Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 147 system is affected by the parameter variation. Compared with the nominal system, the position trajectory is different, bigger overshoot and the relative stability degrades. In summery, the robust optimal SMC system owns the optimal performance and global robustness to uncertainties. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 time(sec) position Robust Optimal SMC Optimal Control (a) Position responses 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 time(sec) J(t) Robust Optimal SMC Optimal Control (b) Performance indexes Fig. 2. Simulation results in Case 2 2.4 Conclusion In this section, the integral sliding mode control strategy is applied to robustifying the optimal controller. An optimal robust sliding surface is designed so that the initial condition is on the surface and reaching phase is eliminated. The system is global robust to uncertainties which satisfy matching conditions and the sliding motion minimizes the given quadratic performance index. This method has been adopted to control the rotor position of an electrical servo drive. Simulation results show that the robust optimal SMCs are superior to optimal LQR controllers in the robustness to parameter variations and external disturbances. Robust Control, Theory and Applications 148 3. Optimal sliding mode control for uncertain nonlinear system In the section above, the robust optimal SMC design problem for a class of uncertain linear systems is studied. However, nearly all practical systems contain nonlinearities, there would exist some difficulties if optimal control is applied to handling nonlinear problems (Chiou & Huang, 2005; Ho, 2007, Cimen & Banks, 2004; Tang et al., 2007).In this section, the global robust optimal sliding mode controller (GROSMC) is designed based on feedback linearization for a class of MIMO uncertain nonlinear system. 3.1 Problem formulation Consider an uncertain affine nonlinear system in the form of () () (,), (), xfx gxudtx yHx =+ + =  (19) where n xR∈ is the state, m uR∈ is the control input, and () f x and ()gx are sufficiently smooth vector fields on a domain n DR⊂ .Moreover, state vector x is assumed available, ()Hx is a measured sufficiently smooth output function and T 1 () ( , , ) m Hx h h=  . (, )dtx is an unknown function vector, which represents the system uncertainties, including system parameter variations, unmodeled dynamics and external disturbances. Assumption 5. There exists an unknown continuous function vector (, )tx δ such that (, )dtx can be written as (, ) ( ) (, )dtx gx tx δ = . This is called matching condition. Assumption 6. There exist positive constants 0 γ and 1 γ , such that 01 (, )tx x δγγ ≤+ where the notation ⋅ denotes the usual Euclidean norm. By setting all the uncertainties to zero, the nominal system of the uncertain system (19) can be described as () () , (). xfx gxu yHx =+ =  (20) The objective of this paper is to synthesize a robust sliding mode optimal controller so that the uncertain affine nonlinear system has not only the optimal performance of the nominal system but also robustness to the system uncertainties. However, the nominal system is nonlinear. To avoid the nonlinear TPBV problem and approximate linearization problem, we adopt the feedback linearization to transform the uncertain nonlinear system (19) into an equivalent linear one and an optimal controller is designed on it, then a GROSMC is proposed. 3.2 Feedback linearization Feedback linearization is an important approach to nonlinear control design. The central idea of this approach is to find a state transformation ()zTx = and an input transformation Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 149 (,)uuxv= so that the nonlinear system dynamics is transformed into an equivalent linear time-variant dynamics, in the familiar form zAzBv = +  , then linear control techniques can be applied. Assume that system (20) has the vector relative degree { } 1 ,, m rr and 1 m rrn++= . According to relative degree definition, we have () () 1 1 ,0 1 (), ii i k k fi i i m rr r i j i j i ff j yLh kr y Lh g L h u − = =≤≤− =+ ∑ (21) and the decoupled matrix 11 1 1 11 11 11 () () () ( ) () () m mm m rr gg ff ij m m rr gm g m ff LLh LLh Ex e LL h L L h −− × −− ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ == ⎢ ⎥ ⎢ ⎥ ⎣ ⎦    is nonsingular in some domain 0 xX∀∈ . Choose state and input transformations as follows: () ,1,,;0,1,,1 jj j ii ii f zTxLhi mj r = == = − (22) 1 ()[ ()],uE xvKx − =− (23) where 1 T 1 () ( , , ) m r r m ff Kx Lh L h=  , v is an equivalent input to be designed later. The uncertain nonlinear system (19) can be transformed into m subsystems; each one is in the form of 00 11 11 1 0 010 0 0 0 001 0 0 . 0 000 1 0 000 0 1 ii i ii ii i rr r ii di f zz zz v zz LL h −− − ⎡ ⎤ ⎡⎤⎡⎤ ⎡⎤ ⎡⎤ ⎢ ⎥ ⎢⎥⎢⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥⎢⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥⎢⎥ =++ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥⎢⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥⎢⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥⎢⎥ ⎣⎦ ⎣⎦ ⎣⎦⎣⎦ ⎢ ⎥ ⎣ ⎦              (24) So system (19) can be transformed into the following equivalent model of a simple linear form: () () () (, ),zt Azt Bvt tz ω =++  (25) where n zR∈ , m vR∈ are new state vector and input, respectively. nn A R × ∈ and nm BR × ∈ are constant matrixes, and (,)AB are controllable. ( , ) n tz R ω ∈ is the uncertainties of the equivalent linear system. As we can see, (,)tz ω also satisfies the matching condition, in other words there exists an unknown continuous vector function (,)tz ω  such that (, ) (, )tz B tz ω ω =  . Robust Control, Theory and Applications 150 3.3 Design of GROSMC 3.3.1 Optimal control for nominal system The nominal system of (25) is () () ().zt Azt Bvt = +  (26) For (26), let 0 vv= and 0 v can minimize a quadratic performance index as follows: TT 00 0 1 [()() () ()] 2 JztQztvtRvtdt ∞ =+ ∫ (27) where nn QR × ∈ is a symmetric positive definite matrix, mm RR × ∈ is a positive definite matrix. According to optimal control theory, the optimal feedback control law can be described as 1T 0 () ()vt RBPzt − =− (28) with P the solution of the matrix Riccati equation T1T 0.PA A P PBR B P Q − + −+= (29) So the closed-loop dynamics is 1T () ( )().zt A BR B Pzt − =−  (30) The closed-loop system is asymptotically stable. The solution to equation (30) is the optimal trajectory z*(t) of the nominal system with optimal control law (28). However, if the control law (28) is applied to uncertain system (25), the system state trajectory will deviate from the optimal trajectory and even the system becomes unstable. Next we will introduce integral sliding mode control technique to robustify the optimal control law, to achieve the goal that the state trajectory of uncertain system (25) is the same as that of the optimal trajectory of the nominal system (26). 3.3.2 The optimal sliding surface Considering the uncertain system (25) and the optimal control law (28), we define an integral sliding surface in the form of 1T 0 () [() (0)] ( )( ) t st Gzt z G A BR B Pz d τ τ − =−− − ∫ (31) where mn GR × ∈ , which satisfies that GB is nonsingular, (0)z is the initial state vector. Differentiating (31) with respect to t and considering (25), we obtain 1T 1T 1T () () ( )() [() () (,)] ( )() () () (, ) st Gzt GA BR B Pzt GAzt Bvt tz GA BR B Pzt GBv t GBR B Pz t G t z ω ω − − − =−− =++−− =+ +  (32) Let () 0st =  , the equivalent control becomes Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 151 11T eq () ( ) () (, )vt GB GBRBPzt Gtz ω −− ⎡ ⎤ =− + ⎣ ⎦ (33) Substituting (33) into (25), the sliding mode dynamics becomes 11T 1T 1 1T 1 1T ()( ) () () () z Az B GB GBR B Pz G Az BR B Pz B GB G Az BR B Pz B GB GB B ABRBPz ω ω ωω ω ω − − −− −− − = −++ =− − + =− − + =−   (34) Comparing (34) with (30), we can see that the sliding mode of uncertain linear system (25) is the same as optimal dynamics of (26), thus the sliding mode is also asymptotically stable, and the sliding motion guarantees the controlled system global robustness to the uncertainties which satisfy the matching condition. We call (31) a global robust optimal sliding surface. Substituting state transformation ()zTx= into (31), we can get the optimal switching function (,)sxt in the x -coordinates. 3.3.3 The control law After designing the optimal sliding surface, the next step is to select a control law to ensure the reachability of sliding mode in finite time. Differentiating (,)sxt with respect to t and considering system (20), we have (() ()) . sss s sx fxgxu xtx t ∂ ∂∂ ∂ =+= + + ∂ ∂∂ ∂  (35) Let 0s =  , the equivalent control of nonlinear nominal system (20) is obtained 1 () ( ) ( ) . eq sss ut gx fx xxt − ∂ ∂∂ ⎡ ⎤⎡ ⎤ == − + ⎢ ⎥⎢ ⎥ ∂ ∂∂ ⎣ ⎦⎣ ⎦ (36) Considering equation (23), we have 1 0 ()[ ()] eq uExvKx − =−. Now, we select the control law in the form of con dis 1 con 1 dis 0 1 () () (), () () () , () ( ) ( ( ) ( ))s g n( ), ut u t u t sss ut gx fx xxt ss ut gx x gx s xx ηγ γ − − =+ ∂∂∂ ⎡⎤⎡ ⎤ == − + ⎢⎥⎢ ⎥ ∂∂∂ ⎣⎦⎣ ⎦ ∂∂ ⎡⎤ =− + + ⎢⎥ ∂∂ ⎣⎦ (37) where [] T 12 s g n( ) s g n( ) s g n( ) s g n( ) m sss s=  and 0 η > . con ()ut and dis ()ut denote continuous part and discontinuous part of ()ut , respectively. The continuous part con ()ut , which is equal to the equivalent control of nominal system (20), is used to stabilize and optimize the nominal system. The discontinuous part dis ()ut provides the complete compensation of uncertainties for the uncertain system (19). Theorem 2. Consider uncertain affine nonlinear system (19) with Assumputions 5-6. Let u and sliding surface be given by (37) and (31), respectively. The control law can force the system trajectories to reach the sliding surface in finite time and maintain on it thereafter. Robust Control, Theory and Applications 152 Proof. Utilizing T (1/2)Vss= as a Lyapunov function candidate, and taking the Assumption 5 and Assumption 6, we have TT T 01 TT T 01 01 11 01 1 (( ) ) ()sgn() ()sgn() () () ss Vsss f gud xt sss s ss sf f xg s d xxt x xt ss ss sxgssdsxgssg xx xx s sx x ηγ γ η γγ η γγ δ ηγγ ∂∂ == +++= ∂∂ ⎧ ⎡⎤ ⎛⎞∂∂∂ ∂ ∂∂ ⎪ ⎫ =−++++ ++= ⎨⎬ ⎢⎥ ⎜⎟ ∂∂∂ ∂ ∂∂ ⎭ ⎝⎠ ⎪ ⎣⎦ ⎩ ⎧⎫ ⎡⎤∂∂ ∂∂ ⎪⎪ =−++ + =− −+ + ⎨⎬ ⎢⎥ ∂∂ ∂∂ ⎪⎪ ⎣⎦ ⎩⎭ ∂ ≤− − + ∂   1 01 01 11 () () s gs gs x ss sxgsxgs xx δ ηγγ γγ ∂ +≤ ∂ ∂∂ ≤− − + + + ∂∂ (38) where 1 i denotes the 1-norm. Noting the fact that 1 ss≥ , we get T 0,for 0.Vss s s η = ≤− < ≠   (39) This implies that the trajectories of the uncertain nonlinear system (19) will be globally driven onto the specified sliding surface 0s = despite the uncertainties in finite time. The proof is complete. From (31), we have (0) 0s = , that is the initial condition is on the sliding surface. According to Theorem2, we know that the uncertain system (19) with the integral sliding surface (31) and the control law (37) can achieve global sliding mode. So the system designed is global robust and optimal. 3.4 A simulation example Inverted pendulum is widely used for testing control algorithms. In many existing literatures, the inverted pendulum is customarily modeled by nonlinear system, and the approximate linearization is adopted to transform the nonlinear systems into a linear one, then a LQR is designed for the linear system. To verify the effectiveness and superiority of the proposed GROSMC, we apply it to a single inverted pendulum in comparison with conventional LQR. The nonlinear differential equation of the single inverted pendulum is 12 2 1211 1 2 2 1 , sin sin cos cos (), (4/3 cos ) xx gxamLx x xaux xdt Lamx = −+ =+ −   (40) where 1 x is the angular position of the pendulum (rad) , 2 x is the angular speed (rad/s) , M is the mass of the cart, m and L are the mass and half length of the pendulum, respectively. u denotes the control input, g is the gravity acceleration, ()dt represents the external disturbances, and the coefficient /( )am Mm=+ . The simulation parameters are as follows: 1k g M = , 0.2 k g m = , 0.5 mL = , 2 9.8 m/sg = , and the initial state vector is T (0) [ /18 0]x π =− . Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 153 Two cases with parameter variations in the inverted pendulum and external disturbance are considered here. Case 1: The m and L are 4 times the parameters given above, respectively. Fig. 3 shows the robustness to parameter variations by the suggested GROSMC and conventional LQR. Case 2: Apply an external disturbance ( ) 0.01sin 2dt t = to the inverted pendulum system at 9ts= . Fig. 4 depicts the different responses of these two controllers to external disturbance. 0 2 4 6 8 10 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 t (s) Angular Position x1 (t) m=0.2 kg,L=0.5 m m=0.8 kg,L=2 m 0 2 4 6 8 10 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 t (s) Angular Position x1(t) m=0.2 kg,L=0.5 m m=0.8 kg,L=2 m (a) By GROSMC (b) By Conventional LQR. Fig. 3. Angular position responses of the inverted pendulum with parameter variation 0 5 10 15 20 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 t (s) angular position x1 (t) Optimal Control GROSMC Fig. 4. Angular position responses of the inverted pendulum with external disturbance. From Fig. 3 we can see that the angular position responses of inverted pendulum with and without parameter variations are exactly same by the proposed GROSMC, but the responses are obviously sensitive to parameter variations by the conventional LQR. It shows that the proposed GROSMC guarantees the controlled system complete robustness to parameter variation. As depicted in Fig. 4, without external disturbance, the controlled system could be driven to the equilibrium point by both of the controllers at about 2ts = . However, when the external disturbance is applied to the controlled system at 9ts = , the inverted pendulum system could still maintain the equilibrium state by GROSMC while the LQR not. Robust Control, Theory and Applications 154 The switching function curve is shown in Fig. 5. The sliding motion occurs from the beginning without any reaching phase as can be seen. Thus, the GROSMC provides better features than conventional LQR in terms of robustness to system uncertainties. 0 5 10 15 20 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 t (s) s (t) Sliding Surface Fig. 5. The switching function s(t) 3.5 Conclusion In this section, the exact linearization technique is firstly adopted to transform an uncertain affine nonlinear system into a linear one. An optimal controller is designed to the linear nominal system, which not only simplifies the optimal controller design, but also makes the optimal control applicable to the entire transformation region. The sliding mode control is employed to robustfy the optimal regulator. The uncertain system with the proposed integral sliding surface and the control law achieves global sliding mode, and the ideal sliding dynamics can minimized the given quadratic performance index. In summary, the system designed is global robust and optimal. 4. Optimal sliding mode tracking control for uncertain nonlinear system With the industrial development, there are more and more control objectives about the system tracking problem (Ouyang et al., 2006; Mauder, 2008; Smolders et al., 2008), which is very important in control theory synthesis. Taking the robot as an example, it is often required to follow some special trajectories quickly as well as provide robustness to system uncertainties, including unmodeled dynamics, internal parameter variations and external disturbances. So the main tracking control problem becomes how to design the controller, which can not only get good tracking performance but also reject the uncertainties effectively to ensure the system better dynamic performance. In this section, a robust LQR tracking control based on intergral sliding mode is proposed for a class of nonlinear uncertain systems. 4.1 Problem formulation and assumption Consider a class of uncertain affine nonlinear systems as follows: () () ()[ (,,)] () xfx fx gxu xtu yhx δ =+Δ+ + ⎧ ⎨ = ⎩  (41) Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization 155 where n xR∈ is the state vector, m uR∈ is the control input with 1m = , and yR∈ is the system output. () f x , ()gx , () f x Δ and ()hx are sufficiently smooth in domain n DR⊂ . (,,)xtu δ is continuous with respect to t and smooth in (,)xu . ()fxΔ and (,,)xtu δ represent the system uncertainties, including unmodelled dynamics, parameter variations and external disturbances. Our goal is to design an optimal LQR such that the output y can track a reference trajectory r ()yt asymptotically, some given performance criterion can be minimized, and the system can exhibit robustness to uncertainties. Assumption 7. The nominal system of uncertain affine nonlinear system (41), that is () () () x f x g xu yhx = + ⎧ ⎨ = ⎩  (42) has the relative degree ρ in domain D and n ρ = . Assumption 8. The reference trajectory r () y t and its derivations () r () i y t (1,,)in=  can be obtained online, and they are limited to all 0t ≥ . While as we know, if the optimal LQR is applied to nonlinear systems, it often leads to nonlinear TPBV problem and an analytical solution generally does not exist. In order to simplify the design of this tracking problem, the input-output linearization technique is adopted firstly. Considering system (41) and differentiating y , we have () (), 0 1 k k f yLhx kn = ≤≤− () 11 () () ()[ (,,)]. n nn n fff gf y LhxLLhxLLhxu xtu δ − − Δ =+ + + According to the input-out linearization, choose the following nonlinear state transformation T 1 () () () . n f zTx hx L hx − ⎡ ⎤ == ⎣ ⎦  (43) So the uncertain affine nonlinear system (40) can be written as 1 11 ,1,,1 () () ()[ (,,)]. ii nn n nf ff gf zz i n z Lhx L L hx LL hx u xtu δ + −− Δ = =− =+ + +    Define an error state vector in the form of 1r (1) r , n n zy ez zy − ⎡⎤ − ⎢⎥ = =−ℜ ⎢⎥ ⎢⎥ − ⎣ ⎦  where T (1) rr n yy − ⎡⎤ ℜ= ⎣⎦  . By this variable substitution ez = −ℜ, the error state equation can be described as follows: 1 () 11 1 r ,1,,1 () () ()() ()(,,) (). ii n nn n n nf ff gf gf ee i n e Lhx L L hx LL hxut LL hx xtu y t δ + −− − Δ ==− =+ + + −    Robust Control, Theory and Applications 156 Let the feedback control law be selected as () r 1 () () () () () n n f n gf Lhx vt y t ut LL hx − −++ = (44) The error equation of system (40) can be given in the following forms: 11 00 010 0 0 00 001 0 0 00 0 () () () . 000 1 000 0 0 1 () ()(,,) nn ff gf et et vt L L hx LL hx xtu δ −− Δ ⎡ ⎤⎡ ⎤ ⎡⎤ ⎡⎤ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ =+++ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥⎢ ⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎣ ⎦⎣ ⎦        (45) Therefore, equation (45) can be rewritten as () () () .et Aet A Bvt δ = +Δ + +Δ  (46) where 010 0 0 001 0 0 0 ,, 000 1 000 0 0 1 AB ⎡ ⎤⎡⎤ ⎢ ⎥⎢⎥ ⎢ ⎥⎢⎥ ⎢ ⎥⎢⎥ == ⎢ ⎥⎢⎥ ⎢ ⎥⎢⎥ ⎢ ⎥⎢⎥ ⎣ ⎦⎣⎦      11 00 00 00 ,. () ()(,,) nn ff gf A L L hx LL hx xtu δ δ −− Δ ⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ Δ= Δ= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦  As can be seen, n eR∈ is the system error vector, vR ∈ is a new control input of the transformed system. nn A R × ∈ and nm BR × ∈ are corresponding constant matrixes. AΔ and δ Δ represent uncertainties of the transformed system. Assumption 9. There exist unknown continuous function vectors of appropriate dimensions A Δ  and δ Δ  , such that A BA Δ =Δ  , B δ δ Δ=Δ  Assumption 10. There exist known constants m a , m b such that m A aΔ≤  , m b δ Δ≤  Now, the tracking problem becomes to design a state feedback control law v such that 0 e → asymptotically. If there is no uncertainty, i.e. (,) 0te δ = , we can select the new input as vKe=− to achieve the control objective and obtain the closed loop dynamics ()eABKe=−  . Good tracking performance can be achieved by choosing K using optimal [...]... -0.1819 25. 5383 142.4423 -118.0289 ⎦ ⎡98.3 252 8 .57 37 ⎤ ⎡ 0.0102 -0.0003 ⎤ BTPB = ⎢ ; invBTPB = ⎢ ⎥ 8 .57 37 301.9 658 ⎥ ⎣ ⎦ ⎣ -0.0003 0.0033 ⎦ 174 Robust Control, Theory and Applications ⎡ -41. 456 6 -29.87 05 -0.6169 -2. 356 4 -0.0008 0.01 05 -0.0016 0.1633 ⎤ ⎢ -29.87 05 -51 .6438 -3.8939 0.8712 -0.0078 0.10 15 -0.0 15 1 .57 28 ⎥ ⎢ ⎥ ⎢ -0.6169 -3.8939 -38.2778 32.1696 -0.0002 0.0028 -0.0004 0.043 ⎥ ⎢ ⎥ -2. 356 4 0.8712... 32.1696 -51 .6081 0 -0.0002 0 -0.0038 ⎥ gnorm = 0. 854 5lhs = ⎢ ; ⎢ -0.0008 -0.0078 -0.0002 0 -52 .59 3 -29 .54 5 -0.3864 -2 .56 7 ⎥ ⎢ ⎥ 0.0028 -0.0002 -29 .54 5 -62.333 -3.6228 0.4 852 ⎥ ⎢0.01 05 0.10 15 ⎢ -0.0016 -0.0 15 -0.0004 0 -0.3864 -3.6228 -48.33 32.703 ⎥ ⎢ ⎥ ⎢0.1633 1 .57 28 0.043 -0.0038 -2 .56 7 0.4 852 32.703 -61. 255 ⎥ ⎣ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ eigsLHS = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -88. 959 2 ⎤ -86.9820 ⎥ ⎥ -78.9778 ⎥ ⎥ - 75. 8961 ⎥... ⎤ 39. 451 5 392 .59 68 10.8368 -1.4649 ⎥ ⎥ ; eigP= [ 57 .3 353 66.3033 397.7102 402.2 156 ] -2.3218 10.8368 67.2609 -56 .4314 ⎥ ⎥ 24.7039 -1.4649 -56 .4314 390.7773 ⎥ ⎦ ⎡ 52 .59 26 29 .54 52 0.3864 2 .56 70 ⎤ ⎢ 29 .54 52 62.3324 3.6228 -0.4 852 ⎥ ⎥ ; eigR1 = [ 21.3032 27.3683 86.9363 88.9010] R1 = ⎢ ⎢ 0.3864 3.6228 48.3292 -32.7030 ⎥ ⎢ ⎥ ⎢ 2 .56 70 -0.4 852 -32.7030 61. 254 8 ⎥ ⎣ ⎦ ⎡ -84 .50 15 -36.7711 6.6 350 -31.9983 ⎤ BTP... ⎥ 5. 4812 -12 .58 12 ⎥ ⎦ ⎣ -4.2000 - 0.6000i ⎦ ⎡ -0.1 858 ⎤ ⎡ eigA0hat = ⎢ ; eigA1hat = ⎢ 0.0000 ⎥ ⎣ ⎦ ⎣ ⎡ ⎢ ⎢ ⎢ lhs = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ P= ⎢ ⎣ 0 ⎤ ⎡ ; eigA2hat = ⎢ 0.0393 ⎥ ⎦ ⎣ -0.70 85 -0 .57 11 -0.00 85 -0.00 85 0.0169 -0.00 85 ⎤ -0 .57 11 -0.8 257 0.0084 0.0084 -0.0167 0.0084 ⎥ ⎥ ⎥ -0.00 85 0.0084 -1.0414 -0.2 855 0 0 ⎥ ; eigsLHS = -0.00 85 0.0084 -0.2 855 -1.1000 0 0 ⎥ 0.0169 -0.0167 0 0 -1.0414 -0.2 855 ⎥ ⎥ -0.00 85. .. 1. 750 0 0. 250 0 0.8000 ⎥ ; A1hat = ⎢ ⎥ ⎢ -7.0038 -0.6413 -3.30 95 ⎥ ⎣ ⎦ 0 0 ⎡ -1.0000 ⎤ ⎢ ⎥ ⎢ -0.1000 0. 250 0 0.2000 ⎥ ⎢ 1 .51 39 -0.6413 -0 .51 30 ⎥ ⎣ ⎦ ⎡ -0 .52 98 + 0 .53 83i ⎤ eigA0hat = ⎢ -0 .52 98 - 0 .53 83i ⎥ ; eigA1hat = ⎢ ⎥ ⎢ 0.0000 ⎥ ⎣ ⎦ -0.2630 -0.0000 -1.0000] G= [ 3.3240 10. 758 3 3.24 05] ; Geq = [ [ 1. 257 3 0 1.0000 ⎡ 2.0000 ⎤ ⎢ A0til = ⎢ 1. 750 0 0. 250 0 0.8000 ⎥ ; eigA0til = ⎥ ⎢ -10.3278 -11.3996 -6 .55 00... -0.00 85 0.0084 0 0 -0.2 855 -1.1000 ⎥ ⎦ 2.0633 0.7781 0.7781 ⎤ ⎡ ; eigP= ⎢ 0. 459 2 ⎥ ⎦ ⎣ ⎡ eigR1 = ⎢ ⎣ BTP = [ 0.1438 ⎤ ⎡ ; R1= ⎢ 2.3787 ⎥ ⎦ ⎣ 0.7837 ⎤ ⎡ ; eigR2 = ⎢ 1. 357 8 ⎥ ⎦ ⎣ 1.0414 0.2 855 0.0000 ⎤ 0.1304 ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.2 855 ⎤ ⎡ ; R2= ⎢ 1.1000 ⎥ ⎦ ⎣ -1. 358 1 ⎤ -1. 357 8 ⎥ ⎥ -1.3412 ⎥ ⎥ -0.7848 ⎥ -0.7837 ⎥ ⎥ -0.1916 ⎥ ⎦ 1.0414 0.2 855 0.2 855 ⎤ 1.1000 ⎥ ⎦ 0.7837 ⎤ 1. 357 8 ⎥ ⎦ 2.8414 1.2373] ;... Linearization 161 2 1 .5 1 control input 0 .5 0 -0 .5 -1 -1 .5 -2 -2 .5 -3 0 5 10 15 time/s Fig 8 The control input τ 5 Acknowledgements This work is supported by National Nature Science Foundation under Grant No 60940018 6 References Basin, M.; Rodriguez-Gonzaleza, J.; Fridman, L (2007) Optimal and robust control for linear state-delay systems Journal of the Franklin Institute Vol.344, pp.830–8 45 Chen, W D.; Tang,... Systems with Time-delay Information and Control Vol.38, No.1, pp.87-92 Tang, G Y.; Gao, D X (20 05) Feedforward and feedback optimal control for nonlinear systems with persistent disturbances Control and Decision Vol.20, No.4, pp 366371 Tang, G Y (20 05) Suboptimal control for nonlinear systems: a successive approximation approach Systems and Control Letters Vol 54 , No .5, pp.429-434 Tang, G Y.; Zhao, Y... Modelling Practice and Theory Vol. 15, pp.801-816 Laghrouche, S.; Plestan, F.; Glumineau, A (2007) Higher order sliding mode control based on integral sliding mode Automatica Vol 43, pp .53 1 -53 7 162 Robust Control, Theory and Applications Lee, J H (2006) Highly robust position control of BLDDSM using an improved integral variable structure system Automatica Vol.42, pp.929-9 35 Lin, F J.; Chou, W D (2003)... 0.3 452 0.04 85 ⎦ ⎡ -0.7162 0.1038 0.1187 0. 056 1 ⎤ ⎢ -0.9910 -0.0 454 -0.0024 0.0379 ⎥ ⎥ A0til = ⎢ ⎢ -0.1130 0.0134 -0.7384 -0.1 056 ⎥ ⎢ ⎥ 0 1.0000 0 ⎢ 0 ⎥ ⎣ ⎦ eigA0til = [ -0 .5+ 0.082i -0 .5- 0.082i -0.3 -0.2 ] eigA0hat = [ -0.0621 -0.0004 -0.0000 -0.0000 ] ⎡ 0. 257 7 ⎤ ⎢ -0.0000 + 0.0000i ⎥ ⎥ eigA1hat = 1.0e-003 * ⎢ ⎢ -0.0000 - 0.0000i ⎥ ⎢ ⎥ ⎢0 ⎥ ⎣ ⎦ ⎡ ⎢ P= ⎢ ⎢ ⎢ ⎢ ⎣ 72.9293 39. 451 5 -2.3218 24.7039 ⎤ 39. 451 5 . maintain on it thereafter. Robust Control, Theory and Applications 152 Proof. Utilizing T (1/2)Vss= as a Lyapunov function candidate, and taking the Assumption 5 and Assumption 6, we have. Systems Based on Feedback Linearization 161 0 5 10 15 -3 -2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 time/s control input Fig. 8. The control input τ 5. Acknowledgements This work is supported by. sliding mode control based on integral sliding mode. Automatica. Vol. 43, pp .53 1 -53 7. Robust Control, Theory and Applications 162 Lee, J. H. (2006). Highly robust position control of BLDDSM