State-Space Representation u0026 Analysis Quiz Quiz

This quiz assesses your understanding of state-space representation, system properties, and fundamental analysis methods in control theory. Improve your knowledge on modeling, controllability, observability, and solution techniques commonly used in dynamic system analysis.

1. Understanding State Variables

   In the state-space model of a dynamic system, what do the 'state variables' primarily represent?

   1. The smallest set of variables that captures the complete system status at any given time
   2. Arbitrary parameters chosen for mathematical convenience
   3. All input and output variables present in the system
   4. Only the measurable quantities of the system

   Explanation: State variables are defined as the minimal set required to represent the status of the system so its future behavior can be determined, given the current state and inputs. Not all input and output variables serve as state variables; only those strictly necessary for describing internal dynamics are included, which makes the second option incorrect. Arbitrary parameters are not equivalent to state variables, ruling out option three. Measurable quantities might not include hidden or unmeasurable states, so the last option is incorrect.

2. Identifying State-Space Matrices

   Given the standard state-space equations dx/dt = Ax + Bu and y = Cx + Du, which matrix directly relates the input vector u to the system output y without involving the state x?

   1. C
   2. A
   3. D
   4. B

   Explanation: The D matrix establishes the direct relationship between the input and the output in the output equation, y = Cx + Du, bypassing the state vector x. The A matrix relates states to their derivatives, B connects inputs to state changes, and C links states to outputs. Therefore, A, B, and C do not directly tie the input to the output in the same direct manner as D does.

3. Controllability Assessment

   Consider a system described by dx/dt = Ax + Bu. When is the system said to be controllable?

   1. When it is possible, via input u, to transfer the state vector from any initial state to any desired final state in finite time
   2. When the system has no zero eigenvalues
   3. When the matrix A is invertible
   4. When the output y can be measured directly at all times

   Explanation: Controllability means being able to direct the system from one state to another in finite time with a suitable control input. Measuring outputs relates to observability, not controllability. The invertibility of A is not required for controllability, as some controllable systems have non-invertible A matrices. Having no zero eigenvalues is not a sufficient or necessary condition for controllability.

4. State-Space and Transfer Function

   For a single-input, single-output linear time-invariant system, what is the primary method to obtain the transfer function from its state-space representation?

   1. Take only the Laplace transform of y
   2. Multiply A and B directly
   3. Compute C(sI - A)⁻¹B + D
   4. Calculate the determinant of C

   Explanation: To derive the transfer function from the state-space model, the formula C(sI - A)⁻¹B + D is used, which incorporates system dynamics through matrices A, B, C, and D. Multiplying A and B alone does not yield the transfer function and ignores the effects of the output and state matrices. Taking the Laplace transform of only y will not provide information about input-output dynamics. The determinant of C is unrelated to transfer function calculation.

5. Observability Criteria

   Which of the following statements best describes the condition for observability in a state-space system?

   1. Every state can be inferred from the system outputs over a finite interval of time
   2. The system has more outputs than states
   3. It is impossible to reconstruct any state from the outputs
   4. The matrix B is full rank

   Explanation: Observability refers to the ability to deduce the entire set of state variables from knowledge of the outputs and inputs over a finite period. Having more outputs than states does not guarantee observability, as it depends on the structure of the output matrix. The third option is the opposite of observability. The rank of matrix B is relevant for controllability, not observability.