lpv approaches for sampling period dependent control design: …o.sename/docs/lpv... · 2021. 1....

LPV approaches for sampling period dependent control design:application to AUVs

Olivier Sename, Daniel Simon

Master Course - January 2021

Olivier Sename, Daniel Simon () LPV AUV Master Course - January 2021 1/73

1. Control/Computing Co-design overviewObjectivesNCS constraintsControl and Scheduling IntegrationComputing-aware control

2. A polytopic approach discrete-time control with varying samplingObjectiveDiscrete polytopic LPV modeling with variable samplingA polytopic method with reduced complexityLPV polytopic control design method

3. An LFT/LPV approachIntroduction and ObjectiveA new LFT Formulation for varying sampling systemsH∞/LFT control designExtension to LPV systems

4. Application to AUVsLTI modelsLPV modelsAUV controlGeneral control schemeLPV polytopic sampling dependent controller for an LTI system (case 1)LPV/LFR varying sampling controller for a LPV model (case 2)

5. ConclusionRemarks on LPV toolsFuture works


Control/Computing Objectives

Context

• Control/Computation Codesign• Development of sampling dependent controllers: PhD thesis of D. Robert (2007) and E.

Roche (2011)• application to AUVs (European IST FeedNetBack project)

Main contribution : LPV control of an AUV with the sampling interval as varying parameter


Control/Computing Objectives

Control/computing co-design objectives

Goal : reaching a specified control performance under execution resources constraints Stankovicet al., 1999; Cervin, 2003; Sename et al., 2003

• control systems implementation over networks, using digital computers and real-timeoperating systems.

• cross coupling between closed-loop control of continuous processes and the controllerimplementation;


Control/Computing NCS constraints

NCS induced computing constraints

• Non-linear and uncertain models• Limited on board computing power• Scheduled U.S. sensors• Varying time/delays• Noise, data-loss

• limited computing power→ computing duration are not negligible• variable computing power→ they are not constant• distributed information sources→ delays, data loss or corruption• many sources of induced delays: computing, preemption, networking. . .• implementation induced uncertainties (in addition to the usual process uncertainties)


Control/Computing Control and Scheduling Integration

Overview

Traditional design : separation of concerns between control and computing :• perfectly periodic, no jitter, no data-loss -> over-constrained• perfect implementation (worst case) -> under-used resources• dedicated hardware/software -> costly components and tools

=⇒ formally provable, but costly and inflexible/unrealisticIn Progress• dynamic environments -> flexible and adaptive implementation• NCS connects asynchronous nodes -> avoid useless synchronizations• cheap off-the-shelf hardware/software components

⇒ Need for a co-design approach

• holistic view : process + algorithm + execution resources• trade off : performance/robustness/energy/costs. . .


Control/Computing Control and Scheduling Integration

Pioneering works 1999/2000

Eker, Hagander and Arzen, 2000

control of n identical pendulums under CPU power constraint

n LQ controllers, period hi, exec. time Ci, single CPU

Cost function J(h) =1h

∫ h

0

(xT (t) uT (t)

)Q(

x(t)u(t)

)dt

Problem: minhi

J =n

∑i=1

Ji(hi) under constraintn

∑i=1

Ci/hi ≤Ud

theoretic feedback-scheduler : h(k+1) = f (∆c(k),Ure f (k))on-line solving of Riccati and Lyapunov equationstoo costly for real-time implementation. . .

. . . although quite simple : LTI process, no delay, no jitter, no data loss, no uncertainty


Control/Computing Computing-aware control

New perspective and recent works: Control & Computing Co-Design

Benefits of feedback• adaptive w.r.t. operating conditions• robustness w.r.t. process modeling uncertainties• robustness w.r.t. timing uncertainties

Control under computing constraints

−

Uk+

Process state estimates

Y

RTOS

Scheduler

Process objectives

(QoC)

Scheduling

Scheduling Policy

WCET

(priority, period)Parameters

SAMP

Controller

ProcessProcessZOH

• reinforced specific robustness issues• delay margin, jitter margin,. . .• time-delay systems : varying, bounded. . .

• adaptive control rate• effective actuator for CPU load/bandwidth• optimal period selection, gain scheduling. . .• stability during interval switching

see... Stankovic et al., 1999; Cervin, 2003; Sename et al., 2003; Sala, 2005...



Computer load aware control

Method : Design of a controller• Which period may vary in [hmin, ...,hmax]

• Preserving of the stability during timing change• Meeting a specified performance

Possible solutions :Bank of n correctors for n periods :• Tedious when n large• a posteriori study of the transitions between controllers• Cost of the implementation

Ojective :

Synthesis of a variable sampling rate controller using a Linear Parameters Varying (LPV) method.Robert, 2007; Robert et al., 2006, 2007 & 2010



LPV/H∞ with varying intervals

ctrl2ctrl3

CPU activity hcontrolinterval

Controlled output

reference

ProcessController

sensor schedule Controller

Process

SchedulingX(k+2)

X(k) X(k+1)

t

k

U[x(k)]

U[x(k+1)]

state

control

k+1 k+2

h(k+1)h(k)

X(k+1) = A(h(k)) X(k) + B(h(k)) U(k)

• LPV where h is the varying parameter• CPU load adaptation using varying control intervals• accommodation with scheduled sensors• stability whatever h change h ∈ [hmin,hmax]

• specified performance level



IEEE TCST Robert et al., 2010

Polytopic approach:• h as the single varying parameter• Taylor’s expansion of matrix exponentials• H∞ with sampling dependent performance templates

Application to a T-inverted pendulum:

0 10 20 30 40 50 60 70 80 90−0.5

0

0.5

θ [r

ad]

Pendulum angle

0 10 20 30 40 50 60 70 80 90−5

0

5

u []

Control input

0 10 20 30 40 50 60 70 80 900

1

2

3

4x 10

−3

Time [s]

h [s

]

h

u

rθ

0 5 10 15−0.5

0

0.5

θ [r

ad]

Pendulum angle

rθ

0 5 10 15−4

−2

0

2Control input

u []

0 5 10 151

2

3x 10

−3 Control interval

h [s

]

Time [s]

7.4 7.42 7.44 7.46 7.48 7.5 7.52 7.54 7.56 7.58 7.6−0.5

0

0.5

θ [

rad]

Pendulum angle

rθ

7.4 7.42 7.44 7.46 7.48 7.5 7.52 7.54 7.56 7.58 7.6−5

0

5Control input

u [

]

7.4 7.42 7.44 7.46 7.48 7.5 7.52 7.54 7.56 7.58 7.61

2

3x 10

−3 Control interval

h [

s]

Time [s]



This talk - AUV Motivation

Objective : to control the altitude z of an AUVMeasurement of z by an ultrasonic sensor : induced delay in the measurement interval→ variablesampling

sensorcontrollerreference output

variable sampling

system

Sensor acting on the sampling period of the controller

• Development of both main approaches for LPV control design: the polytopic approach andthe LFR one.

• Application to AUVs: a LPV model to account for some non linearities. Design, of LPVcontrollers with two set of parameters, representing either the sampling interval or the modelnon linearities.

Roche et al., 2010, 2011


A polytopic approach Objective

Discrete polytopic modeling with variablesampling



General discrete polytopic system

{xk+1 = A(θk)xk +B(θk)ukyk =C(θk)xk +D(θk)uk

• A(θk), B(θk), C(θk) et D(θk) affine in θk

• θk evolves inside a polytope Θ defined by vertices ω1, . . . ,ωN

θk ∈ Co{ω1, . . . ,ωN} (1)

• θk: convex combination of vertices ωi

θk =N

∑i=1

αi(k)ω1, αi(k)≥ 0,N

∑i=1

αi(k) = 1 (2)



From the continuous-time state space representation of the system :

G :

{x = Ax+Bu

y =Cx+Du(3)

The exact discretization of this system is :

Gd :

{xk+1 = Ad(h)xk +Bd(h)uk

yk =Cdxk +Dduk(4)

with Ad(h) ∈Rns×ns and

Ad(h) = eAh Bd(h) =∫ h

0eAτ dτB

Cd =C Dd = D(5)

Objective : schedule the discrete-time system with the variation of sampling interval


A polytopic approach Discrete polytopic LPV modeling with variable sampling

A polytopic model

Usual numerical method :(

Ad(h) Bd(h)0 I

)= exp(M×h) = exp

((A B0 0

)h)

(6)

With h ∈ [hmin,hmax] and defined by : h = h0 +δ

(Ad(h) Bd(h)

0 I

)= eMh0 eMδ =

(Ah0 Bh00 I

)(Aδ Bδ

0 I

)(7)

Finally the matrices of the discretized plant are :

Ad = Ah0 Aδ

Bd = Bh0 +Ah0 Bδ

(8)

Aδ and Bδ can be approximated by Taylor series expansion:

Aδ ≈ I +k

∑i=1

Ai

i!δ

i Bδ ≈k

∑i=1

Ai−1Bi!

δi (9)

The obtained model (combination 8 - 9) is affine in δ which is bounded (h ∈ [hmin,hmax])⇒ polytopic model scheduled by δ


A polytopic approach Discrete polytopic LPV modeling with variable sampling

Polytopic controller synthesis using LMIs


A polytopic approach A polytopic method with reduced complexity

Recall : LPV design methods

LPV system : discrete time system with θk ∈Θ⊂ Rn vector of varying parameters :

{xk+1 = A(θk)xk +B(θk)uk

yk =C(θk)xk +D(θk)uk

zw

K(z, δ)

P (z, δ)

u yδ

LPV method : synthesis of a controller parametrised by θk

[Apkarian et Gahinet (1995)]

• Consider the period h as the varying parameter;• Continuous linear system⇒ discrete time LPV model;• Synthesis of a h-dependent controller.



Recall : Polytopic model

Discrete time LPV polytopic model :{

xk+1 = A(θk)xk +B(θk)uk

yk =C(θk)xk +D(θk)uk

with :

• A(θk), B(θk), C(θk), D(θk) affine en θk ∈Θ

• Θ is a polytope with k vertices ωi

• ωi = θ mink ,θ max

k

• θk = ∑ki=1 αi(k)ωi with αi(k)≥ 0, ∑

ki=1 αi(k) = 1

(A(θk) B(θk)C(θk) D(θk)

)=

k

∑i=1

αi(k)(

Ai BiCi Di

)



Parametrised discretization

Continuous time system :{

x = Ax+Buy = Cx+Du

Discrete time system :{

xk+1 = Ad(h) xk +Bd(h)ukyk = Cd xk +Dd uk

withAd(h) = Ah0 Aδ

Bd(h) = Bh0 +Ah0 Bδ

h = h0 +δ

Aδ = eAδ

Affine form (via a Taylor’s expansion) in δ i, i = 1 . . .N :

Ad(h) = Ah0 Aδ = Ah0 eAδ ≈ Ah0 (N

∑i=1

Ai

i!δ

i)

Bd(h) = Bh0 +Ah0 Bδ = Bh0 +Ah0 (N

∑i=1

Ai−1Bi!

δi)

with h = h0 +δ

N parameters⇒ 2N vertices!



Approximation error

Study of the order N for the series :• Error criterion JM(h) = ‖Gd(h,z)− GdN (h,z)‖∞

• Example on a stable 2nd order system

0.04

0.06

0.08

0.1

0.12 1

2

3

4

5

−300

−250

−200

−150

−100

−50

0

M

Erreur d’approximation, h0 = h

min

h (s)

Err

eur

J (d

B)



Polytope size reduction

Transform into a polytopic model with :

• θk = (δ ,δ 2, . . . ,δ N)T

• State matrices affine in θk

Exploiting the parameters dependency to reduce the polytope :

N=2

0 δh

δh

δ2h

0

δ2h

0 δh

δh

δ2h

0

δ2h

0 δh

δh

δ2h

0

δ2h �

��

��

ω1 ω2

ω3

Θ



Polytope size reduction: general case

h = hmin +δ , 0≤ δ ≤ δmax, δmax = hmax−hmin

(0,0,0, . . . ,0)

(δmax,0,0, . . . ,0)

(δmax,δ2max,0, . . . ,0) (10)

. . .

(δmax,δ2max,δ

3max, . . . ,δ

Nmax)

N +1 vertices rather than 2N !

k influences the conservatism and complexity• of the synthesis (2k + 1) LMIs, k = N +1 or 2N

• of the real-time implementation : convex combination of k controllers


A polytopic approach LPV polytopic control design method

Recall : Polytopic LPV controller

Apkarian et al., 1995; Scherer and Wieland, 2004:Synthesis of a controller K which guarantees :• Internal stability of the closed loop• ‖z‖2 < γ‖w‖2

for all trajectory h in H

zw

K(z, δ)

P (z, δ)

u yδ

Polytopic controller :

K(H) :

{xk+1 = AK(H)xk +BK(H)yk

uk =CK(H)xk +DK(H)yk

with : (AK(H) BK(H)CK(H) DK(H)

)=

N

∑i=1

αi(k)(

AKi BKiCKi DKi

)



Performance specification

• H∞ framework• find a controller K such internal stability is achieved• ‖z‖2 < γ‖w‖2, where γ is the H∞ attenuation level

HWoWi

K

z

y

w

u

zw

P

• closed-loop performance strongly depends on the sampling period→ make the performancespecification vary with the period

• weighting functions are sampling rate dependent Wi(h), Wo(h)• smaller the period, faster the system : ωS = α/h



H∞ design

Kθ G

We Wu

-+

θref ε

β1β2

y

Structure chosen for the control design

• Wu constant: to account for actuators limitations• We: weight on the tracking error

Choice of sampling dependent weighting function (1st order, continuous) :

We(s, f ) =s

Ms +ωs( f )s+ εsωs( f )

(11)



discretization of We

Parameter dependent weighting function :

We(s, f ) :

{x = (a× f )x+(a× f −b× f )u

y = x+u(12)

Discretisation→ constant discrete weight :

Wed :

{xk+1 = Adxk +Bdu

y = x+u(13)

with

Ad = ea f h = ea

Bd = (a f )−1(Ad − I)b f = a−1(Ad − I)b

Simplification between h and f → discrete LTI weight



Generalized plant P(z,δ ): gathering polytopic model Gd(δ ), and weighting function (Wu and Wed )

zw

K(z, δ)

P (z, δ)

u yδ

Figure: General control configuration

Controller computation: tools derived from the bounded real lemma applied to LPV polytopicsystems



Discrete LPV synthesis

• Solving (2N + 1) LMI to synthesise k fixed gains controllers vertex controllers (off-line)• On-line convex combination of the vertex controllers giving :

K(h) = K(α1, . . .αk) =N

∑j=1

α jK j

• αi(h) polytopic coordinates (explicit recursive solution for the reduced polytope)• size of K(h) = size of K j



LMIs to solve

(NR 0

0 I

)T

AiRATi −R AiRCT

1i B1iC1iRAT

i −γI +C1iRCT1i D11i

BT1i DT

11i −γI

(

NR 00 I

)

< 0, i = 1 . . .2n (14)

(NS 0

0 I

)T

ATi SAi −S AT

i SB1i CT1i

BT1iSAi −γI +BT

1iSB1i DT11i

C1i D11i −γI

(

NS 00 I

)

< 0, i = 1 . . .2n (15)(

R II S

)≥ 0 (16)

NR and NS bases of null space of (BT2 , DT

12) and (C2, D21) respectivelyTo guaranty stability and performances: same R, S and γ at each vertex of the polytopeMatrices of the controller derived from R and S


An LFT/LPV approach

LFT (Linear Fractional transformation) model forvarying sampling period systems


An LFT/LPV approach Introduction and Objective

Objective

• Taking the sampling interval into account• Formulation of a discretized model

Tool : Coming from Robust control theoryFirst step : LFT transformation of the system :

N(z)

∆(k)

yu

y∆u∆

System under LFT representation

∆ : "uncertainties" depending on the sampling interval hAim : gain scheduling design w.r.t the sampling interval


An LFT/LPV approach Introduction and Objective

From the continuous-time state space representation of the system :

G :

{x = Ax+Bu

y =Cx+Du(17)

The exact discretization of this system is :

Gd :

{xk+1 = Ad(h)xk +Bd(h)uk

yk =Cdxk +Dduk(18)

(Ad(h) Bd(h)

0 I

)=

(Ah0 Bh00 I

)(Aδ Bδ

0 I

)(19)

Finally the matrices of the discretized plant are :

Ad = Ah0 Aδ

Bd = Bh0 +Ah0 Bδ

(20)

with Aδ and Bδ obtained by Taylor series expansion :

Aδ ≈ I +k

∑i=1

Ai

i!δ

i Bδ ≈k

∑i=1

Ai−1Bi!

δi (21)


An LFT/LPV approach A new LFT Formulation for varying sampling systems

A new LFT Formulation for varying sampling systems

First, an LPV/LFT model of a system, considering the sampling period as varying parameter ispresented, as in [Robert,2007].Objective: transform the system under the LFT form where ∆ represents the uncertaintiesdepending on the sampling period h.

N(z)

∆(k)

yu

y∆u∆

Figure: System under LFT representation

Variation w.r.t the nominal system→ 2 unstructured uncertainties :

Ad(h)−Ah0 = Ah0 (Aδ − I)︸︷︷︸

∆1

, Bd(h)−Bh0 = Ah0 Bδ = ∆2 (22)



The matrices of the discrete time system (38) are rewritten as :

xk+1 = [Ah0 +Ah0 (Aδ − I︸︷︷︸∆1

)]xk +[Bh0 +Ah0 Bδ︸︷︷︸∆2

]u

y =Cdxk +Ddu

The system is transformed into the form described in Figure 2 by defining :

u∆ = [u∆1 , u∆2 ]T , y∆ = [y∆1 , y∆2 ]

T and ∆ =

(∆1 00 ∆2

)

Finally, the LFT form of the model with unstructured uncertainties (with no approximation) is asfollows :

xk+1 = A xk +B

(u∆

u

)

(y∆

y

)= C xk +D

(u∆

u

) ∆ =

(∆1 00 ∆2

)(23)

A = Ah0 B =(Ah0 I Bh0

),C =

I0

Cd

D =

0 0 00 0 I0 0 Dd



The LPV/LFT model of the system is then given by the upper LFT interconnection

Tu,y = Fu(P(z),∆) = P22 +P21∆(I−P11∆)−1P12 (24)

This approach considers ∆ as a full block diagonal with full matrices ∆1 and ∆2. However matrices∆1 and ∆2 depend on a single parameter δ , so this uncertainty definition may be too conservative.



Structured uncertainty case

In order to reduce the conservatism, the Taylor series expansion of order k for matrices Aδ and Bδ

is used to obtain a structured uncertainty matrix.

∆1 ≈ Aδ − I =k

∑i=1

Ai

i!δ

i

≈ Aδ (I +Aδ

2(I +

Aδ

3(I + . . .)) (25)

∆2 ≈ Ah0 Bδ = Ah0

k

∑i=1

Ai−1Bi!

δi

≈ Ah0 (I +Aδ

2(I +

Aδ

3(I + . . .)B δ (26)

Polynomial dependence on δ → structured uncertainties (reduce the conservatism)



Structured uncertainty case

With this structure the LFT representation of the system becomes :

xk+1 = A xk +B

(u∆

u

)

(y∆

y

)= C xk +D

(u∆

u

) , ∆ = δ I2×k×ns ,A = Ah0

with B =(B∆ Bh0

), B∆ =

(AaA 0ns∗[ns∗(k−1)] Aa 0ns∗[ns∗(k−1)]

)

C =

(C∆Cd

), C∆ =

(C1C2

)withC1 =

Ins...

.

.

.Ins

k times ; C2 =

0ns...

.

.

.0ns

k times

D =

(D D20 Dd

), D =

(D 00 D

)with D =

0nsA2 . . . 0ns

.

.

.. . .

. . ....

.

.

.. . .

. . . Ak

0ns . . . . . . 0ns

D2 =

0B

.

.

.B

This allows to express the parameter block ∆ as a direct dependence on the varying samplingperiod δ .



A new LFT representation

∆ = δ I2×k×ns

A = Ah0 B =(B1 B2 Bh0

)(27)

C =

C1C2Cd

D =

D 0 00 D D2u0 0 Dd

(28)


An LFT/LPV approach H∞ /LFT control design

H∞/LFT control design



H∞ / LFT control design

Objective : LPV controller function of the set of parameter ∆ that ensures internal stability and‖Tzw‖∞

≤ γ∞, withTzw = Fl(K,∆) := K11 +K12K(I−K22∆)−1K21 (29)

The LFT control scheme is presented in Figure 41, where the dependency on the parameters ∆ ofthe generalized plant P and of the controller K is described in an LFT form.

z∆w∆

z

y

w

u

z∆ w∆

∆(k)

P (z)

K(z)

∆(k)

LPV process

LPV controller

∆(t)zw

∆(k) 00 ∆(k)

P (z)

K(z)

P ′(z)



Problem statement

Kθ G

We Wu

-+

θref ε

β1β2

y

Structure chosen for the control design

• Wu = I2 (normalized)• We : weight on the tracking error

Choice of sampling dependent weighting function (1st order, continuous) :

We(s, f ) =s

Ms +ωs( f )s+ εsωs( f )

(30)



discretization of We

Parameter dependent weighting function :

We(s, f ) :

{x = (a× f )x+(a× f −b× f )u

y = x+u(31)

Discretisation→ constant discrete weight :

Wed :

{xk+1 = Adxk +Bdu

y = x+u(32)

with

Ad = ea f h = ea

Bd = (a f )−1(Ad − I)b f = a−1(Ad − I)b

zw

∆(k) 00 ∆(k)

P ′(z)

K(z)

Wed -WedGd

0 Wu

I Gd

Simplification between h and f → LTI system




The problem is now to find a discrete controller K for which the system described in Figure 41 isstable and respect some specifications defined by weighting functions set in the control loop(H∞ design).The methodology used here for the discrete-time controller synthesis is based on theH∞ framework, by the resolution of LMIs (derived from the bounded real Lemma), as described inApkarian and Gahinet, 1995 and Gauthier, 2007.To reduce the conservatism due to the presence of uncertainties, the problem is reformulatedusing some scaling variables L∆.




The solution of this problem relies on the bounded real Lemma, as follow :Lemma Consider a parameter structure Θ, the associated scaling setLΘ = {L > 0 : L∆ = ∆L,∀∆ ∈Θ}, and a discrete-time square transfer function T (z) of realizationT (z) = Dcl +Ccl (zI−Acl)

−1 Bcl , then the following statements are equivalent.i) Acl is stable and there exists L ∈ LΘ such that ||L1/2

(Dcl +Ccl (zI−Acl)

−1 Bcl

)L−1/2||∞ < 1.

ii) There exist positive definite solutions Xcl and L ∈ LΘ to the matrix inequality

−X−1cl Acl Bcl 0

ATcl −Xcl 0 CT

clBT

cl 0 −L DTcl

0 Ccl Dcl −L−1

< 0 (33)




Theorem

Consider a discrete-time LPV/LFT plant P∆(z). Let Nr and Ns denote bases of null spaces of(BT

2 ,DTθ2,D

T12,0) and (C2,D2θ ,D21,0) . The gain-scheduled H∞ LFT control problem is solvable if

and only if there exist pairs of symmetric matrices (R,S) ∈ Rna×na , (L3,J3) ∈ Rnθ×nθ and γ > 0 s.t

N Tr

ARAT −R+Bθ J3BTθ

?

Cθ RAT +Dθθ J3BTθ

Cθ RCTθ+Dθθ J3DT

θθ− J3

C1RAT +D1θ J3BTθ

C1RCTθ+D1θ J3DT

θθ

BT1 DT

θ1

? ?? ?

C1RCT1 +D1θ J3DT

1θ− γI ?

DT11 −γI

Nr < 0

Ns

AT SA−S+CTθ

L3Cθ ?

BTθ

SA+DTθθ

L3Cθ BTθ

SBθ +DTθθ

L3Dθθ −L3BT

1 SA+DTθ1L3Cθ BT

1 SBθ +DTθ1L3Dθθ

C1 D1θ

? ?? ?

BT1 SB1 +DT

θ1L3Dθ1 − γI ?

D11 −γI

Ns < 0

(R II S

)> 0

L3Θ = ΘL3, J3Θ = ΘJ3,(

L3 II J3

)> 0

This LMI is then solved using Yalmip interface (Loftberg, 2004) and Sedumi solver (SEDUMI,1999).


An LFT/LPV approach Extension to LPV systems

Extension to LPV models

Σ(ρ(.)) :

xzy

=

A(ρ(.)) B1(ρ(.)) B2(ρ(.))C1(ρ(.)) D11(ρ(.)) D12(ρ(.))C2(ρ(.)) D21(ρ(.)) D22(ρ(.))

xwu

(34)

ρ(.) is a varying parameter vector that takes values in the parameter space Pρ such that:

Pρ :={

ρ(.) :=[

ρ1(.) . . . ρn(.)]T ∈Rn

ρi(.) ∈[

ρi

ρ i]∀i = 1, . . . ,n

}

where n is the number of varying parameters. Pρ is a convex set.



LPV to LFR model

z∆w∆

zw P

∆

u y

Figure: System under LFT form

The equation of the system under LFR representation is asfollows:

xz∆

zy

=

A B∆ B1 B2C∆ D∆∆ D∆1 D∆2C1 D1∆ D11 D12C2 D2∆ D21 D22

xw∆

wu

(35)



Parameterized discretisation

The LFR model in figure 3 can be discretised, assuming that the uncertain input/outputs (w∆ andz∆) are sampled and hold at the sampling frequency. The previous methodology proposed canthen be applied in that context to get a sampling dependent LFR discrete-time model.Let us consider the continuous LTI part of the LFR given in equation 35, and define:

A = A B = [B∆ B1 B2]

C =

C∆

C1C2

D =

D∆∆ D∆1 D∆2D1∆ D11 D12D2∆ D21 D22

(36)

The inputs and outputs are gathered in a single vector as:

u =

w∆

wu

and y =

z∆

zy

(37)



Parameterized discretisation

The exact discretisation of the LTI system (A,B,C,D) is described in equation (38)

xk+1 = Adxk +Bd uk

yk =Cdxk +Dd uk(38)

with:

Ad = eAh Bd =∫ h

0eAτ dτB

Cd = C Dd = D(39)

Matrices Ad and Bd depend both on the constant part h0 and on the varying part δ of the samplingperiod:

Ad = Ah0 Aδ (40)

Bd = Bh0 +Ah0 Bδ (41)



Following the previous method with Taylor expansion ...

Pd :

xk+1 = A xk +B

w∆

uδ

u

z∆

yδ

y

= C xk +D

u∆

uδ

u

(42)

with:

Θ =

[∆ 00 δ I2×k×ns

](43)

A = Ah0 B =(B1 B2 Bh0

)(44)

C =

C1C2Cd

D =

D 0 00 D D2u0 0 Dd

(45)

where k is the order of the Taylor serie expansion in δ and ns is the number of states.



z∆w∆

z

y

w

u

z∆ w∆

LPV process

LPV controller

∆ 00 δ2kns

Pd

K

∆ 00 δ2kns

Wi Wo

Figure: LFR control structure

The dependence between the discrete-time LTIplant Pd , the controller K, the parameter block(including the system and sampling parameters)and the weighting functions Wi and Wo is pre-sented in figure 4.

Finally, the LFR controller is computed using the previous Theorem.


Application to AUVs

AUV models


Application to AUVs LTI models

Description of the AUV

c© Copyright Ifremer

YX

Z

φ

ψ

θ

propellerone horizontal fin : δ

2 inclined fins : β2 and β′2

2 fins : β1 and β′1

12 state variables :• 6 positions : η = [x y z φ θ ψ]T

• 6 speeds : ν = [u v w p q r]T

6 control inputs : β1, β ′1, δ , β2, β ′2 and Qc



Nonlinear model

Dynamical equations representing the physical system :

Mν = G(ν)ν +D(ν)ν +Γg +Γu (46)

η = Jc(η2)ν (47)

See Fossen, 1994)

• M the inertial matrix• G(ν) Coriolis and centrifugal forces• D(ν) hydrodynamics damping coefficients• Γg gravity effort and hydrostatic forces• Γu forces and moments due to vehicle actuators

• Jc(η2) change of referential



Linearized and reduced model

Tangential linearization (around equilibrium point : νe = [ueq, 0, 0, 0, 0, 0]T because constantcruising speed needed for the cartography) yields to :

{x = Ax(t)+Bu(t)y =Cx(t)+Du(t) (48)

Motion in the vertical plane⇒ model reduction :• 4 state variables [z, w, θ , q]T

• 2 control inputs [β1, β2]T (pair of fins actuated together : β1 = β ′1 and β2 = β ′2)


Application to AUVs LPV models

Towards a pitch oriented LPV model

Objective: keep θ as a varying parameter.Considered NL model with 2 state variables (the pitch angle θ an d velocity q):

θ =cos(φ)q− sin(φ)r

Mq =− p× r(Ix− Iz)−m[Zg(q×w− r× v)]− (Zqm−Z f µV )gsin(θ)

− (Xqm−X f µV )gcos(θ)cos(φ)+Mwqw|q|+Mqqq|q|+Ff ins (49)

where M interia matrix, m AUV mass, V volume and µ mass density. The other parameters(Ix, Iz,Zg,Zq,Z f ,Xq,X f ,Mwq) appear in the dynamical and hydrodynamical functions.Ff ins : forces and moments due to the fins.Tangent linearization around Xeq = [0 ueq 0 0 0 0 0 0 θeq 0 0 0]:

˙θ = q

M ˙q = [−(Zgm−Z f µV )gcos(θeq)+(Xgm−X f µV )gsin(θeq)]θ +Ff inseq

where θ and q are the variations of θ and q.



A LFR model

An LFR model of the vehicle described in the form:

z∆w∆

zkwk P (z)

∆(ρ)

uk yk

The ∆ block contains the varying part of the model, which depends on the linearization point (θeq).The LFR form is defined by:

xk+1 = A xk +[

B∆ B1](u∆

u

)

(y∆

y

)=

[C∆

C1

]xk +

[D∆ 00 D1

](u∆

u

) (50)



A LFR model

with

A =

[0 10 0

], B∆ =

[0 0

−(Zqm−Z f ρV )g (Xqm−X f ρV )g

]

B1 =

[0

Ff ins

], C∆ =

[1 10 0

], C1 =

[1 00 1

]

D∆ =

[0 00 0

], D1 =

[0 00 0

](51)

z∆ =

[∆θ

∆θ

]; u∆ = ∆z∆; ∆ =

[ρ1 00 ρ2

](52)

with: {ρ1 = cos(θeq)ρ2 = sin(θeq)

(53)



A LFR model

Following the previous described method, an LFR discrete-time model can be obtained,depending on ρ1, ρ2, and on the sampling period h as well:

z

y

w

u

δIkns 0 0 00 δIkns

0 00 0 ρ1 00 0 0 ρ2

P

wΘ =

wδ1wδ2wρ1wρ2

zδ1zδ2zρ1zρ2

= zΘ


Application to AUVs AUV control

AUV control


Application to AUVs General control scheme

Simulation scenario

Ku

KθKz(h)θref

zref β1, β2∑

NL

z

u

θ+

+

+

−

−

−

uref Qc

Figure: Simulation scheme

• Complete non-linear model of the AUV• Follow the sea bottom at constant speed (Ku is an LTI discrete-time H∞ controller with

h = 0.1 sec.)• Sinusoïdal variation of the sampling interval for the altitude ∈ [0.05, 0.3]s• Altitude reference: filtered step of 5m


Application to AUVs General control scheme

Depth cascade varying sampling control structure

Decoupling of the altitude and the pitch angle

∑NLKθKz(h)

θrefzref β1, β2

θ

z

Figure: Depth cascade control structure

Synthesis of 2 controllers for the vertical axis:

Case 1: Pitch angle controller Kθ : based on the reduced linearised model of the AUV

Altitude controller Kz(h): based on a geometrical model, and scheduled by h

Case 2: Pitch angle controller Kθ (h,ρ): based on the LPV/LFR pitch model, withh ∈ [0.005, 0.03] sec.

Altitude controller Kz(h): LFR controller scheduled by h, with h ∈ [0.05, 0.3] sec.


Application to AUVs LPV polytopic sampling dependent controller for an LTI system (case 1)

Altitude control: an simple LPV model

Z

l

θ

~u

Figure: Relation z θ

z = l sinθ

u = l(54)

z ' uθ

Gz = zθ= 1

p(55)

Inner loop approximated by an integratorDiscrete LPV polytopic model derived from this geometrical relations:• Discretisation leads to: xk+1 = xk +h.uk with h ∈ [hmin,hmax]

• By considering h = h0 +δ ⇒ xk+1 = xk +(h0 +δ ).uk

⇒ LPV model in δ (easy to put in polytopic or LFR forms)



Altitude control: LPV polytopic synthesis (case 1)

Application of the methodology previously described

K G

We(h) Wu

-+

r ε y

Figure: Structure chosen for the control design

• We(s,h) =s

Ms +ω×hs+ω×h×ε

• Wu = 0.1

• sampling period h ∈ [hmin, hmax] = [0.05, 0.3]s, (h0 = hmin)• as shown before We(z) is a discrete-time LTI system.• Polytopic approach



Polytopic Sensitivity functions (case 1)

10-4

10-3

10-2

10-1

100

101

102

-140

-120

-100

-80

-60

-40

-20

0

20

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

weight We

h=0.03

h=0.05s

Figure: S sensitivity function

10-8

10-6

10-4

10-2

100

102

-30

-20

-10

0

10

20

30

40

50

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

h=0.3s

h=0.05s

Figure: Controller Bode diagram



Simulation results (case 1)

0 50 100 150 200 250 300 350 400 450 5000.05

0.1

0.15

0.2

0.25

0.3

Sampling period

0 50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

Longitudinal speed

0 50 100 150 200 250 300 350 400 450 500-1

0

1

2

3

4

5

6

reference

altitude

Altitude

0 50 100 150 200 250 300 350 400 450 500-0.2

-0.1

0

0.1

time(s)

pitc

h an

gle

(rad

)

pitch angle

reference

Pitch angle


Application to AUVs LPV/LFR varying sampling controller for a LPV model (case 2)

LFR altitude controller (case 2)

Using the previous simple LPV model, the LFR approach is considered with the followingperformance weights:• Wze (h) defined in continuous-time, and function of the sampling period:

Wze (h) =s

MSze+wSze h

s+wSze hεSze

(56)

where MSze = 2, wSze = 5 and εSze = 0.001. After discretization, the corresponding discrete-timeLTI weight is part of the generalized plant.

• Wzu = 5



LFR design for Kθ (h,ρ) (case 2)

The previous LFR model is used, that depends on the sampling δ , and on the system parameterscos(θeq) and sin(θeq).As previously the mixed sensitivity problem is considered with:• Wθu on the control inputs [β1, β2]:

Wθu =

[0.8 00 0.8

](57)

• Wθe on the tracking error θ −θre f , defined as previously:

Wθe (h) =

sMSθe

+wSθeh

s+wSθehεSθe

(58)

where MSθe= 2, wSθe

= 10, εSθe= 0.01 and h varies in [h, h] = [0.05, 0.3]

leads to γopt = 38



Simulation results (case 2)

0 50 100 150 200 250 300 350 400 450 5000.05

0.1

0.15

0.2

0.25

0.3

Sampling period0 50 100 150 200 250 300 350 400 450 500

0

0.2

0.4

0.6

0.8

1

1.2

Longitudinal speed

0 50 100 150 200 250 300 350 400 450 500-1

0

1

2

3

4

5

6

zref

et z

Altitude

0 50 100 150 200 250 300 350 400 450 500-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Pitch angle



0 50 100 150 200 250 300 350 400 450 500-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Control Inputs

0 50 100 150 200 250 300 350 400 450 500-0.2

0

0.2

0.4

0.6

0.8

1

1.2

cos(θeq

)

hsin(θ

eq)

Parameters variations

• Good tracking of zre f and θre f with fast responses.• Longitudinal speed: almost always equals 1m/s with light oscillations during the altitude

change (t = 210s). The coupling between the state variables is mitigated.• The control inputs are saturated when the altitude reference is changing (at 10 and 210

secondes), but are becoming quickly normal. The remaining oscillations could be attenuatedto improve the energy efficiency.


Conclusion Remarks on LPV tools

Interest of the LPV framework

• Sampling dependent discrete-time model• Use of affine and LFT models• Performance adaptation function of the sampling period (available CPU resources)• A single framework when the system is non linear and represented as a LPV model.• Brings an interesting methodology for the design of H∞ controllers dor non linear systems,

under computational constraints


Conclusion Future works

Perspectives

• Reduce the conservatism of the method to improve the problem solvability for complexsystem. Use of other multipliers for the LFT method, parameter-dependent Lyapunov functionfor the polytopic approach

• To be consider: controller with reduced complexity in order to cope with other computationalrequirements: low order, stable controller, quantification errors....

•



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