# NonfragileH∞ Stabilizing Nonlinear Systems Described by Multivariable Hammerstein Models

Authors: Zeineb Rayouf; Chekib Ghorbel; Naceur Benhadj Braiek
Journal: Complexity
 (2021)

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## 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 robustH∞ 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, H∞ 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.


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## Body

## 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 H∞ 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 H∞ 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 nonfragileH∞ 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 Section2. Section 3 is reserved to the stability analysis and nonfragile H∞ 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 Figure1. As mentioned obviously, the model’s structure consists of m-static nonlinearities fi⋅ followed by a linear dynamic submodel.Figure 1
Multivariable Hammerstein model configuration.More precisely, each nonlinear componentfi⋅, for i=1,2,…,m, is characterized by the following form:(1)vi,k=fiui,k=λi1ui,k+λi2ui,k2+⋯+λiυui,kυ,and the linear dynamic submodel is described by the following state-space representation:(2)xk+1=Axk+B1Vk+B2wk,Yk=Cxk+DVk+εk,where xk∈ℝn is the state vector, Vk=v1,kv2,k…vm,k⊥ is the unmeasurable output, wk∈ℝn is the external disturbance vector, εk∈ℝq is the measurement noise vector, uk=u1,ku2,k…um,k⊥ is the input vector, Yk=y1,ky2,k…yq,k⊥ is the output vector, and the notation ⊥ denotes the transposed element. A∈ℝn×n, B1=B11B12…B1m∈ℝn×m, B2∈ℝn×n, C∈ℝq×n, and D=D1D2…Dm∈ℝq×m are unknown state-space matrices.By substituting (1) into (2), we obtain the following open-loop model:(3)xk+1=Axk+B1λUk+B2wk,Yk=Cxk+DλUk+εk,where Uk=U1,kU2,k…Um,k⊥, Ui,k=ui,kui,k2…ui,kυ⊥, B1λ=B11λB12λ…B1mλ, Dλ=D1λD2λ…Dmλ, B1iλ=B1iλivec, Diλ=Diλivec, and λivec=λi1λi2…λiυ, for i=1,2,…,m.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 thatλivec2=1 and λi1=1, for i=1,2,…,m [23, 28, 41].(ii)
Step 2: we determine the estimatesA^, B^1λ, B^2, C^, and D^λ of matrices A, B1λ, B2, C, and Dλ directly from the available input-output data.(iii)
Step 3: we compute a matrix estimateΘ^i of Θi, which is defined as(4)Θi≜B1iλDiλ=B1iDiλivec,by solving the following optimization problem:(5)B^1i,D^i,λ^ivec=argminB1i,Di,λivecΘ^i−B1iDiλivec22.(iv)
Step 4: based on the singular value decomposition (SVD) theorem, as is detailed in [42], we calculate the partition of Θ^i as follows:(6)Θ^i=U1U2Σ100Σ2V1⊥V2⊥,whereU1, V1, and Σ1 are specific matrices of appropriate dimensions.(v)
Step 5: we compute finally the estimatesB^1i, D^i, and λ^ivec so that the following system of equations is satisfied:(7)B^1iD^i=U1Σ1,λ^ivec=V1,fori=1,2,…,m.


## 3. NonfragileH∞ Control Scheme Design

In this section, we discuss sufficient conditions that guarantee the global asymptotic stability in closed loop of the following system:(8)xk+1=Axk+B1Vk+B2wk,zk=Czxk+D1Vk+D2wk,where Cz∈ℝp×n is the output matrix of the controlled output vector zk∈ℝp, D1∈ℝp×m, and D2∈ℝp×n.In what follows, we assume that them-nonlinearities:(9)vi,k=fiui,k=ui,k+λi2ui,k2+⋯+λiυui,kυ,are not bijective and approximated of the following form:(10)ui,k=fi,app−1v^i,k=βi1v^i,k+βi2v^i,k2+⋯+βiυv^i,kυ+hot.where hot denotes the higher order terms. As the nonlinearities (9) and (10) are in series, we may write(11)vi,k=ψiv^i,k=fifi,app−1v^i,k.In addition, the parametersβij are determined by solving v^i+∞=vi+∞, for i=1,2,…,m. An example of calculation for the order υ=3 is detailed in Appendix A.With the above approximation, system (8) is transformed as follows:(12)xk+1=Axk+B1ρV^k+B2wk,zk=Czxk+D1ρV^k+D2wk,where B1ρ=B11ρ1B12ρ2…B1mρm, D1ρ=D11ρ1D12ρ2…D1mρm, B1iρi=B1iρi,kv^i,k, D1iρi=D1iρi,kv^i,k, for i=1,2,…,m, ρi,kv^i,k=ψiv^i,k/v^i,kifv^i,k≠01else, and V^k=v^1,kv^2,k…v^m,k⊥.Using then the polytopic transformation method, them−nonlinearities ρiv^i,k are decomposed as follows:(13)ρi,kv^i,k=Hi1v^i,kσ¯i+Hi2v^i,kσ¯i,with(14)Hi1v^i,k=ρi,kv^i,k−σ¯iσ¯i−σ¯i,(15)Hi2v^i,k=1−Hi1v^i,k,where σ¯i and σ¯i are the maximum and minimum of ρiv^i,k, respectively.For the convenience of notations, we defineHij=Hijv^i,k, wi=wiv^j,k, and hi=hiv^j,k.Thereafter, we construct the following fuzzy subsystems:(16)rule1:ifv^1,kisH11andv^2,kisH21and…v^m,kisHm1,thenxk+1=Axk+B˜11V^k+B2wk,zk=Czxk+D˜11V^k+D2wk,(17)rule2:ifv^1,kisH11andv^2,kisH21and…v^m,kisHm2,thenxk+1=Axk+B˜12V^k+B2wk,zk=Czxk+D˜12V^k+D2wk,⋮(18)ruleς=2m:ifv^1,kisH12andv^2,kisH22and…v^m,kisHm2,thenxk+1=Axk+B˜1ςV^k+B2wk,zk=Czxk+D˜1ςV^k+D2wk,where B˜11=B¯11ρ1B¯12ρ2…B¯1mρm, D˜11=D¯11ρ1D¯12ρ2…D¯1mρm, B˜12=B¯11ρ1B¯12ρ2…B¯1mρm, D˜12=D¯11ρ1D¯12ρ2…D¯1mρm, B˜1ς=B¯11ρ1B¯12ρ2…B¯1mρm, D˜1ς=D¯11ρ1D¯12ρ2…D¯1mρm, B¯1iρi=σ¯iB1i, B¯1iρi=σ¯iB1i, D¯1iρi=σ¯iD1i, and D¯1iρi=σ¯iD1i, for i=1,2,…,m.As a consequence, the final system is inferred as follows:(19)xk+1=∑i=1ςhiAxk+B˜1iV^k+B2wk,zk=∑i=1ςhiCzxk+D˜1iV^k+D2wk,where the nonlinear weighting functions are hi=wi/∑i=1ςwi, such that 0≤hi≤1 and ∑i=1ςhi=1, and(20)w1=H11H21…Hm1,w2=H11H21…Hm2,⋮wς=H12H22…Hm2.For stabilizing system (19) in closed loop, we assume that each subsystem A,B˜1i is fully controllable and measurable. Then, we use the nonfragile control law:(21)Vk^=−K¯xk,with the uncertainty:(22)K¯=K+ΔK=I+μϕkK,where K∈ℝm×n is the nominal controller, μ>0 is a scalar to be assigned, I∈ℝm×m denotes the identity matrix, and ϕk∈ℝm×m, such that ϕk⊥ϕk≤I.Substituting (21) into (19), we obtain the final controlled system:(23)xk+1=∑i=1ςhiG¯ixk+B2wk,zk=∑i=1ςhiF¯ixk+D2wk,where G¯i=Gi+ΔGi, F¯i=Fi+ΔFi, Gi=A−B˜1iK, ΔGi=−μB˜1iϕkK, Fi=Cz−D˜1iK, and ΔFi=−μD˜1iϕkK.The closed-loop system (23) is globally asymptotically stable with decay rate α and achieves a prescribed attenuation level γ if(24)Tzw∞=supw2≠0z2w2<γ.As a consequence, we announce the following theorem.Theorem 1.
The equilibriumx=0ℝn of the closed-loop system (23) is quadratically and globally asymptotically stable with decay rate α satisfying the control performance objective (24) if there exist positive scalars μ, τ1, τ2, δ12=τ1−1+τ2−1, and β∈0,1,a common symmetric positive definite matrix X∈ℝn×n, and M∈ℝm×n verifying the following LMI formulation:(25)minimizeX,M,γβsubject to:−βX∗∗∗∗0−γ2I∗∗∗AX−B˜1iMB2−ℓ33,i∗∗CzX−D˜1iMD20−ℓ44,i∗M000−δ12−1I<0,for i=1,2,…,ς, where ℓ33,i=X−τ1μ2B˜1iB˜1i⊥ and ℓ44,i=I−τ2μ2D˜1iD˜1i⊥.Then, the feedback gainK, as is shown in (22), is calculated by using the following relation:(26)K=MX−1.Proof.
The controlled system (23) is globally asymptotically stable with decay rate α if there exist a discrete-time quadratic Lyapunov function VLyapxk=xk⊥Pxk>0 and a positive scalar 0<α<1 such that(27)ΔVLyapxk≤α2−1VLyapxk,where ΔVLyapxk=VLyapxk+1−VLyapxk and P∈ℝn×n is a symmetric positive definite matrix. By considering (27) in (24), we may write(28)ΔVLyapxk−α2−1VLyapxk+zk⊥zk−γ2wk⊥wk<0.By, respectively, substituting the dynamics of xk+1 and zk into (28), it becomes(29)∑i=1ςhixkwk⊥Γixkwk<0,where Γi=G¯i⊥PG¯i−α2P+F¯i⊥F¯i∗B2⊥PG¯i+D2⊥D2B2⊥PB2+D2⊥D2−γ2I and the symbol ∗ represents the transposed element in the symmetric position.
As the nonlinear functionshi∈0,1, matrix inequality (29) is equivalent to Γi<0, for i=1,2,…,ς. Using the Schur Complement, as is presented in Appendix B, we get(30)−α2P∗∗∗0−γ2I∗∗PG¯iPB2−P∗F¯iD20−I<0.
DenotingX=P−1, M=KX, and β=α2 and, respectively, premultiplying and postmultiplying (30) by positive definite matrix diagX,I,X,I yields(31)−βX∗∗∗0−γ2I∗∗G¯iXB2−X∗F¯iXD20−I<0.
As the above matrix inequality contains certain termsΨi,k and uncertain ones ΔΨi,k, (31) can be transformed as follows:(32)Ψi,k+ΔΨi,k<0,with(33)Ψi,k=−βX∗∗∗0−γ2I∗∗GiXB2−X∗FiXD20−I,(34)ΔΨi,k=00∗∗0000−μB˜1iϕkM000−μD˜1iϕkM000.
We notice that there are antidiagonal terms inΔΨi,k. However, we use the Separation Lemma, as is defined in Appendix C, to transform them into diagonal terms as follows:(35)Δψi,k≤δ12M⊥M000000000τ1μ2B˜1iB˜1i⊥0000τ2μ2D˜1iD˜1i⊥,where τ1, τ2, and δ12=τ1−1+τ2−1 are positive scalars to be assigned.
Referring to relations (33) and (35), we obtain(36)−βX+δ12M⊥M∗∗∗0−γ2I∗∗GiXB2−X+τ1μ2B˜1iB˜1i⊥∗FiXD20−I+τ2μ2D˜1iD˜1i⊥<0.
After some manipulations, we get the LMI formulation (25).Remark 1.
Consider system (19) with no uncertainty, i.e., ΔK=0. Then, the origin of the closed-loop system (26) is globally asymptotically stable if [27](37)minimizeX,R,γβsubject to:−βX∗∗∗0−γ2I∗∗AX−B˜1iMB2−X∗CzX−D˜1iMD20−I<0,fori=1,2,…,ς.
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 Figure2. 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 u1V and u2V applied, respectively, to the main and tail rotors. The first output is called pitch angle y1rad when the main rotor is free to rotate in the horizontal plane. The second output is called yaw angle y2rad 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]:(38)I1ψ¨=M1−MFG−MBψ−MG,I2ϕ¨=M2−MBϕ−MR,where M1=a1η12+b1η1, MFG=Mgsinψ, MBψ=B1ψψ˙, MG=KgyM1ϕ˙cosψ−Kgxϕ˙2sin2ψ, M2=a2η22+b2η2, MBϕ=B1ϕϕ˙, η˙1=−T10/T11η1+k1/T11u1, η˙2=−T20/T21η2+k2/T21u2, MR=kcη11+T0s/1+Tps, and s is the Laplace variable. All parameters are defined in Appendix D.


### 4.1. Identification Result

From an identification point of view, we assume that the input-output data are available. Then, we consider that the sampling period isT=0.1s and the inputs are u1,k=2.5sin0.6πkT and u2,k=2sin0.8πkT. Figures 3 and 4 depict the responses of the true (solid line) and estimated (dashed line) outputs of the open-loop system.Figure 3
Response of the pitch angle of the open-loop system.Figure 4
Response of the yaw angle of the open-loop system.It is then clear that the nonlinear TRMS is accurately identified by 2-input 2-output Hammerstein state-space model, which is described by(39)xk+1=Axk+B1Vk+B2wk,Yk=Cxk+DVk+εk,v1,k=u1,k+0.0381u1,k2−0.0457u1,k3,v2,k=u2,k+0.0237u2,k2−0.0118u2,k3,where(40)xk=x1,kx2,kx3,kx4,k,Vk=v1,kv2,k,Yk=y1,ky2,k,wk∈ℝ4,εk∈ℝ2,A=0.9709−0.33800.12320.03060.11790.97450.0093−0.0109−0.03750.00870.9974−0.07980.0109−0.02260.02700.8693,B1=0.01110.0547−0.0131−0.00270.21720.30440.08650.0520,B2=−0.22920.56557.72930.29781.44450.2131−5.64350.0878−0.4621−0.0928−2.43610.1226−0.23240.09149.9696−0.7869,C=−0.08530.2915−0.11470.00930.0121−0.1440−0.3784−0.1127,D=−0.03260.00630.0284−0.0713.


### 4.2. Control Results

From a control point of view, we assume that the inverse functionsui,k=fi,app−1v^i,k, for i=1,2, are approximated as follows:(41)u1,k=v^1,k−0.0381v^1,k2+0.1219v^1,k3+hot,u2,k=v^2,k−0.0237v^2,k2+0.0592v^2,k3+hot.Choosing the scalarsσ¯1=0.4, σ¯1=1.7, σ¯2=0.3, and σ¯2=2, Figure 5 depicts the evolution of nonlinearities vi,k=ψiv^i,k, for i=1,2. This leads to obtain, for the control scheme, the bounded control signals v^1,k∈−5.87,5.73 and v^2,k∈−2.33,2.24.Figure 5
Evolution of nonlinear functions. Up Polt:v1,k=ψ1v^1,k. Down Polt: v2,k=ψ2v^2,k.Afterwards, we assume that the controlled outputzk is expressed by(42)zk=Czxk+D1Vk+D2wk,where zk=z1,kz2,k, Cz=C, D1=D, and D2=10000010.By considering the pairsσ¯1,σ¯2, σ¯1,σ¯2, σ¯1,σ¯2, and σ¯1,σ¯2, we construct the following fuzzy subsystems:(43)rule1:ifv^1,kisH11andv^2,kisH21,thenxk+1=Axk+B˜11V^k+B2wk,zk=Czxk+D˜11V^k+D2wk,(44)rule2:ifv^1,kisH11andv^2,kisH22,thenxk+1=Axk+B˜12V^k+B2wk,zk=Czxk+D˜12V^k+D2wk,(45)rule3:ifv^1,kisH12andv^2,kisH21,thenxk+1=Axk+B˜13V^k+B2wk,zk=Czxk+D˜13V^k+D2wk,(46)rule4:ifv^1,kisH12andv^2,kisH22,thenxk+1=Axk+B˜14V^k+B2wk,zk=Czxk+D˜14V^k+D2wk,where(47)B˜11=0.01870.1094−0.0223−0.00540.36920.60880.14700.1040,B˜12=0.01870.0164−0.0223−0.00080.36920.09130.14700.0156,B˜13=0.00440.1094−0.0052−0.00540.08690.60880.03460.1040,B˜14=0.00440.0164−0.0052−0.00080.08690.09130.03460.0156,D˜11=−0.05540.01260.0483−0.1426,D˜12=−0.05540.00190.0483−0.0214,D˜13=−0.01300.01260.0114−0.1426,D˜14=−0.01300.00190.0114−0.0214.As the pairsA,B˜1i, for i=1,2,3,4, are controllable, the resulting fuzzy system can be described as follows:(48)xk+1=∑i=14hiAxk+B˜1iV^k+B2wk,zk=∑i=14hiCzxk+D˜1iV^k+D2wk,where the nonlinear weighting functions hi=wi/∑i=14wi are depicted in Figure 6.Figure 6
Weighting functions for the four fuzzy sets considered.Using the LMI formulation (25) with μ=0.85 and γ=0.7, we obtain α=0.794, β=0.63, and(49)P=6.44970.8651−0.26910.13270.86514.7781−0.32050.1191−0.2691−0.32055.02220.48600.13270.11910.48604.9015,K=−0.5072−0.22512.30941.40590.5327−0.06531.4641−0.4069.Hence, by considering the nonfragile control law (21) subject to the uncertainty (22) with ϕk=0.5sinπk000.5cosπk, the inferred controlled system can be described as follows:(50)xk+1=∑i=14hiG¯ixk+B2wk,zk=∑i=14hiF¯ixk+D2wk,where G¯i=Gi+ΔGi, F¯i=Fi+ΔFi, ΔGi=−μB˜1iϕkK, ΔFi=−μD˜1iϕkK, for i∈1,2,3,4, and(51)G1=0.9221−0.3266−0.08020.04880.10950.96910.06860.0182−0.17450.1316−0.7467−0.35120.03010.0173−0.46490.7049,G2=0.9716−0.33270.05600.01100.10700.96940.06190.02010.10110.09780.0110−0.56180.07720.0115−0.33540.6689,G3=0.9149−0.3299−0.04710.06890.11810.97300.0293−0.0057−0.31770.0680−0.09460.0458−0.0269−0.0080−0.20520.8630,G4=0.9644−0.33590.08900.03110.11570.97330.0226−0.0039−0.04210.03420.6631−0.16480.0201−0.0138−0.07570.8270,(52)F1=−0.12010.2798−0.00520.09230.1125−0.1425−0.2811−0.2386,F2=−0.11440.27910.01050.08800.0480−0.1345−0.4586−0.1893,F3=−0.09860.2894−0.10300.03280.0938−0.1508−0.1959−0.1867,F4=−0.09290.2887−0.08740.02840.0293−0.1428−0.3733−0.1374.Figures7–10 show the simulation results of applying the designed nonfragile H∞ controller to the TRMS (dashed line) and its identified Hammerstein model (solid line) with null initial conditions and the exogenous disturbance signal wk=rand0rand0⊥, where rand is a uniform number with a uniform distribution on the interval 0,0.01, which is added by wk+=−0.15if50≤kT≤100and0otherwise.Figure 7
Control signalu1V.Figure 8
Control signalu2V.Figure 9
Response of the pitch angle of the closed-loop system.Figure 10
Response of the yaw angle of the closed-loop system.The obtained results indicate that the designed robust and nonfragileH∞ controller shows good results. However, the responses of the pitch and yaw angles of the controlled nonlinear system and its identified Hammerstein model can rapidly achieve the origin despite the presence of external disturbances and uncertainty in the control law.


## 4.1. Identification Result

From an identification point of view, we assume that the input-output data are available. Then, we consider that the sampling period isT=0.1s and the inputs are u1,k=2.5sin0.6πkT and u2,k=2sin0.8πkT. Figures 3 and 4 depict the responses of the true (solid line) and estimated (dashed line) outputs of the open-loop system.Figure 3
Response of the pitch angle of the open-loop system.Figure 4
Response of the yaw angle of the open-loop system.It is then clear that the nonlinear TRMS is accurately identified by 2-input 2-output Hammerstein state-space model, which is described by(39)xk+1=Axk+B1Vk+B2wk,Yk=Cxk+DVk+εk,v1,k=u1,k+0.0381u1,k2−0.0457u1,k3,v2,k=u2,k+0.0237u2,k2−0.0118u2,k3,where(40)xk=x1,kx2,kx3,kx4,k,Vk=v1,kv2,k,Yk=y1,ky2,k,wk∈ℝ4,εk∈ℝ2,A=0.9709−0.33800.12320.03060.11790.97450.0093−0.0109−0.03750.00870.9974−0.07980.0109−0.02260.02700.8693,B1=0.01110.0547−0.0131−0.00270.21720.30440.08650.0520,B2=−0.22920.56557.72930.29781.44450.2131−5.64350.0878−0.4621−0.0928−2.43610.1226−0.23240.09149.9696−0.7869,C=−0.08530.2915−0.11470.00930.0121−0.1440−0.3784−0.1127,D=−0.03260.00630.0284−0.0713.


## 4.2. Control Results

From a control point of view, we assume that the inverse functionsui,k=fi,app−1v^i,k, for i=1,2, are approximated as follows:(41)u1,k=v^1,k−0.0381v^1,k2+0.1219v^1,k3+hot,u2,k=v^2,k−0.0237v^2,k2+0.0592v^2,k3+hot.Choosing the scalarsσ¯1=0.4, σ¯1=1.7, σ¯2=0.3, and σ¯2=2, Figure 5 depicts the evolution of nonlinearities vi,k=ψiv^i,k, for i=1,2. This leads to obtain, for the control scheme, the bounded control signals v^1,k∈−5.87,5.73 and v^2,k∈−2.33,2.24.Figure 5
Evolution of nonlinear functions. Up Polt:v1,k=ψ1v^1,k. Down Polt: v2,k=ψ2v^2,k.Afterwards, we assume that the controlled outputzk is expressed by(42)zk=Czxk+D1Vk+D2wk,where zk=z1,kz2,k, Cz=C, D1=D, and D2=10000010.By considering the pairsσ¯1,σ¯2, σ¯1,σ¯2, σ¯1,σ¯2, and σ¯1,σ¯2, we construct the following fuzzy subsystems:(43)rule1:ifv^1,kisH11andv^2,kisH21,thenxk+1=Axk+B˜11V^k+B2wk,zk=Czxk+D˜11V^k+D2wk,(44)rule2:ifv^1,kisH11andv^2,kisH22,thenxk+1=Axk+B˜12V^k+B2wk,zk=Czxk+D˜12V^k+D2wk,(45)rule3:ifv^1,kisH12andv^2,kisH21,thenxk+1=Axk+B˜13V^k+B2wk,zk=Czxk+D˜13V^k+D2wk,(46)rule4:ifv^1,kisH12andv^2,kisH22,thenxk+1=Axk+B˜14V^k+B2wk,zk=Czxk+D˜14V^k+D2wk,where(47)B˜11=0.01870.1094−0.0223−0.00540.36920.60880.14700.1040,B˜12=0.01870.0164−0.0223−0.00080.36920.09130.14700.0156,B˜13=0.00440.1094−0.0052−0.00540.08690.60880.03460.1040,B˜14=0.00440.0164−0.0052−0.00080.08690.09130.03460.0156,D˜11=−0.05540.01260.0483−0.1426,D˜12=−0.05540.00190.0483−0.0214,D˜13=−0.01300.01260.0114−0.1426,D˜14=−0.01300.00190.0114−0.0214.As the pairsA,B˜1i, for i=1,2,3,4, are controllable, the resulting fuzzy system can be described as follows:(48)xk+1=∑i=14hiAxk+B˜1iV^k+B2wk,zk=∑i=14hiCzxk+D˜1iV^k+D2wk,where the nonlinear weighting functions hi=wi/∑i=14wi are depicted in Figure 6.Figure 6
Weighting functions for the four fuzzy sets considered.Using the LMI formulation (25) with μ=0.85 and γ=0.7, we obtain α=0.794, β=0.63, and(49)P=6.44970.8651−0.26910.13270.86514.7781−0.32050.1191−0.2691−0.32055.02220.48600.13270.11910.48604.9015,K=−0.5072−0.22512.30941.40590.5327−0.06531.4641−0.4069.Hence, by considering the nonfragile control law (21) subject to the uncertainty (22) with ϕk=0.5sinπk000.5cosπk, the inferred controlled system can be described as follows:(50)xk+1=∑i=14hiG¯ixk+B2wk,zk=∑i=14hiF¯ixk+D2wk,where G¯i=Gi+ΔGi, F¯i=Fi+ΔFi, ΔGi=−μB˜1iϕkK, ΔFi=−μD˜1iϕkK, for i∈1,2,3,4, and(51)G1=0.9221−0.3266−0.08020.04880.10950.96910.06860.0182−0.17450.1316−0.7467−0.35120.03010.0173−0.46490.7049,G2=0.9716−0.33270.05600.01100.10700.96940.06190.02010.10110.09780.0110−0.56180.07720.0115−0.33540.6689,G3=0.9149−0.3299−0.04710.06890.11810.97300.0293−0.0057−0.31770.0680−0.09460.0458−0.0269−0.0080−0.20520.8630,G4=0.9644−0.33590.08900.03110.11570.97330.0226−0.0039−0.04210.03420.6631−0.16480.0201−0.0138−0.07570.8270,(52)F1=−0.12010.2798−0.00520.09230.1125−0.1425−0.2811−0.2386,F2=−0.11440.27910.01050.08800.0480−0.1345−0.4586−0.1893,F3=−0.09860.2894−0.10300.03280.0938−0.1508−0.1959−0.1867,F4=−0.09290.2887−0.08740.02840.0293−0.1428−0.3733−0.1374.Figures7–10 show the simulation results of applying the designed nonfragile H∞ controller to the TRMS (dashed line) and its identified Hammerstein model (solid line) with null initial conditions and the exogenous disturbance signal wk=rand0rand0⊥, where rand is a uniform number with a uniform distribution on the interval 0,0.01, which is added by wk+=−0.15if50≤kT≤100and0otherwise.Figure 7
Control signalu1V.Figure 8
Control signalu2V.Figure 9
Response of the pitch angle of the closed-loop system.Figure 10
Response of the yaw angle of the closed-loop system.The obtained results indicate that the designed robust and nonfragileH∞ controller shows good results. However, the responses of the pitch and yaw angles of the controlled nonlinear system and its identified Hammerstein model can rapidly achieve the origin despite the presence of external disturbances and uncertainty in the control law.


## 5. Conclusion

In this paper, a nonfragileH∞ feedback controller was designed for nonlinear systems described as multivariable Hammerstein model with separate nonlinearities. The parameters of the linear and nonlinear blocks characterizing the multivariable Hammerstein model structure were separately estimated using the MOESP identification algorithm. Unlike the normal control scheme, the inverses of the static nonlinearities were supposed not bijective and approximated by polynomial functions. The T-S fuzzy representation was used to simplify the nonlinear system description and reject the nonlinearity effect in the controller design. Based on the quadratic Lyapunov function and H∞ criterion, robust H∞ was then proposed to robustly stabilize the controlled nonlinear system and its identified Hammerstein model and guarantee the attenuation of disturbance effect in spite of controller gain variations. A TRMS was considered to illustrate the validity and the effectiveness of the designed stabilization scheme.



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