Electrical Engineering (EE) Exam > Electrical Engineering (EE) Questions > The following set of differential equations d...
Start Learning for Free
The following set of differential equations describes a linear second-order single input continuous-time system.
X1'(t) = -2X1(t) + 4X2(t)
X2'(t) = 2X1(t) - X2(t) + u(t)
X2'(t) = 2X1(t) - X2(t) + u(t)
Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:
- a)Uncontrollable and unstable
- b)Controllable but unstable
- c)Controllable and stable
- d)Uncontrollable and stable
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
The following set of differential equations describes a linear second-...
We know,
Here,
The product of matrix A and B is,
Thus, the rank of the system is 2.
The determinant of the system is non-zero. Hence, the system is controllable.
Now, we will check for stability.
SI - A = 0
The above matrix can be written in the form of an equation:
s2 + 3s -6 = 0
Now, we will find the roots using Routh's array table, as shown below:
There are two sign changes in the first column. Thus, the system is unstable.
Hence, the correct answer is an option (b).
Free Test
FREE
| Start Free Test |
Community Answer
The following set of differential equations describes a linear second-...
System Description:
The given set of differential equations represents a linear second-order single input continuous-time system with two state variables X1(t) and X2(t) and one control variable u(t). The system dynamics can be represented as:
X1(t) = -2X1(t) + 4X2(t)
X2(t) = 2X1(t) - X2(t) + u(t)
Controllability:
Controllability is a property that determines whether it is possible to find a control input u(t) that can drive the system from any initial state to any desired final state in a finite time. It is an important property for determining the ability to control a system.
Controllable System:
A system is said to be controllable if and only if the controllability matrix is full rank. The controllability matrix for a linear time-invariant system can be constructed by assembling the columns of the matrix [B, AB, A^2B, ..., A^(n-1)B], where A is the system matrix and B is the input matrix.
System Matrix:
The system matrix A can be obtained by rearranging the given differential equations as follows:
X1(t) = -2X1(t) + 4X2(t)
X2(t) = 2X1(t) - X2(t) + u(t)
This can be written in matrix form as:
| X1_dot | | -2 4 | | X1 | | 0 |
| | = | | * | | + | |
| X2_dot | | 2 -1 | | X2 | | u |
Therefore, the system matrix A is given by:
A = | -2 4 |
| 2 -1 |
Input Matrix:
The input matrix B can be obtained by rearranging the given differential equations as follows:
X1(t) = -2X1(t) + 4X2(t)
X2(t) = 2X1(t) - X2(t) + u(t)
This can be written in matrix form as:
| X1_dot | | -2 4 | | X1 | | 0 |
| | = | | * | | + | |
| X2_dot | | 2 -1 | | X2 | | u |
Therefore, the input matrix B is given by:
B = | 0 |
| 1 |
Controllability Matrix:
The controllability matrix can be constructed by assembling the columns of the matrix [B, AB], where A is the system matrix and B is the input matrix.
CB = | B AB |
| 0 B |
CB = | 0 1 -2 4 |
| 1 -2 2 -1 |
Rank of Controllability Matrix:
To check the controllability of the system, we need to calculate the rank of the controllability matrix CB. If the rank of CB is equal to the number of state variables, then the system is controllable.
Rank(CB) = 2
The rank of the controllability matrix
The given set of differential equations represents a linear second-order single input continuous-time system with two state variables X1(t) and X2(t) and one control variable u(t). The system dynamics can be represented as:
X1(t) = -2X1(t) + 4X2(t)
X2(t) = 2X1(t) - X2(t) + u(t)
Controllability:
Controllability is a property that determines whether it is possible to find a control input u(t) that can drive the system from any initial state to any desired final state in a finite time. It is an important property for determining the ability to control a system.
Controllable System:
A system is said to be controllable if and only if the controllability matrix is full rank. The controllability matrix for a linear time-invariant system can be constructed by assembling the columns of the matrix [B, AB, A^2B, ..., A^(n-1)B], where A is the system matrix and B is the input matrix.
System Matrix:
The system matrix A can be obtained by rearranging the given differential equations as follows:
X1(t) = -2X1(t) + 4X2(t)
X2(t) = 2X1(t) - X2(t) + u(t)
This can be written in matrix form as:
| X1_dot | | -2 4 | | X1 | | 0 |
| | = | | * | | + | |
| X2_dot | | 2 -1 | | X2 | | u |
Therefore, the system matrix A is given by:
A = | -2 4 |
| 2 -1 |
Input Matrix:
The input matrix B can be obtained by rearranging the given differential equations as follows:
X1(t) = -2X1(t) + 4X2(t)
X2(t) = 2X1(t) - X2(t) + u(t)
This can be written in matrix form as:
| X1_dot | | -2 4 | | X1 | | 0 |
| | = | | * | | + | |
| X2_dot | | 2 -1 | | X2 | | u |
Therefore, the input matrix B is given by:
B = | 0 |
| 1 |
Controllability Matrix:
The controllability matrix can be constructed by assembling the columns of the matrix [B, AB], where A is the system matrix and B is the input matrix.
CB = | B AB |
| 0 B |
CB = | 0 1 -2 4 |
| 1 -2 2 -1 |
Rank of Controllability Matrix:
To check the controllability of the system, we need to calculate the rank of the controllability matrix CB. If the rank of CB is equal to the number of state variables, then the system is controllable.
Rank(CB) = 2
The rank of the controllability matrix
Attention Electrical Engineering (EE) Students!
To make sure you are not studying endlessly, EduRev has designed Electrical Engineering (EE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Electrical Engineering (EE).
|
Explore Courses for Electrical Engineering (EE) exam
|
|
Similar Electrical Engineering (EE) Doubts
Top Courses for Electrical Engineering (EE)View all
The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer?
Question Description
The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer?.
The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer?.
Solutions for The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Electrical Engineering (EE).
Download more important topics, notes, lectures and mock test series for Electrical Engineering (EE) Exam by signing up for free.
Here you can find the meaning of The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The following set of differential equations describes a linear second-order single input continuous-time system.X1(t) = -2X1(t) + 4X2(t)X2(t) = 2X1(t) - X2(t) + u(t)Here, X1(t) and X2(t) are the state variables, and u(t) is the control variable. Check for the system, if it is:a)Uncontrollable and unstableb)Controllable but unstablec)Controllable and stabled)Uncontrollable and stableCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Electrical Engineering (EE) tests.
|
|
Explore Courses for Electrical Engineering (EE) exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup