离散时间控制系统课件

1、Computer-Controlled Systems 8/19/2022Course InformationTime: 13:30-15:10pm, Wednesday 10:00-11:40am, Friday (even weeks)Venue: 综B204References:离散时间控制系统(英文版第2版)，Katsuhiko Ogata，机械工业出版社，2004离散时间控制系统(中文版)，Katsuhiko Ogata，陈杰，蔡涛等译，机械工业出版社，2006Grading Procedure:in-term evaluationexperimental resultsfinal

2、paper examination. 2Chapter 1Introduction to Discrete-Time 34Contents1-1 Introduction1-2 Digital Control Systems1-3 Quantizing and Quantization Error1-4 Data Acquisition, Conversion and Distribution Systems1-5 Concluding Comments41-1 IntroductionDigital ControllerA rapid increase in the use of digit

3、al controllers5Digital ControllerThe trend is due toDigital control can achieve optimal perfor- mance, have Decision-making capability and flexibility in the control programAvailablity of low-cost digital computers Advantages of digital signals1-1 Introduction6Types of SignalsContinuous-time signal:

4、 A signal defined over a continuous range of timeAnalog signal: A signal defined over a continuous range of time whose amplitude can assume a continuous range of valuesA continuous-time quantized signalDiscrete-time signal: A signal defined only at discrete instants of time1-1 Introduction7Sampled-d

5、ata signal: A discrete-time signal if the amplitude can assume a continuous range of valuesDigital signal: A discrete-time signal with quantized amplitudeComparisonDiscrete-time, digital, sampled data signal (control system)Continuous-time, analog signal (control system)1-1 Introduction81-1 Introduc

6、tion9Systems Dealt With in This BookLinear and time invariant Discrete-Time Control Systemsone or more variables can change only at discrete instants of time. These instants may specify the times at which some physical measurement is performed or the memory of a digital computer is read outDescribed

7、 in linear difference equations with constant coefficients1-1 Introduction101-2 Digital Control SystemsFigure 1-2 Block diagram of a digital control system11S/H and A/D (A/D)Sample-and-Hold (S/H)Sampling Processes, replace original continuous-time signal by a sequence of values at discrete-time time

8、 pointsa circuit that receives analog input signal and holds this signal at a constant value for a specified period of time.Analog-to-Digital Converter (A/D)Also called an encoder, is a device that converts an analog signal into a digital signal, usually a numerically coded signal. A S/H circuit is

9、often an integral part of a commercially available A/D converter.1-2 Digital Control Systems12Types of Sampling OperationsPeriodic samplingtk = kT (k = 0, 1, 2, )Multiple-order samplingtk+r - tk = constantMultiple-rate samplingA digital control system have different sam- pling periods in different f

10、eedback pathsRandom samplingtk is a random variable1-2 Digital Control Systems13Signal Forms in a Digital Control System1-2 Digital Control SystemsFigure 1-3 Block diagram of a digital control system showing signals in binary or graphic form14D/A and holdDigital-to-Analog Converter (D/A)Also called

11、a decoder, is a device that converts a digital signal into an sampled-data signal.Holdreconstruct the analog signal that has been transmitted as a train of pulse samples, i.e. fill in the spaces between sampling periods and thus roughly reconstruct the original analog signal1-2 Digital Control Syste

12、ms15Plant or ProcessA plant is an physical object to be controlled. We call any operation to be controlled a process.Accurate modeling is perhaps the most difficult part in the design of control systemTransducerIs a device that converts an input signal into an output signal of another form, such as

13、a device that converts a pressure signal into an voltage output.Analog transducer, sampled-data transducer, digital transducer1-2 Digital Control Systems16The main functions involved in A/D conversion are sampling, amplitude quantizing and codingAmplitude quantizing Represent a continuous or analog

14、signal by a finite number of discrete states is called amplitude quantizationCoding or EncodingRepresent a sample value by a numerical code1-3 Quantizing and Quantization Error17QuantizingThe standard number system is the binary number system. The code group consists of n pulses each indicating eith

15、er on (1) or off (0). In the case of quantizing, n on-off pulses can represent 2n amplitude levels or output states.The quantization level Q: the range between two adjacent decision points: Q = FSR/2n ,FSR is the full-scale range.MSB is the most significant bit, has the most weight (one half of the

16、full scale)LSB is the least significant bit, has the least weightLSB = FSR/2n1-3 Quantizing and Quantization Error18Quantization ErrorSince digital output can assume only a finite number of levels, an analog number must be rounded off to the nearest digital level.Quantization error varies between 0

17、and 1/2Q.Quantization error depends on fineness of the Q, and can be made as small as desired by making Q smaller.1-3 Quantizing and Quantization Error191-3 Quantizing and Quantization ErrorTo determine the desired size of the quantization level in a given digital control system, the engineer must h

18、ave a good understanding of the relationship between the size of the quantization level and the resulting error. 20For an analog input x(t), the output y(t) takes on only a finite number of levels, which are integral multiples of the quantization level Q1-3 Quantizing and Quantization ErrorFigure 1-

19、4(a) Block diagram of a quantizer and its input-output characteristics21Round-off errorThe error resulting from neglecting the remaining digits is called round-off error.Quantization error is a round-off errorThe finer the quantization level is, the smaller the round-off error.1-3 Quantizing and Qua

20、ntization Error22Round-off error1-3 Quantizing and Quantization ErrorFigure 1-4(b) Analog input x(t) and discrete output y(t)23Quantization noise: the uncertainty present in the quantization process.For a small quantization level Q, the quan-zation error is similar to that of noise. So quantization

21、process acts as a source of random noise.The variance of the quantization noise is 241-4 Data Acquisition, Conversion and Distribution SystemsFigure 1-5 (a) Block diagram of a data-acquisition system; 25TransducerA physical variable such as position, velocity, acceleration, temperature is first conv

22、erted into an electrical signal (a voltage or current)Amplifier Amplifies the voltage output of the transducerConverts a current signal into a voltage signalBuffers the signal1-4 Data Acquisition, Conversion and Distribution Systems26Low-pass filterAttenuates the high-frequency signal components, su

23、ch as noise (electronic noises are random in nature and may be reduced by low-pass filters. However, such common electrical noises as power-line interference are generally periodic and may be reduced by means of notch filters.)Analog MultiplexingA device that performs the function of time-sharing an

24、 A/D converter among many analog channels.1-4 Data Acquisition, Conversion and Distribution Systems27If many signals are to be processed by a single A/D and a digital controller, then these input signals must be fed to the controller through a multiplexer.Is a multiple switch that sequentially switc

25、hes among input channels in some prescribed fashion.At a given instant of time, only one switch is in the “on” position. When the switch is on in a given input channel, the input signal is connected to the output of the multiplexer for a specified period of time.1-4 Data Acquisition, Conversion and

26、Distribution Systems281-4 Data Acquisition, Conversion and Distribution Systems29DemultiplexerSeparates the composite output digital data from the digital controller into the original channelsSample-and-Hold CircuitsSampler: covert an analog signal into a train of amplitude-modulated pulses. Hold ci

27、rcuit: hold the value of the sampled pulse signal over a specified period of time1-4 Data Acquisition, Conversion and Distribution Systems30two operation modesThe tracking mode: the switch is closed, i.e., the input signal is connectedThe hold mode: the switch is open, i.e., the input signal is disc

28、onnectedWhen the sampling duration is negligible, the sampler may be considered an ideal sampler1-4 Data Acquisition, Conversion and Distribution Systems311-4 Data Acquisition, Conversion and Distribution SystemsFigure 1-7 Sample-and-hold circuit321-4 Data Acquisition, Conversion and Distribution Sy

29、stemsFigure 1-8 Tracking mode and hold mode33Analog-to-Digital ConvertersThe process by which a sampled analog signal is quantized and converted to a binary number.Types of frequently used A/D ConvertersSuccessive-approximation typeIntegrating typeCounter typeParallel typeSelection criterions of A/D

30、 converters Conversion speed, accuracy, size and cost1-4 Data Acquisition, Conversion and Distribution Systems34Counter type A/D (the simplest A/D)Clock pulses are applied to the digital counter in such a way that the output voltage of the D/A converter (that is, part of the feedback loop in the A/D

31、 converter) is stepped up one least significant bit (LSB) at a time.Then the output voltage is compared with the analog input voltage once for each pulse. When the output voltage has reached the magnitude of the input voltage, the clock pulses are stopped.The counter output voltage is then the digit

32、al output. 1-4 Data Acquisition, Conversion and Distribution Systems35Successive-approximation type (most frequently used)The principle is: The successive-approximation register (SAR) first turns on the most significant bit (half the maximum) and compares it with the analog input. The comparator dec

33、ides whether to leave the bit on or turn it off. If the analog input voltage is larger, the most significant bit is set on.Next, turn on bit 2 and then compare the analog input voltage with three-fourths of the maximum. 1-4 Data Acquisition, Conversion and Distribution Systems36After n comparisons a

34、re completed, the digital output of the successive-approximation register indicates all those bits that remain on and produces the desired digital code. Thus, this type of A/D converter sets 1 bit each clock cycle, and so it requires only n clock cycles to generate n bits, where n is the resolution

35、of the converter in bits. (The number n of bits employed determines the accuracy of conversion.) The time required for the conversion is approximately 2 sec or less for a 12-bit conversion.1-4 Data Acquisition, Conversion and Distribution Systems371-4 Data Acquisition, Conversion and Distribution Sy

36、stemsFigure 1-9 Schematic diagram of a successive-approximation-type of A/D converter38Errors in A/D ConvertersThe input-output characteristics of A/D Converters change with time and temperature.Actual analog-to-digital signal converters always have some errors, such as offset error, linearity error

37、, and gain error.Commercial converters are specified for three basic temperature ranges Commercial (0C to 70C)Industrial (-25C to 85C)Military (-55C to 125C)1-4 Data Acquisition, Conversion and Distribution Systems39Errors in A/D Converters1-4 Data Acquisition, Conversion and Distribution SystemsFig

38、ure 1-10 Errors in A/D converters (a) offset error; (b) linearity error; (c) gain error40The reverse of the data-acquisition A data-distribution system consists of registers, a demultiplexer, digital-to-analog converters, and hold circuits. It converts the signal in digital form (binary numbers) int

39、o analog form. The output of the hold circuit is fed to the analog actuator, which, in turn, directly controls the plant under consideration.1-4 Data Acquisition, Conversion and Distribution SystemsFigure 1-5 (b) block diagram of a data distribution system41Digital-to-Analog ConvertersFor the full r

40、ange of the digital input, there are 2n corresponding different analog values, including 0. For the digital-to-analog conversion, there is a one-to-one correspondence between the digital input and the analog outputTwo common D/A methodsWeighted resistors: simple in circuit configuration, but its acc

41、uracy may not be very goodR-2R ladder network: a little more complicated in circuit configuration, but is more accurate.1-4 Data Acquisition, Conversion and Distribution Systems42Weighted resistors1-4 Data Acquisition, Conversion and Distribution SystemsFigure 1-11 Schematic diagram of a D/A convert

42、er using weighted resistors43Weighted resistorsThe input resistors of the operational amplifier have their resistance values weighted in a binary fashion.When the logic circuit receives binary 1, the switch connects the resistor to the reference voltage; when the logic circuit receives binary 0, the

43、 switch connects the resistor to groundNotice that: as the number of bits is increased, the range of resistor values becomes large and consequently the accuracy becomes poor1-4 Data Acquisition, Conversion and Distribution Systems44R-2R ladder network1-4 Data Acquisition, Conversion and Distribution

44、 SystemsFigure 1-12 n-bit D/A converter using an R-2R ladder circuit45Note that: with the exception of the feedback resistor (which is 3R), all resistors involved are either R or 2R. This means that a high level of accuracy can be achieved1-4 Data Acquisition, Conversion and Distribution Systems46Ho

45、ld CircuitsFill in the spaces between sampling periods and thus roughly reconstruct the original analog signalThe hold circuit: to extrapolate the output signal between successive points according to some prescribed manner Zero-order-hold: produces a staircase waveform1-4 Data Acquisition, Conversio

46、n and Distribution Systems471-4 Data Acquisition, Conversion and Distribution SystemsFigure 1-13 Output from a zero-order hold48First-order-hold: generates an output slope equal to the slope of a line segment connecting previous and present samples and projecting it from the value of the present sam

47、ple.More accurately than a zero-order hold. If the slope of the original signal does not change much, the prediction is good. If, however, the original signal reverses its slope, then the prediction is wrong. And the output goes in the wrong direction, thus causing a large error for the sampling per

48、iod considered.1-4 Data Acquisition, Conversion and Distribution Systems491-4 Data Acquisition, Conversion and Distribution SystemsFigure 1-14 Output from a first-order hold50Interpolative first-order hold: generates a straight-line output whose slope is equal to that joining the previous sample val

49、ue and the present sample value, but the projection is made from the prediction point.Its accuracy is better than that of other hold circuits, but there is a one-sampling period delay. From the viewpoint of the stability of closed-loop systems, such a delay is not desirable, and so the interpolative

50、 first-order hold (polygonal hold) is not used in control system applications.1-4 Data Acquisition, Conversion and Distribution Systems511-4 Data Acquisition, Conversion and Distribution SystemsFigure 1-14 Output from an interpolative first-order hold (polygonal hold)52Digital Controllers and Analog

51、 ControllerAnalog Controllersrepresent the variables in an equation by continuous physical quantities. can easily be designed to serve satisfactory as non-decision-making controllersthe cost increases rapidly as the complexity of the computations increases1-5 Concluding Comments53Digital Controllersoperate only on numbersdecision making