Applied Sliding Mode Control e Paulo V. S. Cunha 1 Jos 1 Department - - PowerPoint PPT Presentation

Applied Sliding Mode Control

Jos´ e 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:

m F p

◮ Dynamic model: d2p dt2 = 1 m F .

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

Motivating Example

◮ State-space model: ˙ x =

• 1
• x +
• 1

m

• F ,

y =

• 1
• 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) : disturbance

elimination.

◮ 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 ,

• r

u = −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!

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Motivating Example

◮ Phase portrait: u = −sgn(σ(x)) , σ(x) = x1 + 1 3x2 .

−0,8 −0,6 −0,4 −0,2 0,2 0,4 0,6 0,8 −1 −0,8 −0,6 −0,4 −0,2 0,2 0,4 0,6 0,8 1 x1(m) x2(m/s)

t1

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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).

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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 designed

for 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 ˆ e = CM ˆ ζ ˜ e ˆ 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

y ep=10.7 u Cart Rail A/D D/A y Data acquisition system Power amplifier Signal conditioning Potentiometer voltage Motor voltage Linear gear

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

SMC Based on High-Gain Observer

m

1 2 3 4 −20 −15 −10 −5 5 10 15 20 2 3 4 −20 −15 −10 −5 5 10 15 20 t (s) y, y (mm) 1

Linear control

m

1 2 3 4 −20 −15 −10 −5 5 10 15 20 1 2 3 4 −20 −15 −10 −5 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

1 2 3 4 −1,0 −0,5 0,0 0,5 1,0 t (s) u (V)

Linear control

1 2 3 4 −8 −5 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):

σ ^ sgn( ) −ρ

ρ

d/m ^ x m ^= x S

ξ ^

^ x

Tξ( ) ^ σ

d d/m d/m d/m d/m − + − − + +

u y

Plant Nonlinear 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.

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Impedance/Admittance Control

◮ Impedance/admittance control of an active load (Cunha &

Costa 2016);

◮ Model-reference control approach; ◮ Model reference with unstable poles & nonminimum phase

zeros is allowed: unlike usual MRAC!

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

Impedance/Admittance Control

vl il Y

s

Zp is i = k u

c ci

iYs

+

−

Active load Source

Impedance control:

Zl(s) = vl(s)

il(s)

il vs

cv

v = k u

c

vl Yp

s

Z vZs

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 m2

G (s)

m1

G (s) G (s)

s

r1 uc kc us r1 ys

Controller

y r e

− +

y e r 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:

S

1

S

4

S

3

S

2

Vcc vc Yp vl y vs

s

Z 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: R 1 R 2 S

L1

L 1 L 2 ◮ 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 .

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Example: Admittance Control

0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −100 100 −50 50

time (s)

0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −100 100 0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −400 −200 200 400 0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −10 10 −5 5

MRAC: transient

9.9 9.92 9.94 9.96 9.98 9.91 9.93 9.95 9.97 9.99 −100 100 −50 50

time (s)

9.9 9.92 9.94 9.96 9.98 9.91 9.93 9.95 9.97 9.99 −100 100 9.9 9.92 9.94 9.96 9.98 9.91 9.93 9.95 9.97 9.99 −400 −200 200 400 9.9 9.92 9.94 9.96 9.98 9.91 9.93 9.95 9.97 9.99 −10 10 −5 5

MRAC: steady-state

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

Example: Admittance Control

0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −2 −1 −1.5 −0.5

time (s)

0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −0.006 −0.004 −0.002 0.002 0.004 0.006 0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −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

0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −100 100

time (s)

0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −100 100 0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −200 200 0.1 0.02 0.04 0.06 0.08 0.01 0.03 0.05 0.07 0.09 −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 underwater

vehicles (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.

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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.

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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.

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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 (Cunha

et al. 2015):

⊲ Matrix A11

is Hurwitz (minimum phase assumption);

⊲ Exist P = P T > 0

and

Q = QT > 0

such that

KT

p (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

2

m m4

1

m v v

1

v

3

k

31 4

l31 Actuator 1 Actuator 2

1

F

2

F F

2 2

b

23

b

24

k

24 42

l 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 − ρ e ||e|| . ◮ Modulation function: ρ = δ + c2yp + c3r + 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) = −γ1¯ x1(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 5 5 10 15 20 30 35 40 25

FOAF

y

g (t) || ||

y

g (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) Velocities (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 5.0 0.5 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.5 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)

1

F (N)

2

time (s) time (s)

Fourth trailer is connected

500 250 −250 −500 0.5 1.0 1.5 2.0 2.5 3 3.5 4.0 4.5 5.0 0.0 500 250 −250 −500 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0

+ +

Average control Average control

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Conclusion

◮ SMC: ⊲ Robust to parametric uncertainties; ⊲ Rejects disturbances; ⊲ Invariance property guarantees good performance. ◮ VSC: ⊲ Suitable for switched actuators such as solenoid valves and

power 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.

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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).

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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 state

• bservers’, Autom. Remote Control 46(6), 679–684. Pt. 1.

Cheng, J., Yi, J. & Zao, D. (2007), ‘Design of a sliding mode controller 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 observers for global stabilisation of a class of nonlinear systems with

• utput 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-reference impedance 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 high performance variable structure position control of ROV’s’, IEEE

• J. Oceanic Eng. 20(1), 42–55.

Cunha, J. P . V. S., Costa, R. R. & Hsu, L. (2008), ‘Design of first-order approximation filters for sliding-mode control of uncertain systems’, IEEE Trans. Ind. Electronics

55(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 subjected to actuator and internal dynamics failures’, IET Control Theory

• Appl. 9(4), 637–647.

Cunha, J. P . V. S., Costa, R. R., Lizarralde, F . & Hsu, L. (2009), ‘Peaking free variable structure control of uncertain linear systems based on a high-gain observer’, Automatica

45(5), 1156–1164.

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

Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.54/59

References

Hsu, L., Araújo, A. D. & Costa, R. R. (1994), ‘Analysis and design of I/O based variable structure adaptive control’, IEEE Trans. Aut.

• Contr. 39(1), 4–21.

Hsu, L., Nunes, E. V. L., Oliveira, T. R., Peixoto, A. J., Cunha, J. P .

• V. S., Costa, R. R. & Lizarralde, F

. (2011), Output feedback sliding mode control approaches using observers and/or differentiators, in L. Fridman, J. Moreno & R. Iriarte, eds, ‘Sliding Modes after their First Decade of the 21st Century: State of the Art’, Springer-Verlag, pp. 269–292. Também referenciado como Lecture Notes in Control and Information Sciences vol. 412.

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References

Molina, N. I. C. & Cunha, J. P . V. S. (2017), A distributed parameter approach for sliding mode control of soil irrigation, in ‘Proc. of the 20th IFAC World Congress’, Vol. 50, Toulouse, France,

• pp. 2714–2719.

Oliveira, T. R., Cunha, J. P . V. S. & Hsu, L. (2016), Adaptive sliding mode control for disturbances with unknown bounds, in ‘Proc. 14th Int. Workshop on Variable Structure Sys.’, Nanjing, Jiangsu, China, pp. 59–64. Oh, S. & Khalil, H. K. (1997), ‘Nonlinear output-feedback tracking using high-gain observer and variable structure control’, Automatica 33(10), 1845–1856.

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References

Oliveira, T. R., Melo, G. T., Hsu, L. & Cunha, J. P . V. S. (2017), Monitoring functions applied to adaptive sliding mode control for disturbance rejection, in ‘Proc. of the 20th IFAC World Congress’, Vol. 50, Toulouse, France, pp. 2684–2689. Rosario, R. V. C. & Cunha, J. P . V. S. (2017), Experimental variable structure trajectory tracking control of a surface vessel with a motion capture system, in ‘Proc. 43rd Conf. IEEE Ind. Electronics Soc. (IECON)’, Beijing, China, pp. 2864–2869. Shtessel, Y., Edwards, C., Fridman, L. & Levant, A. (2014), Sliding Mode Control and Observation, Springer, New York.

Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.57/59

References

Utkin, V., Guldner, J. & Shi, J. (2009), Sliding Mode Control in Electro-Mechanical Systems, 2nd edn, CRC Press, Boca Raton. Utkin, V. I. (1978), Sliding Modes and Their Application in Variable Structure Systems, MIR Publishers, Moscow. Utkin, V. I. (1992), Sliding Modes in Control and Optimization, Springer-Verlag, Berlin.

Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.58/59

References

Yan, L., Hsu, L. & Xiuxia, S. (2006), ‘A variable structure MRAC with expected transient and steady-state performance’, Automatica 42(5), 805–813. Yoerger, D. N., Newman, J. B. & Slotine, J. J. E. (1986), ‘Supervisory control system for the Jason ROV’, IEEE J. Oceanic Eng. 11(3), 392–400.

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