Grenoble INP / Gipsa-labo.sename/docs/LPVtheory.pdf · 2021. 1. 26. · Grenoble INP / Gipsa-lab Master Course - January 2021 Olivier Sename (Grenoble INP / Gipsa-lab) LPV theory

Linear Parameter Varying systems: from modelling to control

Olivier Sename

Grenoble INP / Gipsa-lab

Master Course - January 2021

Olivier Sename (Grenoble INP / Gipsa-lab) LPV theory Master Course - January 2021 1/81

1. What is a Linear Parameter Varying systems?

2. Classes (models) of LPV systems

3. How to approximate a nonlinear system by an LPV one ?

4. Identification of LPV systems

5. Some properties of LPV systems

6. Stability of LPV systems

7. LPV Control designThe Static State feedback caseThe Dynamic Output feedback case

8. LPV observer designLPV polytopic observer with H∞ performance methodLPV polytopic observer with pole placement method

9. Summary of LPV approach interests

10. Conclusion

O. Sename [GIPSA-lab] 2/81

Grenoble’s studies on LPV systems and approaches


Former and current PhD students on LPV approaches

A. Zin, D. Robert, C. Gauthier, C. Briat, C. Poussot-Vassal, S. Aubouet, E. Roche, D. Hernandez,J. Lozoya, A-L Do, M. Rivas, S. Fergani, J-C Tudon, N. Nwesaty, M-Q Nguyen, D. Hernandez, K.Yamamoto, V-T Vu, D. Dubuc, T-P Pham M. Menezes, D. Kapsalis, H. Atoui, A. Medero

Complex systems

• Non linear models as qLPV ones• Account for various operating conditions

using a variable "equilibrium point":• Modelling of several constraints:

actuator, computation...• LPV Time-Delay Systems

Integration with Fault Diagnosis

LPV Adaptive Fault-scheduling Tolerant Control

LPV control = adaptation

• Real-time performance adaptation usingparameter dependent weighting functions

• Control allocation of MIMO systemsthrough a parameter for the controlactivation (of each actuator)

• Use parameter for monitoring and controlreconfiguration

LPV observer

State, fault and parameter estimation

Applications

Engine, Vehicle Dynamics, Electric Power Steering, Autonomous Underwater Vehicle,

Fuel Cell, Electrical vehicle

What is a Linear Parameter Varying systems?

LPV systems

Definition of an Linear Parameter Varying system

Σ(ρ) :

xzy

=

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

xwu

x(t) ∈Rn, ...., ρ = (ρ1(t),ρ2(t), . . . ,ρN(t)) ∈Ω, is a vector of time-varying parameters (Ω convex set),assumed to be known ∀t



LPV systems


Σ(ρ) :

xzy

=


xwu


System (ρ)

ExogeneousInputs

Controlledoutputs

Measuredoutputs

z

ControlInputs

w

u y

(Scherer, ACC Tutorial 2012)

10/60

Ma

them

ati

cal

Sys

tem

sT

heo

ryExample

Dampened mass-spring system:

p+ c _p+ k(t) p = u; y = x

First-order state-space representation:

d

dt

0@ p

_p

1A =

0@ 0 1

k(t) c

1A0@ p

_p

1A+

0@ 0

1

1Au;

y =1 0

0@ p

_p

1A

Only parameter is k(t)

System matrix depends affinely on this parameter

Could view c as another parameter - keep it simple for now ...



LPV systems


Σ(ρ) :

xzy

=


xwu


The frozen Bode plotsfor c = 1 and k ∈ [1,3]

(Scherer, ACC Tutorial 2012)

10/60

Ma

them

ati

cal

Sys

tem

sT

heo

ryExample

Dampened mass-spring system:

p+ c _p+ k(t) p = u; y = x

First-order state-space representation:

d

dt

0@ p

_p

1A =

0@ 0 1

k(t) c

1A0@ p

_p

1A+

0@ 0

1

1Au;

y =1 0

0@ p

_p

1A

Only parameter is k(t)

System matrix depends affinely on this parameter

Could view c as another parameter - keep it simple for now ...



LPV systems (2)

Let the LPV system be:

Σ(ρ) :

xzy

=


xwu

x(t) ∈ Rn, ...., ρ = (ρ1(t),ρ2(t), . . . ,ρN(t)) ∈Uρ , is a vector of time-varying parameters (Uρ convexset)• ρ(.) varies in the set of continuously differentiable parameter curves ρ : [0,∞)→ RN .

It is assumed to be known or measurable.• The parameters ρ are always assumed to be bounded:

ρ ∈Uρ ⊂ RN and Uρ compact (1)

defined by the minimal ρi, and maximal ρi values of ρi(t)

ρi(t) ∈ [ρi, ρi], ∀i

• The system matrices A(.) .... are continuous on Uρ



LPV systems (3): about the parameters

• Parameters are exogenous if they are external variables. The system is in that case nonstationary.See the previous damped mass-spring system.

• Parameters are endogenous if they are function of the state variables, ρ = ρ(x(t), t), and, inthat case, the LPV system is referred to as a quasi-LPV system.This case is encountered when approximating Nonlinear systems.For instance:

x(t) = x2(t) = ρ(t)x(t)

with ρ(t) = x(t).• It is sometimes required that the derivative of the parameters are bounded, i.e:

ρ ∈Uρ ⊂ RN and Uρ compact (2)

defined by the minimal νi, and maximal νi values of ρi(t)

ρi(t) ∈ [νi, νi], ∀i

This corresponds to the case of slow varying parameters• Other representations can be considered if ρ is piecewise-constant, or varies in a finite set of

elements (ρ(t) ∈ 0,1 for switching systems)

Next, several classes of LPV models are presented, and some ways to go from one class toanother are given.



Some comments

• LPV systems can model uncertain systems (ρ fixed but unknown) or parameter-varyingmodels (ρ(t))

LPV=linear or nonlinear?

• What is often referred to as gain-scheduling control, corresponds to Jacobian linearisation ofthe nonlinear plant about a family of equilibrium points Shamma (90), Rugh & Shamma (2000)In terms of control design this means, linearisation around operating conditions, design (ateach operating points) of a LTI controller, and interpolation of the LTI controllers in betweenoperating conditions (often used in Aerospace and Automotive industries).Pros: Simplicity of design for a non linear systemCons: No a priori guarantee of stability nor robustness

• But: this differs from quasi-LPV representations where nonlinearities are hidden in someparameter descriptions (as seen later in the course)

LPV=LTV

• Theoretical analysis of LPV system properties (stability, controllability, observability), oftenfalls into the framework of LTV systems or of nonlinear ones (for quasi-LPV representations),see (Blanchini).



Discrete-time LPV systems

In discrete-time an LPV can be written asxk+1 = Ad(ρk)xk +Bd(ρk)uk

zk =Cd(ρk)xk +Dd(ρk)uk(3)

where the parameter vector ρk is defined for each time instant k.

ρk =[

ρk1 . . . ρkn

]T ∈ Rn , ρki ∈[

ρki

ρki

]∀i = 1, . . . ,n (4)

Example from (Heemels, Daafouz,Milleroix, 2010)

Ad(ρk) =

0.25 1 00 0.1 00 0 0.6+ρk

, B =

101

, C =(

1 0 2), D = 0

where ρk ∈ [0,0.5] with k ∈ N.

Remarks

Most of the course is dedicated to continuous-time systems. Many tools can be directly extendedto the discrete-time case.Pay attention to the discretization of A(ρ(t)): Ad(ρk) = eA(ρ(kTs))Ts (not affine in ρ)



Some references

Those not to be ignored

• Modelling, identification : (Bruzelius, Bamieh, Lovera, Toth) + 2011 TCST Special Issue on"Applied LPV modelling and identification"

• Control (Shamma, Apkarian & Gahinet, Adams, Packard, Beker, Seiler, Grigoriadis ...)• Stability, stabilization (Scherer, Wu, Blanchini ...)• Geometric analysis (Bokor & Balas)• Survey paper: Hoffmann & Werner, 2015• Fault tolerant control: special issues by

• Balas, 2012: in International Journal of Adaptive Control and Signal Processing• Casavola, Rodrigues & Theilliol, 2015: in International Journal of Robust and Nonlinear Control

Some recent books

• R. Toth, Modeling and identification of linear parameter-varying systems, Springer 2010• J. Mohammadpour, C. Scherer, (Eds), Control of Linear Parameter Varying Systems with

Applications, Springer-Verlag New York, 2012• O. Sename, P. Gaspar, J. Bokor (Eds), Robust Control and Linear Parameter Varying

Approaches: Application to Vehicle Dynamics, Springer, 2013


Classes (models) of LPV systems

Outline










10. Conclusion



Class 1: Affine parameter dependence

In this case, the system matrices are such that:

A(ρ) = A0 +A1ρ1 + ...+ANρN

This is the case of the damped mass-sping system:

p+ cp+ k(t)p = u

considering the state space representationx(t) = A(k(t))x(t)+Bu(t),

y(t) =Cx(t)+Du(t)(5)

with x(t) ∈ R2 and

A(k(t)) =(

0 1−k(t) −c

), B =

(01

), C =

(1 0

), D = 0

Denoting the varying parameter ρ(t) = k(t), we get:

A0 =

(0 10 −c

), A1 =

(0 0−1 0

)



Class 2: Polynomial parameter dependence


A(ρ) = A0 +A1ρ +A2ρ2 + ...+ASρ

S

Example 1

A famous case is the sampling-dependent discrete-time system representation (Robert et al,2010):

Gd :

xk+1 = Ad(h) xk +Bd(h)ukyk = C(h) xk +D uk

(6)

where Ad(h) = eAh (A begin the state matrix of the continuous-time system and h begin thesampling period) is approximated using Taylor expansion at order N, such as:

Ad(h)≈ I +N

∑i=1

Ai

i!hi

So defining ρ = h, we get:

Ad(ρ) = I +N

∑i=1

Ai

i!ρ

i



Class 3: Rational parameter dependence


A(ρ) = [An0 +An1ρn1 + ...+AnNρnN ][I +Ad1ρd1 + ...+AdNρdN ]−1

The bicycle model

To study lateral dynamics the so-called bicycle model considers the dynamics of the ya rate ψ andof the vehicle lateral velocity vy, as follows:

vy

ψ

=

−

C f +Cr

mvx−vx +

CrLr−C f L f

mvx

−L f C f +LrCr

Izvx−

L2f C f +L2

rCr

Izvx

vy

ψ

+

C f

m

L f C f

Iz

δ

δ being the controlled steering angle.Considering the vehicle longitudinal velocity v as a varying parameter, i.e ρ = vx, this model can berecast into an LPV system with rational dependency on ρ.



Class 4: Polytopic models

A polytopic system is represented as

Σ(ρ) =Z

∑k=1

αk(ρ)

[Ak BkCk Dk

], with

2N

∑k=1

αk(ρ) = 1 , αk(ρ)> 0

where[

Ak BkCk Dk

]are LTI systems.

This representation is often used to rewrite an affine LPV system. Indeed, assuming that theparameters are bounded (ρi ∈

[ρ

iρ i]), the vector of parameters evolves inside a polytope

represented by Z = 2N vertices ωi, as

ρ ∈ Coω1, . . . ,ωZ (7)

It is then written as the convex combination:

ρ =Z

∑i=1

αiωi, αi ≥ 0,Z

∑i=1

αi = 1 (8)

where the vertices are defined by a vector ωi = [νi1, . . . ,νiN ] where νi j equals ρ j or ρ j.

The LTI system[

Ak BkCk Dk

]here corresponds to the LPV system frozen at the vertex k.



Let consider the discrete-time system with 2 state variables (x1, x2) and 2 parameters (ρ1, ρ2)(x1

x2

)k+1

=

(x1 +ρ1x2

ρ2x1 + x2

)k=

[1 ρ1ρ2 1

]k

(x1

x2

)k

:= Ad(ρ1,2)

(x1

x2

)k

(9)

The parameters are bounded by[

ρ1,2

ρ1,2

], so the parameter space is a polytope (rectangle)

with 4 vertices defined as :

Pρ =(ρ

1,ρ

2),(ρ

1,ρ2),(ρ1,ρ2

),(ρ1,ρ2)

(10)

The polytopic coordinates are (αi) are computed from the values of ρ1 and ρ2 at time k :

ω1 = (ρ1,ρ

2), α

(k)1 =

(ρ1−ρ

(k)1

ρ1−ρ1

)×

(ρ2−ρ

(k)2

ρ2−ρ2

)

ω2 = (ρ1,ρ2), α

(k)2 =

(ρ1−ρ

(k)1

ρ1−ρ1

)×

(ρ(k)2 −ρ2

ρ2−ρ2

)

ω3 = (ρ1,ρ2), α

(k)3 =

(ρ(k)1 −ρ1

ρ1−ρ1

)×

(ρ2−ρ

(k)2

ρ2−ρ2

)

ω4 = (ρ1,ρ2), α(k)4 =

(ρ(k)1 −ρ1

ρ1−ρ1

)×

(ρ(k)2 −ρ2

ρ2−ρ2

)(11)

The LPV system is then rewritten under the polytopic representation:(A(ρ1,2) B(ρ1,2)C(ρ1,2) D(ρ1,2)

)= α

(k)1

(A(ω1) B(ω1)C(ω1) D(ω1)

)+α

(k)2

(A(ω2) B(ω2)C(ω2) D(ω2)

)+α

(k)3

(A(ω3) B(ω3)C(ω3) D(ω3)

)+α

(k)4

(A(ω4) B(ω4)C(ω4) D(ω4)

)(12)



Implementation issues

When a matrix M(ρ) is affinely dependent on ρ, it also belongs to a polytope defined as:

M(ρ) ∈M :=CoM1, . . . ,M2N (13)

where the vertices Mi = M(ωi). Therefore,

M(ρ) =2N

∑i=1

µi(ρ)Mi (14)

Here we will explain the interpolation procedure for polytopic LPV systems (more details in thePhD thesis of D. Dubuc) i.e:

1 How to compute the vertices Mi.2 How to compute the interpolation functions µi(ρ).

A systematic procedure is given, easily implementable for any number N of parameters.Let denote (bi

N ,biN−1, . . . ,b

i1), the binary representation with N bits, of the integer i. bi

1 is the leastsignificant bit and bi

N the most significant one. Then, each coordinate νk for all k ∈ 1, . . . ,N inωi = [νi1 . . . νiN ]

T can be computed as:

νk =

ρk when bi−1

k = 0

ρk when bi−1k = 1

(15)

Each vertex of M in (13) are then deduced as: Mi = M(ωi). The full procedure for the computationof the vertices is summarized in Algorithm 1.



Computation of the vertices

Figure: Matrix polytopic interpolation



Computation of the interpolation coefficients

The interpolation functions µi(ρ) are such that µi(ρ)≥ 0 and2N

∑i=1

µi(ρ) = 1, and are given by:

µi(ρ) =N

∏k=1

αikρk +βik

ρk−ρkwhere αik =

1 when bi−1

k = 0

−1 when bi−1k = 1

and βik =

−ρk when bi−1k = 0

ρk when bi−1k = 1

Figure: Matrix polytopic interpolation



Class 5: LFT (or LFR) models

z∆w∆

zw P

∆

u y

Figure: System under LFT form

∆ is defined such as: z∆ = ∆(.)w∆.It represents the parameter variations.

∆(.) is a linear function of theparameter vector.

The equation of the system under LFR representation is asfollows:

xz∆

zy

=

A B∆ B1 B2

C∆ D∆∆ D∆1 D∆2C1 D1∆ D11 D12C2 D2∆ D21 D22

xw∆

wu

Denoting the transfer matrix N(s) as:[

z∆

z

]=

[N11(s) N12(s)N21(s) N22(s)

][w∆

w

]The Linear Fractional Representation (LFR) gives then thetransfer matrix from w to z, and is referred to as the upperLinear Fractional Transformation (LFT) :

Fu(N,∆) = N22 +N21∆(I−N11∆)−1N12

This LFT exists and iswell-posed if (I−N11∆)−1 is invertible



About LFT definition

Parameter properties

• ∆(ρ) (L2 → L2 ) is bounded (i.e ∆(ρ)T ∆(ρ)≤ 1) and causal.• it can be always chosen as ∆(ρ) = diag(ρ1In1, . . . ,ρN InN), where ni is the number of

occurrences of parameter ρi, where |ρi|< 1, ∀i

LFT vs LPV

Let consider the LPV part of the global LFR system,[xz∆

]=

[A B∆

C∆ D∆∆

][x

w∆

]with w∆ = ∆(ρ)z∆. If the system is well-posed, i.e if (I−∆(ρ)D∆∆) is non singular ∀ρ, then the LFTform equals:

x = (A+B∆∆(ρ)(I−∆(ρ)D∆∆)−1C∆)x(t)

which is nothing else than the original LPV system:

x = A(ρ)x



Example 1: LFT modelling

Let us consider the discrete-time system(x1

x2

)k+1

=

(x1 +ρ1x2

ρ2x1 + x2

)k

(16)

The parameter matrix ∆ is defined as a diagonal matrix from the 2 parameters :

∆ =

[ρ1k 00 ρ2k

](17)

The system can then be written in the LFR form given in Fig 4, with

z∆ =

(x2k

x1k

); w∆ = ∆× z∆ (18)

where P is defined as the LTI state space representation :(x1

x2

)k+1

=

[1 00 1

](x1

x2

)k+

[1 00 1

]w∆ (19)

zΘwΘ

zw P

ρ1 00 ρ2

Figure: LFR representationO. Sename [GIPSA-lab] 21/81


Example 2: LFT modelling

Let us consider the following nonlinear system:

G :

x1 = −x1 + x2sin(x1)x2 = −x2 +wz = x1

(20)

In order to use an LFT, let us define the input:

u∆ = ∆(ρ)x2 := ρx2, with ρ = sin(x1)


Then the previous system can berewritten into the following LFR:

∆

∆zw

uy

where ∆ and y∆ are given as:

∆(ρ) =[ρ], y∆ =

(x2)

and N given by the state space representation:

N :

x1x2

==

−x1 +u∆

−x2 +wy∆ = x2z = x1

(21)


Example 2: LFT modelling of Autonomous Underwater Vehicle

Considered the NL model with 2 state variables (the pitch angle θ and 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 (22)

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 : control input 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.



Example 3: LFT modelling of Autonomous Underwater Vehicle (cont.)


LFR model of the vehicle :

z∆w∆

zkwk P (z)

∆(ρ)

uk yk

The ∆ block contains the varying partof the model, which depends on thelinearization point (θeq). The LFR form(without external inputs w andcontrolled output z) is defined by:

xk+1 = A xk +[

B∆ B1](w∆

u

)(

z∆

y

)=

[C∆

C1

]xk +

[D∆ 00 D1

](w∆

u

)

with

z∆ =

[θ

θ

]; w∆ = ∆z∆; ∆ =

[ρ1 00 ρ2

]The parameters are

ρ1 = cos(θeq)ρ2 = sin(θeq)

and

A =

[0 10 0

], B1 =

[0

Ff ins

]B∆ =

[0 0

−(Zqm−Z f µV )g (Xqm−X f µV )g

]C∆ =

[1 10 0

],C1 =

[1 00 1

]D∆ =

[0 00 0

], D1 =

[0 00 0

]


Few facts on LPV models under Linear Fractional Representations

LFT modelling

• Such a representation is the direct extension of LFT forms (originally used for uncertainties inthe context of µ-analysis), to model systems with time-varying parameters.

• It can be used to model nonlinear systems (with an adhoc definition of the parameters), andalso linear system scheduled by the operating conditions (i.e. with a varying equilibrium)

The "gain scheduling" approach

• One interest is that it allows a systematic synthesis of robust controllers, in the framework ofthe integral quadratic constraint (IQC) approach (Megretski & Rantzer, 1997), (Scherer,2001), (Veenman & Scherer, 2014).Moreover, through the defintion of particular IQC-multipliers, it allows to model a large verietyof problems including time-varying parametric uncertainties, passive nonlinearities,Norm-bounded uncertainties/nonlinearities, sector bounded nonlinearities, delays...

• This framework allows to design (robust) gain-scheduled controllers/observers, in particular inthe context of L2 -induced performances


How to approximate a nonlinear system by an LPV one ?

Outline










10. Conclusion



Nonlinear vs Linear Differential Inclusion: the clue for qLPV modelling

See (Boyd et al, 1994).Let consider the nonlinear system

ΣN L :

x = f (x(t),w(t))z = g(x(t),w(t)) (23)

Suppose that, for each x, w and t, there is a matrix G(x,w, t) ∈Ω s.t.:[f (x,w)g(x,w)

]= G(x,w, t)

[xw

](24)

where Ω ∈ R(nx+nz)×(nx+nu).As said in (Boyd et al, 1994):"Then of course every trajectory of the nonlinear system (24) is also a trajectory of the LDI definedby (25). If we can prove that every trajectory of the LDI defined by (25) has some property (e.g.,converges to zero), then a fortiori we have proved that every trajectory of the nonlinear system(24) has this property."



LPV modelling of a quarter car vehicle suspension model


Figure: Simple quarter vehicle model for semi-active suspension control

Quarter vehicle dynamicsms zs =−kszde f −Fdampermus zus = kszde f +Fdamper− kt (zus− zr)

(25)

zde f = zs− zus : damper deflection, zde f = zs− zus : deflection velocity.

• The damper’s characteristics : Force-Deflection-Deflection Velocity relation

Fdamper = g(zde f , zde f

)(26)

where g can be linear or nonlinear.


LPV modelling of a quarter car vehicle suspension model (cont.)

A Semi-active nonlinear MR damper model [Gu et al., 2006, Nino-Juarez et al., 2008]

Fdamper = c0 zde f + k0zde f + fI tanh(c1 zde f + k1zde f

)(27)

• The tanh function allows to model the bi-viscous behavior.• (c0, k0, c1, k1)are constant parameters. k0, k1 are dedicated to the hysteresis behavior.• fI is a controllable force and depends on input current or voltage.

LPV model

Choosing ρ = tanh(c1 zde f + k1zde f

), and denoting u = fI the control input, the quarter car model

can be represented as: x(t) = Ax(t)+B(ρ)u(t),

y(t) =Cx(t)+Du(t)(28)



Example of a one-tank model

Usually the hydraulic equation is non linear and of the form

SdHdt

= Qe−Qs

where H is the tank height, S the tank surface, Qe the input flow, and Qs the output flow defined byQs = kt

√H. In steady state : Qe = Q0, H = H0

Definition the state space model

The system is represented by an Ordinary Differential Equation whose solution depends on H(t0)and Qe. Clearly H is the system state, Qe is the input, and the system can be represented as:

x(t) = f (x(t),u(t)), x(0) = x0 (29)

with x = H, f =− ktS√

x+ 1S u

A LPV model

Following the LDI method, we define:

ρ(x) =√

xx

which leads to the LPV model

x(t) =− ktS .ρ(x).x+

1S u, x(0) = x0 (30)



Example of a one-tank model - Exercice session 1

Consider the variations qe,qs,h around the steady state as:Qe = Q0 +qe; Qs = Q0 +qs; H = H0 +h.This leads to the equation :

Sdhdt

= qe− kt(√

H0 +h−√

H0)

Denoting the state variable x = h, the control input u = qe, the output y = h, the tangent linearizationgives

x = Ax+Bu (31)

y = Cx (32)

with A =− kt2S√

H0, B = 1

S and C = 1.Now implement 3 types of models and compare with the nonlinear one:• The linear model around a fixed operating point• A LPV model obtained defining an adequate LDI• A LPV model defined in the LFR framework


Identification of LPV systems

Outline










10. Conclusion



Models definition

Identification often requires discrete-time models than can be of 2 forms:

Discrete-time state space model

ΣLPV

x(k+1) = A(ρ(k))x(k)+B(ρ(k))u(k),y(k) = C(ρ(k))x(k)+D(ρ(k))u(k) (33)

Input-output models

y(k) =−na

∑i=1

ai(ρ(k))y(k− i)+nb

∑j=1

bi(ρ(k))u(k− j) (34)

where ai’s and bi’s can be affine, polynomial, rational, of LFR, functions of the paramteter vector ρ



Derivation of LPV models (Lovera, Grenoble summer school 2011)

How to derive control-oriented LPV models?

From non linear models (see before)

Analytical/numerical approaches based on a non linear model or simulator - see (Marcos andBalas 2004) for an overview

Identification from input/output data

Two broad classes of methods can be defined:• Global approaches:

• a single experiment→ the parameter is also excited• a parameter-dependent model is directly obtained• Do exist for input/output and state space models

• Local approaches:• multiple experiments→ constant parameter values• many LTI models are obtained, which have to be interpolated



About model transformation

Many approaches need a conversion from input-output to state space models (to go throughcanonical forms), which is easy for LTI systems....

for LTI systems

Both systems (A,B,C,D) and (T−1AT,T−1B,CT,D) have equivalent properties (stability,observability, controllability, transfer functions...).

But for LPV systems

The state transformation may depend on time, i.e. T (ρ(k)).

In that case, the transformation x(k) = T (ρ(k))xnew(k) applied to

ΣLPV

x(k+1) = A(ρ(k))x(k)+B(ρ(k))u(k),y(k) = C(ρ(k))x(k)+D(ρ(k))u(k) (35)

leads to (denoting for simplicity all matrices parameter dependency Mk for M(ρ(k))):

ΣnewLPV

xnew

k+1 = (T−1k+1AkTk)xnew

k +(T−1k+1Bk)uk,

yk = (CkTk)xnewk +(Dk)uk

(36)

The change of basis changes locally according to the parameter values and variations !In particular T (ρ(k)) must be invertitble ∀k.



Some literature

Global approaches - input/output models

(Bamieh & Giarre 99, 02): characterisation of persistency of excitation conditions for input-outputLPV models

Previdi & Lovera 03, 04): NLPV model class (LFT feeded by a neural network model for thescheduling policy)

(Toth, 07 + book 2010): An LPV system can be viewed as a collection of "local" behaviours(associated with constant parameter values

Global approaches - state space models

(Lee & Poolla): maximum likelihood (ML) algorithm for the identification of MIMO LPV-LFTmodels (PEM algorithm)

(Verhaegen et al, 02, 07, 09...): Supspace methods

Local approaches

Interpolation of locally identified LTI models... need to pay attentin to:• Input/output form (Toth, 07 + book 2010): interpolating transfer function coefficients• State space form (Steinbuch et al, 03): consistency of state space basis


Some properties of LPV systems

Outline










10. Conclusion



LPV systems properties

Let consider the LPV syetsm

Σρ

x(t) = A(ρ(t))x(t)+B(ρ(t))u(t), x(0) = x0y(t) =C(ρ(t))x(t)+D(ρ(t))u(t) (37)

What kind of properties we should pay attention to?

When ρ is fixed (constant) the previous system is LTI and• controllability, observability, stability, are uniquely defined• controllability⇔ reachabillity, observability⇔ reconstructibility• these properties are equivalent by a state change of basis.

But when ρ(t) is time varying .....

• these facts may not be true (asymptotic and exponential stabilty may differ)• need to study properies of Linear Time-Varying systems.• A generalization of the exp(At) is needed, defining the state transition matrix Φ(t, t0,ρ(t))• For a change of basis T (t) with x(t) = T (t)xnew(t) then, x(t) = T (t)xnew(t)+T (t)xnew(t)



A brief insight on observability

Let us consider the system

Σρ

x(t) = A(ρ(t))x(t), x(0) = x0y(t) =C(ρ(t))x(t) (38)

Observability refers to the ability to estimate the initial state variable.

Definition

The system (39) is completely observable if, given the control and the output over the intervalt0 ≤ t ≤ T , one can determine any initial state x(t0).

Following some studies on observability for Linear Time-Varying Systems (LTV) (Silverman &Meadows, SIAM 67), to characterize the observability we need to define the fundamental matrixΨ, satisfying Ψ(t) = A(ρ(t))Ψ(t) (with Ψ(t0) = In) allowing to state that the solution of the stateequation can be deduced as:

x(t) = Ψ(t, t0,ρ(t))x(t0)

which is not easy to compute. Note that, for LTI systems, Ψ(t, t0) = eA(t−t0)



A brief insight on observability (2)

In an analogous way the unobservability property is defined as : a state x(t) is not observable ifthe corresponding output vanishes, i.e. if the following holds: y(t) = y(t) = y(t) = . . .= 0. In thecase of LTV systems it corresponds to:

Definition

The LPV system (39) is completely observable if rankO = n ∀t, where

O =[

oT1 oT

2 . . . oTn]T

where o1 =C(ρ) and oi+1 = oiA+ oi, i > 1 (for instance o2 = ρ∂C(ρ)

∂ρ+C(ρ)A(ρ)).


A weaker notion of observability can bedefined for the LPV systems (39) in thefunctional sense O function of ρ(t).

Definition

The LPV system (39) is structurallyobservable if rankO = n

This does not guarantee that O is invertible∀t and for all parameter values.

Finally the above notion differ from the directextension of the observability matrix for LTI

systems, i.e O =

C(ρ)

C(ρ)A(ρ)...

C(ρ)An−1(ρ)

.This definition is ONLY valid if ρ is constant,i.e. it corresponds to the observability of theLTI systems frozen at the values of theconstant parameter vector ρ.


A few examples


Σ1(ρ) :

x(t) = A(ρ)x(t)y(t) = C(ρ)x(t)

with

A =

(1 1

ρ(t) 2

),C =

(ρ(t) 1

)Observability matrix with ρ f is a frozen valueof ρ(t):

O =

(ρ f 12ρ f ρ f +2

)which is of rank 2 apart for ρ f = 0.Therefore the LTI frozen systems areobservable.

However the observaility matrix of theconsidered time-varying system is given by:

O =

(ρ(t) 1

ρ(t)+2ρ(t) ρ(t)+2

)which is of rank 2 in the functional sense.Therefore the structural rank of Σ1(ρ) is 2.However it is of rank 1 if ρ satisfiesρ(t) = ρ(t)2. The system is then notcompletely observable.Therefore, for some specific parameterdefinitions, the parameter variations maytherefore induce a loss of observability.


A few examples (2)

Let us consider now


Σ(ρ) :

x(t) = A(ρ)x(t)y(t) = C(ρ)x(t)

with

A =

(1 0

ρ(t) 2

),C =

(ρ(t) 1

)Observability matrix with ρ f is a frozen valueof ρ(t):

O =

(ρ f 1

2ρ f 2

)which is of rank 1.Therefore the LTI frozen systems are notobservable.

However the observaility matrix of theconsidered time-varying system is given by:

O =

(ρ(t) 1

ρ(t)+2ρ(t) 2

)which is of rank 2 in the functional sense(apart if ρ(t) 6= 0). Therefore the structuralrank of Σ1(ρ) is 2.In that case the time-variations of ρ(t)provide a persistent excitation on the systemoutput.

Stability of LPV systems

Outline










10. Conclusion



Problem stament and facts

Recall: for LTI systems all notions of stability are equivalent: global/local, asymptotic/exponential,time-domain (Lyapunov)/frequency-domain (Bode, poles...).

Why stability analysis fo LPV systems is not an easy task?

Let consider x = A(ρ(t))x. Stability analysis is more involved (as for LTV systems) since:• there is a set of solutions for a given x0 (family of systems from ρ variations)• the system may be stable for frozen parameter values and unstable for varying parameters

(as for switching systems)• asymptotic and exponential stability are no more equivalent and cannot be characterized by

eigenvalues ( A(ρ(t)) is a time-varying matrix).• In term of design, we will often rely on the notion of quadratic stability (using quadratic

Lyapunov function V (x) = xT Px) which is stronger but easier to check for stability and simplerto use for control and observer design, see (Wu, PhD 95)

Uncertain or LPV? (Blanchini,00 & 07)

• Robust analysis and control: dedicated to LTI systems subject to time-varying (unknown)uncertainties

• LPV (or gain-scheduling) analysis and control: dedicated to LTV systems or to linearizationsof non linear systems along the trajectory of ρ, where ρ is known.



Recall: robust stability with uncertainties

This concept is very useful for the stability analysis of uncertain systems.Let us consider an uncertain system

x = A(δ )x

where δ is an unknown parameter vector that belongs to an uncertainty set ∆.

Problem statement

Is the system asymptotically stable for all δ in ∆ (with δ unknown) ?

Definition

The considered system is said to be quadratically stable for all uncertainties δ ∈ ∆ if there exists a(single) Lyapunov function V (x) = xT Px with P = PT > 0 s.t

A(δ )T P+PA(δ )< 0, for all δ ∈ ∆ (39)

Computation

For polytopic uncertaities (convex set), i.e. if ρ ∈ Coω1, . . . ,ωZ, then, the problem becomesfeasible since it remains to find P = PT > 0 such that:

A(ωi)T P+PA(ωi)< 0, i = 1, . . . ,Z



Quadratic stability for time-varying parameters

Let us consider the LPV systemx = A(ρ(t))x

where ρ(t) is an time-varying parameter vector that belongs to an uncertainty set Ω.

Use of a single Lyapunov function

If there exists P = PT > 0 such that:

A(ρ(t))T P+PA(ρ(t))< 0,∀ρ(t) ∈Ω

then the system is stable for arbitrarily fast time-varying uncertainties

Remarks

• Quadractic stability imples exponential stability (Wu, 95)• It is an infinite dimension problem (can be relaxed for polytopic uncertainties)• It could be conservative since stability is checked for any variation of the parameters !

Pay attentation in what follows: LPV system means TIME-VARYING parameters so a polytopicLPV system is not an uncertain polytopic system (in the latter case the coefficient αi of thepolytopic description are constant even if unknown)



Robust stability or Parameter Varying Lyapunov stability

Let consider now a parameter dependent Lyapunov function Vρ (x(t)) = x(t)T P(ρ)x(t)> 0 for everyx 6= 0 and V (0) = 0.

Uncertain systems (ρ is time-invariant)

The uncertain system x = A(ρ)x is exponentially stable if there exists Vρ such that (classicalapproach for polytopic uncertain systems):

A(ρ)T P(ρ)+P(ρ)A(ρ)< 0, ∀ ρ ∈

LPV systems (ρ is time-varying)

The uncertain system x = A(ρ(t))x is exponentially stable if there exists Vρ such that:

A(ρ)T P(ρ)+P(ρ)A(ρ)+N

∑i=1

ρi∂P(ρ)

∂ρi< 0 ∀ ρ(t) ∈

which, in addition to bounded parameters, needs to consider rate-bounded parameter variations.Such a condition is more complex since:• It involves the partial differentiation of P• it has to be checked for all ρ(t) ∈Uρ

• It implies to choose a parametrization of P(ρ): from affine to polynomial



Stability of LFR models of LPV systems

See Scherer, Apkarian & Gahinet).In that case, the method corresponds to the generalization of Robust stability tools for LFR modelsto time-varying parameters block ∆(ρ(t)), which is often referred to as the Scaled Small GainTheorem.It is generally assumed, in the LPV framework, that the matrix has a diagonal structure

∆(ρ) = diag(In1 ⊗ρ1, . . . , Inp ⊗ρp) (40)

Let us introduce the set of D-scalings defined by

L∆ := L > 0 : L∆(ρ) = ∆(ρ)L, ∀∆ (41)

where n = ||col(n1, . . . ,np)||1 and

Lemma

The LFR system (A,B∆,C∆,D∆∆) is asymptotically stable and ||L1/2(D∆∆ +C∆(sIn−A)−1B∆)L−1/2||∞if there exist P = PT 0 and L ∈ L∆ such thatAT P+PA PB∆ CT

∆L

? −L DT∆∆

L? ? −L

≺ 0 (42)



L2 stability of LPV systems (Wu, 95)

Definition

Given a parametrically dependent stable LPV system Σρ = (A(ρ),B(ρ),C(ρ),D(ρ)) for zero initialconditions x0. The induced L2 norm is defined as:

||Σρ ||i,2 = supρ(t)∈Ω

supw(t)6=0∈L2

‖y‖2‖u‖2

which is often referred to as (by abuse of langage) the H∞ gain ||Σρ ||∞ of the LPV system.

Theorem

A sufficient condition for the L2 stability of system Σρ is the generalized BRL, using parameterdependent Lyapunov functions, i.e assuming |ρi|< νi, ∀i, if there exists P(ρ)> 0, ∀ρ s.t A(ρ)T P(ρ)+P(ρ)A(ρ)+∑

Ni=1 νi

∂P(ρ)∂ρi

P(ρ) B(ρ) C(ρ)T

B(ρ)T P(ρ) −γ I D(ρ)T

C(ρ) D(ρ) −γ I

< 0, ∀i. (43)

then ||Σρ ||i,2 ≤ γ


LPV Control design

Outline










10. Conclusion


LPV Control design

Towards LPV control

The "gain scheduling" approach

System (ρ)Controller (ρ)

Adaptationstrategy

Measured or estimatedParameters

References ControlInputs

Outputs

Externalparameters

Some references

• Modelling, identification : (Bruzelius, Bamieh, Lovera, Toth)• Control (Shamma, Apkarian & Gahinet, Adams, Packard, Beker ...)• Stability, stabilization (Scherer, Wu, Blanchini ...)• Geometric analysis (Bokor & Balas)


LPV Control design

LPV approach=linear or nonlinear? (Shamma, Apkarian & Gahinet,Balas & Seiler, Grigoriadis ...)

Figure: DLR German Aerospace Center (ESA LPV Workshop 2014)


LPV Control design The Static State feedback case

Outline










10. Conclusion



State feedback control design: pole placement (1)

Let consider the system Σ(ρ):

x(t) = A(ρ)x(t)+B(ρ)u(t) (44)

y(t) = C(ρ)x(t)+D(ρ)u(t)

The objective is to find a state feedback control law u =−F(ρ)x+G(ρ)r, where r is the referencesignal s.t:• the closed-loop system is stable• the output y tracks the reference r (unit closed-loop gain y/r)

The closed-loop system is

x(t) = (A(ρ)−B(ρ)F(ρ))x(t)+B(ρ)G(ρ)r (45)

y(t) = (C(ρ)x(t)−D(ρ)F(ρ))x(t)+D(ρ)G(ρ)r

Then we must consider the following issues:• What controllability property shall we consider?• What parameter dependency should we define for (F(ρ),G(ρ))?• How to choose the poles (dynamics) of the closed-loop system?



State feedback control design: pole placement (2)

Here we can tackle several problems:

Robust SF control:• Design a nominal state feedback control (F,G) (for Σ(ρnominal))• Check quadractic stability and performances of the closed-loop system, with

(A(ρ)−B(ρ)F) ...

LPV SF control with fixed performances:• Choose some desired poles (p1, p2, . . . , pn) for the CL system• Design F(ρ) such that eig(A(ρ)−B(ρ)F(ρ)) = (p1, p2, . . . , pn). Could be

analytic design or ’frozen-type’.• Check stability (quadractic?) of the CL system when ρ is time-varying

LPV SF control with varying (adaptive) performances:• Choose some desired poles (p1(ρ), p2(ρ), . . . , pn(ρ)) for the CL system• Design F(ρ) such that eig(A(ρ)−B(ρ)F(ρ)) = (p1(ρ), p2(ρ), . . . , pn(ρ)).

Could be analytic design or ’frozen-type’.• Check stability (quadractic?) of the CL system when ρ is time-varying• This latter case allows to schedule the performances according to the

parameter changes, so to handle the trade-off, function of the parametervariations, between closed-loop system transient dynamics and control costlimitations



The H∞ state feedback control problem

Let consider the system:

x(t) = A(ρ)x(t)+B1(ρ)w(t)+B2(ρ)u(t) (46)

y(t) = C(ρ)x(t)+D11(ρ)w(t)+D12(ρ)u(t)

The objective is to find a state feedback control law u =−K(ρ)x s.t:∥∥Tyw(s)∥∥

∞≤ γ

The method consists in applying the Bounded Real Lemma to the closed-loop system, and thentry to obtain some convex solutions (LMI formulation).Following the framework of quadratic stabilty, this is achieved if and only is there exists a positivedefinite symmetric matrix P (i.e P = PT > 0 s.t (A(ρ)−B2(ρ)K(ρ))T P+P (A(ρ)−B2(ρ)K(ρ)) P B1(ρ) (C(ρ)−D12(ρ)K(ρ))T

? −γ I DT11(ρ)

? ? −γ I

< 0. (47)



Solution of the state feedback control problem

Use of change of variables

First, left and right multiplication by diag(P−1, In, In), and use Q = P−1 and Y(ρ) =−K(ρ)P−1. Itleads to Q > 0 and A(ρ)Q+B2(ρ)Y(ρ)+QAT (ρ)+Y(ρ)T BT

2 (ρ) B1(ρ) QCT (ρ)−YT (ρ)DT12(ρ)

? −γ I DT11(ρ)

? ? −γ I

< 0. (48)

The state feedback controller is then:

K(ρ) =−Y(ρ)Q−1

How to solve (49)?

• It is indeed not an LMI in ρ due to B2(ρ)Y(ρ) and to D12(ρ)Y(ρ), and still infinite dimensional• For polytopic systems, a solution exists if either we choose Y time-invariant or if B2 and D12

are parameter independent• In the general case this requires to impose some parameter dependency on Y(ρ) (affine,

polynomial ...) and to solve the problems trying to linearize it (or using gridding techniques(Balasz, Packard, Seiler)).


LPV Control design The Dynamic Output feedback case

Outline










10. Conclusion



The H∞/LPV control problem

Definition

Find a LPV controller C(ρ) s.t the closed-loop system is stable and for γ∞ > 0, sup ‖z‖2‖w‖2< γ∞,

• Unbounded set of LMIs (Linear Matrix Inequalities) to be solved (ρ ∈Ω)• Some approaches: polytopic, LFT, gridding. See Arzelier [HDR, 2005], Bruzelius [Thesis,

2004], Apkarian et al. [TAC, 1995]...

A solution: The "polytopic" approach [C. Scherer et al. 1997]

• Problem solved off line for each vertex of a polytope (convex optimisation) (using here asingle Lyapunov function i.e. quadratic stabilization).

• On-line the controller is computed as the convex combination of local linear controllers

C(ρ) =2N

∑k=1

αk(ρ)

[Ac(ωk) Bc(ωk)Cc(ωk) Dc(ωk)

],

2N

∑k=1

αk(ρ) = 1 , αk(ρ)> 0

6

-

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)

• Easy implementation !!



The H∞/LPV control problem

Definition

Find a LPV controller C(ρ) s.t the closed-loop system is stable and for γ∞ > 0, sup ‖z‖2‖w‖2< γ∞,

• Unbounded set of LMIs (Linear Matrix Inequalities) to be solved (ρ ∈Ω)• Some approaches: polytopic, LFT, gridding. See Arzelier [HDR, 2005], Bruzelius [Thesis,

2004], Apkarian et al. [TAC, 1995]...

A solution: The "polytopic" approach [C. Scherer et al. 1997]

• Problem solved off line for each vertex of a polytope (convex optimisation) (using here asingle Lyapunov function i.e. quadratic stabilization).

• On-line the controller is computed as the convex combination of local linear controllers

C(ρ) =2N

∑k=1

αk(ρ)


],

2N

∑k=1

αk(ρ) = 1 , αk(ρ)> 0

6

-

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)

• Easy implementation !!



LPV control design

Generalized plant

w z

y u

Controller S (ρ)

∑ (ρ)

Problem on standard form

CL (ρ)

w: exagenous input u: control input

z: output to minimize y: measurement

Dynamical LPV generalized plant:

Σ(ρ) :

xzy

=


xwu

(49)

LPV controller structure:S(ρ) :

[xcu

]=

[Ac(ρ) Bc(ρ)Cc(ρ) Dc(ρ)

][xcy

](50)

LPV closed-loop system:

C L (ρ) :[

ξ

z

]=

[A (ρ) B(ρ)C (ρ) D(ρ)

][ξ

w

](51)



LPV control design

Generalized plant

w z

y u

Controller S (ρ)

∑ (ρ)


CL (ρ)




Σ(ρ) :

xzy

=


xwu

(49)


[xcu

]=


][xcy

](50)


C L (ρ) :[

ξ

z

]=

[A (ρ) B(ρ)C (ρ) D(ρ)

][ξ

w

](51)



LPV control design

Generalized plant

w z

y u

Controller S (ρ)

∑ (ρ)


CL (ρ)




Σ(ρ) :

xzy

=


xwu

(49)


[xcu

]=


][xcy

](50)


C L (ρ) :[

ξ

z

]=

[A (ρ) B(ρ)C (ρ) D(ρ)

][ξ

w

](51)



LPV control design

Generalized plant

w z

y u

Controller S (ρ)

∑ (ρ)


CL (ρ)




Σ(ρ) :

xzy

=


xwu

(49)


[xcu

]=


][xcy

](50)


C L (ρ) :[

ξ

z

]=

[A (ρ) B(ρ)C (ρ) D(ρ)

][ξ

w

](51)



LPV control design

H∞ criteria Apkarian et al. [TAC, 1995]

Stabilize system CL(ρ) (find K > 0) while minimizing γ∞. A (ρ)T K +KA (ρ) KB∞(ρ) C∞(ρ)T

B∞(ρ)T K −γ2

∞I D∞(ρ)T

C∞(ρ) D∞(ρ) −I

< 0

Infinite set of LMIs to solve (ρ ∈Ω) (Ω is convex)

LPV control designs Arzelier [HDR, 2005], Bruzelius [Thesis, 2004]

LFT, Gridding, Polytopic



LPV control design

H∞ criteria Apkarian et al. [TAC, 1995]

Stabilize system CL(ρ) (find K > 0) while minimizing γ∞. A (ρ)T K +KA (ρ) KB∞(ρ) C∞(ρ)T

B∞(ρ)T K −γ2

∞I D∞(ρ)T

C∞(ρ) D∞(ρ) −I

< 0

Infinite set of LMIs to solve (ρ ∈Ω) (Ω is convex)

LPV control designs Arzelier [HDR, 2005], Bruzelius [Thesis, 2004]

LFT, Gridding, Polytopic



LPV control design

Polytopic approach

Solve the LMIs at each vertex of the polytope formed by the extremum values of each varyingparameter, with a common K Lyapunov function.

C(ρ) =2N

∑k=1

αk(ρ)


]where,

αk(ρ) =∏

Nj=1 |ρ j−C c(ωk) j|∏

Nj=1(ρ j−ρ

j)

,

where C c(ωk) j = ρ j if (ωk) j = ρj

or ρj otherwise.

2N

∑k=1

αk(ρ) = 1 , αk(ρ)> 0



LPV control design

Polytopic approach

Solve the LMIs at each vertex of the polytope formed by the extremum values of each varyingparameter, with a common K Lyapunov function.

C(ρ) =2N

∑k=1

αk(ρ)


]

6

-

ρ2

ρ1ρ1

ρ2

ρ2

ρ1

C(ω1)

C(ω2) C(ω4)

C(ω3)

C(ρ)



LPV/H∞ control synthesis

Proposition - feasibility (brief) Scherer et al. (1997)

Solve the following problem at each vertices of the parametrized points (illustration with 2parameters):

γ∗ = min γ

s.t. (54) |ρ1 ,ρ2

s.t. (54) |ρ1 ,ρ2

s.t. (54) |ρ1 ,ρ2

s.t. (54) |ρ1 ,ρ2

(52)

AX+B2C(ρ1,ρ2)+(?)T (?)T (?)T (?)T

A(ρ1,ρ2)+AT YA+ B(ρ1,ρ2)C2 +(?)T (?)T (?)T

BT1 BT

1 Y+DT21B(ρ1,ρ2)

T −γI (?)T

C1X+D12C(ρ1,ρ2) C1 D11 −γI

≺ 0

[X II Y

] 0

(53)



LPV/H∞ control synthesis

Proposition - reconstruction (brief) Scherer et al. (1997)

Reconstruct the controllers as,solve (56) |ρ1 ,ρ2

(56) |ρ1 ,ρ2

(56) |ρ1 ,ρ2

(56) |ρ1 ,ρ2

(54)

Cc(ρ1,ρ2) = C(ρ1,ρ2)M−T

Bc(ρ1,ρ2) = N−1B(ρ1,ρ2)

Ac(ρ1,ρ2) = N−1(A(ρ1,ρ2)−YAX−NBc(ρ1,ρ2)C2X− Y B2Cc(ρ1,ρ2)MT )M−T

(55)

where M and N are defined such that MNT = I−XY which may be chosen by applying a singularvalue decomposition and a Cholesky factorization.


LPV observer design

Outline










10. Conclusion


LPV observer design

Definition LPV observers

Definition

Let consider the LPV system:

x(t) = A(ρ)x(t)+B(ρ)u(t)y(t) = C(ρ)x(t) (56)

The following LPV state space representation

˙x(t) = A(ρ)x(t)+B(ρ)u(t)+L(ρ)(y(t)−C(ρ)x(t))x0to be defined (57)

is said to be an observer for (57) if

limt→∞

(x(t)− x(t))→ 0 ∀ρ(t) ∈Ω

where x(t) ∈ Rn is the estimated state of x(t) and L(ρ) is the n× p observer gain matrix to bedesigned.


LPV observer design

Some issues for LPV observer design

The estimated error, e(t) := x(t)− x(t), satisfies:

e(t) = (A−LC)(ρ)e(t) (58)

The two main problems to be handle are then• What observability property shall we consider?• What parameter dependency should we define for L(ρ)?

Quadractic detectability (Wu, 95)

A simple solution is to consider a single Lyapunov function in order to guarantte the quadraticdetectability, i.e:

(A(ρ)−L(ρ)C(ρ))T P+P (A(ρ)−L(ρ)C(ρ))< 0

Some remarks:• The previous problem can be solved using a polytopic approach only if C(ρ) =C, a constant

matrix• If this is not solvable, one can try using Parameter dependent Lyapunov functions, but the

coupling between L(ρ) and P(ρ) will lead to soved non affine LMIs (a polynomial or a griddingapproach is then needed).


LPV observer design

Some issues for LPV observer design (2)

On key issue in observer implementation concerns the knownledge of ρ(t). While previously theresult is valid if ρ(t) is perfectly known, such a following observer description must be used if ρ(t)is estimated:

˙x(t) = A(ρ)x(t)+B(ρ)u(t)+L(ρ)(y(t)−C(ρ)x(t)) (59)

Denoting ∆A = A(ρ)−= A(ρ), ∆B = B(ρ)−B(ρ), ∆C =C(ρ)−C(ρ), and ∆L = L(ρ)−L(ρ), thisleads for the estimation error equation:

e(t) = (A−LC)(ρ)e(t)+(∆A+L(ρ).∆C)x+∆Bu(t) (60)

If C(ρ) =C and B(ρ) are constant matrices, then we get the uncertain estimated error system

e(t) = (A(ρ)−L(ρ)C)e(t)+∆Ax(t) (61)

The stability analysis is indeed more involved due to the state vector x (see (Daafouz et al, 2010)for the discrete-time case). Either ∆Ax(t) should be considered as a disturbance, or a stateaugmentation approach is to be used (which has to be done in closed-loop control).

Observer-based control

For control design in the latter case, the following state feedback should be used:

u(t) =−F(ρ)x(t)


LPV observer design LPV polytopic observer with H∞ performance method

Outline










10. Conclusion



Observer design with H∞ performance method

Let us consider the following LPV polytopic system subject to a disturbance:

x =2N

∑i=1

µi(ρ)(Aix+Biu+Ew)

y =Cx

z =Czx

(62)

where x ∈ Rnx is the state vector, u ∈ Rnu is the known input vector, w ∈ Rnw is the additivedisturbance, y ∈ Rny is the measurement vector, z ∈ Rnz is the signal to be estimated, and ρ ∈Pρ .To estimate the variable z, the Luenberger-like LPV polytopic observer is proposed hereunder:

˙x =2N

∑i=1

µi(ρ)(Aix+Biu+Li(y− y))

y =Cx

z =Czx

(63)

Thus, from (63) and (64), the dynamic estimation error e = x− x is governed by:

e =2N

∑i=1

µi(ρ)((Ai−LiC)e+Ew)

ez =Cze

(64)



Observation problem

Problem

Find matrices Li ∈ Rnx×ny , i = 1, . . . ,2N , of the observer (64) for the polytopic LPV system (63) suchthat:

1 the estimation error system (65) is asymptotically stable (e(t)→ 0 when t→ ∞) for w≡ 0.2 the upper bound γ∞ of the induced-L2 norm from the disturbance w to ez is minimized, i.e

supw6=0,w∈L2

‖ez‖2

‖w‖2≤ γ∞ (65)

Solution (from the Bounded Real Lemma)

Consider the LPV polytopic observer (64) for the polytopic LPV system (63). The problem issolved, if there exist matrices Yi ∈ Rnx×ny , i = 1, . . . ,2N , and P = PT 0 ∈ Rnx×nx such that thefollowing LMIs hold for all i = 1, . . . ,2N :

min γ∞

s.t

PAi−YiC+ATi P−CTY T

i E PCTz

∗ −γ∞Inw 0∗ ∗ −γ∞Inz

≺ 0(66)

The gains of the observer (64) are deduced as Li = P−1Yi.

It is worth noting that, when the disturbance matrix E is not known, it can be arbitrarly fixed.Besides, for a loop shaping design, in order to have a finer synthesis, it is possible to add aperformance weight on the input, like it has been done in for example.


LPV observer design LPV polytopic observer with pole placement method

Outline










10. Conclusion



LPV polytopic observer

Let us consider the following LPV polytopic system:

x =2N

∑i=1

µi(ρ)(Aix+Biu)

y =Cx

(67)

where x ∈ Rnx is the state vector, u ∈ Rnu is the known input vector, y ∈ Rny is the measurementvector, and ρ ∈Pρ defined in (69).

Pρ :=

ρ = [ρ1 . . . ρN ]T ∈ RN and ρi ∈ [ρi, ρi], for all i = 1 . . .N

(68)

To estimate the state vector, the following Luenberger-like LPV polytopic observer is proposed:

˙x =2N

∑i=1

µi(ρ)(Aix+Biu+Li(y− y))

y =Cx

(69)

Thus, from (68) and (70), the dynamic estimation error e = x− x is governed by:

e =2N

∑i=1

µi(ρ)(Ai−LiC)e (70)



Solution to LPV polytopic observer design


Problem

Find matrices Li ∈ Rnx×ny , i = 1, . . . ,2N , of the observer (70) for the polytopic LPV system(68) such that:

1 the estimation error system (71) is asymptotically stable i.e, e(t)→ 0 when t→ ∞.2 the eigenvalues of Ai−LiC, i = 1, . . . ,2N , are in a chosen area of the complex plane.

LMI Solution

Based on Chilali 1996, one can establish the following proposition.Consider the LPV polytopic observer (70) for the polytopic LPV system (68). The problemis solved, if there exist matrices Yi ∈Rnx×ny , i = 1, . . . ,2N , and P = PT 0 ∈Rnx×nx such thatthe following LMIs hold for all i = 1, . . . ,2N , and the gains of the observer (70) are deducedas Li = P−1Yi.

given two positive scalars xα and xβ ,


i +2xα P≺ 0


i +2xβ P 0(71)

given a positive scalar θ ,

sinθ(PAi−YiC+ATi P−

CT Y Ti )

cosθ(PAi−YiC−ATi P+

CT Y Ti )

∗ sinθ(PAi−YiC+ATi P−

CT Y Ti )

≺ 0 (72)

Summary of LPV approach interests

Outline










10. Conclusion



Interest of the LPV approach

LPV is a key tool to the control of complex systems.

Some examples :

Modelling of complex systems (non linear)

• Use of a quasi-LPV representation to include non linearities in a linear state space model(even delays)

• Transformation of constraints (e.g. saturation) into an ’external’ parameter• Modelling of LTV, hybrid (e.g. switching control)

BUT :

A q-LPV system is not equivalent to the non linear one:• stability: ρ = ρ(x(t), t) is assumed to be bounded... so are the state trajectories• controllability: some non controllable modes of a non linear system may vanish according to

the LPV representation• observability: unobservability may occur for some specific parameter variations





Some examples :

Modelling of complex systems (non linear)

• Use of a quasi-LPV representation to include non linearities in a linear state space model(even delays)

• Transformation of constraints (e.g. saturation) into an ’external’ parameter• Modelling of LTV, hybrid (e.g. switching control)

BUT :

A q-LPV system is not equivalent to the non linear one:• stability: ρ = ρ(x(t), t) is assumed to be bounded... so are the state trajectories• controllability: some non controllable modes of a non linear system may vanish according to

the LPV representation• observability: unobservability may occur for some specific parameter variations




Some of works using LPV approaches - former PhD students

Gain-scheduled control

• Account for various operating conditions using a variable "equilibrium point": (Gauthier 2007)• Control with real-time performance adaptation using parameter dependent weighting

functions from endogenous or exogenous parameters (Poussot 2008, Do 2011)• Analysis and control of LPV Time-Delay Systems: delay-scheduled control Briat 2008• Control under computation constraints: H∞ variable sampling rate controller with sampling

dependent performances (Robert 2007, Roche 2011, Robert et al., IEEE TCST 2010))

Coordination of several actuators for MIMO systems

• An LPV structure for control allocation Poussot et al. (CEP 2011)• Selection of a specific parameter for the control activation (of each actuator) Poussot et al.

(VSD 2011), Doumiati et al (EJC 2013), Fergani et al (IEEE TVT 2015)

Incorporate fault-(diagnosis, accomodation, tolerant control) properties

• LPV fault-scheduling control: see Sename et al (Systol 2013, ICSTCC 2015).



LPV approach and applications


Aerospace

Marcos, Balas, Seiler,Biannic

Benani, Falcoz

Others

Automotive

Gaspar Poussot,Doumiati ,

Werner, Mohama-madpour, Zhu

Mechatronics, Robotics

Theilliol, Puig

Roche & Simon

Conclusion

Conclusion


Advantages

• Linear approach: easy design and implementation• No on-line optimization• Generic approach: from modelling to control and diagnosis• Can account for many things: MIMO systems, delays, saturations, performance objectives

Drawbacks

• Could be seen as less convenient than a dedicated case-dependent approach• Compared to nonlinear optimal control, constraints are not explicitly considered• May lead to high-order controllers (need for model reduction)


Conclusion

Some references

• W. Nwesaty, A.I. Bratcu, O. Sename, "Power sources coordination through multivariable LPV/H∞ control with application to multi-source electricvehicles", to appear in IET Control theory and Applications, 2016;

• S. Fergani, O. Sename and L. Dugard, "An LPV/H∞ Integrated Vehicle Dynamic Controller," in IEEE Transactions on Vehicular Technology, vol. 65,no. 4, pp. 1880-1889, April 2016.

• O. Sename, "The LPV approach: the key to controlling vehicle dynamics ?", pleanry talk at 19th International Conference on System Theory, Controland Computing (ICSTCC), Cheile Gradistei, Romania, 2015.

• Doumiati, M., Sename, O., Dugard, L., Martinez-Molina, J.-J., Gaspar, P., Szabo, Z., "Integrated vehicle dynamics control via coordination of activefront steering and rear braking", (2013) European Journal of Control, 19 (2), pp. 121-143.;Among the 3 Most Cited Articles published since 2011, extracted from Scopus.

• O. Sename, J. C. Tudon-Martinez and S. Fergani, "LPV methods for fault-tolerant vehicle dynamic control," plenary paper, 2013 Conference onControl and Fault-Tolerant Systems (SysTol), Nice, 2013, pp. 116-130.

• C. Poussot-Vassal, O. Sename, L. Dugard, P. Gaspar, Z. Szabo & J. Bokor, "Attitude and Handling Improvements Through Gain-scheduledSuspensions and Brakes Control", Control Engineering Practice (CEP), Vol. 19(3), March, 2011, pp. 252-263.

• C. Poussot-Vassal, O. Sename, L. Dugard, S.M. Savaresi, "Vehicle Dynamic Stability Improvements Through Gain-Scheduled Steering and BrakingControl", Vehicle System Dynamics (VSD), vol 49, Nb 10, pp 1597-1621, 2011

• Briat, C.; Sename, O., "Design of LPV observers for LPV time-delay systems: an algebraic approach", International Journal of Control, vol 84, nb 9,pp 1533-1542, 2011

• Robert, D., Sename, O., and Simon, D. (2010). An H∞ LPV design for sampling varying controllers: experimentation with a T inverted pendulum.IEEE Transactions on Control Systems Technology, 18(3):741–749.

• C. Briat, O. Sename, and J.-F. Lafay, ""Memory-resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays"Systems & Control Let., vol. 59, pp. 451-459, 2010.

• C. Briat, O. Sename, and J. Lafay, "Hinf delay-scheduled control of linear systems with time-varying delays," IEEE Transactions in Automatic Control,vol. 42, no. 8, pp. 2255-2260, 2009.

• Poussot-Vassal, C.; Sename, O.; Dugard, L.; Gaspar, P.; Szabo, Z. & Bokor, J. "A New Semi-active Suspension Control Strategy Through LPVTechnique", Control Engineering Practice, 2008, 16, 1519-1534

• Zin A., Sename O., Gaspar P., Dugard L., Bokor J. "Robust LPV - Hinf Control for Active Suspensions with Performance Adaptation in view of GlobalChassis Control", Vehicle System Dynamics, Vol. 46, No. 10, 889-912, October 2008

• Gauthier C., Sename O., Dugard L., Meissonnier G. "An LFT approach to Hinf control design for diesel engine common rail injection system", Oil &Gas Science and Technology, vol 62, nb 4, pp. 513-522 (2007)


Conclusion

PhD students on LPV control

• Thanh Phong PHAM, "LPV observer and Fault-tolerant control of vehicle dynamics: application to an automotive semi-active suspension system", PhDGIPSA-lab, Univertisté Grenoble Alpes, 2020.

• Donatien DUBUC, "Observation and diagnosis for trucks", PhD GIPSA-lab, Univertisté Grenoble Alpes, 2018.

• Kazusa YAMAMOTO, " Control of Electromechanical Systems, Application to Electric Power Steering Systems", PhD GIPSA-lab, Univertisté GrenobleAlpes, 2017.

• Van Tan VU, "Enhancing the roll stability of heavy vehicles by using an active anti-roll bar system", PhD GIPSA-lab, Univ. Grenoble Alpes, 2017.

• Manh Quan NGUYEN, "LPV approaches for modelling and control of vehicle dynamics: application to a small car pilot plant with ER dampers", PhDGIPSA-lab, Univ. Grenoble Alpes, 2016.

• Waleed NWESATY, " LPV/H∞ control design of on-board energy management systems for electrical vehicles", PhD GIPSA-lab, Univ. Grenoble Alpes,2015.

• Soheib FERGANI, "Robust LPV/H∞ MIMO control for vehicle dynamics, PhD GIPSA-lab, Univ. Grenoble Alpes, 2014.

• Maria RIVAS, "Modeling and Control of a Spark Ignited Engine for Euro 6 European Normative", PhD, GIPSA-lab / RENAULT, Grenoble INP, 2012.

• Ahn-Lam DO, "LPV Approach for Semi-active Suspension Control & Joint Improvement of Comfort and Security", PhD, GIPSA-lab, Grenoble INP,2011.

• David HERNANDEZ, "Robust control of hybrid electro-chemical generators", PhD, GIPSA-lab / G2Elab, Grenoble INP, 2011.

• Emilie ROCHE, "Commande Linéaire à Paramètres Variants discrète à échantillonnage variable : application à un sous-marin autonome", PhD,GIPSA-lab, Grenoble INP, 2011.

• Sébastien AUBOUET, "Semi-active SOBEN suspensions modelling and control", PhD, GIPSA-lab / SOBEN, Grenoble INP , 2010.

• Charles POUSSOT-VASSAL, "Robust LPV Multivariable Global Chassis Control", PhD , GIPSA-lab, Grenoble INP, 2008.

• Corentin BRIAT, "Robust control and observation of LPV time-delay systems", PhD, GIPSA-lab, Grenoble INP, 2008.

• Christophe GAUTHIER, "Commande multivariable de la pression d’injection dans un moteur Diesel Common Rail", PhD, LAG / DELPHI, Grenoble INP,2007.

• David ROBERT, "Contribution à l’interaction commande/ordonnacement", PhD, LAG, Grenoble INP, 2007.

• Alessandro ZIN, "Sur la commande robuste de suspensions automobiles en vue du contrôle global de châssis", PhD, LAG / Grenoble INP, 2005.


Documents

Grenoble INP / Gipsa-labo.sename/docs/LPVtheory.pdf · 2021. 1. 26. · Grenoble INP / Gipsa-lab Master Course - January 2021 Olivier Sename (Grenoble INP / Gipsa-lab) LPV theory