Abstract

This paper presents the problem of robust and nonfragile stabilization of nonlinear systems described by multivariable Hammerstein models. The objective is focused on the design of a nonfragile feedback controller such that the resulting closed-loop system is globally asymptotically stable with robust disturbance attenuation in spite of controller gain variations. First, the parameters of linear and nonlinear blocks characterizing the multivariable Hammerstein model structure are separately estimated by using a subspace identification algorithm. Second, approximate inverse nonlinear functions of polynomial form are proposed to deal with nonbijective invertible nonlinearities. Thereafter, the Takagi–Sugeno model representation is used to decompose the composition of the static nonlinearities and their approximate inverses in series with the linear subspace dynamic submodel into linear fuzzy parts. Besides, sufficient stability conditions for the robust and nonfragile controller synthesis based on quadratic Lyapunov function, criterion, and linear matrix inequality approach are provided. Finally, a numerical example based on twin rotor multi-input multi-output system is considered to demonstrate the effectiveness.

1. Introduction

The nonlinear modeling of real-world processes, which are complex in nature, remains a challenging problem. However, much research works remain to be done for realization on nonlinear mathematical models that accurately represent these processes [1–4]. One way to cope with this difficulty is to use the block-oriented nonlinear models [5–7], which represent a combination of static nonlinear components and linear dynamic submodels. These models are popular in nonlinear modeling because of their advantages to be quite simple to understand and easy to use [8], for instance, the Hammerstein model (a static nonlinear component followed by a linear submodel), Wiener model (a linear submodel followed by a static nonlinear component), and Hammerstein–Wiener model (a linear submodel sandwiched by two static nonlinearities or vice versa). In particular, the simplest model structure of them is the Hammerstein model, which has been extensively used for modeling electrical generators [9], chemical processes [10], and biological processes [11] and is also used in signal processing applications [12].

Over the past decades, various parametric subspace identification methods have been very successful for the modeling of multivariable Hammerstein models. These methods include the iterative identification approach [13, 14], the overparameterization method [15], the blind approach [16], the instrumental variables method [17], the stochastic approach [18], and the least square support vector machines [19]. Most of them are based on the numerical subspace state-space system identification algorithm [20], the canonical variate analysis approach [21], and the multivariable output error state-space (MOESP) algorithm [10, 22].

From a control point of view, the conventional control scheme of a Hammerstein model has introduced the inverse of the nonlinear block into the appropriate place. This leads to reject the nonlinearity effect in the controller design [23]. Hence, the nonlinear control strategy problem is converted into a new linear one; also, any standard linear controller for a linear dynamic submodel can be applied. It should be a strong assumption that this nonlinearity is supposed to be exactly invertible [24–26]. In contrast, the performance of this strategy becomes limited when the nonlinear component function is not bijective. In this view, many algorithms and approximations are used in the literature to determine the corresponding nonlinearity inverse. One may refer to latest research works based on polynomial form approximation [23, 27, 28], Bernstein–Bezier neural network [29], De Boor algorithm [30], and rational B-spline model approximation [31].

On the other hand, many studies have been devoted to the robust and nonfragile controller design problem for complex systems. Indeed, it is clear that relatively small perturbations in controller gain parameters can result in instability of the controlled system [32, 33]. Hence, it is necessary that any controller should be able to tolerate some bounded uncertainty in its parameters [3, 34, 35]. For instance, a nonfragile controller for uncertain nonlinear networked control systems was proposed in [36]. In [37], a nonfragile robust controller was investigated for uncertain large-scale systems. Lee et al. [38] proposed a nonfragile fuzzy controller for nonlinear systems described in Takagi–Sugeno (T-S) fuzzy model, and so on. To our best knowledge, the nonfragile control problem for Hammerstein models has not been treated yet.

In this framework, we use the MOESP subspace identification algorithm, which mainly involves two aspects: (i) determining the order of the system and obtaining the structure of the estimated state-space model and (ii) identifying the mathematical model’s unknown parameters from the available input-output data [10]. Afterwards, we propose a new control strategy for multivariable Hammerstein model including approximate inverse nonlinearities of polynomial form. Using then the T-S fuzzy model representation [1, 2, 34, 39], the composed nonlinear functions of the considered static nonlinearities and their approximate inverses in series with the linear dynamic submodel are decomposed into linear fuzzy parts. The resulting model is finally obtained by interpolating the constructed linear fuzzy parts through nonlinear fuzzy membership functions [2, 4, 35, 40]. In this regard, a nonfragile feedback controller is designed subject to controller gain variations guaranteeing both the stability and disturbance attenuation of the controlled nonlinear system.

The main contributions of this paper are listed as follows:(i)A modified subspace-based algorithm is used to identify nonlinear systems described by multivariable Hammerstein models.(ii)Compared with the existing results using the normal nonlinearity inversion method, we derive a new control strategy based on approximate inverse nonlinear functions of polynomial form. Furthermore, we appeal the T-S fuzzy model representation to decompose the existing nonlinearities and facilitate the controller synthesis.(iii)From a control point of view, we develop a robust and nonfragile controller with variation in the control law that guarantees both the asymptotic stability and disturbance attenuation of the controlled nonlinear system and its identified multivariable Hammerstein model.(iv)Besides, sufficient controller design conditions in terms of linear matrix inequalities (LMIs) are established, which can be efficiently solved by convex optimization techniques.

Following the introduction, this paper is organized as follows. The subspace identification method for multivariable Hammerstein model is presented in Section 2. Section 3 is reserved to the stability analysis and nonfragile control synthesis. A numerical example based on twin rotor multi-input multi-output system (TRMS) is considered in Section 4 to demonstrate the effectiveness.

2. MOESP Algorithm-Based Subspace Identification

We consider the multi-input multi-output (MIMO) Hammerstein model configuration, as depicted in Figure 1. As mentioned obviously, the model’s structure consists of -static nonlinearities followed by a linear dynamic submodel.

Figure 1 

Multivariable Hammerstein model configuration.

More precisely, each nonlinear component , for , is characterized by the following form:and the linear dynamic submodel is described by the following state-space representation:where is the state vector, is the unmeasurable output, is the external disturbance vector, is the measurement noise vector, is the input vector, is the output vector, and the notation denotes the transposed element. , , , , and are unknown state-space matrices.

By substituting (1) into (2), we obtain the following open-loop model:where , , , , , , and , for .

In order to estimate the system order and determine the unknown elements, as are presented in system (3), we use the MOESP algorithm, which is basically defined by the following steps:(i)Step 1: for sake of simplicity and feasibility, we assume that and , for [23, 28, 41].(ii)Step 2: we determine the estimates , , , , and of matrices , , , , and directly from the available input-output data.(iii)Step 3: we compute a matrix estimate of , which is defined as by solving the following optimization problem:(iv)Step 4: based on the singular value decomposition (SVD) theorem, as is detailed in [42], we calculate the partition of as follows: where , , and are specific matrices of appropriate dimensions.(v)Step 5: we compute finally the estimates , , and so that the following system of equations is satisfied:

3. Nonfragile Control Scheme Design

In this section, we discuss sufficient conditions that guarantee the global asymptotic stability in closed loop of the following system:where is the output matrix of the controlled output vector , , and .

In what follows, we assume that the -nonlinearities:are not bijective and approximated of the following form:where denotes the higher order terms. As the nonlinearities (9) and (10) are in series, we may write

In addition, the parameters are determined by solving , for . An example of calculation for the order is detailed in Appendix A.

With the above approximation, system (8) is transformed as follows:where , , , , for , , and .

Using then the polytopic transformation method, the nonlinearities are decomposed as follows:withwhere and are the maximum and minimum of , respectively.

For the convenience of notations, we define , , and .

Thereafter, we construct the following fuzzy subsystems:where , , , , , , , , , and , for .

As a consequence, the final system is inferred as follows:where the nonlinear weighting functions are , such that and , and

For stabilizing system (19) in closed loop, we assume that each subsystem is fully controllable and measurable. Then, we use the nonfragile control law:with the uncertainty:where is the nominal controller, is a scalar to be assigned, denotes the identity matrix, and , such that .

Substituting (21) into (19), we obtain the final controlled system:where , , , , , and .

The closed-loop system (23) is globally asymptotically stable with decay rate and achieves a prescribed attenuation level if

As a consequence, we announce the following theorem.

Theorem 1. The equilibrium of the closed-loop system (23) is quadratically and globally asymptotically stable with decay rate satisfying the control performance objective (24) if there exist positive scalars , , , , and common symmetric positive definite matrix , and verifying the following LMI formulation:for , where and .

Then, the feedback gain , as is shown in (22), is calculated by using the following relation:

Proof. The controlled system (23) is globally asymptotically stable with decay rate if there exist a discrete-time quadratic Lyapunov function and a positive scalar such thatwhere and is a symmetric positive definite matrix. By considering (27) in (24), we may writeBy, respectively, substituting the dynamics of and into (28), it becomeswhere and the symbol represents the transposed element in the symmetric position.
As the nonlinear functions , matrix inequality (29) is equivalent to , for . Using the Schur Complement, as is presented in Appendix B, we getDenoting , , and and, respectively, premultiplying and postmultiplying (30) by positive definite matrix yieldsAs the above matrix inequality contains certain terms and uncertain ones , (31) can be transformed as follows:withWe notice that there are antidiagonal terms in . However, we use the Separation Lemma, as is defined in Appendix C, to transform them into diagonal terms as follows:where , , and are positive scalars to be assigned.
Referring to relations (33) and (35), we obtainAfter some manipulations, we get the LMI formulation (25).

Remark 1. Consider system (19) with no uncertainty, i.e., . Then, the origin of the closed-loop system (26) is globally asymptotically stable if [27]In the following, a numerical example is provided to demonstrate the validity and the effectiveness of the proposed control scheme.

4. Application to a TRMS

The objective of this simulation study is to stabilize the controlled TRMS and its identified multivariable Hammerstein model at the origin, as an asymptotically stable equilibrium point. More precisely, its system behaviour resembles that of a helicopter, as is seen in Figure 2. It consists of two rotors (main and tail), which are situated on a beam together with a counterbalance. The inputs of the open-loop system are the voltages and applied, respectively, to the main and tail rotors. The first output is called pitch angle when the main rotor is free to rotate in the horizontal plane. The second output is called yaw angle when the tail rotor is free to rotate in the vertical plane.

Figure 2 

TRMS [43].

The studied system is described by the following continuous-time nonlinear equations [43]:where , , , , , , , , , and is the Laplace variable. All parameters are defined in Appendix D.