Observer-Based Robust Control of Uncertain Fuzzy Models with Pole Placement Constraints

Practical systems are often modelled by nonlinear dynamics. Controlling nonlinear systems are still open problems due to their complexity nature. This problem becomes more complex when the system parameters are uncertain. To control such systems, we may use the linearization technique around a given operating point and then employ the known methods of linear control theory. This approach is successful when the operating point of the system is restricted to a certain region. Unfortunately, in practice this approach will not work for some physical systems with a time-varying operating point. The fuzzy model proposed by Takagi-Sugeno (T-S) is an alternative that can be used in this case. It has been proved that T-S fuzzy models can effectively approximate any continuous nonlinear systems by a set of local linear dynamics with their linguistic description. This fuzzy dynamic model is a convex combination of several linear models. It is described by fuzzy rules of the type If-Then that represent local input output models for a nonlinear system. The overall system model is obtained by “blending” these linear models through nonlinear fuzzy membership functions. For more details on this topic, we refer the reader to (Tanaka & al 1998 and Wand & al, 1995) and the references therein. The stability analysis and the synthesis of controllers and observers for nonlinear systems described by T-S fuzzy models have been the subject of many research works in recent years. The fuzzy controller is often designed under the well-known procedure: Parallel Distributed Compensation (PDC). In presence of parametric uncertainties in T-S fuzzy models, it is necessary to consider the robust stability in order to guarantee both the stability and the robustness with respect to the latter. These may include modelling error, parameter perturbations, external disturbances, and fuzzy approximation errors. So far, there have been some attempts in the area of uncertain nonlinear systems based on the T-S fuzzy models in the literature. The most of these existing works assume that all the system states are measured. However, in many control systems and real applications, these are not always available. Several authors have recently proposed observer based robust controller design methods considering the fact that in real control problems the full state information is not always available. In the case without uncertainties, we apply the separation property to design the observer-based controller: the observer synthesis is designed so that its dynamics are fast and we independently design the controller by imposing slower dynamics. Recently, much effort has been devoted to observer-based control for T-S fuzzy models. (Tanaka & al, 1998) have studied the fuzzy observer design for T-S fuzzy control systems. Nonetheless, in


Plant rule i :
If 1 () ztis M 1i and …and () The structured uncertainties considered here are norm-bounded in the form: () , () , 1 , . . .,  In this paper we assume that all of the state variables are not measurable.Fuzzy state observer for T-S fuzzy model with parametric uncertainties (1) is formulated as follows: Observer rule i: The fuzzy observer design is to determine the local gains in the consequent part.
Note that the premise variables do not depend on the state variables estimated by a fuzzy observer.
The output of ( 5) is represented as follows: { } To stabilize this class of systems, we use the PDC observer-based approach (Tanaka & al, 1998).The PDC observer-based controller is defined by the following rule base system: Controller rule i : The overall fuzzy controller is represented by: Let us denote the estimation error as: The augmented system containing both the fuzzy controller and observer is represented as follows: where The main goal is first, to find the sets of matrices i K and i G in order to guarantee the global asymptotic stability of the equilibrium point zero of (10) and secondly, to design the fuzzy controller and the fuzzy observer of the augmented system (10) separately by assigning both "observer and controller poles" in a desired region in order to guarantee that the error between the state and its estimation converges faster to zero.The faster the estimation error will converge to zero, the better the transient behaviour of the controlled system will be.

Main results
Given (1), we give sufficient conditions in order to satisfy the global asymptotic stability of the closed-loop for the augmented system (10).
Lemma 1: The equilibrium point zero of the augmented system described by ( 10) is globally asymptotically stable if there exist common positive definite matrices 1 P and 2 P , matrices Proof: using theorem 7 in (Tanaka & al, 1998), property (3), the separation lemma (Shi & al, 1992)) and the Schur's complement (Boyd & al, 1994), the above conditions ( 12) and ( 13) hold with some changes of variables.Let us briefly explain the different steps… From (11), in order to ensure the global, asymptotic stability, the sufficient conditions must be verified: Let: where 0 is a zero matrix of appropriate dimension.From (14), we have: 12 (,) where 11 1 From (15), we have:  18), ( 19) and ( 20) and by using the separation lemma (Shi & al, 1992)), we finally obtain: Where: From ( 15), ( 16), ( 17) and ( 21), we have: In order to verify ( 14), we must have: Which implies: First, from (24), by using (3), using the Schur's complement (Boyd & al, 1994) as well as the introduction of the new variable: Where I is always the identity matrix of appropriate dimension and Then, from (24), by using (3), using the Schur's complement (Boyd & al, 1994) as well as the introduction of the new variable: Recent Advances in Robust Control -Novel Approaches and Design Methods 46 Thus, conditions ( 12) and ( 13) yield for all i, j from ( 25) and ( 26) and by using theorem 7 in (Tanaka & al, 1998) which is necessary for LMI relaxations.
Remark 1: In lemma 1, the positive scalars ij ε are optimised unlike (Han & al, 2000), (Lee & al, 2001), (Tong & Li, 2002), (Chadli & El Hajjaji, 2006).We do not actually need to impose them to solve the set of LMIs.The conditions are thus less restrictive.Remark 2: Note that it is a two-step procedure which allows us to design the controller and the observer separately.First, we solve (12) for decision variables 1 (, ,) PK ε and secondly, we solve (13) for decision variables 2 (,) i PG by using the results from the first step.Furthermore, the controller and observer gains are given by: Remark 3: From lemma 1 and (10), the location of the poles associated with the state dynamics and with the estimation error dynamics is unknown.However, since the design algorithm is a two-step procedure, we can impose two pole placements separately, the first one for the state and the second one for the estimation error.In the following, we focus in the robust pole placement.We hereafter give sufficient conditions to ensure the desired pole placements by using the LMI conditions of (Chilali & Gahinet (1996) and (Chilali & al, 1999) Definition 2 (Chilali and Gahinet, 1996): Let D be a subregion of the left-half plane.A dynamical system described by: xA x = is called D-stable if all its poles lie in D. By extension, A is then called D-stable.From the two previous definitions, the following theorem is given.Theorem 1 (Chilali and Gahinet , 1996): Matrix A is D-stable if and only if there exists a symmetric matrix 0 X > such as where ⊗ denotes the Kronecker product.From ( 10) and ( 11), let us define: ( ) ( ) We hereafter give sufficient conditions to guarantee that 11 ( ( )) ( ( )) Using the separation lemma (Shi & al, 1992) and (3), we obtain: Where, of course, , By using the Schur's complement (Boyd & al, 1994), Thus, conditions (29) easily yield for all i, j.
Proof: Same lines as previously can be used to prove this lemma.Let: (,) Using the separation lemma (Shi & al, 1992), by pre-and post-multiplying by 1 IX − ⊗ , we obtain: Where, of course, , ij ij λ ∈ ℜ∀ Thus, by using the Schur's complement (Boyd & al, 1994) as well as by defining By using , conditions (38) easily yield for all i, j.The lemma proof is given.

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Remark 4: Any kind of LMI region (disk, vertical strip, conic sector) may be easily used for D S and T D .From lemma 2 and lemma 3, we have imposed the dynamics of the state as well as the dynamics of the estimation error.But from (10), the estimation error dynamics depend on the state.If the state dynamics are slow, we will have a slow convergence of the estimation error to the equilibrium point zero in spite of its own fast dynamics.So in this paper, we add an algorithm using the H ∞ approach to ensure that the estimation error converges faster to the equilibrium point zero.We know from (10) that: This equation is equivalent to the following system: 11 (() ) (() ) 0 The objective is to minimize the 2 L gain from () xt to () et in order to guarantee that the error between the state and its estimation converges faster to zero.Thus, we define the following H ∞ performance criterion under zero initial conditions: Then, the dynamic system: 11 (() ) (() ) 0 satisfies the H ∞ performance with a L 2 gain equal or less than γ (44) .
Proof: Applying the bounded real lemma (Boyd & al, 1994), the system described by the following dynamics: satisfies the H ∞ performance corresponding to the 2 L gain γ performance if and only if there exists 22 0 Using the Schur's complement, (Boyd & al, 1994) By using the separation lemma (Shi & al, 1992) With substitution into ij Θ and defining a variable change: where Thus, from the following condition and using the Schur's complement (Boyd & al, 1994), theorem 7 in ( Tanaka & al, 1998) and (3), condition ( 46) yields for all i,j.
Remark 5: In order to improve the estimation error convergence, we obtain the following convex optimization problem: minimization γ under the LMI constraints (46).
From lemma 1, 2, 3 and 4 yields the following theorem: Theorem 2: The closed-loop uncertain fuzzy system (10) is robustly stabilizable via the observer-based controller (8) with control performances defined by a pole placement constraint in LMI region T D for the state dynamics, a pole placement constraint in LMI region S D for the estimation error dynamics and a 2 L gain γ performance (45) as small as possible if first, LMI systems ( 12) and ( 29) are solvable for the decision variables 1 (, , , ) PKε μ and secondly, LMI systems ( 13), ( 38) , ( 46) are solvable for the decision variables 2 (,,, ) ii j ij PGλ β .Furthermore, the controller and observer gains are , respectively, for ,1 , 2 , . . . ,.ij r = Remark 6: Because of uncertainties, we could not use the separation property but we have overcome this problem by designing the fuzzy controller and observer in two steps with two pole placements and by using the H ∞ approach to ensure that the estimation error converges faster to zero although its dynamics depend on the state.Remark 7: Theorem 2 also proposes a two-step procedure: the first step concerns the fuzzy controller design by imposing a pole placement constraint for the poles linked to the state dynamics and the second step concerns the fuzzy observer design by imposing the second pole placement constraint for the poles linked to the error estimation dynamics and by minimizing the H ∞ performance criterion (18).The designs of the observer and the controller are separate but not independent.

Numerical example
In this section, to illustrate the validity of the suggested theoretical development, we apply the previous control algorithm to the following academic nonlinear system (Lauber, ( ) y ∈ℜ is the system output, u ∈ ℜ is the system input,

[ ]
12 t xxx = is the state vector which is supposed to be unmeasurable.What we want to find is the control law u which globally stabilizes the closed-loop and forces the system output to converge to zero but by imposing a transient behaviour.
Since the state vector is supposed to be unmeasurable, an observer will be designed.
The idea here is thus to design a fuzzy observer-based robust controller from the nonlinear system (57).The first step is to obtain a fuzzy model with uncertainties from (57) while the second step is to design the fuzzy control law from theorem 2 by imposing pole placement constraints and by minimizing the H∞ criterion ( 46).Let us recall that, thanks to the pole placements, the estimation error converges faster to the equilibrium point zero and we impose the transient behaviour of the system output.

First step:
The goal is here to obtain a fuzzy model from (57).
By decomposing the nonlinear term  ( ) m=-0.2172, b=-0.5, a=2 and i=1,2 Second step: The control design purpose of this example is to place both the poles linked to the state dynamics and to the estimation error dynamics in the vertical strip given by: The obtained H ∞ criterion after minimization is: Tables 1 and 2 give some examples of both nominal and uncertain system closed-loop pole values respectively.All these poles are located in the desired regions.Note that the uncertainties must be taken into account since we wish to ensure a global pole placement.That means that the poles of (10) belong to the specific LMI region, whatever uncertainties (2), (3).From tables 1 and 2, we can see that the estimation error pole values obtained using the H ∞ approach are more distant (farther on the left) than the ones without the ∞ approach.-5.38+5.87i -5.38 -5.87i -3.38 + 3.61i -3.38 -3.61i 22 2 2 2 2 bb AG CH E K +− -5.55 +6.01i -5.55 -6.01i -3.83 + 3.86i -3.83 -3.86iTable 2. Pole values (extreme uncertain models).
Figures 1 and 2 respectively show the behaviour of error 1 () et and 2 () et with and without the H ∞ approach and also the behaviour obtained using only lemma 1.We clearly see that the estimation error converges faster in the first case (with H ∞ approach and pole placements) than in the second one (with pole placements only) as well as in the third case (without H ∞ approach and pole placements).At last but not least, Figure 3 and 4 show respectively the behaviour of the state variables with and without the H ∞ approach whereas Figure 5 shows the evolution of the control signal.From Figures 3 and 4, we still have the same conclusion about the convergence of the estimation errors.With the

Conclusion
In this chapter, we have developed robust pole placement constraints for continuous T-S fuzzy systems with unavailable state variables and with parametric structured uncertainties.
The proposed approach has extended existing methods based on uncertain T-S fuzzy models.The proposed LMI constraints can globally asymptotically stabilize the closed-loop T-S fuzzy system subject to parametric uncertainties with the desired control performances.Because of uncertainties, the separation property is not applicable.To overcome this problem, we have proposed, for the design of the observer and the controller, a two-step procedure with two pole placements constraints and the minimization of a H ∞ performance criterion in order to guarantee that the estimation error converges faster to zero.Simulation results have verified and confirmed the effectiveness of our approach in controlling nonlinear systems with parametric uncertainties.
meaning of symbol M ij .
be minimized.Note that the signal () xt is square integrable because of lemma 1.We give the following lemma to satisfy the H ∞ performance.Lemma 4: If there exist symmetric positive definite matrix 2 . The choice of the same vertical strip is voluntary because we wish to compare results of simulations obtained with and without the H ∞ approach, in order to show by simulation the effectiveness of our approach.The initial values of states are chosen: of theorem 2, we obtain the following controller and observer gain matrices respectively: Fig. 1.Behaviour of error 1 () et.
(Boyd & al, 1994)certain T-S fuzzy systems with unavailable state variables.Let us recall the definition of an LMI region and pole placement LMI constraints.Definition 1(Boyd & al, 1994): A subset D of the complex plane is called an LMI region if there exists a symmetric matrix
H ∞ approach Without the H ∞ approach Using lemma 1 www.intechopen.comObserver-Based Robust Control of Uncertain Fuzzy Models with Pole Placement Constraints Behaviour of the state vector and its estimation with the H ∞ approach.Behaviour of the state and its estimation without the H ∞ approach.Observer-Based Robust Control of Uncertain Fuzzy Models with Pole Placement Constraints 57