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triangular systems
Mondher Farza, Mohammed M’Saad, Olivier Gehan, Eric Pigeon, Sofien Hajji
To cite this version:
Mondher Farza, Mohammed M’Saad, Olivier Gehan, Eric Pigeon, Sofien Hajji. State feedback con- troller for a class of MIMO non triangular systems. The 18th IFAC World Congress on Automatic Control, Aug 2011, Milan, Italy. �hal-01060149�
State feedback controller for a class of MIMO non triangular systems
M. Farza∗ , M. M’Saad∗, O. Gehan∗ , E. Pigeon∗ and S. Hajji∗∗
∗GREYC UMR 6072 CNRS, Universit´e de Caen, ENSICAEN 6 Boulevard Mar´echal Juin, 14050 Caen Cedex, France (e-mail:{mfarza,msaad,gehan,pigeon}@greyc.ensicaen.fr)
∗∗Unit´e de Recherche UCA, ENIS, D´epartement de G´enie ´electrique, BP W, 3038 Sfax, Tunisia (e-mail: hjjisfin@yahoo.fr)
Abstract: This paper presents a controller design for a class of MIMO nonlinear systems involving some uncertainties. The latter is particularly composed by cascade subsystems and each subsystem is associated to a subset of the system outputs and assumes a triangular dependence on its own state variables and may depend on the state variables of all other subsystems. The main contribution consists in extending the available control results to allow more interconnections between the subsystems. Of fundamental interest, it is shown that the underlying tracking error exponentially converges to zero in the absence of uncertainties, and can be made as small as desired by properly specifying the control design parameter in the presence of uncertainties.
Keywords: Nonlinear systems, MIMO systems, Tracking, State feedback high gain control.
1. INTRODUCTION
In this paper, one aims at addressing the output tracking problem for MIMO uniformly observable systems which dynamical behavior can be described by the following state representation:
{𝑥(𝑡) =˙ 𝐴𝑥(𝑡) +𝜑(𝑢(𝑡), 𝑥(𝑡)) +𝐵(𝑢(𝑡) +𝜈(𝑡))
𝑦(𝑡) =𝐶𝑥(𝑡) (1)
where
∙ 𝑥denotes the state of the system and is composed as follows
𝑥=
⎛
⎜
⎜
⎜
⎝ 𝑥1 𝑥2 ... 𝑥𝑞
⎞
⎟
⎟
⎟
⎠
∈IR𝑛 𝑤𝑖𝑡ℎ 𝑥𝑘 =
⎛
⎜
⎜
⎜
⎝ 𝑥𝑘1 𝑥𝑘2 ... 𝑥𝑘𝜆𝑘
⎞
⎟
⎟
⎟
⎠
∈IR𝑛𝑘
where 𝑥𝑘𝑖 =
⎛
⎜
⎝ 𝑥𝑘𝑖,1
... 𝑥𝑘𝑖,𝑝𝑘
⎞
⎟
⎠ ∈ IR𝑝𝑘 with 𝑥𝑘𝑖,𝑗 ∈ IR for 𝑘 = 1, . . . , 𝑞, 𝑖 = 1, . . . , 𝜆𝑘, 𝑗 = 1, . . . , 𝑝𝑘 with
𝑞
∑
𝑘=1
𝑛𝑘=
𝑞
∑
𝑘=1
𝑝𝑘𝜆𝑘 =𝑛; 𝑝𝑘 ≥1𝑎𝑛𝑑 𝜆𝑘≥2
∙ 𝑢and𝑦respectively denotes the input and the output of the system
𝑢=
⎛
⎜
⎜
⎝ 𝑢1
𝑢2
... 𝑢𝑞
⎞
⎟
⎟
⎠ , 𝑦=
⎛
⎜
⎜
⎝ 𝑦1
𝑦2
... 𝑦𝑞
⎞
⎟
⎟
⎠
∈IR𝑝
where 𝑢𝑘 =
⎛
⎜
⎝ 𝑢𝑘,1
... 𝑢𝑘,𝑝𝑘
⎞
⎟
⎠, 𝑦𝑘 =
⎛
⎜
⎝ 𝑦𝑘,1
... 𝑦𝑘,𝑝𝑘
⎞
⎟
⎠ ∈ IR𝑝𝑘 for 𝑘= 1, . . . , 𝑞with𝑢𝑘,𝑖, 𝑦𝑘,𝑖∈IR and hence
𝑞
∑
𝑘=1
𝑝𝑘=𝑝
∙ The matrices A and C are respectively given by
𝐴=𝑑𝑖𝑎𝑔(𝐴1, . . . , 𝐴𝑞), 𝐴𝑘 =
⎡
⎢
⎢
⎣
0 𝐼𝑝𝑘 0 ... . ..
0 . . . 0 𝐼𝑝𝑘
0 . . . 0 0
⎤
⎥
⎥
⎦ (2)
𝐶=𝑑𝑖𝑎𝑔(𝐶1, . . . , 𝐶𝑞), 𝐶𝑘= [𝐼𝑝𝑘 0 . . . 0 ] (3) and
𝐵=𝑑𝑖𝑎𝑔(𝐵1, . . . , 𝐵𝑞), 𝐵𝑇𝑘 = [ 0 . . . 0 𝐼𝑝𝑘] (4)
∙ 𝜑(𝑢, 𝑥) denotes the nonlinear function field which is composed as follows
𝜑(𝑢, 𝑥) =
⎛
⎜
⎜
⎜
⎝
𝜑1(𝑢, 𝑥) 𝜑2(𝑢, 𝑥)
... 𝜑𝑞(𝑢, 𝑥)
⎞
⎟
⎟
⎟
⎠
, 𝜑𝑘(𝑢, 𝑥) =
⎛
⎜
⎜
⎜
⎝
𝜑𝑘1(𝑢, 𝑥) 𝜑𝑘2(𝑢, 𝑥)
... 𝜑𝑘𝜆𝑘(𝑢, 𝑥)
⎞
⎟
⎟
⎟
⎠
with 𝜑𝑘(𝑢, 𝑥) ∈ IR𝑛𝑘 and each function 𝜑𝑘𝑖(𝑢, 𝑥) ∈ IR𝑝𝑘 is differentiable with respect to 𝑥and assumes the following structural dependence on the state variables.
∙ for 1≤𝑖≤𝜆𝑘−1:
𝜑𝑘𝑖(𝑢, 𝑥) =𝜑𝑘𝑖(𝑥)
=𝜑𝑘𝑖(𝑥1, 𝑥2, . . . , 𝑥𝑘−1, 𝑥𝑘1, 𝑥𝑘2, . . . , 𝑥𝑘𝑖) (5)
Copyright by the 11097
∙for𝑖=𝜆𝑘:
𝜑𝑘𝜆𝑘(𝑢, 𝑥) =𝜑𝑘𝜆𝑘(𝑢1, . . . , 𝑢𝑘−1, 𝑥1, 𝑥2, . . . , 𝑥𝑞) (6)
∙ 𝜈(𝑡) is an unknown function that can be approxi- mated (more or less precisely) by a known function that shall be denoted by ˆ𝜈(𝑡). If no approximation is available, one takes ˆ𝜈(𝑡) = 0. The function 𝜈(𝑡) (as well as ˆ𝜈(𝑡)) assumes the following structure:
𝜈(𝑡) =
⎛
⎜
⎜
⎜
⎝ 𝜈1(𝑡) 𝜈2(𝑡)
... 𝜈𝑞(𝑡)
⎞
⎟
⎟
⎟
⎠
∈IR𝑛, 𝜈𝑘(𝑡) =
⎛
⎜
⎜
⎜
⎝ 𝜈1𝑘(𝑡) 𝜈2𝑘(𝑡)
... 𝜈𝜆𝑘𝑘(𝑡)
⎞
⎟
⎟
⎟
⎠
∈IR𝑛𝑘
where the functions𝜈𝑘𝑖(𝑡)∈IR𝑝𝑘 are given by 𝜈𝑖𝑘 =
⎧
⎨
⎩
0 𝑓 𝑜𝑟 𝑘= 1, . . . , 𝑞 𝑎𝑛𝑑 𝑖= 1, . . . , 𝜆𝑘−1 𝜈𝑘(𝑡) 𝑓 𝑜𝑟 𝑘= 1, . . . , 𝑞 𝑎𝑛𝑑 𝑖=𝜆𝑘
For𝑘= 1, . . . , 𝑞, let
𝜀𝑘 = ˆ𝜈𝑘−𝜈𝑘 (7)
Then one assumes that the 𝜀𝑘’s, are unknown bounded functions, i.e.
∃𝛽 >0; ∀𝑡≥0; ∥𝜀𝑘(𝑡)∥ ≤𝛽 (8) The control problem to be addressed consists in an admis- sible asymptotic tracking of an output reference trajectory that will be noted ¯𝑦 =
⎛
⎜
⎜
⎝
¯ 𝑦1
¯ 𝑦2
...
¯ 𝑦𝑞
⎞
⎟
⎟
⎠
∈IR𝑝 where ¯𝑦𝑘 ∈IR𝑝𝑘 for 𝑘= 1, . . . , 𝑞is assumed to be smooth enough, i.e.
∥ lim
𝑡→∞ 𝑦(𝑡)−𝑦(𝑡)∥ ≤¯ 𝛼 (9)
where 𝛼is a non negative scalar with is equal to zero if
𝑡lim→∞ 𝜈(𝑡)−𝜈(𝑡) = 0 and henceforthˆ 𝛽= 0.
The problems of observation and control of nonlinear systems have received a particular attention throughout the last four decades. Considerable efforts were dedicated to the analysis of the structural properties to understand better the concepts of controllability and of observability of nonlinear systems (Hammouri and Farza [2003], Raja- mani [1998], Isidori [1995], Gauthier and Kupka [1994], Fliess and Kupka [1983]). Several control and observer design methods were developed thanks to the available techniques, namely feedback linearisation, flatness, high gain, variable structure, sliding modes and backstepping (Farza et al. [2005], Agrawal and Sira-Ramirez [2004], Fliess et al. [1999], Sepulchre et al. [1997], Fliess et al.
[1995], Isidori [1995], Krstiˇc et al. [1995]). The main differ- ence between these contributions lies in the design model, and henceforth the considered class of systems, and the nature of stability and performance results. A particular attention has been devoted to the design of state feed- back control laws incorporating an observer satisfying the separation principle requirements as in the case of linear systems (Mahmoud and Khalil [1996]).
In this paper, one proposes a state feedback controller for a class of non triangular MIMO nonlinear systems described by (1). The following features are worth to be mentioned
∙ The last equation of each subsystem, where the input intervenes, may depend on the whole state. The sys- tem cannot be henceforth considered as a cascade of subsystems that can be sequentially controlled. This is the main issue in the control design.
∙ The absence of zeros in the considered class of systems is particularly motivated by pedagogical purposes as one seeks to emphasize the rational behind the control design method. Nevertheless, the proposed method can be straightforwardly extended to minimum phase systems (see for instance Praly [2003]).
∙ For the sake of space limitation, only state feedback control is considered in this paper. Nevertheless, as the system (1) is uniformly observable, the proposed state feedback control law can be combined with an appropriate high gain observer to get an output state feedback controller as in Hajji et al. [2008]. A high gain state observer for a class of systems similar to (1) has been proposed in Farza et al. [2011].
Notice that the authors in Liu et al. [1999] considered a stabilization problem for a class of non uniformly observ- able systems that has not a triangular form. The proposed method cannot be extended to tackle neither the output feedback stabilization, nor the tracking problem.
The state feedback control design under consideration is particularly suggested from the high gain observer design bearing in mind the control and observation duality. Of particular interest, the controller gain involves a well de- fined design function which provides a unified framework for the high gain control design, namely several versions of sliding mode controllers are obtained by considering par- ticular expressions of the design function. Furthermore, it is shown that an integral action can be simply incorporated into the control design to carry out a robust compensation of step like disturbances.
This paper is organized as follows. The reference model is introduced in section 2 with an explicit procedure for the determination of its state and input. Some useful preliminaries, notations and definitions are given in section 3 with a technical lemma which is crucial for the controller design. Section 4 is devoted to the state feedback control design with a full convergence analysis of the tracking error. The possibility to incorporate an integral action into the control design is shown in section 5. Simulation results are given in section 6 in order to show the effectiveness of the proposed control design method.
2. REFERENCE MODEL
System (1) can be written under the following more developed form as follows
{𝑥˙𝑘=𝐴𝑘𝑥𝑘+𝜑𝑘(𝑢, 𝑥) +𝐵𝑘(𝑢𝑘+𝜈𝑘)
𝑦𝑘=𝐶𝑘𝑥𝑘 𝑘= 1, . . . , 𝑞 (10) Taking into account the class of systems, it is possible to determine the system state trajectory ¯𝑥 =
⎛
⎜
⎜
⎜
⎝
¯ 𝑥1
¯ 𝑥2
...
¯ 𝑥𝑞
⎞
⎟
⎟
⎟
⎠
∈
11098
IR𝑛 and the system input sequence ¯𝑢 ∈ IR𝑝 with ¯𝑥𝑘 ∈ IR𝑛𝑘 and ¯𝑢𝑘 ∈ IR𝑝𝑘, 𝑘 = 1, . . . , 𝑞, corresponding to the output trajectory ¯𝑦(𝑡). This allows to define an admissible reference model as follows
{𝑥˙¯𝑘 =𝐴𝑘𝑥¯𝑘+𝜑𝑘(¯𝑢,¯𝑥) +𝐵𝑘(¯𝑢𝑘+ ˆ𝜈𝑘)
¯
𝑦𝑘 =𝐶𝑘𝑥¯𝑘 for 𝑘= 1, . . . , 𝑞 (11) where ˆ𝜈𝑘 is the available approximation of𝜈𝑘. The deter- mination of the components of the reference model state,
¯
𝑥𝑘 as well as those of its input, ¯𝑢𝑘 is described in the next subsection.
2.1 Determination of the state and input of the reference model
One shall firstly detail the determination of ¯𝑥1𝑖, 𝑖 = 1, . . . , 𝜆1. Indeed, one has:
¯
𝑥11= ¯𝑦1 (12)
¯
𝑥1𝑖 = ˙¯𝑥1𝑖−1−𝜑1𝑖−1(¯𝑥11, . . . ,𝑥¯1𝑖−1) for 𝑖= 2, . . . , 𝜆1(13) By assuming that the reference trajectory ¯𝑦 is smooth enough, one can recursively determine the components of ¯𝑥1 from ¯𝑦1 and its (𝜆1−1) first time derivatives, i.e.
¯
𝑦(𝑖)=𝑑𝑖𝑦¯1
𝑑𝑡𝑖 ,𝑖= 1, . . . , 𝜆1−1, as follows:
¯
𝑥1𝑖 =𝛽𝑖1(¯𝑦1,𝑦˙¯1, . . . ,𝑦¯(𝑖1−1)), 𝑖= 1, . . . , 𝜆1 (14) where the functions 𝛽𝑖1 are defined as follows:
⎧
⎨
⎩
𝛽11(¯𝑦1) = ¯𝑦1 for 𝑖= 2, . . . , 𝜆1: 𝛽𝑖1=
𝑖−2
∑
𝑗=0
∂𝛽𝑖1−1
∂𝑦¯(𝑗)1 𝑦¯(𝑗+1)1 −𝜑1𝑖−1(¯𝑥11, . . . ,𝑥¯1𝑖−1)
=
𝑖−2
∑
𝑗=0
∂𝛽𝑖1−1
∂𝑦¯(𝑗)1 𝑦¯(𝑗+1)1
−𝜑1𝑖−1
(
𝛽11(¯𝑦1), 𝛽21(¯𝑦1,𝑦˙¯1), . . . , 𝛽𝑖1−1
(𝑦¯1, . . . ,𝑦¯(𝑖1−2))) Now, one can proceed in a similar manner in order to de- termine recursively ¯𝑥𝑘𝑖, for𝑘= 2, . . . , 𝑞and 𝑖= 1, . . . , 𝜆𝑘, as follows:
¯
𝑥𝑘𝑖 =𝛽𝑖𝑘(¯𝑦1, . . . ,𝑦¯(𝑖−1+
∑𝑘−1 𝑙=1(𝜆𝑙−1))
1 , . . . , (15)
¯
𝑦𝑘−1, . . . ,𝑦¯𝑘(𝑖−−11+𝜆𝑘−1−1),𝑦¯𝑘, . . . ,𝑦¯(𝑖𝑘−1)) 𝑘= 2, . . . , 𝑞 and 𝑖= 1, . . . , 𝜆𝑘 (16) where the functions 𝛽𝑖𝑘 are defined as follows:
⎧
⎨
⎩
for 𝑘 = 2, . . . , 𝑞: 𝛽1𝑘= ¯𝑦𝑘
for 𝑖 = 2, . . . , 𝜆𝑘: 𝛽𝑖𝑘 =
𝑘−1
∑
𝑙=1 𝑖−2
∑
𝑗=0
∂𝛽𝑘𝑖−1
∂𝑦¯𝑙(𝑗)𝑦¯𝑙(𝑗+1)+
𝑖−2
∑
𝑗=0
∂𝛽𝑘𝑖−1
∂𝑦¯𝑘(𝑗)𝑦¯𝑘(𝑗+1)
−𝜑𝑘𝑖−1(¯𝑥1, . . . ,𝑥¯𝑘−1,𝑥¯𝑘1, . . . ,𝑥¯𝑘𝑖−1)
=
𝑘−1
∑
𝑙=1 𝑖−2
∑
𝑗=0
∂𝛽𝑘𝑖−1
∂𝑦¯𝑙(𝑗)𝑦¯𝑙(𝑗+1)+
𝑖−2
∑
𝑗=0
∂𝛽𝑘𝑖−1
∂𝑦¯𝑘(𝑗)𝑦¯𝑘(𝑗+1)
−𝜑𝑘𝑖−1
(
𝛽11, . . . , 𝛽𝜆11, . . . , 𝛽𝜆𝑘𝑘−1−1, 𝛽1𝑘, . . . , 𝛽𝑘𝑖−1
) (17)
Once all the components of the reference model state ¯𝑥 are determined, one can compute the components of the reference model input using system (11) as follows:
¯
𝑢1= ˙¯𝑥1𝜆1−𝜑1𝜆1(¯𝑥)−𝜈ˆ1(𝑡) and for 𝑘= 2, . . . , 𝑞
¯
𝑢𝑘= ˙¯𝑥𝑘𝜆𝑘−𝜑𝑘𝜆𝑘(¯𝑢1, . . . ,𝑢¯𝑘−1,𝑥)¯ −ˆ𝜈𝑘(𝑡) (18) The tracking problem (9) can be hence turned to a state trajectory tracking problem defined by
𝑡→lim+∞∥𝑒(𝑡)∥ ≤𝛼 where 𝑒(𝑡) =𝑥(𝑡)−𝑥(𝑡)¯ (19) Such a problem can be interpreted as a regulation problem for the tracking error system obtained from the system and model reference state representations (10) and (11), respectively, namely
⎧
⎨
⎩
˙
𝑒𝑘 = 𝐴𝑘𝑒𝑘+𝜑𝑘(𝑢, 𝑥)−𝜑𝑘(¯𝑢,𝑥)¯ +𝐵𝑘(𝑢𝑘−𝑢¯𝑘+𝜈𝑘−𝜈ˆ𝑘)
˜
𝑦𝑘 = 𝑦𝑘−𝑦¯𝑘 =𝐶𝑘𝑒𝑘 for 𝑘= 1, . . . , 𝑞
(20)
3. SOME DEFINITIONS AND NOTATIONS For𝑘= 1, . . . , 𝑞, let Δ𝑘(𝜌) be the diagonal matrix defined by:
Δ𝑘(𝜌) =𝑑𝑖𝑎𝑔 (
𝐼𝑝𝑘, 1
𝜌𝛿𝑘𝐼𝑝𝑘, . . . , 1 𝜌𝛿𝑘(𝜆𝑘−1)𝐼𝑝𝑘
)
(21) where 𝜌 > 0 is a real number and one defines 𝛿𝑘 which indicates the power of𝜌as follows:
⎧
⎨
⎩
𝛿𝑘 = 2𝑞−𝑘 ( 𝑞
∏
𝑖=𝑘+1
( 𝜆𝑖−3
2 ))
for 𝑘= 1, . . . , 𝑞−1 𝛿𝑞 = 1
(22) One also defines for 𝑘 = 1, . . . , 𝑞 and 𝑖 = 1, . . . , 𝜆𝑘, the following sequence of scalar numbers:
𝜎𝑘𝑖 =𝜎𝑘1+ (𝑖−1)𝛿𝑘
with𝜎𝑘1=−(𝜆𝑘−1)𝛿𝑘+ (𝜆1−1)𝛿1+𝜂 (
1− 1 2𝑘−1
) (23) where 0 < 𝜂 ≤ 1 is an arbitrarily small non negative number.
One can check that 𝜎𝑖𝑘 ≥ 0 for 𝑘 = 1, . . . , 𝑞 and 𝑖 = 1, . . . , 𝜆𝑘.
Similarly to the Δ𝑘’s, one defines for 𝑘 = 1, . . . , 𝑞 the diagonal matrices Λ𝑘’s as follows:
Λ𝑘(𝜌) =𝜌−𝜎𝑘1Δ𝑘(𝜌) (24) Notice that, according to the definition of the𝜎𝑖𝑘’s given by (23), one has𝜎𝑘𝜆
𝑘= (𝜆1−1)𝛿1+𝜂(
1−2𝑘−11
)and therefore one has
𝜎𝜆𝑘𝑘−𝜎𝜆𝑙𝑙=𝜂 ( 1
2𝑙−1 − 1 2𝑘−1
)
for 𝑘, 𝑙= 1, . . . , 𝑞(25) This means that whatever is the difference between 𝜆𝑘
and 𝜆𝑙, the difference between 𝜎𝜆𝑘𝑘 and 𝜎𝜆𝑙𝑙 (which are the powers of 𝜌 on the last rows of Λ𝑘(𝜌) and Λ𝑙(𝜌), respectively) can be made as small as desired by choosing 𝜂 small enough (very close to zero).
Now, taking into account the structures of Λ𝑘, Δ𝑘(𝜌) and 𝐴𝑘, respectively given by (24), (21) and (2), one can show that the following identities hold:
∙ Λ𝑘(𝜌)𝐴𝑘Λ−𝑘1(𝜌) = Δ𝑘(𝜌)𝐴𝑘Δ−𝑘1(𝜌) =𝜌𝛿𝑘𝐴𝑘
∙ 𝜌−𝜎𝑘1𝐶𝑘Λ−𝑘1(𝜌) =𝐶𝑘Δ−𝑘1(𝜌) =𝐶𝑘 (26) Two other set of matrices that shall be used in the controller design are 𝑆𝑘 and 𝑃𝑘, 𝑘 = 1, . . . , 𝑞 which are defined as follows. Indeed, let consider the following algebraic Lyapunov equation
𝑆𝑘+𝐴𝑇𝑘𝑆𝑘+𝑆𝑘𝐴𝑘=𝐶𝑘𝑇𝐶𝑘 (27) where 𝐴𝑘 and𝐶𝑘 are defined in (2) and (4), respectively.
It has been shown that such this equation has a unique solution 𝑆𝑘 that is symmetric positive definite and one has (Gauthier et al. [1992])
𝑆𝑘−1𝐶𝑘𝑇 = (𝐶𝜆1𝑘𝐼𝑝𝑘, . . . , 𝐶𝜆𝜆𝑘𝑘𝐼𝑝𝑘)𝑇 (28) where 𝐶𝜆𝑖𝑘 = 𝜆𝑘!
𝑖!(𝜆𝑘−𝑖)! for𝑖= 1, . . . , 𝜆𝑘
Similarly, consider the following algebraic Riccatti equa- tion:
𝑃𝑘+𝐴𝑇𝑘𝑃𝑘+𝑃𝑘𝐴𝑘 =𝑃𝑘𝐵𝑘𝐵𝑘𝑇𝑃𝑘 (29) Again, equation (29) admits a unique solution 𝑃𝑘 that is symmetric positive definite and it can be expressed as follows:
𝑃𝑘=𝑇𝑘𝑆𝑘−1𝑇𝑘 with 𝑇𝑘 =
⎡
⎢
⎢
⎢
⎣
0 0 𝐼𝑝𝑘 ... 0 𝐼𝑝𝑘 0
0 ...
𝐼𝑝𝑘 0 . . . 0
⎤
⎥
⎥
⎥
⎦
, (30)
where 𝑆𝑘 is the solution of (27). This can be checked by multiplying the left and right sides of equation (27) by 𝑇𝑘𝑆𝑘−1and𝑆𝑘−1𝑇𝑘, respectively and by using the facts that 𝑇𝑘𝐴𝑘𝑇𝑘 =𝐴𝑇𝑘, 𝑇𝑘𝑇𝑘 =𝐼𝑘 and 𝐵𝑘 =𝑇𝑘𝐶𝑘𝑇. As a result, one has:
𝐵𝑘𝑇𝑃𝑘 =𝐶𝑘𝑆𝑘−1𝑇𝑘 = (𝐶𝜆𝜆𝑘
𝑘𝐼𝑝𝑘, . . . , 𝐶𝜆1𝑘𝐼𝑝𝑘) (31) In the sequel, one shall denote by𝜆𝑚𝑖𝑛(⋅) and𝜆𝑚𝑎𝑥(⋅) the smallest and largest eigenvalues of (⋅), respectively. One also defines the following block diagonal matrix:
𝑃 =𝑑𝑖𝑎𝑔(𝑃1, 𝑃2, . . . , 𝑃𝑞) (32) Before ending this section, one shall give a technical lemma, given in [Faral11] needed in the proof of our main result. This lemma allows to provide a sequence of reals that reflects in some sense the interconnections between the blocks nonlinearities.
Lemma 3.1. Let 𝜒𝑘,𝑖𝑙,𝑗 =
⎧
⎨
⎩
0 𝑖𝑓 ∂𝜑𝑘𝑖
∂𝑥𝑙𝑗(𝑢, 𝑥)≡0 1 otherwise
(33) for 𝑘, 𝑙 = 1, . . . , 𝑞, 𝑖 = 1, . . . , 𝜆𝑘 and 𝑗 = 2, . . . , 𝜆𝑙. Then, the sequence of real numbers 𝜎𝑘𝑖 defined by (23) are such that
if𝜒𝑘,𝑖𝑙,𝑗 = 1 then𝜎𝑗𝑙−𝜎𝑖𝑘−𝛿𝑙
2 −𝛿𝑘
2 ≤ − 𝜂
2𝑞 (34)
4. STATE FEEDBACK CONTROL
As it was early mentioned, the proposed state feedback control design is particularly suggested by the duality from the high gain observer design proposed in Farza et al.
[2011]. The underlying state feedback control law is then given by
⎧
⎨
⎩ 𝑣𝑘(
𝑒𝑘)
=−𝐾𝑘(
𝜌𝜆𝑘𝛿𝑘𝐵𝑘𝑇𝑃𝑘Δ𝑘(𝜌)𝑒𝑘) 𝑢𝑘(𝑥) = ¯𝑢𝑘+𝑣𝑘(𝑒𝑘)
(35) where 𝑒𝑘 = 𝑥𝑘 −𝑥¯𝑘, 𝜌 > 0 is the controller design parameter, Δ𝑘(𝜌) and 𝑃𝑘 are given by (21) and (29), respectively and finally 𝐾𝑘 : IR𝑝𝑘 7→ IR𝑝𝑘 is a bounded design function satisfying the following property
∀𝜉𝑘∈Ω𝑘 𝑜𝑛𝑒 ℎ𝑎𝑠 𝜉𝑘𝑇𝐾𝑘(𝜉𝑘) ≥ 1
2 𝜉𝑇𝑘𝜉𝑘 (36) where Ω𝑘 is an arbitrarily (fixed) compact subset of IR𝑝𝑘. Many functions satisfying condition (36) are given in Farza et al. [2005].
Before stating the fundamental result of the paper, one needs the following technical assumption, usually adopted when using high gain approaches, namely
Assumption1. 𝜑(𝑢, 𝑥) is a globally Lipschitz nonlinear function with respect to𝑥uniformly in𝑢.
One can then states the following fundamental result Theorem4.1. Assume that system (1) satisfies assump- tion (1), then
∃𝜌0 >0;∀𝜌 > 𝜌0;∃𝜆𝜌 >0;∃𝜇𝜌 >0; ∃𝛼𝜌 >0 such that for𝑘∈ {1, . . . , 𝑞}
∥𝑥𝑘(𝑡)−¯𝑥𝑘(𝑡)∥ ≤𝜆𝜌𝑒−𝜇𝜌𝑡∥𝑥(0)−𝑥(0)∥¯ +𝛼𝜌𝛽 where 𝛽 is the upper bound of ∥𝜈 −𝜈ˆ∥. Moreover, 𝜆𝜌 is polynomial in𝜌, lim
𝜌→∞𝜇𝜌= +∞and lim
𝜌→∞𝛼𝜌= 0.
Sketch of the proof of Theorem 4.1.Set the tracking error𝑒(𝑡) =𝑥(𝑡)−𝑥(𝑡) and let¯ 𝑒𝑘(𝑡) be the𝑘′𝑡ℎ subcom- ponent of𝑒(𝑡). One has:
˙
𝑒𝑘=𝐴𝑘𝑒𝑘+𝜑𝑘(𝑢, 𝑥)−𝜑𝑘(𝑢,𝑥)¯
−𝐵𝑘𝐾𝑘(
𝜌𝜆𝑘𝛿𝑘𝐵𝑘𝑇𝑃𝑘Δ𝑘(𝜌)𝑒𝑘)
−𝐵𝑘𝜀𝑘 (37) where𝜀𝑘= ˆ𝜈𝑘−𝜈𝑘 as defined by (7).
For𝑘= 1,⋅ ⋅ ⋅, 𝑞, set
¯
𝑒𝑘 = Λ𝑘(𝜌)𝑒𝑘 (38)
where Λ𝑘(𝜌) is given by (24).
From equation (37) and using identities (26), one gets
˙¯
𝑒𝑘=𝜌𝛿𝑘𝐴𝑘𝑒¯𝑘+ Λ𝑘(𝜌)(𝜑𝑘(𝑢, 𝑥)−𝜑𝑘(𝑢,𝑥))¯ −Λ𝑘(𝜌)𝐵𝑘𝜀𝑘
− 1
𝜌2𝜎𝑘1+2(𝜆𝑘−1)𝛿𝑘𝜌𝜎𝑘1+𝜆𝑘𝛿𝑘𝐵𝑘𝐾𝑘
(
𝜌𝜎1𝑘+𝜆𝑘𝛿𝑘𝐵𝑘𝑇𝑃𝑘𝑒¯𝑘) Set
𝑉𝑘(¯𝑒𝑘) = ¯𝑒𝑘𝑇𝑃𝑘¯𝑒𝑘 (39) where𝑃𝑘is given by (29) and let𝑉(¯𝑒) =
𝑞
∑
𝑘=1
𝑉𝑘(¯𝑒𝑘) be the candidate Lyapunov function.
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