New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems 229 ( ) , 0 T Vx xPx P = > (43) The derivative of (43) becomes 00 0 0 () [ (,) (,)] (,) (,) TT TT T Vx x f xt P P f xt x u g xtPx xP g xtu=+++  (44) By means of the Lyapunov control theory(Khalil, 1996), take the control input as 0 (,) TT u g xtP y BP y =− =− (45) and (,) 0Qxt > and (,) 0 c Qxt> for all n xR∈ and all 0t ≥ is 00 (,) (,) (,) T f xtP P f xt Qxt+=− (46) 00 (,) (,) (,) T ccc f xtP P f xt Q xt+=− (47) then {} 00 min () (,) [ ( , ) ] [ ( , ) ( , )] ( , ) ( , ) TTTTTT TTTT TT cc T c c V x x Q x t x x C PBB Px x PBB PCx x Q x t C PBB P PBB PC x xfxtPPfxtx xQ xtx Qxt x λ =− − − =− + + =− + =− ≤−  (48) Therefore the stable gain is chosen as 1 110 ( ) or ( ) ( , ) T G y BP HCB HC f xt − == (49) 2.3.2 Output feedback discontinuous control input A corresponding output feedback discontinuous control input is proposed as follows: 01020 () ( )uGyyGyGSGsignS = −−Δ−− (50) where ()Gy is a nonlinear output feedback gain satisfying the relationship (37) and (49), GΔ is a switching gain of the state, 1 G is a feedback gain of the output feedback integral sliding surface, and 2 G is a switching gain, respectively as [ ] 1, , i Gg i qΔ=Δ = (51) { } {} {} {} 11 11 110 0 11 11 110 0 max ( ) ''( . ) ( ) ( , ) ( ) 0 min min ( ) ''( . ) ( ) ( , ) ( ) 0 min i i i i i HCB HC f xt IHCB H f xt sign S y II g HCB HC f xt IHCB H f xt sign S y II −− −− ⎧ Δ+Δ ≥> ⎪ +Δ ⎪ Δ= ⎨ Δ+Δ ⎪ ≤< ⎪ +Δ ⎩ (52) 1 0G > (53) Recent Advances in Robust Control – Novel Approaches and Design Methods 230 { } 2 max | ''( , )| min{ } dxt G II = +Δ (55) The real sliding dynamics by the proposed control (50) with the output feedback integral sliding surface (35) is obtained as follows: 1 01 1 0 1 110 1 1 1 0 1 110 1 0 1 11 1 1 11 ()[ ] ( )[ (,) (,) ( (,)) (,) ] ( ) [ ( , ) ( ) ] ( ) [ ( , ) ( , ) ( ) ] ( ) [ SHCBHyHy HCB HC f xtx HC f xt HCB g xt u HCdxt H y HCB HCf xtx HCBGyy Hy HCB HC f xt HC gxtKy y HCB HC − − − − − =+ =+Δ++Δ++ =−+ +Δ+Δ +   10 2 0 1 1 11 1 10 2 0 11 11 110 ( ( , ))( ( ) ( , )] ( ) [ ''( , ) ( , ) ( ) ] ( ) ( ) [( )( ( )) ''( , )] ( ) ''( , ) ( ) ( , ) ( ) Bgxt GyGSGsignS HCdxt HCB HC f xtCx HC gxtG y y I I Gyy IIGSGsignS dxt HCBHCfxty IHCBHfxty I I − −− +Δ −Δ − − + =Δ+Δ−+ΔΔ ++Δ− − + =Δ+Δ −+Δ 10 2 0 () ( )( ( )) ''( , ) Gyy IIGSGsignS dxt Δ ++Δ− − + (56) The closed loop stability by the proposed control input with the output feedback integral sliding surface together with the existence condition of the sliding mode will be investigated in next Theorem 1. Theorem 2: If the output feedback integral sliding surface (35) is designed to be stable, i.e. stable design of ()Gy , the proposed control input (50) with Assumption A1-A10 satisfies the existence condition of the sliding mode on the output feedback integral sliding surface and closed loop exponential stability. Proof; Take a Lyapunov function candidate as 00 1 () 2 T Vy SS= (57) Differentiating (57) with respect to time leads to and substituting (56) into (58) 00 11 01 1 1 10 0 010200 2 10 10 () [( ) ''( , ) ( ) ( , )] ( ) ( ) ( )( ( )) ''( , ) || || , min{|| ||} T T T TT T Vy S S S HCB HC f xt I HCB H f xt y S I I Gyy SI I GS GsignS Sdxt GS I I GSS εε ε −− = =Δ+Δ −+ΔΔ ++Δ−− + ≤− = +Δ =−   0 1 2 ( )GV y ε =− (58) From (58), the second requirement to get rid of the reaching phase is satisfied. Therefore, the reaching phase is clearly removed. There are no reaching phase problems. As a result, the real output dynamics can be exactly predetermined by the ideal sliding output with the matched uncertainty. Moreover from (58), the following equations are obtained as 1 () 2 () 0Vy GVy ε + ≤  (59) New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems 231 1 2 (()) ((0)) Gt Vyt Vy e ε − ≤ (60) And the second order derivative of ()Vx becomes 21 00 00 0 0 1 1 0 ( ) || || ( ) ( ) TT Vy SS SS S S HCB HCx HCx − = += + + <∞       (61) and by Assumption A5 ( )Vx  is bounded, which completes the proof of Theorem 2. 2.3.3 Continuous approximation of output feedback discontinuous control input Also, the control input (50) with (35) chatters from the beginning without reaching phase. The chattering of the discontinuous control input may be harmful to the real dynamic plant so it must be removed. Hence using the saturation function for a suitable 0 δ , one make the part of the discontinuous input be continuous effectively for practical application as 0 01020 00 () { ( )} || c S uGyyGSGyGsignS S δ =− − − Δ + + (62) The discontinuity of control input of can be dramatically improved without severe output performance deterioration. 3. Design examples and simulation studies 3.1 Example 1: Full-state feedback practical integral variable structure controller Consider a second order affine uncertain nonlinear system with mismatched uncertainties and matched disturbance 2 11 1 12 1 0.1 sin ( ) 0.02sin(2.0 )xx x xx xu=− + + +  2 222 2 sin ( ) (2.0 0.5sin(2.0 )) ( , )xxx x tudxt=+ + + +  (63) 22 1212 ( , ) 0.7sin( ) 0.8sin( ) 0.2( ) 2.0sin(5.0 ) 3.0dxt x x x x t=−+++ + (64) Since (63) satisfy the Assumption A1, (63) is represented in state dependent coefficient form as 2 1 11 1 2 2 2 2 0 0.02sin( ) 10.1sin() 1 2.0 0.5sin(2.0 ) 01sin() (,) xxx x u xxt x dxt ⎡ ⎤ −+ ⎡⎤ ⎡⎤ ⎡⎤⎡ ⎤ =⋅++ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥⎢ ⎥ + + ⎣⎦ ⎣⎦⎣ ⎦ ⎣⎦ ⎣ ⎦   (65) where the nominal parameter 0 (,) f xt and 0 (,)gxt and mismatched uncertainties (,) f xtΔ and (,)gxtΔ are 2 1 00 2 2 1 11 0 0.1sin ( ) 0 (,) , (,) , (,) 01 2.0 0sin() 0.02sin( ) (,) 0.2sin(2.0 ) x fxt gxt fxt x x gxt t − ⎡ ⎤ ⎡⎤ ⎡⎤ ==Δ= ⎢ ⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎣ ⎦ ⎡⎤ Δ= ⎢⎥ ⎣⎦ (66) Recent Advances in Robust Control – Novel Approaches and Design Methods 232 To design the full-state feedback integral sliding surface, (,) c f xt is selected as 00 11 (,) (,) (,)() 70 21 c fxt fxt gxtKx − ⎡ ⎤ =− = ⎢ ⎥ −− ⎣ ⎦ (67) in order to assign the two poles at 16.4772 − and 5.5228 − . Hence, the feedback gain ()Kx becomes [ ] ( ) 35 11Kx = (68) The P in (14) is chosen as 100 17.5 0 17.5 5.5 P ⎡⎤ = > ⎢⎥ ⎣⎦ (69) so as to be 2650 670 (,) (,) 0 670 196 T cc fxtPPfxt −− ⎡⎤ + =< ⎢⎥ −− ⎣⎦ (70) Hence, the continuous static feedback gain is chosen as [ ] 0 () (,) 35 11 T Kx g xtP== (71) Therefore, the coefficient of the sliding surface is determined as [ ] [ ] 11112 10 1LLL== (72) Then, to satisfy the relationship (8a) and from (8b), 0 L is selected as [ ] [ ] [ ] 010 0 1 11121112 (,) (,)() (,) 70 21 80 11 c LLfxtgxtKxLfxtLLLL=− − =− = + − + = (73) The selected gains in the control input (21), (23)-(25) are as follows: 1 1 1 4.0 if 0 4.0 if 0 f f Sx k Sx + > ⎧ ⎪ Δ= ⎨ − < ⎪ ⎩ (74a) 2 2 2 5.0 if 0 5.0 if 0 f f Sx k Sx + > ⎧ ⎪ Δ= ⎨ − < ⎪ ⎩ (74b) 1 400.0K = (74c) 22 212 2.8 0.2( )Kxx=+ + (74d) The simulation is carried out under 1[msec] sampling time and with [] (0) 10 5 T x = initial state. Fig. 1 shows four case 1 x and 2 x time trajectories (i)ideal sliding output, (ii) no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv)unmatched New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems 233 uncertainty and matched disturbance. The three case output responses except the case (iv) are almost identical to each other. The four phase trajectories (i)ideal sliding trajectory, (ii)no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv) unmatched uncertainty and matched disturbance are depicted in Fig. 2. As can be seen, the sliding surface is exactly defined from a given initial condition to the origin, so there is no reaching phase, only the sliding exists from the initial condition. The one of the two main problems of the VSS is removed and solved. The unmatched uncertainties influence on the ideal sliding dynamics as in the case (iv). The sliding surface ( ) f St (i) unmatched uncertainty and matched disturbance is shown in Fig. 3. The control input (i) unmatched uncertainty and matched disturbance is depicted in Fig. 4. For practical application, the discontinuous input is made be continuous by the saturation function with a new form as in (32) for a positive 0.8 f δ = . The output responses of the continuous input by (32) are shown in Fig. 5 for the four cases (i)ideal sliding output, (ii)no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv)unmatched uncertainty and matched disturbance. There is no chattering in output states. The four case trajectories (i)ideal sliding time trajectory, (ii)no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv) unmatched uncertainty and matched disturbance are depicted in Fig. 6. As can be seen, the trajectories are continuous. The four case sliding surfaces are shown in fig. 7, those are continuous. The three case continuously implemented control inputs instead of the discontinuous input in Fig. 4 are shown in Fig. 8 without the severe performance degrade, which means that the continuous VSS algorithm is practically applicable. The another of the two main problems of the VSS is improved effectively and removed. From the simulation studies, the usefulness of the proposed SMC is proven. Fig. 1. Four case 1 x and 2 x time trajectories (i)ideal sliding output, (ii) no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv)unmatched uncertainty and matched disturbance Recent Advances in Robust Control – Novel Approaches and Design Methods 234 Fig. 2. Four phase trajectories (i)ideal sliding trajectory, (ii)no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv) unmatched uncertainty and matched disturbance Fig. 3. Sliding surface ( ) f St (i) unmatched uncertainty and matched disturbance New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems 235 Fig. 4. Discontinuous control input (i) unmatched uncertainty and matched disturbance Fig. 5. Four case 1 x and 2 x time trajectories (i)ideal sliding output, (ii) no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv)unmatched uncertainty and matched disturbance by the continuously approximated input for a positive 0.8 f δ = Recent Advances in Robust Control – Novel Approaches and Design Methods 236 Fig. 6. Four phase trajectories (i)ideal sliding trajectory, (ii)no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv) unmatched uncertainty and matched disturbance by the continuously approximated input Fig. 7. Four sliding surfaces (i)ideal sliding surface, (ii)no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv) unmatched uncertainty and matched disturbance by the continuously approximated input New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems 237 Fig. 8. Three case continuous control inputs f c u (i)no uncertainty and no disturbance (ii)matched uncertainty/disturbance, and (iii) unmatched uncertainty and matched 3.2 Example 2: Output feedback practical integral variable structure controller Consider a third order uncertain affine nonlinear system with unmatched system matrix uncertainties and matched input matrix uncertainties and disturbance 2 11 1 2 2 22 32 33 1 33sin() 1 0 0 0 011 0 0 1 0.5sin ( ) 0 2 0.4sin ( ) 2 0.3sin(2 ) (,) xx x xxu xx xxt dxt ⎡ ⎤ −− ⎡⎤ ⎡⎤ ⎡⎤⎡ ⎤ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥⎢ ⎥ =− ++ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥⎢ ⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥⎢ ⎥ ++ + ⎣⎦ ⎣⎦⎣ ⎦ ⎣⎦ ⎣ ⎦    (75) 1 2 3 100 001 x y x x ⎡ ⎤ ⎡⎤ ⎢ ⎥ = ⎢⎥ ⎢ ⎥ ⎣⎦ ⎢ ⎥ ⎣ ⎦ (76) 22 11213 ( , ) 0.7sin( ) 0.8sin( ) 0.2( ) 1.5sin(2 ) 1.5dxt x x x x t=−++++ (77) where the nominal matrices 0 (,) f xt , 0 (,)gxt B= and C , the unmatched system matrix uncertainties and matched input matrix uncertainties and matched disturbance are 2 1 0 22 23 310 0 3sin()0 0 100 (,) 0 1 1, 0, C , 0 0 0 001 102 2 0.5sin()00.4sin() x fxt B f xx −− ⎡ ⎤ ⎡⎤⎡⎤ ⎡⎤ ⎢ ⎥ ⎢⎥⎢⎥ =− = = Δ= ⎢⎥ ⎢ ⎥ ⎢⎥⎢⎥ ⎣⎦ ⎢ ⎥ ⎢⎥⎢⎥ ⎣⎦⎣⎦ ⎣ ⎦ Recent Advances in Robust Control – Novel Approaches and Design Methods 238 1 00 (,) 0 , (,) 0 0.3sin(2 ) (,) gxt dxt t dxt ⎡ ⎤ ⎡⎤ ⎢ ⎥ ⎢⎥ Δ= = ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎣⎦ ⎣ ⎦ . (78) The eigenvalues of the open loop system matrix 0 (,) f xt are -2.6920, -2.3569, and 2.0489, hence 0 (,) f xt is unstable. The unmatched system matrix uncertainties and matched input matrix uncertainties and matched disturbance satisfy the assumption A3 and A8 as 2 1 1 22 23 3sin ( ) 0 1 " 0 0 , 0.15sin(2 ) 0.15 1, "( , ) ( , ) 2 0.5sin ( ) 0.4sin ( ) x f It dxtdxt xx ⎡⎤ − ⎢⎥ Δ= Δ= ≤ < = ⎢⎥ ⎢⎥ ⎣⎦ (79) disturbance by the continuously approximated input for a positive 0.8 f δ = To design the output feedback integral sliding surface, (,) c f xt is designed as 00 31 0 (,) (,) () 0 1 1 19 0 30 c fxt fxt BGyC − ⎡ ⎤ ⎢ ⎥ =− =− ⎢ ⎥ ⎢ ⎥ −− ⎣ ⎦ (80) in order to assign the three stable pole to (,) c fxt at 30.0251 − and 2.4875 0.6636i − ± . The constant feedback gain is designed as [ ] { } 1 () 2 [1 0 2] 19 0 30GyC − =−− (81) [ ] () 10 16Gy∴= (82) Then, one find [ ] 11112 Hhh= and [ ] 00102 Hhh= which satisfy the relationship (37) as 11 01 12 02 12 0, 19 , 30hhhhh = == (83) One select 12 1h = , 01 19h = , and 02 30h = . Hence 112 22HCB h = = is a non zero satisfying A4. The resultant output feedback integral sliding surface becomes [] [ ] 101 0 202 1 0 1 19 30 2 yy S yy ⎧ ⎫ ⎡ ⎤⎡⎤ ⎪ ⎪ =+ ⎨ ⎬ ⎢ ⎥⎢⎥ ⎪ ⎪ ⎣ ⎦⎣⎦ ⎩⎭ (84) where 01 1 0 () t yy d τ τ = ∫ (85) 02 2 2 0 () (0)/30 t yydy ττ =− ∫ (86) The output feedback control gains in (50), (51)-(55) are selected as follows: [...]... unmatched uncertainty and matched disturbance by the continuously approximated input Fig 16 Three case continuous control inputs u0 c (i)no uncertainty and no disturbance (ii)matched uncertainty/disturbance, and (iii) unmatched uncertainty and matched disturbance by the continuously approximated input for a positive δ 0 = 0.02 244 Recent Advances in Robust Control – Novel Approaches and Design Methods... Linear Case," KIEE, vol. 59, no .9, pp.1680-1685 Lee, J H., (2010c) A MIMO VSS with an Integral-Augmented Sliding Surface for Uncertain Multivariable Systems ," KIEE, vol. 59, no.5, pp .95 0 -96 0 Lijun, L & Chengkand, X., (2008) Robust Backstepping Design of a Nonlinear Output Feedback System, Proceeding of IEEE CDC 2008, pp.5 095 -5 099 Lu, X Y & Spurgeon, S K ( 199 7) Robust Sliding Mode Control of Uncertain... Regulation via Sliding Mode for Nonlinear Systems, System & Control Letters, vol.24, pp.361-371 Utkin, V I ( 197 8) Sliding Modes and Their Application in Variable Structure Systems Moscow, 197 8 246 Recent Advances in Robust Control – Novel Approaches and Design Methods Vidyasagar, M ( 198 6) New Directions of Research in Nonlinear System Theory Proc of the IEEE, Vol.74, No.8, ( 198 6), pp.1060-1 091 Wang, Y.,... controllers without the reaching phase in this chapter can be practically applicable to the real dynamic plants 240 Recent Advances in Robust Control – Novel Approaches and Design Methods Fig 9 Four case two output responses of y1 and y2 (i)ideal sliding output, (ii) with no uncertainty and no disturbance, (iii)with matched uncertainty and matched disturbance, and (iv) with ummatched uncertainty and. .. Slottine, J J E & Li, W., ( 199 1) Applied Nonlinear Control, Prentice-Hall Sun, Y M (20 09) Linear Controllability Versus Global Controllability," IEEE Trans Autom Contr, AC-54, no 7, pp.1 693 -1 697 Tang, G Y., Dong, R., & Gao, H W (2008) Optimal sliding Mode Control for Nonlinear System with Time Delay Nonlinear Analysis: Hybrid Systems, vol.2, pp 891 - 899 Toledo, B C., & Linares, R C., ( 199 5) On Robust. .. 241 242 Recent Advances in Robust Control – Novel Approaches and Design Methods Fig 13 Four case y 1 and y 2 time trajectories (i)ideal sliding output, (ii) no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv)unmatched uncertainty and matched disturbance by the continuously approximated input for a positive δ 0 = 0.02 Fig 14 Four phase trajectories (i)ideal sliding trajectory,... control scheme with an integral action (Seraj and Tarokh, 197 7), (Freeman and Kokotovic, 199 5) and they are based on the choice of a 248 Recent Advances in Robust Control – Novel Approaches and Design Methods suitable set of reference poles, on a proportionality parameter of these poles and on the theory of externally positive systems (Bru and Romero-Vivò, 20 09) The utility and efficiency of the proposed... experimentally by using e.g Matlab command invfreqs); on the contrary, it is easy to transform the interval uncertainties of A , B, C into the ones (even if more conservative) of ai , b 250 Recent Advances in Robust Control – Novel Approaches and Design Methods Moreover note that almost always the controller is supplied with an actuator having gain g a In this case it can be posed b ← bg a and also consider... values of s , ts , ta , ωs , K v (intensively studied in the optimization theory) are well-known and/ or easily computing (Butterworth, 193 0), (Paarmann, 2001) 4 Second main result The following fundamental result, that is the key to design a robust controller satisfying the required specifications 2., is stated 256 Recent Advances in Robust Control – Novel Approaches and Design Methods Theorem 7 Consider... parametric uncertainties and non standard disturbances, which need to be efficiently controlled Indeed, e.g consider the numerous manufacturing systems (in particular the robotic and transport systems,…) and the more pressing requirements and control specifications in an ever more dynamic society Despite numerous scientific papers available in literature (Porter and Power, 197 0)-(Sastry, 199 9), some of which . Application in Variable Structure Systems. Moscow, 197 8. Recent Advances in Robust Control – Novel Approaches and Design Methods 246 Vidyasagar, M. ( 198 6). New Directions of Research in Nonlinear. uncertainty/disturbance, and (iii) unmatched uncertainty and matched disturbance by the continuously approximated input for a positive 0 0.02 δ = Recent Advances in Robust Control – Novel Approaches. 0.8 f δ = Recent Advances in Robust Control – Novel Approaches and Design Methods 236 Fig. 6. Four phase trajectories (i)ideal sliding trajectory, (ii)no uncertainty and no disturbance