State Space Analysis Questions and Answers

Answer: B

Both statements are false. The state variable technique is a time domain approach: it writes the system as first order differential equations in the state vector, \(\dot{x} = Ax + Bu\), \(y = Cx + Du\), whereas the transfer function method works in the frequency (s) domain. It is not limited to linear time invariant systems, since the state variable method can describe time varying and nonlinear systems as well, and it handles multiple inputs and outputs.

So the pair false and false is correct. The answer is option B.

Answer: B

The minimum number of states is 2.

An nth-order differential equation needs n state variables, one for each independent initial condition. A second-order equation such as \(\ddot{y} + a\dot{y} + by = u\) needs two, usually \(x_{1} = y\) and \(x_{2} = \dot{y}\), giving

\[ \begin{aligned} \dot{x}_{1} &= x_{2} \\ \dot{x}_{2} &= -b x_{1} - a x_{2} + u \end{aligned} \]

One state cannot capture both the value and the rate of change. Three or four states would be more than the minimum, a non-minimal realisation.

Answer: B

State variable analysis applies to all three, so the answer is (1), (2) and (3).

The state-space method describes a system by first-order differential equations in its state variables, \(\dot{x} = f(x, u, t)\). Unlike the transfer function, which is defined only for linear time-invariant systems with zero initial conditions, this form can handle linear time-invariant systems, time-varying systems (where the matrices depend on t) and non-linear systems (where f is a non-linear function), as well as multiple inputs and outputs.

The options that limit it to LTI systems, or leave out one class, describe the transfer-function approach rather than state variable analysis.

Answer: B

Statements (i), (ii) and (iv) are the true ones. The state space method is a time-domain approach suited to systems with many inputs and many outputs, and it reduces the mathematical complexity because the system is written as first order matrix equations.

Statement (iii) is false: initial conditions are part of the state space description, since the state at the start fixes the response, so the method does not mean "with no initial conditions". Options with (iii) are therefore wrong.