padè approximation, smith predictor, circle and popov...

Pade approximationSmith predictor

Absolute stability

Pade approximation, Smith predictor, Circle and Popov Criteria

Daniele Carnevale

Dipartimento di Ing. Civile ed Ing. Informatica (DICII),

University of Rome “Tor Vergata”

Corso di Controlli Automatici, A.A. 2014-2015

Testo del corso: Fondamenti di Controlli Automatici,

P. Bolzern, R. Scattolini, N. Schiavoni.

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Pade approximationSmith predictor

Absolute stabilityFirst order delay approximation

Pade: delay approximation

Given a signal x(t) and its transfer function x(s) = L[x(t)](s), the transferfunction of the delayed signal x(t− T ) is

L[x(t− T )](s) =

∫ ∞0

x(t− T )e−stdt =︸︷︷︸τ=t−T

∫ ∞0

x(τ)e−(τ+T )sx(τ)dτ

=︸︷︷︸x(τ)=0,∀s≤0

e−sT∫ ∞

0

x(τ)e−τsx(τ)dτ = e−sTx(s). (1)

This is a transcendental function in s and, for example, the poles can not beevaluated in closed form. The Taylor expansion of the delay around s is

e−sT = e−sT∞∑k=0

(−(s− s)T )k

k!= e−sT

(1− T (s− s) +

T 2(s− s)2

2+ . . .

)(2)

can be considered when approximating e−sT with the n−order Padeapproximation such as

Pade, n(s) = µ1 + b1s+ · · ·+ bns

n

1 + a1s+ · · ·+ ansn, (3)

where the coefficients ai and bi are picked to match the first n coefficients of(4) with the long division of (5). 2 / 14

Pade approximationSmith predictor

Absolute stabilityFirst order delay approximation

First order delay approximation

The Taylor expansion at low frequency is usually centered in s = 0 yielding

e−sT =∞∑k=0

(−sT )k

k!= 1− Ts+

T 2s2

2+ . . . (4)

and the long division of first order Pade approximation

Pade, 1(s) = µ1 + b1s

1 + a1s= µ(1 + (b1 − a1)s− a1(b1 − a1)s2 + . . . (5)

is matched with the Taylor expansion as

µ = 1,µ(b1 − a1) = −T,−µ(a1(b1 − a1)) = T2

2,

⇒ {µ = 1, b1 = −0.5T, a1 = 0.5T}. (6)

Then

e−sT ≈︸︷︷︸s=0

Pade, 1(s) =1− 0.5s

1 + 0.5s

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Pade approximationSmith predictor

Absolute stabilityFirst order delay approximation

First order delay approximation

−1

−0.5

0

0.5

1M

agni

tude

(dB)

100 101 102−360

−270

−180

−90

0

90

180

270

360

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)

DelayPade1

Figure : First order Pade approximation at s = 0. 4 / 14

Pade approximationSmith predictor

Absolute stability

Approximate Smith predictorConclusions

The Smith predictor control scheme

r e u y+

−C

+

P

+z

M

Figure : Smith predictor control scheme.

Let P (s) = P0(s)e−sT with P0(s) a rational transfer function. The aim of theSmith predictor scheme is to replace the output y(s) = P (s)u(s) withy(s) = P0(s)u(s).It holds that

z(s) = M(s)u(s)+P (s)u(s) =(M(s) + P0(s)e−sT

)u(s) =︸︷︷︸

forced

P0(s)u(s) = y(s)

⇓M(s) = P0(s)(1− e−sT )

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Pade approximationSmith predictor

Absolute stability

Approximate Smith predictorConclusions

Pade approximation in the Smith predictor

The selectionM(s) = P0(s)(1− e−sT )

is impossible to realize in practice, then a Pade approximation can be exploitedand

M(s) = P0(s)(1− Pade,n(s)).

The overall (approximated) controller C1(s), u(s) = C1(s)e(s) is then

C1(s) =C(s)

1 + C(s)M(s)=

C(s)

1 + C(s)P0(s)(1− Pade,n(s)), (7)

=Nc(s)DP0(s)DPade,n(s)

DC(s)DP0(s)DPade,n(s) +NC(s)NP0(s)DPade,n(s)−NC(s)NP0(s)NPade,n(s)

Plant zero-pole cancellation (C1(s)P (s)): the plant P0(s) has to beasymptotically stable to use the Smith predictor control scheme.

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Pade approximationSmith predictor

Absolute stability

Approximate Smith predictorConclusions

Smith predictor: conclusions

The output is delayed

Wyr(s) =C(s)P (s)

1 + C(s)M(s) + C(s)P (s)=

C(s)P0(s)

1 + C(s)P0(s)e−sT ,

stability and performances have been improved but it is not possible tocancel out the delay effect on the output.

The Pade approximation has to be considered.

The plant P0(s) has to be asymptotically stable...and exactly known...

It would be possible to provide an inner loop to pre-stabilize the plant andconsequently apply the Smith predictor.

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Pade approximationSmith predictor

Absolute stability

IntroductionDefinitionResults

Introduction

Consider a block N(·) that is characterized by a static nonlinear functionϕ(·) : R→ R such that

k1 ≤ϕ(ε)

ε≤ k2, ε 6= 0, k1 ≤ k2 ⇒ ϕ(·) ∈ Φ[k1, k2].

Figure : Static nonlinearity N(·).

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Pade approximationSmith predictor

Absolute stability

IntroductionDefinitionResults

Introduction

Consider a block N(·) that is characterized by a static nonlinear functionϕ(·) : R→ R such that

k1 ≤ϕ(ε)

ε≤ k2, ε 6= 0, k1 ≤ k2 ⇒ ϕ(·) ∈ Φ[k1, k2].

Figure : Class of nonlinear functions Φ[k1, k2] with: a) [k1 > 0, k2 > 0], b)[k1 = 0, k2 > 0] and c) [k1 < 0, k2 > 0].

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Pade approximationSmith predictor

Absolute stability

IntroductionDefinitionResults

Introduction

Consider a block N(·) that is characterized by a static nonlinear functionϕ(·) : R→ R such that

k1 ≤ϕ(ε)

ε≤ k2, ε 6= 0, k1 ≤ k2 ⇒ ϕ(·) ∈ Φ[k1, k2].

Figure : Example of nonlinear functions ϕ(·) with: a) [k1 = 0, k2 > 0], b)[k1 = 0, k2 = +∞] and c) [k1 = 0, k2 = 1].

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Pade approximationSmith predictor

Absolute stability

IntroductionDefinitionResults

The equivalent plant

The usual feedback scheme can be transformed into:

r e ε y+

−C(s) P (s)

ξN(·)

ε χ

−Γ(s)

ξN(·)

Figure : The equivalent control scheme.

Where Γ(s) = C(s)P (s) and (one of) its minimal state space representation is:

η = AΓη +BΓξ, (8)

χ = CΓη, (9)

Γ(s) = CΓ(sI −AΓ)−1BΓ. (10)

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Pade approximationSmith predictor

Absolute stability

IntroductionDefinitionResults

Absolute stability

Absolute stability

The system Γ is absolutely stable within the sector [k1, k2] if theequilibrium η = 0 is globally stable for all function ϕ(·) ∈ Φ[k1, k2]

If the system Γ is absolutely stable within the sector [k1, k2], no matter thestatic function ϕ(·) ∈ Φ[k1, k2] is, the closed loop system is globally (robustly)asymptotically stable.There are not necessary and sufficient conditions.

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Pade approximationSmith predictor

Absolute stability

IntroductionDefinitionResults

Closed loop well-posedness

Consider the system Γ where χ = CΓη +DΓε. Then at the initial time t = 0the following equality has to hold

ε(0) = −CΓη(0)−DΓ(ϕ(ε(0))).

As an example, consider simply ϕ(η)→ η, i.e. there is not the nonlinearity,then it has to hold

ε(0) = −CΓη(0)−DΓε(0)

⇓

(1 +DΓ)ε(0) = −CΓη(0) ⇒︸︷︷︸for any η(0)

DΓ 6= −1.

Well-posedness (linear case)

The negative closed-loop of the system Γ is well-posed iff DΓ 6= −1.

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Pade approximationSmith predictor

Absolute stability

IntroductionDefinitionResults

A necessary condition

Define the graph of generalized segments

ρ[k1, k2] := {x ∈ R : x ∈ [−1/k1,−1/k2]},

and circles σ[k1, k2] as shown in the Figure.

Figure : The generalized segments ρ[k1, k2] and circles σ[k1, k2].

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