From Continuous-time To Discrete Time: Some Sampling Theory
From Continuous-time to Discrete time: some Sampling Theory Tim Wescott Wescott Seminars www.wescottdesign.com/seminars.html November 2, 2012
Sampling Theory I To date we've treated everything from the perspective of the processor: all time is sampled time, and everything (nearly) can be reduced to the z domain. I But in the real world, time is continuous. I This introduces three questions that we must answer before we can do real-world problems: 1. How do we get feedback (and possibly command) data into the controller? 2. How do we get command data out of the controller? 3. How do we model the real-world process that we want to control? I Sampling theory answers the rst two questions; the Laplace transform (and experience) answer the last
Outline Sampling Noise and Signals The Laplace Transform Modeling the Real World in the z Domain
Outline Sampling Sampling Aliasing Reconstruction Noise and Signals The Laplace Transform Modeling the Real World in the z Domain
De nition of Sampling I Sampling is the process that converts continuous-time signals into discrete-time signals y (t ) in continuous time and a sample Ts , the sampled-time feedback is y (k ) y (k Ts ) I For a feedback signal interval I Embedded control systems will need to take the appropriate plant measurements and turn them into digital signals: sampling is inherent in this process.
The Sampling Process I The lower trace shows no signal between samples because it simply isn't there I All information between samples is irrevocably lost.
Real-World Sampling I The most common sampling device is probably the analog to digital converter I Other methods for acquiring feedback, such as rotary encoders or timers may exist but all of them are capturing an analog quantity in continuous-time, and converting it to discrete time. I Most ADCs can be modeled as a sampler followed by a gain:
Outline Sampling Sampling Aliasing Reconstruction Noise and Signals The Laplace Transform Modeling the Real World in the z Domain
Aliasing I Sampling throws away all information about what's happening between samples. I By doing so, the sampling process throws away information about the original signal. I This information loss must be taken into account during system design. I The e ect is called aliasing. I Like many e ects we need to deal with, aliasing can be best explained in the frequency domain.
An Aliasing Example I The gure shows an example of aliasing. I Two signals, one at 110 kHz Hz, one at 1110Hz are sampled at 1 I The dots show the value of the sine waves at the sampling instants for both signals
Aliasing in the Frequency Domain I Consider a signal in the continuous-time domain, and its sampled counterpart: I x (t ) cos (ωt ), x (k ) cos (ω Ts k ) I If you know ω, the frequency of the original signal, you can exactly predict the sampled signal. I But if all you have is the sampled signal, you can't say what the original signal was: there is an ambiguity in the signal frequency that is equal to the sampling rate. I For example, consider this continuous-time signal, and its n: sampled counterpart for any integer I x (t ) cos ω 2Tπs n t , x (k ) cos I If you know your trig, you'll know that ω 2Tπs n Ts k x (k ) cos (ω Ts k ) no matter what integer value n takes on. I This is what we mean by aliasing
Another Aliasing Example I The gure shows an example of aliasing in the frequency domain. I The central plot is for the real signal I The dashed plots are for all the signals that same output could result in the
Aliasing and Control Systems I Aliasing is very important for systems: I Whose sampling rates are constrained to small multiples of their bandwidths I for which it is not economical to increase sampling rates I and for which delity of high-frequency reproduction is important I In general, control systems do not exhibit these traits I The primary reasons that aliasing, by itself, can be a problem is I if you have high-frequency noise that you shift down to a frequency that your system is sensitive to. I or if you are receiving a command in the analog domain, and that command has high-frequency components
The Problem with Anti-Aliasing Filters I Let us say that you want to be diligent about anti-aliasing, so you determine that you will put in anti-alias lters. I You are going to be sampling at 500Hz, so you choose a 5th order Butterworth lter with a 100Hz cuto . I Nice attenuation but the phase shift!!
Anti-Alias Filters for Control Systems I Don't use 'normal' anti-alias lters if you can avoid it! I Consider comb lters: I Sample at a high rate, average, then resample at your control loop rate. I Or use a Sigma-Delta converter, as these generally have comb lters built in. I A comb lter has nulls at every non-zero harmonic of its length I If it is synchronized to your sampling rate, that's every non-zero harmonic of the sampling rate
Outline Sampling Sampling Aliasing Reconstruction Noise and Signals The Laplace Transform Modeling the Real World in the z Domain
Reconstruction De ned I A sampled-time signal isn't de ned in between the samples, so you can't directly use it in a continuous-time system. I To use a sampled-time signal in a continuous-time system, it must be converted into continuous-time. I This conversion is called reconstruction.
The Zero-Order Hold I Reconstruction is done by interpolating the value of the output signal in continuous time from the sampled-time input. I In embedded control system, this interpolation is done with digital to analog converters (or PWM generators), which behave as zero-order hold blocks. I A zero-order hold is simply a device which takes on the value of the most recent sample and holds it until the next sample time: y (t ) y t Ts I In a DAC, the zero-order hold action comes about because the processor writes a value to the DAC, which remembers this value until the next write. I That memory provides the hold function.
Zero-Order Hold (cont) I This shows the sampled-time signal (top) and the output of the zero-order hold I You can see the typical stair-step response of a DAC in it.
Reconstruction and Aliasing I Reconstruction causes a phenomenon quite similar to aliasing: I Consider the signal y (k ) cos (2π Ts f k ) 0 I On reconstruction, the resulting signal will have components at f f ,f 0 0 Ts f 1 , 0 2 Ts , · · ·
Zero-Order Hold Frequency Response I The zero-order hold has an e ect on frequency response: it has an amplitude response A sinc (πTs f ) ( 1 (π Ts f ) π Ts f sin f 0 otherwise I The phase response is shown in the Bode plot above, and is a consequence of the 1 2 cycle delay.
Zero-Order Hold Frequency Response, Cont I In audio and video circuits, these e ects are highly undesirable: I the unwanted high-frequency components means that highs appear in the low frequencies I the rollo of the zero-order hold a ects the frequency response, or the pixel appearance I Designers of such systems will work hard at reconstruction lters to solve these issues I As with aliasing, however, in a control system the added ltering is usually unnecessary, and adds unwanted phase shift.
Outline Sampling Noise and Signals The Laplace Transform Modeling the Real World in the z Domain
Outline Sampling Noise and Signals Orthogonal Signals and Power Random Noise Nonideal Sampling The Laplace Transform Modeling the Real World in the z Domain
Orthogonal Signals I Any two signals, whether random or not, have some measure (perhaps zero) of linear dependence, known as correlation. I Correlation is calculated or measured by comparing the product of the signals with their power: ρ r limN limN 1 N P N 1 N I The correlation coe cient, PN k 0 k 0 ρ, x (k ) x (k ) y (k ) 2 P N 1 N k 0 y (k ) 2 can range from -1 (for perfectly anti-correlated signals) to 1 (for perfectly correlated signals) I Signals x and y are perfectly orthogonal if ρ 0
Orthogonal Signals and Power I If you have two signals that are perfectly orthogonal, then the power of their sum is equal to the sum of their power: lim 1 N N X N k x k y k ( ( ) ( ))2 0 ρ 0 lim N 1 N X N k x (k ) 2 y k ( )2 0 I Real world examples of well-correlated signals are I the input and output of an ampli er I the command and response (we hope!) of a servo loop I Real world examples of uncorrelated signals are I sine waves at two di erent frequencies (which is why radio works) I two sinusoidal signals at exactly 90 degrees phase from one another I truly random noise and just about anything else (even other random noise)
Outline Sampling Noise and Signals Orthogonal Signals and Power Random Noise Nonideal Sampling The Laplace Transform Modeling the Real World in the z Domain
Random Noise in Control Systems I Many processes that in uence the operation of a control system take on the appearance of random signals. I Even if these signals aren't truly random, it is often convenient to treat them as such
Random Signal, Random Process I You need to use care when talking about random signals. A step is a step, a ramp is a ramp, a sine wave is a sine wave. I But a random signal can be anything; we need a meaningful way to talk about it without getting lost in speci cs. I The operative distinction is to separate a random the process that generates it. signal from For example I If I ip a coin, and if I assign a value of 1 to heads and -1 to tails, then the results of any one session of coin- ipping might go something like x {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, · · · } I Here, ipping the coin is the random process. The results of the coin tosses is my signal.
Characterizing Random Process I A random process has: I A probability distribution, such as Gaussian, uniform, etc. I A mean value, that is, the expected value of the average of the signal I A variance, that is, the expected value of the average power of the signal I A spectral density, which describes how that noise power is distributed in frequency I If a random variable is Gaussian, then it can be completely characterized by its mean and variance. I If a random process is Gaussian and stationary, it can be completely characterized by its spectral density (which tells you a lot about its mean and density)
Dealing With Random Noise I How to predict what will happen in a system with random noise is a study in itself I But the short story is simple, if the noise is Gaussian: I The noise will have a spectrum, with a power spectral density at each given frequency I The total noise will be a function of this power spectral density and the bandwidth of the noise that gets through the system. I You can lter noise just like any other signal I If you have noise in a frequency band that's a don't care for your desired signal, you can lter it out. I If you have noise in a frequency band that's congruent with your desired signal, you need to nd other ways to beat down the noise.
Random Processes and Power I The RMS level of a signal is the square root of its average power for example, North American power is 120 actual peak-peak value is around 340 V. VRMS ; the I If two noise processes are uncorrelated, then their respective noise powers will add: I if one noise process generates noise power of 9% of full scale, and another has power of 16% of full scale, their sum will have power of 25% of full scale. I In RMS terms, the rst signal would have a RMS level of 30% of full scale, the second would have an RMS level of 40% of full scale, and the third would have an RMS level of 25% of full scale. I Any time two uncorrelated signals are added, their power adds, which means that to nd their RMS level you need to square their levels, add, then take the square root.
Outline Sampling Noise and Signals Orthogonal Signals and Power Random Noise Nonideal Sampling The Laplace Transform Modeling the Real World in the z Domain
When Sampling Isn't Perfect I So far, our description of sampling has been the ideal case. I Sampling can be less-than-ideal in an in nite number of ways, but it usually boils down to some variation of these three: I E ects that add noise, bias or other undesired signals to the desired signal I E ects that act to lter the desired signal in undesired ways I E ects that act to make the sampling point jitter around I Most of these e ects are electronic in nature, so the more circuit theory you know, or the more you stay on the good side of your EE colleagues, the better. I Most of the ones that aren't electronic pertain to the plant that you're controlling, so don't neglect your mechanical (or chemical, or medical, or whatever) colleagues, either.
Measurement Noise & Bias I Noise, bias and unwanted signals can get into your measurement in a number of ways I Signal conditioning electronics can pick up ambient electromagnetic interference I Sensors can pick up undesired signals I (Not always just unwanted versions of what they're supposed to be picking up, either a project manager for a gyro vendor once held up his ber optic gyro and said this is a gyro. And a thermometer, an accelerometer, a microphone, a clock. ) I Electronics both the signal conditioning and the analog to digital converters themselves can be a source of signi cant random noise as well as signi cant measurement bias
System response to measurement noise I Once the noise is in your measurement, your system will respond to it just as if it were feedback. I This means that any measurement noise that is in band (in the frequency range your system responds to) will show up full strength in your system's intended output. at I This can be particularly bad if the measurement has bias or slow-moving noise: I a well designed system will faithfully keep the measured plant output on track I So if there is bias in the measurement, the actual plant output will be biased to the same degree the measurement is. I Bias that changes over time is particularly insidious, because you can't calibrate it out.
Dealing with measurement noise I Make it not happen. I This is really a job for the mechanical and circuit designers, I but it is your job to make them aware of the degree to which noise is a problem, I and to be aware of the strengths and limitations of various sensors and circuit approaches I Filter it out where possible I This has obvious problems (see my previous anti-aliasing example) I But sometimes there are things that one can do to bandlimit a system's response to noise without killing performance I Make it not reach your controller I I'm a big fan of sampling fast and decimating down to the control loop speed I You add a bit of delay, but also honkin' deep notches at all the frequencies that can alias to steady state
Filtering I Analog to digital converters lter data. I Every one introduces some amount of frequency shaping and delay into a signal. I This e ect ranges from insigni cant (in a ash converter) to heavy (in a sigma-delta converter) I Signal conditioning electronics and sensors lter data I Sometimes the e ect is insigni cant I Sometimes the e ect is sharp, and unavoidable due to sensor physics
Filtering in Sensor & Signal Conditioning I The eld is too large to address in speci cs here, but things to consider are: I Does the sensor have inherent delay, or does the signal path from plant to sensor have delay? I Does the sensor have an inherent low-pass nature (with the accompanying phase shift)? I Does the sensor have any resonances, either notches or peaks at certain frequencies? I All of the above questions pertain to the signal conditioning electronics, too.
Filtering in ADCs ADC type Filtering Delay Other insigni cant low bit count Less than can have one sample 1LSB noise E ects Flash little to none SAR Very little Pipelined Very little SAR Can be many Same or sample times more noise as plain ol' SAR Single-slope Dual-slope Some Up to 1 2 the low-pass sample time Comb As much as Lots of issues Slow one sample time Sigma-Delta Low-pass multiple Noise, check sample times for bias
Sampling Jitter I Sampling jitter is unique to the sampling process, and hard to analyze. I With sampling jitter, the sampling equation becomes x (k ) x (Ts k k ) where k is the sampling time error for the I In general (but not always!) k k th sample. will be a random process, where each number will be unknown but the ensemble will have some (hopefully) known statistics.
Modeling Sampling Jitter I You can't really model it within the framework of the z transform I With care, however, you can approximate the e ect as x (k ) ' x (Ts k ) k dtd x (t ) t Ts k I To get this into the z-domain, boil it down to x (k ) ' x (Ts k ) n (k ) then estimate the statistics of n (k ) from the expected rate of change of x (k ) and the statistics of .
Sources of Sampling Jitter I Jitter in the internal timing of the ADC. I This is usually an issue with SAR and ash converters, I and it is usually insigni cant for control loops I Jitter in the external timing of the ADC I Triggering the conversion from software I Hardware design I If you can drive your ADC conversion start from a hardware timer, you are generally safe.
When Sampling Jitter is a Problem I When you have a controller that must have high gain at high frequencies (i.e., derivative action) I When your plant output must slew rapidly I Put those together, and you can overwhelm your drive signal with high frequency noise. At worst I your drive signal will be slamming the rails at close to the sampling rate I your plant will never get the drive that it needs I ampli ers and/or your plant may use excess power, heat up, or even be mechanically damaged
Outline Sampling Noise and Signals The Laplace Transform Modeling the Real World in the z Domain
The Laplace Transform I The z transform is ne and dandy, and you can do most or all, if you're determined of your work in it. I But interfacing with the real world means working with continuous-time systems. I And continuous time systems means di erential equations I And to do formal analysis on transform those, we need the Laplace
Getting to the Laplace Transform I This shows part of a mixed-time system. I The continuous-time part implements a di erential equation: d y a (u y ) c dt (1) I Which you can approximate in sampled-time as: y (Ts k ) y (Ts (k 1)) ' a (u (T k ) y (T (k 1))) s s Ts (2)
Getting to the Laplace Transform (2) I If you look at (1) and (2), you see that the closer that Ts gets to zero, the more accurate the approximation is. I The z transform of (2) is Y (z ) Y (z ) Ts Y (z ) z a U (z ) z 1 or z 1 Y (z ) ' aU (z ) a Y (z ) Ts z z (3)
Getting to the Laplace Transform (3) I If we make a huge leap and declare a new variable sTs , then (3) becomes z e s such that e sTs 1 Y (s ) ' aU (s ) a Y (s ) Ts e sTs e sTs (4) I Now, (4) is just the z transform of (2); this means that as we bring Ts to zero it should become exact: e sTs 1 Y (s ) lim aU (s ) a Y (s ) lim Ts Ts Ts e sTs e sTs 0 or 0 sY (s ) aU (s ) aY (s )
Getting to the Laplace Transform (3) I This is starting to look an awful lot like the z transform. I To nish the thought, let's solve for Y (s ): Y (s ) s a a U (s ) (5) In fact, (5) is the Laplace transform of (1) I This is a special case. But we can extend it to derive the Laplace transform from the z transform: x t L { ( )} lim t 0 T X k 0 x (T k ) e s T k ˆ 0 x (t ) e st dt
Outline Sampling Noise and Signals The Laplace Transform Using the Laplace Transform A Frequency-Domain Interpretation Modeling the Real World in the z Domain
Laplace and Di erential Equations I The Laplace transform is the continuous-time equivalent of the z transform I Using it is similar, too: I it has handy properties which you use to turn a di erential equation into a Laplace-domain algebra problem, I and it has handy transforms that you use to convert the solution back into the time domain I And it only works with time-invariant, linear systems
Laplace Properties Property k x (t )} k X (s ) x (t ) x (t )} X (s ) X (s ) d L dt x (t ) sX (s ) L{ L{ 1 2 1 Comments k must be a constant The Laplace transform is linear 2 As the unit delay is to the z transform, so is the derivative L L dt n x (t ) dn n t 0 x (τ ) d τ x t X s 1 s X (s ) 1 ( ) ( ) s limt limt 0 o s n X (s ) operator to the Laplace just apply the above more than once Integration is just the opposite of the derivative, so . This is the nal value theorem for the Laplace transform
Common Laplace Transforms x((t ) u (t ) 10 tt 00 t n u (t ) e at u (t ) t n e at u (t ) sin (ω t ) u (t ) cos (ω t ) u (t ) e at sin (ωt ) u (t ) e at cos (ωt ) u (t ) x t L { ( )} 1 s 1 sn 1 s a 1 (s a)n 1 ω s 2 ω 2 s s 2 ω 2 ω s 2 2a s a2 ω 2 s a s 2 2a s a2 ω 2
Solving Di erential Equations I Consider a motor with an electromechanical time constant and a voltage-speed constant voltage v (t ). τ kT , which is being driven by a I The di erential equation for this system is d dt 2 2 t θ( ) d θ (t ) 1 v (t ) τ dt τ kT 1 I We go to the Laplace domain by recalling that s 2 s Θ( ) L d dt θ s: Θ 1 s Θ (s ) V (s ) τ τk 1 T I Finally, the idea of transfer functions works just as well in the Laplace domain as in z: s V s Θ( ) ( ) 1 τ kT s s τ 1
Outline Sampling Noise and Signals The Laplace Transform Using the Laplace Transform A Frequency-Domain Interpretation Modeling the Real World in the z Domain
Laplace and the Frequency Domain I Just as you can nd the frequency response of a system in the z domain, you can nd the frequency response of a system in the Laplace domain. Ae j φ H e j θ j φ H (j ω) For the Laplace domain, we nd Ae I For the z domain, we nd I I Here, ω is a frequency in radians per second
Magnit ude (dB) Laplace and the Frequency Domain (2) 20 0 -20 -40 -60 -80 -100 -120 -1 0 10 10 1 10 2 10 3 10 Phase (degree) -90.0 -112.5 -135.0 -157.5 -180.0 -1 0 10 10 I This is for our motor with I H (s ) s (s 250 50 ) 1 10 τ 20 ms 2 10 and 3 10 kT 0.2Nm/A
Outline Sampling Noise and Signals The Laplace Transform Modeling the Real World in the z Domain
Outline Sampling Noise and Signals The Laplace Transform Modeling the Real World in the z Domain Going from the Laplace Domain to the z Domain Going from Di erential Equations to the z Domain Going from the Real World to Math
By Approximation I If you have a Laplace-domain model of your plant, you can use the identity z e sTs (6) to generate a z-domain model. I In general, this involves nding a suitable approximation for equation (6) z ' 1 s Ts z 1 ' 1 s T s s T 2 The Tustin approximation splits the di erence: z ' s T 2 I The rst backward di erence uses I The rst forward di erence uses I s s
First Backward Di erence I First, observe that for I Then solve for s: s ' sT s z 1 Ts . I Then, everywhere that replace it with z 1 . Ts s 1, z ' 1 s Ts . occurs in the transfer function, I So, given a Laplace domain model G (z ) ' k z 1 Ts τ z 1 Ts 1 z G (s ) s (τ ks ) , we get 1 k Ts2/τ 2 I We can then use this in our analysis. 2 Tτs z 1 Tτs
First Forward Di erence I First, observe that for I Then solve for s: s ' s Ts 1, z z 1 . z Ts s k z 1 z Ts τ z 1 z Ts s Ts . occurs in the transfer function, I So, given a Laplace domain model G (z ) ' '1 z 1 Ts z . I Then, everywhere that replace it with 1 1 z G (s ) s (τ ks ) , we get 1 2 I We can then use this in our analysis. z k Ts2 2 Ts τ 2τ Ts 1 Ts τ z Tsτ τ
Tustin Approximation I Tustin approximation is equivalent to trapezoidal integration I Split the di erence between rst forward and rst backward s ' T2 ((zz 11)) di erence: s I Then, everywhere that replace it with 2(z 1) Ts (z 1) . s occurs in the transfer function, I So, given a Laplace domain model G (z ) ' 2(z 1) Ts (z 1) k τ T2s((zz 11)) 1 G (s ) s (τ ks ) , we get I We can then use this in our analysis. 1 z 2 z 1) z Tτs Tτs k Ts2 (Ts 2τ ) ( Ts4 τ2τ 2 2 2 2
The Exact Transform I If we take the zero-order hold action of the ADC into account (and assuming that there are no other such e ects in the system), then we can go from a Laplace domain plant transfer function to a z-domain transfer function directly I I won't go into the math here, but in Scilab you get the result by using the dscr function: H syslin('c', 10 / (%s * (%s 10)); dscr(H, 0.01)
Outline Sampling Noise and Signals The Laplace Transform Modeling the Real World in the z Domain Going from the Laplace Domain to the z Domain Going from Di erential Equations to the z Domain Going from the Real World to Math
Via the Laplace Transform I The best way to go from a di erential equation is via the Laplace Transform: I Develop a Laplace-transform model of the system from the di erential equations, I Then convert that model to the z domain using your favorite method
By Approximation I This is really just a shortcut for going through Laplace, using one of the approximate methods above. I For the rst forward di erence method, look at the di erential xk xk 1 dx equation, and convert every occurrence of . dt into Ts Repeat as necessary for higher-order derivatives, i.e. dx dt 2 2 dxp dt p k dxp dt p k 1 xk xk 1 Ts x x k 1Ts k 2 Ts Ts d x xk 2xk kk dt Ts 2 1 2 2 2 I Reduce the equation down. I This is a possible way to do the job but going through the Laplace transform makes the steps more explicit, and helps you keep from making errors.
Outline Sampling Noise and Signals The Laplace Transform Modeling the Real World in the z Domain Going from the Laplace Domain to the z Domain Going from Di erential Equations to the z Domain Going from the Real World to Math
Mathematical Modeling I This subject is worth its own lecture series I The short story: I study the physics, I make a di erential equation, I go from there to Laplace then z, I or go straight to z I You can go a long way with a domain expert, by seat of the pants, or by measurement.
Linearization I Laplace domain analysis only works with plants that are linear and time-invariant. I Real plants are neither I The challenge then is to nd an approximate model for the plant that is linear and time-invariant I The two most common methods for linearization are to linearize around an operating point, and to use describing function analysis.
Linearize around an Operating Point I This is the most common method of linearization to the point where it is often done unconsciously. I The idea is that you take a set of characteristics of the plant at one operating condition (torque, speed, etc.), and construct an a ne system around it. I The most common example of this is when you have a system (such as a DC motor) that is linear to the edges of its operating region I In this case then you can treat the whole system as linear, with the caviat that you never exceed those operating limits.
A Linearization Example I For example, a shunt-wound (universal) motor has a torque equal to the square of the motor current: Ta ks i 2 I If you know this motor is going to be operated at one torque only, say Top , then you nd the rst two terms of the Taylor's expansion around that point. i I Let op q Top ks . Then for su ciently small i i iop : Ta ' Top 2ks (i iop ) I Now you can ignore the bias torque (which is presumably absorbed by whatever the motor is driving) and write a di erential equation around (7): d ω (t ) 2ks i (t ) dt Ja (7)
Example: A DC Motor I A DC motor has a torque proportional to current: T kT i Ea kT ω (that's kT , by the way, if you keep your units straight). I A back-emf voltage proportional to speed: the same I An armature voltage that's the sum of the back-emf voltage and the resistive loss: va ea Ra i I And an acceleration that depends on the moment of inertia T d and the torque: dt ω Ja I Put this together, and you get a di erential equation: d kT dt ω (t ) Ja Ra (va (t ) kT ω (t )) (8)
A DC Motor I equation (8) describes the motor behavior in continuous time. I Take its Laplace transform: Ω ( ) kT ( a ( ) T Ω ( )) Ja Ra I Get its transfer function: V s s s T (s ) V Ω a s k kt Ja Ra k2 Ja Rt a I Use any of the approximations (or the exact transform) to convert. s
I Sampling is the process that converts continuous-time signals into discrete-time signals I For a feedback signal y (t ) in continuous time and a sample interval T s, the sampled-time feedback is y (k ) y (k T s) I Embedded control systems will need to take the appropriate plant measurements and turn them into digital signals:
2.1 Discrete-time Signals: Sequences Continuous-time signal - Defined along a continuum of times: x(t) Continuous-time system - Operates on and produces continuous-time signals. Discrete-time signal - Defined at discrete times: x[n] Discrete-time system - Operates on and produces discrete-time signals. x(t) y(t) H (s) D/A Digital filter .
2.1 Sampling and discrete time systems 10 Discrete time systems are systems whose inputs and outputs are discrete time signals. Due to this interplay of continuous and discrete components, we can observe two discrete time systems in Figure 2, i.e., systems whose input and output are both discrete time signals.
What is Discrete Mathematics? Discrete mathematics is the part of mathematics devoted to the study of discrete (as opposed to continuous) objects. Calculus deals with continuous objects and is not part of discrete mathematics. Examples of discrete objects: integers, distinct paths to travel from point A
Calculus tends to deal more with "continuous" mathematics than "discrete" mathematics. What is the difference? Analogies may help the most. Discrete is digital; continuous is analog. Discrete is a dripping faucet; continuous is running water. Discrete math tends to deal with things that you can "list," even if the list is infinitely .
6 POWER ELECTRONICS SEGMENTS INCLUDED IN THIS REPORT By device type SiC Silicon GaN-on-Si Diodes (discrete or rectifier bridge) MOSFET (discrete or module) IGBT (discrete or module) Thyristors (discrete) Bipolar (discrete or module) Power management Power HEMT (discrete, SiP, SoC) Diodes (discrete or hybrid module)
Definition and descriptions: discrete-time and discrete-valued signals (i.e. discrete -time signals taking on values from a finite set of possible values), Note: sampling, quatizing and coding process i.e. process of analogue-to-digital conversion. Discrete-time signals: Definition and descriptions: defined only at discrete
Computation and a discrete worldview go hand-in-hand. Computer data is discrete (all stored as bits no matter what the data is). Time on a computer occurs in discrete steps (clock ticks), etc. Because we work almost solely with discrete values, it makes since that
2 Nonlinear Continuous Discrete State Estimation 2.1 State Space Model The continuous-discrete state space representation (Jazwinski [4]) turns out to be very useful in systems, in which the underlying models are continuous in time and only discrete observa-tions are available. It consists of a continuous state equation for the state y(t) and .
