Chapter 2

# 2.1 General Definition of Stability

Stability is an implicitly stated control objective. Intuitively, a closed loop system is stable if it does not “blow up”. For an introduction, see Online Tutorials – sections on Basic Concepts and on Stability.

Recall from ELE532 that mathematically, stability is related to the location of the closed loop system transfer function poles.

Definition: A system is stable in BIBO sense if, for every bounded input, the output remains bounded. |

Consider now the transfer function of a basic closed loop system:

[latex]G_{cl}(s) = \frac{G(s)}{1+G(s)H(s)}[/latex] | Equation 2-1 |

[latex]G(s) = \frac{N_{G}(s)}{D_{G}(s)}[/latex], [latex]H(s) = \frac{N_{H}(s)}{D_{H}(s)}[/latex] | |

[latex]G_{cl}(s) = \frac{\frac{N_{G}(s)}{D_{G}(s)}}{1+\frac{N_{G}(s)N_{H}(s)}{D_{G}(s)D_{H}(s)}} = \frac{N_{G}(s)D_{H}(s)}{D_{G}(s)D_{H}(s) + N_{G}(s)N_{H}(s)} = \frac{N(s)}{Q(s)}[/latex] |

Characteristic equation of the closed loop system is:

[latex]Q(s) = 0[/latex] | Equation 2‑2 |

[latex]Y(s) = G_{cl}(s)\cdot U(s)[/latex] | |

[latex]Y(s) = \frac{N(s)}{Q(s)}\cdot U(s) = \sum_{i=1}^{n}\frac{K_{i}}{s-p_{i}} + \sum_{j}\frac{K_{j}}{s-p_{j}}[/latex] | |

[latex]Y(s) = Y_{natural}(s) + Y_{forced}(s)[/latex] | |

[latex]y(t) =L^{-1}\{Y(s)\} = y_{natural}(t) + y_{forced}(t)[/latex] | |

[latex]y_{natural}(t) = \sum_{i=1}^{n} K_{i}e^{p_{i}t}[/latex] | Equation 2‑3 |

Definition:
Suppose that the closed loop system has a transfer function [latex]G_{cl}(s)[/latex] . The system is stable if, and only if, the poles [latex]G_{cl}(s)[/latex]of shown in Equation 2‑3 have strictly negative real parts: [latex]Re\{p_{i}\} < 0[/latex]. |

To determine this condition analytically (as opposed to a numerical solution, such as provided by MATLAB) a Criterion of Stability needs to be defined – it will be the Routh-Hurwitz Criterion of Stability.