Lecture13 controls
Download as PPT, PDF0 likes33 views
This document discusses control theory and its application to adaptive optics systems. It begins with definitions of open-loop and closed-loop control systems, then discusses how block diagrams can represent control systems. The Laplace transform is introduced as a tool for analyzing systems in both the time and frequency domains. An integrator is presented as one choice for the transfer function C(s) in closed-loop control systems. Its properties allow high gain at low frequencies, improving disturbance rejection. The document explores how an integrator in the feedback loop can stabilize control and optimize performance for adaptive optics.
Lecture13 controls
- 1. Page 1 Lecture 13 AO Control Theory Claire Max with many thanks to Don Gavel and Don Wiberg UC Santa Cruz February 19, 2013
- 2. Page 2 What are control systems? • Control is the process of making a system variable adhere to a particular value, called the reference value. • A system designed to follow a changing reference is called tracking control or servo.
- 3. Page 3 Outline of topics •What is control? - The concept of closed loop feedback control •A basic tool: the Laplace transform - Using the Laplace transform to characterize the time and frequency domain behavior of a system - Manipulating Transfer functions to analyze systems •How to predict performance of the controller
- 4. Page 4 3 Aberrated wavefront phase surfaces Telescope Aperture Imaging dectorWavefront corrector (Deformable Mirror) Adaptive Optics Control Wavefront sensor Corrected Wavefront Computer
- 5. Page 5 Differences between open-loop and closed-loop control systems • Open-loop: control system uses no knowledge of the output • Closed-loop: the control action is dependent on the output in some way • “Feedback” is what distinguishes open from closed loop • What other examples can you think of? OPEN CLOSED
- 6. Page 6 More about open-loop systems • Need to be carefully calibrated ahead of time: • Example: for a deformable mirror, need to know exactly what shape the mirror will have if the n actuators are each driven with a voltage Vn • Question: how might you go about this calibration? OPEN
- 7. Page 7 Some Characteristics of Feedback • Increased accuracy (gets to the desired final position more accurately because small errors will get corrected on subsequent measurement cycles) • Less sensitivity to nonlinearities (e.g. hysteresis in the deformable mirror) because the system is always making small corrections to get to the right place • Reduced sensitivity to noise in the input signal • BUT: can be unstable under some circumstances (e.g. if gain is too high)
- 8. Page 8 Historical control systems: float valve • As liquid level falls, so does float, allowing more liquid to flow into tank • As liquid level rises, flow is reduced and, if needed, cut off entirely • Sensor and actuator are both “contained” in the combination of the float and supply tube Credit: Franklin, Powell, Emami-Naeini
- 9. Page 9 Block Diagrams: Show Cause and Effect • Pictorial representation of cause and effect • Interior of block shows how the input and output are related. • Example b: output is the time derivative of the input Credit: DiStefano et al. 1990
- 10. Page 10 “Summing” Block Diagrams are circles • Block becomes a circle or “summing point” • Plus and minus signs indicate addition or subtraction (note that “sum” can include subtraction) • Arrows show inputs and outputs as before • Sometimes there is a cross in the circle X Credit: DiStefano et al. 1990
- 11. Page 11 A home thermostat from a control theory point of view Credit: Franklin, Powell, Emami-Naeini
- 12. Page 12 Block diagram for an automobile cruise control Credit: Franklin, Powell, Emami-Naeini
- 13. Page 13 Example 1 • Draw a block diagram for the equation
- 14. Page 14 Example 1 • Draw a block diagram for the equation Credit: DiStefano et al. 1990
- 15. Page 15 Example 2 • Draw a block diagram for how your eyes and brain help regulate the direction in which you are walking
- 16. Page 16 Example 2 • Draw a block diagram for how your eyes and brain help regulate the direction in which you are walking Credit: DiStefano et al. 1990
- 17. Page 17 Summary so far • Distinction between open loop and closed loop – Advantages and disadvantages of each • Block diagrams for control systems – Inputs, outputs, operations – Closed loop vs. open loop block diagrams
- 18. 18 The Laplace Transform Pair ( ) ( ) ( ) ( )ò ò ¥ ¥- ¥ - = = i i st st dsesH i th dtethsH p2 1 0 • Example 1: decaying exponential ( ) ( ) ( ) ( ) ( ) s s s s s s -> + = + - = = = ¥ +- ¥ +- - ò s s e s dtesH eth ts ts t Re; 1 1 0 0 Re(s) Im(s) x - t0 Transform:
- 19. 19 The Laplace Transform Pair Inverse Transform: ( ) t t t i i st e di i e die i ds s e i th s p p s p p s q p qrr p sp - - - - -- ¥ ¥- = = = + = ò ò ò 2 1 2 1 1 2 1 1 Re(s) Im(s) x - qr r s rs q q q deids e s es i i i = = + +-= --11 The above integration makes use of the Cauchy Principal Value Theorem: If F(s) is analytic then ( ) ( )aiFds as sF p2 1 = -ò Example 1 (continued), decaying exponential
- 20. 20 The Laplace Transform Pair Example 2: Damped sinusoid Re(s) Im(s) x - x - t0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s wsws w wsws wws s ->÷ ø ö ç è æ ++ + -+ = += += = +--- -- - s isis sH ee eee teth titi titit t Re; 11 2 1 2 1 2 1 cos
- 21. 21 Laplace Transform Pairs h(t) H(s) unit step 0 t s 1 (like lim s0 e-st ) t e s- s+s 1 ( )te t ws cos- ÷ ø ö ç è æ ++ + -+ wsws isis 11 2 1 ( )te t ws sin- ÷ ø ö ç è æ ++ - -+ wsws isisi 11 2 1 delayed step 0 tT s e sT- unit pulse 0 tT s e sT- -1 1
- 22. 22 Laplace Transform Properties (1) L ah t( )+ bg t( ){ }=aH s( )+ bG s( ) L h t + T( ){ }= esT H s( ) L d t( ){ }=1 L h ¢t( )g t - ¢t( )d ¢t 0 t ò ì í î ü ý þ = H s( ) G s( ) L h ¢t( )d t - ¢t( )d ¢t 0 t ò ì í î ü ý þ = H s( ) h t - ¢t( ) eiw ¢t d ¢t 0 t ò = H iw( ) eiwt Linearity Time-shift Dirac delta function transform (“sifting” property) Convolution Impulse response Frequency response ( )0£T
- 23. 23 Laplace Transform Properties (2) h(t) x(t) y(t) H(s) X(s) Y(s) convolution of input x(t) with impulse response h(t) product of input spectrum X(s) with frequency response H(s) (H(s) in this role is called the transfer function) System Block Diagrams Power or Energy ( ) ( ) ( )ò òò ¥ ¥- ¥ ¥- ¥ = = ww p p diH dssH i dtth i i 2 2 0 2 2 1 2 1 “Parseval’s Theorem” L h ¢t( )x t - ¢t( )d ¢t 0 t ò ì í î ü ý þ = H s( )X s( ) = Y s( )
- 24. 24 Closed loop control (simple example, H(s)=1) E s( )= W s( )- gC s( )E s( ) solving for E(s), E s( ) = W s( ) 1+ gC s( ) Our goal will be to suppress X(s) (residual) by high-gain feedback so that Y(s)~W(s) W(s) + gC(s) - E(s) Y(s) residual correction disturbance where g = loop gain What is a good choice for C(s)?... Note: for consistency “around the loop,” the units of the gain g must be the inverse of the units of C(s).
- 25. 25 The integrator, one choice for C(s) ( ) ( ) î í ì =« ³ < = s sC t t tc 1 01 00 A system whose impulse response is the unit step acts as an integrator to the input signal: ( ) ( ) ( ) ( ) tdtxtdtxttcty tt ¢¢=¢¢¢-= òò 00 t’ 0 c(t-t’) x(t’) t 0 t c(t) x(t) y(t) that is, C(s) integrates the past history of inputs, x(t)
- 26. Page 26 26 Control Loop Arithmetic Y s( )= A(s)W s( )- A s( )B s( )Y s( ) Solving for Y(s), Y s( )= A s( )W s( ) 1+ A s( )B s( ) Instability if any of the roots of the polynomial 1+A(s)B(s) are located in the right-half of the s-plane Y(s)W(s) + B(s) - input A(s) output
- 27. 27 An integrator has high gain at low frequencies, low gain at high frequencies. C(s) X(s) Y(s) ( ) ( ) s sX sY = In Laplace terminology: The Integrator (2) Write the input/output transfer function for an integrator in closed loop: + gC(s) - X(s) Y(s) W(s) HCL(s) W(s) X(s) º The closed loop transfer function with the integrator in the feedback loop is: C s( ) = 1 s Þ X s( ) = W s( ) 1+ g s = sW s( ) s + g = HCL s( )W s( ) input disturbance (e.g. atmospheric wavefront) closed loop transfer function output (e.g. residual wavefront to science camera)
- 28. Page 28 Block Diagram for Closed Loop Control 9 Our goal will be to find a C(f) that suppress e(t) (residual) so that ! DM tracks ! ² where ! = loop gain H(f) = Camera Exposure x DM Response x Computer Delay C(f) = Controller Transfer Function ! (t) + H(f) - e(t) ²DM(t) residual correction disturbance gC(f) We can design a filter, C(f), into the feedback loop to: a) Stabilize the feedback (i.e. keep it from oscillating) b) Optimize performance ( ) ( )fHfgC fe )(1 1 + = ! ( f ) g
- 29. Page 29 Disturbance Rejection Curve for Feedback Control Without Compensation 17 Increasing Gain (g) Not so great rejection Starting to resonate ( )fHfgC )(1 1 + = ! ( f ) e( f ) C(f) =1
- 30. 30 The integrator, one choice for C(s) ( ) ( ) î í ì =« ³ < = s sC t t tc 1 01 00 A system who’s impulse response is the unit step acts as an integrator to the input signal: ( ) ( ) ( ) ( ) tdtxtdtxttcty tt ¢¢=¢¢¢-= òò 00 t’ 0 c(t-t’) x(t’) t 0 t c(t) x(t) y(t) that is, C(s) integrates the past history of inputs, x(t)
- 31. 31 An integrator has high gain at low frequencies, low gain at high frequencies. C(s) X(s) Y(s) ( ) ( ) s sX sY = In Laplace terminology: The Integrator Write the input/output transfer function for an integrator in closed loop: + gC(s) - X(s) Y(s) W(s) HCL(s) W(s) X(s) º The closed loop transfer function with the integrator in the feedback loop is: C s( ) = 1 s Þ X s( ) = W s( ) 1+ g s = sW s( ) s + g = HCL s( )W s( ) input disturbance (e.g. atmospheric wavefront) closed loop transfer function output (e.g. residual wavefront to science camera)
- 32. 32 ( ) s s sHCL + = g HCL(s), viewed as a sinusoidal response filter: DC response = 0 (“Type-0” behavior)( ) 0as0 ®® wwiHCL ( ) ¥®® ww as1iHCL High-pass behavior and the “break” frequency (transition from low freq to high freq behavior) is around w ~ g The integrator in closed loop (2) where
- 33. 33 The integrator in closed loop (3) At the break frequency, w = g, the power rejection is: HCL ig( ) 2 = ig g +ig( ) -ig g -ig( ) = g2 2g2 = 1 2 Hence the break frequency is often called the “half-power” frequency w ( )2 giHCL 1 1/2 g ~w2 (log-log scale) Note also that the loop gain, g, is the bandwidth of the controller. Frequencies below g are rejected, frequencies above g are passed. By convention, g is thus known as the gain-bandwidth product. HCL (ig) 2
- 34. Page 34 Disturbance Rejection Curve for Feedback Control With Compensation 19 Increasing Gain (g) Much better rejection Starting to resonate ( )fHfgC )(1 1 + = ! ( f ) e( f ) C(f) = Integrator = e-sT / (1 – e-sT)
- 35. Page 35 Residual wavefront error is introduced by two sources 1. Failure to completely cancel atmospheric phase distortion 2. Measurement noise in the wavefront sensor Optimize the controller for best overall performance by varying design parameters such as gain and sample rate
- 36. Page 36 Atmospheric turbulence ! Temporal power spectrum of atmospheric phase: S! (f) = 0.077 (v/r0)5/3 f -8/ 3 ! Power spectrum of residual phase Se(f) = | 1/(1 + g C(f) H(f)) |2 S! (f) 22 Increasing wind Uncontrolled Closed Loop Controlled
- 37. Page 37 Noise ! Measurement noise enters in at a different point in the loop than atmospheric disturbance ! Closed loop transfer function for noise: 23 ! (t) + H(f) - e(t) ! DM(t) residual correction disturbance gC(f) n(t) noise ( ) ( )fHfgC fe )(1 gC( f )H( f ) + = n( f ) Noise feeds through Noise averaged out
- 38. Page 38 Residual from atmosphere + noise ! Conditions ! RMSuncorrected turbulence: 5400 nm ! RMSmeasurement noise: 126 nm ! gain = 0.4 ! Total Closed Loop Residual = 118 nm RMS 24 Residual Turbulence Dominates Noise Dominates
- 39. Page 39 Increased Measurement Noise ! Conditions ! RMSuncorrected turbulence: 5400 nm ! RMSmeasurement noise: 397 nm ! gain = 0.4 ! Total Closed Loop Residual = 290 nm RMS 25 Residual Turbulence Dominates Noise Dominates
- 40. Page 40 Reducing the gain in the higher noise case improves the residual ! Conditions ! RMSuncorrected turbulence: 5400 nm ! RMSmeasurement noise: 397 nm ! gain = 0.2 ! Total Closed Loop Residual = 186 nm RMS 26 Residual Turbulence Dominates Noise Dominates
- 41. 41 Quick Review h(t) x(t) y(t) H(s) X(s) Y(s) convolution of input x(t) with impulse response h(t) product of input spectrum X(s) with frequency response H(s) (H(s) in this role is called the transfer function) System Block Diagrams L h ¢t( )x t - ¢t( )d ¢t 0 t ò ì í î ü ý þ = H s( )X s( ) = Y s( )
- 42. 42 ( ) t h t e s- = t0 Re(s) Im(s) x -s ( ) 1 H s s s = + Re(s) Im(s) x -s x -w w ( ) 1 1 1 2 H s s i s is w s w æ ö = +ç ÷ + - + +è ø t0 ( ) ( )cost h t e ts w- = Stable input-output property: system response consists of a series of decaying exponentials Time Domain Transform Domain
- 43. 43 Re(s) Im(s) x -s ( ) 1 H s s s = - t0 ( ) t h t e s+ = What Instability Looks Like Re(s) Im(s) x -s x-w w ( ) 1 1 1 2 H s s i s is w s w æ ö = +ç ÷ - - - +è ø t0 ( ) ( )cost h t e ts w+ = Time Domain Transform Domain
- 44. 44 Control Loop Arithmetic ( ) ( ) ( ) ( ) ( )( )Y s A s W s A s B s Y s= - solving for Y(s), ( ) ( ) ( ) ( ) ( )1 A s W s Y s A s B s = + W(s) + B(s) - input Instability if any of the roots of the polynomial 1+A(s)B(s) are located in the right-half of the s-plane A(s) Y(s) output
- 45. Page 45 Summary • The architecture and problems associated with feedback control systems • The use of the Laplace transform to help characterize closed loop behavior • How to predict the performance of the adaptive optics under various conditions of atmospheric seeing and measurement signal-to-noise • A bit about loop stability, compensators, and other good stuff
- 46. Page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