{"id":378,"date":"2019-05-22T17:41:09","date_gmt":"2019-05-22T17:41:09","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=378"},"modified":"2021-01-12T18:39:03","modified_gmt":"2021-01-12T18:39:03","slug":"2-1-general-definition-of-stability","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/2-1-general-definition-of-stability\/","title":{"raw":"2.1 General Definition of Stability","rendered":"2.1 General Definition of Stability"},"content":{"raw":"

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.<\/p>\r\n

Recall from ELE532 that mathematically, stability is related to the location of the closed loop system transfer function poles.<\/p>\r\n\r\n\r\n\r\n\r\n
Definition:<\/strong> A system is stable in BIBO sense if, for every bounded input, the output remains bounded.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Consider now the transfer function of a basic closed loop system:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]G_{cl}(s) = \\frac{G(s)}{1+G(s)H(s)}[\/latex]<\/td>\r\nEquation 2-1<\/td>\r\n<\/tr>\r\n
[latex]G(s) = \\frac{N_{G}(s)}{D_{G}(s)}[\/latex], [latex]H(s) = \\frac{N_{H}(s)}{D_{H}(s)}[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[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]<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Characteristic equation of the closed loop system is:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]Q(s) = 0[\/latex]<\/td>\r\nEquation 2\u20112<\/td>\r\n<\/tr>\r\n
[latex]Y(s) = G_{cl}(s)\\cdot U(s)[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[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]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex]Y(s) = Y_{natural}(s) + Y_{forced}(s)[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex]y(t) =L^{-1}\\{Y(s)\\} = y_{natural}(t) + y_{forced}(t)[\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex]y_{natural}(t) = \\sum_{i=1}^{n} K_{i}e^{p_{i}t}[\/latex]<\/td>\r\nEquation 2\u20113<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
Definition:<\/strong>\r\n\r\nSuppose 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\u20113 have strictly negative real parts: [latex]Re\\{p_{i}\\} < 0 [\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

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.<\/p>\r\n

<\/p>","rendered":"

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.<\/p>\n

Recall from ELE532 that mathematically, stability is related to the location of the closed loop system transfer function poles.<\/p>\n\n\n\n
Definition:<\/strong> A system is stable in BIBO sense if, for every bounded input, the output remains bounded.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Consider now the transfer function of a basic closed loop system:<\/p>\n\n\n\n\n\n
[latex]G_{cl}(s) = \\frac{G(s)}{1+G(s)H(s)}[\/latex]<\/td>\nEquation 2-1<\/td>\n<\/tr>\n
[latex]G(s) = \\frac{N_{G}(s)}{D_{G}(s)}[\/latex], [latex]H(s) = \\frac{N_{H}(s)}{D_{H}(s)}[\/latex]<\/td>\n<\/td>\n<\/tr>\n
[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]<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Characteristic equation of the closed loop system is:<\/p>\n\n\n\n\n\n\n\n\n
[latex]Q(s) = 0[\/latex]<\/td>\nEquation 2\u20112<\/td>\n<\/tr>\n
[latex]Y(s) = G_{cl}(s)\\cdot U(s)[\/latex]<\/td>\n<\/td>\n<\/tr>\n
[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]<\/td>\n<\/td>\n<\/tr>\n
[latex]Y(s) = Y_{natural}(s) + Y_{forced}(s)[\/latex]<\/td>\n<\/td>\n<\/tr>\n
[latex]y(t) =L^{-1}\\{Y(s)\\} = y_{natural}(t) + y_{forced}(t)[\/latex]<\/td>\n<\/td>\n<\/tr>\n
[latex]y_{natural}(t) = \\sum_{i=1}^{n} K_{i}e^{p_{i}t}[\/latex]<\/td>\nEquation 2\u20113<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
Definition:<\/strong><\/p>\n

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\u20113 have strictly negative real parts: [latex]Re\\{p_{i}\\} < 0[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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.<\/p>\n

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