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/ December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Chapter SYSTEM STABILITY ANALYSIS December 2013 © H T Hồng - ÐHBK TPHCM 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 loci l i Stability analysis using root locus Frequency response analysis Bode criterion Nyquist Nyquist’s s stability criterion December 2013 © H T Hồng - ÐHBK TPHCM Stability concept December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 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) y(t) Stable system December 2013 System y(t) y(t) System at stability boundary © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ Unstable system 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 n1 an1s an ((TF’s denominator)) B ( s ) b0 s m b1s m1 bm1s bm (TF’ numerator) Poles: P l are the h roots off the h denominator d i off the h transfer f function, i.e the roots of the equation A(s) = Since A(s) is of order n,, the system y has n p poles 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) = Since B(s) is of order m, the system has m zeros denoted as zi, i =1,2,…m December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Pole – zero plot Pole – zero plot is a graph which represents the position of poles and zeros in the complex s-plane Pole Zero December 2013 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 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 December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Characteristic equation Characteristic Ch t i ti equation: ti i the is th equation ti A(s) A( ) = Characteristic polynomial: is the denominator A(s) Note: Feedback systems R(s) +_ G(s) H(s) Y(s) Systems described by state equations x (t ) Ax(t ) Bu (t ) y (t ) Cx(t ) Characteristic equation Characteristic equation G(s) H (s) det sI A December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Algebraic stability criteria December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 10 Graphical representation of frequency response (cont’) Bode diagram Nyquist plot Gain margin Gain margin Phase margin Phase margin December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 58 Nyquist stability criterion Consider a unity feedback system shown below, below suppose that we know the Nyquist plot of the open loop system G(s), the problem is to determine the stability of the closed-loop system Gcl(s) ( ) R(s) + G(s) Y(s) Nyquist criterion: The closed-loop system Gcl(s) is stable if and only if the Nyquist plot of the open open-loop loop system G(s) encircles the critical point (1, j0) l/2 times in the counterclockwise direction when changes from to + (l is the number of poles of G(s) lying in the right-half right half s-plane) s plane) December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 59 Nyquist stability criterion – Example Consider an unity negative feedback system, system whose openopen loop system G(s) is stable and has the Nyquist plots below (three cases) Analyze the stability of the closed-loop system December 2013 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 60 Nyquist stability criterion – Example Solution The number of poles of G(s) lying in the right-half s-plane is because G(s) ( ) is stable Then according g to the Nyquist yq criterion, the closed-loop system is stable if the Nyquist plot G(j) does not encircle the critical point (1, j0) Case : G(j) does not encircle (1, j0) the close close-loop loop system is stable stable Case : G(j) pass (1, j0) the close-loop p system y is at the stabilityy boundary; Case : G(j) encircles (1, j0) the th close-loop l l system t i unstable is t bl December 2013 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 61 Nyquist stability criterion – Example Analyze the stability of a unity negative feedback system whose open loop transfer function is: K G ( s) s (T1s 1)(T2 s 1)(T3 s 1) Solution: Nyquist plot: Depending on the values of T1, T2, T4 and d K, K the h N Nyquist i plot of G(s) could be one of the three curves 1,, or December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 62 Nyquist stability criterion – Example (cont’) The number Th b off poles l off G(s) G( ) lying l i in i the th right-half i ht h lf s-plane l i is because G(s) is stable Then according to the Nyquist criterion,, the closed-loop p system y is stable if the Nyquist yq plot p G(j) does not encircle the critical point (1, j0) Case : G(j) does not encircle (1, j0) the close-loop system is stable Case : G(j) pass (1, j0) the close-loop system is at the stability boundary; Case : G(j) encircles (1, j0) the close-loop system is unstable December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 63 Nyquist stability criterion – Example Given an unstable open open-loop loop systems which have the Nyquist plot as below In which cases the closed-loop system is stable? Stable December 2013 Unstable © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 64 Nyquist stability criterion – Example (cont’) Given an unstable Gi t bl open-loop l systems t which hi h have h th Nyquist the N i t plot as below In which cases the closed-loop system is stable? Unstable December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 65 Nyquist stability criterion – Example (cont’) Given an unstable open-loop systems which have the Nyquist plot as below In which cases the closed-loop system is stable? St bl Stable December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ U t bl Unstable 66 Nyquist stability criterion – Example Given a open open-loop loop system which has the transfer function: K G( s) (Ts 1) n (K>0, T>0, n>2) Find the condition of K and T for the unity negative feedback closed-loop system to be stable Solution: Frequency response of the open-loop system: K G ( j ) (Tj 1) n Magnitude: M ( ) K T 1 Phase: December 2013 n ( ) ntg 1 (T ) © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 67 Nyquist stability criterion – Example (cont’) N Nyquist i plot: l Stability condition: the Nyquist plot of G(j) does not encircle the critical point (1,j0) According to the Nyquist plot, this requires: M ( ) December 2013 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 68 Nyquist stability criterion – Example (cont’) 1 We have: ( ) ntg (T ) Then: tg (T ) 1 n g tg T n M ( ) December 2013 K K tg g n (T ) tg n n T tg t T n © H T Hồng - www4.hcmut.edu.vn/~hthoang/ n 1 69 Bode criterion Consider a unity feedback system, system suppose that we know the Nyquist plot of the open loop system G(s), the problem is to determine the stability of the closed-loop system Gcl(s) R(s) + G(s) Y(s) Bode criterion: criterion The closed-loop closed loop system s stem Gcl(s) is stable if the gain margin and phase margin of open-loop system G(s) are positive GM The closed - loop system is stable M M December 2013 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 70 Bode criterion – Example Consider a unityy negative g feedback system y whose open-loop p p system has the Bode diagram as below Determine the gain margin, phase margin of the open-loop system Is the closedloop system stable or not? Bode diagram: c L( ) GM L( ) 35dB ( c ) 2700 GM 35dB M 180 (270 ) 900 180 (C) December 2013 M C Because GM