Presentation on theme: "Higher Order Sliding Mode Control"— Presentation transcript:
1Higher Order Sliding Mode Control Department of EngineeringHigher Order Sliding Mode ControlM. Khalid KhanControl & Instrumentation group
2References Levant, A.: ‘Sliding order and sliding accuracy in sliding mode control’, Int. J. Control, 1993,58(6)pp2. Bartolini et al.: ‘Output tracking control of uncertainnonlinear second order systems’, Automatica, 1997,33(12) ppH. Sira-ranirez, ‘On the sliding mode control ofnonlinear systems’, Syst.Contr.Lett.1992(19) pp4. M.K. Khan et al.: ‘Robust speed control of anautomotive engine using second order sliding modes’,In proc. of ECC’2001.
3Review: Sliding Mode Control Consider a NL systemDesign consists of two stepsSelection of sliding surfaceMaking sliding surface attractive
4High frequencyswitching of controlRobustnessChattering
5Pros and cons Order reduction Full state availability Robust to matched uncertaintiesChattering at actuatorSliding error = O(τ)Simple to implement
6Sliding variable must have relative degree one w.r.t. control. Isn’t it restrictive?Sliding variable must have relative degree one w.r.t. control.
7Higher Order Sliding Modes Consider a NL systemSliding surfacerth-order sliding set: -rth-order sliding mode:- motion in rth-order sliding set. Sliding variable (s) has relative degree r
8What about reachability condition? So traditional sliding mode control is now 1st order sliding mode control!ButWhat about reachability condition?There is no generalised higher orderreachability condition available
91-sliding vs 2-sliding s ds 1-sliding τ s ds 2-sliding τ τ2 Sliding error = O(τ)Sliding error = O(τ2)
10Sliding variable dynamics Selected sliding variable, s, will haverelative degree, p= 1relative degree, p 21-sliding design ispossible.r-sliding (r p) is thesuitable choice.2-sliding design is doneto avoid chattering.
112-sliding algorithms: examples Consider system represented in sliding variable asFinite time converging 2-sliding twisting algorithm < 1Sliding set:
12Pendulum The model: Sliding variable: Sliding variable dynamics: Twisting Controller coefficients: α = 0.1, VM = 7
14Examples continue … Finite time 2-sliding super-twisting algorithm Consider a system of the typeFinite time 2-sliding super-twisting algorithmSliding set:
15Review: 2-sliding algorithms Twisting algorithm forces sliding variable (s) of relative degree 2 in to the 2-sliding set but usesSuper Twisting algorithm do not uses but sliding variable (s) has relative degree only one.
16Question:Is it possible to stabilise sliding surface with relative degree 2 in to 2-sliding set using only s, not its derivative?Answer: yes!by designing observer2. using modified super-twisting algorithm.
17Modified super-twisting algorithm System type:Where λ, u0 , k and W arepositive design constantsSinusoidal oscillations for = u0Unstable for < u0Stable for > u0
22Conclusions HOSM can be used to avoid chattering The restriction over choice of sliding variable can be relaxed by HOSM.HOSM can be used to avoid chatteringA new 2-sliding algorithm which uses only sliding variable s (not its derivative) has been presented together with sufficient conditions for stability.The algorithm has been applied to ABS system and simulation results presented
23Future Work The algo can be extended for MIMO systems. Possibility of selecting control dependentsliding surfaces is to be investigated.Stability results are local, need to find globalresults.