Closed-Loop Stability

1. Closed-Loop Stability • Motivating example • General stability criterion • Routh stability criterion • Direct substitution method • Simulink example

2. Motivating Example • Transfer functions • Step response for setpoint change

3. Motivating Example cont.

4. Linear Stability • Linear system stability • A linear system is stable if the output response is bounded for all bounded inputs. Otherwise the system is unstable. • Open-loop stability • System that have poles with positive real part are unstable. • Integrating systems are unstable. • The closed-loop stability problem • Determine the range of controller parameter values for which the closed-loop system.

5. The Characteristic Equation • Closed-loop transfer function for setpoint change • Step response for distinct poles • Characteristic equation

6. General Linear Stability Criterion • A feedback control system is stable if and only if all the roots of the characteristic equation has negative real part.

7. Applying the Stability Criterion • Transfer functions • Characteristic equation • Roots

8. Routh Stability Criterion • Disadvantage of general stability criterion • Must explicitly compute roots in terms of controller parameters • Must deduce range of controller parameter values that produce roots with positive real parts • Routh method • Allows stable controller parameter values to be determined without explicitly calculating roots of characteristic equation • Only applicable to characteristic equations that are polynomial functions of s • Time delays must be approximated (e.g. Pade approximation)

9. Routh Stability Criterion cont. • Characteristic equation • Assume: an > 0 (otherwise make multiply equation by -1) • Necessary condition for stability is that all the coefficients (a0, a1,…, an-1, an) are positive • Routh array elements • Routh stability criterion: A necessary and sufficient condition for the feedback control system to be stable is for all the elements (b1, c1, …) in the left-hand column of the Routh array to be positive.

10. Routh Stability Criterion Example • Characteristic equation (n = 3) • Necessary condition • Elements of Routh array • Stability range: -1 < Kc < 12.6

11. Direct Substitution Method • Any point in the complex plane can be represented as: s = a+jw • Along the imaginary axis that divides the stable and unstable regions: s = jw • Substitute s = jw into characteristic equation to determine stability limits in terms of controller parameters • Generates two equations for real and imaginary parts of equation

12. Direct Substitution Example • Characteristic equation • Substitute s = jw • Separate real and imaginary parts • Stability range: -1 < Kc < 12.6

13. Simulink Example • Transfer functions >> gp=tf([1],[5 1]); >> gv=tf([1],[2 1]); >> gm=tf([1],[1 1]); >> gc=6; >> km=1; >> gcl=km*gc*gv*gp/(1+gc*gv*gp*gm) Transfer function: 60 s^3 + 102 s^2 + 48 s + 6 ------------------------------------------------ 100 s^5 + 240 s^4 + 209 s^3 + 143 s^2 + 57 s + 7 >>isstable(gcl) ans = 1 >> pole(gcl) ans = -1.4791 -0.1105 + 0.6790i -0.1105 - 0.6790i -0.5000 -0.2000 >> zero(gcl) ans = -1.0000 -0.5000 -0.2000

14. Pole-Zero Map >> pzmap(gcl)

15. Root Locus >> g=gp*gv*gm Transfer function: 1 ------------------------- 10 s^3 + 17 s^2 + 8 s + 1 >> rlocus(g)

16. Root Locus cont.