Modeling and Analysis of Signal Estimation for Stepper Motor Control. Dan Simon Cleveland State University October 8, 2003

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1 Modeling and Analysis of Signal Estimation for Stepper Motor Control Dan Simon Cleveland State University October 8, 23

2 Outline Problem statement Simplorer and Matlab Optimal signal estimation Postprocessing Simulation results Conclusion

3 Problem statement Speed control for PM DC motor Aerospace applications Flywheel energy storage Flight control trim surfaces Hydraulics Fans Thrust vector control Fuel pumps

4 Problem statement DC Machine Permanent Excitation I& = V / L RI / L kω / L ω& = ki / J T L / J I = armature current V = armature voltage L = inductance R = resistance k = motor constant ω = rotor speed θ = rotor angle J = moment of inertia T L = load torque

5 Problem statement State assignment: x 1 = I x 2 = ω x 3 = θ x 4 = T L / J Measurements: y = current (and possibly position)

6 Problem statement Estimate velocity x 2 noise 1 1 y noise 1/ 1 1 / / / = = x V L x J k L k L R x&

7 Simplorer and Matlab Simplorer Circuit element models Electric machine models Data analysis tools Interfaces with Matlab / Simulink

8 Simplorer and Matlab Matlab Powerful math and matrix capabilities Co-Simulation Link Simplorer and Matlab Plot and analyze data in either environment

9 Simplorer and Matlab Use the SiM2SiM tool in Add Ons / interfaces6 Begin the simulation in Simulink

10 Simplorer and Matlab Define Simplorer inputs and outputs in the property dialog of the SiM2SiM component Simplorer? Simulink

11 Simplorer and Matlab Use the S-function property dialog in Matlab to link Simplorer / Matlab signals

12 Simplorer and Matlab Begin the simulation in Matlab Couple Simplorer s and Matlab s strengths Simplorer: power electronics, electromechanics, data analysis, state diagrams Matlab: matrix algebra, toolboxes Data analysis / viewing can be done in either Simplorer or Matlab

13 Optimal signal estimation Given a linear system: x&= Ax B u u B w w y = Cx Dv x = state y = measurement u = control input w, v = noise Find the best estimate for the state x

14 Optimal signal estimation Suppose w ~ N(, Q) and v ~ N(, R). The Kalman filter solves the problem min E { ( x xˆ) ( x xˆ) dt} T P& = AP PA T B w QB T w KCP K = PC T D T R 1 D 1 x& ˆ = Axˆ K( y Cxˆ)

15 Optimal signal estimation The H filter solves the problem x&= y = Ax Cx x() xˆ() B u u Dv 2 P 1 x xˆ B w 2 S 2 Q dt 1 dt This is a game theory approach. w Nature tries to maximize the estimation error. The engineer tries to minimize the error. w v 2 R 1 dt < 1 θ

16 Optimal signal estimation Rewrite the previous equation: J 1 θ < x() xˆ() 2 P 1 x xˆ 2 S dt 1 θ w 2 Q 1 v 2 R 1 dt < Game theory: nature tries to maximize J and the engineer tries to minimize J min xˆ max w, v, x() J

17 Optimal signal estimation The H filter is given as follows: P& = AP PA T B w QB T w KCP θpsp T T 1 K = PC D R D xˆ & = Axˆ K( y Cxˆ) 1 Note this is identical to the Kalman filter except for an extra term in the Riccati equation.

18 Optimal signal estimation The Kalman filter is a least-mean-squares estimator The H filter is a worst-case estimator The Kalman filter is often made more robust by artificially increasing P The H filter shows exactly how to increase P in order to add robustness P& = T T AP PA B QB KCP θpsp w w

19 Optimal signal estimation Steady state: P& T = AP PA B = T T 1 K = PC D R D x& ˆ = Axˆ K( y Cxˆ) w QB 1 T w KCP θpsp Jacopo Riccati This is an Algebraic Riccati Equation Real time computational savings

20 Transfer Matlab data to Simplorer for plotting and analysis Postprocessing

21 Postprocessing Start the Matlab postprocessor interface before starting the co-simulation After running the co-simulation, the Day postprocessor can exchange data with Matlab

22 Postprocessing Drag data between Day and Matlab Matlab variables Matlab output Matlab commands

23 Postprocessing Day cannot handle arrays with more than two dimensions use Matlab s squeeze command Make sure Matlab data is not longer than Simplorer s time array

24 Postprocessing Simplorer data Matlab data

25 Postprocessing Analysis Characteristics to view statistical information Select the desired output variable Export to table

26 Simulation results motor parameters control input measurements Ouput from Matlab

27 Simulation results motor PI Controller implement equations P & and x& ˆ

28 Simulation results Simulation parameters: 1.2 ohms, 9.5 mh,.544 Vs,.4 kg m 2 Initial speed = cmd speed = 1 RPM External load torque changes from to.1 Measurement errors.1 A,.1 rad (1 σ)

29 Simulation results

30 Simulation results Steady state parameters: Initial speed = 1 commanded speed = 1 RPM External load torque = Measurement error =.1 A,.1 rad (1 σ)

31 Simulation results

32 Simulation results

33 Simulation results L = 2 L J = 2 J k = 2 k RMS Estimation Errors (RPM) current and position measurements Nominal R = R / 2 L = L / 2 J =.6 J k = k / 2 R = 2 R Kalman filter H filter

34 Simulation results Now suppose we measure winding current but not rotor position. Can we still get a good estimate of motor velocity? [ ] sigma) one noise (.1amps, 1 y noise 1/ 1 1 / / / = = x V L x J k L k L R x&

35 Simulation results Nominal R = R / 2 L = L / 2 J =.6 J k = k / 2 R = 2 R L = 2 L J = 2 J k = 2 k RMS Estimation Errors (RPM) current measurement only Kalman filter H filter

36 Conclusion Motor state estimation is required for motor control Kalman filtering and H filtering can be used for motor state estimation Steady state filtering saves time Simplorer / Matlab co-simulation Estimate motor parameters R, L, J, k

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