EXAM IN MODELING AND SIMULATION (TSRT62)

Transcription

1 EXAM IN MODELING AND SIMULATION (TSRT62) SAL: ISY:s datorsalar TID: Wednesday 4th January 2017, kl KURS: TSRT62 Modeling and Simulation PROVKOD: DAT1 INSTITUTION: ISY ANTAL UPPGIFTER: 5 ANTAL BLAD (inkl försättsblad): 10 ANSVARIG LÄRARE: Claudio Altafini, , BESÖKER SALEN: cirka kl. 9 och kl. 10 KURSADMINISTRATÖR: Ninna Stensgård , ninna.stensgard@liu.se TILLÅTNA HJÄLPMEDEL: 1. L. Ljung & T. Glad Modellbygge och Simulering (English title Modeling and Identification of Dynamical Systems ) 2. T. Glad & L. Ljung: Reglerteknik. Grundläggande teori 3. Tabeller (t ex L. Råde & B. Westergren: Mathematics handbook, C. Nordling & J. Österman: Physics handbook, S. Söderkvist: Formler & tabeller ) 4. Miniräknare Normala inläsningsanteckningar i läroböckerna är tillåtet. Notera att kommunikation med andra personer och informationshämtning via nätverket eller Internet inte är tillåtet under tentamen. LANGUAGE: you can write your exam in both English (preferred) or Swedish LÖSNINGSFÖRSLAG: Finns på kursens websida efter skrivningens slut. VISNING av tentan äger rum kl :00 i Ljungeln, B-huset, ingång 25, A-korridoren, room 2A:514. PRELIMINÄRA BETYGSGRÄNSER: betyg 3 23 poäng betyg 4 33 poäng betyg 5 43 poäng OBS! Lösningar till samtliga uppgifter ska presenteras så att alla steg (utom triviala beräkningar) kan följas. Bristande motiveringar ger poängavdrag. Lycka till!

2 COMPUTER TIPS: To open Matlab: open a terminal (right-click on the background and choose open terminal) type module add prog/matlab matlab & Print out the model description and the plots requested Remember to write your AID number on each printed page you include In the identification exercise using the System Identification toolbox: To print the model description: Right-click on the icon of the model you have computed and then click Present. The model description appears then on the matlab main window. Copy it into a file and print it. the SysId plots cannot be directly printed. You have to choose File Copy figure, which gives an ordinary matlab plot you can print. Printing in Linux: A file called file.pdf can be printed out for instance typing in a terminal lp -d printername file.pdf (replace printername with the name of the printer in the room you sit in). It is possible to print using File Print in a matlab plot, but one must select the printer name writing -Pprintername in the Device option (again printername is the name of your printer). 2

3 Phase (deg) Phase (deg) Magnitude (db) Magnitude (db) 1. (a) In a numerical integration method, what is the difference between a one-step and a multistep method? When is a multistep method said implicit? (b) You are asked to identify a system with black box models, knowing that the true transfer function from input to output has the Bode plot shown in Fig. 1(a), while the transfer function from noise to output has the Bode plot of Fig. 1(b). i. What sampling time would you choose to collect the measurements? ii. What class of black-box models and with how many poles/zeros would you choose? 20 input -> output 0 noise -> output Frequency (rad/s) Frequency (rad/s) (a) (b) Figure 1: Bode plot of Ex. 1(b) (c) Consider the circuit of Fig. 2, where φ 1 ( ) and φ 2 ( ) are nonlinear resistors. i. Show that there is more than one conflict-free causality markings. ii. What causality marking would you choose if φ 1 is invertible but φ 2 is not? Why? Figure 2: Circuit of Ex. 1(c) 3

4 2. (a) Given the true system y(t) = 0.5y(t 1) + u(t 1) + w(t) with w(t) a white noise (of zero-mean, variance λ w ), which of the following 3 models y(t) = αy(t 1) + βy(t 2) + γu(t) + e(t) (1) y(t) = αy(t 1) + βu(t 1) + e(t) (2) y(t) = αy(t 1) + βu(t) + γu(t 1) + e(t) (3) (e(t) is always a zero-mean white noise) can give an unbiased estimation of its parameter vector if u(t) is chosen as white noise uncorrelated with e(t)? Motivate your answer. [3p] (b) Consider the model (2), and assume u(t) is obtained through the feedback law u(t) = y(t) + v(t) (4) where v(t) is a zero-mean white noise of variance λ v, uncorrelated with w(t) and e(t). For the model (2) with the feedback (4) use the least-squares method to compute the asymptotic value of the parameters α and β when N. [5p] (c) For the model (2) with the feedback (4), compute the covariance matrix of the estimates for N large enough. 4

5 3. The data for this exercise are in a file called sysid_data_ mat located in the directory /site/edu/rt/tsrt62/exam/. To load it into your Matlab workspace use any of the following: type in the Matlab window load /site/edu/rt/tsrt62/exam/sysid_data_ mat copy the file to your current directory and then load it into your Matlab workspace (typing load sysid_data_ mat at the Matlab prompt). Inside sysid_data_ mat you will find the sampled signals u and y (the sample time is T s = 0.1). Construct one or more appropriate black-box models. For one or more of these models report plot of the fitted model vs. validation data parameter values and uncertainty quality of the fit poles and zeros placement Discuss and comment your choices and results. [10p] 5

6 4. Consider the mechanical-hydraulic system of Fig. 3. Call m 1 and m 2 u 1 m 1 k 1 k 2 m 2 u 2 Figure 3: the two frictionless masses, on which the two forces u 1 and u 2 act. The masses are connected, via elastic springs (of spring constants k 1 and k 2 ), to hydraulic pistons of area A. Call p 1 the pressure at the left piston and p 2 that at the right piston. Between the pistons, the fluid exercises a friction. If Q is the fluid flow from left to right, then the friction is expressed by the law where φ is an invertible function. Q = φ(p 1 p 2 ) (a) Draw the bond graph of the system and check its causality. [4p] (b) Compute the state space equations. [4p] (c) Assuming that the flow in the hydraulic system is friction-free (p 1 = p 2 ), what kind of complication (if any) appears in the bond graph? 6

7 5. Consider a friction-free pendulum in cartesian coordinates, with mass m and weightless rope of length L. The equations are: ẋ 1 (t) = x 2 (t) ẋ 2 (t) = τ(t)x 1 (t) ẋ 3 (t) = x 4 (t) ẋ 4 (t) = τ(t)x 3 (t) g 0 = x 2 1(t) + x 2 3(t) L 2 where x 1 (t) and x 2 (t) are position and velocity along the x-axis, x 3 (t) and x 4 (t) are position and velocity along the y-axis, τ(t) is the tension along the rope and g is the gravitational constant. The system can be written in semi-explicit form as ẋ(t) = f ( x(t), y(t) ) 0 = g ( x(t) ) where the state variables have been split into x = (x 1, x 2, x 3, x 4 ) T and y = τ. Compute the index of this DAE. [10p] 7