Global Exponential Stabilization for a Class of Uncertain Nonlinear Control Systems Via Linear Static Control

1. International Journal of Trend in International Open Access Journal International Open Access Journal | www.ijtsrd.com International Journal of Trend in Scientific Research and Development (IJTSRD) Research and Development (IJTSRD) www.ijtsrd.com ISSN No: 2456 ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov 1 | Nov – Dec 2018 Global Exponential Stabilization Uncertain Nonlinear Control Systems Uncertain Nonlinear Control Systems Via Linear Static Control Global Exponential Stabilization for a Class Class of Linear Static Control Yeong-Jeu Sun Professor, Department of Electrical Engineering Electrical Engineering, I-Shou University, Kaohsiung Kaohsiung, Taiwan ABSTRACT In this paper, the robust stabilization for a class of uncertain nonlinear systems is investigated. Based on the Lyapunov-like approach inequalities, a simple linear static control is offered to realize the global exponential stability of such uncertain nonlinear systems. guaranteed exponential convergence rate can be correctly estimated. Finally, simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Key Words: Global exponential uncertain nonlinear systems, exponential convergence rate, linear static control 1.INTRODUCTION Control design with implementation nonlinear dynamical systems is one of the most challenging areas in systems and control theory. well known that uncertainties and nonlinearities appear in various physical systems and a systems are essentially nonlinear in nature. control has been active for many years, and many results concerning with nonlinear control have been proposed. However, it is still difficult to implement nonlinear controllers for practical systems. Recently, there have techniques and methodologies for analyzing nonlinear systems, such as back stepping approach fuzzy adaptive control linearization, H-infinity control approach, mode control methodology, LMI approach, perturbation method, Lyapunov approach, equation approach, center manifold theorem sliding mode control, adaptive fuzzy-neural and others; see, for example, [1-8] and the references therein. In this paper, the stabilizability for a class of uncertain nonlinear systems will been considered Lyapunoe-like approach with differential inequality, a linear static control will be established global exponential stability of such Moreover, the guaranteed exponential convergence rate can be correctly calculate simulations will also be provided to illustr of the main results. The layout of the rest of this paper is organized as follows. The problem formulation, main result, and controller design are presented in Section 2. Section 3, numerical simulations realization are given to illustrate the effectiveness of the developed results. Finally, drawn in Section 4. In what follows, n-dimensional real space, x norm of the vector x ℜ ∈ value of a real number a, and of the matrix A. 2.PROBLEM FORMULATION AND MAIN RESULTS In this paper, we explore the following nonlinear systems: x a x a x a x a x ∆ + ∆ + ∆ + ∆ = & a x a x a x a x ∆ + ∆ + , 2 1 9 3 12 2 11 3 x x a x a x a x ∆ − ∆ + ∆ = & x x a x a x a x ∆ + ∆ − ∆ + ∆ = & ( ) [ 40 30 20 10 x x x x = where ( ) 3 2 1 := t x t x t x t x vector, ( ) 2 1 : ℜ ∈ = t u t u t u stabilization for a class of the stabilizability for a class of uncertain nonlinear systems will been considered. Based on the approach with differential inequality, a uncertain nonlinear systems is investigated. Based on like approach inequalities, a simple linear static control is offered to realize the global exponential stability of such systems. guaranteed exponential convergence rate can be correctly estimated. Finally, simulations are given to demonstrate the feasibility and effectiveness of the obtained results. with with differential differential established to realize the of such uncertain systems. Meanwhile, Meanwhile, the the the guaranteed exponential convergence orrectly calculated. Several numerical simulations will also be provided to illustrate the use some some numerical numerical The layout of the rest of this paper is organized as The problem formulation, main result, and controller design are presented in Section 2. In , numerical simulations with circuit o illustrate the effectiveness of Finally, some conclusions are In what follows, Global exponential stabilization, stabilization, nonlinear systems, exponential convergence of uncertain ℜ denotes the n nonlinear dynamical systems is one of the most challenging areas in systems and control theory. It is denotes the Euclidean a denotes the absolute A denotes the transport , n onlinearities often and all physical in nature. Nonlinear T control has been active for many years, and many results concerning with nonlinear control have been owever, it is still difficult to implement nonlinear controllers for practical systems. ROBLEM FORMULATION AND MAIN the following uncertain 4 (1a) , x several several for analyzing uncertain stepping approach, ol approach, infinity control approach, sliding methodology, LMI approach, singular , Lyapunov approach, Riccati enter manifold theorem, adaptive well well-developed 1 1 1 2 2 3 4 4 3 x 4 = ∆ + ∆ + ∆ + ∆ & (1b) 2 5 a 1 x 6 2 a 7 3 8 4 , x u 9 1 3 10 1 approach, feedback (1c) 15u (1d) , a 4 13 x 1 ( ) 0 14 2 4 0 1 ] 3 15 2 [ ( ) 0 ( ) T 0 x x x (1e) 1 2 3 4 ] , T [ ( )] t ( ) ( ) ( ) ( ) ( ) neural-network, ] and the references is the state T ∈ ℜ 4 x is the system control, 4 [ ] T 2 @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1227

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 ]T x x 40 30 20 is the initial value, { 15 , , 2 , 1 L i indicate uncertain parameters of the system. The hyper-chaotic Pan system case of systems (1) with ( ) , 0 = t u 8 12 = ∆a , ∆a { 15 , 13 , 11 , 10 , 7 , 6 , 4 , 3 , 0 ∈ ∀ = ∆ i ai . The global exponential stabilization exponential convergence rate of the system (1) are defined as follows. Definition 1. The uncertain systems (1) are said to be exponentially stable if there exist a control positive number α satisfying ( ) 0 , 0 ≥ ∀ ⋅ ≤ t e x t x . In this case, the positive number α exponential convergence rate. The aim of this paper is to find a simple linear control such that the global exponential stabili of uncertain systems (1) can be guaranteed. Meanwhile, an estimate of the exponential convergence rate of such stable systems is also explored. Throughout this paper, we make th assumption: (A1) There exist constants { 15 , , 2 , 1 , L i a a a i i i , with b { 15 , 10 , 0 ∈ ∀ > i ai and 12 , 1 , 0 ∈ ∀ < i ai Now we present the main result for exponential stabilization of uncertain systems (1) via Lyapunov-like theorem with the differential and integral inequalities. Theorem 1 The uncertain systems (1) with (A1) globally exponential stabilization under the static control [ x k x k u u u 4 2 2 1 2 1 − − = = , (2a) where 1 1 × ≥ a (2b) International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 [ ∆ with exponential convergence rate is  − = , 3 Proof. Let ( ) x t x t x V 2 1 : + = The time derivative of the closed-loop systems (1) given by ( ) 2 2 2 x x x x & & + + = ( ( 6 1 5 2 2 k a x x a ∆ + ∆ + and . In this this case, the guaranteed is given by the initial value, and 1> 2> δ δ 0 0 x x 10 ai } parameters of the ∀ ∈ , −  a a chaotic Pan system is a special . (3) α δ δ 1 12 : min , ,   1 2 2 ,   ∆ = − ∆ = 10 a a 2 1 − , , , and 14= = − ∆a 5= ∆ = − ∆ = 10 28 1 a a ( ) ( ) ( ) t ( ) t ( ) t ( ) t . . (4) 8 9 + + V 2 2 2 3 2 4 3 x x } ( ) along the trajectories of with (2) and (A1), is x exponential stabilization and the rate of the system (1) are ( ) & V x t + 2 x x x & ∆ x x & ∆ 1 1 ∆ 2 2 3 3 4 4 ) = + ∆ + + 2 x a ∆ x a ∆ x a ∆ a ∆ x x 1 x 1 a 1 2 2 3 4 4 a 3 3 4 said to be globally + + + + x − a x a x x x exponentially stable if there exist a control u and 2 7 3 8 4 ) + ∆ ∆ x 9 1 3 10 1 2 x ( ( ) − − ∆ 2 x a x a a x x ( ) − α t 3 11 2 12 3 9 1 2 ) + ∆ + + ∆ − ∆ ∆ ∆ ∆ ∆ 2 x a x a x a x x a k x is called the 4 13 2 1 14 2 2 4 1 3 + 15 2 x 4 x ≤ + 2 1 2 2 a x b x x b x x a x 1 b 2 1 2 2 x 3 b 1 x 4 4 1 3 4 + + + + 2 2 2 2 2 2 x x b x b x x 5 1 x 2 6 7 2 3 8 2 4 simple linear static + ∆ − 2 2 2 2 a x x a k x 9 1 2 3 10 1 such that the global exponential stabilization (1) can be guaranteed. , an estimate rate of such stable systems is also + + − ∆ 2 3 2 2 2 b x x a x a x x x 11 2 3 12 9 2 1 2 3 + + − ∆ of the exponential 2 2 b x x b x x a x x x x 13 1 4 14 2 4 4 1 3 4 − 2 4 2 a k x 15 2   a a a ( ) ≤ + + + + 2 1     1 1 1 2 2 x b b x x x 2 5 1 2 3 3 3 the following ( ) ( ) + + + + + 2 2 2 2 2 b b x x b x b b b x x 3 13 1 4 6 7 11 2 3 ia and ia such that {   a a ( ) + + + + 2 3 3 2     12 2 12 2 2 2 b b x x x } 8 14 2 4 } , ≤ ∆ ≤ ∀ ∈ } := max , b a a i i i ( ) ( )   2 2 − + − + 3 b b b b { }. − + + +δ + 2 2 2 5 7 2 11 2 1 b x   6 6 1 4 a a     1 12 ( ) ( )     2 2 − + + 3 b b b b Now we present the main result for the globally − + + δ 2 4 3 4 13 8 14 2 x  2 4 a       uncertain systems (1) via the differential and 1 ( )  2 + − − 3 b b a ( ) = − − + + 2 1 2 2 2 2 5 1 2 x x b b x x     2 5 1 2 3 4 4 a 1 ( )   2 + − − 3 b b a ( ) − − + + 2 1 2 4 3 4 4 3 13 1 2 x x b b x x     with (A1) realize the under the linear 3 13 1 4 3 a 1 ( )   2 + − − b b a ( ) − − + + 2 3 2 2 7 7 2 2 11 12 2 x x b b x x     7 11 2 3 ] [ ]T 2 a T 12 ( )   2 + b b ( ) − − + + 2 2 2 2 8 14 2 x x b b x x     8 14 2 4 4 k  −  −   a a 10 − − δ − − − δ 2 1 2 3 2 2 2 4    −       −    1 12 2 2 2 2 x x x x 1 2 3 2 a ( ) ( )   2 2 − + a − + a ) 3 b b b b + + + δ + 2 5 7 11 1 , b      a 6 1 4 2 ≤ − − − ) δ − − δ 2 1 2 3 2 2 2 4     +   α + . 0   α 1 12 2 2 2 2 x x x x 1 12 1 2 3 2 ( ( )  2 2 − + a + 3 ( b b b b 1 (2c) ≥ + + δ 3 13 8 14 , k     ≤ = − − α V α ∀ + 2 1 , 2 2 ≥ 2 3 2 4 2 2 α x x t x x 2 2 4 4 a 15 1 @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1228

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 4.CONCLUSION In this paper, the robust stabilization for a class of uncertain nonlinear systems has been explored on the Lyapunov-like approach inequalities, a simple linear presented to realize the global exponential stability of such uncertain nonlinear systems. guaranteed exponential convergence rate can be correctly calculated. Finally, simulations have been offered feasibility and effectiveness of the obtained results. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 106 MOST 106-2813-C-214-025-E E-214-030. Furthermore, the author Chair Professor Jer-Guang Hsieh comments. REFERENCES 1.A. Bayas, I. Škrjanc, and fuzzy robust control strategies for a distributed solar collector field,” Applied Soft Computin vol. 71, pp. 1009-1019, 201 2.Z.M. Li and X.H. Chang, “ networked control systems with randomly occurring uncertainties: Observer ISA Transactions, vol. 83, pp. 3.R.X. Zhong, C. Chen, Y.P. Huang, A. Sum and D.B. Xu, “Robust perimeter control for two urban regions with macroscopic fundamental diagrams: A control approach,” Transportation Research Part B: Methodological, vol. 117, pp. 4.F. Yue and X. Li, “Robust adapt backstepping control for opto system based on modified LuGre friction model ISA Transactions, vol. 80, pp. 5.N. Adhikary and C. Mahanta control of position manipulators,” Control Engineering Practice 81, pp. 183-198, 2018. 6.A. Ayadi, M. Smaoui, S. Farza, “Adaptive sliding mode control with moving surface: Experimental validation for electropneumatic system,” and Signal Processing, vol. , vol. 109, pp. 27-44, 2018. Hence, one has [ ] In this paper, the robust stabilization for a class of d & α α α ⋅ + ⋅ α = ⋅ 2 2 2 t t t 2 e V e V e V has been explored. Based approach with differential static control has been dt ≤ ∀ ≥ , 0 . 0 t It results d ∫ τ to realize the global exponential stability of systems. Meanwhile, the guaranteed exponential convergence rate can be Finally, have been offered to demonstrate the ffectiveness of the obtained results. [ ] t ( ( ) t ) ατ ⋅ τ 2 e V x d d 0 ( ( ) t ) ( ( ) 0 ) α = ⋅ − 2 t e V x V x some some numerical numerical t ≤∫ (5) τ = ∀ ≥ 0 , 0 . 0 d t 0 From (4) and (5), it follows ( ) ( ) 0 = x e As a consequence, we conclude that ( ) , 0 ∀ ≤ x e t x This completes the proof. 3.NUMERICAL SIMULATIONS Consider the uncertain systems (1) with 9 , 10 11 1 ∆ ≤ − ≤ ∆ ≤ − a 0 , 28 27 8 5 ∆ ≤ ≤ ∆ ≤ a a ( ( ) t ) ( ( ) 0 ) 2 − α = ≤ 2 t x t V x e V x Ministry of Science and 2 − α ∀ ≥ 2 t , . 0 t of Republic of China for supporting this work under grants MOST 106-2221-E-214-007, ( ) − α ≥ t . 0 t E, and MOST 107-2221- , the author is grateful to Guang Hsieh for the useful □ (6a) (6b) ≤ 10 8 − , a 2 , 1 ≤ and D. Sáez, “Design of fuzzy robust control strategies for a distributed Applied Soft Computing, , 2018. (6c) − ≤ ∆ ≤ − ≤ ∆ ≤ 1 , 1 3 , a a 9 12 3 (6d) (6e) − ∆ By comparing (A1) and (6) with parameters of ( , 1 , 10 , , , 15 12 10 1   ( 28 , 10 , , , , 6 11 7 5 2 = b b b b b ( 10 , 1 , 0 , 0 , , , 14 8 13 3 = b b b b (A1) is evidently satisfied. With the choice in (2), it can be obtained that [ 111 2 2 1 x u u u − − = = As a consequence, by Theorem 1, we conclude that the uncertain systems (1) with (6) exponentially stable under the linear static (7). Besides, from (3), the guaranteed exponential convergence rate is given by trajectories of the uncontrolled system and the feedback-controlled system are depicted in Fig and Figure 2, respectively. In addition, signals and the electronic circuit to realize such a control law are depicted in Figure 3 respectively. ≤ = ∆ ≤ , 9 − ∈ ≤ ∆ ∆ ≤ 10 1 , , 2 a a 11 a 14 10 15 . { , 7 , 6 , 4 , 3 } , “Robust H∞ control for ∀ , 0 13 , ai i networked control systems with randomly occurring uncertainties: Observer-based case,” , pp. 13-24, 2018. selecting the ) − R.X. Zhong, C. Chen, Y.P. Huang, A. Sumalee, Robust perimeter control for two urban regions with macroscopic fundamental diagrams: A control-Lyapunov Transportation Research Part B: 7, pp. 687-707, 2018.   8 = ) ( ) ( −   1 , , a a a a 3 ), 0 , 0 , 0 , ), function With the choice =δ = δ 1 1 2 Robust adaptive integral ] [ ] . (7) T T 32 x backstepping control for opto-electronic tracking system based on modified LuGre friction model,” , pp. 312-321, 2018. 4 we conclude that are globally static control of guaranteed exponential Mahanta, “Sliding mode position trol Engineering Practice, vol. control of commanded commanded robot robot . The typical s The typical state system and the = α 1 Aloui, S. Hajji, and M. controlled system are depicted in Figure 1 In addition, the control and the electronic circuit to realize such a Adaptive sliding mode control with moving surface: Experimental validation for ,” Mechanical Systems and Figure 4, @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1229

4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 7.Y. Solgi and S. Ganjefar, “Variable structure fuzzy wavelet neural network controller for complex nonlinear systems,” Applied Soft Computing, vol. 64, pp. 674-685, 201 8.J.F. Qiao, Y. Hou, L. Zhang, and “Adaptive fuzzy neural network control of wastewater treatment process with multiobjective operation,” Neurocomputing, vol. 27 393, 2018. 9.A. Erfani, S. Rezaei, M. Pourseifi, “Optimal control in teleoperation systems with time delay: A singular perturbation approach Journal of Computational Mathematics, vol. 338, pp. 168-184, 201 10.H. Gritli and S. Belghith, “Robust feedback control of the underactuated Inertia Wheel Inverted Pendulum under parametric uncertainties and subject to external disturbances: LMI formulation,” Journal of the Franklin Institute 355, pp. 9150-9191, 2018. Variable structure 200 u1: the Blue Curve fuzzy wavelet neural network controller for 100 u2: the Green Curve Applied Soft , 2018. 0 -100 and H.G. Han, u1(t); u2(t) tive fuzzy neural network control of wastewater treatment process with multiobjective -200 -300 275, pp. 383- -400 and H. Derili, -500 Optimal control in teleoperation systems with A singular perturbation approach,” Journal of Computational -600 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (sec) t (sec) Figure 3: Control signals Control signals of ( ) t ( ) t and and , 2018. Applied Applied and . u1 u2 Robust feedback 2u control of the underactuated Inertia Wheel Inverted Pendulum under parametric uncertainties d subject to external disturbances: LMI Journal of the Franklin Institute, vol. 4x The uncertain systems (1) with (6) he uncertain systems (1) 1u 2x 100 x1: the deep blue curve x2: the green curve x3: the red curve x4: the light blue curve x4: the light blue curve x3: the red curve 2 R 1 R 50 x1(t); x2(t); x3(t); x4(t) 0 4 R -50 0 5 10 15 20 25 30 35 40 3 R t (sec) Figure 1: Typical state trajectories of the system (1) with (6) and 5 Typical state trajectories of the system (1) = . 0 u Figure 4: The diagram of implementation of controller, where 1 = R R 32 4 = R The diagram of implementation of x1: the deep blue curve x1: the deep blue curve 4 and = 1 1 Ω k . Ω k = Ω 3 , 2 111 , R k x2: the green curve x2: the green curve 3 x3: the red curve x3: the red curve k 2 x4: the light blue curve x4: the light blue curve x1(t); x2(t); x3(t); x4(t) 1 0 -1 -2 -3 -4 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (sec) Figure 2: Typical state trajectories of the feedback controlled system of (1) with (6) and (7) f the feedback- and (7). @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1230