Robust Control
Roland Longchamp
Laboratoire d’automatique
Ecole polytechnique federale de Lausanne
1015 Lausanne
Switzerland
e-mail: [email protected]
ÉCOLE POLYTECHNIQUEFÉDÉRALE DE LAUSANNE
Outline
1. Basic Concepts
2. Plant Uncertainty
3. Robust Stability
4. Robust Performance
5. Design Constraints
6. Standard Loopshaping
7. Concluding Remarks
Basic Concepts
SISO, continuous-time systems
Loop transfer function L(s) = K(s)G(s)
Sensitivity function
S(s) =1
1 + L(s)=
E(s)
Yc(s)= −E(s)
V (s)=U ′(s)W (s)
Complementary sensitivity function
T (s) =L(s)
1 + L(s)=
Y (s)
Yc(s)= −Y (s)
V (s)= − U(s)
W (s)
S(s) + T (s) = 1 S(G) =
dT (G)
dGT (G)
G
Infinity-Norms
∞-signal norm
g : → , t 7→ g(t)
‖g‖∞ = supt|g(t)|
∞-system normG : → , s 7→ G(s)
‖G‖∞ = supω|G(jω)|: Exists iff G(s) is proper and
has no poles on the imaginary axis
∞-system norm is submultiplicative
‖KG‖∞ ≤ ‖K‖∞ ‖G‖∞
Nominal Performance
|S(jω)| < σ ω ∈ [0, ωb], σ ∈
ωb: Closed-loop bandwidth
maxω∈[0,ωb]
1
σ|S(jω)| < 1
W1(jω) =
1
σif ω ∈ [0, ωb]
0 if ω 6∈ [0, ωb]
supω|W1(jω)S(jω)| < 1
|W1(jω)| is a roughly decreasing function of ω:
Performance decreases with increasing frequency.
Nominal Performance (cont.)
Nominal performance
‖W1 S‖∞ < 1 W1(s) : Weight
⇔∣∣∣W1(jω) 1
1+L(jω)
∣∣∣ < 1 ∀ω
⇔ |W1(jω)| < |1 + L(jω)| ∀ω
At each frequency, the point L(jω) on the Nyquist
plot lies outside the disk of center -1, ra-
dius |W1(jω)|.
Plant Uncertainty
Parametric uncertainty
=
B(s)
sn + a1sn−1 + . . .+ an: ai ≤ ai ≤ ai,i = 1,2, . . . , n
Finite set of models
= {G1(s), G2(s), . . . , Gn(s)}Unstructured uncertainty (multiplicative form)
={G(s) = (1 + ∆(s)W2(s)) G(s) : ‖∆‖∞ ≤ 1
}G(s) : Nominal plant transfer function
W2(s) : Fixed stable transfer function (weight)∆(s) : Variable stable transfer function
Multiplicative Perturbation
s = jω, ω given
GG − 1 = ∆W2
⇔
∣∣∣∣GG − 1
∣∣∣∣ = |∆| |W2| ∈ [0, |W2|]
Arg(GG − 1
)= Arg∆ + ArgW2 ∈ (−π, π]
Im
Re1
1 : No perturbation (G = G)GG − 1 : Can be everywhere in the disk with
center 1, radius |W2|:∣∣∣∣GG − 1
∣∣∣∣ ≤ |W2||W2(jω)| is a roughly increasing function of ω:Uncertainty increases with increasing frequency.
Example: Time Constant
Neglected time constant (e.g. the inductance inan electrical drive)
G(s) = G(s)1
τs+ 1τ ∈ [0, τ ]
G(s)
G(s)− 1 =
1
τs+ 1− 1 = − s
s+ 1τ
= ∆(s)W2(s)
∣∣∣∣∣∣ jω
jω + 1τ
∣∣∣∣∣∣ ≤ |W2(jω)| ∀ω, τ
W2(s) = τ sτs+1: |W2(jω)| increases with
increasing frequency{G(jω)
G(jω): 0 ≤ τ ≤ τ
}⊂ {s ∈ : |s− 1| ≤ |W2(jω)|} :
Conservatism
Example: Time Delay
Neglected time delay (e.g. in power electronics)
G(s) = G(s) e−sT T ∈ [0,0.1]G(s)
G(s)− 1 = e−sT − 1 = ∆(s)W2(s)∣∣∣e−jω T − 1
∣∣∣ ≤ |W2(jω)| ∀ω, T
W2(s) = 2.1 ss+10: |W2(jω)| increases with
increasing frequency{G(jω)
G(jω): 0 ≤ T ≤ 0.1
}⊂ {s ∈ : |s− 1| ≤ |W2(jω)|} :
Conservatism
Example: Uncertain Gain
Uncertain gain (parametric uncertainty)
G(s) = G(s) γ γ ∈ [0.1,10]
{G(jω)
G(jω): 0.1 ≤ γ ≤ 10
}⊂{s ∈ : |s− 1| ≤ 4.95
5.05
}:
Conservatism
Example: Multiple Models
Finite set of models
G1(s): Nominal plant transfer function
|W2(jω)| = max
{∣∣∣∣∣Gi(jω)
G1(jω)− 1
∣∣∣∣∣ : 2 ≤ i ≤ n}
{Gi(jω)
G1(jω): 2 ≤ i ≤ n
}⊂ {s ∈ : |s− 1| ≤ |W2(jω)|} :
Conservatism
Robust Stability
Classical measures of stability margin
Modulus margin
infω|1 + L(jω)| =
1
supω
1
|1 + L(jω)|=
1
‖S‖∞
Absolute Stability
x = A x + Buy = C x
}: Controllable and observable
u = −ψ(t, y)
ψ : [0,∞)× → : Piecewise continuous in t and
Lipschitz in y
α y2 ≤ y ψ(t, y) ≤ β y2, α < β: Sector condition
Absolute stability: The origin is globally uniformly
asymptotically stable for any nonlinearity in the
sector.
Absolute Stability (cont.)
Theorem (circle criterion): The system is ab-solutely stable if one of the following conditions issatisfied:
(i) If 0 < α < β, the Nyquist plot of L(jω) doesnot enter the disk D(α, β) and encircles it P
times in the counterclockwise direction, whereP is the number of poles of L(s) with positivereal parts.
(ii) If 0 = α < β, L(s) is Hurwitz and the Nyquistplot of L(jω) lies to the right of the lines = −1
β.
(iii) If α < 0 < β, L(s) is Hurwitz and the Nyquistplot of L(jω) lies in the interior of the diskD(α, β).
Absolute Stability (cont.)
Modulus margin: ‖S‖−1∞
α =1
1 + ‖S‖−1∞β =
1
1− ‖S‖−1∞
Robust Stability
Robust stability: Internal stability for all G(s) ∈ .
Theorem: The controller K(s) provides robust
stability iff ‖W2 T‖∞ < 1.
‖W2 T‖∞ < 1
⇔∣∣∣∣∣W2(jω)
L(jω)
1 + L(jω)
∣∣∣∣∣ < 1 ∀ω
⇔ |W2(jω)L(jω)| < |1 + L(jω)| ∀ω
At each frequency, the critical point lies outside
the disk of center L(jω), radius |W2(jω)L(jω)|.
Robust Performance
Robust performance: Internal stability and perfor-mance for all G(s) ∈ .
s 7→ |W1(s)S(s)|+ |W2(s)T (s)| = |W1 S|+ |W2 T |Theorem: The controller K(s) provides robustperformance iff ‖ |W1 S|+ |W2 T | ‖∞ < 1.
‖ |W1 S|+ |W2 T | ‖∞ < 1m∣∣∣∣∣W1(jω)
1
1 + L(jω)
∣∣∣∣∣+
∣∣∣∣∣W2(jω)L(jω)
1 + L(jω)
∣∣∣∣∣ < 1 ∀ω
m|W1(jω)|+ |W2(jω)L(jω)|
|1 + L(jω)|< 1 ∀ω
m|W1(jω)|+ |W2(jω)L(jω)| < |1 + L(jω)| ∀ω
At each frequency, the disks of center L(jω),radius |W2(jω)L(jω)|, and of center −1, radius|W1(jω)|, are disjoint.
Design Constraints
S(s) + T (s) =1
1 + L(s)+
L(s)
1 + L(s)= 1
Robust performance
⇒ min {|W1(jω)|, |W2(jω)|} < 1 ∀ωTheorem (area formula): Assume that the rel-ative degree of L(s) is at least 2 and let {pi :i = 1,2, . . . , n} denote the set of poles of L(s) inRe s > 0. Then∫ ∞
0log |S(jω)|dω = π (log e)
n∑i=1
Re pi
Waterbed Effect
|L(jω)| < 1ω2 ∀ω ≥ ω0 > 1
|S(jω)| = 1
|1 + L(jω)|≤ 1
1− |L(jω)|<
1
1− 1ω2
=ω2
ω2 − 1
∫ ∞ω0
log |S(jω)|dω ≤∫ ∞ω0
logω2
ω2 − 1dω = c <∞
Standard Loopshaping
G(s): Stable and minimum phase
|W1(jω)| À 1 > |W2(jω)| : |L(jω)| > |W1(jω)|1− |W2(jω)|
|W1(jω)| < 1¿ |W2(jω)| : |L(jω)| < 1− |W1(jω)||W2(jω)|
|W1(jω)S(jω)|+ |W2(jω)T (jω)| < 1
Concluding Remarks
Robust stability, robust performance, design con-straints, loopshaping, etc., nicely extend classicalcontrol theory.
Extension to MIMO systems:
‖G‖∞ = supωσ(G(jω))
σ(G(jω)): Largest singular value of G(jω)
ConservatismHigh order controllersSmall set of performance specificationsProvides what can be achievedetc.
Numerous other approaches to robust control
Kharitonov polynomialsMultimodel approachSingular perturbationSliding mode controlLyapunov methodsetc.
