SLIDING MODE CONTROLLER

SLIDING MODE CONTROLLER

Pierre Duysinx

LTAS - Automotive Engineering Academic year 2009-2010

Introduction

† ABS system has a lot of uncertainties:

„ Friction coefficient

„ Vehicle mass and inertia

„ Wheel sizes

„ Braking system characteristics

† ABS is a strongly non linearsystem

„ Slip ratio friction curve

† Sliding mode controllers are standard nonlinear control strategy to tackle with nonlinear problems with

uncertainties

Introduction

† Sliding mode controller was originally introduced in Robotics by Slotine and Asada (1986), Slotine and Li (1990,1991)

† Sliding mode control approach is to transform the higher-order system into a first order system

† Control law can include bounds over uncertainties in the model to become robust

SLIDING SURFACE

Sliding surface

† Let’s consider a single input nonlinear system

„ u(t) the control command

„ x, the scalar output, the state variable

„ X, the state variable vector

„ d(t) disturbances

† Generally

„ f(X) is nonlinear and not exactly known, but the imprecision is bounded by a continuous function of X

„ b(X) is nonlinear and not exactly known but its sign is constant and the imprecision is bounded by a continuous function

( )ⁿ ( ) ( ) ( ) ( ) ( ) x t = f X +b X u t +d t

( 1)

[ , , ⁿ ] X = x x…x ⁻

Sliding surface

† The problem consists in finding a state vector to follow a given state vector

† With some uncertainties on the modeling of f(x), b(x) and some disturbances d(t)

† To be achievable using finite control u, assume that X_d(t) is such that:

† Let’s define the tracking error

( 1)

[ , , ⁿ ]

d d d d

X = x x …x ⁻

( 0) ( 0)

Xd t= =X t=

Sliding surface

† Let’s define the time-varying sliding surfaces(t) in the state space Rⁿ by the scalar equation

„ Where δis a positive constant, which will be interpreted as a desired control bandwidth

( 1)

( , ) : d

s X t x

dt δ

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠

Sliding surface

† With the initial conditions X_d(0)=X(0), the problem of tracking X=X_dis equivalent to that of remaining on the surface s(t) for all t>0

„ s(t)=0 is a differential equation whose solution is X-X_d=0

„ Verifying the differential equation s(t)=0 is referred to as a sliding surface

„ The regime of the system’s behavior once on the surface is called sliding modeor sliding regime.

„ One has to differentiate only once s(t) to make appear x⁽ⁿ⁾ and then the control u

Sliding surface

† The tracking problem can be reduced to that of keeping the scalar quantity S at zero

† This can be achieved by choosing a control law that is such that outside of s(t)=0

„ With ηa positive constant

† This inequality is called sliding condition.

† The sliding condition means that trajectories point towards the surface s(t)=0

1 ²( , ) 2

d s X t s

dt ≤ −

η

S(t)

Sliding surface

† Idea: pick-up a well-behave function of the tracking error s(t) and then select the feedback control law u such that s² remain a Lyapunov function of the closed-loop system despite the presence of uncertainties and disturbances

† Satisfying the sliding condition guarantees that if initial conditions are not satisfied, then the surface s(t)=0 will be reached in finite time smaller than

† Definition of sliding surface implies that once on the surface, the tracking error tends exponentially to zero with a time constant (n-1)/δ.

( 0) /

reach

t ≤ s t =

η

Sliding surface

S=0 dx/dt

x Finite

reaching time

Sliding mode:

exponential convergence

Sliding mode control

† The controller design procedure consists in two steps

† Step 1:Select a feedback control law u to verify the sliding condition

† However in order to account for the presence of

modeling imprecision and disturbances, the control law has to be discontinuous across s(t), which leads to chattering

1 ²( , ) 2

d s X t s

dt ≤− η

Sliding mode control

† Chattering is undesirable because of excessive activity of the controller and excitation of high frequency neglected dynamics (e.g. elastic modes)

† Step 2:the discontinuous control law u is suitably smoothed to achieve an optimal trade-off between control bandwidth and tracking precision

† The second step achieves robustness to high-frequency unmodeled dynamics

PERFECT TRACKING USING

SWITCHED CONTROL LAW

Trajectory tracking

† Given the bounds on f(X) and b(X) and on the disturbances

† Constructing a control law to verify sliding conditions is straightforward

† Illustration of the procedure on simple applications

Controller design

† Let’s consider a second order system

„ Function f is estimated by

„ Estimation error is bounded:

† Define sliding surface s=0

x = + f u

fˆ− f ≤F fˆ

: d

s x x x

dt δ δ

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠ = +

Controller design

† Sliding surface equation gives

† Best approximation of a continuous control law that achieve

† To satisfy the sliding condition despite uncertainties on the dynamic f, we add the discontinuousterm across the surface s=0

d d

s = − x x + δ x = + − f u x + δ x 0

s =

ˆ ˆ

u = − + f x − δ x

: ˆ ( )

u = − u k sign s ( ) 1 if 0

( ) 1 if 0

sign s s

sign s s

= + >

⎧ ⎨ = − <

⎩

+η

-η

S Us

Controller design

† To guarantee the sliding condition, we have to choose k large enough

† Sliding condition implies

† Let

† We get

1 ²( , ) [ ˆ ( )] ( ˆ )

2

d s X t s s f f k sign s s f f s k s

dt = = − − = − −

( , ) : ( , ) k x x = F x x + η

{ }

1 ² (ˆ ) ˆ

2

d s f f s k s f f F s s

dt = − − = − − −

η

Controller design

† An equivalent result can be obtained by using integral control, i.e. the variable of interest becomes

† Sliding condition: third order system

† We get

0 T

x dt

∫

( )

2

0 0

: d

2 ²

s x d x x x d

dt δ τ δ δ τ

⎛ ⎞

= ⎜ + ⎟ = + +

⎝ ⎠ ∫ ∫

ˆ ˆ

2 ²

u = − + f x − δ x − δ x

Controller design

† Assume now that dynamic system is

† Where the control gain b is known only to within a certain margin β(X):

† With the estimate of the control gain b(X)

( ) x = + f b X u

1

b b ˆ / β

≤ ≤ β

b ˆ

Controller design

† One can show that the control law

† To satisfy the sliding condition, one introduces the discontinuity

† The control discontinuity has been increased in order to account for the uncertainty on the control gain b(X).

[ ]

ˆ

ˆ ( )

u = b

u − k sign s

( )

ˆ

ˆ

ˆ

u = b

− + f x − δ x

( ) ^{( 1) ˆ}

k = β F + η + β − u

CONTINUOUS CONTROL LAW

TO APPROXIMATE SWITCHED

CONTROL

Continuous control law

† The control laws that satisfy sliding

conditions and that lead to perfect tracking are discontinuous across the surface s(t)

† Discontinuity implies:

„ Chattering

„ High control activity

„ Excitation of high frequency, generally unmodeled

† To circumvent the problem, smoothing out the control discontinuity in a thin boundary layerneighboring the switching surface

S=0 dx/dt

x Chatering behavior

Continuous control law

† Boundary layerneighboring the switching surface

† With

„ The boundary layerthickness

„ The boundary layer width

{ }

( ) , ( , ) B t = X s X t ≤ Φ

Φ

: /

ε = Φ δ

X dx/dt

φ ε

Continuous control law

† Outside of B(t),the control law u as before is selected

„ Guarantees boundary layer attractiveness and positive invariance

„ All trajectories starting inside B(t) remain inside B(t)

† Inside of B(t),interpolate u by replacing the expression of sign(s) by a linear interpolation s/Φ.

( ) if

- s if

sign s s

u s

η η

⎧− > Φ

= ⎨⎪

⎪ Φ ≤ Φ

⎩

+η

-η

S U_s

Φ

−Φ

Continuous control law

† This reduces the chattering BUT

† This skips the perfect tracking ability of the sliding mode controller algorithm since s can not be driven to zero exactly

† This reduces the precision to the guaranteed precision ε

† More generally it guarantees that for all trajectories starting inside B(t)

( )ⁱ

( ) (2 )

1, , 1

x t ≤ δ ε i = … n −

Continuous control law

† The smoothing of the control discontinuity inside B(t) essentially assigns a low-pass filter structure to the local dynamics of the variable s

† It is possible to tune up the control law to achieve a trade-off between tracking precision and robustnessto un-modeled dynamics

† Boundary layer Φcan be made time varyingand monitored to exploit the maximum control bandwidth

Continuous control law

† Let’s consider again the following dynamic equation

† To maintain the attractiveness of the boundary layer while Φ is allowed to vary with time, we have to modify the sliding condition

† The additional term means that the boundary layer attraction is more stringent during BL contraction and

x = + + f u d

1 ²( , ) ( )

2

s d s X t s

dt

η

≥ Φ ⇒ ≤ Φ −

Continuous control law

† In order to satisfy the sliding condition, the control discontinuity gain must be modified

† If the saturation function is defined as

† Then the control law becomes

( ) : ( ) k X = k X − Φ

( ) if 1

( ) ( ) if 1

sat y y y

sat y sign y y

⎧ = ≤

⎨ = >

⎩

: ˆ ( ) ( / )

u = − u k X sat s Φ

APPLICATION

TO ABS CONTROL

Sliding mode controller for ABS

† ABS is a strongly non linear problem

† Sliding mode controller are well suited to control ABS

† One generally wants to maintain a prescribed longitudinal slip s_L* (typically s_L* =0,02)

† The tracking error writes

x = λ

− λ

V R

V λ = ⁻ ω

Sliding mode controller for ABS

† The sliding surface is thus defined as:

† Its derivatives can be evaluated by finite differences between two sampling time steps

( 1)

slide slip slip slip

d d

s x x x

dt dt

δ δ

= + = +

( 1) ( 1)

s k s k

d + − +

Sliding mode controller for ABS

† The surface s_slip=0 divides the state space into 2 parts

† If a sliding mode controller with discontinuous part is selected, the control signals commute from one region to the other.

† This can introduce chattering

Sliding mode controller for ABS

† In order to avoid a high controller activity, one introduces a smooth approximation in the boundary layer [-Φ, +Φ]

Sliding mode controller for ABS

† Smooth transition in the boundary layer can be realized using for instance PWM (power width modulation)

† In the two diagrams, +1 means that the valve is fully open and pressure rises while -1 the valve is fully open an pressure drops.

† If U is in between -1 and +1, this means that pressure increases or decreases progressively. This can be achieved when the opening and closing times times are proportional to the absolute value of U