# Integral sliding mode

In 1996, V. Utkin and J. Shi proposed an improved sliding control method named integral sliding mode control (ISMC). In contrast with conventional sliding mode control, the system motion under integral sliding mode has a dimension equal to that of the state space. In ISMC, the system trajectory always starts from the sliding surface. Accordingly, the reaching phase is eliminated and robustness in the whole state space is promised.

## Control Scheme

For a system ${\displaystyle {\overset {\cdot }{x}}=f(x)+B(x)u+h(x)}$, an integral sliding surface can be designed as

${\displaystyle \sigma (t)=Gx(t)-Gx(0)-\int _{0}^{t}[GBu_{0}(\tau )+Gf(x(\tau ))]d\tau }$

Doing so, ${\displaystyle \sigma (0)=0}$ is guaranteed and the reaching mode during which the system is sensitive to noise in normal sliding mode is eliminated.

## References

1. V, I, Utkin, Sliding Modes in Control and Optimization, Springer-Verlag, 1992, ISBN, 978-0387535166
2. Yugang Niua, Daniel. W. C. Ho, and James Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay. [1]
3. Christopher Edwards, Sarah K. Spurgeon, Sliding mode control: theory and applications, CRC Press, 1998, ISBN, 0-7484-0601-8.