5 chapter 5 stability exploration methods

SYSTEM DYNAMICS & CONTROL CHAPTER SYSTEM STABILITY EXPLORATION METHODS Dr Vo Tuong Quan HCMUT - 2011 Stability Exploration Methods Definitions A stable system is a system with a bounded response to a bounded input  2011 – Vo Tuong Quan Stability Exploration Methods Examples  2011 – Vo Tuong Quan Stability Exploration Methods Poles and Zeros Method  2011 – Vo Tuong Quan Stability Exploration Methods Poles and Zeros Method Zeros and Poles plot: a graph which represents the position of poles and zeros in the complex s-plane  2011 – Vo Tuong Quan Stability Exploration Methods Poles and Zeros Method The stability of a system depends on the locations of Its poles: - If all the poles of the system lie in the left-half s-plan then the system is stable - If any of the poles of the systems lie in the right-half s-plane then the system is unstable - If some of the poles of the system lie in the imaginary axis and the others lie in the left-half s-plane then the system is at the stability boundary  2011 – Vo Tuong Quan Stability Exploration Methods Routh Method  2011 – Vo Tuong Quan Stability Exploration Methods Routh Method  2011 – Vo Tuong Quan Stability Exploration Methods Routh Method The necessary and sufficient condition for a system to be stable is that all the coefficients of the characteristic equation are positive and all parameters in the first column of the Routh table have positive signs The number of sign changes in the first column of the Routh table is equal the number of roots lying in the right-half s-plane  2011 – Vo Tuong Quan Stability Exploration Methods Routh Method The system is stable because all the parameters in the first column are positive 10  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Example 32  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Example 33  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Example 34  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example Consider the negative feedback system shown in Figure Plot root locus 35  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example clc; clear all; a=[1 0]; b=[1 16]; c=conv(a,b); den=c; num=[1 3]; rlocus(num,den) v=[-6 -6 6]; axis('square') grid on; 36  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example Root Locus 6 0.64 0.5 0.38 0.24 0.12 0.78 4 0.88 2 Imaginary Axis 0.97 1 0.97 -2 0.88 -4 0.78 0.64 -6 -6 -5 0.5 -4 0.38 -3 0.24 -2 Real Axis  2011 – Vo Tuong Quan 0.12 -1 60 37 Stability Exploration Methods Roots Locus Method – Using Matlab - Example K G s H s s s s2 0.6s 10 38  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example num=[1]; a=[1 0]; b=[1 0.5]; den1=conv(a,b); c=[1 0.6 10]; den=conv(den1,c); r=roots(den); r=rlocus(num,den) plot(r,'+') v=[-6 -6 6]; axis(v); axis('square') grid; 39  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example Root-Locus Plot of G(s) = K/[s(s + 0.5)(s + 0.6s + 10)] Imag Axis -2 -4 -6 -6  2011 – Vo Tuong Quan -4 -2 Real Axis 40 Stability Exploration Methods Roots Locus Method – Using Matlab - Example num=[1]; a=[1 0]; b=[1 0.5]; den1=conv(a,b); c=[1 0.6 10]; den=conv(den1,c); r=roots(den); K1=0:0.2:20; K2=20:0.1:30; K3=30:5:1000; K=[K1 K2 K3]; r=rlocus(num,den,K) plot(r,'+') v=[-6 -6 6]; axis(v); axis('square') grid;  2011 – Vo Tuong Quan 41 Stability Exploration Methods Root-Locus Plot of G(s) = K/[s(s + 0.5)(s + 0.6s + 10)] Roots Locus Method – Using Matlab - Example Imag Axis -2 -4 -6 -6 -4 -2 Real Axis 42  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example Consider the negative feedback system shown in Figure Plot root locus 43  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example num = [1 4]; den = conv(conv([1 0],[1 6]), [1 1.4 1]); rlocus(num, den) v = [-7 -5 5]; axis(v); axis('square') grid title('Root-Locus Plot of G(s) = K(s^2 + 2s + 4)/[s(s + 4)(s + 6)(s^2 + 1.4s + 1)]') text(1.0, 0.55,'K = 12') text(1.0,3.0,'K = 73') text(1.0,4.15,'K = 154') 44  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method – Using Matlab - Example Root-Locus Plot of G(s) = K(s + 2s + 4)/[s(s + 4)(s + 6)(s + 1.4s + 1)] 0.76 0.64 0.5 0.34 0.16 0.86 K = 154 K = 73 0.94 Imaginary Axis 0.985 -1 K = 12 0.985 -2 0.94 -3 -4 0.86 0.76 -5 -7 -6 0.64 -5 -4 0.5 -3 0.34 -2 Real Axis  2011 – Vo Tuong Quan 0.16 -1 45 Stability Exploration Methods Roots Locus Method: Types 46  2011 – Vo Tuong Quan ... – Vo Tuong Quan Stability Exploration Methods Routh Method 14  2011 – Vo Tuong Quan Stability Exploration Methods Routh Method 15  2011 – Vo Tuong Quan Stability Exploration Methods Routh Method... in the left-half s-plane then the system is at the stability boundary  2011 – Vo Tuong Quan Stability Exploration Methods Routh Method  2011 – Vo Tuong Quan Stability Exploration Methods Routh... Tuong Quan Stability Exploration Methods Roots Locus Method 17  2011 – Vo Tuong Quan Stability Exploration Methods Roots Locus Method 18  2011 – Vo Tuong Quan Stability Exploration Methods 19