Chapter 11: Feedback and PID Control Theory

Transcription

1 Chapter 11: Feeback an D Control Theory Chapter 11: Feeback an D Control Theory. ntrouction Feeback is a mechanism for regulating a physical system so that it maintains a certain state. Feeback works by measuring the current state of a physical system, etermining how far the current state is from the esire state, an then automatically applying a control signal to bring the system closer to the esire state. This process is repeate iteratively to bring the system to the esire state an keep it there. Feeback can be use very effectively to stabilize the state of a system, while also improving its performance: Engineers use feeback to control otherwise unstable esigns; op-amps use feeback to stabilize an linearize their gain; an physicists use feeback to stabilize an improve the performance of their instruments. A. Feeback in engineering Feeback is ubiquitous in engineering. ts application has le to evice features an machines which woul not otherwise function. Here are few examples: Climate control: A sensor measures the temperature an humiity in a room an then heats or cools an humiifies or ehumiifies accoringly. Automobile cruise control: The car measures its spee an then applies the accelerator or not epening on whether the spee must be increase or ecrease to maintain the target spee. Highly maneuverable fighter jets: The F-16 Falcon fighter jet is an inherently unstable aircraft (i.e. the airframe will not glie on its own). The F-16 oes fly because 5 onboar computers constantly measure the aircraft s flight characteristics an then apply corrections to the control surfaces (i.e. ruer, flaps, ailerons, etc ) to keep it from tumbling out of control. The avantage of this technique is that the aircraft has the very rapi response an maneuverability of a naturally unstable airframe, while also being able to fly. B. Feeback in electronics: Op-amps use feeback to achieve very high linearity an preictability for their close-loop gain by sacrificing some of their extremely high open-loop gain. Another common application of feeback in electronics is in precision, fast- response power supplies. Constant current an constant voltage power supplies which have a high egree of stability use feeback to regulate their current or their voltage, by measuring the current an voltage across a precision shunt resistors an then using feeback to automatically correct for any eviations from the esire output. Feeback also allows the power supply to ajust its voltage or current very quickly an controllably in response to a change in loa

2 Chapter 11: Feeback an D Control Theory C. Feeback in physics Feeback has become a familiar tool for experimental physicists to improve the stability of their instruments. n particular, physicists use feeback for precise control of temperature, for stabilizing an cooling particle beams in accelerators, for improving the performance of atomic force microscopes, for locking the optical frequency of lasers to atomic transitions, an referencing quartz oscillators to groun state atomic hyperfine microwave transitions in atomic clocks, to name just a few example. Temperature control: Many elicate physics evices, such as crystals, lasers, RF oscillators, an amplifiers, require their temperature to be very stable in orer to guarantee their performance. For example, the wavelength of ioe lasers generally has a temperature epenence on the orer of. nm/ C, but requires a stability of 1-6 nm for experiments. tochastic cooling: n a particle accelerator, the transverse momentum sprea of particles must be reuce to a minimum. The reuce momentum sprea increases the particle ensity, or beam luminosity, an consequently the probability of collisions with a similar counter-propagating particle beam in the etector area. tochastic cooling works by measuring the transverse positions an momenta of the particles as they pass through a section of the accelerator, an then applying appropriate momentum kicks to some of the particles at other points in the accelerator ring to reuce the overall transverse momentum sprea. The process is repeate until the momentum sprea is sufficiently reuce. The 1984 Nobel rize in hysics was aware in part to imon van er Meer for his invention of stochastic cooling which contribute to the iscovery of the W an Z bosons (weak force meiators) at CERN. Atomic force microscope: An atomic force microscope uses a very sharp tip (just a few nanometers in size at the very tip) which is scanne back an forth just a few nanometers above the surface to be image. nstea of scanning the tip at a constant height above the surface, which coul lea to the tip actually running into a bump on the surface, the microscope uses feeback to ajust the tip height such that the force (from the surface atoms) on the tip is constant. Laser locking: Many experiments in atomic an optical physics require lasers which have a very stable optical frequency. The optical frequency of the laser is locke by measuring the optical frequency ifference between the laser an an atomic transition an using feeback to set this ifference to a constant value. Lasers can be routinely stabilize with feeback to better than 1 MHz out of 3x1 14 Hz (about 1 part per billion), though stabilities close to 1 Hz have been reporte after heroic efforts. Atomic clocks: n an atomic clock, the frequency of an RF oscillator (a quartz crystal for example) is compare to that of a groun state atomic hyperfine microwave transition (6.8 or 9. GHz). The frequency ifference is measure an the frequency of the RF oscillator is correcte by feeback. The process is constantly repeate to eliminate any rift in the frequency of the RF oscillator. Atomic fountain clocks can achieve accuracies in the range of 1 part in 1 15, an plans are unerway to construct optical atomic clocks with accuracies an stabilities of about 1 part in

3 Chapter 11: Feeback an D Control Theory. Feeback n this section we introuce the main elements of a generic feeback moel. A. ystem Consier a simple system characterize by a single variable. Uner normal conitions the system has a steay state value of = which may vary an rift somewhat over time ue to the variation of environmental variables v which we cannot measure or are unaware of. We possess a mechanism for measuring the state of the system as well as a control input u with which we can use to moify the state of the system. n summary, the system has the following functional form (u; v;. We will make the final assumption that is monotonic with u in the vicinity of (i.e. that the plot of vs. u oes not have any maxima or minima, an that /u is either always positive or always negative). Figure 1 shows a conceptual schematic of the relationship between the system, the variables u an v, an the measurement of the system state. unknowns v moify ystem tate: = Measurement of Control u Moifies Figure 1: Conceptual schematic of system B. Objective Our objective is to set or lock the state of the system to a esire value = an keep it there without letting it rift or vary over time, regarless of variations in the environmental variables v. C. Feeback moel We will set or lock the state of the system to = with the following proceure (see also figure ): 1. Measure the state of the system

4 Chapter 11: Feeback an D Control Theory. Determine how far the system is from its esire set point by efining an error variable, e=-. 3. Calculate a trial control value u=u(e). 4. Fee the calculate control value, u(e), back into the control input of the system. 5. The state of the system changes in response to the change in the control value. 6. Return to step. f we repeat this feeback cycle inefinitely with an appropriately calculate control value u(e), then the system will converge to the state = an remain there even uner the influence of small changes to other variables (i.e. v) which influence the value of the state. This feeback moel can be aapte to inclue several state variables an several feeback variables. unknowns v moify ystem tate: Measurement of Calculate error e=- Control u Moifies Calculate u=u(e) Figure : Conceptual schematic of system with the feeback loop. n section, we iscuss a frequently use expression for calculating the feeback control variable u(e).. D Feeback Control The most popular type of feeback stabilization control, u(e), is roportional-ntegral- Derivative (D) gain feeback. D is very effective an easy to implement. The expression for u(e) epens only on the error signal e=- an is given by u( e; = g e( + g t e( t + g D t e( (1) - 4 -

5 Chapter 11: Feeback an D Control Theory where g, g, an g D are respectively the proportional, integral, an erivative gains. We also note that g, g, an g D o not have the same units. We will assume for simplicity that g is imensionless in which case u(e) has the same units as. A. Time evolution of the system with D feeback control We are now in a position to calculate the time evolution of the system uner the influence of feeback. Without feeback, the system woul remain in the state : ( no feeback = () may vary in time, but we will ignore this effect until part.c. n the presence of feeback, the state of the system at time t+δt (step 5) epens on the state of the system without feeback,, which has been moifie by the control input variable u(e). We now make the following simplifying assumption that the control input variable, u(e), controls or moifies the state of the system through the process of aition. n this case, the system state variable evolves accoring to the following equation: ( t Δ = + u( e; + (3) We can convert this equation to an integro-ifferential equation, if we assume that the system has a characteristic reponse time τ (small). n this case, equation 3 becomes ( + τ ( = + ge( + g e( t + gd e( (4) t t t B. pecial case: pure proportional gain feeback As a limiting case we consier pure proportional gain feeback (g = an g D =). We stuy this special case, because it is the basis for op-amp feeback an is also the simplest form of feeback. For g = an g D =, equation 4 becomes ( ( = + g e( t +τ (5) We can solve this 1 st orer ifferential equation, for the initial conition (t=)=, with the same technique we use in chapter 3 (equations 17-1). After a little bit of integration an algebra, which is left as an exercise to the reaer, we fin the following solution: 1 g g t τ ( = e + g 1 g 1 g (6) - 5 -

6 Chapter 11: Feeback an D Control Theory Equation 6 shows that the system will converge to the state =( -g )/(1-g ) when feeback control is applie, so long as the exponential exponent is negative (i.e. g <1), otherwise will iverge. We note that g < correspons to negative feeback. Figure 3 shows the response of a system for a imensionless gain of g =-1 an state values =.5 an =1, with time measure in units of τ (the system characteristic response time). ( Figure 3: ystem response with pure proportional gain feeback control an parameters g =-1, =.5, an =1. Time is measure in units of τ (the system characteristic response time). As figure 3 makes clear, the system oes not converge to the esire state =, though it oes reach its final steay state value relatively quickly. f we restrict ourselves to negative feeback, then accoring to equation 1, the system will converge to the steay state value ss of ss g = 1 g Equation 7 inicates that the system can be mae to converge to a steay state value ss which is arbitrarily close to = just by increasing the gain. n fact, for infinite proportional gain (i.e g - ) the system oes converge to ss = : This is the limit in which op-amps feeback operates. A note of caution: On its own, equation 7 is a little misleaing since it woul seem to imply that large positive feeback, g -, woul also prouce ss =. Of course, this is (7) - 6 -

7 Chapter 11: Feeback an D Control Theory not true since accoring to equation 6, the system will never achieve a steay state, but instea will iverge forever. C. olution for feeback control A large majority of D feeback controllers are actually just controllers (i.e. proportional an integral gain, but no erivative gain), an so for simplicity we solve equation 4 without the erivative gain term (g D =). The inclusion of the erivative gain term is conceptually simple an follows the same treatment as feeback an is left as an exercise to reaer. Derivative gain is use to improve the time response of the feeback, so that the system converges more quickly to its steay state value. With the erivative gain term omitte, equation (4) becomes ( + τ ( = + g e( + g e( t (8) t We can convert this integro-ifferential equation to a n orer linear ifferential equation with constant coefficients by taking the time erivative of equation (8) to obtain t τ = g + g g, (9) t t + where we employe the substitution e(=-. After combining similar terms, equation (9) becomes τ + (1 g ) g = g (1) t t Equation 1 is an inhomogeneous n orer ifferential equation with constant coefficients. The full solution to equation (7) is given by + λ+ t λ t ( = A+ e + A e (11) ( g 1) ± ( g 1) + 4g τ λ = τ ± (11a) The first two terms of equation 11 represent the homogeneous solution to equation 1, while the 3 r term is the inhomogeneous solution to the equation (it oes not epen on the initial contions). A + an A - are constants to be etermine from the initial conitions. Equation 11 shows that the system will converge to the state = when feeback control is applie, so long as λ + an λ - are negative (i.e. negative feeback), otherwise will iverge (exactly the opposite of what we want to accomplish with feeback). t - 7 -

8 Chapter 11: Feeback an D Control Theory f we choose (t=)= an (t=)/t= as our initial conitions, we can calculate the constants A + an A -. After a little bit of algebra, we fin that A + λ = λ λ+ ( ) (1a) A λ + = λ λ+ ( ) (1b) n figure 4, the behavior of the system uner feeback control is plotte for several ifferent parameters configurations. ( ( Figure 4: Time-evolution of a generic system with control feeback for =.5 an =1. For the left han plot the gain parameters are g =-1, g =-3, while for the right han plot the parameters are g =-1, g =-4. The small overshoot in the left han plot is ue to a small imaginary part in the exponential exponent of equation 11(a). The primary purpose of integral gain is to provie essentially infinite gain at DC ( Hz), which guarantees that ss =, as can be seen in figure 4. Figure 4 also shows that the larger the gain, the faster the correction time of the feeback control loop. D. Fourier space analysis of noise suppression One of the primary objectives of feeback is to make the system insensitive to noise on the system state, so that the system state stays locke to = regarless of external influences. n the absence of corrective feeback, external noise will cause the system state to eviate from =. External noise at a frequency ω will cause the system state to oscillate aroun its natural steay state such that = + N cos(ω, where N is the amplitue of the oscillations. Following the stanar Fourier space recipe of chapter 3, we replace cos(ω with exp(iω, an then take the real part at the en of our calculations. n essence, we must re-solve equations 8 with the following moification: - 8 -

9 Chapter 11: Feeback an D Control Theory e iωt + N (13) Using the substitution of equation 13, equation 1 becomes τ + (1 g ) t t g = iω N e iωt g Equation 14 has the same homogeneous solution as equation 1, but the inhomogeneous solution, ih (, iffers an is given by the following expression iω t = e + (14) iωt ih ( ) N (15) (1 g ) iω τω g We see that the noise term is present in the inhomogeneous solution, but with an aitional factor moifying the amplitue of the noise. n the case of negative feeback the moulus of this suppression factor, which we will call A N, is always less than unity an is given by the following expression A N = ω ( τω + g ) + (1 g ) ω (16) The plot in figure 5 shows the epenence of the suppression factor, A N, on frequency for ifferent feeback schemes. The plot shows that a combination of proportional an integral control gives the best suppression of noise, except in the vicinity of the resonant frequency ω = g / τ. The high frequency rop-off of the suppression factor is not ue to feeback but simply the natural response time τ of the system which also suppresses noise. ω, frequency (ras/τ) A N, suppression factor Figure 5: Comparison of the suppression factor, A N, for ifferent feeback schemes. The feeback control loop parameters are g =-1 an g =

10 Chapter 11: Feeback an D Control Theory V. Reality n practice, feeback is not quite as straightforwar as presente in the previous section. A. Gain vs, Frequency n the theoretical treatment of part, we assume that the proportional gain was inepenent of frequency. n practice, gain will generally fall off at higher frequencies ue to natural low-pass RC filtering in an amplifier an the larger circuit. As an example, figure 6 shows a plot of the open-loop gain of an op-amp as function of frequency, which has a clear rop-off in gain at higher frequencies. Figure 6: Open-loop gain of the O7 op-amp (Analog Devices O7 atasheet revision F, p. 1 (6)). B. hase shifts an positive feeback The natural or stray RC filtering of an amplifier not only rolls off the gain at high frequencies, but also introuces a -π/ phase shift. f the feeback loop has a secon stray unintentional RC filter present (for example, the natural time response of the system), then a secon -π/ phase shift is introuce. f the feeback gain is larger than 1 at the frequency at which the total accumulate phase is -π, then the feeback loop goes into positive feeback which causes the state of the system to iverge or sometimes oscillate out of control

11 Chapter 11: Feeback an D Control Theory C. tray RC positive feeback compensation One way to avoi having the system go into positive feeback is to purposely introuce an aitional RC low-pass filter into the feeback loop. f this RC filter has a f 3B frequency which is sufficiently smaller than the frequency at which the positive feeback occurs then the attenuation of the filter can bring the gain below 1 when the -π phase shift occurs. This way the feeback loop will no longer go into positive feeback above a certain frequency (of course there will not be any noise suppression or feeback action above this frequency either). Design Exercises: Design exercise 11-1: Consier an LED facing a photoioe in a manner similar to what you i last week in lab 1. Design a circuit which will maintain a constant optical power incient on the photoioe, even in the presence of external fluctuations in the room lighting. You shoul use the op-amp circuit of chapter 1, figure 11 an use feeback control to stabilize the intensity of the LED. Your circuit shoul be able to provie at least 1 ma at ~ V to the LED. Design exercise 11-: The non-inverting op-amp amplifier with finite open-loop gain. n this exercise you will NOT use the op-amp golen rules to solve the problem, unless explicitly inicate. Consier the non-inverting op-amp amplifier in the circuit below. V N + V OUT R 1 R n the following parts, V + an V - refer respectively to the voltages at the non-inverting an inverting terminals of the op-amp. The open-loop gain of the op-amp is A. a. Write own the funamental op-amp relation between V +, V -, V out, an A when no feeback is present (i.e. when R 1 an R are not presen. b. Assuming that the V - input raws no current, erive an expression for V - in terms of V out, R 1, an R

12 Chapter 11: Feeback an D Control Theory c. Obtain an expression for V out in terms of V in, A, R 1, an R, an etermine the gain G of the amplifier. Calculate V out in the limit of A +. Calculate V out in the limit A - an comment on its physical meaning.. uppose that the open-loop gain, A, is not very constant with frequency an changes by ΔA between frequencies f 1 an f. Derive an expression for the resulting relative variation in the amplifier gain ΔG/G in terms of the relative variation in the openloop gain ΔA/A. Calculate ΔG/G an ΔA/A for A=1 6, ΔA=1 5, R =1 kω, an R 1 =1 kω. e. Most op-amps feature a significant rop-off in their open-loop gain at frequencies above ~1 Hz. The rop-off follows a well establishe curve given by Af=constant, where the constant is calle the gain-banwith prouct (f is frequency in Hz). The gainbanwith prouct of the O-7 is 8 MHz. On a same log-log graph, plot the open-loop gain A vs frequency an the close loop gain G vs. frequency (for R =1 kω an R 1 =1 kω) for an O-7-base non-inverting amplifier