CRONE Control of LPV Systems: Application to Air-Fuel Ratio Control of an Internal Combustion Engine Mathieu Moze, Jocelyn Sabatier, Alain Oustaloup Laboratoire de l’Intégration, du Matériau au Système 351 cours de la Libération F33405 TALENCE cedex FRANCE Tel : +33 (0)540 006 607 Fax : +33 (0)540 006 644 [email protected] www.laps.u-bordeaux1.fr Abstract: The paper presents an extension of the CRONE control method to LPV systems. This extension uses small gain theorem on a LFT formulation of the system to determine a LPV controller whose parameters are scheduled on those of the system. This approach permits open loop insensitivity, enabling its optimal parameterization for robustness purposes. The method is finally applied for air/fuel ratio control of internal combustion engine. Results in simulation validate the approach. Keywords: CRONE control, LPV systems, air-fuel ratio, internal combustion engine.

1. INTRODUCTION

2. CRONE CONTROL PRINCIPLES

Fractional differentiation is now a well known tool for controller synthesis. Several presentations and applications of fractional PID controller [Podlubny, 1999], [Monje et al., 2004], [Caponetto et al., 2004], [Chen et al., 2004] and of CRONE controller [Oustaloup and Mathieu, 1999] [Lanusse et al., 2005] (CRONE is the French acronym of “Commande Robuste d’Ordre Non Entier” which means Robust Control using Fractionnal Order) demonstrate their efficiency. Fractional differentiation also permits a simple representation of some high order complex integer systems [Bataglia et al., 2001]. Consequently, basic properties of fractional systems have been investigated these last ten years. Criteria and theorems are now available in the literature concerning stability [Matignon, 1996], observability, and controllability [Matignon and D’AndreaNovel, 1996] of fractional systems. Whereas CRONE methodology have been developed for a large variety of plants such as linear non minimum phase plants, linear unstable plants [Oustaloup and Mathieu, 1999], linear plants with low damped modes [Oustaloup et al., 1995], linear sampled plants, multivariable plants [Lanusse et al., 2000], and non linear plants [Pommier et al., 2001], time variant plants have been only studied through consideration of periodic coefficients [Sabatier et al., 1998], [Sabatier et al., 2001]. General case of variable parameters plants is still to be developed. One particularity of CRONE synthesis is that uncertainties are taken into account through convex hulls in Nichols chart, computed at given frequencies. Transfer function and frequency response concepts being not available for time varying systems, a new formulation of CRONE control is required for such an extension. The paper presents an extension of the method presented in [Moze et al., 2006] to LPV systems case.

The CRONE control design procedure is a well defined tool that ensures robustness properties to a controlled system using complex fractional differentiation in the definition of the open loop. Three generations of CRONE control have been developed. The most advanced is the third one, which is now presented.

In the sequel, the notation A* is used to refer to transpose conjugate of matrix A . For any hermitian matrix B , notation B > 0 ( B ≥ 0 ) means that B is positive (semi positive) definite. For any operator G , G L denotes its L 2 gain. If 2

G is Linear Time Invariant, its L 2 gain equals the H ∞ norm of the associated transfer matrix G (s ) noted G (s ) ∞ .

2.1 Template definition

The third generation of CRONE control is based on the definition of a template which can be represented in the Nichols chart by an any-direction straight line segment around open loop gain crossover frequency ωu [Oustaloup and Mathieu, 1999]. This template is based on the real part with respect to imaginary unit i of a fractional integrator [Lanusse et al., 2005]: −1  ω  n    π  (1) βT (s ) =  cosh b   Re / i  u   ,  s    2   where n = a + ib , n ∈ C i , and s = σ + jω , s ∈ C j , C i and C j being respectively time-domain and frequency-domain complex planes. In the Nichols chart, the real order a determines the phase placement of the template, and the imaginary order b determines its angle to the vertical, at frequency ωu imposed by the designer to obtain nominal performance. 2.2 Open loop transfer function including the template

As the design of a third generation CRONE controller takes into account specifications at low and high frequencies, the open loop transfer function is given by β (s ) = β l (s )β n (s )β h (s ) , (2) where s   1+ ωh sign(b )  C β n (s ) = Y0 C  0 s  1+ ωb 

a    1 + s  ωh  Re   C /i  0  s  1 +    ωb  

     

ib 

     

− sign (b )

(3)

defines a limited frequency band template. C and C0 are such that the gain of β n (s ) is Y0 at any desired resonant frequency ωr . At low frequency is added a proportional integrator nb

  s  s   , (4)      ωb  ωb  where Cb is such that ωr is the gain crossover frequency of β l (s ) , and nb ∈ N + enables the steady state error to be nullified. At high frequency is added a low-pass filter

β l (s ) =  Cb 1 +

n

h   s    β h (s ) =  Ch 1 + (5)  ,   ωh    where Ch is such that ωr is the gain crossover frequency of β h (s ) , and nh ∈ N + enables the constancy or the decrease of the control effort sensitivity function.

2.3 Parameterization of optimal open loop CRONE control design guaranties the robustness of both stability and performance through the robustness of the maximum Q of the complementary sensitivity function magnitude. An open loop Nichols locus is considered as optimal if the generalized template around ωr tangents the Qr Nichols magnitude contour for the nominal state of the plant, Qr being the desired magnitude peak, and minimizes the variations of Q for the other parametric states. The open loop Nichols locus optimization thus consists in determining optimal values of the following five parameters: the optimal real integration order a , the optimal imaginary integration order b , the optimal corner frequencies ωb and ωh , and the optimal gain Y0 . Figure 1 shows how these parameters affect the Nichols locus. ωb β ( jω ) Y0 Qr −3π/2

−π ω r

b

a −π/2

Y0

(7)

where Qr is the desired sensitivity function magnitude peak,

Qmax and Qmin being the extreme values of this peak when the plant is affected by perturbations. Figure 2 shows how minimisation of criterion (6) positions the template so that the uncertainty domains overlap as little as possible the low stability degree area of the Nichols chart. Constraints Associated with criterion (6) are constraints on S (s ) = (1) (1 + β (s )) , complementary sensitivity function

sensitivity function T (s ) = β (s )S (s ) and control effort function CS (s ) = C (s )S (s ) , through: sup CS ( jω ) < CS max (ω ) (8) to limit the solicitation level of the plant input, sup S ( jω ) < S max (ω ) (9) to ensure a satisfactory rejection of plant output disturbances, and inf T ( jω ) > Tmin (ω ) (10) and (11) sup T ( jω ) > Tmax (ω ) , to limit the sluggishness of the responses, and to ensure a satisfactory rejection of measurement noise. b

Qr

Qr

a

Qmax (-180°,0dB)

Qmax (-180°,0dB)

ωu

ωu ωcg

ωcg

(a)

(b)

Fig. 2. Open-loop placement by optimal approach: (a) unspecified template (b) optimal template 2.5 Parametric synthesis of the controller

0 arg (β ( j ω ))

Once the optimal nominal open loop transfer function is determined, fractional controller C F (s ) is defined by its frequency response: β ( jω ) C F ( jω ) = , (12) G0 ( jω )

ωh Fig. 1. Effect of parameters a , b , ωb , ωh and Y0 on the asymptotic Nichols locus of β ( jω )

In fact, the tangency condition of the Nichols locus of β (s ) with a Qr Nichols magnitude contour reduces parameterization to three independent high level parameters [Oustaloup and Mathieu, 1999]: the open loop gain Y0 at ωr , and the corner frequencies ωb and ωh . The controller tuning is thus reduced to the computation of three parameter optimal values, which is the number of parameters required for standard PID tuning. The fact that only three parameters are needed for controller synthesis is essential from an industrial point of view and of major interest for computational time during optimization. 2.4 Optimization for optimal open loop determination Objective function Generally, the optimal template is achieved through minimization of the cost function

J = (Qmax − Qr )2 + (Qmin − Qr )2 , or

J reduced = (Qmax − Qr )2 ,

(6)

where G0 ( jω ) is the frequency response of the nominal plant. The synthesis of the rational controller C R (s ) is then achieved through identification of the ideal frequency response C F ( jω ) by that of a low order transfer function using any frequency domain system identification technique [Oustaloup and Mathieu, 1999].

3. CRONE CONTROL OF LPV SYSTEMS 3.1 Principles Control of LPV systems is often obtained in industry by freezing the time variations of the system around some operating points, performing a synthesis for each system and then interpolating the controller parameters. Such an approach does not guarantee performance or stability outside the operating points, nor even when dynamically passing on them. The approach proposed in the paper derives from methods presented by Gahinet, Apkarian and Biannic (see [Apkarian et al., 1995], [Apkarian and Gahinet, 1995]) in which the controller

is LPV, its parameters being automatically scheduled on those of the system thus ensuring global stability and performance. The CRONE approach enhances this method as it leads to robust controller that can handle the system perturbations as well as the perturbations of the measured varying parameters. The general interconnection structure for the LPV CRONE controller design is presented on fig. 3. measures controller

θ

θ

C C(ps ))

G G((ps))

Note that a similar approach can be used with a polytopic description of the plant [Moze, 2007]. 3.3 LPV CRONE controller structure

LPV controller is directly obtained from the definition of the open loop β = Gc (θ ) o C (θ ) , (20) where o stands for the series connection operator:

plant

C (θ ) = Gc (θ )−1 o β .

−

Fig. 3. General interconnection structure for the design of LPV CRONE controller The principle is thus to associate: • the CRONE approach that consists in dealing with the plant perturbations through an optimal open loop parameterization, • and the LPV controller formulation whose varying parameters are scheduled on the plant varying parameters. The idea is first to calculate the LPV controller from the open loop and the nominal LPV plant, then to verify that the open loop behaviour is robust. In the sequel, distinction will hence be made between the perturbed varying plant G (θ ) and the nominal plant Gc (θ ) whose perturbations are considered in a nominal state. The latter is used to obtain the controller structure while the former is used for optimization. 3.2 LPV plant representation In the paper, the plant is supposed to admit a LFT description such that: G (θ ) = Fu (Gm (s ), Θ(t )) , (13) and Gc (θ ) = Fu (Gm (s ), Θ 0 (t )) = Fu (Gcm (s ), θ (t )) , where s is the Laplace variable, Fu denotes the upper linear fractional transformation defined by   M 11 Fu    M   21

perturbations that conjointly act on the plant phase and gain. ∆ m 0 is then composed of the nominal values of the perturbations.

M 12   , K  = M 22 + M 21K (I − M 11 K )−1 M 21 , M 22  

 Gmθθ (s ) Gm∆θ (s ) GmGθ (s )    Gm (s ) =  Gmθ∆ (s ) Gm∆∆ (s ) GmG∆ (s )  , G   mθG (s ) Gm∆G (s ) GmGG (s )

(14)

(15)

and

(21)

The inverted transfer in (21) can be directly obtained when the direct transfer GcmGG (s ) in (16) is invertible. A constraint on open loop high frequency order is added otherwise. The direct transfer is invertible. Combined with (21) and (13), the relation

Gc (θ )−1 = Fu (G 'cm (s ), θ m (t )) , where

−1 G − GcmGθ GcmGG GcmθG G ' cm (s ) =  cmθθ − 1  − G G cmGG cmθG 

(

)

−1

C (θ ) = β h−11 o Gc (θ ) o β 2 , (27) where β 2 is the open loop of high frequency order nh 2 . Relation (24) then holds with 0  1  , Ccm (s ) = G ' 'cm (s ) (28) 0 β 2 (s ) 

Θ(t ) = θ m (t )T , ∆ mT , and Θ 0 (t ) = θ m (t )T , ∆ m0T where θ m (t ) = diag (θ1 (t ), ...,θ1 (t ), ..., θ nθ (t ), ..., θ nθ (t )) , and ∆ m = diag ∆1 (s ), ..., ∆ q (s ), δ1I , ..., δ r I , ε1I , ..., ε c I ,

3.4 Objectives and constraints

T

(

(

)

), T

(17) (18) (19)

in which

∆ i ∈ C k i × k i , δ i ∈ R and ε i ∈ C respectively represent the neglected dynamics, the parametric perturbations, and the

(23)

The direct transfer is not invertible. This case happens when the plant is strictly proper. Open loop low pass filter is then decomposed into β h (s ) = β h1 (s ) β h 2 (s ) , (26) β h1 (s ) and β h2 (s ) being low pass filters of the form (5) of order nh1 and nh 2 respectively such that β h1 (s )−1 GcmGG (s ) is invertible and nh 2 = nh − nh1 . LPV controller is then obtained using

where

)

−1  GcmGθ GcmGG , −1  GcmGG 

in which any dependence to Laplace variable s is omitted for brevity, then permits the following controller expression: C (θ ) = Fu (Ccm (s ), θ m (t )) , (24) where 0  1 . Ccm (s ) = G 'cm (s ) (25) 0 β (s ) 

(s ) GcmGθ (s ) G Gcm (s ) =  cmθθ (16) .  GcmθG (s ) GcmGG (s ) The nθ varying parameters and the uncertainties are taken into account through

(

(22)

(

)

−1  − GcmGθ β h−11GcmGG β h−11GcmθG G G ' cm (s ) =  cmθθ −1  − β h−11GcmGG β h−11GcmθG 

(

)

(

GcmGθ β h−11GcmGG

(β

)

−1 −1 h1 G cmGG

)

−1 

 .  

(29)

As in LPV case frequency responses of sensitivity functions are not available, minimisation of criterion (7) is performed through minimization of input-output L 2 gain of the closed loop augmented systems that meets the small gain theorem. Theorem 1 (Small Gain) [Khalil, 2000]: The system formed by interconnecting two causal operators M and Θ with finite L 2 gains γ 1 and γ 2 respectively as presented on figure 4 has finite L 2 gain given by

γ=

γ1

.

1 − γ 1γ 2

(30)

W2 (s ) and W3 (s ) are added to frequency constrain the sensitivity functions.

Furthermore, closed loop is stable if γ 1γ 2 < 1 . θm

Θ

θm eθC

sθC

eθG

M

This theorem permits to deduce that L 2 gain of SISO system

(

)

H = Fu M aug (s ), Θ where Θ is any causal operator verifying

Θ

L

2

≤ 1 , verifies H

M ' (s )

H∞

L

2

(31)

(

)

(32)

or equivalently if ∀ω , M ' ( jω )* M ' ( jω ) − I < 0 . (33) In order to reduce pessimism associated with small gain theorem, scaling matrices D and G are added. The scaling D takes into account the structure of Θ and verifies DΘ = ΘD . The scaling G takes into account the nature (real or complex) of the stationary part of Θ . Scalings associated with Θ given by (17) are thus of the form: D = diag (DLPV , DLTI ) , and G = diag (0, G LTI ) , (34) with  d1 (ω )I ,..., d d (ω )I ,    , (35) DLTI (ω ) = diag  D1 (ω ),..., Dr (ω ),   c c   D1 (ω ),..., Dc (ω )  GLTI (ω ) = diag (0 , G1 (ω ),..., Gr (ω ), 0) , where di ∈ R , di > 0 , Di ∈ C ri × ri , Di = Di * > 0 and Dic ∈ C ci × ci , Dic = Dic* > 0 , and where Gi ∈ C ri × ri , Gi = Gi* , DLPV being of the form DLPV = diag (L1 ,..., Lnθ ) , Li = L*i . Relation (33) in this case reduces to the problem: max min α < 1 (36) D, G

(

)

∀ω , M ' ( jω )* DM ' ( jω ) + j GM ' ( jω ) − M ' ( jω )* G − α 2 D < 0 ,

where ω is taken in a discrete set ω = {ω1 , ..., ω2 } for computational feasibility, as dense as possible.

Objective The CRONE robustness objective is now obtained by minimizing γ in relation (36), M aug ( jω ) in (32) being replaced by transfer matrix M crit ( jω ) between input and output

(

sθTC , sθTG , s∆T , z

(

)

T eθTC , eθTG , e∆T , r

) , as presented on figure 5. T

Constraints Constraints are similarly taken into account by ensuring bounds on L 2 gain of transfers associated with sensitivity functions presented in section 2.4. This approach leads to verify relation (36) with transfer matrix M const ( jω ) between

(

input

)

T sθTC , sθTG , s∆T , e1 , e2

(eθ

Cθ 1

C11

− d

u

−

W3 (s )

s∆

Gθθ Gθ∆

G∆θ G∆∆

G1θ G1∆

Gθ 1

G∆1

G11

sθG

z

M crit (s )

Pp W1 (s )

M ' (s ) = M aug (s ) . diag I , γ −1 ,

ω

C1θ

< γ if