1 Hiroshima University, Japan http://home.hiroshima-u.ac.jp/kikuuwe/ http://www.youtube.com/kikuuwe/ Ryo Kikuuwe a Anti-Noise and Anti-Disturbance Anti-Noise and Anti-Disturbance Properties of Properties of Differential-Algebraic Relaxation Differential-Algebraic Relaxation Applied to Applied to a Set-Valued Controller a Set-Valued Controller 

2 Outline Outline  [1] Overview of “Proxy-based Sliding Mode Control” (PSMC)  [2] Frequency Domain Properties  [3] Time Domain Properties  [4] Conclusions main results 

3 [1] [1] Overview of Overview of Proxy-based Sliding Mode Control Proxy-based Sliding Mode Control (PSMC) (PSMC) [1] [1] Overview of Overview of Proxy-based Sliding Mode Control Proxy-based Sliding Mode Control (PSMC) (PSMC) 

4 Proxy-based Sliding Mode Control Proxy-based Sliding Mode Control (PSMC) (PSMC)  PID Control  A position (joint angle) controller for manipulators.  As accurate as PD (PID) control in normal operation (usually robust enough) .  Slow and non-overshooting after torque saturation (with slowness explicitly designed by a parameter.) [Kikuuwe & Fujimoto: ICRA2006] [Kikuuwe et al., 2010, IEEE-TRO ] 

5 It’s been applied to many devices It’s been applied to many devices [Van Damme et al., 2007]  Pneumatic actuators  Rehabilitation devices  Piezoactuators  Motion platform  Fuel cells, etc [Kashiri et al. 2016] [Jin et al. 2016] [Gu et al. 2015] [Tanaka et al., 2010] [Yoshimoto et al. 2015] [Prieto et al. 2013] [Hasturk et al. 2011] [Chen et al., 2016] [Liao et al. 2015] [Nishi & Katsura, 2015] 

6 Control law of PSMC Control law of PSMC  It is written as DAI (differential algebraic inclusions)  for 2nd-order plants written as:  pd : desired position  p : position (measured by sensors)  u : actuator force command  q : state variable of the controller (proxy’s position)  B < H.  It realizes overdamped convergence with the slowness designed by H. proxy controlled object desired position SMC PD q p pd u u slope H stiffness K (should be high) damping time constant B 

7 It includes “SMC-B” as a special case It includes “SMC-B” as a special case  SMC with boundary layer (SMC-B) is a special case of PSMC with B = H. F ¡F ¾ u stiffness K  PSMC (in DAI form, B < H)  PSMC (in ODE form, B < H) algebraically equivalent (No approximation) 

8 Our Empirical Results in 2006 Our Empirical Results in 2006  Both controllers produce the slowness designated by H.  But SMC-B results in noisy actuator sound and bad tracking.  Our claim: PSMC is robust to noise and disturbance.  SMC-B (B = H)  PSMC (B < H)  position p  torque u  position error || p - p d || power down 

9 Contribution of this paper Contribution of this paper  Justifications of our empirical claims. noise response noise response disturbance response disturbance response …. in frequency and time domains 

10 [2] [2] Frequency Domain Properties Frequency Domain Properties [2] [2] Frequency Domain Properties Frequency Domain Properties 

11 Setup for Freq-Domain Analysis Setup for Freq-Domain Analysis  As long as u is not saturated, it is written as:  2nd-Order Plant: where:  with disturbance ³ and noise  PSMC: B < H: Design parameters are K and B  SMC-B: B = H. The only design parameter is K  Controller: B = H equivalent (noise response) (disturbance response) (initial-value response) 

12 Gain Plots: PSMC vs SMC-B Gain Plots: PSMC vs SMC-B  Noise response  Disturbance response  With SMC-B, smaller K reduces the influence of noise but increases the influence of disturbance  Performance of PSMC cannot be achieved by SMC-B  PSMC: B = 0.04, H = 1  SMC-B: B = H = 1 KH/M 1/K 

13 Effects of Noise Filtering + SMC-B Effects of Noise Filtering + SMC-B  Stronger filtering may improve the noise response.  But it causes peaky resonance and can even destabilize the system.  Noise response  Disturbance response  PSMC: B = 0.04, H = 1, K = 5000  SMC-B: B = 1, H = 1, K = 5000  Noise filter: 

14 Summary of Freq. Domain Results Summary of Freq. Domain Results  PSMC realizes better balance of noise reduction and disturbance rejection than SMC-B.  Parameter tuning of SMC-B cannot achieve the performance of PSMC.  Low-pass filtering cannot either. [ Disturbance response ] [ Noise response ] 

15 [3] [3] Time Domain Properties Time Domain Properties [3] [3] Time Domain Properties Time Domain Properties 

16 Setup for Numerical Simulations Setup for Numerical Simulations  Ideal (set-valued) SMC  F = 100, H = 1  It realizes the Fillipov solution:  2nd-Order Plant  initial value: p 0 = 0.03  disturbance: ³ = 50 for t Î [1.0,1.5]  noise: | | < 0.001  Controller  F = 100, H = 1  Timestep size: h = 0.001  SMC-B: B = H, ¸ = 0  SMC-BF: B = H, ¸ = 0.05  PSMC: B < H, ¸ = 0 equivalent B = H 

17 Disturbance Response: SMC-B vs PSMC Disturbance Response: SMC-B vs PSMC  Disturbance results in deviation from Fillipov solution with both controllers.  It’s temporary with PSMC but permanent with SMC-B.  SMC-B:  B = H = 1, K = 100  K is set optimal so that chattering does not happen.  PSMC:  B = 0.04, K = 5000 period of disturbance Fillipov solution (the ideal SMC) 

18 Disturbance Response: PSMC vs SMC-B Disturbance Response: PSMC vs SMC-B  The “proxy” stays on the sliding mode even when the “real” object is displaced.  SMC-B:  B = H = 1, K = 100  K is set optimal so that chattering does not happen  PSMC:  B = 0.04, K = 5000 period of disturbance Fillipov solution (the ideal SMC) 

19 Effects of Noise & Filtering Effects of Noise & Filtering  PSMC is much more robust to noise than SMC-B.  Filtering improves SMC-B but it does not achieve the accuracy of PSMC. without noise SMC-B SMC-B with filtering PSMC with noise 

20 Results with Different Results with Different K K and and ¸ ¸  Chattering intensity  Error from Fillipov solution  PSMC results in smaller chattering and higher tracking accuracy than SMC-B with noise filtering.  SMC-B: H = B = 1  PSMC: H = 1, B = 0.005 

21 [4] [4] Conclusions Conclusions [4] [4] Conclusions Conclusions 

22 Conclusions Conclusions  We have investigated noise and disturbance responses of a controller, “PSMC”.  Frequency-domain results have shown that PSMC has better balance of anti-noise and anti-disturbance properties as long as it is unsaturated.  Time-domain results have shown that its behavior is quite different from conventional implementations of SMC.