STABILITY

STABILITY

1. Most important characteristic of control systems

2. A control system is stable if before any bounded entry, the system has a bounded output

   • To check, you have to analyze the transfer function
     • We will have two equations, one in the numerator and one in the denominator. The denominator equation will be called characteristic equation and to study the stability of the system we will have to find out its roots

3. It can be analyzed

   • Types in LTI systems (Linear Time-Invariant)
     • Stable
       • A system is stable if for each defined entry, there is a defined output within limits
         • It must not tend to infinity nor should it oscillate
         • A control system is in equilibrium if, in the absence of any disturbance or input, the output remains in the same state
     • Unstable
       • A system is unstable if at least there is a simple root of the characteristic equation of the system in the S.P.D., or a double root on the imaginary axis
         • The output varies and diverges without limit
         • If there is a disturbance in the system, it will hardly reach its point of equilibrium
     • Marginally stable
       • A system is marginally stable if there are no roots of the characteristic polynomial in the S.P.D. and at most there are simple roots on the imaginary axis
         • Its natural response does not decay or grow and tends to remain constant when time tends to infinity
   • Degrees
     • Absolute stability
       • It refers to the condition of whether the system is stable
     • Relative stability
       • Characterize the degree of stability, studying the relative situations of the poles

4. Tools for analysis

   • Criteria of Routh-Hurwitz
   • Nyquist criterion
   • Bode diagram
   • Diagram of Nichols