On Practical Applications of Active Disturbance Rejection Control

2010 Chinese Control Conference On Practical Applications of Active Disturbance Rejection Control Qing Zheng Gannon University Zhiqiang Gao Cleveland State University

Outline Ø Introduction Ø Active Disturbance Rejection Control Ø Practical Applications Ø Conclusions 2

Introduction: The Three Paradigms Industry Control Practice Modern Control Active Theory Disturbance PID + Rejection Feedforward Model Based

The Industry Paradigm y& = p( y, y&, w, u, t) u= ( K e+ K e+ K e& ) + u p I D ff Disturbance Rejection: PID tuning

The Model Paradigm Modeling Plant: y& = p( y, y&, w, u, t) q( y, y& ) + u Design Goal: y& = g( y&, y) Known Control Law: u = qyy (&, ) + gyy (&, ) Examples: pole placement; feedback linearization; etc.

Between knowing none and knowing all u Plant y& = p( y, y&, w, u, t) u Primary Dynamics k y&= bu, b= T m u Secondary Dynamics y& = f( y, y&, w, t) + bu f( y, y&, w, t) = p( y, y&, w, u, t) bu

The Dist. Rej. Paradigm Total Disturbance Plant: y& = f ( yy, &, wt, ) + bu b = 1 Design Goal: y& = g( y&, y) Dist. Estimation: Control Law: f ˆ( t) f( y, y &, w,) t u = f ˆ( t) + g( y &, y)

Active Disturbance Rejection x& = x 1 2 x& = f + u 2 y = x1 Uncertain Nonlinear Time Varying Complex u= u fˆ 0 x& = x x& u y = x1 Fixed 1 2 2 0 Linear Time Invariant Simple

The Extended State Observer Augmented plant in state space: y& = f( y, y&, w, t) + u x = y, x = y&, x = f 1 2 3 x& 1 = x2 x& 2 = x3 + u, x& 3 = f& y = x1 Extended State Observer [Han,95] z& 1 = z2 β1g1( z1 y) z& 2 = z3 β2g2( z1 y) + u z& 3 = β3g3( z1 y) z x z x z x = f 1 1 2 2 3 3

Active Disturbance Rejection Control r Controller ADRC Structure z 3 u0 + = fˆ z, z 1 2 - u 1 ˆ 1/b u + ESO Idea u Estimate and cancel the generalized disturbance d + Plant u Reduce the plant to double integrators u Use a PD controller to control the double integrator plant + + y n u= u / bˆ x x 1 2 x3 z z z k k = 1 y = y& x 1 1 x 2 2 x 3 3 P D = f u1 = u0 z3 = ω 2 c = 2ω c y& = f( y, y&, d, t) + bu y& = f( y, y&, d, t) + u 0 1 0 0 x& = x+ u 1 + f 0 0 1 1 y = 0 1 0 0 3 ω o 2 z& = 0 0 1 z 1 + u 1+ 3 ω o ( y z 1) 3 0 0 0 0 ω o fˆ = 0 0 1 z [ ] y& u 0 [ 1 0] x u = k ( r z ) + k ( r& z ) + r& 0 P 1 D 2 1 10

Practical Applications Simulation and Hardware Tests 11

Application 1: Motion Control Ø In a typical application using motor as the power source, y& = f(, t y, y&, w) + bu (1) Ø In most motion control literature, the linear time-invariant approximation is used: a b y& = y& + u J J t t (2) 12

Application 1: Motion Control Control System Design Objectives: Ø Track the desired trajectory quickly and accurately; Ø Smoother control signal and lower level of wear and tear of actuators; Ø High degree of robustness; Ø Better external disturbance rejection capability; Ø Simplification of controller design and tuning 13

Application 1: Motion Control Output Tracking error Control signal 1.5 LADRC performance 1 0.5 nominal plant with disturbance and increased inertia 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Time(s) 0.2 0.1 0 4 2 0-2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time(s) -4 0 0.5 1 1.5 2 2.5 3 3.5 4 Time(s) 14

Application 2: Web Tension Regulation A Typical Web Winding System

Application 2: Web Tension Regulation 1 v & c = ( Ntc Ff + uc ) g M c 1 2 2 v& e = ( Bfve + R ( tc tr ) + RKeue + R δe ) J 1 ( 2 ( ) 2 v& p = Bfvp + R tc tr + RKpup + R δ p ) J AE 1 t& = v c c ( v v e p ) x () t + N c Parameter variations and uncertainties External Disturbances Tension-velocity Coupling Nonlinear, sensitive to velocity variations n Nonlinear tension dynamics; n External Disturbance; n Large amount of parameter variations and uncertainties; n Tension-velocity strong coupling;

Application 2: Web Tension Regulation Two different ADRC solutions for the tension loop are investigated: open-loop (ADRC1) and closed-loop (ADRC2). 5 x 10-4 IC IC LBC ADRC1 Carriage Velocity Error(m/sec) 0 ADRC1 LBC -5 0 50 100 150 200 250 300 350 400 Time sec Velocity tracking errors for IC, LBC and ADRC1. IC: the PID based industry controller, LBC: Lyapunov Based Controller 17

Application 2: Web Tension Regulation 14 12 LBC ADRC1 ADRC2 Error of Tensions 10 8 6 4 2 0 Lyapunov Based Controller (LBC) ADRC1 ADRC2-2 0 50 100 150 200 250 300 350 400 Time sec Tension tracking error for LBC, ADRC1, ADRC2 18

Application 3: DC-DC Power Converter Digitally Controlled Power Converter

Application 3: DC-DC Power Converter Linear model of H-bridge converter 20

Application 3: DC-DC Power Converter Load step-up (3A 36A) disturbance rejection. 21

Application 4: Continuous Stirred Tank Reactor (CSTR) F, C, T w in A, in in F, T j ρ, V, Cp mixer ρ, V, C w j pw F, T w w x& C x A, in 1 0 rx 1 V V VHrx + UA( x x ) T x u VρCp V UA( x x ) T x 0 1 3 2 in 2 = + 0 w 3 2 3 V V j w jρwc ρ pw F, C, T out A T CA, in x 1 y1 y2 = x2 CA, in [ ] T Ø MV: the reactant feed flow rate the coolant water mass rate Ø CV: the reactor concentration C A the reactor temperature T F in F w r = k x 0 E exp( ) Rx [ ] T [ u, u ] [ F, F ] 2 T T = x1, x2, x3 = CA, T, T j u = = 1 2 in w T 22

Application 4: CSTR Conversion 0.75 0.7 0.65 0.6 0.55 0.5 Setpoint DDC 0.45 0 200 400 600 800 1000 1200 1400 1600 1800 2000 time (second) 312 310 Temperature(K) 308 306 304 302 300 0 200 400 600 800 1000 1200 1400 1600 1800 2000 time (second) Output Response 23

Application 4: CSTR 0.03 0.025 Fin(m 3 s -1 ) 0.02 0.015 0.01 0.005 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 time (second) 30 25 Fw(kgs -1 ) 20 15 10 5 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 time (second) Control Signals 24

Application 5: MEMS Gyroscopes Mechanical gyroscope and Micro machined gyroscope

Application 5: Dynamics of MEMS Gyroscopes u Mathematical Model 2 K x& + 2ζωnx& + ωnx + ωxyy 2 Ω y& = ud( t) m 2 K y& + 2ζ yωyy& + ωyy + ωxyx + 2 Ω x& = us( t) m ω x, ω y are Quadrature errors caused by spring xy xy coupling terms 2Ωx&, 2Ωy& are Coriolis acceleration terms

Application 5: Control of MEMS Gyroscopes q Control Objectives Ø Force the drive axis to resonance; Ø Force the sense axis output to zero; Ø Rotation rate estimation. q Challenges Ø Structure uncertainty; Ø Mechanical couplings (stiffness and damping); Ø Time-Varying

Application 5: MEMS Gyroscopes 400 The output of the drive axis Output x 200 0-200 -400 0 1 2 3 4 5 6 7 8 200 The steady state drive axis output x 10-3 Output x 100 0-100 -200 7 7.1 7.2 7.3 7.4 7.5 Time(s) x 10-3 The output of the drive axis with the ADRC. 28

Application 5: MEMS Gyroscopes 400 The tracking error of the drive axis 200 Error 0-200 -400 0 1 2 3 4 5 6 7 8 0.5 The steady state tracking error of the drive axis x 10-3 Error 0-0.5 7 7.1 7.2 7.3 7.4 7.5 Time(s) x 10-3 The tracking error of the drive axis. 29

Application 5: MEMS Gyroscopes 400 The tracking error of the drive axis 200 Error 0-200 -400 0 1 2 3 4 5 6 7 8 0.5 The steady state tracking error of the drive axis x 10-3 Error 0-0.5 7 7.1 7.2 7.3 7.4 7.5 Time(s) The tracking error of the drive axis with parameter variations. x 10-3 30

Application 5: MEMS Gyroscopes The output of the drive axis Output x(mv) 200 0-200 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 The steady state drive axis output Output x(mv) 200 100 0-100 -200 0.044 0.0441 0.0442 0.0443 0.0444 0.0445 0.0446 0.0447 0.0448 0.0449 0.045 Time(s) reference output The drive axis output of the FPGA implementation. 31

Practical Applications Assembly Line 32

Application 1: Industrial Servo Drive 81% Reduction in maximum position error. 33

Application 1: Industrial Servo Drive 41% Reduction in RMS torque. 34

Application 1: Industrial Servo Drive 71% Reduction in jerk. 35

Application 2: Temperature Control in Hose Extrusion Energy savings in a Hose Extruder Line: over 200% in product quality and 58% in energy reduction. 36

Conclusions q ADRC Ø Does Not Require an Accurate Mathematical Model Ø Strong Disturbance Rejection Ability Ø Highly Robust Ø Easy to Use (after parameterization) Ø A Transformative Control Technology 37

Thank You! Questions? 38