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    MID EXAM - ELECTRICAL - Linear System and Control - Muhammad Zubair - SU-20-02-048-004

    Uploaded by

    M Xubair Yousaf Xai
    1. The document provides instructions for a midterm exam in linear systems and control for an electrical engineering student. It includes questions to answer and spaces for the instructor to provide marks. 2. The first question asks the student to obtain the state-space representation of a system using canonical form. The second question asks the student to compute the controllability matrix and check if a system is controllable. 3. For the third question, the student is asked to use block diagram algebra to derive a transfer function relating the input R(s) to the output C(s) of a system with multiple paths. Steps are shown to perform the derivation.

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    MID EXAM - ELECTRICAL - Linear System and Control - Muhammad Zubair - SU-20-02-048-004

    Uploaded by

    M Xubair Yousaf Xai
    43 views13 pages
    1. The document provides instructions for a midterm exam in linear systems and control for an electrical engineering student. It includes questions to answer and spaces for the instructor to provide marks. 2. The first question asks the student to obtain the state-space representation of a system using canonical form. The second question asks the student to compute the controllability matrix and check if a system is controllable. 3. For the third question, the student is asked to use block diagram algebra to derive a transfer function relating the input R(s) to the output C(s) of a system with multiple paths. Steps are shown to perform the derivation.

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    MID EXAM_ELECTRICAL_Linear System and Control_Muhammad Zubair_SU-20-02-048-004

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    1. The document provides instructions for a midterm exam in linear systems and control for an electrical engineering student. It includes questions to answer and spaces for the instructor to provide marks. 2. The first question asks the student to obtain the state-space representation of a system using canonical form. The second question asks the student to compute the controllability matrix and check if a system is controllable. 3. For the third question, the student is asked to use block diagram algebra to derive a transfer function relating the input R(s) to the output C(s) of a system with multiple paths. Steps are shown to perform the derivation.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    43 views13 pages

    MID EXAM - ELECTRICAL - Linear System and Control - Muhammad Zubair - SU-20-02-048-004

    Uploaded by

    M Xubair Yousaf Xai
    1. The document provides instructions for a midterm exam in linear systems and control for an electrical engineering student. It includes questions to answer and spaces for the instructor to provide marks. 2. The first question asks the student to obtain the state-space representation of a system using canonical form. The second question asks the student to compute the controllability matrix and check if a system is controllable. 3. For the third question, the student is asked to use block diagram algebra to derive a transfer function relating the input R(s) to the output C(s) of a system with multiple paths. Steps are shown to perform the derivation.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    of 13

    Page |1

    Sarhad University, Peshawar


    Department: Electrical Engineering Program: M.S Electrical Engineering
    Course & Course Code : Linear Systems and Control Examination : Mid Term Spring 2020
    *Student Name: Muhammad Zubair
    Total Marks: 30 (30% Weightage)
    *Registration # SU-20-02-048-004
    Instructor: Engr. Dr. Salman Ahmed
    *Student Email:zubairksk@gmail.com

    IMPORTANT INSTRUCTIONS FOR THE STUDENTS:


    Before uploading, save the same file in this format MID EXAM_DEPARTMENT_COURSE_Student Name_Reg#.doc

    Upload it at your SUIT PORTAL before the deadline. Answer Sheet sent through EMAIL will not be accepted.

    The total size of this file should not exceed 1MB at the time of uploading it at portal

    Just type the answers under each question and draw the figures yourself in MS-word, Try to avoid inserting
    images. Follow the instructions at SUIT Portal.

    Table: To be filled by the Instructor

    Question Number 1 2 3 Grand Total


    Obtained Marks 0 0 0 0
    Total Marks 10 10 10 30
    Page |2

    Sarhad University, Peshawar

    Department: Electrical Engineering Program: M.S Electrical Engineering

    Student Name: Muhammad Zubair

    Registration No SU-20-02-048-004

    Student Email: zubairksk@gmail.com

    Table: To be filled by the Instructor

    Question Number 1 2 3 Grand Total


    Obtained Marks 0 0 0 0
    Total Marks 10 10 10 30
    Page |3

    Contents
    No. Title Page

    1 Answer no 1 4

    2 Answer no 2 7

    3 Answer no 3 part a 10

    4 Answer no 3 part b 11
    Page |4

    Question #1:

    Using the canonical form to obtain state-space representation of the


    system shown in the given Figure:

    Answer1:

    Step 1

    U(t) Σ 16 S4 −32 S 2+16


    y(t)

    Step 2
    Page |5

    16(s4 −2 s 2+1)
    1+(16 s4 −32 s 2+16)(1)
    u(t) y(t)

    16(s4 −2 s 2+1)
    1+(16 s4 −32 s 2+16)(1)

    16 (s 4 −2 s2 +1)
    1+(16 s4 −32 s 2+16)

    16(s 4−2 s 2 +1)


    (16 s4 −32 s 2+17)

    16(s4 −2 s 2+ 1)
    16 ¿ ¿
    y ( t ) s 4−2 s 2+ 1
    =
    u (t ) 17
    s 4−2 s2 +
    16
    b 0=1 ,b 1=0 , b2=−2 ,b 3=0 , b 4=1

    a 0=1.06 , a1=0 ,a 2=−2 , a3=0 , a 4=1

    Change into S.S


    Page |6

    b4s 4 +¿
    b3 s ¿
    ¿
    G(s)= 3+¿
    b
    2s 2+¿
    b
    1s
    1+¿
    b0 ¿
    ¿
    0s5 + a s4 +a s 3 +a s2 +a s 1+ a
    4 3 2 1 0

    0 1 0 0

    0 0 1 0

    A= 0 0 0 1

    -a0 -a1 -a2 -a3

    0 1 0 0
    Or A= 0 0 1 0

    0 0 0 1

    -1.06 0 2 0

    1
    0
    B= −2 ⌉
    ⌈ C= ⌈ 10 0 0 0 ⌉
    0
    1
    Page |7

    Question #2

    A capacitive displacement sensor is used to monitor small changes in


    work-piece position. The two metal cylinders are separated by a distance
    of 1 mm and dielectric constant at 1KHz of 2.5. This sensor is installed
    in a system where the sensor outputs the position of the system. The
    system has the following state-space equations:

    A newly inducted engineer wants to design a controller for this system.


    Compute the controllability matrix and check if this system is
    controllable.

    Answer 2:
    First we check Eigen value of Matrix A so we check that the
    system is stable or unstable.
    det ( λI −A )=0

    [ 10 ]
    I= 0 1

    λ I= [ 0λ 0λ]
    [ λ 0] [ 0.5 1]
    ( λI − A ¿= 0 λ ‒ 0.8 2

    Or [ λ−0.5−1 ]
    ( λI − A ¿= −0.8 λ−2
    Page |8

    det ( λI −A )= λ−0.5−1
    [ −0.8 λ−2 ]
    det ( λI −A )=¿

    det ( λI −A )=(λ2 −2.5 λ+1)−0.8

    det ( λI −A )=(λ2 −2.5 λ+1−0.8)

    det ( λI −A )= λ2−2.5 λ+ 0.2

    λ=−b ± √b 2−4 ac
    2a

    λ=−(−2.5)± √(−2.5)2 −4 (1)( 0.2)


    2(1)

    λ=2.5± √ 6.25−0.8
    2

    λ 1=2.5 ± √ 5.45
    2

    λ 1=2.5 ± 2.33
    2

    λ 1=2.5+2.33
    2

    λ 1=2.415

    2.5−2.33
    λ 2=
    2
    Page |9

    λ 2=0.085

    Here it is clear that Eigen Values are positive so the system is unstable

    And we need to stabilize it, following the pre-requisite steps for Matrix
    C, and the matrix C must be Identity so

    Matrix C= ⌊ 0.5 0.1 ⌋

    Matrix C is not identity so we cannot make controller for it.


    P a g e | 10

    STEP 1:
    P a g e | 11

    C( s) G 1+G 2
    =
    R( s) 1+G 1+G 2(G 4−G3)

    Q3(B)
    G1

    R(s) G2
    C(s)

    H1

    H2

    STEP 1:

    G1
    P a g e | 12

    R(s) G2
    C(s)

    H1

    H2

    STEP 2:
    G1

    R(s)
    G2
    C(s)

    H2-H1

    STEP 3:

    R(s) G2 C(s)
    1+ G2(H 2−H 1)

    STEP 4:

    G2
    R(s) 1+G1 1+ G2(H 2−H 1)
    P a g e | 13

    C(S)
    STEP 5:

    (1+ G1)G 2
    R(s) 1+ G2(H 2−H 1)
    C(S)

    STEP 6:
    ¿¿
    R(s)
    C(S)

    C( s)
    =¿ ¿
    R( s)

    End the paper

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