Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
68 views
in Electronics by (85.3k points)
closed by

The second order dynamic system

\(\frac{{dx}}{{dt}} = Px + Qu\)

y = Rx

has the matrices, P, Q and R as follows:

\(P = \left[ {\begin{array}{*{20}{c}} { - 1}&1\\ 0&{ - 3} \end{array}} \right]\;Q = \left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right]\;R = \left[ {0\;1} \right]\)

The system has the following controllability and observability properties:
1. Controllable and observable
2. Not controllable but observable
3. Controllable but not observable
4. Not controllable and not observable

1 Answer

0 votes
by (88.5k points)
selected by
 
Best answer
Correct Answer - Option 3 : Controllable but not observable

Concept:

Controllability:

It is the internal states of the system are changed from one value to another value in a finite time by a finite input, then we can say the system is controllable otherwise it is not controllable.

To check Controllability, consider the controllability matrix (ϕc)

\(\left[ {{\phi _c}} \right] = {\left[ {{A^0}B\;\;\;{A^1}B} \right]_{2 \times 2}}\;\;\;;\left[ {{\phi _c}} \right] = {\left[ {{A^0}B\;\;\;\;\;{A^1}B\;\;\;\;{A^2}B} \right]_{3 \times 3}}\)

If |ϕc| = 0; then the system is uncontrollable

If |ϕc| ≠ 0, then the system is controllable.

Observability:

If the internal states of the system can be evaluated from the output of the system of any time, then we can say, the system is observable otherwise it is not observable.

To check observability, consider the observability matrix

\(\left[ {{\phi _0}} \right] = {\left[ {{C^T}\;\;\;\;A{C^T}} \right]_{2 \times 2}}\; \to {2^{nd}}\;order\:\:\:;\:\:\:\left[ {{\phi _0}} \right] = {\left[ {{C^T}\;\;\;\;\;A{C^T}\;\;\;\;\;{A^2}{C^T}} \right]_{3 \times 3}}\)

If |ϕo| = 0; Then system is not-observable

If |ϕo| ≠ 0; then the system is observable.

Calculation:

Controllability matrix, C = [Q PQ]

\(PQ = \left[ {\begin{array}{*{20}{c}} { - 1}&1\\ 0&{ - 3} \end{array}} \right]\;\left[ {\begin{array}{*{20}{c}} 0\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ { - 3} \end{array}} \right]\)

\(C = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&{ - 3} \end{array}} \right]\)

|C| = -1 ≠ 0

System is controllable.

Observability matrix, \(O = \left[ {\begin{array}{*{20}{c}} R\\ {RP} \end{array}} \right]\)

\(RP = \left[ {0\;1} \right]\left[ {\begin{array}{*{20}{c}} { - 1}&1\\ 0&{3 - } \end{array}} \right] = \left[ {0\; - 3} \right]\)

\(O = \left[ {\begin{array}{*{20}{c}} 0&1\\ 0&{ - 3} \end{array}} \right]\)

|O| = 0

System is not observable.

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...