Hossni Zebiri

Benjamin Mourllion

Michel Basset

Frequency Limited H ∞ -Controller Order Reduction for Linear Parameter Varying Systems

Keywords: linear parameter varying system, H∞-control, generalised reachability and observability Gramians, balanced truncation, semi-active suspension control

In this paper, a controller order-reduction method for linear parameter varying systems is presented. The proposed method is based on the frequency weighted balanced truncation technique, which has the advantage to reduce the order in a specific frequency range. The approach is discussed and is proved to preserve the closed-loop stability with a guaranteed upper error bound. Effectiveness and performance of the obtained reduced-order controller, are investigated by applying it to an automotive semi-active suspension control. The obtained simulation results show that objectives such as the road handling and the passenger comfort realised with the reduced-order controller are kept in the same performance level as with the full order controller. Moreover, a comparison with an other order reduction method is shown and confirms the advantage of the developed method.

Notations

R

: fields of real numbers ≺ ( ), ( ) : negative (semi-negative), positive (semi-positive) definite S n 0 : cone of n × n symmetric positive definite matrices R, O : reachability and observability Gramians P (ρ), Q(ρ)

: generalised reachability and observability Gramians P Ω (ρ), Q Ω (ρ) : frequency limited generalised reachability and observability Gramians σ i : ith generalised Hankel singular value γ i : ith frequency limited generalised Hankel singular value S * : complex conjugate transpose of S A -T : (A -1 ) T = (A T ) -1 I n

: n × n identity matrix

Introduction

Most of physical systems are inherently non-linear and almost all of them have parameter dependent representations. Considering those aspects and the advantages of linear control theory, more and more control strategies use Linear Parameter Varying (LPV) models. In the same way, and as operating conditions may change, the closed-loop performance designed by an LTI controller may be degraded. To overcome this problem, the design of parameter dependent controller is more suitable. Consequently, a major part of the recently developed control strategies is based on optimal and/or robust control. The H ∞ -control strategies have the advantage to design controllers achieving stabilisation with guaranteed performance. However, these techniques usually produce high order controllers. The design and the use of such high-order controllers can lead to numerical difficulties. This is why their use is still limited in the engineering field. Thus, for these practical reasons, low-order controllers are particularly preferred: their simple dynamics are easier to manage, they require less computing cost and simpler software can be finally implemented. Then, an order reduction step appears of great interest. Simultaneously, the reduction process should always preserve the closed-loop stability and should guarantee a level of performance close to the one obtained with the full-order controller. In a general way, a low-order controller can be obtained through the direct method or the indirect ones (see Figure 1). Considering the direct method, the final low-order controller is obtained directly from the high-order plant and the order is generally fixed beforehand [START_REF] Ankelhed | A partially augmented lagrangian method for low order h-infinity controller synthesis using rational constraints[END_REF][START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers. Automatic Control[END_REF].

In the indirect ways, the low-order controller is the result of a model or a controller order reduction step: either a reduced-order plant is found and then a controller is designed or a high-order controller is first designed and then a controller order reduction step is performed. The last method is widely used to obtain reduced-order controllers [START_REF] Kong | Controller reduction for linear systems[END_REF][START_REF] Goddard | Performance-Preserving Controller Approximation[END_REF]. For LTI systems, a large investigation has been made on model order reduction procedures. In this domain, methods based on balanced realisation are extensively used [START_REF] Gugercin | A survey of model reduction by balanced truncation and some new results[END_REF]. First introduced by [START_REF] Mullis | Synthesis of minimum roundoff noise fixed point digital filters[END_REF] and later in the systems and control literature by [START_REF] Moore | Principal component analysis in linear systems: Controllability, observability, and model reduction[END_REF], the balanced truncation has been a significant contribution to system theory. In [START_REF] Pernebo | Model reduction via balanced state space representations[END_REF], the stability of the reduced-order model is preserved.

Then, [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their l ∞ error bounds[END_REF] has proposed an H ∞ -norm upper bound of the approximation error.

Based on this, BT method has been proposed to reduce the order of controllers for LTI systems [START_REF] Liu | Singular perturbation approximation of balanced systems[END_REF][START_REF] Zhou | Structurally balanced controller order reduction with guaranteed closed loop performance[END_REF]. The extension of these reduction techniques to LPV-systems is still in progress.

Generally, these methods substitute the use of LTI Gramians by using parameter/time varying equivalents [START_REF] Sandberg | Balanced truncation of linear time-varying systems[END_REF][START_REF] Wood | Approximation of linear parameter-varying systems[END_REF]. A generalised method with unbounded rate parameter model is given in [START_REF] El-Zobaidi | Robust control and model and controller reduction of linear parameter varying systems[END_REF]. In the same way, an effective BT method for H ∞ -LPV-controller order reduction is proposed in [START_REF] Widowati | Controller reduction of parameter dependent systems[END_REF].

The balanced truncation produces good approximations in high frequency, whereas for control propose, an acceptable reduction is often required at intermediate and/or low frequencies. To solve this problem, the so-called Frequency Weighted Balancing Truncation FWBT has been proposed firstly for LTI systems in [START_REF] Enns | Model reduction with balanced realizations: An error bound and a frequency weighted generalization[END_REF]. In this approach, two weighting functions are included in the procedure to reduce the model in a certain frequency range. However, the stability preservation is not guaranteed anymore. To tackle this problem, a modified version assuming that the full-order model does not have any common pole/zero with the weighting functions is proposed in [START_REF] Lin | Model reduction via frequency weighted balanced realization[END_REF].

Then, [START_REF] Wang | A new frequency-weighted balanced truncation method and an error bound. Automatic Control[END_REF][START_REF] Varga | Accuracy enhancing methods for the frequencyweighted balancing related model reduction[END_REF] have developed a new approach based on the Enns method without the previous assumption to guarantee the stability. It gives also an upper error bound. Others upper bounds have also been established by [START_REF] Sreeram | Frequency weighted balanced reduction technique: a generalization and an error bound[END_REF], [START_REF] Kim | Error bound for transfer function order reduction using frequency weighted balanced truncation[END_REF] and [START_REF] Zhou | A comparative study of H ∞ controller reduction methods[END_REF]. The frequency range is explicitly defined instead of the two frequency weights in [START_REF] Gawronski | Model reduction in limited time and frequency intervals[END_REF]. More recently, in [START_REF] Gugercin | A survey of model reduction by balanced truncation and some new results[END_REF], the method is modified and an error upper bound to the relative error is given. Note that the stated techniques are developed for the stable LTI-systems and no guarantee is given to preserve the passivity of the systems. Recent work in this direction are given in [START_REF] Li | Passivity-preserving model reduction with finite frequency H ∞ approximation performance[END_REF][START_REF] Li | Frequency-limited H -inf inity model reduction for positive systems[END_REF] where the H ∞ -norm of the error is bounded and the positive realness of the obtained model is guaranteed.

Based on these recent results in the literature, the aim of this paper is to adapt the FWBT method ( [START_REF] Gugercin | A survey of model reduction by balanced truncation and some new results[END_REF]) in order to reduce the order of an H ∞ -LPV controller. For the development of this approach, the generalised Gramian framework is used [START_REF] Shaker | Generalized gramian framework for model/controller order reduction of switched systems[END_REF]. The paper is organised as follows: the LPV-systems, their properties and the the full-order H ∞ -LPV-controller design are introduced in section 2. In Section 3. the Generalised Gramian framework for LPV-systems is given and the main proposed method is detailed and discussed.

To emphasize the effectiveness of the developed method, the obtained reduced-order controller is evaluated on a practical engineering problem: the control of an automotive semi-active suspension.

The performance of the obtained reduced-order controller are investigated and compared with another reduced-order method based on BT approach [START_REF] Widowati | Model reduction of LPV control with bounded parameter variation rates[END_REF] in Section 4. Final conclusion is given in Section 5.

Problem Statement

LPV-Systems

Linear Parameter Varying LPV models represent systems whose sate-space descriptions are known functions of varying parameters. This notion of LPV-system has been introduced in context of gain scheduling [START_REF] Shamma | Analysis of gain scheduled control for nonlinear plants[END_REF] and has been also widely used in control design for non-linear systems [START_REF] Packard | Gain scheduling the LPV way[END_REF].

Considering a compact subset Δ ρ ⊂ R s , an LPV-system can be described by the following the state-space realisation

ẋ(t) = A(ρ(t))x(t) + B(ρ(t))u(t) y(t) = C(ρ(t))x(t) + D(ρ(t))u(t) ∀ t ≥ 0 (1)
where

A : Δ ρ → R n×n , B : Δ ρ → R n×m , C : Δ ρ → R p×n , D : Δ ρ → R p×m .
The state-space representation (1) can be also written as

G(ρ) = A(ρ) B(ρ) C(ρ) D(ρ) . (2)
The exogenous parameter vector ρ is varying in a bounded set P ρ defined as

P ρ = {ρ : R + → Δ ρ | ρ i (t) ≤ ρ i (t) ≤ ρ i (t), i = 1, 2, . . . , s} t → ρ(t) (3)
where ρ i is the ith component of the ρ vector. The ρ function is not known in advance but is assumed to be bounded and measurable on-line. This assumption ensures that all trajectories of the system are contained in those ones of the LPV-model.

It is noted that some LPV-models could be written as function of system states [START_REF] Sename | Robust Control and Linear Parameter Varying approaches: Application to Vehicle Dynamics[END_REF]. Such models known as quasi-LPV -systems are used for the modelling of non-linear systems.

Definition 1 (Quadratic Stability): System (1) is said to be quadratically stable if the positive definite quadratic form

V ρ : x → x T P 0 x, P 0 ∈ S n 0 (4)
is a Lyapunov function for (1). A such Lyapunov function is often referred to a common Lyapunov function or a parameter-independent Lyapunov function.

Proposition 1 ([Bri15]): The system (1) is quadratically stable if and only if there exists a matrix P ∈ S n 0 such that the LMI A T (ρ)P + P A(ρ) ≺ 0 (5) holds for all ρ ∈ Δ ρ .

Moreover, the induced L 2 -gain notion is of a big interest for the proposed approach. Indeed, an upper bound is given to the L 2 -norm of error between the full and the reduced order models. It can be defined as Definition 2 (Induced L 2 -gain [START_REF] Goddard | Performance-Preserving Controller Approximation[END_REF]): Given a quadratic-stable LPV-system (1) with zero initial conditions, the induced L 2 -gain is defined as

G ρ i,2 = sup ρ∈P ρ sup u∈L 2 u =0 y 2 u 2 (6)
The induced L 2 -norm represents the largest induced gain from inputs in L 2 to outputs in L 2 over the set of all causal linear operators described by the LPV system.

LPV-Controller Synthesis

Consider a general LPV-system represented under the following state-space realisation

   ẋ(t) = A(ρ(t))x(t) + B 1 (ρ(t))w(t) + B 2 (ρ(t))u(t) z(t) = C 1 (ρ(t))x(t) + D 11 (ρ(t))w(t) + D 12 (ρ(t))u(t) y(t) = C 2 (ρ(t))x(t) + D 21 (ρ(t))w(t) + D 22 (ρ(t))u(t) ∀t. (7)
where

x(t) ∈ R n , u(t) ∈ R n u , w(t) ∈ R n w , z(t) ∈ R n z , y(t) ∈ R n y
are respectively the state, the input, the disturbance, the controlled output and the measured output. Then, A :

Δ ρ → R n×n , B 1 : Δ ρ → R n×n w , B 2 : Δ ρ → R n×n u , C 1 : Δ ρ → R n z ×n , C 2 : Δ ρ → R n y ×n , D 11 : Δ ρ → R n z ×n w , D 12 : Δ ρ → R n z ×n u , D 21 : Δ ρ → R n y ×n w and D 22 : Δ ρ → R n y ×n u .
The H ∞ -LPV-controller associated to the LPV-system (7) is defined by

K(ρ) : ẋK (t) = A K (ρ(t))x K (t) + B K (ρ(t))y(t) u(t) = C K (ρ(t))x K (t) + D K (ρ(t))y(t) (8)
where x K ∈ R nK , y(t) ∈ R ny and u(t) ∈ R nu are respectively the states, the inputs and outputs of the controller K(ρ).

A K : Δ ρ → R nK×nK , B K : Δ ρ → R nK×ny , C K : Δ ρ → R nu×nK and D K : Δ ρ → R nu × ny .
The H ∞ -LPV-controller synthesis concerns the design of an LPV global controller that guarantees both stability and performance for all parameters variations defined in the set Δ ρ . To guarantee the closed-loop system quadratic stability and to satisfy the H ∞ -performance criteria, the approach developed in [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF] is used to design the H ∞ -LPV-controller. It is assumed that

• The matrices B 2 , D 12 , C 2 and D 21 are parameter independent and D 22 = 0. • The parameter-dependent matrix pair [A(ρ), B 2 (ρ)] is stabilisable and detectable ∀ρ ∈ Δ ρ . • The matrix [C 2 (ρ), D 21 (ρ)] has full row rank ∀ρ ∈ Δ ρ .
Then, for a given real positive scalar γ and a parameter dependent quadruplet matrices ( ÂK , BK , ĈK , DK ), there exist two parameter independent symmetric matrices X and Y such that the sufficient condition that solves the H ∞ -LPV problem is given by the following LMIs

∀ρ ∈ Δ ρ     A(ρ)X + B 2 ĈK (ρ) + ( * ) * * * ÂK (ρ) + A T (ρ) + C T 2 DT K (ρ)B T 2 Y A(ρ) + BK (ρ)C 2 + ( * ) * * B T 1 (ρ) + D T 21 DT K (ρ)B T 2 B T 1 (ρ)Y + D T 21 BT K (ρ) -γI nw * C 1 (ρ)X + D 12 ĈK (ρ) C 1 (ρ) + D 12 DK (ρ)C 2 D 11 (ρ) + D 12 DK (ρ)C 2 -γI nz     ≺ 0, (9)
X I n I n Y 0, (10)
where terms denoted ( * ) are induced by symmetry, for example

M + N + ( * ) * K L = M + N + M T + N T K T K L
The LPV-controller of the form (8) is then reconstructed as

D K (ρ) = DK (ρ) C K (ρ) = ( ĈK (ρ) -D K (ρ)C 2 X)M -T B K (ρ) = N -1 ( BK (ρ) -Y B 2 D K (ρ)) A K (ρ) = N -1 ( ÂK (ρ) -Y A(ρ)X -Y B 2 D K (ρ)C 2 X)M -T -B K (ρ)C 2 XM -T -N -1 Y B 2 C K (ρ), (11)
where M and N are defined such that M N T = I n -XY and which can be solved through a singular value decomposition and a Cholesky factorisation.

Full-Order Controller Design

Before introducing the approximation method, the full-order controller should be designed as shown in Section 2.2. Then, let

T zw (ρ) = A(ρ) B(ρ) C(ρ) D(ρ) = LFT A(ρ) B(ρ) C(ρ) D(ρ) , A K (ρ) B K (ρ) C K (ρ) D K (ρ) , (12)
be the state-space realisation of the closed-loop system considering the augmented model ( 7) and its H ∞ -controller defined in (8) with:

A(ρ) = A(ρ) + B 2 D K (ρ)C 2 B 2 C K (ρ) B K (ρ)C 2 A K (ρ) , B(ρ) = B 1 (ρ) + B 2 D K (ρ)D 21 B K (ρ)D 21 , C(ρ) = C 1 (ρ) + D 12 D K (ρ)C 2 D 12 C K (ρ) and D(ρ) = D 11 (ρ) + D 12 D K (ρ)D 21 .
Note that the closed-loop system T zw (ρ) is an LPV-system in its own right which is quadratic stable by construction. In order to apply the proposed method, the T zw (ρ) state space representation is assumed to be minimal.

Main Contribution

In this section, the new controller order reduction method for LPV-controllers is proposed. The method is based on the Frequency Weighted Balanced Truncation FWBT approach used for the LTI-models order reduction. The FWBT is proposed to reduce the model-order over a known frequency range

[ω 1 , ω 2 ] in [GJ90]
. This approach uses the classical reachability and observability Gramians.

Inspired by this work, the proposed method is an extension in two directions:

• Extended to the LPV-case.

• Applied to reduce the controller order (rather than the model order).

To this end, the classical reachability and observability Gramians for LTI systems are introduced first. Then, the equivalent ones of the LPV-case are defined. Based on this, the so-called generalised

Gramians are introduced then used to perform the proposed controller order reduction approach. The relation between the Gramians and generalised Gramians is given in Lemma 1.

Frequency Limited Generalised Gramians

The proposed method is based on the FWBT method where the key notions are the so-called reachability and observability Gramians.

Definition 3 (Gramians): Considering the quadratic stable and minimal realisation of the closedloop LPV-system T zw (ρ) given in (12), Let us define R(ρ) and O(ρ), the related reachability and observability Gramians respectively, they are defined as the solution of the two parameter dependent Lyapunov equations

- s i=1 v i ∂R(ρ) ∂ρ i + A(ρ)R(ρ) + R(ρ)A T (ρ) + B(ρ)B T (ρ) = 0, (13)
s i=1 v i ∂O(ρ) ∂ρ i + A T (ρ)O(ρ) + O(ρ)A(ρ) + C T (ρ)C(ρ) = 0. ( 14
)
For a given parameter trajectory ρ, let Φ ρ (t, 0) be the state-transition matrix. Then, the functional R(ρ) and O(ρ), respectively the reachability and observability Gramians of the closed-loop LPVsystem (12), are expressed such that

R(ρ) = 0 -∞ Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) dt, (15)
O(ρ) = ∞ 0 Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) dt. (16)
Definition 4 (Generalised Gramians): Let P(ρ) and Q(ρ) be respectively the generalised reachability and observability Gramians of the quadratic stable and minimal closed-loop LPV-system (12) given as

∀(ρ, v) ∈ Δ ρ × Δ v - s i=1 v i ∂P(ρ) ∂ρ i + A(ρ)P(ρ) + P(ρ)A T (ρ) + B(ρ)B T (ρ) ≺ 0 (17) s i=1 v i ∂Q(ρ) ∂ρ i + A T (ρ)Q(ρ) + Q(ρ)A(ρ) + C T (ρ)C(ρ) ≺ 0. (18)
Indeed, Lyapunov inequalities (rather than Lyapunov equations) are solved to compute generalised Gramians. This linear matrix inequality (LMI) approach to the model reduction problem is particularly useful when some structures need to be preserved in the process of model reduction. Controller reduction is a typical example of this type of problems [START_REF] Shaker | Generalized gramian framework for model/controller order reduction of switched systems[END_REF]. Note that the physical interpretations of generalised Gramians are similar to ordinary Gramians. Considering T zw (ρ), R(ρ) and O(ρ), the following lemma introduces useful results about the relation between Gramians and generalised Gramians.

Lemma 1: Let T zw (ρ) be a minimal state-space realisation of the quadratic stable and minimal closed-loop system defined by (12). Then,

∀ρ ∈ P ρ R(ρ) ≺ P(ρ(0)) O(ρ) ≺ Q(ρ(0)). ( 19
)
where R(ρ), O(ρ), P(ρ) and Q(ρ) are the solutions of (15), ( 16), ( 17) and (18) respectively.

Proof: By multiplying on the left by Φ ρ (0, t) and on the right by Φ T ρ (0, t), the relation (17) yields

-Φ ρ (0, t) s i=1 v i ∂P ∂ρ i Φ T ρ (0, t) + Φ ρ (0, t)A(ρ(t))P(ρ)Φ T ρ (0, t) + Φ ρ (0, t)P(ρ)A T (ρ(t))Φ T ρ (0, t) + Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) ≺ 0. ( 20
) Using fact that ∂ ∂t0 Φ ρ (t, t 0 ) = -Φ ρ (t, t 0 )A(ρ(t 0 )), then (20) yields - d dt Φ ρ (0, t)P(ρ)Φ T ρ (0, t) + Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) ≺ 0. ( 21
)
which on integrating over the semi-infinite time axis (-∞, 0] and considering that lim t→-∞ Φ ρ (0, t) = 0, gives

P(ρ(0)) 0 -∞ Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) dt R(ρ) (22)
Similarly, let us multiply on the right by the closed-loop state-transition matrix Φ ρ (t, 0) and on the left by Φ T ρ (t, 0), the relation (18):

Φ T ρ (t, 0) s i=1 v i ∂Q ∂ρ i Φ ρ (t, 0) + Φ T ρ (t, 0)A T (ρ(t))Q(ρ)Φ ρ (t, 0) + Φ T ρ (t, 0)Q(ρ)A(ρ(t))Φ ρ (t, 0) + Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) ≺ 0 (23) Using the fact that ∂ ∂t Φ ρ (t, t 0 ) = A(ρ(t))Φ ρ (t, t 0 ), then (23) yields d dt Φ T ρ (t, 0)Q(ρ)Φ ρ (t, 0) + Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) ≺ 0. ( 24
)
Integrating the last expression from 0 to +∞ gives

Q(ρ(0)) +∞ 0 Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) dt O(ρ) (25)
3.2 Frequency-Limited Order Reduction

For a given parameter trajectory ρ, let consider Φ ρ (t, 0) the state-transition matrix of the closed-loop system (12). Then, let us define

f ρ (t) = Φ ρ (0, t)B(ρ(t))H(-t) g ρ (t) = C(ρ(t))Φ ρ (t, 0)H(t) ∀t, ( 26
)
where H is the heaviside step function.

By considering Ω = [ω 1 ; ω 2 ] the frequency range where order-reduction is desired to be better, the following definition is given Definition 5 (Frequency Limited Gramians): Let R Ω (ρ) and O Ω (ρ) two frequency dependent terms defined as

R Ω (ρ) = R ω 2 (ρ) -R ω 1 (ρ) and O Ω (ρ) = O ω 2 (ρ) -O ω 1 (ρ)
where

R ω (ρ) = +ω -ω F ρ (w)F * ρ (w)dw (27) O ω (ρ) = +ω -ω G * ρ (w)G ρ (w)dw (28)
with F ρ (resp. G ρ ) is the Fourier transform of f ρ (resp. g ρ ). Then, the functional RΩ (ρ) and ÔΩ (ρ), respectively the frequency limited reachability and observability Gramians of the closed-loop LPVsystem (12), are defined as the solutions of

∀(ρ, v) ∈ Δ ρ × Δ v - s i=1 v i ∂ RΩ (ρ) ∂ρ i + A(ρ) RΩ (ρ) + RΩ (ρ)A T (ρ) + R Ω (ρ) = 0 (29) s i=1 v i ∂ ÔΩ (ρ) ∂ρ i + A T (ρ) ÔΩ (ρ) + ÔΩ (ρ)A(ρ) + O Ω (ρ) = 0. ( 30
)
The quantities R Ω (ρ) and O Ω (ρ) have the following eigenvalues decomposition

R Ω (ρ) = U Ω (ρ)diag (λ 1 (ρ), . . . , λ n K (ρ)) U T Ω (ρ) (31) O Ω (ρ) = V Ω (ρ)diag (δ 1 (ρ), . . . , δ n K (ρ)) V T Ω (ρ) (32)
with

|λ 1 | ≥ . . . |λ n K | ≥ 0 and |δ 1 | ≥ . . . |δ n K | ≥ 0. Let u ρ ≤ n K and v ρ ≤ n K be respectively the ranks of R Ω (ρ) and O Ω (ρ).
Based on these definitions, let us define the two quantities:

B Ω (ρ) = U Ω (ρ)diag(|λ 1 (ρ)| 1 2 , . . . , |λ u ρ (ρ)| 1 2 , 0, . . . , 0), and (33)
C Ω (ρ) = diag(|δ 1 (ρ)| 1 2 , . . . , |δ vρ (ρ)| 1 2 , 0, . . . , 0)V T Ω (ρ). ( 34
)
Definition 6 (Modified Frequency Limited Generalised Gramians): Consider K(ρ), the full-order stabilising LPV-controller given in (8). Let PΩ (ρ) = P1 (ρ) 0 0 P2 (ρ) 0 and

QΩ (ρ) = Q1 (ρ) 0 0
Q2 (ρ) 0 be the modified frequency limited reachability and observability Gramians defined as the solutions of the following Lyapunov equations

∀(ρ, v) ∈ Δ ρ × Δ v - s i=1 v i ∂ PΩ (ρ) ∂ρ i + A(ρ) PΩ (ρ) + PΩ (ρ)A T (ρ) + B Ω (ρ)B T Ω (ρ) = 0 (35) s i=1 v i ∂ QΩ (ρ) ∂ρ i + A T (ρ) QΩ (ρ) + QΩ (ρ)A(ρ) + C T Ω (ρ)C Ω (ρ) = 0. ( 36
)
For the generalisation, we have the following inequalities:

-

s i=1 v i ∂P Ω (ρ) ∂ρ i + A(ρ)P Ω (ρ) + P Ω (ρ)A T (ρ) + B Ω (ρ)B T Ω (ρ) ≺ 0 (37) s i=1 v i ∂Q Ω (ρ) ∂ρ i + A T (ρ)Q Ω (ρ) + Q Ω (ρ)A(ρ) + C T Ω (ρ)C Ω (ρ) ≺ 0. ( 38
)
with

P Ω (ρ) = P 1 (ρ) 0 0 P 2 (ρ) 0 and Q Ω (ρ) = Q 1 (ρ) 0 0 Q 2 (ρ) 0.
If the block diagonal solutions P Ω (ρ) and Q Ω (ρ) exist, then let T 1 (ρ) and T 2 (ρ) be two nonsingular matrices given such that

T -1 1 (ρ)P 1 (ρ)T -T 1 (ρ) = T T 1 (ρ)Q 1 (ρ)T 1 (ρ) = Σ 1 (ρ) = diag(ξ 1 (ρ), . . . , ξ n (ρ)), (39)
with ξ 1 (ρ) ≥ ξ 2 (ρ) ≥ • • • ≥ ξ n (ρ), and
T -1 2 (ρ)P 2 (ρ)T -T 2 (ρ) = T T 2 (ρ)Q 2 (ρ)T 2 (ρ) = Σ 2 (ρ) = diag(γ 1 (ρ), . . . , γ r (ρ) Σ21 , γ r+1 (ρ), . . . , γ nK (ρ) Σ22 ), (40)
with

γ 1 (ρ) ≥ γ 2 (ρ) ≥ • • • ≥ γ r (ρ) > γ r+1 (ρ) ≥ γ r+2 (ρ) ≥ • • • ≥ γ n K (ρ)
are the frequency limited generalised Hankel singular values of K(ρ) and r is the desired order for the reduced-order controller. The balanced realisation of K(ρ) can be written as

K(ρ) = T -1 2 (ρ)A K (ρ)T 2 (ρ) T -1 2 (ρ)B K (ρ) C K (ρ)T 2 (ρ) D K (ρ) = ÃK (ρ) BK (ρ) CK (ρ) DK (ρ) . (41)
Further, K(ρ) is partitioned as conformably with Σ 2 (ρ) as

K(ρ) =    ÂK (ρ) ÃK12 (ρ) ÃK21 (ρ) ÃK22 (ρ) BK (ρ) BK2 (ρ) ĈK (ρ) CK2 (ρ) DK (ρ)    . (42)
Finally, a truncation step is performed to obtain a reduced-order controller.

Definition 7: Given the balanced realisation K(ρ) defined in (42), let K(ρ) be the truncated realisation to the rth order and denoted as follows

K(ρ) = ÂK (ρ) BK (ρ) ĈK (ρ) DK (ρ) (43)
Furthermore, the reduced-order parameter dependent closed-loop system is given as

Tzw (ρ) = Â(ρ) B(ρ) Ĉ(ρ) D(ρ) =    A(ρ) + B 2 DK (ρ)C 2 B 2 ĈK (ρ) B 1 (ρ) + B 2 DK (ρ) DK (ρ)D 21 BK (ρ)C 2 ÂK (ρ) BK (ρ)D 21 C 1 (ρ) + D 12 DK (ρ)C 2 D 12 ĈK (ρ) D 11 (ρ) + D 12 DK (ρ)D 21    . ( 44
)
Theorem 1: Suppose K(ρ) is the stabilising parameter dependent controller defined in (8) such that the closed-loop transfer T zw (ρ) defined in (12) is minimal, quadratic stable and there exist Lyapunov inequality solutions P Ω (ρ) and Q Ω (ρ) such that (37) and (38) are satisfied. Let K(ρ) be the reduced-order controller defined in (43) and obtained by truncation. Then, the closed-loop system with the reduced-order controller Tzw (ρ) defined in (44) is stable. If in addition

rank[B(ρ), B Ω (ρ)] = rank[B Ω (ρ)], (45)
and

rank[C T (ρ), C T Ω (ρ)] = rank[C T Ω (ρ)], (46)
then, Tzw (ρ) is quadratic stable and satisfies

T zw (ρ) -Tzw (ρ) i,2 ≤ 2 J B (ρ) ∞ J C (ρ) ∞ nK i=r+1 γ i,ρ (47)
where

J B (ρ) := diag(|λ 1 | -1 2 (ρ), . . . , |λ uρ | -1 2 (ρ), 0, . . . , 0)U Ω (ρ)B(ρ) and J C (ρ) := C(ρ)V Ω (ρ)diag(|δ 1 | -1 2 (ρ), . . . , |δ vρ | -1 2 (ρ), 0, . . . , 0), Proof:
The reachability and the observability Gramians given in (15),( 16) can be expressed as

R(ρ) = +∞ -∞ f ρ (τ )f * ρ (τ ) dτ (48) O(ρ) = ∞ -∞ g * ρ (τ )g ρ (τ ) dτ (49)
where f ρ (τ ) and g ρ (τ ) are given in (26). Then, using Parseval relationship, the reachability and the observability Gramians could be expressed as follows

R(ρ) = 1 2π +∞ -∞ F ρ (w)F * ρ (w)dw (50)
O(ρ) = 1 2π +∞ -∞ G * ρ (w)G ρ (w)dw. (51)
By considering F ρ (resp.g ρ ) the Fourier transform of f ρ (resp.g ρ ), we can define R ω (ρ) and O ω (ρ) as the limited reachability and observability Gramians given in ( 28) and (28). Then, by setting Ω = [ω 1 ; ω 2 ] the frequency range where approximation is desired to be better. Then, a new modified terms R Ω (ρ) and O Ω (ρ) are defined. Since R Ω (ρ) and O Ω (ρ) are not guaranteed to be positive definite, stability of the reduced-order controller is not guaranteed. Then, an idea based on eigenvalues decomposition presented in [START_REF] Gugercin | A survey of model reduction by balanced truncation and some new results[END_REF] is proposed here to guarantee stability by providing an upper error bound. Indeed, the solution of (37) and ( 38) is performed instead of ( 29) and (30). In addition, the modified frequency limited Gramians in ( 35) and ( 36) are used instead of the ordinary ones defined in ( 35) and (36). In fact, Lemma 1 shows that the generalised Gramians could be an approximation of the ordinary Gramians. The transition to generalised Gramian framework might induce less accurate approximation but the order-reduction error still bounded. Finally the expression of the upper error bound given in (47) is found according to this Let Tzw (ρ) be the frequency limited balanced realisation of the full-order closed-loop system. Then, Tzw (ρ) is defined by

Tzw (ρ) = T -1 (ρ)A(ρ)T (ρ) T -1 (ρ)B(ρ) C(ρ)T (ρ) D(ρ) =    Â(ρ) Ã12 (ρ) Ã21 (ρ) Ã22 (ρ) B(ρ) B2 (ρ) Ĉ(ρ) C2 (ρ) D(ρ)    , ( 52
)
where

T (ρ) = diag(T 1 (ρ), T 2 (ρ)).
By considering the assumptions ( 45) and ( 46), there exist J B (ρ) and

J C (ρ) such that B(ρ) = B Ω (ρ)J B (ρ) and C(ρ) = J C (ρ)C Ω (ρ).
On the other hand, the reduction error is expressed as

T zw (ρ) -Tzw (ρ) i,2 = C(ρ)(sI -A(ρ)) -1 B(ρ) -Ĉ(ρ)(sI -Â(ρ)) -1 B(ρ) i,2 = J C (ρ) C Ω (ρ)(sI -A(ρ)) -1 B Ω (ρ) -Ĉ(ρ)(sI -Â(ρ)) -1 B(ρ) .J B (ρ) i,2 ≤ 2 J B (ρ) ∞ J C (ρ) ∞ nK i=r+1 γ i,ρ
Remark 1: The Assumptions (45) and (46) mean that there exist J B (ρ) and J C (ρ) such that

B(ρ) = B Ω (ρ)J B (ρ) and C(ρ) = J C (ρ)C Ω (ρ).
In addition, by following the steps in [WSL99, AC81, IGS14] it was shown that assumptions (45) and ( 46) are almost always true. Hence we expect that our approach will apply in most of the cases. Indeed, during our simulations, the assumptions have always been satisfied.

Algorithm 1 H ∞ -LPV-Controller Order Reduction Considering the LPV-plant described in (7) the reduced-order controller can be computed as follows Inputs: (A(ρ), B(ρ), C(ρ), D(ρ)).

Outputs: ( Â(ρ), B(ρ), Ĉ(ρ), D(ρ)). Assumptions: (A(ρ), B(ρ), C(ρ), D(ρ)) minimal. Algorithm:

(1) Compute K(ρ) the full-order controller according Section 2.2.

(2) Compute the closed-loop system T zw given in ( 12).

(3) Compute the generalised Gramians P(ρ) and Q(ρ) solutions of ( 17) and ( 18) respectively.

(4) Compute the balanced realisation K(ρ) of the full-order controller K(ρ) by: (a) Find T 2 , the basis change matrix according to (40).(For instance, Procedure in [START_REF] Moore | Controller reduction methods maintaining performance and robustness[END_REF] can be used). (b) Compute the balanced realisation K(ρ) as defined in (42).

(5) Compute the reduced-order controller K(ρ) from K(ρ) by truncation.

Case Study: Semi-Active Suspension Control

LPV-Model

In this section, the given method to reduce the LPV-controller is applied on a semi-active automotive suspension presented in [START_REF] Do | Multi-objective optimization by genetic algorithms in H ∞ /LPV control of semi-active suspension[END_REF]. Actually, when suspension modelling and control are considered, the vertical quarter car model is often used. This model allows to study the vertical behaviour of a vehicle according to the suspension characteristic (passive or controlled). Figure 3 shows the so-called vertical quarter car. Then, the dynamical equations of the system are given by

m s zs = -k s z def -F mr m us zus = k s z def + F mr -k t (z us -z r ) (53)
where F mr is the magneto-rheological force generated by the semi active suspension. According to the non-linear model of Guo [START_REF] Guo | Dynamic modeling of magnetorheological damper behaviors[END_REF], F mr can be expressed as follows

F mr = a 2 żdef + v 0 x 0 z def + a 1 tanh a 3 żdef + v 0 x 0 z def ( 54
)
with z def = z s -z us is the damper deflection (must be measured or estimated) and żdef = żs -żus is the damper velocity. Parameters a 2 , a 3 , v 0 and x 0 are constant, and a 1 is the controllable force such that

a 1 ∈ [a 1min ; a 1max ]. By defining ρ 1 = tanh a 3 żdef + v0 x0 z def , c mr = a 2 : MR damping coefficient, k mr = a 2 v0
x 0 : MR stiffness coefficient, Then, a state-space representation can be given by considering the state vector x s = [z s żs z us żus ] T and the exogenous input w = z r , as follows

     ẋs = A s x s + B s ρ 1 a 1 + B s1 w z = C s1 x s + D s1 ρ 1 a 1 y = C s x s (55)
with

A s =     0 1 0 0 -k s +k mr m s -c mr m s k s +k mr m s c mr m s 0 0 0 1 ks+kmr mus cmr mus -ks+kmr+kt mus -cmr mus     , B s =     0 -1 m s 0 -1 m us     , B s1 =     0 0 0 -k t mus     .
The measurement output is y = z s -z us , and the controlled outputs are chosen as z = [z s z s ] T , respectively the acceleration and the displacement of the sprung mass. Then

C s1 = -k s +k mr ms -c mr ms k s +k mr ms c mr ms 1 0 0 0 , D s1 = -1 ms 0 , C s = 1 0 -1 0 .
However, two constraints must be satisfied

(1) The control signal a 1 must be positive (dissipative constraint)

(2) The input matrices B s ρ 1 and D s1 ρ 1 must be constant to satisfy the LPV-H ∞ synthesis assumption.

The passivity problem is solved by defining a new control signal u = a 1 -F 0 where F 0 is the mean value of a 1 (F 0 = (a 1max -a 1min )/2). Then, the problem of the passivity on a 1 is recast to a simple saturation problem on u (u ∈ [-F 0 ; F 0 ]). With these modifications, (56) yields

       ẋs = (A s + B s2 ρ 1 C s2 x s C s2 )x s + B s ρ 1 u + B s1 w z = C s1 x s + D s1 ρ 1 u y = C s x s (56)
where

B s2 = 0 -F0 ms 0 F0 mus T and C s2 = -a3v0 x 0 a 3 -a3v0 x 0 -a 3 T .
To overcome the second problem, [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers. Automatic Control[END_REF] proposes to add a strictly proper filter F to make the controlled output matrices independent of the scheduling parameters

F : ẋf u = A f B f C f 0 x f u c (57)
Then, by defining ρ 2 = ρ1 Cs2xs and x = (x s x f ) T the system (56) can be represented as

     ẋ = A(ρ 1 , ρ 2 )x + Bu c + B 1 w z = C 1 (ρ 1 , ρ 2 )x y = Cx (58)
where

A(ρ 1 , ρ 2 ) = A s + ρ 2 B s2 C s2 ρ 1 B s C f 0 A f , B = 0 B f , B 1 = B s1 0 , C 1 (ρ 1 , ρ 2 ) = C s1 ρ 1 D s1 C f and C = C s 0 .

H ∞ -Controller Synthesis

By considering the LPV-model ( 58), an H ∞ controller is designed to guarantee the internal closedloop stability and to satisfy some required performance. In fact, the main objective and challenge of a controlled suspension system is to improve the comfort for car passengers simultaneously to the performance on road holding. The passenger comfort can be improved by isolating the vibrations transmitted from the road surface. Then, the "frequency response" from the road profile z r to the vehicle chassis acceleration zs must be kept small in the low frequency range. Then, a weighting function is designed as

W zs = w zs s 2 + ξ 11 w 11 s + w 2 11 s 2 + 2ξ 12 w 12 s + w 2 12
Furthermore, the road holding is evaluated from the unsprung mass (wheel) oscillations with respect to the road profile. This transfer should be kept small at high frequencies. Then, W z us is designed as W zr = 5 × 10 -3 is the road profile gain. Finally, the filter introduced in (57) is given as: F = wf s+wf . It is designed with a large bandwidth to decouple the input and the varying parameters, where w zs = 1, ξ 11 = 0.1, ξ 12 = 1, w 11 = 2π × 1 rad.s -1 , w 12 = 2π × 3 rad.s -1 , w zus = 10, ξ 21 = 0.3, ξ 22 = 1, w 21 = 2π × 9 rad.s -1 , w 22 = 2π × 9 rad.s -1 , w f = 90.34.

W zus =
Then, an interconnection between the LPV-model and these weighting functions are presented in Figure 4. Model parameters are obtained by considering experimental data in [TMFV + 13] and given as Table 3To carry out a controller satisfying these objectives, the H ∞ -LPV-synthesis is designed by using solution for polytopic systems: it consists in finding a global LPV-controller K(ρ 1 , ρ 2 ) which is a convex combination of local controllers obtained by solving the LMIs set at each vertex (formed by limits values of the varying parameters). All varying parameters are bounded:

ρ 1 ∈ [-1 ; 1], ρ 2 ∈ [0 ; 1].
For more details and explanation on H ∞ -LPV-control synthesis, see [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers. Automatic Control[END_REF][START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF]. The design method for LPV-systems is used like in [START_REF] Apkarian | Self-scheduled H ∞ linear parameter-varying systems[END_REF].

Numerical Issue

The proposed method requests the solution of two Lyapunov inequalities with an infinite number of constraints. These sets of infinite LMIs can be solved by gridding techniques. Then, some ap-proximations must be made by gridding the set Δ ρ with finite number L of points {ρ i } L i=1 [START_REF] Lee | Identification and Robust Control of Linear Parameter-Varying Systems[END_REF]. Moreover, the infinite variables P Ω (ρ) and Q Ω (ρ) in LMIs (37),(38) are approximated by combinations of scalar basis functions such as

P Ω (ρ) = N P j=1 φ j (ρ)P j 0 Q Ω (ρ) = NQ j=1 ϕ j (ρ)Q j 0 where P j = P T j , Q j = Q T j .
There is a large freedom in the choice of basis functions [START_REF] Wood | Control of Parameter-Dependent Mechanical Systems[END_REF]. For this example, the following choice is made:

{φ j } 13 j=1 = {ϕ j } 13 j=1 = {1, ρ 1 , ρ 2 , ρ 2 1 , ρ 2 2 , ρ 1 ρ 2 , ρ 1 ρ 2 2 , ρ 2 1 ρ 2 , ρ 2 1 ρ 2 2 , ρ 3 1 , ρ 3 2 , ρ 1 ρ 3 2 , ρ 3 1 ρ 2 }.
The main consequence of this approximation is that the number of LMIs to be solved is finite and is 2L(2 s+1 + 1) where s is the number of parameters, i.e. s = 2. The full-order controller is designed using the procedure developed in [START_REF] Scherer | Multiobjective output-feedback control via lmi optimization[END_REF]. Then, an 8-order controller K full is obtained. The proposed method is used to reduce K full . Then, K FWBT is produced.

To test its effectiveness, a comparison with the method developed in [START_REF] Widowati | Controller reduction of parameter dependent systems[END_REF] (LPV balanced truncation) namely K BT , is performed. The feasibility problems (37)(38) are convex. Using (Matlab LMI Control Toolbox), controllers are reduced to the 5th order obtained heuristically by trial-anderror approach. Therefore, a frequency and time analysis are performed.

Results and Discussion

The first evaluation is represented in Table 1 where first the assumption (45) and ( 46) are checked. Note that for 25 frozen values of (ρ 1 , ρ 2 ), the rank assumption is satisfied for all these points. This fact confirms Remark 1 and allows us compute an upper bound. Precisely, Table 2 express the upper bounds and H ∞ -norm of the error values. These results evince that the upper bound is correctly positioned (the gap is positive). However, we note that this upper bound is not tight to error H ∞ -norm.

Frequency analysis:

The Bode diagrams at several frozen values of ρ 1 and ρ 2 (25 points) of the three transfer function T zs , T zs and T zus are shown respectively in Figure 5, 6 and 7. In fact, the frequency behaviours of the chassis position z s and acceleration zr are chosen to be analysed in order to observe the comfort performance regarding the road profile input z r . The wheel position signal z us is also analysed to test the road holding. Then, the weighting functions W zs and W zs designed in Section 4.2, limit the amplification of the previously cited transfers in low frequency range (around [1 ; 10] Hz). In fact, the human sensitivity to vertical vibrations is important in this frequency range [DSS + 10]. For this reason, the frequency interval of the proposed frequency limited FWBT method is chosen as [1 ; 8] Hz. In Figure 5, note the reduced-order closed-loop system produced by FWBT approximate well the full-order closed-loop system in the chosen frequency range [1 ; 10] Hz. In this interval, the reducedorder closed-loop system produced by the unweighted BT fails. In fact, an important gap appears a 2 Hz and 3 Hz which is exacted as th BT is known to guarantee good approximation at high frequency. The same comment is given in Figure 6 where FWBT fits the full order closed-loop system in all the shown range unlike BT method that miss the peak around 2 (a resonance frequency). These results are more explicit when observing Figure 7. Indeed, FWBT gives a good approximation when BT fails (2 Hz and 8 Hz). The other important fact stated by these results, is the sensitivity against the parameters variation. Actually, a dispersion of ρ 1 and ρ 2 values is induced by the reduction step. This degradation is expected since the given application is a qLPV system where ρ 1 and ρ 2 are depending on the state vector. Then, every decrease in the order (states) affects the ρ 1 and ρ 2 values. However, this loss when reducing is under control for two reasons: the first one, the stability of reduced-order closed-loop is preserved and the error is guaranteed limited. The second one is that this dispersion is weak in the required frequency range. Thus, performance are not affected.

Remark 2: As Bode diagram for non-linear systems is not possible, a pseudo-Bode plot is then proposed in [START_REF] Poussot-Vassal | Robust LPV multivariable Automotive Global Chassis Control[END_REF] Time analysis: In the time domain, the several controlled suspensions are travelling a bump of 0.01 m x 2 m for a vehicle speed 8.3 m/s (i.e. 30 km/h). It is observed that the time response confirms the contribution of the FWBT reduction method. In fact, in Figure 8, the chassis is stabilised rapidly ( 1 sec. after the perturbation) without overtake on the suspension unlike the suspension with K BT . This observation, preserves the required 'comfort' performance. Moreover, for z us , the wheel equipped with a K FWBTsuspension, keeps almost the same profile of the road although its variations which which respects the 'road handling' performance. The K BT -suspension generate an infinite perturbation just after the bump (after 1 sec.). Note also, that temporal test draw two output signals (chassis and wheel positions) regarding the input (the road profile) and by the way there will be just one plot of each transfer besides the several plots in the frequency responses.

Conclusion

In this paper, the problem of LPV-Controller order reduction is investigated, and the approximation in a limited frequency range is considered. based on a similar work on the LTI-case, a new frequency limited method for LPV-controller order reduction is derived. The obtained reduced-order controller is proven stable and the degradation in the closed-loop performance is guaranteed bounded. An application of the proposed method to reduce the order of a semi-active suspension-controller is performed. The obtained results show its effectiveness to preserve the closed-loop stability and to respect the required performance specifications. The comparison with another existent LPV order-reduction technique based on the the classical BT, confirms the contribution of the proposed method.

Acknowledgement

The author would like to address their acknowledgement to the Associate Editor and the anonymous reviewers for their valuable comments that have helped significantly improve this paper.

Figure 5 .Figure 7 .Figure 8 .

 578 Figure 3.: Quarter-vehicle model

  w zus s 2 + ξ 21 w 21 s + w 2 21 s 2 + 2ξ 22 w 22 s + w 2 22

Table 1 .

 1 : values of rank[B(ρ), B Ω (ρ)]/rank[B Ω (ρ)] ; rank[C T (ρ), C T Ω (ρ)]/rank[C T Ω (ρ)]for the several frozen values of (ρ 1 , ρ 2 )

	r r r r r r ρ 1 ρ 2	-1.0	-0.5	0.0	0.5	1.0
	0	(14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14)
	0.25	(14/14 ; 14/14) (13/13 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14)
	0.5	(11/11 ; 14/14) (13/13 ; 14/14) (11/11 ; 14/14) (12/12 ; 14/14) (11/11 ; 14/14)
	0.75	(12/12 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14)
	1.0	(14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14) (14/14 ; 14/14)

Table 2 .

 2 : values of the T zw (ρ) -Tzw (ρ) i,2 and the upper bound for several frozen values of (ρ 1 , ρ 2 ) ≤ 5.07 1.35 ≤ 3.32 1.33 ≤ 4.08 1.31 ≤ 3.37 1.30 ≤ 3.72 0.25 0.32 ≤ 9.88 0.32 ≤ 9.93 0.32 ≤ 10.38 0.31 ≤10.39 0.31 ≤ 9.80 0.5 0.00 ≤ 1.79 0.00 ≤ 0.13 0.00 ≤ 2.22 0.00 ≤11.81 0.00 ≤ 2.03 0.75 0.31 ≤ 9.97 0.31 ≤ 9.92 0.30 ≤10.58 0.30 ≤9.86 0.29 ≤ 9.89 1.5 1.22 ≤ 3.77 1.20 ≤ 3.04 1.19 ≤ 4.46 1.18 ≤ 2.88 1.17 ≤ 2.90

	r r r r r r ρ 1 ρ 2	-1.0	-0.5	0.0	0.5	1.0
	0.0	1.36 Table 3.: Parameter values
		Parameter			Value
		Sprung mass		(m s )	470 [kg]
		Unsprung mass	(m us )	110 [kg]
		Tyre stiffness coefficient	(k t )	86378 [N/m]
		Spring stiffness coefficient (k s )	270000 [N/m]

Full order plant

Full order controller Reduced order controller Reduced order plant


Bruno Ricaud
email: bruno.ricaud@mines-paristech.fr

Robin Lietar
email: robin.lietar@mines-paristech.fr

Cyril Joly
email: cyril.joly@mines-paristech.fr

Are Virtual Reality headsets efficient for remote driving?

Keywords: ADAS, Oculus Rift, Virtual Reality, Headset, Driving, Safety 2"

Are Virtual Reality headsets efficient for remote driving?

INTRODUCTION

Vehicle automation is a great challenge for future products, but today users, especially armies, are not ready for deployment of fully automated vehicles. A gate to make them accept, is spreading of remotely guided vehicles. But, existing remotely guided products are not yet used at their maximum capacity because of driving difficulties. A lot of crashes occur due to human errors. These crashes, not including casualties, are not classified as major, but avoiding loss of robots can be an important financial benefit and an increasing way of automation acceptance. To decrease number of accidents we work on defining and setting up a driving assistance system for remote driving, or indirect driving. Indirect driving being the fact of driving without direct line of sight to exterior environment, like in a fully armoured vehicle without window driven with screen display. This work has two axes. First axe is software solutions development for path planning and hazard detection like the preliminary work treated in [RICAUD2014]. In it, we improved by 32 times an existing pathways detection solution for rough environment, such as dirt road or snow environment. This solution was a tool used to extract road in order to detect possible hazards on it. Second axe is hardware experiments for remote driving improvements. Researches on Oculus Rift are part of it. In our point of view displaying information on monitor is one of the biggest problems of actual remote driving interfaces. Actually, displaying driver viewpoint on a monitor bring an important lack of information. Two problems occur: -Viewpoint restriction due to small screens, about 14 inches (Figure 1), that can display one camera by one camera with a field of view of 60° each.

-Lack of depth perception due to 2D cameras displaying essentially black and white images that downgrade context and do not give any depth.

Whereas lab researches did more and more improvement in remote driving interfaces [FONG2003, BRADY2002], actual industrial solutions try to answer the viewpoint restriction by adding monitors (Figure 1) displaying exterior cameras images positioned around the vehicle. This solution is helpful but not sufficient. Undeniably, adding monitors helps operator to perceive his environment but it also decreases operator on-board space that is already very limited.

We decided to work on virtual reality headset for several reasons. First, it is as cheap as monitors, so it does not add cost to current systems. Second, this technology appears to be one of tomorrow-mainstreamed products, this means fast improvements brought by competition spirit. Third, by its conception it allows a fully 360 degrees viewpoint freedom in system capable of recreating a complete 3D environment around vehicle from sensors acquisitions. Fourth, this viewpoint freedom is accessible without adding more equipment (Joystick) to be used by operators due to included tracking done with IMU and infrared camera.

By manufacturing Oculus Rift, Oculus matures virtual reality Headsets and increases interest about it. Researches from all kinds appear last years, as tracking improvements [RABET2014] or medical game developments [BLAHA2014]. On another hand, Norway Army showed recently (Figure 2) that it is looking after solution for improving armoured vehicle driving also known as transparent shielding. All these topics manifest the growing

3"

interest for virtual reality and virtual reality headsets. Some researches concern remote driving [BUG2015], but none compares the current integrated solution, screen display, to VR headset. It convinced us to verify by our own that interest to VR headsets is justified. As it is the most developed and easily accessible virtual reality headset, we used Oculus Rift, but we are not limited to it. A lot of other products will appear this year on the developers market: Valve-HTC Vive, Oculus-Samsung Gear, Sony Morpheus [WAREABLE2015].

This study was preliminary to large-scale researches with such technology. Its focus was to determine if viewpoint freedom brought by Oculus Rift by correlating head motion and viewpoint motion, improves performances in several tasks, especially driving in complex environment.

METHODOLOGY -Hardware & Software

In our experiment the following hardware is used (Figure 3): The virtual reality headset Oculus Rift Developer Kit 2, a 14 inches screen, a Windows 7 computer, Euro-Truck Simulator 2 video game, a Logitech G27 racing wheel and a Thrustmaster T16000M joystick.

As explained in Introduction section, Oculus Rift was at time of experiment the most advanced virtual reality headset on market. Windows 7 computer was mandatory as Oculus Rift DK2 was not fully supported on Unix OS. Moreover, simulators game library it offers gave us a lot of possibilities. We have chosen Euro-Truck Simulator 2 for several reasons:

-It comes with a map editor (Figure 4), which means that we were able to create our own track with its obstacles and hazard. -It is fully supporting Oculus Rift (Figure 6). Euro Truck Simulator 2 (ETS2) offers a realistic cockpit (Figure 5) that gives visual references about head motion. These essential references are necessary to reduce sickness generated by VR systems.

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Oculus Rift is a VR headset system composed by an OLED screen displaying two stereoscopic highly defined (960p) images (Figure 6) at 75Hz. Head pose is computed by capturing data from an internal IMU and an infrared camera tracking infrared LED positioned all over the headset.

METHODOLOGY -Units of Measure

Users performances have been measured following several criteria: 1) Full circuit completion time, 2) Head motion frequency and joystick handling, 3) Penalties caused by colliding environment such as walls and road signs, or platform falls. See next sections for details about environment and penalties.

1) Full circuit completion time computation consisted in adding all exercise times (explained in next section) to possible penalties. Exercise time was the time needed (in seconds) by users to complete defined tasks mandatory to evolve to next phase.

2) 3D environment allows users a fully 360° viewpoint freedom (in our case, 180° composed of window and the rest was truck's interior). We measured frequency with which users moved their head to take information from environment. A joystick allowed users to change screen viewpoint when they were experimenting screen display as well as head motion with Oculus Rift. Joystick handling was measured too. Scoring was executed by an exterior operator, which was observing experimenting user. This criteria was evaluated from 0 to 5, 0 when no viewpoint change was detected and 5 when user took a lot of information about surrounding environment.

3) Giving penalties in some cases encouraged driving precision (like collision with obstacles). In the following, precision is quantified with penalties. More details concerning their description are given in next section.

METHODOLOGY -Exercises

The circuit (map) was divided into four exercises defining a range of tasks users had to complete.

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The First exercise (Figure 7) aimed to show the interest of taking advantage of viewpoint freedom to achieve precise manoeuvres without knowledge of following turns. We built this exercise with several very sharp turns between 90° and 180°. Moreover, high walls were surrounding the way, forbidding user to see what was coming next. Lack of visibility should forced user to look around to gain information to turn without losing precious time. In this exercise penalties were the collisions with walls, it costs 5 seconds each to user.

As the first one, the second exercise (Figure 8 Top Left) has been designed to exhibit necessity of looking around and use viewpoint freedom to achieve precise manoeuvres. In this case users had knowledge of future environment since the entire platform was in sight. Driving was done at high altitude, about 5m from ground on a platform not wider than truck chassis. In this exercise, penalties were determined as falls. Falling from a platform provoked heavy time penalty: to go back into circuit users had to search then take a ramp that brought them back onto the platform, this operation could cost them until 30 seconds.

Third exercise (Figure 8 Top Right) called "Search and Destroy" asked user to search for particular road signs hidden in or around a dense vegetated environment. When found, these road signs had to be rolled over without touching the forbidden road signs. This exercise evidenced interest of having viewpoint freedom in case of research for information in environment where reigns low visibility. This case of application is very common when using robot. De facto, for security or militaries purposes, robots are often used to detect explosives, or retrieve information in dangerous places. In this exercise, the forbidden road signs touched counted as penalties, it costs 5 seconds each to user.

The fourth and last exercise (Figure 8 Bottom Left) was a slalom at high speed. In this one, we tried to spot head movements made by users to prepare their following turns. This exercise has objective to demonstrate that the key to high speed remote driving is the necessity to give user the ability to prepare next turn by looking at it. In this exercise, a missed or collided gate costs a 5 seconds penalty.

METHODOLOGY -Protocol

The experiment was divided into two phases of 5 minutes each. The only differences between these two phases were the display method and viewpoint control method. One phase was dedicated to Oculus Rift when the other one was dedicated to monitor and joystick test. To avoid bias caused by users learning the track, we alternated beginning display method. Once experiment began with Oculus Rift and follows with monitor whereas for the following user it There is an overview of the elevated platform, road signs of the "Search and Destroy" exercise -central parking road sign has to be rolled over avoiding forbidden parking road signs and slalom.

Credit to Euro-Truck simulator 2.

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was monitor to be used first and then Oculus Rift. Each phase began with a tutorial teaching user what was expected during the four exercises with easier tasks. Of course, time measurement began after tutorial.

At the end of experiment a form was given to users to know their feeling about experiment. This form contained several questions: it asked them to score their sickness from 0 to 5, what benefit brings VR Headset for them, and what they think of the tutorial.

METHODOLOGY -User sample

Users sample is 44. Sample is very heterogenic, ages are from 13 to 60 years old, women, men, boys and girls are part of it. Students, teachers, technicians and bus drivers compose it.

It is important to note that this experiment was, for most of the users, the first experiment with Oculus Rift and VR headsets. Unfortunately 49% of users were not able to complete the whole circuit because of sickness. Indeed, "Simulator Sickness" is well known by virtual reality users [HOWARTH1997]. It varies from a user to another and creates nausea and malaise. Societies such as Oculus are working on these pains and try to decrease it with hardware improvements. These improvements concern latency reduction between images, persistence lowering or head motion prediction. This paper does not tackle the problem of simulator sickness. However, it is a major subject to solve for tomorrow transportation as [SIVAK2015] demonstrated it for automated cars.

RESULTS AND ANALYSIS

As exposed in last section, our sample was of 44 users. On this sample 49% of users quitted because of sickness. These sicknesses appear only when using Oculus Rift. In the following analysis users that have ended up experiment will be called "finishers".

In the following paragraphs, we expose detailed results concerning each kind of measurements. Each exercise describes a use case for remote driving of vehicle or robot.

In spite of overall sickness, as shows on Figure 9, 61% of finishers did a better completion time on all the circuit. But this score as to be minored. Indeed, on average time, users did 4 min 44 sec to end the circuit with Oculus Rift when they did 4min 43s with monitor display.

In the first exercise, 74% of finishers did a better time when using Oculus Rift (Figure 10). Average time with Oculus Rift is 57s whereas it is 1min 5s with monitor display. This improvement of 13% appears as a real benefit for this kind of situation that can be compared to robot navigation in indoor environment or vehicle driving in narrow urban environment. "Penalties Exercise 1" Histogram (Figure 14) clearly exposes that there are more penalties with monitor than with Oculus Rift. In this exercise penalties were defined as collisions with walls. This result is notable as it indicates that users experienced less difficulty to execute sharp turns without damaging the truck. It could be explained by the Head motion histogram (Figure 18) where it appears that majority of users benefited of head movement to negotiate their curves and avoid penalties by analyzing environment before executing their manoeuvres (Figure 14 and18).

In the second exercise, 44% of finishers did a better time when using Oculus Rift (Figure 11). Average time with Oculus Rift is 1min 49s whereas it is 1min 35s with monitor display. It seems that the low number of visual references in this exercise perturbed users, who made mistakes trying to negotiate turns. The second exercise diagrams (Figure 15) exposes that whether for monitor display or Oculus Rift, a lot of falls occurred (8/11). Lack of platform visibility seems to be the main cause of penalties. To be effective users must had to evaluate well when they had to turn and with what intensity. This evaluation was too complicated to realize with so low truck awareness, they tried to estimate the path (Figure 19) and hope to pass the platform.

In the third one, 57% of finishers did a better time when using Oculus Rift (Figure 12). Average time with Oculus is 1min 22s whereas it is 1min 24s with monitor display. This exercise was not as hard as the previous one as shown on the penalties histograms (Figure 16). Oculus Rift still is superior (3%) but considering overwhelming head motions compared to joystick use (Figure 20) it is a surprising result. We expected to see the highest time gap on this

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exercise. A higher complexity might have increase the gap between completion times, a point that we will study in future experiments.

In the last exercise, 53% of finishers did a better time when using Oculus Rift (Figure 13). Average time with Oculus is 36s whereas it is 39s with screen display. Again, Oculus Rift distinguishes itself by an 8% gain. This gain could be explained by the ease Oculus Rift brought to users compared to joystick (Figure 21) when they had no time to evaluate and prepare their manoeuvres, this observation is also true concerning penalties (Figure 17) where Oculus Rift driving resulted in less gate miss or gate collisions. This situation equivalent to all day driving demonstrates that comfort brought by hand free viewpoint movement improves curves estimation and path negotiation.

Concerning nausea, with a median value equal to 2 and a mean of 1.8, it seems that finishers were not so much affected by sickness. But some finishers give a value of 4 that can explain their low score in head motion measurements. Users described us that they did not move their head to do not increase their nausea. Considering the fact that some users reported us that they did not used the headset very much at the end to limit nausea, we look at head motion results to analyse if there were outliers to be considered. Results demonstrate that the finishers with a sickness score inferior or equal to 2 made an overall head motion result of 3.25 on all exercises, whereas all finishers made a result of 3.23. These close scores prove that finishers with nausea were not outliers, because only few stop moving their head to limit sickness.

Considering that exercise 2 was too complex, results on exercises 1, 3 and 4 lay out greater overall return. Completion time for exercise 1+3+4 is 2min 55s for Oculus Rift and 3min 8s for screen display (Figure 22). This 7% time improvement indicates that despite mitigated supporting results (median value equal to 2), finishers were able to score a close or better (61%, Figure 22) completion time. It is a very interesting result, because it lets us think that with hardware improvements, VR headsets could be a lot better than monitor display. So, it justifies continuing research using this technology to acquire knowledge relative to where and how to use it when it will be at its top level of manufacturing. 00:00:00" 00:02:53" 00:05:46" 00:08:38" 00:11:31" 1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23"

Comple'on)Time)

#User)

Comple'on)Time)on)all)exercises)

Oculus"

Screen" 00:00:00" 00:00:17" 00:00:35" 00:00:52" 00:01:09" 00:01:26" 00:01:44" 1" 2" 3" 4" 5" 6" 7" 8" 9" 10" 11" 12" 13" 14" 15" 16" 17" 18" 19" 20" 21" 22" 23"

Comple'on)Time)

#User)

Comple'on)Time)Exercise)1)

Oculus"

Screen" 11"

DISCUSSION

These results are notable for several reasons. First, concerning sickness, it indicates that about 50% of people is able to support the current version of Oculus Rift for driving simulation; which is better than expected but far from widely deployable. We expect to see great improvements brought by video game industry on this kind of display. In case they did not or if they fail, taking these results into account, VR headsets will not be exploitable in industrial product for driving or remote driving.

Second, results show better performances of Oculus Rift in regards of driving in simulated environment. It is very promising for further research we will engage about this subject. In near future we will try to highlight benefits of Oculus Rift in real remote driving situation by making experiments with robots. We are developing a robotic platform based on Wifibot Lab V4[WIFIBOT2015] to achieve it. Moreover, we work on multi-sensors environment reconstruction, and later we will merge both works to create a complete and safe system following the idea of [TERRIEN2000], with adding hazard detection, path planning and hand free viewpoint change.

The drawback of this experiment is the second exercise. Falling penalty was too punitive; in addition of slowing down user in his progression, most withdrawal due to sickness occurred during this exercise. We learned with this exercise that whereas obvious tasks are not suitable to extract information harsh ones could bias results too.

Other topics to take into account in our next experiments are the stereoscopy and the multiple monitors configuration. Oculus Rift displays stereoscopic images that give user depth information. Does it influence results compared to the monoscopic images of screen display? We cannot answer yet. In the future experiment in real environment, sensors are monoscopic cameras. In this case, Oculus Rift display will be degraded compared to simulated environment. It will be interesting to see if Oculus Rift keeps its first place.

CONCLUSIONS

These results demonstrate that users take advantage of viewpoint freedom to increase manoeuvring, being more reactive and look for information. This shows that VR headset could be a great alternative to monitor for remote driving. We are working on the next step, which is making an immersive driving system on a mobile robot platform, and complete this analysis with real driving tests on this platform.

Figure 2 :

 2 Figure 2: Driving an armored vehicle with an Oculus Rift. Credit to Norway Army

Figure 3 :

 3 Figure 3: User installation, with Oculus Rift (Left), with Monitor and Joystick (Right)

Figure 4 :

 4 Figure 4: Euro-Truck simulator 2, map construction. Here a view of tutorial zone 2. Credit to Euro-Truck simulator 2.

Figure 6 :

 6 Figure 6: Image of Euro-Truck Simulator displayed on Oculus Rift Monitor inside headset. Credit to Euro-Truck simulator 2.

Figure 7 :

 7 Figure 7: Image of Euro-Truck Simulator, giving preview of viewpoint freedom. Image taken during the first phase. Credit to Euro-Truck simulator 2.

Figure 8 :

 8 Figure 8: Second (top left), third (top right) and fourth (bottom left) phases.There is an overview of the elevated platform, road signs of the "Search and Destroy" exercise -central parking road sign has to be rolled over avoiding forbidden parking road signs and slalom.

Figure 9 :

 9 Figure 9: Completion Time for each user with Oculus Rift (Dark) and with Monitor (Clear).

Figure 10 :

 10 Figure 10: Completion Time for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 1.

Figure 14 :

 14 Figure 14: Penalties for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 1.

Figure 12 :

 12 Figure 12: Completion Time for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 3.

Figure 11 :

 11 Figure 11: Completion Time for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 2.

Figure 13 :

 13 Figure 13: Completion Time for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 4.

Figure 18 :

 18 Figure 18: Head moves for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 1.

Figure 15 :

 15 Figure 15: Penalties for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 2.

Figure 16 :

 16 Figure 16: Penalties for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 3.

Figure 17 :

 17 Figure 17: Penalties for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 4.

Figure 19 :

 19 Figure 19: Head moves for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 2.

Figure 20 :

 20 Figure 20: Head moves for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 3.

Figure 21 :

 21 Figure 21: Head moves for each user with Oculus Rift (Dark) and with Monitor (Clear) in exercise 4.

Figure 22 :

 22 Figure 22: Completion Time for each user with Oculus Rift (Dark) and with Monitor (Clear) for exercises 1+3+4.

ACKNOWLEDGMENTS

This work is part of research lead by Nexter Systems. We want to thank our subjects that give us some of their time, interesting remarks, and accepted to suffer the pain this experiment could give to their stomach.