International Journal of Control Science and Engineering

p-ISSN: 2168-4952    e-ISSN: 2168-4960

2017;  7(1): 1-10

doi:10.5923/j.control.20170701.01

 

Separation Principle for a Class of Takagi-Sugeno Descriptor Models

Ilham Hmaiddouch1, 2, El Mahfoud El Bouatmani1, 2, Jalal Soulami1, 2, Abdellatif El Assoudi1, 2, El Hassane El Yaagoubi1, 2

1Laboratory of High Energy Physics and Condensed Matter, Faculty of Science, Hassan II University of Casablanca, Casablanca, Morocco

2ECPI, Department of Electrical Engineering, ENSEM, Hassan II University of Casablanca, Casablanca, Morocco

Correspondence to: Jalal Soulami, Laboratory of High Energy Physics and Condensed Matter, Faculty of Science, Hassan II University of Casablanca, Casablanca, Morocco.

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Abstract

This paper studies the design problem of an observer-based controller for a class of nonlinear descriptor systems described by Takagi-Sugeno (T-S) fuzzy structure [1, 2] where the premise variables are unmeasurable. Based on the combination of the control law established in [3] and the fuzzy observer in explicit structure given in [4], a new procedure to investigate the separation principle problem for the considered class of Takagi-Sugeno descriptor models (TSDMs) is then proposed. The guarantee of the global asymptotic stability of the closed-loop augmented system is proved by using the Lyapunov theory and the conditions of asymptotic stability are given in terms of linear matrix inequalities (LMIs). Finally, an application based on rolling disc process is presented as an illustrative example to show the performance of the proposed dynamic controller design.

Keywords: Takagi-Sugeno descriptor model, Fuzzy observer, State feedback controller, Lyapunov method, LMI technique, Separation principle

Cite this paper: Ilham Hmaiddouch, El Mahfoud El Bouatmani, Jalal Soulami, Abdellatif El Assoudi, El Hassane El Yaagoubi, Separation Principle for a Class of Takagi-Sugeno Descriptor Models, International Journal of Control Science and Engineering, Vol. 7 No. 1, 2017, pp. 1-10. doi: 10.5923/j.control.20170701.01.

Article Outline

1. Introduction
2. Takagi-Sugeno Descriptor Models
3. Main Result
4. Illustrative Example
5. Conclusions

1. Introduction

It is well-known that in order to realize in practice a stabilisation by static state feedback for most industrial applications, the separation principle variously called dynamic output feedback controller or observer-based controller plays a key role since instead of a full-state feedback, we use signals reconstructed by a state observer from the on-line measurements of the input and output of the process. The aim of the paper consists to design an observer-based controller for a class of nonlinear descriptor systems described by T-S representation. Notice that, descriptor models representation has been widely used in the dynamic modeling of many chemical and physical processes [5-8]. The numerical simulation of such dynamic models usually combines an ODE numerical method together with an optimization algorithm. In the literature, there have been several studies concerning the issue of stability and the stabilisation of T-S models [9-11]. Likewise, many works have been carried out to investigate the observer design of T-S models [12-15]. The separation property of T-S observers and controllers was discussed by [16-20]. For TSDMs which are defined by extending the ordinary T-S representation [3, 21], several research works concerning the problem of control and observation design and applications have been developed [3, 4, 21-25]. Notice that, generally an interesting way to solve the various fuzzy observer and controller problems raised previously is to write the convergence conditions on the LMI form [26].
Based on the use of the state feedback given in [3] and the fuzzy observer proposed in [4], the main contribution of this paper consists to propose a new result of separation principle for a class of TSDMs with unmeasurable premise variables. The global asymptotic stability of the closed-loop system is studied by using the Lyapunov theory and the stability conditions are given in terms of LMIs. Moreover, the proposed result is given without the use of an optimization algorithm.
The outline of the paper is structured as follows. In Section 2, the considered class of TSDMs with unmeasurable premise variables is presented. In Section 3, the main result about fuzzy observer-based controller design is established. The control and observer gains are found directly from LMI formulation. Finally, in Section 4, an illustrative application to show the good performance of the proposed result is given.
Throughout the paper, some notations used are fair standard. For example, means the matrix is symmetric and positive definite. denotes the transpose of . The symbol I (or 0) represents the identity matrix (or zero matrix) with appropriate dimension.
At first, we recall some basic lemmas that are frequently used in the setting the proof of the main result of this paper.
Lemma 1 (Congruence): Let two matrices and , if is positive definite and if is a full column rank matrix, then the matrix is positive definite.
Lemma 2 (Young's inequality): For any matrices and with appropriate dimensions, the following property holds for any invertible matrix and scalar :
(1)
Lemma 3: Considering matrices , and a scalar , the following holds:
(2)

2. Takagi-Sugeno Descriptor Models

The following class of TSDMs is considered in this paper:
(3)
where is the state vector, is the control input, is the measured output. , and such that , are real known constant matrices. is the number of sub-models. is the premise variable which can be here depend on completely or partially unmeasured state of the system and the are the weighting functions that ensure the transition between the contribution of each sub model:
(4)
They verify the so-called convex sum properties:
(5)
The main object of this paper is to design an observer-based controller for the class of TSDMs (3). For this objective, we suppose that [5]:
H1) is regular, i.e.
H2) Sub-models (4) are impulse controllable and stabilisable.
H3) Sub-models (4) are impulse observable and detectable.
Moreover, the following hypothesis which is necessary for the observer design given in [4] is assumed to be verified:
H4)
Note that from the consideration of the hypothesis H4), there exists a non-singular matrix such that:
(6)
where, , , are constant matrices which can be found by the singular value decomposition of .
In order to investigate the separation principle problem for the considered class of TSDMs (3), firstly the following control law established in [3] is adopted:
(7)
where the gains can determined in order that the closed-loop system (3) is asymptotically stable.
The following result has been studied in [3].
Theorem 1 [3]: The closed-loop system (3)-(7) is globally exponentially stable if there exist matrices, verifying the following LMIs:
(8)
where
(9)
The fuzzy local feedback gains are given by:
(10)
On the other hand, the following fuzzy observer given in [4] is employed:
(11)
where is the state of fuzzy observer and is the estimate of . denotes the estimated premise variables vector. The matrices , , , and are unknown matrices of appropriate dimensions to be determined such that converges asymptotically to . and are such that equation (6) is satisfied.
The convergence condition of the observer (11) can be formulated by the following Theorem.
Theorem 2 [4]: The state error between the T-S descriptor model (3) and its observer (11) converges asymptotically towards zero, if there exist matrices , , and such that the following LMIs hold :
(12)
where
(13)
and
(14)
The observer gains are given by:
(15)

3. Main Result

As mentioned above, based on the use of the state feedback given by (7) and the fuzzy observer (11), this section studies the problem of observer-based controller design for a class of TSDMs (3). Indeed, the following augmented system in closed-loop is considered:
(16)
In order to establish the conditions for the asymptotic convergence of the system (16), we define the state estimation error:
(17)
Then, by substituting (6) and (16) into (17) we obtain:
(18)
So, error dynamics will be:
(19)
It follows from (16) and (17) that the error dynamics can be written as:
(20)
where
(21)
By substituting (18), equation (20) becomes:
(22)
where
(23)
Provided the matrices gains and satisfy:
(24)
(25)
Then, from (6), (24) and (25), we have:
(26)
Take:
(27)
Then:
(28)
It follows that the system (22) reduces to:
(29)
Note that:
(30)
where and .
Then, from (21), the equation (29) becomes:
(31)
By (5), equation (31) can be reduced to the following:
(32)
where
(33)
Thus, the global asymptotic stability of system (16) can be proved in an equivalent fashion for the following system:
(34)
which can be rewritten as follows:
(35)
where
(36)
The following Theorem constitutes the main result of the paper.
Theorem 3: The system (35) is globally asymptotically stable if for predefined scalars ; there exist matrices , verifying the following LMIs:
(37)
where
(38)
with
(39)
and
(40)
with and are defined in (9) and (13) respectively.
The dynamic controller gains and are given by:
(41)
where are such that (6) is satisfied.
Proof of Theorem 3: To prove the convergence to zero of the state variable system (35), let us consider the candidate Lyapunov function as follows:
(42)
with
(43)
The time derivative of the Lyapunov function (42) along the trajectories of the system (35) is obtained as:
(44)
Therefore, we have the following stability conditions:
(45)
which become from (36) and (43):
(46)
Applying Lemma 1 to (46) by pre-multiplying and post-multiplying to both sides and defining new variables and , thus we obtain:
(47)
which become from (33):
(48)
where
(49)
By considering the following:
(50)
Matrix can be rewritten again as:
(51)
Applying Lemma 2, the equation (51) becomes:
(52)
where
(53)
and are defined in (39).
Hence, using the Schur complement [26], the inequality hold if the following matrix inequality is satisfied:
(54)
Now, in order to establish the LMI conditions (37) of Theorem 3, we rewrite the inequality (54) as follows:
(55)
where
(56)
with
(57)
and we consider the following matrix defined by:
(58)
where
(59)
with
(60)
Applying Lemma 1, is equivalent to:
(61)
which can be rewritten as follows:
(62)
Hence, applying Lemma 3 on leads to:
(63)
where .
Using the Schur complement, we obtain:
(64)
Then, the replacement of the various terms given in (64) by their expressions given before and the use of the changes of variables:
(65)
We establish the LMI conditions given in Theorem 3. Finally, the dynamic controller gains given by (41) can be determined from (25), (27), (28) and (65).

4. Illustrative Example

To illustrate the efficiency of the proposed dynamic controller, we consider a rolling disc process described by the following TSDM with unmeasurable premise variable given in [25] such that the hypothesis H4) is satisfied:
(66)
where is the state vector with is the position of the center of the disc, is the translational velocity of the same point, is the angular velocity of the disc, is the contact force between the disc and the surface. is the applied input force to the disc and is the vector of output measurement.
The weighting functions are defined by:
(67)
where is the premise variable having for expression:
(68)
The physical parameters definition and their numerical values are given in [25].
As mentioned previously, the objective of this section is to use the TSDM (66) as an illustrative example to demonstrate the ability of Theorem 3 to be applied.
Therefore, the following augmented system in closed-loop is considered:
(69)
where are the gains of the proposed dynamic controller. Notice that to apply the theorem 3 in order to calculate such dynamic controller gains; it suffices to rewritten system (69) into its equivalent representation (35) as mentioned in Section 3.
Thus, in a first time, we calculate the matrices satisfying (6) from singular value decomposition of . Indeed, under hypothesis H4), the following numerical expressions of were obtained:
In a second time, we resolve the LMIs (37) in . Indeed, with and , we obtain the following dynamic controller gains calculated by (41):
In order to illustrate the performances of the proposed observer-based controller, numerical simulations of system (69) were performed using a Runge-Kutta method combined with the Newton-Raphson algorithm.
The following initial conditions of system (69) were used:
The simulation results of the system (69) with the dynamic controller gains and are given in Figures 1, 2, 3 and 4 where the dashed lines denote the state variables estimated by the fuzzy observer.
These simulation results show the good performance of the proposed observer-based controller designed. Indeed, the global asymptotic stability of the augmented system (69) in closed-loop with the proposed observer-based controller is satisfied.
Figure 1. with fuzzy observer-based controller
Figure 2. with fuzzy observer-based controller
Figure 3. with fuzzy observer-based controller
Figure 4. with fuzzy observer-based controller
Moreover, these results of simulation show a good estimation of the unknown states of the process which is necessary for the real-time implementation of the control law (7).

5. Conclusions

In this paper, a new result of separation principle for a class of TSDMs with unmeasurable premise variables is suggested. More precisely, the proposed result concerns the real-time implementation without the use of an optimization algorithm of the static state feedback given in [3] by using the fuzzy observer proposed in [4]. The global asymptotic stability of the augmented system in closed-loop is studied by using the Lyapunov method and the stability conditions are given in terms of LMIs. The good performance of the proposed observer-based controller design is illustrated in simulation through a rolling disc process as an illustrative example.

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