INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2004; 14:1283–1297 (DOI: 10.1002/rnc.947)
A parametric insensitive H2 control design approach Philippe Chevrel1,2,n,y and Mohamed Yagoubi1,2 1
IRCCyN, UMR C.N.R.S. 6597, 1 rue de la Noe., BP 92101, 44321 Nantes Cedex 3, France 2 EMN, 4 rue Alfred Kastler, La chantrerie, BP 20722, 44307 Nantes Cedex, France
SUMMARY H2 and H1 control design methodologies are known to be efficient to deal with multivariable control problems. However, most of them do not take explicitly the parametric uncertainties into account. This paper proposes a low parametric sensitivity H2 control design method as an alternative to m-synthesis or robust H2 control design. In addition to the standard H2 criterion, the H2 norm of the parametric sensitivity function is introduced in order to improve the robustness of the resulting controller. Unfortunately, this problem is a difficult one. Its equivalence to structured feedback H2 control problem will be shown. The underlying BMI will be solved by making use of an iterative LMI procedure. Two examples will illustrate the interest of the approach. Copyright # 2004 John Wiley & Sons, Ltd. KEY WORDS:
linear systems; robust control; H2 control; parametric sensitivity; bilinear matrix inequalities; automotive control
1. INTRODUCTION Starting from a physical model, the modelling of physical parameter uncertainties through a real and structured model of uncertainties (rather than a global non-structured one), is of particular interest. The question is: how to deal, in the context of H2 control, with such a precise model of uncertainties during the control design stage? There is in fact two possible ways to answer. The first one consists mainly in finding out H2 performances in the worst case of uncertainties. Some methods such as robust H2 control design [1, 2] or m-synthesis [3, 4] attempt to solve this difficult problem. Although this formulation of the problem is attractive, it is not always coherent with industrial needs [5, 6]. The knowledge of a priori bounds on parameter deviation often fails during the process design stage. The second way, which will be considered in this paper, consists in quantifying the potential degradation included when the uncertain parameters deviate from their nominal values, in order to make it small enough. The theory of sensitivity is not a new one [7]. Reducing the sensitivity is the main goal of feedback control. Some notable efforts have been made to enrich classical LQG/H2 control in that direction. One can quote the parametric LQG/LTR method proposed in [8] and the
n
Correspondence to: P. Chevrel, IRCCyN, UMR C.N.R.S. 6597, 1 Rue de la No.e, BP 92101, 44321 Nantes Cedex 3, France. y E-mail:
[email protected]
Published online 12 July 2004 Copyright # 2004 John Wiley & Sons, Ltd.
Received 21 May 2003 Revised 17 November 2003 Accepted 2 March 2004
1284
P. CHEVREL AND M. YAGOUBI
‘desensitised LQG’ control [5, 9]. The method presented in this paper seems to the authors to be an original way to design robust control laws. In fact, we are interested in a design approach able to deal classically with specifications on nominal performance and robust stability (as some H2 control design methodologies do), and also to reduce the sensitivity of the performance with regard to the system parametric variations. In this paper, the insensitive H2 control ðIH2 ) is considered. Its general principle has been introduced in Reference [10]. The contribution of the present work concerns the reformulation of the IH2 control method as a structured feedback H2 control problem and the solution of the underlying BMI. The paper is organized as follows: The IH2 problem is first presented in Section 2. In Section 3, the IH2 problem is shown to be equivalent to a structured H2 problem derived from an auxiliary model. It is then reformulated as a linear objective optimization problem under some BMI constraints. Finally, an original method and an associated iterative LMI based algorithm are proposed. Section 4 applies the proposed method to some automotive design problems and compares the results obtained with existing ones.
2. THE IH2 CONTROL PROBLEM
----------
Consider the scheme of Figure 1 in which G is an LTI operator with partitioned inputs and outputs and D is an unknown operator related to the parametric uncertainties. Let the transfer matrix GðsÞ associated with G be defined by 3 2 A Bg Bw Bu 6- - - - - - - - - - - - - - -7 6 Cz Dzg Dzw Dzu 7 7 6 ð1Þ GðsÞ :¼ 6 7 6 Cz Dzg Dzw Dzu 7 5 4 Cy
Dyg
Dyw
Dyu
where A 2 R ; Bg 2 R ; Bw 2 R ; Bu 2 R ; Cz 2 Rnz n ; Cz 2 Rnz n and Cy 2 Rny n : Let also D be defined by DðzÞ ¼ D:z with the particular form D ¼ diag ðy1 IP1 ; y2 IP2 ; . . . ; yq IPq Þ in which yi 2 R; i 2 f1; . . . ; qg; are the uncertain parameters. Finally, the feedback transfer matrix K is introduced according to Figure 2. Hzg Hzw The closed-loop transfer matrix Fl ðGðsÞ; KðsÞÞ has the partitioned form : Hzg Hzw Indeed, all the transfers Hzw ; Hzg ; Hzg and Hzw depend on the feedback K: Closing the ‘D-loop’, the final transfer H depends on both s and y and can be written as nn
nng
nnw
nnu
H ¼ Hzw þ Hzg ðI DHzg Þ1 DHzw
ð2Þ
Figure 1. Linear fractional representation of parametric uncertainties. Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
PARAMETRIC INSENSITIVE DESIGN
1285
Figure 2. The closed-loop system.
The IH2 control problem consists in searching for a control law that minimizes a special H2 criterion. In addition to the standard H2 norm, this criterion involves the H2 norm of the parametric sensitivity function @H=@y: In the following sections, the sensitivity function will be considered in the neighbourhood of D ¼ 0 (the nominal model). The parametric sensitivity function is given by @H @D ¼ ½Iq fHzg ðI DHzg Þ1 g ðI DHzg Þ1 Hzw @y @y
ð3Þ
@H @D Hzw ¼ Iq Hzg @y y¼0 @y
ð4Þ
Definition 1 The IH2 control problem ðIH2 PÞ consists in minimizing with respect to KðsÞ; the following criterion under the constraint of internal stability: 2 2 q X @D @D Hzw ¼ jjHzw jj22 þ s2i Hzg Hzw ð5Þ JIH2 ðKÞ ¼ jjHzw jj22 þ S Hzg @y @yi 2 2 i¼1 with S ¼ diagðs1 ; . . . ; sq Þ; each si 2 R being considered as a weighting parameter associated with @H=@yi : This criterion is given in a weighted form to allow a more selective action on the parametric sensitivity reduction with respect to one or several parameters. The minimization of the criterion above is not a standard H2 optimization problem because of the additional term that corresponds to jj@H=@yjj22 : This term contains the product of the two transfer matrices Hzg and Hzw both depend on KðsÞ:
3. AN IH2 P SOLUTION USING CONVEX OPTIMIZATION TOOLS 3.1. Equivalence to a structured H2 control problem The starting point of the solution is Theorem 1 stated in References [11, 12]. Assumption 1 For the sake of simplicity, it is supposed that S ¼ Iq : Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1286
P. CHEVREL AND M. YAGOUBI
Let us first derive from GðsÞ and a given D the state space representation of Gy ðsÞ ¼ Fu ðG; DÞ: This is given by 2
A%
6 6 Gy ðsÞ :¼ 6 C% z 4 C% y
B% u
B% w D% zw D% yw
3
7 7 D% zu 7 5 D% yu
ð6Þ
where A% ¼ A þ Bg DfðDÞCz ;
C% z ¼ Cz þ Dzg DfðDÞCz ;
C% y ¼ Cy þ Dyg DfðDÞCz
B% w ¼ Bw þ Bg DfðDÞDzw ;
D% zw ¼ Dzw þ Dzg DfðDÞDzw ;
D% yw ¼ Dyw þ Dyg DfðDÞDzw
B% u ¼ Bu þ Bg DfðDÞDzu ;
D% zu ¼ Dzu þ Dzg DfðDÞDzu ;
D% yu ¼ Dyu þ Dyg DfðDÞDzu
ð7Þ
and fðDÞ ¼ ðI Dzg DÞ1 : The parametric sensitivity of the trajectory signals is taken to be given by 4
xy ¼
@x ; @y
zy ¼
@z ; @y
yy ¼
@y @y
and
uy ¼
@u @y
ð8Þ
The exogenous input signal w is assumed to be independent of y: Thus, differentiating the state space equations of (2) with respect to y yields: x’ y ¼ Ay x þ ðIq A% Þxy þ Bwy w þ Buy u þ ðIq B% u Þuy zy ¼ Czy x þ ðIq C% z Þxy þ Dzwy w þ Dzuy u þ ðIq D% zu Þuy
ð9Þ
yy ¼ Cyy x þ ðIq C% y Þxy þ Dywy w þ Dyuy u þ ðIq D% yu Þuy where Ay ; Bwy ; Buy ; Czy ; Dzwy ; Dzuy ; Cyy ; Dywy and Dyuy are given below: @D Cz ; @y @D Dzw ; Bwy ¼ ðIq Bg Þ @y @D Dzu ; Buy ¼ ðIq Bg Þ @y Ay ¼ ðIq Bg Þ
@D Cz ; @y @D Dzw ; Dzwy ¼ ðIq Dzg Þ @y @D Dzu ; Dzuy ¼ ðIq Dzg Þ @y Czy ¼ ðIq Dzg Þ
@D Cz @y @D Dzw Dywy ¼ ðIq Dyg Þ @y @D Dzu Dyuy ¼ ðIq Dyg Þ @y Cyy ¼ ðIq Dyg Þ
ð10Þ
Assumption 2 It is supposed throughout the rest part of the paper that Dyu ¼ Dzw ¼ 0: In the following, only small variations in the neighbourhood of D ¼ 0 (the nominal model) are considered. Let us now introduce the augmented model G1a shown in Figure 3. Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
PARAMETRIC INSENSITIVE DESIGN
1287
Figure 3. A structured H2 control problem.
----------------
G1a has ½wT ; uT ; uTy T as inputs and ½zT ; zTy ; yT ; yTy T as outputs. Using Assumption 2, G1a is related to G as follows: 3 2 Bu 0 A 0 Bw 7 6 6 Ay ðIq AÞ Bwy Buy ðIq Bu Þ 7 7 6 6- - - - - - - - - - - - - - - - - - - - - - - - - - - -7 7 6 Cz 0 0 D 0 zu 7 6 ð11Þ G1a ðsÞ :¼ 6 7 6 Czy ðIq Cz Þ 0 Dzuy ðIq Dzu Þ 7 7 6 7 6 7 6 Cy 0 Dyw 0 0 5 4 Cyy ðIq Cy Þ Dywy 0 0
The fact that u and y are linked through the dynamic feedback u ¼ KðsÞy induces the relation uy ¼ ðIq KðsÞÞyy : Let us state the result. Theorem 1 The IH2 control problem (cf. Definition 1) is equivalent to the structured H2 control problem (associated with Figure 3), which consists in finding a stabilizing controller KðsÞ that minimizes 4 the criterion JSH2 ðKÞ ¼ jjFl ðG1a ; Iqþ1 KÞjj22 : Proof The result is a consequence of the definition of G1a : The 2-norm property yields Tzw 2 2 2 1 JSH2 ðKÞ ¼ jjFl ðGa ; Iqþ1 KÞjj2 ¼ jjTza w jj2 ¼ ¼ jjTzw jj22 þ jjTzy w jj22 Tz w y 2
ð12Þ
Moreover, it can be shown after some calculations that Tzw ¼ Hzw and Tzy w ¼ @H=@y (cf. (2) and (3)). Consequently, JSH2 ðKÞ ¼ jjHzw jj22 þ jj@H=@yjj22 ¼ JIH2 ðKÞ: & Using this result it is possible to reformulate the IH2 control problem as a linear objective optimization problem with BMI and LME (linear matrix equality) constraints. Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1288
P. CHEVREL AND M. YAGOUBI
Let KðsÞ be defined by the system matrix
# AK BK KðsÞ :¼ - - - - - - - CK DK -----
"
ð13Þ
where AK 2 Rnn ; BK 2 Rnny ; CK 2 Rnu n and DK 2 Rnu ny : Theorem 2 The IH2 P (cf. Definition 1) is equivalent to the following linear objective optimization problem under BMI and LME constraints: minX¼X T 50;Y;K# TraceðYÞ ðA# þ B# 2 K# C# 2 ÞX þ XðA# þ B# 2 K# C# 2 ÞT þðB# 1 þ B# 2 K# D# 21 ÞðB# 1 þ B# 2 K# D# 21 ÞT 50
ð14Þ
Y ðC# 1 þ D# 12 K# C# 2 ÞXðC# 1 þ D# 12 K# C# 2 ÞT > 0 " # Iqþ1 AK Iqþ1 BK K# ¼ Iqþ1 CK Iqþ1 DK
--------
----------------------
A# ; B# 1 ; B# 2 ; C# 1 ; C# 2 ; D# 12 and D# 21 are related to the state space matrices of GðsÞ (cf. (1)) through the definition of the augmented model G2a ðsÞ: 3 2 Bu 0 0 A 0 0 Bw 7 6 6 Ay ðIq AÞ 0 Bwy Buy ðIq Bu Þ 0 7 7 6 7 6 7 2 6 0 3 0 0 0 0 0 I 6- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -n -7 B# 2 A# B# 1 7 6 6 Cz 6- - - - - - - - - - -7 0 0 0 Dzu 0 07 746 # 6 7 G2a ðsÞ :¼ 6 0 D# 12 7 ð15Þ 7 ¼ 6 C1 7 6 Czy ðIq Cz Þ 0 4 5 0 D ðI D Þ 0 zuy q zu 7 6 # 2 D# 21 7 6 0 C 6 Cy 0 0 Dyw 0 0 07 7 6 7 6 7 6 Cyy ðIq Cy Þ 0 Dywy 0 0 0 5 4 0 0 In 0 0 0 0 Proof From Theorem 1, it is known that the IH2 P is equivalent to the structured H2 control problem of Figure 3 where KðsÞ is a dynamical feedback. This last problem may be reformulated as the static output feedback problem: Find K# 2 Rðqþ1Þðnþnu Þðqþ1Þðnþny Þ with the special structure " # " # A# K B# K Iqþ1 AK Iqþ1 BK K# ¼ ¼ ð16Þ Iqþ1 CK Iqþ1 DK C# K D# K such that Fl ðG2a ðsÞ; K# Þ is internally stabilized and jjFl ðG2a ðsÞ; K# Þjj2 is minimized ðG2a ðsÞ is the augmented model defined in (15)). The matrix inequality formulation of the H2 norm leads to optimization problem (14). & Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
PARAMETRIC INSENSITIVE DESIGN
1289
Because of the equality constraint on the static output feedback K# ; the BMI (14) cannot be simplified (as it is usually done for the classical H2 problem) to a convex optimization problem with LMI constraints. In fact, solving (14) for X ¼ X T > 0 and K# simultaneously is a nonconvex programming problem as other close problems in control (see for e.g. Reference [13]). It is then possible to consider a two-stage convex optimization approach (V-K iteration [14]) by fixing either the controller gain or the Lyapunov function. However, such an algorithm does not give satisfactory results for the problem considered here. Alternatively, an iterative algorithm will be used here. This algorithm described below has shown its efficiency in different situations [15, 16] and converges monotonically. The proposed algorithm is based on a close approximation (not too conservative) of some bilinear terms involved in (14). 3.2. An iterative LMI algorithm The algorithm deals with a ‘generic’ BMI constraint of the form 9X ¼ X T > 0 and K# s:t: LðX; K# Þ þ BLðX; K# Þ50
ð17Þ
where terms LðX; K# Þ and BLðX; K# Þ are, respectively, linear and bilinear with respect to variables X and K# such that 4 BLðX; K# Þ ¼ ðM1 K# M2 ÞT XM3 þ M T XðM1 K# M2 Þ ð18Þ 3
where M1 ; M2 and M3 are some given matrices. It is straightforward to show that the first constraint in (14) is equivalent to the following: " 1 # " 1 T # " 1 # X A# þ A# X 1 X 1 B# 1 X X B# 2 K# ½ C# 2 D# 21 þ ½ C# 2 D# 21 K# T B# T2 50 ð19Þ þ T 1 0 0 B# 1 X I which is a particular case of the ‘generic’#BMI constraint (17) (with respect to variables X 1 " X 1 A# T þ A# X 1 X 1 B# 1 and K# ) if: L ¼ ; M1 ¼ B# 2 ; M2 ¼ ½ C# 2 D# 21 and M3 ¼ ½ I 0 : B# T1 X 1 I According to the Schur lemma, the second constraint in (14) can be written as a linear constraint (with respect to variables X 1 and K# ) as follows: " # Y ðC# 1 þ D# 12 K# C# 2 Þ >0 ð20Þ ðC# 1 þ D# 12 K# C# 2 ÞT X 1 Our purpose is to approach the bilinear inequality (17) by a set of parameterised LMIs. To precise more the original strategy proposed here, let us define an LMI parameterized by a matrix Q of appropriate dimension. 2 3 LðX; KÞ þ FQ ðX; KÞ ðXM3 ÞT ðM1 KM2 ÞT 6 7 4 6 7 LMIQ ðX; KÞ ¼ 6 ð21Þ ð*ÞT I 0 750 4 5 ð*ÞT 0 I with FQ ðX; KÞ ¼ QT Q QT ðXM3 M1 KM2 Þ ðXM3 M1 KM2 ÞT Q
ð22Þ
where M1 ; M2 and M3 are some given matrices of appropriate dimensions. Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1290
P. CHEVREL AND M. YAGOUBI
Theorem 3 Consider a pair ðX; KÞ such that X 2 Rnn ; X ¼ X T > 0 and K 2 Rmp : Then the following holds: (i) 8Q 2 Rnn ; LMIQ ðX; KÞ50 ) LðX; KÞ þ BLðX; KÞ50 (ii) LðX; KÞ þ BLðX; KÞ50 ) 9Q 2 Rnn ; LMIQ ðX; KÞ50
Proof Part (i) It can be easily verified that the following holds: BLðX; KÞ ¼ ðM1 KM2 ÞT XM3 þ M3T XðM1 KM2 Þ ¼ ðXM3 M1 KM2 ÞT ðXM3 M1 KM2 Þ þ ðXM3 ÞT XM3 þ ðM1 KM2 ÞT ðM1 KM2 Þ ð23Þ and 8Q 2 Rnn ðQ ðXM3 M1 KM2 ÞT ðQ ðXM3 M1 KM2 Þ50
ð24Þ
It is clear then from (23) and (24) that 8Q 2 Rnn ; BLðX; KÞ4 QT Q QT ðXM3 M1 KM2 Þ ðXM3 M1 KM2 ÞT Q þ ðXM3 ÞT ðXM3 Þ þ ðM1 KM2 ÞT ðM1 KM2 Þ
ð25Þ
Finally, according to (22) and (25) the following proposition is true: 8Q 2 Rnn ; LðX; KÞ þ BLðX; KÞ4 LðX; KÞ þ FQ ðX; KÞ þ ðXM3 ÞT ðXM3 Þ þ ðM1 KM2 ÞT ðM1 KM2 Þ
ð26Þ
Applying the Schur lemma twice to the right term of inequality (26) one obtains the result 8Q 2 Rnn ; LðX; KÞ þ BLðX; KÞ4LMIQ ðX; KÞ
ð27Þ
Part (ii) If LðX; KÞ þ BLðX; KÞ50 holds then for the particular choice Q ¼ XM3 M1 KM2 one obtains LMIQ ðX; KÞ50: & Let us now assume that we are looking for pair ðXopt ; Kopt Þ that minimizes a linear objective JðX; KÞ under the BMI constraint LðX; KÞ þ BLðX; KÞ50: For a given Q; it is obvious from Theorem 3 that pair ðXQ ; KQ Þ obtained by minimization of J under LMIQ ðX; KÞ50 is such that JðXQ ; KQ Þ > JðXopt ; Kopt Þ: The problem consists then in finding Q such that XQ ’ Xopt and KQ ’ Kopt : The algorithm considered next uses this idea. The ILMI algorithm Initialization step (a convex problem) Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
PARAMETRIC INSENSITIVE DESIGN
*
Compute K0 the H2 nominal controller (obtained for S ¼ 0), compute X0 such that X0 ¼ arg minX JðX; K0 Þ under LðX; K0 Þ þ BLðX; K0 Þ50
*
1291
ð28Þ
Compute Q0 : Q 0 ¼ X 0 M3 M 1 K 0 M2
ð29Þ
kth iteration (a convex problem) ðXk ; Kk Þ ¼ arg minX;K JðX; KÞ
ð30Þ
under LMIQk1 ðX; KÞ50 *
Compute Qk : Qk ¼ Xk M3 M1 Kk M2
*
ð31Þ
Test jjQk Qk1 jj5e for a given tolerance bound e: If the test is true, then the algorithm is stopped and X# opt ¼ Xk ; K# opt ¼ Kk : If the test fails go to the ðk þ 1Þth iteration. &
The local convergence of this algorithm is proved in Appendix A by showing that the algorithm produces a monotonically decreasing criterion sequence of positive terms. In fact, the successive parameterised LMIs used by the algorithm are such that the optimal solution at step k is also a feasible solution at step k þ 1. 4. EXAMPLES 4.1. Example1: vehicle dynamics control The practical interest of the new numerical algorithm proposed in this paper is shown in this example through a robust vehicle dynamics control as considered in Reference [6]. The lateral ’ have to be controlled through two control inputs: the yaw velocity Vy and the yaw velocity c moment Cz that can be obtained by differential braking and the rear steering ar (see Figure 4). The vehicle must stay near to the desired trajectory as shown in Figure 5. Disturbance efforts acting on the vehicle can be summarized into lateral force F and yaw moment M:
Figure 4. The desired trajectory. Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1292
P. CHEVREL AND M. YAGOUBI
Figure 5. The ‘bicycle’ model. +
R1/2 c
−
1/2
Ro w
1/2
Qo
z
Predictive Model
D(s)dαr= 0 D(s)dCz= 0
dα r dCz αr Cz
Nominal Model
⊕
⊕
=1
u
Vy . Vy .
S1/2 c ⊕
y
Controller
Figure 6. Standard problem.
The well known ‘bicycle model’ given by (32) is used to describe the vehicle motions. " # " # Vy V’ y ¼A þ Br ðar þ dar Þ þ Bc ðCz þ dCz Þ ’ . c c
ð32Þ
with 2 6 A¼6 4
2m ðCyv þ Cyr Þ mVx
2m ðl2 Cyr l1 Cyv Þ CVx
3 2 3 2m 2mCyr " # ðl2 Cyr l1 Cyv Þ 0 7 6 7 Vx m 7; B r ¼ 6 7 and Bc ¼ : 5 4 5 2m 2 2ml2 Cyr 1 2 ðl Cyv þ l2 Cyr Þ CVx 1 C
Vx þ
In this model, m denotes the weight, C the inertia, l1 the front wheelbase, l2 the rear wheelbase and ðCyv ; Cyr Þ the nominal cornering stiffness. Note also that the model is parameterized by the road friction parameter m which is, indeed, uncertain. The standard H2 problem to be minimized to meet the control requirements is built following the Standard State Control methodology [17] as adopted in Reference [6]. The augmented plant is of order 4 as it includes the disturbances model DðsÞ as shown in Figure 6. Moreover, the weighting matrices R0 ; Q0 ; Rc and Qc allow the controller dynamics to be tuned (see Reference [6] for details). The nominal model has been taken for a constant longitudinal speed Vx ¼ 80 km=h: The H2 optimal controller is denoted by K0 ðsÞ: For s ¼ 0:75 (a sensitivity weighting parameter), two controllers, namely K1 ðsÞ and K2 ðsÞ; are derived using a heuristic introduced in Reference [6] Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1293
PARAMETRIC INSENSITIVE DESIGN
Table I. Numerical results. The controller K0 ðsÞ K1 ðsÞ K2 ðsÞ
jjHzw jj22 2.58 6.13 3.34
jjS Hzg
@D Hzw jj22 @y
10.30 4.30 5.37
JIH2
The controller order
12.88 10.43 8.71
4 20 4
Figure 7. Performances obtained with K0 : ðm ¼ 1Þ ½} and ðm ¼ 0:6Þ ½ :
(limited to the monoparametric case) and the new iterative LMI algorithm proposed in this paper, respectively. Table I summarizes the results obtained. Controller K1 ðsÞ is obtained after only one iteration to have a controller of admissible order (this heuristic suffers from an augmentation of the controller order). In fact, for two iterations of the first heuristic, the controller order would be 52. Additional iterations are possible (making use of controller reduction at each step) but do not improve significantly the result. It appears that the proposed ILMI algorithm gives better results in terms of the IH2 criterion with a loworder controller (the same as the standard model system). The computational time is comparable for the two methods. It is also clear that the norm of the parametric sensitivity function has been decreased at the expense of the standard H2 criterion. A simulation test has been performed for two values of m; ðm ¼ 1 and 0:6Þ in order to observe the effect of the parametric sensitivity reduction. A lateral force step occurs at t ¼ 1 s and a yaw moment step occurs at t ¼ 4 s: Figures 7 and 8, respectively, report the results obtained with the standard H2 controller and those of the IH2 controller K2 ðsÞ: The (1,1) and (1,2) plots show the outputs and the (2,1) and (2,2) plots the inputs with respect to the H2 optimal controller K0 ðsÞ: Controller K2 ðsÞ clearly improves the parametric robustness in comparison with controller K0 ðsÞ: Step responses obtained with K2 ðsÞ are clearly less sensitive to the road friction parameter. This example shows the interest of the IH2 methodology together with the efficiency of the proposed ILMI algorithm. Note also that the resulting controller is of the same order as the H2 controller. Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1294
P. CHEVREL AND M. YAGOUBI
Figure 8. Performances obtained with K2 : ðm ¼ 1Þ ½} and ðm ¼ 0:6Þ ½ :
Figure 9. A simplified model of the powertrain.
4.2. Example 2: active control of vehicle longitudinal oscillations This example deals with the simplified model (33) of the powertrain (cf. Figure 9) introduced in Reference [18]. It is parameterized with a few parameters such as the equivalent stiffness Keq (aggregating different stiffness of the complete model), the engine inertia Imot ; the equivalent vehicle inertia Iveq and the friction coefficients Am ; Av : 2
Am 6 Imot ’m o 6 6 7 6 6o 7 ¼6 ’ v 4 5 6 0 6 4 ’ Gr Keq 2
3
0
Av Iveq
Keq
3
3 2 2 3 1 7 Imot 7 om 7 6 76 7 6 Imot 7 7Gmot ; omot ¼ 30 1 76 ov 7 þ 6 74 5 6 0 7 p 5 4 Iveq 7 5 Gr 0 0
1
2 3 om 6 7 7 0 0 6 4 ov 5 Gr
ð33Þ
The vehicle inertia parameter Iveq is, clearly, an uncertain parameter. This parameter depends on the vehicle mass which is, obviously, subject to some variations Iveq 2 ½ 0:84 1:24 ðkg m2 Þ: Figure 10 shows bode diagrams of the system for different values of the vehicle load (0 and 700 kgÞ: Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1295
PARAMETRIC INSENSITIVE DESIGN
Figure 10. The system bode diagrams. Table II. Numerical results. The controller K0 ðsÞ K1 ðsÞ K2 ðsÞ
jjHzw jj22 3.93 5.38 4.45
jjS Hzg
@D Hzw jj22 @y
4.80 2.81 2.27
JIH2
The controller order
8.73 8.19 6.72
4 20 4
The standard H2 problem to be minimized is defined in the following [6]. The conceptual model is of order 4 as it includes the disturbances model DðsÞ: The nominal model has been taken for a constant value of the vehicle load of M ¼ 1480 þ p170 ffiffiffi kg: Let us denote the H2 optimal controller by K0 ðsÞ: For s ¼ 2; two controllers, namely K1 ðsÞ and K2 ðsÞ; are derived using the heuristic proposed in Reference [6] and the new ILMI algorithm proposed in this paper, respectively. Table II summarizes the results obtained. Once again it appears that the proposed ILMI algorithm, initialised by the H2 controller, gives significantly better results in terms of the IH2 criterion with a low-order controller.
5. CONCLUSION The insensitive H2 control is an interesting way to deal with applied control design. Based on it, a powerful multivariable control design methodology can be proposed. This observation has motivated the present work. After having presented, the insensitive H2 control problem, this paper has shown that it is equivalent to an H2 problem for a particular augmented plant, with a particular structure constraint on the feedback loop. The problem has then been reformulated as a linear objective optimization problem under BMI constraints. Unfortunately, it cannot be reduced to a convex optimization problem by the usual techniques. So, the second step has consisted in presenting and using an original iterative LMI algorithm which has already proved its efficiency to solve it. It seems to be both efficient and tractable. Its application to some automotive control design problems gives much better results than the existing ones with a much lower controller order. Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1296
P. CHEVREL AND M. YAGOUBI
APPENDIX A Lemma Let ðXk ; Kk Þ be a solution obtained at the kth iteration of the algorithm (cf. Section 3.2). The following inequalities are satisfied: 8k 2 N;
LðXk ; Kk Þ þ BLðXk ; Kk Þ4LMIQk ðXk ; Kk Þ4LMIQk1 ðXk ; Kk Þ50
Proof The first inequality is clear from Theorem 3 LðXk ; Kk Þ þ BLðXk ; Kk Þ4LMIQk ðXk ; Kk Þ The inequality: LMIQk1 ðXk ; Kk Þ50 is obvious since ðXk ; Kk Þ is a solution of the problem LMIQk1 ðX; KÞ50: Let us now consider the second inequality LMIQk ðXk ; Kk Þ4LMIQk1 ðXk ; Kk Þ: Using the definition of LMIQ ðX; KÞ one obtains LMIQk ðXk ; Kk Þ LMIQk1 ðXk ; Kk Þ ¼ LðXk ; Kk Þ þ FQk þ ðXk M3 ÞT ðXk M3 Þ þ ðM1 Kk M2 ÞT ðM1 Kk M2 Þ LðXk ; Kk Þ FQk1 ðXk M3 ÞT ðXk M3 Þ ðM1 Kk M2 ÞT ðM1 Kk M2 Þ After simplification: LMIQk ðXk ; Kk Þ LMIQk1 ðXk ; Kk Þ ¼ FQk FQk1 : Moreover, from the definition of FQ ðX; KÞ (21) it follows that, FQk ðXk ; Kk Þ FQk1 ðXk ; Kk Þ ¼ ½QTk Qk QTk Qk QTk Qk ½QTk1 Qk1 QTk1 Qk QTk Qk1 ¼ ðQk Qk1 ÞT ðQk Qk1 Þ40 This implies that 8k 2 N; FQk ðXk ; Kk Þ4FQk1 ðXk ; Kk Þ and the final result LMIQk ðXk ; Kk Þ4 LMIQk1 ðXk ; Kk Þ; whatever 8k 2 N: & Theorem Consider the optimization problem consisting in minimizing a linear objective JðX; KÞ under the constraint LðX; KÞ þ BLðX; KÞ50: The iterative LMI algorithm defined in Section 3.2 converges monotonically. Proof Let lk be the optimal value of the criterion JðX; KÞ under LMIQk ðX; KÞ50: According to the previous Lemma, ðXk ; Kk Þ; the solution obtained at the kth iteration of the algorithm, is also a feasible solution of the optimization problem associated with the ðk þ 1Þth iteration since LMIQk ðXk ; Kk Þ50: Consequently, 8k 2 N; lk 4lk1 since lk is the optimal value of criterion JðX; KÞ obtained at the ðk þ 1Þth iteration of the algorithm. It is then clear from the Bolzano–Weierstrass theorem that the proposed algorithm converges monotonically. & Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
1297
PARAMETRIC INSENSITIVE DESIGN
APPENDIX B: NOMENCLATURE MT M > 0; M50 diagð. . .Þ Iq
The ‘system matrix’ notation Fl ð; Þ
transpose of matrix M the matrix M is symmetric and positive definite or semi-definite diagonal matrix formed from the arguments the q q identity matrix the Kronecker product of matrices i.e.: a11 B . . . a1n B am1 B . . . amn B A B GðsÞ ¼ CðsI AÞ1 B þ D , GðsÞ :¼ C D
4
C ¼ A B¼
denotes the lower linear fractional transformation (LFT)
ACKNOWLEDGEMENTS
The authors wish to thank F. Gay, D. Lefebvre and Ph. de Larminat for their help.
REFERENCES 1. Banjerdpongchai D, How JP. Parametric robust H2 control design with generalized multipliers via LMI synthesis. Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, 1996; 265–270. 2. Shirley PO, Lemos JM. An introduction to a nonlinear programming approach to optimal continuous robust parametric H2 control. Proceedings of the European Control Conference, Porto, Portugal, 2001. 3. Doyle JC. Analysis of control systems with structured uncertainty. IEE Proceeding 1982; D129:242–250. 4. Safonov MG. Stability margins of diagonally perturbed multivariable feedback systems. IEE Proceeding 1982; D129:251–256. 5. Heniche A, Bourl"es H. A desensitised controller for voltage regulation of power systems. IEEE Transactions on Power Systems 1995; 10:1461–1466. 6. Gay F, De Larminat Ph. Robust vehicle dynamics control under cornering stiffness uncertainties with insensitive H2 theory. 4th World Multiconference on Circuits, Systems, Communications, Computers, CSCC, Vouliagmeni, Greece, July, 2000. 7. Eslami M. Theory of sensitivity in dynamic systems. An Introduction. Springer: Berlin, 1994. 8. Tahk M, Speyer JL. Modelling of parameter variations and asymptotic LQG synthesis. IEEE Transactions on Automatic Control 1987; AC-32: 9. 9. Begovich O. D!eveloppement et analyse d’outils pour la conception des syst"emes de commande robuste. Ph. D. Thesis, Rennes University, France, 1992. 10. De Larminat Ph. Automatique, commande des syste"mes line!aires (2nd edn). Herm"es: Paris, 1996. 11. Yagoubi M, Chevrel P. A solution to the insensitive H2 problem and its application to automotive control design. IFAC’2002, Barcelona, Spain, 2002. 12. Chevrel P, Yagoubi M. On Insensitive H2 Control Problem Complexity. ACC: Arlington, 2001. 13. Geromel JC, Bernussou JM, de Oliveira MC. H2 norm optimisation with constrained dynamic output feedback controllers: decentralised and reliable control. 36th CDC, San Diego, CA, 1997. 14. El Ghaoui L, Balakrishnan V. Synthesis of fixed-structure controllers via numerical optimisation. Proceedings of CDC’94, New Orleans, LA, 1994. 15. Yagoubi M, Chevrel P. An ILMI approach to structure constrained control design. European Control Conference, Porto, Portugal, 2001. 16. Yagoubi M, Chevrel P. Direct versus indirect observer-based H1 control design. IFAC’2002, Barcelona, Spain, 2002. 17. De Larminat Ph. Contro#le d’e!tat standard. Herm"es: Paris, 2000. 18. Lefebvre D, Chevrel P, Richard S. A hinfinity-based control design methodology dedicated to active control of vehicle longitudinal oscillations. 40th CDC, FL, U.S.A., 2001.
Copyright # 2004 John Wiley & Sons, Ltd.
Int. J. Robust Nonlinear Control 2004; 14:1283–1297
