Sampled Data and Digital Control Systems
Sampled Data and Digital Control Systems ECEN 5458
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This 39 page Class Notes was uploaded by Mrs. Lacy Schneider on Thursday October 29, 2015. The Class Notes belongs to ECEN 5458 at University of Colorado at Boulder taught by Lucy Pao in Fall. Since its upload, it has received 14 views. For similar materials see /class/231780/ecen-5458-university-of-colorado-at-boulder in ELECTRICAL AND COMPUTER ENGINEERING at University of Colorado at Boulder.
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Lucy Y Pao Lecture3 Page 1 ECEN 5458 Analysis of Systems Transfer functions Statespace representations Block diagram manipulation Relation of transfer function to pulse response BIBO stability Lucy Y Pao Lecture3 Page 2 ECEN 5458 Transfer Functions o The Z transform has the same role in discretetime systems that the Laplace transform has in the analysis of continuoustime systems 0 If uk and ek are related by a difference equation the transfer function Hz is defined as the ratio of the transform of the output uk of the system to the transform of the input ek of the system Lucy Y Pao Lecture3 Page 3 ECEN 5458 o For a general difference equation k a1 k 1 azuk z anuk n bOek blek l bmek m 0 Take Z transform and use the timeshift property b0 blz 1 bmz39m 11z 1 azlz391 a2z2 anz39 boz blz 1 bmz39Hquot bz zquot azlz39 1 azzz39 2 an az 11z Lucy Y Pao Lecture3 Page 4 ECEN 5458 Hz z39Hquot bozm blzm1 bm bz z azlz 1 azzz 2 an az Hz is a rational function of a complex variable Solutions of bz 0 are zeros of the transfer function Solutions of az 0 are poles of the transfer function 0 Special example Lucy Y Pao Lecture3 Page 5 ECEN 5458 StateSpace Representations 0 You should know how to convert an nthorder ordinary differential equation ODE into a set of n 1Storder ODE s 0 Similarly an nthorder difference equation can be converted into a set of n 15torder difference equations 0 Often done because set of 15torder equations is easier to solve 0 Consider 3rdorder system as an example There are many statespace representations of this system Lucy Y Pao Lecture3 Page 6 ECEN 5458 Control Canonical Form Uz HzEz Ez Ez az then Uz bzfz and az z Ez Let f z 5k3 ek a1 k2 aze k1 13506 uk b0 k 3 b1 k 2 12250 1 123460 Lucy Y Pao Lecture3 Page 7 ECEN 5458 Block Diagrams of Systems 0 Can be drawn directly from statespace representations using only simple delay z391 blocks gains and summers 3510 1 a1x1k a2x2k a3x3k 90 x2k1 3510 x3k1 x200 Lucy Y Pao Lecture3 Page 8 ECEN 5458 0 Control Canonical Form and Observer Canonical Form which we will discuss much later is a Direct Canonical realization because all the gains in the block diagram are just the coefficients in the transfer function polynomials 0 Will discuss Control and Observer forms more later Lucy Y Pao Lecture3 Page 9 ECEN 5458 NonUniqueness of StateSpace Representations 0 Any nonsingular transformation of state yields another statespace representation of the same transfer function Xk 1 Axk Bek uk CXk Dek Let ik Txk k 1 TAT 1ik TBek uk CT 1 k Dek Lucy Y Pao Lecture3 Page 10 ECEN 5458 Block Diagram Manipulation Four rules of block diagram manipulation analysis and simplification 0 Series Product ampH1z xa o ParallelSum H1z E z 25 U z H2z Lucy Y Pao Lecture3 Page 11 ECEN 5458 0 Single loops E Z 111z lz2 112z Lucy Y Pao Lecture 3 Page 12 ECEN 5458 0 Moving nodes across blocks or blocks across nodes Iiz 172Z gt L 113z Multipath multiloop block diagrams can be simplified using 0 Above techniques 0 Mason s rule Lucy Y Pao Lecture3 Page 13 ECEN 5458 Relation of Transfer Function to Pulse Response In continuoustime systems Es Hs 75 The transfer function of a system is equal to the Laplace Transform of the impulse response We find a similar result in discretetime systems E z 11z lz Lucy Y Pao Lecture3 Page 14 ECEN 5458 3130 Stability 0 Internal vs External stability Internal stability is concerned with all the internal variables states of the system E z Hz External stability is concerned only with the output BIBO stability is the most common type of external stability used o If for every ounded lnput we have a ounded Qutput we say the system is 3130 stable Lucy Y Pao Lecture3 Page 15 ECEN 5458 A Test for 3130 Stability 0 can be given in terms of the unitpulse response First consider a sufficient condition Suppose the input ek is bounded The output can be written as U z H zE z Can we find a bound on the magnitude of the output This condition is also necessary Lucy Y Pao Lecture3 Page 16 ECEN 5458 Consider the input 1 h j lt 0 e sgnh sgnh j 0 h j 0 1 h j gt 0 a bounded input Above provides requirement on unitpulse response hk Lucy Y Pao Lecture3 Page 17 ECEN 5458 o What are the requirements on Hz for 3130 stability That is what are the requirements in the frequency domain Hz bozquotblzquot39l bmzquot m bZ zquot azlzquot391 azzzquot392 0 0Z c c c lz im quotz for mltn zp1 zp2 zpn Lucy Y Pao Lecture3 Page 18 ECEN 5458 Example Suppose in our general difference equation k a1 k 1 azuk z anuk n bOek blek l bmek m that all the coefficients are zero except a1 and b0 o What is the unitpulse response 0 Is the system 3130 stable Or under what specific conditions will the system be 3130 stable Lucy Y Pao Lecture3 Page 19 ECEN 5458 Jury Test 0 Provides a quick way to determine by hand whether the roots of a polynomial are all inside the unit circle 0 Analogous to Routh Test for continuoustime systems 0 Useful for determining stability of a class of systems where certain parameters may not have fixed quantities Lucy Y Pao Lecturel Page 1 ECEN 5458 SampledData and Digital Control Systems Course overview Background knowledge AnalogtoDigital converters Digital signals 0 Approximation of differential equations using difference equations DigitaltoAnalog converters Delay Lucy Y Pao Lecturel Page 2 ECEN 5458 I assume you have a background in introductory feedback control at the level of ECEN 4138 You should know 0 Laplace transforms 0 Block diagram analysis 0 P I D control 0 Lead and lag compensation Stability 0 Root locus 0 Frequency response Bode and Nyquist plots 0 Introductory statespace representations Relationship to transfer functions Statefeedback controllers Lucy Y Pao Lecturel Page 3 ECEN 5458 In practice controllers are often implemented digitally Good performance usually means 0 Output follows or tracks reference well despite Disturbances and sensor noise Modeling errors Parameter variations 0 Feedback systems are more robust than openloop systems Lucy Y Pao Lecturel Page 4 ECEN 5458 AnalogtoDigital A D Converters 0 Convert a continuous physical variable usually a voltage to a stream of numbers Lucy Y Pao Lecture 1 Page 5 ECEN 5458 o A discrete signal can only change at discrete times 0 A sampleddata system is a system having both discrete and continuous signals 0 A D converters not only provide a discrete signal but also a guantized signal that is the signal must be stored in a finite number of bits Quantization is a nonlinear function 0 A signal that is both discrete and quantized is a digital signal Digital computers process digital signals Lucy Y Pao Lecture 1 Page 6 ECEN 5458 Digital controller analysis and design take into account effects of sampling period T and the quantization size q If both Tand q are extremely small the digital signals may be considered nearly continuous and continuous methods of analysis and design can be used and then converted to the digital domain In this course we will try to gain an understanding of the effects of Sample rates fast and slow Quantization large and small word sizes Why not just always make sure the sampling rate is fast and the quantization size is small and then just design Ds and approximate it with Dz Lucy Y Pao Lecturel Page 7 ECEN 5458 In this course we will treat the problem of varying T and q separately 0 11 weeks Consider only the effect of T assuming q 0 Assume linearity and timeinvariance what is LTI o 12 weeks depending on student interest Effects of q i 0 o 1 week depending on student interest Samplerate selection Lucy Y Pao Lecture 1 Page 8 ECEN 5458 In more detail Chapters 0 11 weeks q 0 3 amp 4 Discrete systems linear constant I Ztransform of discrete signals pulse transfer functions Sampleddata systems Discrete transfer functions of continuous systems that are sampled System representations ale Transform methods ale Statespace methods Dynamic response of discrete systems Lucy Y Pao Lecturel Page 9 ECEN 5458 In more detail continued Chapter 5 Intersample ripple Fourier analysis Aliasing sampling theorem Chapter 6 Digital filters Chapter 7 Design of feedback controllers Transform methods Root locus o Frequency response Chapter 8 Statespace methods Lucy Y Pao Lecture 1 Page 10 ECEN 5458 In more detail continued Depending on student interest Chapter 10 o 12 weeks effects of q i 0 Worst case analysis Average effects using random signal analysis Chapter 11 o 1 week samplerate selection Earlier analyzed effects of T Here Consider Tas a design parameter 0 Will use MATLABSimulink in homeworks and labs Lucy Y Pao Lecturel Page 11 ECEN 5458 Digitization o How can we approximate a differential equation using a difference equation 0 Euler s Forward Rectangular Rule or Euler s Method xk 1T xkT T 5ckT 0 You will explore Euler s Backward Rectangular Method in HW 1 Lucy Y Pao Lecturel Page 12 ECEN 5458 Example 31 of text 0 Suppose Ds UsK0sa agt0 bgt0 Es sb General form of a lead or lag compensator o How can we implement this compensator using a difference equa on First what is the differential equation that Ds represents Lucy Y Pao Lecturel Page 13 ECEN 5458 Example continued 0 Apply Euler s Method Libu K0e39ae o Rework to write uk1 the new control as a function of the past control uk and the current and past errors ek1 and ek Lucy Y Pao Lecture 1 Page 14 ECEN 5458 Comments on Example 0 Best to rearrange calculations so that uk1 is output to plant as quickly as possible after current y and r are read See example in Table 31 of text 0 Implementing Ds in this way will work well meaning digital implementation leads to essentially the same performance as Ds if the sample rate is 2 30 x BW of the system where the bandwidth is of the closedloop continuous system with Ds Lucy Y Pao Lecture 1 Page 15 ECEN 5458 c As the sample rate decreases below 30 x BW the closedloop control performance degrades More overshoot Appears to be less damped Longer settling times 0 In this class we will spend some time discussing good methods of converting Ds for discretetime implementation if a continuous Ds design is already available 0 We will also discuss methods of directly designing a discrete controller Lucy Y Pao Lecturel Page 16 ECEN 5458 DigitaltoAnalog DA Converters 0 Single most important impact of implementing a control system digitally is the delay associated with the BIA 0 Each value of ukT is typically held constant until the next value is available from the computer called a ZeroOrder Hold or ZOH The continuous value of ut consists of steps that on average lag ukT by Lucy Y Pao Lecture 1 Page 17 ECEN 5458 o If we simply include a T delay in a continuoustime analysis of a digital system good agreement results 0 Performance of discretetime implementation of Ds with sample time T can be approximated by this system 0 Delay in any feedback system degrades damping and ultimately stability of the system Lucy Y Pao Lecture 1 Page 18 ECEN 5458 0 Could plot a locus of roots as a function of T to understand the behavior of the discretized implementation of Ds on system performance 0 Alternatively the delay effect can be analyzed using frequency response techniques Lucy Y Pao Cover Contents Page 1 ECEN 5458 ECEN 5458 SampledData and Digital Control Systems Fall 2007 Lecture Notes Professor Lucy Y Pao Electrical amp Computer Engineering Department University of Colorado Boulder Colorado Lucy Y Pao Cover Contents Introductory Note I started typesetting these notes for ECEN 5458 when I taught the course in Fall 2004 and I modi ed and updated them in Fall 2005 and Fall 2006 and again for this Fall 2007 Because gures and equations are extremely time consuming to typeset many of these have not been typeset into these notes These and many other details will be lled out during lectures as we go along I have used and will use many sources in the preparation of the lecture notes and I do not claim to be original in the presentation of any of the ideas In fact some portions have been directly taken from various texts etc While I have done my best to check these notes there are probably still errors in them and I hope to catch any remaining errors throughout the semester Lecture PWHF VIPP P NNHHHHHHHHHH PPPP NPMPPPFP Page 2 ECEN 5458 Table of Contents Topic Course Overview Linear Difference Equations and the Z Transform Transfer Functions State Space Representations and Stability Discrete Models of Sampled Data Systems Dynamic Response of Discrete Time Systems Sample and Hold and Block Diagram Analysis Numerical Integration Pole Zero Mapping and Hold Equivalents Performance Speci cations and Root Locus Review Discrete Controller Design Steady State Analysis and Frequency Response Methods Nyquist Stability Criterion Bode Design and Lead Lag amp PID Compensators Ragazzini Method and Robust Control Introduction to State Space Methods State Space Controller and Estimator Design More on Estimator Design and Duality Current and Reduced Order Estimators Combined Control Law and Estimator Separation Principle Introducing a Reference Input Integral Control and Disturbance Estimation