Integral sliding mode control Integral sliding mode control (ISM) is a modification of sliding mode control designed to compensate matched perturbations in nonlinear control systems. The method introduces an integral sliding variable that allows matched perturbations compensation by means of a sliding mode control component ensuring the sliding mode on a virtual integral sliding surface. In this note systems solutions are interpreted in the sense of Filippov solution.
Mathematical formulation Consider a nonlinear control system with matched disturbance
x ˙ = f ( x , t ) + B ( x ) ( u + h ( x , t ) ) , {\displaystyle {\dot {x}}=f(x,t)+B(x)(u+h(x,t)),}
where x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , u ∈ R m {\displaystyle u\in \mathbb {R} ^{m}} , r a n k ( B ) = m {\displaystyle rank(B)=m} , and h ( x , t ) {\displaystyle h(x,t)} is a bounded perturbation entering into the system through the same channel as the control input B ( x ) {\displaystyle B(x)} . The objective is to ensure that the trajectories of the perturbed system converge to the trajectories of the nominal system
x ˙ 0 = f ( x 0 , t ) + B ( x 0 ) u 0 . {\displaystyle {\dot {x}}_{0}=f(x_{0},t)+B(x_{0})u_{0}.}
Control Scheme Matthews and DeCarlo proposed selecting the control input in the form
u = u 0 + u I S M , {\displaystyle u=u_{0}+u_{ISM},}
where u 0 {\displaystyle u_{0}} is a nominal controller for the nominal system and u I S M {\displaystyle u_{ISM}} is a sliding mode controller compensating the disturbance h ( x , t ) {\displaystyle h(x,t)} . They introduced a virtual integral sliding variable
σ ( t ) = G x ( t ) − G x ( 0 ) − ∫ 0 t [ G B ( x ( τ ) ) u 0 ( τ ) + G f ( x ( τ ) ) ] d τ . {\displaystyle \sigma (t)=Gx(t)-Gx(0)-\int _{0}^{t}[GB(x(\tau ))u_{0}(\tau )+Gf(x(\tau ))]\,d\tau .}
The variable σ ( t ) {\displaystyle \sigma (t)} is called virtual integral because it extends real system states and depends on the integral of the system dynamics and value of nominal control.
Control laws If det ( G B ) ≠ 0 {\displaystyle \det(GB)\neq 0} , a sliding mode controller can be constructed. One possible choice is a unit control law
u I S M = − ρ ( x , t ) σ ¯ ‖ σ ¯ ‖ 2 , {\displaystyle u_{ISM}=-\rho (x,t){\frac {\bar {\sigma }}{\|{\bar {\sigma }}\|_{2}}},}
where
σ ¯ = ( G B ) T σ {\displaystyle {\bar {\sigma }}=(GB)^{T}\sigma }
is an auxiliary switching variable and the gain for u I S M {\displaystyle u_{ISM}} should be chosen as
ρ ( x , t ) ≥ ‖ h ( x , t ) ‖ 2 . {\displaystyle \rho (x,t)\geq \|h(x,t)\|_{2}.}
Another option is a relay control law
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