## 1. Introduction

Over the past decades, many advances have been made in the field of control theory which rely on state-space theory. The control design methodology that has been most investigated for the state-feedback control, see for example [1, 2] and the references therein. The state-feedback control design supposes that all the system states are available, which is not always possible in realistic applications. Instead, one has to deal with the absence of full-state information by using observers. From the control point of view, observers can be used as part of dynamical controllers. This observer-based design has been extensively studied in the literature [3, 4]. However, it leads to high-order controllers. As a matter of fact, one has to solve a large problem, which increases numerical computations for large scale systems. Other difficulties may arise, if we consider additional performances, such as disturbance rejection, time delays, uncertainties, etc. Hence, it is more suitable to develop methodologies which involve a design with a low dimensionality. In this context, intensive efforts have been devoted to design low-order controllers [5, 6, 7, 3]. In particular, it has been shown that designing reduced order stabilizing controllers can be cast as a static output-feedback stabilization problem. Also, it is recognized that, in general, the static output-feedback control design may not exist for certain systems. Note that an important advantage of these controllers is that they are easy to implement without significant numerical burden.

In general, the synthesis of static output-feedback stabilizing controllers is known to be a hard task [5, 6, 7]. The main difficulty rises from its nonconvexity. In the literature, some convexification techniques and iterative algorithms have been proposed to handle this problem [5, 7, 3]. A comprehensive survey on static output-feedback stabilization can be found in [6]. The authors show that despite the considerable efforts devoted to solve this problem, there is yet no methodology that can solve it exactly, so it is still an important open topic. However, it has been shown that for SISO (Single-Input Single-Output) systems, this problem can be solved exactly based on an algebraic characterization [8, 9]. Unfortunately, these approaches are valid only for SISO case and cannot be used to take into account additional constraints on the system. In any case, the investigation of this topic within the field of fuzzy control is continuously increasing and leading to many approaches. A most efficient approach is based on the Linear Matrix Inequality (LMI) technique: see for example [10, 11, 12]. Indeed, since the developed interior-point methods [13], LMIs can be solved in polynomial-time, using numerical algorithms [14]. Recently, other approach, based on a projective algorithm has been proposed [15]. Notice that the existing LMI tools have opened an important research area in system and control theory and tackled numerous unsolved problems [14]. Therefore, our main focus in this chapter is the design of static output-feedback controllers using LMI theory for a class of nonlinear systems described by Takagi-Sugeno (T-S) fuzzy models.

Recently, the study of T-S fuzzy models has attracted the attention of many of researchers: see [16] and references therein. Fuzzy models have local dynamics (i.e., dynamics in different state space regions), that are represented by local linear systems. The overall model of a fuzzy system is then obtained by interpolating these linear models through nonlinear fuzzy membership functions. Unlike conventional modeling techniques, which use a single model to describe the global behavior of a nonlinear system, fuzzy modeling is essentially a multi-model approach, in which simple local linear submodels are designed in the form of a convex combination of local models in order to describe the global behavior of the nonlinear system. This kind of models has proved to be a good representation for a certain class of nonlinear dynamic systems.

Since the work by [17] on stability analysis and state feedback stabilization for fuzzy systems, the Parallel Distributed Compensation (PDC) procedure has extensively been used for the control of such systems: for more details see [16]. The basic idea of this procedure is to design a feedback gain for each local model, and then to construct a global controller from these local gains, so that the global stability of the overall fuzzy system can be guaranteed. The most interesting of this concept is that the obtained stability conditions do not depend on the nonlinearities (membership functions), so that this makes possible to use linear system techniques for nonlinear control design.

Up to now, the stabilization control design for T-S systems is successfully investigated based on state-feedback or static/dynamic output-feedback [18, 19]. However, the design of a controller which guarantees an adequate tracking performance for finite-dimensional systems is more general problem than the stabilization one, and is still attract considerable attentions due to demand from practical dynamical processes in electric, mechanics, agriculture,.... One of our main interest in this chapter is solving the static output-feedback tracking problem. Due to the fact that the T-S fuzzy models aggregate a set of local linear subsystems, blended together through nonlinear scalar functions, the static output-feedback control problem can be very complicated to solve. With regard to the literature of fuzzy control, a few recent approaches have dealt with the tracking control design problem for nonlinear systems described by T-S fuzzy model. Generally speaking, the incorporation of linearization techniques and adaptive schemes usually needs system’s perfect knowledge and leads to complicated adaptation control laws. In [20], the author has been shown that the use of the feedback linearization strategy [21] may lead to unbounded controllers, since their stability is not guaranteed. To overcome these drawbacks, LMI-based methodologies have been developed for tracking control problem, using observer-based fuzzy controller to deal with the absence of full-state information [22].

In this context, this chapter will tackle the static output-feedback fuzzy tracking control problem, focusing on an

The main contribution of this chapter is the purpose of a simple procedure reflected by an efficient LMI-based iterative algorithm to solve the fuzzy tracking control problem for nonlinear systems described by discrete-time T-S models. Therefore, since the proposed fuzzy tracking controllers have a low-order character, they are suitable for industrial application. Furthermore, this chapter shows an application to a relevant practical problem, in power engineering and drives field, of the proposed design procedure: guaranteeing a good tracking of the output voltage of DC-DC buck converter [24, 25, 26].

## 2. Problem formulation and preliminaries

## 4. H ∞ fuzzy tracking controller synthesis

Consider a nonlinear system which is approximated by a T-S fuzzy model of the following form:

(1) |

where

where

Thus, the global T-S model is an interpolation of all uncertain subsystems through nonlinear functions [16]:

(4) |

where

Note that using the so-called sector of nonlinearity approach, a wide number of nonlinear systems can be represented exactly by T-S models in a compact set of the state space. However, with the growing complexity of nonlinear systems, it is useful to take into account the approximations in the dynamical process. Thus, the main objective of the next paragraph is to provide stability conditions that ensure the tracking performance for the uncertain T-S model (4).

3.This section gives sufficient stability conditions which ensure an

** Lemma 3.1** [27]

*Let*A , D , S , W and F be real matrices of appropriate dimension such that W > 0 andF F T ≤ I . Then, for any scalar ε > 0 such thatW - ε D D T > 0 , we have ( A + D F S ) T W - 1 ( A + D F S ) ≤ A T ( W - ε D D T ) - 1 A + ε - 1 S T S .

Suppose that the desired trajectory can be generated by the following reference model as follows:

where,

Since we deal with the static output-feedback control design problem, the fuzzy controller can incorporates information from

where the the controller gain

The advantages of the static output-feedback controller (7), is well discussed in the literature [3], [6]. This fact motivates us to use such type of control law avoiding the complex control schemes with an additional observer.

Combining (4), (5) and (7), the following augmented closed-loop system is obtained

where

(9) |

Hence, to meet the required tracking performance, the effect of

The following theorem shows that

**Theorem 3.1** The augmented closed-loop system in (8) achieves the

where

Proof. Consider the following fuzzy Lyapunov function

The stability of (8) is ensured, under zero initial condition, with guaranteed

where

By substituting (13) in(12), we have:

where

Now, let

Then, the inequality (14) can be rewritten as follows

(18) |

By consequence, (16) leads to:

where

Thus, to proof (12), it is sufficient to show that

The first part of (19) can also be rewritten as

where

On the other hand, pre- and post-multiplying (11) by

(24) |

Since

(25) |

Applying Schur complement on (23), it is straightforward to verify that the condition (23) is equivalent to the following inequalities:

Using (20), (24) and Lemma 3.1, we have

By consequence

Hence,

In this section, a cone complementarity formulation [7] is used to solve the bilinearity involved in (11). The idea is based on converting the conditions (11) to convex and nonconvex parts and then casting them into an optimization problem subject to some LMIs. For this, first recall the following lemma, which generalizes the result of [7].

** Lemma 4.1** [12] Let

Using

(31) |

Before giving the final formulation of the problem in hand, we suggest to relax the LMIs (26) from the point of view number of LMIs to be satisfied, for this, we suggest to use the following lemma.

**Lemma 4.2** [12] Consider the following matrix

(33) |

Hence, using Lemma 4.2, (26) can be rewritten as follows:

where,

(36) |

(37) |

where

From Lemma 4.2, It is only sufficient to see that [12]

It should be noted that, Lemma 4.2 is very useful in reducing the number of LMIs to be satisfied. Indeed, (26) leads to

Now, back to our main problem. We suggest to use Lemma 4.1 to handle the nonconvexity involved in (27), as it is clearly shown by the following theorem:

**Theorem 4.1** Given a weight

The following iterative algorithm [7, 12] can be used to linearize the objective function of the optimization problem (29).

**Algorithm 4.1**

give a weight

(28) and

While

,Select

**Remark 4.1** In the optimization problem (29), the attenuation level

## 5. Illustrative example

In this section, the proposed tracking control scheme is applied to regulate the output voltage of DC-DC converter. The model of a buck converter is described in Fig. 1.

Using the Kirchoff laws, the converter of Fig. 1 can be represented by the following discrete-time nonlinear model [24]:

(42) |

where

The nonlinear system (30) can be represented by the following uncertain T-S model:

(44) |

where

with(46) |

In this example, the objective is to make the output voltage of the buck converter, i.e.

The reference system matrices of (5) is selected as follows

Let(49) |

and the

Fig. 2 shows the evolution of the output signal of the nonlinear system (30), using the fuzzy controller, with an external disturbance input

and the reference signal

Fig. 3 and Fig. 4 depict a zoom of Fig. 2 at 0 s and between 5 ms and 10 ms respectively. It can be seen that the designed fuzzy static output-feedback controller ensures the robust stability of the nonlinear system (30) and guarantees an acceptable

## 6. Conclusion

In this chapter, the problem of model reference tracking control with a guaranteed