Klausur Summer 2019 PDF

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Technische Universität München Lehrstuhl für Regelungstechnik Prof. Dr.-Ing. habil. B. Lohmann

Advanced Control Exam, 19 July 2019 Dr.-Ing. P. Kotyczka

Surname, first name: Student ID: Important: • This exam booklet contains 18 pages with questions and space for answers plus 4 extra pages. Check immediately for completeness and readability. • The solutions of the preceding questions are required (or at least helpful) for the solution of questions marked with a *. • The solutions must be written in the fields below the problems. If the space is not sufficient, please use the extra pages at the end of the booklet. Make a clear reference to the extra pages in this case. Use a permanent pen. No pencil, no red or green color. • Briefly justify all your solutions. Unjustified answers will not be counted. (Exception: single choice questions.) • Each single choice question has only one correct answer. • Do not remove pages from the booklet. • Write your full name and your student ID on top of this title page.

Allowed material: • 2 handwritten (DIN A4, double-sided) cheat sheets • No calculators, computers or other electronic devices

Evaluation: Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

/11

/13

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/9

/17

Problem 6 /18

Total

Grade /90

Advanced Control (19 July 2019)

Problem 1 – Linearization and State Representations (11 points) Given the following set of nonlinear differential equations for the states x1 , x2 and x3 , with two control inputs u1 and u2 and one disturbance input z : x˙ 1 = 3x2 + x23 − 4z x˙ 2 = sin x2 + 4x3 − u2 x˙ 3 = x1 x3 + x2 + u1 . Consider an operating point with x∗1 = 3,

x∗2 = 0,

z∗ = 1 and x3∗ ≤ 0.

a) Determine the values of the remaining state x∗3 and the control inputs u∗1 and u∗2 at this equilibrium. (3 points)

1/22

Advanced Control (19 July 2019) b) Linearize the state differential equations at the equilibrium and determine the matrices A, B and e of the resulting linear state differential equation. (4 points)

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Advanced Control (19 July 2019) c) Consider the transfer function description Y (s) = G(s)U (s) of a linear time invariant system with 2s + 3 G1 (s) = 2 . s − 5s + 4 i) Give the state representation in observer canonical form. (3 points) ii) What changes, if you consider instead the transfer function (1 point) G2 (s) =

3/22

s2

2s + 3 + 2? − 5s + 4

Advanced Control (19 July 2019)

Problem 2 – System Properties (13 points) Consider the LTI system S

−3 0 0 0 1 W X W X x˙ = U 4 −2 1V x + U1 0V u 4 −4 3 1 0 C

T

S

T

D

0 1 0 x. y= −1 0 1

The eigenvalues of the matrix A are λ1 = −3, λ2 = −1 and λ3 = 2. a) Determine the invariant zero of the system. (3 points)

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Advanced Control (19 July 2019) b) Examine the system properties controllability and observability. (7 points)

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Advanced Control (19 July 2019)

*c) Based on the answers of a) and b): Is the system stabilizable? Is it detectable? (3 points)

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Advanced Control (19 July 2019)

Problem 3 – Controller and Observer Design (22 points) For parts a) to c) consider the controllable third order SISO system S

−1 1 0 1 W X W X x˙ = U 1 0 1 V x + U0V u 0 1 −2 0 Ë

T

S T

È

y = 1 0 0 x.

a) Using coefficient matching, design a state feedback u = −r T x such that the closed-loop eigenvalues are −3, −4 and −5. (7 points)

7/22

Advanced Control (19 July 2019)

b) What is the correction vector l for a Luenberger observer, whose eigenvalues shall coincide with the closed-loop eigenvalues given in part a)? Give a short explanation. No calculations! (2 points)

8/22

Advanced Control (19 July 2019) c) Compute the feedforward gains mx and mu to achieve steady state gain one. (4 points)

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Advanced Control (19 July 2019) d) Consider the state differential equation in 3a) with the flat output Ë

È

yf = 0 0 1 x. Express the states x1 , x2 , x3 and the input u in terms of this flat output and its time derivatives, i. e. determine their differential parametrizations. (4 points) Hint: – Express the state in the flat output equation in terms of yf . – Then write the differential equation for this state in terms of the flat output and solve for the remaining state. – Repeat this accordingly with the two remaining differential equations.

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Advanced Control (19 July 2019) e) Now consider a general LTI system x˙ = Ax + Bu y = Cx. Write down the state representation of the corresponding Luenberger observer with ˆ the observer state (1 point). Show that the set of resulting eigenvalues under the x observer-based state feedback u = −Rˆx is composed of the closed-loop (“controller”) eigenvalues and the observer eigenvalues (4 points).

11/22

Advanced Control (19 July 2019)

Problem 4 – Disturbance Rejection (9 points) a) Consider the disturbance signal z(t) = e−2t sin(3t). Determine a disturbance model in the form w(t) ˙ = W w(t), z(t) = f (w (t)), with state vector w ∈ Rs of appropriate dimension and a nonlinear output function f : Rs → R. (4 points)

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Advanced Control (19 July 2019) b) What is the condition for disturbance decoupling in a system of the form (2 points) x˙ = Ax + Bu + Ez y = Cx?

c) Given an LTI system with the matrices C

D

1 −2 A= , 1 −4

C D

1 b= , 0

C

D

1 , e= −2

Ë

È

cT = 0 1 .

Compute the constant disturbance feedback nx , nu to achieve steady state rejection of piecewise constant disturbances. (3 points)

13/22

Advanced Control (19 July 2019)

Problem 5 – General Questions (17 points) a) The locations of the closed-loop eigenvalues are degrees of freedom, which have to be chosen by the designer. Draw the sketch of a reasonable target region for the closed-loop eigenvalues in the complex plane, as presented in the lecture. (3 points) Give a short explanation for each part of the boundary of this target region. (3 points)

14/22

Advanced Control (19 July 2019) b) Mention three typical control goals. (3 points)

c) In which cases can stable decoupling control by state feedback not be realized? Why? (2 points)

d) Sketch the block diagram of an LTI system in a two-degrees-of-freedom state feedback controller structure. Consider also a disturbance model, whose state (assumed measurable) is used for constant disturbance feedback. (6 points) Draw the block diagram on the next page, turned by 90 degrees (landscape orientation). • The blocks of system and disturbance model shall contain the corresponding symbolic state representations. Don’t forget initial values. • Mark every signal with the corresponding symbol. • Don’t forget the arrows.

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Advanced Control (19 July 2019)

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Advanced Control (19 July 2019)

Problem 6 – Single Choice Questions (18 points) Tick the correct answers in the following single choice questions. (Only one correct answer per question. No negative points for wrong answers.) Q1 The matrices A and AT have different . . . A

determinant.

B

characteristic polynomial.

C

eigenvalues.

D

eigenvectors.

Q2 The inverse Laplace transform of the transfer function gives . . . A

the impulse response.

B

the step response.

C

the steady state gain.

D

the final value of the output.

Q3 Consider a real-valued matrix A ∈ Rn×n with odd dimension n. A can not have . . . A

an even number of complex eigenvalues.

B

an even number of purely imaginary eigenvalues.

C

an even number of real eigenvalues.

D

an even number of eigenvalues in the origin.

Q4 Which of the following typical components of a block diagram can not be realized as a state space model of the following form? x˙ = ax + bu y = cx + du A

P (proportional controller)

B

PD (proportional-derivative controller)

C

PI (proportional-integral controller)

D

PT1 (first order lag element)

Q5 M x and M u are the matrices in the feedforward part of the 2 DOF controller structure. In which of the following cases can we be sure that these matrices can be computed? A

The state matrix A has only real eigenvalues.

B

The state matrix A has only eigenvalues in the open left half plane.

C

All invariant zeros of the system (A, B, C) lie in the open left half plane.

D

All invariant zeros of the system (A, B, C) have positive real part.

17/22

Advanced Control (19 July 2019) Q6 We consider two different LQR controllers for a single state SISO system x˙ (t) = −x(t) + u(t). The first LQR controller minimizes the cost functional J1 =

1⁄ ∞ q1 x2 (t) + q˜1 u2 (t) dt 2 0

and produces the closed-loop dynamics x˙ (t) = −2x(t). The second LQR controller minimizes J2 =

1⁄ ∞ q2 x2 (t) + q˜2 u2 (t) dt 2 0

and we obtain x˙ (t) = −4x(t). Which conclusion can be drawn about the relations of the weighting factors in the two cases? A

q1 < q 2

B

q˜1 < q˜2

C

q2 q1 /˜ q1 < q2 /˜

D

q2 q1 /˜ q1 > q2 /˜

Q7 An invariant zero coincides with an eigenvalue of a MIMO system. This zero . . . A

is always a transmission zero.

B

can be a transmission zero.

C

is never a transmission zero.

D

is only a transmission zero in the SISO case.

Q8 By constant state feedback u = −Rx, it is not possible . . . A

to make a controllable system uncontrollable.

B

to make an observable system unobservable.

C

to make a stable system unstable.

D

to make a stabilizable system unstable.

Q9 The flat output of a flat linear SISO system of order n . . . A

must be a function of all states.

B

is a function of a single state.

C

is unique.

D

has relative degree n.

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Advanced Control (19 July 2019)

Additional Paper

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