Robust Control

Roland Longchamp

Laboratoire d’automatique

Ecole polytechnique federale de Lausanne

1015 Lausanne

Switzerland

e-mail: roland.longchamp@epfl.ch

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.