Control System
INTRODUCTION
Open- and closed-loop transfer functions have certain basic characteristics that permit transient and steady-state analyses of the feedback-controlled system. Five factors of prime importance in feedback-control systems are stability, the existence and magnitude of the steady-state error, controllability, observability, and parameter sensitivity. The stability characteristic of a linear time-invariant system is determined from the system’s characteristic equation. Routh’s stability criterion provides a means for determining stability without evaluating the roots of this equation. The steady-state characteristics are obtainable from the open-loop transfer function for unity feedback systems (or equivalent unity-feedback systems), yielding figures of merit and a ready means for classifying systems.
ROUTH HURWITZ STABILITY CRITERION
Stability
The response of control systems consists of (1) natural response and forced response, or (2) zero input response and zero state response. For natural (zero input) response, a system is,
1. stable if the natural response approaches zero as time approaches infinity.
2. unstable if the natural response approaches infinity as time approaches infinity.
3. marginally stable if the natural response neither decays nor grows but remians constant or oscillates.
For the total response, a system is
1. stable if every bounded input yields a bounded output.
2. unstable if any bounded input yields an unbounded output.
The Routh Hurwitz Stability Criterion
The Routh Hurwitz stability criterion is a tool to judge the stability of a closed loop system without solving for the poles of the closed loop system. Generating a Routh-Hurwitz table
SOLVED PROBLEMS
1. Given the characteristic equation, is the system described by this characteristic equation stable?
Answer: One coefficient (-2) is negative. Therefore, the system does not satisfy the
Open- and closed-loop transfer functions have certain basic characteristics that permit transient and steady-state analyses of the feedback-controlled system. Five factors of prime importance in feedback-control systems are stability, the existence and magnitude of the steady-state error, controllability, observability, and parameter sensitivity. The stability characteristic of a linear time-invariant system is determined from the system’s characteristic equation. Routh’s stability criterion provides a means for determining stability without evaluating the roots of this equation. The steady-state characteristics are obtainable from the open-loop transfer function for unity feedback systems (or equivalent unity-feedback systems), yielding figures of merit and a ready means for classifying systems.
ROUTH HURWITZ STABILITY CRITERION
Stability
The response of control systems consists of (1) natural response and forced response, or (2) zero input response and zero state response. For natural (zero input) response, a system is,
1. stable if the natural response approaches zero as time approaches infinity.
2. unstable if the natural response approaches infinity as time approaches infinity.
3. marginally stable if the natural response neither decays nor grows but remians constant or oscillates.
For the total response, a system is
1. stable if every bounded input yields a bounded output.
2. unstable if any bounded input yields an unbounded output.
The Routh Hurwitz Stability Criterion
The Routh Hurwitz stability criterion is a tool to judge the stability of a closed loop system without solving for the poles of the closed loop system. Generating a Routh-Hurwitz table
SOLVED PROBLEMS
1. Given the characteristic equation, is the system described by this characteristic equation stable?
Answer: One coefficient (-2) is negative. Therefore, the system does not satisfy the