Input-to-state stability (ISS)[1] [2] [3] is a stability notion widely used to study stability of nonlinear control systems with external inputs. Roughly speaking, a control system is ISS if it is globally asymptotically stable in the absence of external inputs and if its trajectories are bounded by a function of the size of the input for all sufficiently large times.The importance of ISS is due to the fact that the concept has bridged the gap between input–output and state-space methods, widely used within the control systems community.
ISS unified the Lyapunov and input-output stability theories and revolutionized our view on stabilization of nonlinear systems, design of robust nonlinear observers, stability of nonlinear interconnected control systems, nonlinear detectability theory, and supervisory adaptive control. This made ISS the dominating stability paradigm in nonlinear control theory, with such diverse applications as robotics, mechatronics, systems biology, electrical and aerospace engineering, to name a few.
The notion of ISS was introduced for systems described by ordinary differential equations by Eduardo Sontag in 1989.[4]
Since that the concept was successfully used for many other classes of control systems including systems governed by partial differential equations, retarded systems, hybrid systems, etc.[5]
Consider a time-invariant system of ordinary differential equations of the form
where
u:R+\toRm
f
To define ISS and related properties, we exploit the following classes of comparison functions. We denote by
l{K}
\gamma:\R+\to\R+
\gamma(0)=0
l{L}
\gamma:\R+\to\R+
\limr\toinfty\gamma(r)=0
\beta\inl{K}l{L}
\beta( ⋅ ,t)\inl{K}
t\geq0
\beta(r, ⋅ )\inl{L}
r>0
System is called globally asymptotically stable at zero (0-GAS) if the corresponding system with zero inputis globally asymptotically stable, that is there exist
\beta\inl{K}l{L}
x0
t\geq0
System is called input-to-state stable (ISS) if there exist functions
\gamma\inl{K}
\beta\inl{K}l{L}
x0
u
t\geq0
The function
\gamma
Clearly, an ISS system is 0-GAS as well as BIBO stable (if we put the output equal to the state of the system). The converse implication is in general not true.
It can be also proved that if
\limt → |u(t)|=0
\limt → infty|x(t)|=0
For an understanding of ISS its restatements in terms of other stability properties are of great importance.
System is called globally stable (GS) if there exist
\gamma,\sigma\inl{K}
\forallx0
\forallu
\forallt\geq0
System satisfies the asymptotic gain (AG) property if there exists
\gamma\inl{K}
\forallx0
\forallu
The following statements are equivalent for sufficiently regular right-hand side
f
1. is ISS
2. is GS and has the AG property
3. is 0-GAS and has the AG property
The proof of this result as well as many other characterizations of ISS can be found in the papers and. Other characterizations of ISS that are valid under very mild restrictions on the regularity of the rhs
f
An important tool for the verification of ISS are ISS-Lyapunov functions.
A smooth function
V:Rn\toR+
\exists\psi1,\psi2\inl{K}infty
\chi\inl{K}
\alpha
\psi1(|x|)\leqV(x)\leq\psi2(|x|), \forallx\inRn
\forallx\inRn, \forallu\inRm
|x|\geq\chi(|u|) ⇒ \nablaV ⋅ f(x,u)\leq-\alpha(|x|),
The function
\chi
If a system is without inputs (i.e.
u\equiv0
\nablaV ⋅ f(x,u)\leq-\alpha(|x|), \forallx ≠ 0,
which tells us that
V
An important result due to E. Sontag and Y. Wang is that a system is ISS if and only if there exists a smooth ISS-Lyapunov function for it.[8]
Consider a system
| x |
=-x3+ux2.
Define a candidate ISS-Lyapunov function
V:\R\to\R+
| V(x)= | 1 |
| 2 |
x2, \forallx\in\R.
| V |
(x)=\nablaV ⋅ (-x3+ux2)=-x4+ux3.
Choose a Lyapunov gain
\chi
\chi(r):=
| 1 | |
| 1-\epsilon |
r
Then we obtain that for
x,u: |x|\geq\chi(|u|)
| V |
(x)\leq-|x|4+(1-\epsilon)|x|4=-\epsilon|x|4.
This shows that
V
\chi
One of the main features of the ISS framework is the possibility to study stability properties of interconnections of input-to-state stable systems.
Consider the system given by