Differential Neural Networks Observers: development, stability analysis and implementation 73 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10 -3 Time [s] mole x 3 min x 3 DNN Observer without projection x 3 Projectional DNN Observer x 3 Figure 4. Estimation of x 3 (t) (2 s) 0 2 4 6 8 10 12 14 16 18 20 -2 -1 0 1 2 3 4 5 6 7 x 10 -3 Time [s] mol e x 3 x 3 Project ional DNN Observer x 3 DNN Observer without projection Figure 5. Estimation of x 3 (t) (20 s) Systems, Structure and Control 74 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 10 -4 Time [s] g/g soil x 4 DNN Observer without projection x 4 Projectional DNN Observer x 4 x 4 min x 4 max Figure 6. Estimation of x 4 (t) (1 s) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 10 -4 Ti me [ s] g/g soil x 4 x 4 Project ional DNN Observer x 4 DNN Observer without projection Figure 7. Estimation of x 4 (t) (5 s) As it can be seen, the projectional DNNO has significantly better quality in state estimation, especially in the beginning of the process, when negative values and over-estimation have been obtained by a non-projectional DNNO. Differential Neural Networks Observers: development, stability analysis and implementation 75 6. Conclusion and future work The complete convergence analysis for this class of adaptive observer is presented. Also the boundedness property of the adaptive weights in DNN was proven. Since the projection method leads to discontinuous trajectories in the estimated states, a nonstandard Lyapunov - Krasovski functional is applied to derive the upper bound for estimation error (in "average sense"), which depends on the noise power (output and dynamics disturbances) and on an unmodelled dynamic. It is shown that the asymptotic stability is attained when both of these uncertainties are absent. The illustrative example confirms the advantages, which the suggested observers have being compared with traditional ones. Appendix (proof of Theorem 2) Evidently that () () () )tht(tLhtt −− ′ ≤−− ′ δ δδ () () () ηη η η η η ηη η η η ϒ − Λ≤ Λ − Λ ≤ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Λ − ΛΛ= 21 1 2 1 21121 / )t( t / ,t / t ξξ ξ ϒ − Λ≤ 21 1 / )t( () 21 2 1 10 21 1 / f ~ txf ~ f ~ / f ~ )t(f ~ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Λ + − Λ≤ where () () () txtx ˆ :tδ ′ − ′ = ′ is the state estimation error at time t . Consider the next "nonstandard" Lyapunov-Krasovskii ("energetic") function () () () () {} τ dτW ~ τ T W ~ trτk p δ(τ) t tht V(t) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∫ − = 2 where .W ˆ W(τ((ττ)W ~ −= Since the problem under consideration contains uncertainties and external output disturbances we won't demonstrate that the time-derivative of this energetic function is strictly negative. Instead, we will use it to obtain an upper bound for the averaged state estimation error. Taking time derivative of Lyapunov-Krasovski function and considering the property (5), the assumption A2, and in view of (29) we have: Systems, Structure and Control 76 () () () [] () () () () {} () () () () {} () () () {} () () () () {} () () () [] () () ( ) [] () () () () {} () () () () {} () () () {} () () () () {} ) )(+W ˆ +W ˆ +x ++++ ) t )(x- )x ˆ x ˆ tht(W ~ )tht(W ~ thtktW ~ tW ~ tk )tht(W ~ )tht(W ~ thtktW ~ tW ~ tk tht-d)()(f ~ )(u)(x()t(x)t(Ax-))t(h-t( d))(C-)((K)(u)(x ˆ )((W)(x ˆ )(W)(x ˆ A))t(h-t(x ˆ tht(W ~ )tht(W ~ thtktW ~ tW ~ tk )tht(W ~ )tht(W ~ thtktW ~ tW ~ tk )ht( p t d))t(x ˆ C-)(+)(Cx(K+)(u )()((W+)()(W+)(x ˆ A t ht +))t(h-t(x ˆ )t(V dt d TT TT p p t tht t tht TT TT p t X −−−−+ −−−−+ −+ ≤−−−− +−−−− +− − −= ≤ −= −= ∫− ∫ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∫ 222222 111111 2 2 21 21 222222 111111 2 2 21 trtr trtr trtr trtr δττξτττϕσ ττδτηττϕττσττ δ ττητττϕττσττ τ π τ τ Taking into account that () bPa, P b P a P ba 2 222 ++=+ Defining: ()() ()() x(t)(t)x ˆ :(t) ~ x(t)σ(t)x ˆ σ(t):σ ~ , i i W ˆ (t) i W(t): i W ~ KC,A:A ~ ϕϕϕ −= −= =−= −= 21 we derive () () () () () {} ()() () () {} () () () {} ()() () () {} )th(tW ~ )th(t T W ~ trthtktW ~ t T W ~ trtk )th(tW ~ )th(t T W ~ trthtktW ~ t T W ~ trtk tβtαV −−−− +−−−− ++≤ 222222 111111  where: () () () () [ ] () ( ) () () () [ ] ) τττξτηττϕ ττϕττστσττδ τ δβ τττξτηττϕ ττϕττστσττδ τ α d)(f ~ )()(K)(u)( ~ W ˆ )(u)(x ˆ )((W ~ )( ~ W ˆ )(x ˆ )(W ~ )(A ~ t tht ,htP:t P d)(f ~ )()(K)(u)( ~ W ˆ )(u)(x ˆ )((W ~ )( ~ W ˆ )(x ˆ )(W ~ )( A ~ t tht :t −−++ ⎜ ⎜ ⎝ ⎛ +++ ∫ −= −= −−+ ++++ ∫ −= = 2 211 2 2 2 211 Differential Neural Networks Observers: development, stability analysis and implementation 77 The term () tβ is expanded as () ()() () () ()() () () () ()() () () ()() () () ()() () ()() () () ()() ()() () () ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ −= −− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∫ −= −+ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ∫ −= −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ −= −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ −= −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ −= − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∫ −= −= ττ τ δττξτη τ δ τττϕ τ δ τττϕτ τ δ ττσ τ δττστ τ δ ττδ τ δβ df ~ t tht ,thtPdK t tht ,thtP d)(u)( ~ W ˆ t tht ,thtP d)(u)(x ˆ )((W ~ t tht ,thtP d ~ W ˆ t tht ,thtPdx ˆ W ~ t tht ,thtP dA ~ t tht ,thtP t 22 2 2 2 2 2 1 22 2 Similarly, we can estimate t α by the Jensen's inequality we get () () () () [ ] () () () () () () () () ⎪ ⎭ ⎪ ⎬ ⎫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +++ ∫ −= + ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +++ ∫ −= ≤−−++ +++ ∫ −= = ττξττηττϕ τ τττϕττστσττδ τ τττξτηττϕ ττϕττστσττδ τ α d p p f ~ p K p )(u)( ~ W ˆ t tht d p )(u)(x ˆ )((W ~ p )( ~ W ˆ p )(x ˆ )(W ~ p A ~ t tht P d)(f ~ )()(K)(u)( ~ W ˆ )(u)(x ˆ )((W ~ )( ~ W ˆ )(x ˆ )(W ~ )(A ~ t tht :t 2 2 2 2 2 2 2 2 1 2 1 2 8 2 2 211 Each term of t α and )t( β is upper bounded, next facts are used. Norm inequality AB ≤ BA and the matrix inequality T Y Y Λ T X Λ Λ T Y X T X Y 1− +≤+ valid for any sr RYX, × ∈ and any ssT RΛΛ × ∈=<0 (Poznyak, 2001). It also necessary to represents the state estimation error t δ as a function of the available output, the estimation error t e : () () () () () () () () ()() () () () () () () () () () tIC T Ctt T Cte T C tttC T Ct T Cte T C ttC T Cte T C ttCxtx ˆ Ctyty ˆ te δϖϖδη ϖδϖδδη ηδ η ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +=++− −+=+− −=− −−=−=− Giving () () () () ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ++−= tt T Cte T CNt ϖδη ϖ δ Systems, Structure and Control 78 where: 1− ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ += IC T C:N ϖ ϖ and ϖ is a small positive scalar. Taking into account all these facts next estimation is obtained: () () () ( ) () + − ⎥ ⎦ ⎤ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ϒ+++ϒ+ ⎟ ⎠ ⎞ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − + − + − + − ⎢ ⎣ ⎡ ++ − ≤ tht δQΛΛIL u μ σ Lμμ u LΛ σ LΛ PΛΛ T W ˆ ΛW ˆ T W ˆ ΛW ˆ ΛP A ~ PP T A ~ T t ht δth)t(V dt d 0 1 7 1 3 2 321 2 85 1 10 1 92 1 821 1 51 1 1 ϖ ϕϕ () () () () () () () () ()() () ()() () () () ()() dττx ˆ στW ~ PNΛCCΛPNτ T W ~ τx ˆ T σ t thtτ dττx ˆ στW ~ PCNtht T e t thtτ tht δQ T tht δ ηξξ ΛP Diam(x) f ~ Λf ~ f ~ f ~ ΛP η / η ΛPK f ~ Λ txf ~ f ~ f ~ ΛΛ ξ / ξ Λ η / η ΛKΛh(t) δ L u L Λ δ LL u μ δ L μ δ L σ L μ δ L σ L Λ δ L A ~ Λth ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∫ −= + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∫ −= + ⎥ ⎦ ⎤ −− −ϒ+ϒ − ⎟ ⎟ ⎠ ⎞ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − +ϒ − ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ϒ − +ϒ − + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ϒ + ϒ ++++ 1321 1 2 0 2 1 21 10 1 21 1 2 2 1 10 1 10 2 21 1 21 1 9 3 22 8 3 22 3 3 2 1 3 2 2 3 2 5 4 2 2 1 3 ϖ ϖ ϖ ϖ ϕϕ () () () () () () {} ()() () () {} () () () () () ()() () ( ) () () () () () () {} ()() () () {} )th(t 2 W ~ )th(t T 2 W ~ trtht 2 kt 2 W ~ t T 2 W ~ trt 2 k )dτu()(x ˆ )(( 2 W ~ )P( T 2 W ~ T )(x ˆ )(( T u t thtτ )dτu()(x ˆ )(( 2 W ~ PN 7 ΛC 6 Λ T CPNτ T 2 W ~ τx ˆ T )( T u t thtτ dτ)u()(x ˆ )(( 2 W ~ PCNtht T e2 t thtτ )th(t 1 W ~ )th(t T 1 W ~ trtht 1 k t 1 W ~ t T 1 W ~ trt 1 kdτ τ x ˆ σ τ W ~ )P( T 1 W ~ τ x ˆ T σ t thtτ −−−− + ∫ −= + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∫ −= + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∫ −= +−−− −+ ∫ −= + ττϕτττϕτ ττϕτ ϖ ϖ ϖ ϕτ ττϕτ ϖ τ Differential Neural Networks Observers: development, stability analysis and implementation 79 Considering () () () () ( ) () 0 1 7 1 3 2 321 2 85321 1 10 1 92 1 821 1 51 1 1 1 0 321 1 Q IL u L u LL,,,Q T W ˆ W ˆ T W ˆ W ˆ R ,,,QPPRKA ~ PPK T A ~ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − Λ+ − Λ+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ϒ+++ϒΛ+Λ= − Λ+ − Λ+ − Λ+ − Λ+ − Λ= − ≤+ − ++ ϖ ϕ μ σ μμ ϕσ μμμδ μμμδ implies: () () () () () ()() () ()() ] ()() } () () () {} ()( ) () () {} 0 111111 1 123 2 1 =−−−−+ + ⎩ ⎨ ⎧ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Λ+Λ+− ∫ −= )tht(W ~ )tht( T W ~ trthtktW ~ t T W ~ trtk dx ˆ T x ˆ W ~ x ˆ W ~ PNC T CNthte T CNP T W ~ tr t t ht ττστστ τστ ϖ ϖ ϖϖ τ τ that can be obtained selecting () () () () ()() () ()() ] ()() () ⎭ ⎬ ⎫ − ⎩ ⎨ ⎧ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = tW ~ dt (t)dk τx ˆ T στx ˆ στW ~ +τx ˆ στW ~ PNCΛ T +CΛ+Ntt-he T CNP (t)k - tW dt d 1 1 1 123 2 1 2 1 1 ϖ ϖ ϖϖ Analogously, for the second adaptive law () () () () () ( ) () ] ()() } () () () {} ()() () () {} 0 222222 2 176 2 2 =−−−−+ + ⎩ ⎨ ⎧ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Λ+Λ+− ∫ −= )tht(W ~ )tht( T W ~ thtktW ~ t T W ~ tk dx ˆ T )( T u)(u)(x ˆ )((W ~ )(u)(x ˆ (W ~ PNC T C Nthte T CNP T W ~ t tht trtr tr ττϕτττϕτ ττϕτ ϖ ϖ ϖϖ τ τ leading to () () () () ( ) () ] ()() () ⎭ ⎬ ⎫ − ⎩ ⎨ ⎧ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ΛΛ − − = tW ~ dt )t(dk x ˆ T )( T u)(u)(x ˆ )((W ~ +)(u)(x ˆ (W ~ PN+C T CN+th-te T CNP )t(k tW dt d 2 2 2 176 2 1 2 2 2 τϕτττϕτ ττϕτ ϖ ϖ ϖϖ Systems, Structure and Control 80 Finally: () () () () ⎥ ⎦ ⎤ −− −ϒ+ϒ − Λ+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Λ+ − Λ+ϒ − Λ+ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Λ + − ΛΛ+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ϒ − Λ+ϒ − ΛΛ+ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ϒ Λ+ ϒ +++Λ+Λ≤ tht Q T tht P )x(Diam f ~ f ~ f ~ f ~ P / PK f ~ txf ~ f ~ f ~ // K)t(h L u LLL u LLLLLL A ~ th)t(V dt d δδ ηξξ ηη ξξηη δϕδϕ μ δ μ δσ μ δσδ 0 2 1 21 10 1 21 1 2 2 1 10 1 10 2 21 1 21 1 9 3 22 8 3 22 3 3 2 1 3 2 2 3 2 5 4 2 2 1 3 or in the short form: () () () () () () ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−−+≤ thtQtht T bathth)t(V dt d δδ 0 2 where () ηξξηη ξξηη δϕδϕ μ δ μ δσ μ δσδ ϒ+ϒ − Λ+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Λ+ − Λ+ϒ − Λ+ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Λ + − ΛΛ+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ϒ − Λ+ϒ − ΛΛ= ϒ Λ+ ϒ +++Λ+Λ= 2 121 10 1 21 1 2 2 1 10 1 10 2 21 1 21 1 9 3 22 8 3 22 3 3 2 1 3 2 2 3 2 5 4 2 2 1 P)x(Diam f ~ f ~ f ~ f ~ P / PK f ~ txf ~ f ~ f ~ // K:b L u LLL u LLLLLL A ~ :a So, () () () () () ( ) () thdt )t(dV btahthtQtht T 1 2 0 −+≤−− δδ And integrating, we obtain () () () τ τ τ τ τττδττδ τ d )(h dt )t(dV b)(ah T d)(thQ)(h T T ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ∫≤−− = ∫ 1 2 0 0 0 And hence, () )(h V )(h V th t V )(h V d T d)(h )(h V T + )(h V d T - )(h dV T 0 0 0 0 0 2 000 ≤+−= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∫− ≤ = ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∫= = ∫− τ τ τ ττ τ τ τ τ τ τ τ τ τ This implies () () () () () )(h V bTdthQ T ad)(th)(th T T 0 0 2 0 0 0 ++≤∫ = ∫−− = ττδτδ τ τττ τ Dividing by T and taking the upper limit we finally get (30). 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(Venayagamoorthy et al., 2003, Mohagheghi et al., 2007), fuzzy logic (Yousef & Mohamed, 2004) and normal form analysis (Kshatriya, et al., 20 05, Liu et al., 2006) 84 Systems, Structure and Control All of the mentioned controllers provide larger stability margins with respect to traditional ones But these control schemes were designed for reduced order plant The unmodelled electrical dynamics can affect... expensive control algorithm Moreover, this control scheme does not take into account practical limitation on the magnitude of the excitation voltage, and an observer design problem was not solved On the other hand, sliding mode control (SMC), (Utkin, et al., 1999) is one of the most effective strategies to deal with robust nonlinear controllers SMC enables high accuracy and robustness to disturbances and. .. from ⎤ ⎣ equations (17), (18) and (23) as ⎡ ⎛ s1 ⎞ ⎤ ⎢f1 ( x1 ) − k1ψ1 ( x1 ) − ρ1 (x1 ) sigm ⎜ ⎟ ⎥ z 2 = B2x2 + ⎢ ⎝ ε1 ⎠ ⎥ := ψ 2 ( x2 ) ⎢ ⎥ 0 ⎣ ⎦ ( 25) 90 Systems, Structure and Control where x2 = [ x1 x 2 ] The procedure describe above can be achieved in the ith block of ( 15) as T follows Step i At this step, the dynamics of the transformed ith block of the system ( 15) are given by z i = fi ( xi... 1999), and nested sliding mode control (Adhami-Mirhosseini and Yazdanpanah, 20 05) The block control technique is used to design a nonlinear sliding surface in such a way that the sliding mode dynamics are represented by a linear system with desired eigenvalues The integral sliding mode control combined with nested control technique are applied to reject perturbations The controller designed in this way... electrical network and loads models) Section 3 deals with the problem of nonlinear robust controller for the class of the nonlinear systems represented in the nonlinear block controllable form, the Integral Sliding Modes with Block Control technique is analyzed Section 4 shows the design of a nonlinear robust Integral Sliding Modes with Block Control of Multimachine Electric Power Systems 85 control scheme... dynamics and two for mechanical dynamics) has been considered and a nonlinear controller using this model and FL technique has been designed to enhance transient stability (Akhkrif, et al., 1999) The proposed nonlinear control law is a function of all plant parameters and disturbances In practice some of these parameters are subjected to variations as a result of a change in the system loading and/ or... ( x ) gr ( t , x ) where u1eq ( t , x ) is the equivalent control (Utkin et al., 1999) Therefore, the integral control (37) rejects the perturbation term g r ( t , x ) in the last block of (32): z r = fr ( x ) + B r ( x ) u 0 + B r ( x ) u1eq + g r ( t , x ) and we have z r = fr ( x ) + B r ( x ) u 0 Now, choosing 92 Systems, Structure and Control u 0 = −B −1 ( x ) ⎡ fr ( x ) + kr z r ⎤ , kr > 0 r... loading and/ or configuration Then, the EPS are modeled as complex large-scale nonlinear systems and the generators may be interconnected over several kilometers in very large power systems Thus, the controller design is a challenging problem A complete centralized control scheme could be difficult to implement in EPS, due to the reliability and distortion in information transfer On the other hand, accurate... technique are applied to reject perturbations The controller designed in this way is computationally low demanding and takes into account structural constraints of the control input The main feature of the proposed control scheme is robustness with respect to the both matched and unmatched perturbations and only local information is required Moreover, a nonlinear observer for the unmeasureable estates of... problem in the design of feedback controllers for EPS is that of robust stabilizing both rotor angle and voltage magnitude, and achieving a specified transient behavior Robustness implies operation with adequate stability margins and admissible performance level in spite of plant parameters variations and in the presence of external disturbances The EPS have nonlinearities and are subject to variations . x 4 min x 4 max Figure 6. Estimation of x 4 (t) (1 s) 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 10 -4 Ti me [ s] g/g soil x 4 x 4 Project ional DNN Observer x 4 . derivative of Lyapunov-Krasovski function and considering the property (5) , the assumption A2, and in view of (29) we have: Systems, Structure and Control 76 () () () [] () () () () {} () () () () {} (). et al., 20 05, Liu et al., 2006). Systems, Structure and Control 84 All of the mentioned controllers provide larger stability margins with respect to traditional ones. But these control schemes