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

Applied Sliding Mode Control e Paulo V. S. Cunha 1 ⋆ Jos´ 1 Department of Electronics and Telecommunication Engineering State University of Rio de Janeiro, Brazil Beihang University, Beijing, China, November 8 th , 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 0 p ◮ Dynamic model: d 2 p dt 2 = 1 m F . Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.6/59

Motivating Example ◮ State-space model: � � � � 0 1 0 x = ˙ x + F , 1 0 0 m � � y = x , 1 0 where: � � p ⊲ State: x := , 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 ) = K p p ref ( t ) − Kx ( t ) , � � K := . with gain matrix K p K d ◮ Problem: closed-loop transfer function is sensitive to m : p ( s ) K p G f ( s ) := p ref ( s ) = . ms 2 + K d s + K p 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 + ( x, t ) , if σ ( x ) > 0 , u = u − ( x, t ) , if σ ( x ) < 0 , or 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 . σ ( x ) = 0 , ∀ t ≥ t 1 ≥ 0 , ◮ When the state is governed by: p + λp = 0 , ˙ ◮ which has the solution: p ( t ) = e − λ ( t − t 1 ) p ( t 1 ) , ∀ t ≥ t 1 ≥ 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: σ ( x ) = x 1 + 1 u = − sgn( σ ( x )) , 3 x 2 . 1 0,8 0,6 0,4 t 1 0,2 x 2 (m/s) 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 x 1 (m) 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 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 Model y M r W M ( s ) − e − e ˜ d ( t ) u nom Plant + + e = C M ˆ ˆ ζ + + y u + G ( s ) + Observer ˆ ζ Ideal sliding loop σ ¯ U ¯ S ( ε ) − ρ sgn(¯ σ ) ρ Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.14/59

SMC Based on High-Gain Observer u Power amplifier Data acquisition A/D D/A system Motor e p =10.7 y voltage Potentiometer voltage Cart Signal Rail conditioning Linear gear 0 y Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.15/59

SMC Based on High-Gain Observer 20 20 20 20 15 15 15 15 10 10 10 10 y, y (mm) 5 5 5 5 y, y (mm) 0 0 0 0 m m −5 −5 −5 −5 −10 −10 −10 −10 −15 −15 −15 −15 −20 −20 −20 −20 0 0 1 1 2 2 3 3 4 4 0 0 1 1 2 2 3 3 4 4 t (s) t (s) Linear control HGO + VSC Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.16/59

SMC Based on High-Gain Observer 8 1,0 5 0,5 u (V) u (V) 0 0,0 −0,5 −5 −1,0 −8 0 1 2 3 4 0 1 2 3 4 t (s) t (s) Linear control 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): ρ Delay ^ ξ ^ σ y u Nonlinear ^ −ρ σ ^ d T ξ ( ) sgn( ) x S Plant Ideal sliding loop d/m Observer #m − ^= ^ + x x m d/m d/m Observer #m−1 − + Fractional delay d/m d/m Observer #1 − + 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 phase zeros is allowed: unlike usual MRAC! Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.20/59

Impedance/Admittance Control i l i l Active load Source Active load Source − + + Z Y p v Zs s i = k u Z p i Ys i s Y v l v l c ci s + v s − v = k u − c cv Impedance control: Admittance control: Y l ( s ) = i l ( s ) Z l ( s ) = v l ( s ) v l ( s ) i l ( s ) Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.21/59

Impedance/Admittance Control G m ( s ) = G m 1 ( s ) G m 2 ( s ) . ◮ Model reference: Model Reference r 1 G (s) G (s) m2 m1 y m Passive Load − e u s r y + + G (s) p + − + u c k c G (s) s y s r 1 y e r u Controller Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.22/59

Impedance/Admittance Control ◮ Model-reference adaptive control (MRAC): u ( t ) = θ T ( t ) ω ( t ) , ˙ θ ( t ) = Γ e ( 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: i l y S S 1 4 Fonte Y p − + Z r s + v c v l V cc v s − − + S S 2 3 ◮ 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 S L1 ◮ Passive load: R 1 R 2 L 2 L 1 ◮ Reference model: s 2 ( s + 2 πf c ) G m 2 ( s ) = k m [ s 2 + 2 ζ (2 πf r ) s + (2 πf r ) 2 ] 2 1 G m 1 ( s ) = s + 2 πf c k m = 2 kS rad 2 s 2 , ⊲ ζ = 0 . 2 , f r = 60 Hz , ⊲ f c = 300 Hz . Cunha, J. P . V. S. – Applied SMC – Beihang University – 2017 – p.25/59