State-Space Representation and System Analysis Quiz Quiz

Challenge your understanding of state-space models and system analysis concepts, including system properties, canonical forms, controllability, observability, and response calculations. This quiz covers essential state-space methods used in control theory and engineering for modeling and analyzing dynamic systems.

1. Identifying Components in State-Space Equations

   In the continuous-time state-space model given by dx/dt = Ax + Bu and y = Cx + Du, what does the matrix 'A' specifically represent for a given system?

   1. System dynamics or state matrix
   2. Direct transmission matrix
   3. Output matrix
   4. Input matrix

   Explanation: The matrix 'A' in state-space representation is called the system dynamics or state matrix because it characterizes how the current state affects the rate of change of the state. The 'B' matrix is the input matrix that relates external inputs to the state derivatives. The 'C' matrix connects the system state to the output, and 'D' (the direct transmission matrix) connects the input directly to the output. Confusing these terms can lead to errors in modeling and analysis.

2. Controllability Interpretation

   Given a state-space model, which statement correctly defines controllability for a linear time-invariant (LTI) system?

   1. The rate of change of the input equals the rate of change of the output
   2. The output can be measured directly from the state variables
   3. All system states can be driven to any desired value within a finite time using suitable inputs
   4. The system's states cannot be reconstructed from the outputs

   Explanation: Controllability means that, using appropriate control inputs, the system can transition from any initial state to any final state in finite time. Output measurement is related to observability, not controllability. The inability to reconstruct state from output describes lack of observability, again, not controllability. The last option incorrectly mixes input-output rates, which is unrelated to the concept.

3. Canonical Forms Recognition

   Which of the following is a recognized state-space canonical form commonly used for single-input, single-output (SISO) systems?

   1. Laplace canonical form
   2. Controllable canonical form
   3. Stability canonical form
   4. Discrete-time canonical form

   Explanation: Controllable canonical form is a standard way to arrange state-space equations to highlight controllability, especially in SISO systems. The Laplace canonical form is not a recognized canonical structure; Laplace is a transform method. Discrete-time form refers to system type, not canonical structure. Stability canonical form is not a standard term in state-space analysis.

4. Observability Matrix Construction

   When analyzing a 3-state system with state-space matrices A (3×3) and C (1×3), what is the necessary number of rows in the observability matrix to test for full observability?

   1. 2
   2. 1
   3. 6
   4. 3

   Explanation: For an n-state system, the observability matrix is built with n stacked rows, each generated by C, CA, CA², and so on, up to CA^(n-1). Therefore, with 3 states, you need 3 rows (C, CA, CA²). Using 6 or 2 rows is incorrect and would not match the system's dimension. A single row only applies to a 1-state system.

5. Time Response Using State-Space Equations

   Given the state-space equations dx/dt = [-2]x + [3]u and y = [1]x with x(0) = 0, what is the output y(t) when a unit step input u(t) is applied?

   1. y(t) = (3/2)(1 - e^{-2t})
   2. y(t) = 2 - 3e^{-2t}
   3. y(t) = e^{-2t} + 3
   4. y(t) = 3e^{-2t}

   Explanation: Solving this first-order linear system with a unit step input gives y(t) = (3/2)(1 - e^{-2t}). The expression 3e^{-2t} does not match the step response's steady-state value. The form 2 - 3e^{-2t} and e^{-2t} + 3 are not consistent with the solution to the system nor the initial condition x(0) = 0. Only the first option represents the correct unit step response.