Final answer:
To classify the damping condition of the given second-order LTI system, examination of the characteristic equation derived from the state equations is necessary. This will determine if the system is overdamped, underdamped, or critically damped based on the damping ratio indicated by the coefficients.
Step-by-step explanation:
The question asks to determine the damping condition of a second-order Linear Time-Invariant (LTI) system described by a set of state equations. The given system has an output c(t) which equals one of the state variables, x1(t), and an input, r(t). We consider the characteristic equation of the system to determine the type of damping.
The characteristic equation in this case, obtained from the state equations, would give us a quadratic equation that resembles that of a second-order linear differential equation for a damped harmonic oscillator. This type of differential equation is widely studied in the context of mechanical vibrations and electric circuits (e.g., RLC circuits).
An overdamped system has a damping ratio greater than one, causing it to return to equilibrium slowly without oscillating. An underdamped system has a damping ratio less than one, making it oscillate as it returns to equilibrium. A critically damped system has a damping ratio exactly equal to one, enabling it to return to equilibrium as fast as possible without oscillating.