Terminal sliding mode

In the early 1990’s, a new type of sliding mode, named “terminal sliding mode (TSM)” was invented. The main idea of terminal sliding mode control is evoked by the concept of terminal attractors which guarantee finite time convergence of the states. While, in normal sliding mode, asymptotic stability is promised which leads to the convergence of the states to the origin. But this convergence may only be guaranteed within infinite time. In TSM, a nonlinear term is introduced in the sliding surface design so that the manifold is formulated as an attractor. After the sliding surface is intercepted, the trajectory is attracted within the manifold and converges to the origin following a power rule. Terminal sliding mode also has been widely applied to nonlinear process control, for example, rigid robot control etc..

Control Scheme

Consider a continuous nonlinear system in canonical form

${\overset {\cdot }{x}}_{1}(t)=x_{2}(t)$ ......

${\overset {\cdot }{x}}_{n-1}(t)=x_{n}(t)$

${\overset {\cdot }{x}}_{n}(t)=a(x)+b(x)u(t)$

where $x(t)\in R^{n}$ is the state vector, $u\in R$ is the control input, $a(x)$ and $b(x)$ are nonlinear functions in $x(t)$. Then a sequence of terminal sliding surfaces can be designed as follows:

$s_{1}(t)={\overset {\cdot }{s}}_{0}(t)+\alpha _{1}(t)s_{0}^{\gamma _{1}}(t)$

$s_{2}(t)={\overset {\cdot }{s}}_{1}(t)+\alpha _{2}(t)s_{1}^{\gamma _{2}}(t)$ ......

$s_{n-1}(t)={\overset {\cdot }{s}}_{n-2}(t)+\alpha _{n-1}(t)s_{n-2}^{\gamma _{n-1}}(t)$ where $s_{0}(t)=x_{1}(t)$ and $\alpha _{i}(t)={\frac {p_{i}}{q_{i}}},i=1,2,...,n-1$ . $p_{i},q_{i}$ are positive odd numbers and $p_{i}\leq q_{i}$.