Routh Criteria Routh criterion based on regular Routh
Routh Criteria Routh criterion based on regular Routh table: 1) d(s) is A. S. iff 1 st col have same sign 2) # of sign changes in 1 st col = # of roots in right half plane
Routh Criteria Special case 1: one whole row = 0 Solution: 1) use prev. row to form aux. poly. A(s) 2) get: 3) use coeff of to replace 0 -row 4) continue as usual Fact: Roots of A(s) are also roots of d(s) A(s) is a factor of d(s)
Example ←whole row=0
Replace by e
Useful case: variable parameters in d(s) How to use: 1) form table as usual 2) set all 1 st col. >0 3) solve for parameter range for A. S. 2’) set one whole row = 0 3’) solve for parameter that leads to M. S. or leads to sustained oscillation
Steady-state tracking & sys. types • Unity feedback control: r(s) + e C(s) - r(s) + e - Go. l. (s) plant G(s) y(s)
r(t) t
r(t) 0 t
Steady-state tracking & sys. types plant r(s) + e - C(s) G(s) y(s)
Assuming closed-loop system stability
r(t)=us(t) r(s)=1/s r(t)=tus(t) r(s)=1/s 2 r(t)= ½t 2 us(t) r(s)=1/s 3 type 0 (N=0 a 0≠ 0) Kp=b 0/a 0 ess=1/(1+Kp) Kv=0 ess=∞ Ka=0 ess=∞ type 1 (N=1 a 0=0 a 1≠ 0 b 0≠ 0 ) K p= ∞ ess=0 Kv=b 0/a 1 ess=1/Kv Ka=0 ess=∞ type 2, N=2 a 0=a 1=0 a 2≠ 0, b 0≠ 0 K p= ∞ ess=0 K v= ∞ ess=0 Kp=b 0/a 2 ess=1/Ka type≥ 3, N ≥ 3 Kp= ∞ a 0=a 1=a 2=0 ess=0 b 0≠ 0 K v= ∞ ess=0 K a= ∞ ess=0 sys. type ref. input
Example: water tank level control + - C H
r(s) e. g. r(s) + e - Kps+KI ωn 2 1 s s(s+2ξ ωn) Ts+1 +
example r(s) e(s) G(s) y(s)
d 1(s) r(s) + e - d 2(s) Kps+KI ωn 2 1 s s(s+2ζ ωn) Ts+1 A B
Example
Td(s) Vin(s) + Kps+KI V a s e - - Tm 1 1 Las+Ra Js+b s Vb Kb Td(s) Vin(s) + e - Kps+KI V a s Vb Tm 1 Las+Ra Js+b Kb
- Slides: 44