Department of Engineering
Higher Order Sliding Mode Control M. Khalid Khan Control & Instrumentation group
References 1. Levant, A.: ‘Sliding order and sliding accuracy in sliding mode control’, Int. J. Control, 1993,58(6) pp.1247-1263. 2. Bartolini et al.: ‘Output tracking control of uncertain nonlinear second order systems’, Automatica, 1997, 33(12) pp.2203-2212. 3. H. Sira-ranirez, ‘On the sliding mode control of nonlinear systems’, Syst.Contr.Lett.1992(19) pp.303-312 4. M.K. Khan et al.: ‘Robust speed control of an automotive engine using second order sliding modes’, In proc. of ECC’2001.
Review: Sliding Mode Control Consider a NL system x& = f (t , x) + g (t , x)u Design consists of two steps Selection of sliding surface s = s (t , x) = 0 Making sliding surface attractive
High frequency switching of control
Robustness
Chattering
Pros and cons Order reduction
× Full state availability
Robust to matched uncertainties
× Chattering at actuator
Simple to implement
× Sliding error = O(τ)
Isn’t it restrictive? Sliding variable must have relative degree one w.r.t. control.
Higher Order Sliding Modes Consider a NL system Sliding surface
x& = f (t , x, u )
s = s (t , x) = 0
rth-order sliding set: -
s
( r −1)
=s
( r −2)
= L = &s& = s& = s = 0
rth-order sliding mode:- motion in rth-
order sliding set. Sliding variable (s) has relative degree r
So traditional sliding mode control is now 1st order sliding mode control!
But What about reachability condition?
There is no generalised higher order reachability condition available
1-sliding vs 2-sliding
ds
ds τ τ
s
1-sliding
Sliding error = O(τ)
τ2
s
2-sliding Sliding error = O(τ2)
Sliding variable dynamics Selected sliding variable, s, will have relative degree, p= 1 1-sliding design is possible. 2-sliding design is done to avoid chattering.
relative degree, p ≥ 2 r-sliding (r ≥ p) is the suitable choice.
2-sliding algorithms: examples Consider system represented in sliding variable as
&s& = φ (t , s, s&) + γ (t , s, s&)u; | φ |≤ Φ, Γm ≤ γ ≤ ΓM , s ≤ S 0 Finite time converging 2-sliding twisting algorithm − VM sign( s ) ss& > 0 u (t ) = − αVM sign( s ) ss& ≤ 0 α<1
Sliding set: s& = s = 0
Pendulum The model:
&y& = −0.25 sin( y ) + u
Sliding variable:
s = y& + y
Sliding variable dynamics:
s& = −0.25 sin( y ) + y& + u &s& = −0.25 y& cos( y ) − 0.25 sin( y ) + u + u& Twisting Controller coefficients: α = 0.1, VM = 7
Simulation
Examples continue … Consider a system of the type
s& = φ (t , s ) + γ (t , s )u; | φ |≤ Φ, Γm ≤ γ ≤ ΓM , s ≤ S 0 Finite time 2-sliding super-twisting algorithm
u (t ) = −λ | s |sign ( s ) + u1 | u |> u0 − ku u&1 = − Wsign ( s ) | u |≤ u0 Sliding set: s& = s = 0
Review: 2-sliding algorithms Twisting algorithm forces sliding variable (s) of relative degree 2 in to the 2-sliding set but uses s& Super Twisting algorithm do not uses s& but sliding variable (s) has relative degree only one.
Question: 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.
Modified super-twisting algorithm System type:
&s& = φ (t , s, s&) + γ (t , s, s&)u; | φ |≤ Φ, Γm ≤ γ ≤ ΓM , s ≤ S 0 u (t ) = −λsign ( s ) + u1 | u |> u0 − ku u&1 = − Wsign ( s ) | u |≤ u0
Where λ, u0 , k and W are positive design constants
1. Sinusoidal oscillations for λ= u0
2. Unstable for λ< u0 3. Stable for λ> u0
Phase plot Sufficient conditions for stability λ > u0 > Φ / Γm k > 0, W > 0
Application: Anti-lock Brake System (ABS) ABS model: Rw 2 N v N w µ (λ ) x&1 = −0.595 x1 + Mv M v Rw R N 1 &x2 = − w v µ (λ ) + x3 Jw Jw x&3 = −
x3
τ
+
kb
τ
u
Can be written as:
λ&& = f (⋅) + g (⋅)u
λ=
(x2 − x1 ) max( x1 , x2 )
µ (λ ) = µ p λ p
λ λ2 + λ2 p
kb =
0.25k1 x1 k x + k3 x 1 1 + 3 1+ k x + k x 4 1 5 1 2 2 1
λdesired = −0.12
4
2
Simulation Results Controller coefficients:λ = 75, u0 = k = 35, W = 15
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.
