Applied Digital Control: Optimization for System Identification and Controller Design

Transcription

1 Applied Digital Control: Optiization for Syste Identification and Controller Design David G. Wilson Mechanical Engineering University of New Mexico /27/2006

2 Lecture Outline Introduction Optiization: MATLAB Toolbox Syste Identification Control Design Conclusions Probles

3 Continuous Controller Continuous controller r(t) + u(t) y(t) D(s) Mechanical Syste (Plant) - Continuous controller can be built using analog electronics A digital controller can take the place of continuous controller The echanical syste or plant considered analog or continuous (REF: CMU)

4 Digital Controller Digital controller r(t) A/D r(k) e(k) G(z) u(k) u(t) Difference D/A and + Hold Equation - H(s) Plant y(t) clock y(k) A/D Digital controller operates on discrete signals (or saples) rather than continuous signals Ipleentation: Norally as discrete functions either in transfer function or state-space fors (difference equations) (REF: CMU)

5 Servo Control Syste Coponents Motor, aplifier, and sensing: + Controller v θ i - and Aplifier Motor and Load Sensing Device (Encoder,Tachoeter) Exaple: general closed-loop control syste (REF: Electro-Craft)

6 DC Motor Model Fro input voltage to otor angle: J & θ + B & θ + T sgn( & θ ) = K i τ L di dt + Ri + K & θ e The electrical tie constant L/R considered negligible: i = ( vi K & eθ ) / R J && θ + ( B + K K / R) & θ + T f t = v e i t f L sgn( & θ ) Motor Datasheet: Resistance Inductance Torque constant Back EMF constant Siplifications: Direct drive (N=1) No Gear Backlash (hysteresis, deadzone) = K v t i / R τ L (REF: Wen)

7 Aplifier Interaction Controller outputs +/- 10 Volts. Eploy current aplifier to convert voltage to current Current energizes otor to produce torque at shaft Aplifier Options: Linear: output current proportional to input voltage PWM: output current switched between ax/in voltages at high frequency (20-50Khz) with duty cycle proportional to input voltage (such as 50%) (REF: Wen)

8 Siple PWM Aplifier Model Modeling of the aplifier: When PWM frequency is uch larger than the syste frequency of interest then applied Voltage voltage can be reduced to average value established for one PWM cycle Result: PWM and power stage considered as perfect voltage source as long as saller than axiu available DC voltage Current controlled voltage source internal portion of PWM aplifier voltage characteristics: Reference Ka Required Voltage Linearized Applied voltage voltage supply saturation (constant gain) Voltage supply Applied Voltage If required voltage realizable then will be fully applied otherwise liited to axiu available (REF: Friedrich)

9 Ap Current Feedback Loop Previously without current feedback it was noted: J && θ + ( B + K K / R) & θ + T sgn( & θ ) = t e With current feedback the input effectively becoes = then J && θ i K a V D / A & Effectively: i = K a V D is subject to saturation due to / A ax ap current, ax supplied voltage, and back EMF f K v + Bθ + Tf θ) = KtKaVD / A sgn( & t i τ / R L τ L (REF: Wen)

10 Sensor Model - Encoders Rotational encoder generates counts fro the up/down edges of A/B channels. Quadrature decoding gives an effective ultiplier of 4 For a 1000 line encoder the resolution will give 2pi/(1000*4) or rad deg K enc = 1.57 = 0.09 count count Ideally, encoder can be odeled as a gain with a fixed integer resolution of 1 count = 1.57 rad = 0.09 deg Other types of sensors (tachoeters, potentioeters) ay have additional dynaics, such as low pass 1 st or 2 nd order filters used to iniize noise, etc. (REF: Wen)

11 Digital Quantization Effects Quantization: analog signal with D/A and A/D conversions are defined in ters of finite word length For 12-bit word that represents +/- 10 V the resolution becoes 20/2 12 =2.44 v. Effects control output and easured input signals Encoder quantization can effect velocity estiation through finite differencing Therefore a detailed siulation odel would need to take into consideration (REF: Wen)

12 PID Digital Controller Continuous PID Controller: Derivative action: Integral action: to preserve both regions of stability and instability when apping fro s to z eploy the bilinear transforation : Resulting z-transfor odel of PID controller: C( s) / E( s) = KP + KI / s + e( kt ) e[( k 1) T ] C c D ( kt ) = D ( z) z 1 = T E( z) Tz 1 CI ( s) = sys = CI ( z) = c2d( sys, T,' tustin' ) s CI ( z) T ( z + 1) = E( z) 2 ( z 1) C( z) T ( z + 1) ( z 1) = KP + KI + KD E( z) 2( z 1) Tz If sapling is fast enough, even w/o bilinear, no probles should occur. However, when under-sapled the results are instability and inaccuracy. (REF: Background Mat l) K D s

13 Position Servo with PID Control r(t) A/D r(k) + e(k) - More coplete odel for a position servo with PID control: K p 1 ( z 1) K D T z PID u(k) V D / A T ( z + 1) K D/A and a K I 2 ( z 1) Hold sτ a + 1 Current Aplifier T + ( s) T ( s) f + - J T d K 2 s t + DC Motor plus load Bs clock y(k) K enc y( t) = θ ( t) Digital controller

14 Other Potential Probles Control engineer needs to be aware during the design of such things as: What if encoder generates spurious signal creating a large error ter which in ter generates a large correction ter? What happens if the correction ter is larger than the axiu D/A output? Saturation nonlinearity ter coded before the D/A converter for prevention needs further investigation. What about integrator anti-reset windup? Reset windup or integrator saturation can occur if output saturates and the controller continues to integrate the error. Proble autoatically avoided when velocity for of algorith is used, then integration stops autoatically when the output is liited. If using the position for then one way to stop updating the integrator when the output is liited is to use conditional integration, where the integral part of the controller is used only when the error is sufficiently sall. Additional logic and coding ipleentation needs to be considered. (REF: Background Mat l, Klafter)

15 Case Study Project Steps What if asked to design a digital controller for a echanical syste? Optiization applies where? What level of detail is sufficient? Mechanical syste with ebedded controller prototype (hardware) Model developent (physics/echatronics) Plant odel syste identification Digital control syste design Experiental/predicted correlation and perforance results

16 Plant Model SYS-ID -1 The Matherworks cap: Excerpts fro Ebedded Real-Tie Control via MATLAB, Siulink, and xpc Target, by P.J. Mosteran, et.al.: 1. Foral control design ethods invariably rely on a plant odel. The plant odel can be derived fro first principles but often contains unknown paraeters. Experients ust be conducted to gather inforation on the behavior of the plant dynaics to help estiate these paraeters. Typically, values for physical paraeters such as oents of inertia, asses, rod lengths, and gear ratios are well known and can be incorporated into the odel. Paraeters such as friction coefficients, viscosity, and stiction behavior, however, are not precisely known. Measured data fro the rapid prototyping syste can be used with the first-principle description to calibrate these paraeters using Siulink Paraeter Estiation (SPE) SPE allows the selection of a set of paraeters and states to be estiated. Min/ax values of the paraeters and initial states can be set, in addition to the expected values. SPE uses optiization algoriths fro the Optiization Toolbox and, optionally, the Genetic Algorith and Direct Search Toolbox.

17 Plant Model SYS-ID -2 The Matherworks cap: Excerpts fro Ebedded Real-Tie Control via MATLAB, Siulink, and xpc Target, by P.J. Mosteran, et.al.: 2. Linear syste identification tools are useful if the dynaics of a syste behave in a near-linear anner about a given operating condition, but cannot be odeled fro first principles. The Syste Identification Toolbox coputes linear discrete and continuous odels using both tie and frequency doain data. 3. Transfer function estiation is another process that takes experiental data and converts it using spectral estiation techniques to copute the frequency response of a syste. The Signal Processing Toolbox has any functions to support the estiation of a transfer function. 4. If linear odels do not describe the odel accurately and the underlying equations of otion are not well known, a nonlinear black box neural network can be created using the Neural Network Toolbox.

18 Plant Model SYS-ID -3 Several other SYS-ID resources and techniques: Applied SYS-ID (Jer-Nan Juang). The Observer/Kalan filter Identification (OKID) algorith, is forulated entirely in the tie-doain. The ethod coputes the Markov paraeters of a linear syste, fro which the state-space odel and corresponding observer are deterined siultaneously. The OKID ethod can effectively identify statespace odels using tie-doain (and frequency response data) I/O data. Flexible Robot Dynaics and Control, Chapter 4 SYS-ID (Robinett, et.al.). SYS-ID plays a key role in control syste design. The first thing that a controls engineer learns in the real world is that the transfer function is not written on the outside of the H/W container. SYS-ID is used to obtain the transfer function and the critical paraeters of plant odels for controller design. Several SYS-ID techniques are reviewed with exaples: Linear Least Squares, Nonlinear Least Squares (NLS), Frequency Doain NLS Optiization Techniques: VFO2AD Fortran optiization progras fro Harwell Subroutine Library (freeware) for iniizing arbitrary perforance index as a function of paraeter constants and subject to both equality and inequality constraints (UPDATE: LANCELOT code?) MATLAB Optiization Toolbox: fincon constrained nonlinear optiization functions.

19 Optiization: MATLAB Toolbox fincon Constrained NL Opt. Multi-objective optiization is concerned with the iniization of a vector of objectives f(x) that can be the subject of a nuber of constraints or bounds. Min f ( x) x c( x) 0 ceq( x) = 0 A x b Aeq x = beq lb x ub Subject to: Where x, b, beq, lb, and ub are vectors, A and Aeq are atrices, c(x) and ceq(x) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions. (REF: MATLAB Opt. Toolbox)

20 Optiization: MATLAB Toolbox fincon Steps Used for SYS-ID SYS_ID_driver.... initialization experiental data setup paras=fincon( perfindex,paras0,a,b,aeq,beq,lb,up, constraints ) perfindex. si( sysid_opt ) f= yout( ); constraints. si( sysid_opt ) c= []; ceq=yout( ); sysid_opt.dl Open-loop data sets Closed-loop data sets switch Dynaic Plant Model (paras) yout MATLAB Environent (Optiization Toolbox, Siulink Model)

21 Syste Identification: Servo Control Syste Proble J T && θ ˆ τ = K Motor odel and PD controller: + B & θ P ( θ DES + T sgn( & f θ) = KtKaVD / θ ) + K ( & θ & θ ) D DES A = ˆ τ Cost Index: Subject to: 2 J = ( θ θe) iterdt to ~ ~ ~ ( JT ) in JT ( JT ) ~ ~ ~ ( B) in B ( B) ax ~ ~ ~ ( T ) T ( T ) f tf in f f ax ax position Siulink odel encoder Fro Workspace ENCODER DESIRED kp KP proportional gain Zero-Order Hold ka AMP Gain AMP Saturation Kt Jt.s+B DC Motor plus load 1 s Integrator encoderdot Fro Workspace ENCODER RATE DESIRED kd KD derivative gain -K- Tf coefficient Friction sign function velocity Ts=1 sec otor_sysid_opt.dl

22 Syste Identification: SYS-ID Optiized Servo Results Calibration runs used skeleton PD controller to generate experiental closed-loop data sets Model paraeters are frozen for validation run Friction ay be both velocity and position dependent due to offset in position response Friction odel refineents Hub angle (deg) Hub rate (r/s) 45 Degree Calibration Run Tie (sec) opt exp des 45 Degree Calibration Run Tie (sec) opt exp des Hub angle (deg) Hub rate (r/s) Degree Validation Run Tie (sec) opt exp des 90 Degree Validation Run Tie (sec) opt exp des

23 Digital Controller Design: Optiize Controller Design Proble PID control paraeters set-up to be optiized based on a calibrated and validated SYS-ID odel and preliinary controller design encoder_exp Fro Workspace ENCODER EXP encoder Fro Workspace ENCODER DESIRED ki*ts/2 KI position kp KP ZOH 1/z z+1 z-1 Rate TransitionDiscrete Rate Transition2 Transfer Fcn ap_out ka den(s) AMP Gain AMP Saturation DC Motor plus load torque_out Siulink odel 1 s Integrator yout To Workspace encoderdot Fro Workspace ENCODER RATE DESIRED kd/ts KD ZOH 1/z z-1 z Rate Transition1 Discrete Rate Transition3 Transfer Fcn1 friction Tf coefficient Friction sign function ncoderdot_ex Fro Workspace ENCODER RATE EXP velocity pid_sysid_opt.dl

24 Conclusions: Overview/Keypoints of Discussion Discussed servo odeling, assuptions, and liitations Introduced soe SYS-ID ethods Reviewed fincon in MATLAB optiization toolbox Siulated servo syste optiization for odel atching and calibration Set-up controller gain optiization proble

25 Conclusions: Nonlinear Friction Results Slewing single flexible link syste optiization results nd Order Plant Design Response to Input Shaping 1200 High Fidelity with Friction Copensation Input Shaping Theta (counts) input shaped reference in experiental response. siulation response Theta Theta_c (count) Coanded Shaped Input - Experiental Response -- Siulation Response Tie (sec) Tie (sec) Optiization based on high fidelity odel help ID nonlinear friction and generate an input profile that copensated for this nonlinearity (REF: Wilson)

26 Probles: Optiization Ephasis May include an extra credit portion during projects Potential extra credit proble set

27 References/Resources: Lecture Coposition Material Carnegie Mellon University (CMU), Digital Control Tutorial, on internet. J. Wen, ECSE 4962 Control Systes Design, Lecture #4, Dept. ECSE, RPI, Feb. 2003, on internet. Robbins & Myers/Electro-Craft, DC Motors Speed Controls Servo Systes, Electro-Craft. P.G. Friedrich, Modelling of a Wound Rotor Salient Pole synchronous Machine and its Converter in the Constant Power Zone, on internet. P. Mosteran, et. al., Ebedded Real-Tie Control via MATLAB, Siulink, and xpc Target, on internet. R.D. Klafter, et. al., Robotic Engineering an Integrated Approach, Prentice- Hall, Jer-Nan Juang, Applied Syste Identification, Prentice-Hall, R.D. Robinett, III, et. al., Flexible Robot Dynaics and Controls, Chapter 4, Kluwer Acadeic/Plenu Publishers, MATLAB, Optiization Toolbox, User s Guide, The Mathworks, D.G. Wilson, D. Stokes, G.P. Starr, and R.D. Robinett, III, Optiized Input Shaping for a Single Flexible Robot Link, In SPACE 96: 5 th International Conference and Exposition on ECOS, Albuquerque, NM, June 1996.

28 References: Background Material MATLAB website: Several Digital Control Books: R.G. Jacquot, Modern Digital Control Systes, Marcel Dekker, Inc., K.J. Astro and B. Wittenark, Coputer Controlled Systes, Prentice-Hall, 1984 C.L. Phillips and H.Troy Nagle, Digital Control Syste Analysis and Design, 2 nd Ed., Prentice Hall, Other pertinent references called out within the lecture