Extended second order sliding mode control for mismatched uncertain systems with only output measurable

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Extended Second Order Sliding Mode Control for Mismatched Uncertain Systems with Only Output Measurable Van VAN HUYNH1, Bui LE NGOC MINH2, Tam MINH NGUYEN2, Vo HOANG DUY1 1Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, No. 19 Nguyen Huu Tho Street, Tan Phong Ward, District 7, Ho Chi Minh City, Vietnam 2Faculty of Electrical and Electronics Engineering, Ho Chi Minh City University of Technology and Education, No 1 Vo Van Ngan Street, Linh Chieu Ward, Thu Duc District, Ho Chi Minh City, Vietnam huynhvanv[email protected], [email protected], [email protected], v[email protected] DOI: 10.15598/aeee.v16i4.2775 Abstract. Most existing Second Order Sliding Mode Control (SOSMC) approaches are achieved under assumptions that 1) all of state variables must be accessible; 2) the second derivative of all state variables must exist, even though mathematical model of systems uses the first order equations. In this paper, a new adaptive SOSMC scheme is proposed for mismatched uncertain systems in which these above assumptions are required. In this proposed method, only output variables are used in the sliding surface and controller design. The advantage of no need of all state variables in controller design makes the method more useful and realistic since it can be applied to a wider class of systems. Finally, a vertical take-off and landing aircraft at the nominal airspeed of 135 knots is simulated to demonstrate the advantages and effectiveness of the proposed approach. Keywords Adaptive control, linear matrix inequalities, output feedback controller, second order sliding mode control. 1. Introduction Over the past three decades, there has been an increasing research interest in Sliding Mode Control (SMC) theory and application. The main advantages of SMC are fast global convergence, simplicity of implementation, order reduction, high robustness to external disturbances and insensitivity to model errors and system parameter variations [1]. Thanks to these advantages, the SMC theory has been successfully applied to a wide variety of practical engineering systems such as robot manipulators, aircrafts, underwater vehicles, spacecraft, flexible space structures, electrical motors, power systems, and automotive engines [1], [2] and [3]. Although the sliding mode controller guarantees robustness with respect to uncertainties and external disturbances, chattering is its main drawback. Chattering is the high frequency finite amplitude oscillations occurring because of the discontinuous control signal used in the SMC [4]. Such chattering has many negative effects in practical applications since it may damage the control actuator and excite the undesirable unmodeled dynamics, which probably leads to unforeseen instability [4]. Many authors have applied various techniques to reduce chattering problem across the sliding surface. Recently, some good results have been published in high quality journal such as [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] and [21]. The authors of [5] have presented a direct way to reduce chattering problems by inserting a fixed or variable boundary layer near the sliding variable so that a smooth continuous control replaces the discontinuous one when the system is inside the boundary layer. This approach can produce a chattering-free system but a larger boundary layer width results in larger errors in control accuracy and a finite steady-state error may occur. Another approach to eliminate the chattering is carried out by using fuzzy control with sliding mode controller [6], using low-pass filtering [7] or nonlinear reaching law [8]. This method can give a chatteringfree system but a finite steady-state error may remain. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 435 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER One of the most effective methods to avoid chattering problems is to use the Second-Order Sliding Mode Control (SOSMC). The basic idea of the second order sliding mode controller is that the discontinuous sign function is made to act on the time derivative of the control inputs and the actual control signal obtained after integration is continuous and hence chattering is removed [9]. In addition, SOSMC allows driving to zero the sliding variable and its consecutive derivatives in the presence of the disturbances/uncertainties increasing the accuracy of the sliding variable stabilization [10]. Thanks to these advantages, the SOSMC with finite-time convergence has been successfully implemented for solution of real problems [11] and [12]. Another approach given in [13] was to present a modified second-order sliding mode control for single-input nonlinear systems. This study guarantees the finite time reaching of the sliding manifold and chattering reduction. The SOSMC proposed in [13] was extended by [14] for a class of uncertain multi input nonlinear systems but the disturbances were not considered in the above approach. In [15], the second-order sliding mode control approach with additional capabilities of learning and control adaptation was developed to estimate and compensate for the uncertainty affecting the system’s dynamics. This technique is capable of reducing the discontinuous control effort to an arbitrarily small quantity. However, the approach given in [15] could not be applied for systems with unknown upper bounds of uncertainties. The study of [16] proposed a robust adaptive SOSMC scheme for a class of uncertain nonlinear systems where the upper bounds of uncertainties are not required to be known in advance. As a result, a finite-time convergent second-order sliding mode is established and the chattering problem is eliminated. In [17], an adaptive second order sliding mode control law is proposed for the control of an electro pneumatic actuator. In order to reduce the overshoot and the settling time, the adaptive second order sliding mode controller with a nonlinear sliding surface is presented in [18]. The authors of [19] have proposed a chattering free adaptive sliding mode controller for stabilizing a class of multi-input multi-output systems. This approach can ensure asymptotical stability of the overall system and eliminate chattering in the control input. In [20], based on the linear quadratic regulator method, an optimal second order sliding mode controller was proposed for a class of matched uncertain systems. By designing a new sliding surface, the approach given in [21] can solve both the chattering and singularity problems in sliding mode control. A second-order sliding mode control method to handle sliding mode dynamics with mismatched term was presented in [22]. In [23], a robust chattering-free control scheme was proposed using second-order fast terminal sliding mode control technique for the tracking problem of a class of uncertain systems with matched and mismatched uncertainties. However, it is worth pointing out that most of the previous results have been developed under the assumption that all the system states are available for the control law. It may be impossible or prohibitively expensive to measure all of the process variables in some practical systems [24]. For example, it is difficult to measure the variables describing the flexible motion, the modal position, and the velocity of flexible spacecraft [25]. Thus, it is very important to establish a new adaptive SOSMC method to control mismatched uncertain systems via output feedback. Herein, we intent to use the output information completely in the sliding surface and controller design but still remain the advantages of SOSMC such as the chattering-free, the maximum convergence time interval, and the dimension of neighbourhood of the origin to which the controlled trajectory converges [14]. In this paper, we extend the concept of second order sliding mode controller, introduced by [19], [22], [23] and [27], for the aim of stabilizing mismatched uncertain systems where only output variables are accessible. The main contributions of this paper are as follows: •A new Lemma and a novel adaptive law are established for the aim of controller design using only output variables. •New sufficient conditions in terms of Linear Matrix Inequalities (LMI) are derived such that the equivalent reduced-order system in the sliding mode is asymptotically stable. •The two major assumptions by [19], [22], [23] and [27] (that all of state variables must be accessible, and that the second derivative of all state variables must exist) are both eliminated. Therefore, the proposed method can be applied to a wider class of mismatched uncertain systems. 2. System Description and Preliminary Results Consider the following mismatched uncertain systems: ˙x= [A+ ∆A(x, t)] x+B[u+ξ(x, t)] , y=Cx. (1) Here x∈Rn,u∈Rmand y∈Rpdenote the state variables, inputs and outputs, respectively. A∈Rn×n is state matrix, B∈Rn×mis the input matrix and C∈Rp×nis the output matrix. The terms ∆Aand ξ(x, t)represent the system matrix and the input matrix uncertainties, respectively. We assume that: c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 436 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Assumption 1. The matrix pair (A,B)is completely controllable. Assumption 2. The mismatched uncertainty ∆A(x, t)is a norm-bounded time varying uncertainty as follows: ∆A(x, t) = DF(x, t)E,kF(x, t)≤1k, where Dand Eare known constant real matrices with appropriate dimensions that characterize the structure of the uncertainty, and F(x, t)is a norm-bounded unknown matrix. Assumption 3.rank(CB) = m. From [28], Assumption 3 implies that there exists a non-singular linear coordinate transformation z=˜ Tx such that the triple (A,B,C)with respect to the new coordinates has the structure ˜ A=˜ A1˜ A2 ˜ A3˜ A4,˜ B=0 ˜ B2,˜ C=0˜ C2,(2) where ˜ A1∈R(n−m)×(n−m),˜ B2∈Rm×mare nonsingular and ˜ C2∈Rp×pis orthogonal. Assumption 4: The triple (˜ A1,˜ A2,Ξ) is output feedback stabilisable, where Ξ = 0(p−m)×(n−p)I(p−m). Assumption 4 implies that there exist matrix ˜ Ksuch that the matrix A1=˜ A1−˜ A2˜ KΞis stable. From [28], the coordinate transformation z=¯ T˜xwhere ¯ T=I0 −˜ KΞI,(3) will transform the triple (A,B,C)to the following form in the new coordinate system z A1A2 A3A4,0 B2,0C2,(4) where A1=˜ A1−˜ A2˜ KΞ∈R(n−m)×(n−m)is stable and both these matrices B2∈Rm×m,C2∈Rp×pare non-singular. Remark 1: In [19], [22], [23] and [27], all of state variables x∈Rnmust be accessible and the second derivative of all state variables ¨xexist, even though mathematical model of systems is of the first order. The proposed method needs only a subset of state variables y∈Rpto be accessible and the second derivative of output variables ¨yexists. Therefore, the proposed approach can be applied to a wider class of mismatched uncertain systems. Remark 2: The output feedback SOSMC scheme is proposed in [26]. However, there are three major conditions set by [26]: •The system under consideration is assumed to be matched. •The exogenous disturbances are bounded by a known constant value. That is kfk≤ πwhere πis known. This condition is quite restrictive. •The sliding matrix Fsatisfies that the matrix FCAB is invertible to guarantee sliding condition S(t) = Fy(t) + w(t)=0. This limitation is really strong. Remark 3: The SOSMCs using output variables were subject of many recently researches [29] and [30]. However, all these methods require more hardware and increase system dimension. In this paper, an adaptive output feedback SOSMC scheme is proposed for mismatched uncertain systems where above limitations are eliminated. 3. Adaptive Output Feedback Second Order Sliding Mode Control Design In this section, we introduce a systematic design procedure of an adaptive output feedback Second Order Sliding Mode Control (SOSMC) scheme. There are three steps involved in the design of an adaptive output feedback SOSMC for system, see Eq. (1). In the first step, a sliding surface is designed to depend on only output variables. In the second step, we derive appropriate Linear Matrix Inequalities (LMI) stability conditions by the Lyapunov method to guarantee that the system in the sliding mode is asymptotically stable. In the third step, we design an adaptive output feedback second order sliding mode controller in a way such that the system states reach the sliding manifold in finite time and remain it thereafter. 3.1. Sliding Surface Design From Eq. (11), Eq. (12) and Eq. (13) of paper [28], it follows that under Assumptions 3 and 4, there exists a non-singular matrix Tsuch that in the new coordinates z=Tx the system Eq. (1) can be described as: ˙z=A1A2 A3A4+D1 D2FE1E2z+ +0 B2u+ξ(T−1z, t), (5) and y=0C2z, (6) where z=z1z2T,z1∈Rn−m,z2∈Rm,TAT−1= A1A2 A3A4,TDFET−1=D1 D2FE1E2,TB = c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 437 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER 0 B2and CT−1=0C2. The matrices B2∈ Rm×m,C2∈Rp×pare non-singular and A1=˜ A1− ˜ A2˜ KΞ∈R(n−m)×(n−m)is stable. It follows from Eq. (5) that ˙z1= (A1+D1F E1)z1+ (A2+D1FE2)z2,(7) and ˙z2= (A3+D2F E1)z1+ (A4+D2F E2)z2+ +B2[u+ξ].(8) For the systems Eq. (5) and Eq. (6), consider a sliding surface σ(y(t)) = KC−1 2y= 0,(9) where K=K1K2=0m×(p−m)K2. The matrix K2∈Rm×mis the form of K2= ΨPΨT,(10) in which P∈R(n−m)×(n−m)is defined later and the matrix Ψ∈Rm×(n−m)is selected such that the matrix K2∈Rm×mis non-singular. According to Eq. (6), the sliding surface Eq. (9) can be rewritten as: σ(y(t)) = KC−1 2y= =KN0(p−m)×m 0m×(n−m)Im×mz=K2z2= 0,(11) where N=O(p−m)×(n−p)I(p−m)×(p−m). From Eq. (11) and since K2∈Rm×mis non-singular, in the coordinate z, the sliding surface Eq. (9) can be described by: {col(z1, z2)|z2= 0}.(12) Using Eq. (12), the dynamic equation in sliding mode is: ˙z1= (A1+D1F E1)z1.(13) 3.2. Stability Analysis of Sliding Motion In last section, we have designed an output sliding surface. There are still two important tasks that should be done. The first task is to derive appropriate LMI stability conditions by the Lyapunov method to guarantee that the sliding mode dynamics Eq. (13) is asymptotically stable. The second task is to design an adaptive output feedback SOSMC in a way such that the system states reach the sliding manifold in finite time and stay on it thereafter. Now, we are going to do the former task by considering the following LMI:   AT 1G+GTA1+ Θ AT 1G−GT+P+ Θ ET 1 GTA1−G+P+ Θ −G−GT+ Θ 0 E10−ϕI  <0, (14) where G∈R(n−m)×(n−m)is general and non-zero matrix, P∈R(n−m)×(n−m)is any positive matrix, Θ = ϕGTD1DT 1Gand the scalar ϕ > 0. Then, we can establish the following theorem. Theorem 1. Suppose that Assumptions 1-3 hold. Then, the sliding mode dynamics Eq. (13) is asymptotically stable if the matrices P>0and non-zero matrix Gsatisfy Eq. (14). Proof: For the sliding mode dynamics Eq. (13), consider a candidate Lyapunov function function V=zT 1Pz1,(15) where P>0satisfies Eq. (14). The time derivative of Valong the trajectories of Eq. (13) is given by ˙ V=zT 1(A1+D1FE1)T P+P(A1+D1FE1)z1.(16) From Eq. (16), if (A1+D1F E1)T P+P(A1+D1F E1)<0 then ˙ V < 0and the sliding mode dynamics Eq. (13) is asymptotically stable. Before proving ˙ V < 0, we recall the following Lemmas. Lemma 1. [31]: Let X,Yand Fbe matrices of compatible dimension then XF Y +YTFTXT< ϕ−1XXT+ϕY TYfor any Fsatisfying kFk≤ 1and a scalar ϕ > 0. Lemma 2. [31]: Given a symmetric matrix Wand two matrices Γand, Σand consider the problem of finding some matrix Gsuch that W+ ΣGΓ + (ΣGΓ)T<0. Denote Σ⊥and Γ⊥any matrices whose columns form the bases of the null spaces of Σand Γ, respectively. Then the above inequality is solvable for Gif and only if Σ⊥WΣ⊥T<0,ΓT⊥WΓT⊥T<0. Now, we are going to prove ˙ V < 0. Let us first define Γ⊥=I I,W=0P P0,Σ=(A1+D1FE1)T −I, Σ⊥=I(A1+D1FE1)T,ΓT⊥=I−Iand G is defined in LMI Eq. (14). Then, we have Λ=W+ ΣGΓ + (ΣGΓ)T =AT 1G+GTA1AT 1G−GT+P GTA1−G+P−G−GT +GTD1 GTD1FE10+ET 1 0FTDT 1G DT 1G. (17) Applying Lem. 1 to Eq. (17), we achieve Λ≤AT 1G+GTA1+ Θ AT 1G−GT+P+ Θ GTA1−G+P+ Θ −G−GT+ Θ  +ϕ−1ET 1 0E10,(18) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 438 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER where Θ = ϕGTD1DT 1Gand the scalar ϕ > 0. Using Schur complement formula, LMI Eq. (14) can be rewritten as AT 1G+GTA1+ Θ AT 1G−GT+P+ Θ GTA1−G+P+ Θ −G−GT+ Θ  +ϕ−1ET 1 0E10<0.(19) From Eq. (18) and Eq. (19), it can be observed that Λ≤AT 1G+GTA1+ Θ AT 1G−GT+P+ Θ GTA1−G+P+ Θ −G−GT+ Θ  +ϕ−1ET 1 0E10<0.(20) It follows from Eq. (20) and Lem. 2 that Σ⊥WΣ⊥T<0.(21) Since the fact W=0P P0, and Σ⊥=I(A1+D1FE1)T, we obtain: Σ⊥WΣ⊥T= (A1+D1FE1)TP+P(A1+D1FE1)<0. (22) According to Eq. (16) and Eq. (22), it is obvious that ˙ V < 0.(23) Note that Eq. (23) verifies that Eq. (14) holds, which further implies that sliding motion is asymptotically stable. The following new Lemma is derived for controller design using only output variables. Lemma 3. Consider the reduced-order system Eq. (7). If the matrix A1is stable then kz1(t)kis bounded by η(t)for all time, where η(t)is the solution of ˙η(t) = (kkD1kk E1k+λ)η(t)(24) +k(kkA2k+kD1kk E2k)kK−1 2KC−1 2kk yk, where kkD1kk E1k+λ < 0, the scalar k > 0and λ is the maximum eigenvalue of the matrix A1. Proof: Solving Eq. (7) gives kz1(t)k≤ t R 0 kexp {A1(t−τ)}k × [kD1FE1kkz1k + (kA2k+kD1FE2k)kz2k]dτ +kexp(A1t)kk z1(0) k. (25) The stable matrix A1satisfy the constraint kexp(A1t)k≤ kexp(λt),(26) where k > 0and λ < 0are defined in Eq. (3). According to Eq. (25) and Eq. (26) we have kz1(t)k≤ t R 0 kkexp {λ(t−τ)}k × [kD1kkE1kkz1k + (kA2k+kD1kk E2k)kz2k]dτ +kkexp(λt)kk z1(0) k. (27) Multiplying both sides of Eq. (27) by exp(−λt), gives kz1(t)kexp(−λt)≤ t R 0 kexp(−λτ)kD1kkE1kkz1kdτ + t R 0 kexp(−λτ)×(kA2k+kD1kkE2k)kz2kdτ +kkz1(0) k. (28) Letting h(t)is the right term of Eq. (28) and taking the time derivative of h(t), yields d dth(t) = kexp(−λt)kD1kk E1kk z1k +kexp(−λt)×(kA2k+kD1kk E2k)kz2k. (29) According to Eq. (28) and Eq. (29) we achieve d dth(t)≤kkD1kk E1kh(t) +kexp(−λt) (kA2k+kD1kk E2k)kz2k. (30) Multiplying both sides of Eq. (30) by exp(−kkD1kk E1kt), gives d dt{h(t) exp(−kkD1kk E1kt)} ≤ kexp(−λt)· (kA2k+kD1kkE2k)kz2kexp(−kkD1kkE1kt). (31) Integrating both sides of Eq. (31), we obtain h(t)≤kkz1(0) kexp(¯ kt) + exp(¯ kt)× t Z 0 kexp(−λτ) (kA2k+kD1kk E2k)kz2kexp(−¯ kτ)dτ, (32) where ¯ k=kkD1kk E1k. Since kz1(t)kexp(−λt)≤ h(t)and KC−1 2y=K2z2, therefore it can be obtained that kz1(t)k≤ η(0) exp (¯ k+λ)t+Zt 0 k(kA2k+kD1k kE2k)kK−1 2KC−1 2kk ykexp (¯ k+λ)(t−τ)dτ(33) =η(t),if η(0) ≥kkz1(0) k, where η(t)is defined in Eq. (3). Therefore, it is easy to conclude that η(t)≥k z1(t)kfor all time, if η(0) sufficiently large. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 439 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER 3.3. Adaptive Output Feedback Second Order Sliding Mode Controller Design In the last section, we dealt with the first and second elements of the design process. In this section, we design an adaptive output feedback second order sliding mode controller such that the system states reach the sliding manifold in finite time and stay on it thereafter. Let us begin with defining the sliding manifold s(t)such that s(t) = ˙σ+Xσ (34) and ˙s(t) = ¨σ+X˙σ, (35) where X∈diag(χ1, χ2, ...χm)is any diagonal matrix. The main idea behind the second-order sliding mode is to act on the second-order derivative of the sliding variable ¨σrather than the first derivative as in conventional sliding mode. The second-order sliding mode is determined from the basic equality condition ˙σ= ¨σ= 0 reaches in finite time, whereas the proposed controller reaches the condition asymptotically. According to Eq. (9), we obtain s(t) = KC−1 2˙y+Xσ (36) and ˙s(t) = KC−1 2¨y+X˙σ. (37) Since KC−1 2y=K2z2and Eq. (8), Eq. (37) can be rewritten as: ˙s(t) = K2[A3˙z1+A4˙z2] + B2˙u+˙ ψ+X˙σ, (38) where ψ=K2D2FE1z1+K2D2FE2z2+B2ξ. Assumption 5. The disturbance ξ(t)of Eq. (38) in the domain of interest satisfies ˙ ψ(t)≤ r X i=0 aikxki,(39) where aiare unknown positive constants, ris a designed positive integer. The proposed adaptive output feedback second order sliding mode controller for tackling the system uncertainty is designed as follows: u(t) = u(0) −Zt 0 (K2B2)−1[κ(t) +ρη(t) + ¯ρkyk+ˆρk˙yk+α]s(t) ks(t)kdt, (40) where the scalar α > 0, ρ=kK2kk A3k(kA1k+kD1kk E1k), κ(t) = Pr i=0 ˆai(t)(kH1kη(t)+ kH2kk K−1 2KC−1 2kk yk)i, ˆρ=kK2kk A4kk K−1 2KC−1 2k+kXkk KC−1 2k, ¯ρ=kK2kk A3k(kA2k+kD1kk E2k)k K−1 2KC−1 2k, and H1H2=T−1. The adaptive gains ˆai(t)are given by ˙ ˆai(t) = qi[−˘qiˆai+ (kH1kη +kH2kk K−1 2KC−1 2kk yk)i.(41) The time function η(t)is the solution of Eq. (3) and qi, ˘qiare positive constants. The major drawback of sliding mode control is so-called chattering phenomenon. Such a phenomenon consists of the oscillation of the control signal, tied to the discontinuous nature of the control strategy, at a frequency and with an amplitude capable of disrupting, damaging or, at least, wearing the controlled physical system. It should be pointed out that the controller Eq. (40) is a continuous control input and uses only output variables. Hence the undesired high frequency chattering of the control signal is eliminated. This is a new contribution of the proposed method. Now let us discuss the reaching conditions in the following theorem. Theorem 2. Consider the uncertain dynamic system defined by Eq. (1) with the assumptions 1–4. If the sliding manifold and the adaptive sliding mode controller are designed as Eq. (34) and Eq. (40), respectively. Then, the system states reach the sliding manifold s(t)in finite time and stay on it thereafter. Proof: Define a Lyapunov function candidate as follows: V(t) =ksk+1 2 r X i=0 ˜a2 i qi ,(42) where ˜ai(t) = ai−˜ai(t),i= 0,1, ..., r are the estimation errors of the adaptive gains. Now taking the derivative of V(t)yields ˙ V(t) = sT ksk˙s− r X i=0 1 qi ˙ ˆai˜ai.(43) Using Eq. (7), Eq. (38) and Eq. (43) and property kAB k≤k Akk Bk, it generates ˙ V(t)≤k K2kk A3k[(kA1k+kD1kk E1k)kz1k + (kA2k+kD1kk E2k)kz2k] + kK2kk A4kk ˙z2k +sT kskK2B2˙u+k˙ ψk+kXkk ˙σk − r X i=0 1 qi ˙ ˆai˜ai.(44) From Assumption 4 and kxk≤k H1kk z1k+kH2kk z2kwhere H1H2=T−1. We can obtain that ˙ V(t)≤sT kskK2B2˙u−Pr i=0 1 qi ˙ ˆai˜ai+kK2kk A3k [(kA1k+kD1kk E1k)kz1k+(kA2k+kD1k kE2k)kz2k]+kXkk ˙σk+kK2kk A4kk ˙z2k +Pr i=0 ai(kH1kk z1k+kH2kk z2k)i. (45) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 440 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Equation (11) implies that z2=K−1 2KC−1 2y. (46) According to Lem. 3, we have kz1(t)k≤ η(t).(47) From Eq. (41), Eq. (45), Eq. (46) and Eq. (47), it can be observed that ˙ V(t)≤ρη + ¯ρkyk+ˆρk˙yk+ r P i=0 ˘qiˆai(ai−ˆai) + r P i=0 ˆaikH1kη+kH2kk K−1 2KC−1 2kk yki +sT kskK2B2˙u. (48) Substituting the controller Eq. (40) into Eq. (48), we achieve ˙ V(t)≤ −α− r X i=0 ˘qi(ˆai−ai 2)2+ r X i=0 ˘qi a2 i 4.(49) Then, from Eq. (49), it is easy to see that the uniform ultimate boundedness can be guaranteed. The proposed adaptive SOSMC, using the output information completely in the sliding surface and controller design, offers following advantages. Firstly, conservatism is reduced and the robustness is enhanced. Secondly, an improved transient performance can be obtained without the knowledge about the upper bound of the system uncertainties. Finally, the chattering in the control input is removed. Design procedure: The proposed adaptive output feedback SOSMC scheme can be simultaneously designed by the following steps. •Step 1: Find a feasible solution of LMI Eq. (14) and calculate the scaling of sliding surface parameter K2using Eq. (10). •Step 2: Design the sliding surface σ(t)according to Eq. (9). •Step 3: Design the sliding manifold s(t)using Eq. (36). •Step 4: Design the adaptive output feedback SOSMC u(t)according to Eq. (40). 4. Numerical Example In order to demonstrate the validity and effectiveness of the proposed method, in this section, we are going to apply the adaptive output feedback SOSMC given in previous sections for a Vertical Take-Off and Landing (VTOL) aircraft at the nominal airspeed of 135 knots, which is modified from [19]. ˙x=        −0.0366 0.0271 0.0188 −0.0455 0.0482 −1.01 0.0024 −4.0208 0.1002 0.3681 −0.707 1.42 0 0 1 0     +∆A     x+ +    −0.4422 0.1761 3.5446 −7.5922 −5.52 4.49 0 0     (u+ξ),(50) y=  1000 0100 0001  x, (51) where x=x1x2x3x4T,u=u1u2Twith x1 is the horizontal velocity (knots), x2is the vertical velocity (knots), x3is the pitch rate (degrees per second) and x4is the pitch angle (degrees), u1is the collective pitch control, u2is the longitudinal cyclic pitch control. It is assumed that x1,x2, and x4are the output signals. The mismatched uncertainty is given as ∆A=DF E with D=1110T,E=0111and F= sin (x2×t+ 4x1×π×t+x1x3x4×t).(52) The disturbance is assumed to satisfy the following condition k˙ ψ(t)k≤ 2 + 0.1kxk. For this work, the following parameters are selected as follows: α= 300.1, ϕ= 10.0109, and k= 1.001. Assumptions 2 and 3 can be shown to hold. The coordinate transformation z=Tx is given by: T=    −1 0.038 0.10514 0 0 0 0 1 0 0.841 0.540 0.758 0−0.540 0.841 0.539     . Solving LMI Eq. (14), we have the solution P=0.320 −0.045 −0.045 2.254 and G=0.269 0 0 0.942. The matrix K2=0.3204 0.00025 0.00025 0.27545is nonsingular. The sliding surface is given as Σ(x) = 1.64858 0.20519 0.24317 2.20577 −0.23462 0.27545y= 0. The controller for the system Eq. (50) and Eq. (51) is the solution of the following equation: u(t) = u(0) − t Z 0 −0.94575 −0.5525 −0.7877 0.00072κ(t)+ +1.346η(t)+13.376kyk+15.338k˙yk+300s ksk dt, (53) where κ(t) = 1 P i=0 2.999ˆai(t)(1.375η(t) + 10.109 kyk)i, ˙ ˆai(t) = −180ˆai(t) + 2.999(1.375η(t) + 10.109 kyk)i, c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 441 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER 0 2 4 6 8 10 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Time (s) Magnitude x1 x2 x3 x4 Fig. 1: Time responses of states x1(dash-dot), x2(dashed), x3 (dotted), x4(solid). 0 2 4 6 8 10 −3 −2 −1 0 1 2 3 Time(s) Magnitude Fig. 2: Control input u1. i= 0,1and ˙η(t) = −0.048996η(t) + 10.109 kyk, η(0) = 0.1. The initial conditions for the above system are selected to be x(0) = 2−211Tand u(0) = 0 0T. The system states and the control input of the VTOL aircraft system using the proposed adaptive output feedback SOSMC are shown in Fig. 1, Fig. 2 and Fig. 3. It is evident from Fig. 1 that the proposed adaptive output feedback SOSMC produces faster convergence of the system states to equilibrium as compared to the method proposed by [19]. It is observed from Fig. 2 and Fig. 3 that the actual control input obtained by the proposed method is smooth and chattering free. Convergence of the sliding surface is shown in Fig. 4 and Fig. 5. Figure 4 and Fig. 5, clearly show that the proposed sliding surface is smooth and approach to equilibrium point quickly. Remark 4: The method proposed by [19] cannot be applied for the system Eq. (50) and Eq. (51) if the state variable x3is unmeasurable. This limitation has been removed by the proposed adaptive output feedback SOSMC Eq. (53) because the proposed controller Eq. (53) only uses three output variables (x1,x2, and x4). Remark 5: The mismatched parameter uncertainties in the state matrix of the system Eq. (50) and Eq. (51) are non-linear and time-varying. 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