Final answer:
The number of unstable closed loops in the given system G(s) = K / (s(s+1)(s+2)) can be determined by the Nyquist plot generated using MATLAB's nyquist.m function and the Nyquist stability criterion. However, without running the analysis and knowing the value of K, the exact number of unstable loops cannot be accurately predicted.
Step-by-step explanation:
The Nyquist stability criterion is a graphical analysis tool used to determine the stability of a control system by examining the Nyquist plot of the open-loop transfer function. For the given open-loop transfer function G(s) = K / (s(s+1)(s+2)), to determine the number of unstable closed loops, one would employ the Nyquist stability criterion using MATLAB's nyquist.m function.
This function will generate the Nyquist plot, and by applying the criterion, which involves counting the number of encirclements of the critical point (-1,0) in the complex plane, we can assess the stability of the closed-loop system.
The function's structure is not inherently unstable since there are no right-half plane poles; hence, it primarily depends on the value of K. If K > 0, the system will be stable because the Nyquist path does not encircle the critical point.
However, the exact answer for the value of K and the number of unstable closed loops can only be determined by actually creating and analyzing the Nyquist plot for the specific system with MATLAB. In the absence of a specific K value and without performing the Nyquist plot analysis, a definitive answer cannot be given.
Thus, we can say that the system could be stable or unstable, depending on K, but without further computation, the number of unstable poles in the closed loop remains undetermined.