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 $$\overset{\cdot }{x}=f(x)+B(x)(u+h(x,t)), x\in R^n, u \in R^m, rank B =m$$, $$h(x,t)$$ bounded uncertainty.

Mathews and DeCarlo [1] suggested to select an integral sliding surface as $$\sigma (t)=Gx(t)-Gx(0)-\int_0^t [GBu_0(\tau )+Gf(x(\tau ))] \, d\tau$$

In this case there exists a unit or discontinuous sliding mode controller compensating uncertainty $$h(x,t)$$.

Utkin and Shi [2] have remarked that, if $$\sigma (0)=0$$ is guaranteed, the reaching phase is eliminated.

In the case, when unmatched uncertainties occur $$G$$ should be selected as $$G=B^+, $$ where $$B^+=(B^TB)^{-1}B^T$$ is a pseudo inverse matrix [3-5].