Lecture Fundamentals of control systems: Chapter 4 - TS. Huỳnh Thái Hoàng

Lecture "Fundamentals of control systems - Chapter 4: System stability analysis system stability analysis" presentation of content: Stability concept, algebraic stability criteria, root locus method, frequency response analysis.

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Lecture Fundamentals of control systems: Chapter 4 - TS. Huỳnh Thái Hoàng Lecture Notes Introduction of Control Systems Instructor: Assoc. Prof. Dr. Huynh Thai Hoang Department of Automatic Control Faculty of Electrical & Electronics Engineering Ho Chi Minh City University of Technology Email: hthoang@hcmut.edu.vn huynhthaihoang@yahoo.com Homepage: www4.hcmut.edu.vn/~hthoang/6 December 2013 © H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1 Chapter 4 SYSTEM STABILITY ANALYSIS6 December 2013 © H. T. Hoàng - ÐHBK TPHCM 2 Content Stability concept Algebraic stability criteria  Necessary y condition  Routh’s criterion  Hurwitz’s criterion Root locus method  Root locus definition  Rules R l for f drawing d i root locil i  Stability analysis using root locus Frequency response analysis  Bode criterion  Nyquist Nyquist’ss stability criterion6 December 2013 © H. T. Hoàng - ÐHBK TPHCM 3 Stability concept6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 4 BIBO stability A system is defined to be BIBO stable if every bounded input to the system results in a bounded output over the time interval [t0,+∞) for all initial times t0. u(t) y(t) System y(t) y(t) y(t) Stable system System at Unstable stability boundary system 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 5 Poles and zeros C Consider id a system t d described ib d by b the th transfer t f function f ti (TF): (TF) Y ( s ) b0 s m  b1s m 1    bm 1s  bm G(s)   U ( s ) a0 s n  a1s n 1    an 1s  an Denote: A( s )  a0 s n  a1s n1    an1s  an ((TF’s denominator)) B ( s )  b0 s m  b1s m1    bm1s  bm (TF’ numerator) Poles: P l are theh roots off the h denominator d i off the h transfer f function, i.e. the roots of the equation A(s) = 0. Since A(s) is of order n,, the system y has n ppoles denoted as pi , i =1,2,…n. , , Zeros: are the roots of the numerator of the transfer function, i.e. the roots of the equation B(s) = 0. Since B(s) is of order m, the system has m zeros denoted as zi, i =1,2,…m. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 6 Pole – zero plot Pole – zero plot is a graph which represents the position of poles and zeros in the complex s-plane. Pole Zero 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 7 Stability analysis in the complex plane The stability of a system depends on the location of its poles. If all the poles of the system lie in the left-half s-plane then the system t i stable. is t bl If any of the poles of the system lie in the right-half s-plane then the system is unstable. unstable If some of the poles of the system lie in the imaginary axis and the others lie in the left left-half half ss-plane plane then the system is at the stability boundary. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 8 Characteristic equation Characteristic Ch t i ti equation: ti i the is th equation ti A(s) A( ) = 0 Characteristic pol ...