1. What does the state space representation of a system describe?
a) Only the input-output relationship of the system
b) The internal dynamics and state variables of the system
c) The steady-state behavior of the system
d) The transfer function of the system
Answer: b) The internal dynamics and state variables of the system
Explanation: The state space representation describes the internal dynamics of a system using state variables, which represent its internal state at any given time.
2. In a block diagram for a state equation, what do the blocks represent?
a) State variables
b) Input and output variables
c) Transfer functions
d) Differential equations
Answer: c) Transfer functions
Explanation: In a block diagram for a state equation, the blocks typically represent transfer functions, each describing the relationship between state variables.
3. How is the transfer function decomposed from a state equation?
a) By directly solving the state equation
b) By using Laplace transforms
c) By matrix inversion
d) By differentiating the state equation
Answer: b) By using Laplace transforms
Explanation: Transfer function decomposition involves transforming the state equation into the Laplace domain, where the transfer function can be extracted.
4. How is the solution of a state equation typically obtained?
a) By using Laplace transforms
b) By matrix inversion
c) By differentiating the state equation
d) By numerical integration
Answer: d) By numerical integration
Explanation: The solution of a state equation is commonly obtained through numerical integration methods such as Euler’s method or Runge-Kutta methods.
5. What does the transfer matrix represent in a state space system?
a) The Laplace transform of the state equation
b) The matrix of state variables
c) The transfer function matrix
d) The controllability matrix
Answer: c) The transfer function matrix
Explanation: The transfer matrix in a state space system represents the relationship between the input and output variables in matrix form.
6. What is the relationship between a state equation and a transfer function?
a) They are identical representations of a system
b) The state equation describes the dynamics, while the transfer function describes the steady-state behavior
c) The transfer function is derived from the state equation
d) The state equation is derived from the transfer function
Answer: c) The transfer function is derived from the state equation
Explanation: The transfer function is derived from the state equation by applying Laplace transforms to describe the input-output relationship.
7. What does controllability refer to in the context of state space systems?
a) The ability to manipulate state variables to achieve a desired output
b) The stability of the system
c) The ability to observe state variables
d) The uniqueness of the solution to the state equation
Answer: a) The ability to manipulate state variables to achieve a desired output
Explanation: Controllability refers to the capability of influencing the system’s behavior by applying appropriate inputs to control the state variables.
8. What is observability in the context of state space systems?
a) The ability to predict future states of the system
b) The ability to measure state variables from output measurements
c) The degree of system stability
d) The uniqueness of the solution to the state equation
Answer: b) The ability to measure state variables from output measurements
Explanation: Observability refers to the ability to determine the internal state of a system based solely on its outputs over a finite time interval.
9. Which method is commonly used to assess the controllability of a system?
a) Eigenvalue analysis
b) State transition matrix
c) Observability matrix
d) Kalman filter
Answer: a) Eigenvalue analysis
Explanation: Eigenvalue analysis is commonly used to determine the controllability of a system by examining the eigenvalues of the system’s controllability matrix.
10. How does observability differ from controllability?
a) Controllability deals with the system’s internal dynamics, while observability deals with its external behavior.
b) Controllability focuses on the ability to influence the system, while observability focuses on the ability to infer its internal state.
c) Observability is related to stability, while controllability is related to uniqueness of solutions.
d) Controllability refers to input-output relationships, while observability refers to state variables.
Answer: b) Controllability focuses on the ability to influence the system, while observability focuses on the ability to infer its internal state.
Explanation: Controllability is about the ability to manipulate the system’s behavior, while observability is about the ability to determine the system’s internal state from its outputs.