Applied Sliding Mode Control

Jose Paulo V. S. Cunha1⋆

1Department of Electronics and Telecommunication Engineering

State University of Rio de Janeiro, Brazil

Beihang University, Beijing, China, November 8th, 2017

Outline

1. Introduction

2. Motivating Example:

(a) Linear control

(b) Variable structure control

3. Chattering Phenomena

4. SMC based on observer

5. SMC based on high-gain observer

6. SMC of time-delay systems

7. Applications

8. Conclusion

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.1/59

Introduction

◮ Automatic control applications:

⊲ aerospace;

⊲ robotics;

⊲ consumer electronics;

⊲ industrial process control;

⊲ power systems;

⊲ biomedical;

⊲ etc.

◮ Benefits of automatic control:

⊲ improves transient and steady-state performance;

⊲ reduces the effects of uncertainties and disturbances;

⊲ reduces energy consumption ...

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.2/59

Introduction

◮ Some control approaches:

⊲ linear: PIDs, state feedback, etc;

⊲ linear robust: H∞

, QFT, etc;

⊲ adaptive;

⊲ neural networks;

⊲ fuzzy logic;

⊲ learning control;

⊲ sliding mode control (SMC) or

⊲ variable structure control (VSC) ,

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.3/59

Introduction

◮ Some control approaches:

⊲ linear: PIDs, state feedback, etc;

⊲ linear robust: H∞

, QFT, etc;

⊲ adaptive;

⊲ neural networks;

⊲ fuzzy logic;

⊲ learning control;

⊲ sliding mode control (SMC) or

⊲ variable structure control (VSC) ,

⊲ and if nothing works, ...

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.4/59

Introduction

◮ Some control approaches:

⊲ linear: PIDs, state feedback, etc;

⊲ linear robust: H∞

, QFT, etc;

⊲ adaptive;

⊲ neural networks;

⊲ fuzzy logic;

⊲ learning control;

⊲ sliding mode control (SMC) or

⊲ variable structure control (VSC) ,

⊲ and if nothing works, then try voodoo.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.5/59

Motivating Example

◮ Simple mechanical system:

0

mF

p

◮ Dynamic model:

d2p

dt2=

1

mF .

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.6/59

Motivating Example

◮ State-space model:

x =

[

0 1

0 0

]

x +

[

01m

]

F ,

y =[

1 0]

x ,

where:

⊲ State: x :=

[

p

p

]

,

⊲ Input: u :=[

F]

,

⊲ Output: y :=[

p]

.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.7/59

Motivating Example

◮ Linear control:

⊲ proportional (P) : no damping;

⊲ proportional + derivative (PD) : damped oscillations;

⊲ proportional + integral + derivative (PID) : disturbanceelimination.

◮ PD control is equivalent to state feedback:

u(t) = Kppref (t) − Kx(t) ,

with gain matrix K :=[

Kp Kd

]

.

◮ Problem: closed-loop transfer function is sensitive to m:

Gf (s) :=p(s)

pref (s)=

Kp

ms2 + Kds + Kp

.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.8/59

Motivating Example

◮ Variable structure control (VSC):

⊲ based on state feedback;

⊲ damps oscillations;

⊲ rejects disturbances;

⊲ immune to parametric uncertainties.

◮ Control laws:

u =

{

u+(x, t) , if σ(x) > 0 ,

u−(x, t) , if σ(x) < 0 ,

oru = −ρ(x, t) sgn(σ(x)) .

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.9/59

Motivating Example

◮ Sliding surface:

σ(x) = Sx = 0 .

◮ In this case:σ(x) = p + λp .

◮ When σ(x) = 0 , ∀t ≥ t1 ≥ 0 , the state is governed by:

p + λp = 0 ,

◮ which has the solution:

p(t) = e−λ(t−t1)p(t1) , ∀t ≥ t1 ≥ 0 ,

◮ that is immune to parameter uncertainties or disturbances.

◮ This is the invariance property of SMC!

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.10/59

Motivating Example

◮ Phase portrait:

u = −sgn(σ(x)) , σ(x) = x1 +1

3x2 .

−0,8 −0,6 −0,4 −0,2 0 0,2 0,4 0,6 0,8−1

−0,8

−0,6

−0,4

−0,2

0

0,2

0,4

0,6

0,8

1

x1(m)

x 2(m/s

)

t1

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.11/59

Chattering Phenomena

◮ Ideal sliding mode: infinite switching frequency.

◮ Chattering:

⊲ Imperfections cause finite switching frequency:

⋆ time delays;⋆ hysteresis;⋆ etc.

⊲ May lead to:

⋆ power losses;⋆ mechanical wear;⋆ noise;⋆ tracking errors;⋆ other undesirable effects.

◮ Some remedies: (Utkin, Guldner & Shi 2009).

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.12/59

SMC Based on Observer

◮ State observer to avoid chattering in VSC (Bondarev,Bondarev, Kostyleva & Utkin 1985), (Utkin et al. 2009).

◮ Output-feedback SMC:

⊲ Variable structure model-reference adaptive control(VS-MRAC) (Hsu, Araújo & Costa 1994);

⊲ High-gain observer (HGO) robust to uncertainties designedfor output-feedback VSC (Oh & Khalil 1997), (Cunha,Costa, Lizarralde & Hsu 2009);

⊲ Exact differentiators (Shtessel, Edwards, Fridman &Levant 2014, Hsu, Nunes, Oliveira, Peixoto, Cunha, Costa& Lizarralde 2011), ...

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.13/59

SMC Based on High-Gain Observer

+

−

+

+

+

++

−d(t)unom e

e = CM ζ

e

ζ

S(ε)

r

ρ

u

U σ

y

yM

G(s)

WM (s)

Observer

Model

Plant

Ideal sliding loop

−ρ sgn(σ)

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.14/59

SMC Based on High-Gain Observer

yep=10.7

u

Cart

Rail

A/D D/A

0 y

Data

acquisition

system

Power amplifier

Signalconditioning

Potentiometer voltage

Motor

voltage

Linear gear

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.15/59

SMC Based on High-Gain Observerm

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

0 2 3 4−20

−15

−10

−5

0

5

10

15

20

t (s)

y, y

(

mm

)

1

Linear controlm

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

t (s)

y, y

(

mm

)

HGO + VSC

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.16/59

SMC Based on High-Gain Observer

0 1 2 3 4−1,0

−0,5

0,0

0,5

1,0

t (s)

u (

V)

Linear control

0 1 2 3 4−8

−5

0

5

8

t (s)

u (

V)

HGO + VSC

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.17/59

SMC for Time-Delay Systems

◮ Cascade observers + VSC (Coutinho, Oliveira & Cunha 2014):

σsgn( )−ρ

ρ

d/m

xm^=x

Sξ^

xTξ( )σ

d

d/md/m

d/md/m

−+

−

−

+

+

u y

PlantNonlinear

Observer

Observer#m−1

#m

Observer#1

Ideal sliding loop

Fractional delay

Delay

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.18/59

Applications

1. Control of electrical impedance/admittance:

◮ Example: admittance control.

2. Marine control systems:

◮ Experiments: unmanned surface vehicle (USV) control.

3. Fault tolerant control (FTC):

◮ Example: trailer chain.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.19/59

Impedance/Admittance Control

◮ Impedance/admittance control of an active load (Cunha &Costa 2016);

◮ Model-reference control approach;

◮ Model reference with unstable poles & nonminimum phasezeros is allowed: unlike usual MRAC!

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.20/59

Impedance/Admittance Control

vl

il

YsZ

pi

si = k uc ci

iYs

+

−

Active load Source

Impedance control:Zl(s) = vl(s)

il(s)

il

vscv

v = k uc

vl

Yp sZv

Zs

Active load Source

−

+

+

−

Admittance control:Yl(s) = il(s)

vl(s)

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.21/59

Impedance/Admittance Control

◮ Model reference: Gm(s) = Gm1(s)Gm2(s) .

ym

G (s)p

m2G (s)

m1G (s)

G (s)s

r1

uc

kc

us

r1

ys

Controller

yr e−

+

y er u

+

−

+

+

Passive Load

Model Reference

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.22/59

Impedance/Admittance Control

◮ Model-reference adaptive control (MRAC):

u(t) = θT (t)ω(t) ,

θ(t) = Γe(t)ω(t) .

◮ Variable structure model-reference adaptive control(VS-MRAC):

u = −ρ(t) sgn(e) .

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.23/59

Impedance/Admittance Control

◮ H-bridge realization of the active load:

S1

S4

S3

S2

Vccv

c

Yp

vl

y

vs

sZ

il

+

−

−

+

r+

−

Fonte

◮ MRAC: pulse-width modulated (PWM) control signal;

◮ VS-MRAC: drives the power switches directly.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.24/59

Example: Admittance Control

◮ Passive load:

R1

R2

SL1

L1

L2

◮ Reference model:

Gm2(s) = km

s2(s + 2πfc)

[s2 + 2ζ(2πfr) s + (2πfr)2]2

Gm1(s) =1

s + 2πfc

⊲ ζ = 0.2 , fr = 60 Hz , km = 2 kS rad2s2 ,

⊲ fc = 300 Hz .Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.25/59

Example: Admittance Control

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−100

100

−50

50

time (s)

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−100

100

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−400

−200

200

400

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−10

10

−5

5

MRAC: transient

9.9 9.92 9.94 9.96 9.989.91 9.93 9.95 9.97 9.99

0

−100

100

−50

50

time (s)

9.9 9.92 9.94 9.96 9.989.91 9.93 9.95 9.97 9.99

0

−100

100

9.9 9.92 9.94 9.96 9.989.91 9.93 9.95 9.97 9.99

0

−400

−200

200

400

9.9 9.92 9.94 9.96 9.989.91 9.93 9.95 9.97 9.99

0

−10

10

−5

5

MRAC: steady-state

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.26/59

Example: Admittance Control

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−2

−1

−1.5

−0.5

time (s)

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−0.006

−0.004

−0.002

0.002

0.004

0.006

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−0.4

−0.2

0.2

−0.3

−0.1

0.1

0.3

MRAC: parameters

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.27/59

Example: Admittance Control

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−100

100

time (s)

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−100

100

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−200

200

0 0.10.02 0.04 0.06 0.080.01 0.03 0.05 0.07 0.09

0

−10

10

VS-MRAC signals

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.28/59

Marine Control Systems

◮ Characteristics of marine systems:

⊲ Hydrodynamics is unknown or uncertain;

⊲ Large parametric changes (e.g.: load mass);

⊲ Environmental disturbances: currents, waves and winds.

◮ SMC is advantageous!

◮ Applications:

⊲ State-feedback VSC for remotely operated underwatervehicles (ROVs), (Yoerger, Newman & Slotine 1986);

⊲ Output-feedback VS-MRAC for ROVs (Cunha, Costa &Hsu 1995);

⊲ VSC trajectory tracking of unmanned surface vessels(USVs), (Cheng, Yi & Zao 2007), (Rosario & Cunha 2017).

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.29/59

Control of an USV

◮ Feedback linearization:

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.30/59

Control of an USV

◮ Small boat:

⊲ Maximum forward speed: 0.26 m/s ;

⊲ Length: 0.48 m ;

⊲ Mass: 1.4 kg ;

⊲ 3 thrusters with DC motors;

⊲ Commanded by an Arduino + Wi-Fi.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.31/59

Control of an USV

◮ Position and attitude measurement: 4 Vicon MX cameras;

◮ Sampling frequency up to 1 kHz;

◮ Accuracy better than 1 mm.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.32/59

Control of an USV

◮ Zigzag trajectory simulate a scan;

◮ Velocity: 0.1 m/s;

◮ Tests:

1. PD at fs = 30 Hz;

2. VSC at fs = 30 Hz;

3. VSC at fs = 150 Hz.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.33/59

Control of an USV

PD at fs = 30 Hz.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.34/59

Control of an USV

VSC at fs = 30 Hz.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.35/59

Control of an USV

VSC at fs = 150 Hz.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.36/59

Control of an USV

PD at fs = 30 Hz;

VSC at fs = 30 Hz; VSC at fs = 150 Hz.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.37/59

Fault Tolerant Control

◮ Consider the system in regular form:

x1 = A11x1 + A12yp ,

yp = A21x1 + A22yp + Kp(t) [up + df (x1, yp, t)] ,

◮ Output (measured) : yp ∈ Rm ,

◮ Other state variables (not measured) : x1 ∈ Rn−m .

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.38/59

Fault Tolerant Control

◮ Consider the system in regular form:

x1 = A11x1 + A12yp ,

yp = A21x1 + A22yp + Kp(t) [up + df (x1, yp, t)] ,

◮ Output (measured) : yp ∈ Rm ,

◮ Other state variables (not measured) : x1 ∈ Rn−m .

◮ Actuator faults modeled by (Cunha, Costa, Hsu &Oliveira 2015):

⊲ Time-variant control distribution matrix Kp(t) and/or

⊲ Input disturbance df (x1, yp, t).

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.39/59

Fault Tolerant Control

◮ Unit vector control (UVC) stabilizes the system if (Cunhaet al. 2015):

⊲ Matrix A11 is Hurwitz (minimum phase assumption);

⊲ Exist P = P T > 0 and Q = QT > 0 such that

KTp (t)P + PKp(t) − Q ≥ 0 , ∀t ≥ 0 .

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.40/59

Fault Tolerant Control

◮ Example: Trailer Chain

m3 2m m41m

v v1 v3

k31

4l310 0

Actuator 1

Actuator 2

1

F2 F

F

2

2

b23 b24

k24

42l v

Fourth trailer can be disconnected

yp =

[

v1

v2

]

up =

[

F1

F2

]

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.41/59

Fault Tolerant Control

◮ Without fourth trailer: x1 =

[

l31

v3

]

◮ Fourth trailer connected: x1 =

l31

v3

l42

v4

◮ Uncertain parameters:

0.5 ≤ m3 ≤ 1.5 (kg)

2.5 ≤ b23 ≤ 6 (Ns/m)

25 ≤ k31 ≤ 35 (N/m)

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.42/59

Fault Tolerant Control

◮ Model Matching :

⊲ Reference model: yM(t) = −γMyM(t) + r(t) .

⊲ r(t) is piecewise continuous & uniformly bounded.

⊲ Velocity error: e(t) = yp(t) − yM(t) → 0 .

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.43/59

Fault Tolerant Control

◮ Unit vector control: u = unom − ρe

||e||.

◮ Modulation function: ρ = δ + c2‖yp‖ + c3‖r‖ + c4 x1(t) .

◮ The signal x1(t) ≥ ‖gy(t) ∗ yp(t)‖ ,

with gy(t) = A21 exp(A11t)A12 ,

is estimated by the first order approximation filter (FOAF)(Cunha, Costa & Hsu 2008):

˙x1(t) = −γ1x1(t) + c5 ‖yp(t)‖ .

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.44/59

Fault Tolerant Control

◮ FOAF:

1 2 3 4 50

5

10

15

20

30

35

40

0

25

FOAF

yg (t)|| ||

yg (t)|| ||

Impulse response norm

time (s)

Envelope function

without fourth trailer

with fourth trailer connected

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.45/59

Fault Tolerant Control

yM1

v1

yM2

v2

time (s)

time (s)

Ve

loci

ties

(m/s

)5.0

2.5

0.0

−2.5

−5.0 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5

0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5

5.0

2.5

0.0

−2.5

−5.0

+

+

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.46/59

Fault Tolerant Control

F

(N)

1F

(N

)2

time (s)

time (s)

Fourth trailer is connected

500

250

0

−250

−500 0.5 1.0 1.5 2.0 2.5 3 3.5 4.0 4.5 5.00.0

500

250

0

−250

−500 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

+

+

Average control

Average control

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.47/59

Conclusion

◮ SMC:

⊲ Robust to parametric uncertainties;

⊲ Rejects disturbances;

⊲ Invariance property guarantees good performance.

◮ VSC:

⊲ Suitable for switched actuators such as solenoid valves andpower electronics devices;

⊲ Simple realization.

◮ Disadvantages:

⊲ Requires high-frequency switching;

⊲ Chattering;

⊲ Sensitive to measurement noise.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.48/59

Current Works

◮ Control of a vessel pushing an underactuated floating load;

◮ Cooperative control of USVs;

◮ SMC of partial differential equations (PDEs) (Molina &Cunha 2017);

◮ Adaptive SMC:

⊲ Extended equivalent control approach (Oliveira, Cunha &Hsu 2016);

⊲ Monitoring function (Yan, Hsu & Xiuxia 2006) approach(Oliveira, Melo, Hsu & Cunha 2017);

◮ Uncertain time-delay systems.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.49/59

Suggested Books

◮ Classic book: (Utkin 1978).

◮ More recent books: (Utkin 1992), (Edwards & Spurgeon 1998)and (Utkin et al. 2009).

◮ Book with up to date techniques: (Shtessel et al. 2014).

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.50/59

Contact

◮ Prof. José Paulo V. S. Cunha

◮ E-mail: jpaulo@ieee.org

◮ Homepage: http://www.lee.uerj.br/˜jpaulo

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.51/59

References

Bondarev, A. G., Bondarev, S. A., Kostyleva, N. E. & Utkin, V. I.(1985), ‘Sliding modes in systems with asymptotic stateobservers’, Autom. Remote Control 46(6), 679–684. Pt. 1.

Cheng, J., Yi, J. & Zao, D. (2007), ‘Design of a sliding modecontroller for trajectory tracking problem of marine vessels’,IET Control Theory Appl. 1(1), 233–237.

Coutinho, C. L., Oliveira, T. R. & Cunha, J. P. V. S. (2014),‘Output-feedback sliding-mode control via cascade observersfor global stabilisation of a class of nonlinear systems withoutput time delay’, Int. J. Contr. 87(11), 2327–2337.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.52/59

References

Cunha, J. P. V. S. & Costa, R. R. (2016), ‘Model-referenceimpedance and admittance control of linear systems’, Int. J.Adaptive Contr. Signal Process. 30(8–10), 1317–1332.

Cunha, J. P. V. S., Costa, R. R. & Hsu, L. (1995), ‘Design of a highperformance variable structure position control of ROV’s’, IEEEJ. Oceanic Eng. 20(1), 42–55.

Cunha, J. P. V. S., Costa, R. R. & Hsu, L. (2008), ‘Design offirst-order approximation filters for sliding-mode control ofuncertain systems’, IEEE Trans. Ind. Electronics55(11), 4037–4046.

Cunha, J. P. V. S. – Applied SMC – Beihang University – 2017 – p.53/59

References

Cunha, J. P. V. S., Costa, R. R., Hsu, L. & Oliveira, T. R. (2015),‘Output-feedback sliding-mode control for systems subjectedto actuator and internal dynamics failures’, IET Control TheoryAppl. 9(4), 637–647.

Cunha, J. P. V. S., Costa, R. R., Lizarralde, F. & Hsu, L. (2009),‘Peaking free variable structure control of uncertain linearsystems based on a high-gain observer’, Automatica45(5), 1156–1164.

Edwards, C. & Spurgeon, S. K. (1998), Sliding Mode Control:Theory and Applications, Taylor & Francis Ltd., London.

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References

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