {x ( {kT} )} \bigr)e^{h ( {t - kT} )}. $$
(2.5)

In the same way, let \(t \in ( {kT + \tau _{k} , ( {k + 1} )T} ) \), we also have

$$D^{+} \bigl( {V \bigl( {x ( t )} \bigr)} \bigr) \le hV \bigl( {x ( t )} \bigr), $$

which leads to

$$ V \bigl( {x ( t )} \bigr) \le V \bigl( {x ( {kT + \tau _{k} } )} \bigr)e^{h ( {t - kT - \tau _{k} } )}. $$
(2.6)

Let \(t = kT + \tau _{k} \), we obtain

$$\begin{aligned} V \bigl( {x ( t )} \bigr) =& \bigl( { ( {I + Q} )x \bigl( {t^{-} } \bigr) + \phi \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)} \bigr)^{T} P \bigl( { ( {I + Q} )x \bigl( {t^{-} } \bigr) + \phi \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)} \bigr) \\ =& x^{T} \bigl( {t^{-} } \bigr) ( {I + Q} )^{T}P ( {I + Q} )x \bigl( {t^{-} } \bigr)+ \phi ^{T} \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)P\phi \bigl( {x \bigl( {t^{-} } \bigr)} \bigr) \\ &{}+ 2x^{T} \bigl( {t^{-} } \bigr) ( {I + Q} )^{T} P\phi \bigl( {x \bigl( {t^{-} } \bigr)} \bigr) \\ \le& x^{T} \bigl( {t^{-} } \bigr) ( {I + Q} )^{T} P ( {I + Q} )x \bigl( {t^{-} } \bigr) + \phi ^{T} \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)P\phi \bigl( {x \bigl( {t^{-} } \bigr)} \bigr) \\ &{}+ 2\sqrt {x^{T} \bigl( {t^{-} } \bigr) ( {I + Q} )^{T} P ( {I + Q} )x \bigl( {t^{-} } \bigr)\phi ^{T} \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)P\phi \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)} \\ =& \bigl( {\sqrt {x^{T} \bigl( {t^{-} } \bigr) ( {I + Q} )^{T} P ( {I + Q} )x \bigl( {t^{-} } \bigr)} +\sqrt { \phi ^{T} \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)P\phi \bigl( {x \bigl( {t^{-} } \bigr)} \bigr)}} \bigr)^{2} \\ \le& \biggl( {\sqrt {\beta } +\sqrt { \frac{{\beta _{2} }}{{\beta _{3} }}}}l_{2} \biggr)^{2} V \bigl( {x \bigl( {t^{-} } \bigr)} \bigr) \\ =& \gamma V \bigl( {x \bigl( {t^{-} } \bigr)} \bigr). \end{aligned}$$
(2.7)

(2.6) and (2.7) can lead to

$$ V \bigl( {x ( t )} \bigr) \le \gamma V \bigl( {x \bigl( { ( {kT + \tau _{k} } )^{-} } \bigr)} \bigr)e^{h ( {t - kT - \tau _{k} } )}, $$
(2.8)

where \(t \in [ {kT + \tau _{k} , ( {k + 1} )T} ) \).

When \(k=0\), let \(t \in [ {0 ,\tau _{0} } )\), from (2.5), we obtain

$$V \bigl( {x ( t )} \bigr) \le V \bigl( {x ( 0 )} \bigr)e^{ht} . $$

Thus

$$ V \bigl( {x \bigl( {\tau _{0}^{-} } \bigr)} \bigr) \le V \bigl( {x ( 0 )} \bigr)e^{h\tau _{0} } . $$
(2.9)

Let \(t \in [ {\tau _{0} , T} )\), from (2.8) and (2.9), we have

$$\begin{aligned} V \bigl( {x ( t )} \bigr) \le&\gamma V \bigl( {x \bigl( {\tau _{0}^{-} } \bigr)} \bigr)e^{h ( {t - \tau _{0} } )} \le \gamma V \bigl( {x ( 0 )} \bigr)e^{ht}. \end{aligned}$$
(2.10)

When \(k=1\), let \(t \in [ {T , T+\tau _{1} } )\), from (2.5) and (2.10), we have

$$\begin{aligned} V \bigl( {x ( t )} \bigr) \le& V \bigl( {x ( T )} \bigr)e^{h ( {t - T} )} \\ \le& \gamma V \bigl( {x \bigl( {\tau _{0}^{-} } \bigr)} \bigr)e^{h ( {T - \tau _{0} } )} e^{h ( {t - T} )} \\ =& \gamma V \bigl( {x \bigl( {\tau _{0}^{-} } \bigr)} \bigr)e^{h ( {t - \tau _{0} } )} \\ \le& \gamma V \bigl( {x ( 0 )} \bigr)e^{ht}. \end{aligned}$$
(2.11)

Let \(t \in [ {T+\tau _{1},2T } )\), from (2.8) and (2.11), we get

$$\begin{aligned} V \bigl( {x ( t )} \bigr) \le& \gamma V \bigl( {x \bigl( { ( {T + \tau _{1} } )^{-} } \bigr)} \bigr)e^{h ( {t - T - \tau _{1} } )} \\ \le& \gamma ^{2} V \bigl({x \bigl( {\tau _{0}^{-} } \bigr)} \bigr)e^{h ( {T + \tau _{1} - \tau _{0} } )} e^{h ( {t - T - \tau _{1} } )} \\ \le& \gamma^{2} V \bigl( {x ( 0 )} \bigr)e^{ht}. \end{aligned}$$
(2.12)

When \(k=2\), let \(t \in [ {2T , 2T+\tau _{2} } )\), from (2.5) and (2.12), we get

$$\begin{aligned} V \bigl( {x ( t )} \bigr) \le& V \bigl( {x ( 2T )} \bigr)e^{h ( {t - 2T} )} \\ \le& \gamma ^{2} V \bigl( {x \bigl( {\tau _{0}^{-} } \bigr)} \bigr)e^{h ( {2T - \tau _{0} } )} e^{h ( {t - 2T} )} \\ \le& \gamma^{2} V \bigl( {x ( 0 )} \bigr)e^{ht}. \end{aligned}$$
(2.13)

Let \(t \in [\tau _{0} ,T+\tau _{1} )\), from (2.10) and (2.11), we get

$$V \bigl( {x ( t )} \bigr) \le \gamma V \bigl( {x ( 0 )} \bigr)e^{ht}. $$

Let \(t \in [T+\tau _{1} ,2T+\tau _{2} )\), from (2.12) and (2.13), we get

$$V \bigl( {x ( t )} \bigr) \le \gamma ^{2} V \bigl( {x ( 0 )} \bigr)e^{ht} . $$

By induction, for \(t \in [kT+\tau _{k} ,(k+1)T+\tau _{k + 1} )\), we get

$$V \bigl( {x ( t )} \bigr) \le \gamma ^{k + 1} V \bigl( {x ( 0 )} \bigr)e^{ht} . $$

Let \(kT + \tau _{k} = \tau '_{k} \). Since \(\ln \gamma +T ( {h + \varepsilon } ) \le 0\), we get

$$\begin{aligned} V \bigl( {x ( t )} \bigr) \le& \gamma ^{k + 1} V \bigl( {x ( 0 )} \bigr)e^{ht} \\ =& \gamma ^{k + 1} V \bigl( {x ( 0 )} \bigr)e^{ ( {h + \varepsilon } )t} e^{ - \varepsilon t} \\ \le& \gamma ^{k + 1} V \bigl( {x ( 0 )} \bigr)e^{ ( {h + \varepsilon } ){\tau '_{k} } } e^{ - \varepsilon t} \\ \le& \gamma ^{k + 1} V \bigl( {x ( 0 )} \bigr)e^{ ( {h + \varepsilon } )kT } e^{ - \varepsilon t} \\ =& \gamma V \bigl( {x ( 0 )} \bigr)e^{k ( {\ln \gamma + ( {h + \varepsilon } )T} )} e^{ - \varepsilon t} \\ \le& \gamma V \bigl( {x ( 0 )} \bigr)e^{ - \varepsilon t} . \end{aligned}$$
(2.14)

By Lemma 2.3 and (2.14), we obtain

$$\lambda _{m} ( {P } ) \bigl\Vert {x \bigl( {t,\tau _{0} ,x(0) } \bigr)} \bigr\Vert ^{2} \le V \bigl( {x ( t )} \bigr) \le \gamma V \bigl( {x ( 0 )} \bigr)e^{ - \varepsilon t} \le \bigl\Vert {x(0)} \bigr\Vert ^{2} \lambda _{M} (P )e^{ - \varepsilon t}. $$

That is,

$$\bigl\Vert {x \bigl( {t,\tau _{0} ,x(0) } \bigr)} \bigr\Vert \le \sqrt {\frac{{\lambda _{M} ( P )}}{{\lambda _{m} ( P )}}} \bigl\Vert {x(0) } \bigr\Vert e^{\frac{- \varepsilon t}{2}}. $$

This completes the proof. □

Theorem 2.2

Let the assumptions about \(w ( t )\), \(f ( {x ( t )} )\), \(\phi ( {x ( t )} ) \) be satisfied. If there exist positive numbers \(\varepsilon, \eta\), matrices \(H, W\) with \(0 < H \in R^{n \times n}\) satisfying conditions as follows:

$$\begin{aligned} & ( 1 )\quad \begin{bmatrix} {I + ( {AH + CW} )^{T} + ( {AH + CW} ) + \eta ^{ - 1} BB^{T} } & {\sqrt {l^{2} +\eta l_{1}^{2} } H} \\ {\sqrt {l^{2} +\eta l_{1}^{2} } H} & { - I} \end{bmatrix} < 0, \\ & ( 2 )\quad {\sqrt {\beta } +\sqrt { \frac{{\beta _{2} }}{{\beta _{3} }}}}l_{2} < 1, \end{aligned}$$

where \(\beta = \lambda _{M} ( {H ( {I + Q} )^{T} H^{ -1} ( {I + Q} ) } )\), \(\beta _{1} = \lambda _{M} ( H^{ -1} )\), \(\beta _{2} = \lambda _{m} ( H^{ -1} )\), \(h = \lambda _{M} ( H ( H^{ - 1}(A+CG) + (A+CG)^{T} H^{ - 1} + \eta ^{ - 1} H^{ - 1}BB^{T} H^{ - 1} + ( {H^{ - 1} } )^{2} + ( {l^{2} + \eta l_{1}^{2} } )I ) )\), \(\gamma = ( {\sqrt {\beta } +\sqrt { \frac{{\beta _{2} }}{{\beta _{3} }}}}l_{2} )^{2} \). Then system (1.2) is exponentially stable at origin and we have the following linear feedback controller:

$$u ( t ) = Gx ( t ),\qquad G = WH^{ - 1} . $$

Proof

By Lemma 2.2, condition (1) of Theorem 2.2 is equivalent to

$$ I + ( {AH + CW} )^{T} + ( {AH + CW} ) + \eta ^{ - 1} BB^{T} + \bigl( {l^{2} +\eta l_{1}^{2} } \bigr)H^{2} < 0. $$
(2.15)

Let

$$P = H^{ - 1} ,\qquad G = WH^{ - 1}. $$

Multiplying both sides of (2.15) by P, we have

$$P^{2} + P ( {AH + CW} )^{T} P + P ( {AH + CW} )P + \eta ^{ - 1} PBB^{T} P + \bigl( {l^{2}+ \eta l_{1}^{2} } \bigr)I < 0. $$

That is,

$$ P^{2} + ( {A + CG} )^{T} P + P ( {A + CG} ) + \eta ^{ - 1} PBB^{T} P + \bigl( {l^{2} +\eta l_{1}^{2} } \bigr)I < 0. $$
(2.16)

By Lemma 2.2 and (2.16), we have

$$ \begin{bmatrix} { ( {A + CG} )^{T} P + P ( {A + CG} ) + ( {l^{2}+ \eta l_{1}^{2} } )I} & P \\ P & { ( { - I - \eta ^{ - 1} BB^{T} } )^{ - 1} } \end{bmatrix} < 0 . $$

Thus, condition (1) of Theorem 2.1 holds. Since

$${\sqrt {\beta } +\sqrt { \frac{{\beta _{2} }}{{\beta _{3} }}}}l_{2} < 1, $$

which implies

$$\ln \gamma + T ( {h + \varepsilon } )\le 0, $$

namely, condition (2) of Theorem 2.1 is satisfied, too. Then system (2.2) is exponentially stable at origin.

This completes the proof. □

A numerical example

In this section, we demonstrate and verify the effectiveness of our theoretical results employing a nonlinear impulsive system as follows:

$$A = \begin{bmatrix} 2 & 1 \\ 2 & 3 \end{bmatrix},\qquad B = \begin{bmatrix} {1.5} & {1.3} \\ {1.2} & 0 \end{bmatrix},\qquad C = \begin{bmatrix} {1.8} & {0.8} \\ 1 & {1.7} \end{bmatrix},\qquad f \bigl( {x ( t )} \bigr) = \begin{bmatrix} {\sin x_{1} } \\ {\sin x_{2} } \end{bmatrix} $$

and

$$Q = - \begin{bmatrix} {0.58} & 0 \\ 0 & {0.58} \end{bmatrix},\qquad \phi \bigl( {x ( t )} \bigr) = 0.3 \begin{bmatrix} {\sin x_{1} } \\ {\sin x_{2} } \end{bmatrix},\qquad \omega ( t ) = \begin{bmatrix} {x_{1} \sin 20\pi t} \\ {x_{2} \sin 20\pi t} \end{bmatrix}. $$

Then we can choose

$$\eta = l= l_{1} = 1,\qquad l_{2} = 0.3. $$

By condition (1) of Theorem 2.2, we obtain

$$H = \begin{bmatrix} {0.2565} & 0 \\ 0 & {0.2565} \end{bmatrix}, \qquad W = \begin{bmatrix} { - 25.7414} & {-49.9086} \\ {53.3876} & { 27.8104} \end{bmatrix}. $$

Simple calculations show that \(\gamma = 0.72 < 1\). Thus, the nonlinear impulsive system is exponentially stable because the conditions of Theorem 2.2 are satisfied.

Conclusions

In this paper, we discuss exponential stability of nonlinear systems with impulse time window, disturbance input, and bounded gain error. In [3], the authors did not consider the disturbance input and bounded gain error of nonlinear impulsive control systems. In [25], the authors did not consider the disturbance input of nonlinear impulsive control systems. Obviously, system (1.1) is more general and more applicable than [3, 25]. Using Theorem 2.1 and the construction of a linear stabilizing feedback controller, a new criterion of exponential stability is obtained. Finally, a numerical example demonstrates the effectiveness of the theoretical results.

References

  1. Ai, Z., Chen, C.: Asymptotic stability analysis and design of nonlinear impulsive control systems. Nonlinear Anal. Hybrid Syst. 24, 244–252 (2017)

    MathSciNet  Article  Google Scholar 

  2. Boyd, S., Ghaoui, E.I.L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  3. Feng, Y., Li, C., Huang, T.: Periodically multiple state-jumps impulsive control systems with impulse time windows. Neurocomputing 193, 7–13 (2016)

    Article  Google Scholar 

  4. Feng, Y., Li, C., Huang, T.: Sandwich control systems with impulse time windows. Int. J. Mach. Learn. Cybern. 8, 2009–2015 (2017)

    Article  Google Scholar 

  5. Feng, Y., Peng, Y., Zou, L., Tu, Z., Liu, J.: A note on impulsive control of nonlinear systems with impulse time window. J. Nonlinear Sci. Appl. 10, 3087–3098 (2017)

    MathSciNet  Article  Google Scholar 

  6. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  7. Huang, T., Li, C., Duan, S., Starzyk, J.A.: Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects. IEEE Trans. Neural Netw. Learn. Syst. 23, 866–875 (2012)

    Article  Google Scholar 

  8. Li, X., Bohner, M., Wang, C.: Impulsive differential equations: periodic solutions and applications. Automatica 52, 173–178 (2015)

    MathSciNet  Article  Google Scholar 

  9. Li, X., Cao, J.: An impulsive delay inequality involving unbounded time-varying delay and applications. IEEE Trans. Autom. Control 62, 3618–3625 (2017)

    MathSciNet  Article  Google Scholar 

  10. Li, X., Song, S.: Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans. Autom. Control 62, 406–411 (2017)

    MathSciNet  Article  Google Scholar 

  11. Li, X., Wu, J.: Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64, 63–69 (2016)

    MathSciNet  Article  Google Scholar 

  12. Li, Z., Wen, C., Soh, Y.: Analysis and design of impulsive control systems. IEEE Trans. Autom. Control 46, 894–897 (2001)

    MathSciNet  Article  Google Scholar 

  13. Song, Q., Cao, J.: Global exponential stability of bidirectional associative memory neural networks with distributed delays. J. Comput. Appl. Math. 202, 266–279 (2007)

    MathSciNet  Article  Google Scholar 

  14. Song, Q., Cao, J.: Passivity of uncertain neural networks with both leakage delay and time-varying delay. Nonlinear Dyn. 67, 169–1707 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Song, Q., Zhao, Z.: Stability criterion of complex-valued neural networks with both leakage delay and time-varying delays on time scales. Neurocomputing 171, 179–184 (2016)

    Article  Google Scholar 

  16. Sun, J., Wu, Q.: Impulsive control for the stabilization and synchronization of Lorenz systems. Appl. Math. Mech. 25, 322–328 (2004)

    Article  Google Scholar 

  17. Wang, H., Liao, X., Huang, T., Li, C.: Improved weighted average prediction for multi-agent networks. Circuits Syst. Signal Process. 33, 1721–1736 (2014)

    MathSciNet  Article  Google Scholar 

  18. Yang, T.: Impulsive control. IEEE Trans. Autom. Control 44, 1081–1083 (1999)

    MathSciNet  Article  Google Scholar 

  19. Yang, X., Feng, Z., Feng, J., Cao, J.: Synchronization of discrete-time neural networks with delays and Markov jump topologies based on tracker information. Neural Netw. 85, 157–164 (2017)

    Article  Google Scholar 

  20. Yang, X., Lam, J., Ho, D.W.C., Feng, Z.: Fixed-time synchronization of complex networks with impulsive effects via nonchattering control. IEEE Trans. Autom. Control 62, 5511–5521 (2017)

    MathSciNet  Article  Google Scholar 

  21. Yang, Z., Xu, D.: Stability analysis and design of impulsive control systems with time delay. IEEE Trans. Autom. Control 52, 1448–1454 (2007)

    MathSciNet  Article  Google Scholar 

  22. Zhang, Y.: Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities. Nonlinear Dyn. 75, 101–111 (2014)

    MathSciNet  Article  Google Scholar 

  23. Zhou, Y., Li, C., Huang, T., Wang, X.: Impulsive stabilization and synchronization of Hopfield-type neural networks with impulse time window. Neural Comput. Appl. 28, 775–782 (2017)

    Article  Google Scholar 

  24. Zou, L., Peng, Y., Feng, Y., Tu, Z.: Stabilization and synchronization of memristive chaotic circuits by impulsive control. Complexity 2017, Article ID 5186714 (2017)

    Article  Google Scholar 

  25. Zou, L., Peng, Y., Feng, Y., Tu, Z.: Impulsive control of nonlinear systems with impulse time window and bounded gain error. Nonlinear Anal., Model. Control 23, 40–49 (2018)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere thanks to referees and the editor for their enthusiastic guidance and help.

Funding

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11561037, 11801240).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Xingkai Hu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hu, X., Nie, L. Exponential stability of nonlinear systems with impulsive effects and disturbance input. Adv Differ Equ 2018, 354 (2018). https://doi.org/10.1186/s13662-018-1798-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-018-1798-1

MSC

  • 37N35
  • 49N25

Keywords

  • Exponential stability
  • Impulse time window
  • Disturbance input
  • Bounded gain error
  • Linear matrix inequalities