Podcast
Questions and Answers
For the system $\dot{x} = Ax + Bu$, what does the matrix $A$ represent?
For the system $\dot{x} = Ax + Bu$, what does the matrix $A$ represent?
- Input matrix
- State matrix (correct)
- Control matrix
- Output matrix
In the equation $\dot{x} = Ax + Bu$, what does $u$ represent?
In the equation $\dot{x} = Ax + Bu$, what does $u$ represent?
- Input or control vector (correct)
- Output vector
- State vector
- Disturbance vector
What is the purpose of state feedback control?
What is the purpose of state feedback control?
- To reduce noise in the system
- To eliminate disturbances
- To modify the system's dynamics (correct)
- To estimate the system states
If a system is described by $\dot{x} = Ax$, what is the system's input?
If a system is described by $\dot{x} = Ax$, what is the system's input?
What is the role of a state observer?
What is the role of a state observer?
In the context of control systems, what are poles?
In the context of control systems, what are poles?
What is the purpose of pole placement?
What is the purpose of pole placement?
Which of the following represents a state variable?
Which of the following represents a state variable?
In a state-space representation, what is the output equation typically?
In a state-space representation, what is the output equation typically?
What is the purpose of the direct transmission matrix $D$ in state-space representation?
What is the purpose of the direct transmission matrix $D$ in state-space representation?
What does the rank of the controllability matrix indicate?
What does the rank of the controllability matrix indicate?
What is the purpose of a similarity transformation?
What is the purpose of a similarity transformation?
If a system is uncontrollable, what does this indicate?
If a system is uncontrollable, what does this indicate?
What is the relationship between eigenvalues and system stability?
What is the relationship between eigenvalues and system stability?
What is the role of the matrix $C$ in the state-space representation?
What is the role of the matrix $C$ in the state-space representation?
Which of the following must be true for two matrices A and B to be similar?
Which of the following must be true for two matrices A and B to be similar?
What does it mean if a system is 'observable'?
What does it mean if a system is 'observable'?
Which of the following matrices is used to determine if a system is controllable?
Which of the following matrices is used to determine if a system is controllable?
Which of the following is a typical representation for a state-space system?
Which of the following is a typical representation for a state-space system?
What is the function of a controller in a control system?
What is the function of a controller in a control system?
Flashcards
Controllable and observable system
Controllable and observable system
A system is controllable and observable if its states can be controlled and observed from input-output measurements.
Rank of controllability matrix
Rank of controllability matrix
The rank of the controllability matrix C indicates the number of linearly independent states that can be controlled.
Similarity transformation matrix T
Similarity transformation matrix T
A similarity transformation matrix T transforms a system into a specific form while preserving its eigenvalues.
Similar Matrices
Similar Matrices
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Controllable Eigenvalues
Controllable Eigenvalues
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State Variables in Output Feedback
State Variables in Output Feedback
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State Feedback Gain Matrix (K)
State Feedback Gain Matrix (K)
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Observer Feedback Gain(L)
Observer Feedback Gain(L)
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Study Notes
Week 12: Design using State Space
Question 1
- A system is described by state-space equations with matrices A, B, and output equation.
- The system is controllable and observable.
Common Data for Q2-Q7
- Consider the state matrix A and input matrix B given by specific matrices.
2) Rank of Controllability Matrix
- The rank of the controllability matrix C is 2.
3) Similarity Transformation Matrix
- A similarity transformation matrix T is chosen such that T⁻¹ is formed by taking the first two linearly independent columns of C and the last vector [0 0 1]ᵀ.
- T is then given by a specific matrix.
4) Matrices A and B
- The matrices A = T A T⁻¹ and B = T B are given by specific matrices.
5) How Many Controllable Eigenvalues
- There are 1 controllable eigenvalue(s) of A.
6) Largest Controllable Eigenvalue
- The largest controllable eigenvalue is 0.
7) Largest Uncontrollable Eigenvalue
- The largest uncontrollable eigenvalue is 2.
8) Similar Matrices
- If A and B are similar matrices such that A = P⁻¹BP for any transformation matrix P, then A² = P⁻¹BP.
9) System Definition
- Consider the system defined by state-space equations with specific A and B matrices.
- By using the state feedback control u = -Kx, it is desired to have the closed-loop poles of A at -1 ± j2, and -10.
- The state feedback gain matrix K is [49 20 6].
Common Data for Q10-Q12
- System definition: xdot = Az + Bu and y = Cz with specified matrices A, B, and C.
10) Total State Variables
- Total number of state variables in the output feedback (comprising the plant and the observer) system defined by (I) is 6.
11) Controller Gain
- Let K = [k1 k2 k3] be the controller gain. k3 = 0.
12) Feedback Gain
- Let L = [l1 l2 l3]ᵀ be the feedback gain of the observer: l1 = 16.
13) Second-Order System
- The state space representation of a second-order system is given.
- The claim that the states of the system can reach z* = [0.8 -0.48]ᵀ in finite time is wrong since the state z1 is uncontrollable.
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