Lead/lag networks: inversion formulas
• Let us consider the following feedback system:
- - C(s) - G(s) -
6
r e m y
where G(s) is the system to be controlled and C(s) is an lead/lag network having the following structure:
C(s) = 1 + τ1s 1 + τ2s
• You have a Lead network when τ1 > τ2: C(s) = 1 + τ s
1 + ατ s dove τ = τ1, α = τ2 τ1 < 1
• You have a Lag network when τ1 < τ2: C(s) = 1 + ατ s
1 + τ s dove τ = τ2, α = τ1 τ2 < 1
• The dynamic specifications for a feedback system are usually given in terms of phase margin Mϕ and gain margin Mα.
• The parameters τ1 and τ2 of a lead/lag network that introduces an ampli- fication M and a phase shift ϕ at frequency ω can be obtained using the following inversion formulas:
τ1 = M − cos ϕ
ωsin ϕ , τ2 = cos ϕ − M1 ωsin ϕ
Inversion formulas: mathematical details
• Design problem: Determine the parameters τ1 and τ2 of the lead/lag network C(s) such that
C(jω) = 1 + j τ1ω
1 + j τ2ω = M ejϕ
that is, the network amplifies M and provides a phase shift ϕ at frequency ω.
• The design project is solved by using the following inversion formulas:
τ1 = M − cos ϕ
ωsin ϕ , τ2 = cos ϕ − M1 ωsin ϕ
• The inversion formulas are obtained by rewriting the equation C(jω) = 1 + j τ1ω
1 + j τ2ω = M ejϕ = M cos ϕ + jM sin ϕ in the form
(M cos ϕ + jM sin ϕ)(1 + j τ2ω) = 1 + j τ1ω This equation can also be rewritten as follows:
" 1 −M cos ϕ 0 M sin ϕ
# "
τ1ω τ2ω
#
=
"
M sin ϕ M cos ϕ − 1
#
Solving with respect to the variables τ1 and τ2 one obtains
τ1 =
M sin ϕ −M cos ϕ M cos ϕ − 1 M sin ϕ
ωM sin ϕ = M − cos ϕ
ωsin ϕ
τ2 =
1 M sin ϕ 0 M cos ϕ − 1
ωM sin ϕ =
cos ϕ − 1 M ωsin ϕ
• These formulas are valid both for lead networks (M > 1 and ϕ > 0) and lag networks (M < 1 and ϕ < 0).
• From the design specifications to the determination of point B:
B1 Mϕ
B2 1 Mα
B3
Re Im
−1 O
Piano di Nyquist
B1 Mϕ
B2 Mα
B3
ϕ
| · |
−1
Piano di Nichols
• Lead network: admissible regions for points A to be moved in B:
B
A1
A2
A3
Re Im
D1 0
ReteAnticipatrice Piano di Nyquist
π 2
B
A1
A2
A3
ϕ
| · |
−1
D1
ReteAnticipatrice
Piano di Nichols
• Lag network: admissible regions for points A to be moved in B:
B
A1
A2
A3
Re Im
0
D2
ReteRitardatrice
Piano di Nyquist
π 2
B
A1
A2
A3
ϕ
| · |
−1
D2
ReteRitardatrice
Piano di Nichols
• Design of a lead network for system G1(s). Design specification on the phase margin: Mϕ = 60◦.
A
D1 ω
G1(jω) ω1
ω2
Mϕ= 60◦
B
Gc(jω)
−1 O
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Step response
y(t)
Time [s]
• Design of a lag network for system G2(s). Design specification on the phase margin: Mϕ = 60◦.
A B
D2
G2(jω) ω2
ω1 Mϕ= 60◦
ω Gc(jω) O
−1
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Step response
y(t)
Time [s]
• Design of a lag network for system G2(s). Design specification on the gain margin: Mα = 5.
−7 −6 −5 −4 −3 −2 −1 0 1 2
−6
−5
−4
−3
−2
−1 0 1
−1
G3(jω)
Gc(jω) 0
A
B
D2
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Step response
y(t)
Time [s]
