TENTAMEN REGLERTEKNIK TSRT15 (English)

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1 TENTAMEN REGLERTEKNIK TSRT5 (English) SAL: TER TID: 28 april 2, klockan 8-3 KURS: TSRT5 PROVKOD: TEN INSTITUTION: ISY ANTAL UPPGIFTER: 5 ANTAL BLAD: 4 exklusive försättsblad ANSVARIG LÄRARE: Johan Löfberg JOURHAVANDE LÄRARE: Johan Löfberg, tel , BESÖKER SALEN: 9:, :, 2:3 KURSADMINISTRATÖR: Ninna Stensgård, tel , ninna.stensgard@liu.se TILLÅTNA HJÄLPMEDEL: Godtycklig lärobok i reglerteknik (anteckningar i bok tillåtna) tabeller, formelsamling, räknedosa utan färdiga program. LÖSNINGSFÖRSLAG: Anslås efter tentamen på kursens hemsida. VISNING av tentan äger rum kl i Ljungeln, B-huset, ingång 27, A-korridoren till höger. PRELIMINÄRA BETYGSGRÄNSER: betyg 3 23 poäng betyg 4 33 poäng betyg 5 43 poäng Solutions to all problems must be presented in such detail that all steps (except trivial calculations) can be followed. Missing motivations will reduce the points given. Lycka till! Good luck! Viel Glück! Bonne chance!

3 . As always, all answers must be thoroughly motivated! (a) Which ending of the following joke is mathematically motivated? On a flight between Warsaw, Poland and Moscow, Russia, the plane enters an area with turbulence and the plane goes unstable. The Captain grabs his microphone and shouts: i. This is the Captain speaking, would all Poles please move to the left half of the plane! ii. This is the Captain speaking, would all Poles please move to the right half of the plane! (p) (b) Auto-tune is tool used in the music industry to correct pitch, i.e., change the frequency of a sound signal. Would you expect it to be possble to implement an auto-tuner as a linear system, i.e., y(t) = G(s)u(t) where u(t) is the original sound signal, G(s) a linear system (filter) and y(t) the auto-tuned sound signal? (c) A system is given by the following state-space model ( ) ( ) ẋ(t) = x(t)+ u(t) α y(t) = ( α ) x(t) For which α is it possible to create an observer, with arbitrarily placed poles, estimating the two states from the measurement y(t) and u(t)? Try to interpret the condition. (3p) (d) The system Y(s) = G(s)U(s), G(s) = + s+ is driven by the input u(t) = sin(t). What will the output y(t) be after a sufficiently long time (i.e. after transients have disappeared)? (e) Which of the two systems ẏ(t) = y(t)+u(t) and ÿ(t) = y(t).ẏ(t)+u(t) will experience the largest overshoot in a step-response?

4 2. (a) The system Y(s) = is controlled using the PID controller (s+) 2U(s) U(s) = (K P +K I s +K Ds)(R(s) Y(s)) In the figure below, step-responses for four combinations of coefficients are given () K P = K I = K D = (2) K P = K I = 4 K D = (3) K P = K I = K D = (4) K P = K I = K D = 4 A B.5.5 Amplitude Amplitude Time (sec) 5 5 Time (sec) C D.5.5 Amplitude Amplitude Time (sec) 5 5 Time (sec) Combine the figure and coefficients. (4p) 2

5 (b) A system is controlled using a combination of feedback and so called feedforward according to the figure below. F (s) f r u y Σ F(s) Σ G(s) - Verify that the relation between reference signal and control error is given by E(s) = S(s)( F f (s)g(s))r(s) where S(s) is the sensitivity function. (4p) (c) According to the expression in b) it would be possible to achieve zero control error by using the feedforward filter F f (s) = G (s) Give two reasons why this would be impossible in practice. You may study the special case G(s) = b s+a 3

6 3. A simple model for micro-organism growth is given by ṁ = λ(n)m ṅ = µ(m)+q where m is the amount of micro-organisms, n is the amount of nutrition available and q is the supplied amount of nutrition. Let x, x 2 and u be the deviation from an equilibiria of m, n and q. If the model is approximated using a linear model, the following system is obtained for a particular case. ẋ = with transfer function ( ) ( ) x+ u, y = ( ) x s 2 + (a) Can the system be asymptotically stabilized using a P-controller? (b) Let the PD-controller K(s+) control the system. Derive an equation for the poles of the closed-loop system. Draw a root-loci with respect to K and discuss qualitatively, based on the root loci, how K should be picked. (4p) (c) The controller in (b) can not be implemented in practice. The following modified controller is therefor used instead K(s+) s+ What is the purpose of this change? Make a sketch of the root-loci with respect to K for this setup. Which differences are there compared to (b). Study, in particular, the case of small K and K. (4p) 4

7 4. A system is described by the following transfer function G(s) = Ae st (s+) 2 where the static gain A and time-delay T can vary. On the next page, you are given the the Bode plot for the case A = 2,T =. The system shall be controlled using a feedback of the following form U(s) = F(s)(R(s) Y(s)) (a) Derive a controller F(s) such that the closed-loop system satisfies the following requirements for the case T =,A = 2: The output shall be equal to a constant reference signal in stationarity, also in the case when constant disturbances act on the output. The crossover frequency of the open-loop system shall be the same as when the controller F(s) = is used. The phase-margin must be at least 5. (6p) (b) Let us now assume that the coefficient A deviates from our assumed value, but that it still holds that T =. Is there any upper limit to how large A can be before the closed-loop system in (a) goes unstable? The exact limit for A is not required. (c) Let us now assume that A deviates from the assumed value, but we have (the more realistic case) that T >. Is there any upper limit to how large A can be before the closed-loop system in (a) goes unstable? The exact limit for A is not required. 5

9 5. A bicycle is described by (after suitable scaling of time and states) the model [ ] [ ] V ẋ = x+ V 2 u, y = [ ] x /2 Here x is the lean angle, x 2 is the angular momentum, u is the angle of the steering bar and V the forward velocity (assumed constant V > ). (a) Compute the poles of the system. How do they depend on V? (p) (b) Compute a state-feedback for arbitrary V, placing the eigenvalues in 2, 2 (the reference signal is assumed to be ). (c) Show that there exist a positive value V of V such that some of the coefficients computed in b) grow indefinitely when V V. Give a numerical value of V. (d) Why is it impossible to place the poles when V = V? (p) (e) Suppose V = V. Show that although it is impossible to place the poles arbitrarily, it is still possible to stabilize the system. Derive such a stabilizing state-feedback. (4p) 7

10 Robustness criteria Given a feedback controller F(s) stabilizing a model G(s). Let the real system be given by G (s) = G(s)(+ (s)), Now assume that G (s) and G(s) have the same number of poles in the right half plane (including the origin), and F(s)G(s) and F(s)G (s) both tend to when s goes to infinity. Let T(s) denote the complimentary sensitivity function arising when G(s) is controlled using F(s). If (iω) T(iω) ω the closed loop system is stable when G (s) is controlled using F(s) Frequency compensation F lead (s) = K τ Ds+ βτ D s+, F lag = τ Is+ τ I s+γ τ D =, τ I = ω c,desired β ω c,desired β Figur : Maximal phase advancement (in degrees) as a function of β. 8

EXAM IN MODELING AND SIMULATION (TSRT62)

EXAM IN MODELING AND SIMULATION (TSRT62) SAL: ISY:s datorsalar TID: Wednesday 4th January 2017, kl. 8.00 12.00 KURS: TSRT62 Modeling and Simulation PROVKOD: DAT1 INSTITUTION: ISY ANTAL UPPGIFTER: 5 ANTAL