Govt. Exams
Entrance Exams
The state vector x(t) contains all state variables. With 4 state variables, it's a 4×1 column vector, so the dimension is 4.
System order is determined by the highest power of 's' in the characteristic equation denominator. Here, it's (s+2)(s+3)(s+4) = s³+..., so order is 3.
The root locus starts at open-loop poles (K=0) and ends at open-loop zeros (K=∞). This is a fundamental property of root locus construction.
For ζ < 1, the system is underdamped and exhibits oscillatory response. At ζ = 0.5, there are definitely oscillations with exponential decay.
The system type equals the number of poles at origin. Here, there is one pole at origin (s in denominator), making it Type 1.
A proportional controller cannot eliminate steady-state error completely for step inputs in type-0 systems. An integral term is needed for zero steady-state error.
Settling time is the time required for the transient to decay and response to remain within 2% (or 5%) of the steady-state value.
The system has one pole at s=0, making it Type 1. The denominator has s¹ factor representing one integration.
Bode plots use dB (20log₁₀|G(jω)|) for magnitude on a logarithmic scale and phase in degrees on a semi-log plot.
Roots are s = -1 and s = -3 (both negative). All poles are in the left half-plane, making the system stable.
