Presentation on theme: "Higher Order Sliding Mode Control M. Khalid Khan Control & Instrumentation group Department of Engineering."— Presentation transcript:
Higher Order Sliding Mode Control M. Khalid Khan Control & Instrumentation group Department of Engineering
References 1.Levant, A.: Sliding order and sliding accuracy in sliding mode control, Int. J. Control, 1993,58(6) pp Bartolini et al.: Output tracking control of uncertain nonlinear second order systems, Automatica, 1997, 33(12) pp H. Sira-ranirez, On the sliding mode control of nonlinear systems, Syst.Contr.Lett.1992(19) pp M.K. Khan et al.: Robust speed control of an automotive engine using second order sliding modes, In proc. of ECC2001.
Review: Sliding Mode Control Design consists of two steps Selection of sliding surface Making sliding surface attractive Consider a NL system
RobustnessChattering High frequency switching of control
Pros and cons Order reduction Full state availability Robust to matched uncertainties Simple to implement Chattering at actuator Sliding error = O(τ)
Isnt it restrictive? Sliding variable must have relative degree one w.r.t. control.
Higher Order Sliding Modes r th -order sliding mode:- motion in r th - order sliding set. Sliding variable (s) has relative degree r r th -order sliding set: - Consider a NL system Sliding surface
But What about reachability condition? So traditional sliding mode control is now 1 st order sliding mode control! There is no generalised higher order reachability condition available
1-sliding vs 2-sliding s ds 2-sliding τ τ2τ2 s ds 1-sliding τ Sliding error = O(τ)Sliding error = O(τ 2 )
Sliding variable dynamics Selected sliding variable, s, will have relative degree, p= 1 relative degree, p 2 1-sliding design is possible. 2-sliding design is done to avoid chattering. r-sliding (r p) is the suitable choice.
2-sliding algorithms: examples Consider system represented in sliding variable as Finite time converging 2-sliding twisting algorithm Sliding set: < 1
Pendulum The model: Sliding variable: Sliding variable dynamics: Twisting Controller coefficients: α = 0.1, V M = 7
Examples continue … Consider a system of the type Finite time 2-sliding super-twisting algorithm Sliding set:
Review: 2-sliding algorithms Twisting algorithm forces sliding variable (s) of relative degree 2 in to the 2-sliding set but uses Super Twisting algorithm do not uses but sliding variable (s) has relative degree only one.
Is it possible to stabilise sliding surface with relative degree 2 in to 2-sliding set using only s, not its derivative? Answer: yes! 1.by designing observer 2. using modified super-twisting algorithm. Question:
Modified super-twisting algorithm System type: Where λ, u 0, k and W are positive design constants 1. Sinusoidal oscillations for = u 0 2. Unstable for < u 0 3. Stable for > u 0
Phase plot Sufficient conditions for stability
Application: Anti-lock Brake System (ABS) ABS model: Can be written as:
Simulation Results Controller coefficients:
Results continued …
Conclusions The restriction over choice of sliding variable can be relaxed by HOSM. HOSM can be used to avoid chattering A 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
Future Work The algo can be extended for MIMO systems. Possibility of selecting control dependent sliding surfaces is to be investigated. Stability results are local, need to find global results.