High Order Sliding Mode Neurocontrol for Uncertain Nonlinear SISO Systems - Theory and Applications

High Order Sliding Mode Neurocontrol for Uncertain Nonlinear SISO Systems: Theory and Applications Isaac Chairez1 , Alexander Poznyak1, and Tatyana Poznyak2 Department of Automatic Control CINVESTAV-IPN, M´xico, D.F e {ichairez,apoznyak}@ctrl.cinvestav.mx Superior School of Chemical Engineering and Extractive Industries, M´xico, D.F e tpoznyak@ipn.mx Introduction Uncertainties in dynamic systems are common in real applications, provoking substantial troubles in any control realization and being a source of instability or poor performance for tracking or regulation problems Considerable research efforts had been undertaken on control designing for uncertain nonlinear dynamic systems over the last thirty years There are several approaches to design and construct a control in this situation Among them, the more effective are the Artificial Neural Networks (ANN) and the Sliding Mode (SM) technique with all possible variants within (Integral Sliding Mode, Higher Order Sliding Mode, etc.) Such combination seems to be very promising [21], [28] because it provides a new instrument for identification, state estimation and control of many classes of uncertain systems affected by external perturbations This chapter deals with the realization of this idea and suggests an adaptive control designing based on both Differential Neural Network Observation and High Order Sliding Mode Technique Below this approach is referred to as High Order Sliding Mode Neural Control (HOSMNC) 1.1 Classical and Unconventional Sliding Mode Many physical systems naturally require the use of discontinuous terms in their dynamics or in their control actions The keystone of this approach is the theory of differential equations with discontinuous right-hand side [5] Based on it, the switching logic is designed in such a way that a contracting property dominates the controlled dynamics leading thus to stabilization on a desired manifold, which induces desirable trajectories These principles constitute the main idea in developing one of the most effective approach to identify and control a wide class of uncertain nonlinear systems: Sliding Mode Control (SMC) [30], [31] The essential feature of this technique is the application of discontinuous feedback laws to reach and maintain the closed-loop dynamics on a certain manifold in the state space (the switching surface) with some desired properties G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 179–200, 2008 c Springer-Verlag Berlin Heidelberg 2008 springerlink.com 180 I Chairez, A Poznyak, and T Poznyak for the system trajectories This control offers many advantages comparing to other identification and control techniques: good transient behavior, the need for a reduced amount of information in comparison to classical control techniques, unmodeled disturbance rejection capability, insensitivity to plant nonlinearities or parameter variations, remarkable stability and performance robustness There are two important drawbacks in real sliding-mode controllers implementations: - the first one is related to the reaching phase which could induce control actions with high values, this fact could provoke the real controlled system to reach its maximum operation limits (dissipated energy, physical restrictions, etc.) leading to a bad performance on the close-loop dynamics; - the second one deals with the actuators which have to cope with the high frequency bang-bang type of control actions that could produce premature wear or even breaking However the last drawback-phenomenon, called chattering, could be reduced using several techniques such as nonlinear gains, dynamic extensions, dynamic filtering and high-order (or unconventional) sliding modes [3], [6] This chapter exactly describes the application effect of the last technique The standard sliding mode (first order or 1-Sliding Mode) may be implemented only if the relative degree of the sliding variable s is with respective to the measurable variable It is well known that it provokes a high frequency switching (chattering effect) of the designed control Therefore, the traditional notion of sliding mode control has been extended and the concept of 2-Sliding Mode and higher order sliding modes (HOSM) has been developed [2], [15] With these controllers, a sliding control of an arbitrary smoothness order can be achieved Shortly speaking, higher order sliding modes remove the restrictions faced by standard sliding mode, while keeping its main properties 1.2 Differential Neural Networks Artificial Neural Networks (ANN) have shown good identification properties in the presence of some uncertainties or external disturbances There are known two type of NN: static one, using the, so-called, back-propagation technique [8] and dynamic neural networks (DNN) [19], [27] The first one deal with the class of global optimization problems trying to adjust the weights of such NN to minimize an identification error The second type exploits the feedback properties of the applied DNN and permits to avoid many problems related to global extremum search, transforming the learning process to an adaptive feedback design [21] If the mathematical model of a considered process is incomplete or partially known, the DNN-approach provides an effective instrument to attack a wide spectrum of problems such as identification, state estimation, trajectories tracking an etc [17], [21] Dynamic neuro-observers are studied in [21] In [24] the SMC approach is used to obtain the algebraic (non differential) weight-learning procedure for on-line High Order Sliding Mode Neurocontrol 181 identification of a nonlinear plant (a model design) with completely available states DNN observers containing sign-term are considered also in [23] DNN Observation with Stable Learning 2.1 Class of Nonlinear Systems The class of uncertain nonlinear SISO systems considered throughout this chapter is governed by a set of n nonlinear ordinary differential equations (ODE) and the algebraic state-output mapping given by xt = f (xt , ut ) + ξ 1,t , ˙ yt = Cxt + ξ 2,t (1) where xt ∈ n is the system state at time t ≥ 0, yt ∈ is the system output, ut ∈ is the control action, C ∈ 1×n is an a priory known output matrix The vectors ξ 1,t ∈ n and ξ 2,t ∈ represent the state and output deterministic bounded (unmeasurable) disturbances, i.e., ξ j,t Λξj ≤ Υj , Λξ1 ∈ n×n , Λξ1 = Λξ1 > 0, Λξ2 > 0, Υ2 = (2) Suppose that f (x, u) − f (w, v) ≤ L1 x − w + L2 |u − v| (3) f (0, 0) ≤ C1 , w, x ∈ n ; u, v ∈ , ≤ L1 , L2 < ∞ which automatically implies the following property f (x, u) ≤ C1 + C2 x + C3 u , Ck ∈ + , k = 1, (4) valid for any x and u Notice that (1) always could be represented as ˜ ˜ xt = f0 (xt , ut | Θ) + ft + ξ 1,t , ft := f (xt , ut ) − f0 (x, u | Θ) ˙ (5) ˜ where f0 (x, u | Θ) is the nominal dynamics while ft is a vector called the modelling error Here the parameters Θ are suggested to be adjusted to minimize the approximation of the nominal part In particular and according to the DNN approach [21], the nominal dynamics may be defined within the following nonlinear structure ∗ ∗ f0 (x, u | Θ) = Ax + W1 σ (xt ) + W2 ϕ (xt ) u n×n ∗ n×l ∗ A∈ , W1 ∈ , W2 ∈ n×s ∗ ∗ σ (·) ∈ l×1 , ϕ (·) ∈ s×1 , Θ := [W1 | W2 ] ∈ n × (n+l+s) (6) The validation of such approximation is based on the approximative Kolmogorov Theorem [12], the Stone-Weierstrass Theorem [25] on sigmoidal approximation and the Lipschitz property (in fact, ”quasi-linearity”) (3) Here the matrix A is 182 I Chairez, A Poznyak, and T Poznyak selected as a stable one and such that the pair (A, C) is observable The vectorfunctions σ (·) := [σ (·) , , σ l (·)] and ϕ (·) := [ϕ1 (·) , , ϕs (·)] are usually constructed with sigmoid function components (following the standard neural networks design algorithms): ⎞⎞−1 ⎛ ⎛ a ⎝1 + b exp ⎝− n cj xj ⎠⎠ , a, b, cj ∈ + , x = [x1 , , xn ] (7) j=1 The nonlinear functions σ (x) and ϕ (x) satisfy σ (x1 ) − σ (x2 ) ≤ lσ x1 − x2 , ϕ (x1 ) − ϕ (x2 ) ≤ lϕ x1 − x2 (8) − < ϕ ≤ ϕ (x) ≤ ϕ + The admissible control set is supposed to be described by a state estimated feedback controllers defined (in general) by: U adm := u = u (ˆ) :u2 ≤ v0 + v1 x x ˆ Λs u (9) (0 < Λs = [Λs ] , Λs ∈ n×n ) where x ∈ n is a state estimation defined by any ˆ u u u suitable (adaptive and stable) nonlinear observer By (9) and in view of the (4) ˜ property, the following upper bound for the modelling error dynamics ft ∈ n takes place: ˜ f Λf Λf , Λf ∈ ¯ ˜ n×n ˜ ˜ ≤ f0 + f1 x , < Λf = Λf ˜ ˜ Λf ¯ ˜ ˆ + f2 x Λf , ˜ ˜ ˜ ˜ f0 , f1 , f2 ∈ + (10) < Λf = Λf Assumed that: ¯ ¯ A1 A is a stable matrix A2 Any of unknown controlled SISO ODE has solution and it is unique, that is (3) and (4) hold A3 The unmeasured disturbances for the uncertain dynamics ξ 1,t and the output signal ξ 2,t satisfy (2) and they not violate the existence of the ODE solution (1) A4 Admissible controls satisfy the sector condition (9), and again, not violate the existence of the solution to ODE (1) 2.2 DNN Observer with Variable Structure Term Defined the DNN observer which can be used to reproduce the unknown xt vector as follows: d yt − C xt ˆ x x x ˆ xt = Aˆt + W1,t σ (ˆt ) + W2,t ϕ (ˆt ) ut + K1 [yt − C xt ] + K2 ˆ dt yt − C xt ˆ A∈ n×n , K1 , K2 ∈ n×1 , W1,t ∈ n×l , W2,t ∈ n×s ∀t ≥ 0, (11) High Order Sliding Mode Neurocontrol 183 where x0 is fixed and the weight matrices (Wj,t , j = 1, 2) are updated by a ˆ nonlinear learning law ˙ ˆ Wj,t = Φj (Wj,t , xt , yt , ut , t | Θ) (12) to be designed Notice this nonlinear adaptive observer reproduces (as it usually called in the state estimation theory) the nominal plant structure (or its approximation) with two additional output based correction terms: one proportional to the output error and the second one known as a unitary corrector As a consequence, when yt = C xt , the ODE (11) should be attended as a differential ˆ inclusion (see [5]) The pair of correction matrices K1 and K2 should be selected as it is described below 2.3 Problem Statement The main idea is to force the uncertain nonlinear system (1) to track a desired reference signal by means of a sliding mode controller using the estimated states provided by a differential neural network observer This problem can be formulated as the solution of the following two subproblems: • Under the assumptions A1-A4 for any admissible ut control strategy (9), to select the adequate matrices A, K1 , K2 and the update law (12) (including the selection of Wj∗ , j = 1, 2) in such a way that the upper bound for the averaged estimation error β defined as t β := lim t→∞ t + εss xs − xs ˆ Q0 ds εss > 0, Q0 = Q0 > (13) s=0 would be as less as possible • Using the obtained state estimations, to construct a feedback controller ut = ˜ ut (ˆs ) |s∈[0,t] such that the averaged (or, non averaged, if feasible) tracking ˜ x performance index t t→∞ t + εac J (˜) := lim u x∗ − xs ˆ s ˜ Q ˜ ˜ ds, εac > 0, Q = Q > (14) s=0 would be small enough Here x∗ is the state vector of a suitable desired dynamics given by: x∗ = φ (t, x∗ ) , x∗ is fixed, x∗ ∈ ˙t t t 2.4 n , φ (·, ·) : n+1 → n (15) Adaptive Weights Learning Law with Bounded Dynamics To adjust the given neuro-observer (11), let us apply the following learning law: 184 I Chairez, A Poznyak, and T Poznyak ˙ W1,t = −k1,t et C + σ (ˆt ) W1,t P1 Nδ C Λ−1 C + Λ1 Nδ P1 x ˜ ξ2 ˙ ˜ +k1,t [2σ (ˆt ) x P2 ] σ (ˆt ) + 2−1 k −1 k1,t W1,t x ˆ x t σ (ˆt ) x 1,t −2 ˙ ˙ W2,t = 2−1 k2,t k2,t Ψ (t, xt ) W2,t + Ψ (t, xt ) [2ˆt P2 ] [ϕ (ˆt ) ut ] ˆ ˜ ˆ x x −2κΨ (t, xt ) ˆ Ψ (t, xt ) ˆ −2 ˜ W2,t ϕ (ˆt ) x ˜ ˙ x W2,t ϕ (ˆt ) ϕ (ˆt ) − x +λ et C + ut ϕ (ˆt ) W2,t P1 Nδ C Λ−1 C + Λ2 x ˜ ξ2 −1 Ψ (t, x) := k2,t -2κ ˆ ˜ W2,t ϕ (ˆ) x −2 +λ ˆ et := yt − C xt , Nδ := (C C + δI) ϕ (ˆ) ϕ (ˆ) x x −1 Nδ P1 [ϕ (ˆt ) ut ] x −1 , ϕ (ˆt ) = ˙ x ∂ϕ (ˆt ) dˆt x x ∂ xt dt ˆ ˜ , Wj,t := Wj,t − W ∗ , κ, λ ∈ R+ (16) ˙ The time varying paramters kj,t , j = 1, are such that kj,t ≥ and kj,t ∀t ≥ (see the following subsection) Pj , j = 1, are the positive definite solutions for the following algebraic Riccati equations [21]: Pj Aj + Aj Pj + Pj Rj Pj + Qj = 0, j = 1, A1 := (A + K1 C) , A2 := A ∗ ∗ ∗ ∗ R1 := W1 Λ−1 (W1 ) + W2 Λ−1 (W2 ) + Λ−1 + Λ−1 + K1 Λ−1 K1 σ ϕ f ξ1 ξ2 ˜ ¯ Q1 := δ Λ−1 + δ Λ−1 + 2f1 Λf + λmax (Λσ ) lσ + Λ−1 + Q0 , Q0 > K1 R2 := K1 CΛK1 C + Λ−1 ξ2 K1 + (17) 2Λ−1 , K2 ∗ ∗ ˜ ˜ Q2 := 2f1 λmax Λf + f2 λmax Λf +2ϕ+ v1 Λs + +W1 Λ−1 (W1 ) + ¯ ˜ u σ ∗ ∗ +W2 Λ−1 (W2 ) +λmax (Λσ ) lσ ϕ 2.5 Main Result Theorem If there exist positive definite matrices [Λ1 , Λ2 , Q0 , ΛK1 , ΛK2 ] and positive constants δ, v1 such that the matrix Riccati equations (17) have positive definite solutions, then the DNN observer (11), supplied by the learning law (16) with any matrix K1 guarantying that the close-loop matrix A1 := (A + K1 C) is −1 Hurwitz and K2 = kP1 C , k > 0, provides the following upper bound for the state estimation process: T lim T →∞ T Δt t=0 P1 2χ (k) dt ≤ 2k α + ˜ (18) [2k α] + 4αQ χ (k) ˜ √ √ ˜ Υ2 + χ (k) := 4Υ2 + f0 + Υ1 + 2k nλ−1/2 Λξ2 max −1 −1 k λmax CP1 Λ−1 P1 C + 2ϕ+ v0 K2 √ α := αP1 − ˜ −1 δλmax P1 , αQ := λmin −1/2 −1/2 P1 Q0 P1 (19) >0 High Order Sliding Mode Neurocontrol 185 Proof The proofs of this theorem is given in Appendix Remark As it can be seen from (18)-(19) the quality of the suggested neural observer depends on both the power of external perturbations Υ1 , Υ2 and the ˜ approximation error f0 as well Remark If there are no noises in the system dynamics and the output measurements (Υ1 = Υ2 = 0) and if the class of uncertain systems and the control ˜ functions are ”zero-cone” type, i.e., (f0 = v0 = 0), then ρQ =0 and the asymptotic error convergence Δt → (t → ∞) is guaranteed Once the upper bound for the estimation process has been derived, it is possible (independently) to consider the stability analysis on the learning laws obtained during the observer development Define the Lyapunov function V2 (zt , k2,t ) = zt 2 1 ˜ , zt =W2,t ϕ (ˆt ) + [k2,t -k2,min ]2 + x + zt + λ (20) and the constant κ := V2 (zt , k2,t ) |t=0 Theorem The weights time profiles (16) are bounded, and moreover, for the second weight matrix W2,t for any < λ ≤ κ −1 the following properties hold ˜ k2,t ≤ 2κ + k2,min , < κ −1 − λ ≤ W2,t ϕ (ˆt ) ≤ 2κ x (21) c ˜ ˜ Proof a) Suggest the Lyapunov function V1,t : V1,t := tr W1,t W1,t + [k1,t − 2 k1,min ]+ Here the function [zt ]+ is defined as [zt ]+ := z, 0, z≥0 Then z 0, k1,0 > k1,min > 0) ˙ k1,t = ˜ −2k1,t tr W1,t Ξ1 (t, x) ˆ ˜ ˜ tr W1,t W1,t + ck1,t [k1,t − k1,min ]+ + ε0 (22) ˙ that implies V1,t ≤ The last inequality permits the existence of several learning laws schemes depending on the k1,t structure 186 I Chairez, A Poznyak, and T Poznyak Proof b) For the weights W2,t we have ˜ ˙ V2 W2,t ϕ (ˆt ) , k2,t ≤ x 1−4 zt +λ −2 ˙ zt π t + [k2,t − k2,min ]+ k2,t −2 ˙ π t := 2−1 k2,t k2,t Ψ (t, xt ) zt + ˆ ˙ ˜ ˙ x W2,t ϕ (ˆt ) + Ψ (t, xt ) Ξ2,t ϕ (ˆt ) + [k2,t − k2,min ]+ k2,t ˆ x ∂ϕ (ˆt ) x yt − C xt ˆ ϕ (ˆt ) := ˙ x x ˆ Aˆt +W1,t σ (ˆt ) +zt ut +K1 [yt -C xt ] +K2 x ∂ xt ˆ yt − C xt ˆ −2 ˜ 2,t ϕ (ˆt ) ϕ (ˆt ) + W ˙ x x Ξ2,t = 2κ zt + λ et C + ut zt P1 Nδ C Λ−1 C + Λ2 ξ2 Nδ P1 − 2ˆt P2 x [ϕ (ˆt ) ut ] x Following a similar procedure as before, put ˙ k2,t = -2k2,t -4 1−4 zt zt 2 +λ +λ −2 −2 zt zt ˜ ˙ x x W2,t ϕ (ˆt ) + Ψ (t, xt ) Ξ2,t ϕ (ˆt ) ˆ 2 Ψ (t, xt ) zt + 2k2,t [k2,t − k2,min ]+ ˆ ˙ that implies V2,t ≤ 0, and hence, V2,t ≤ V2,0 < ∞ So, one may conclude that ≤ κ, k2,t ≤ 2κ + k2,min zt ≤ 2κ, zt + λ 2.6 Training Process Using the Integral Sliding-Mode Derivative Estimation To realize the learning algorithms (16) one needs the knowledge of the nominal ∗ ˜ matrices Ws , s = 1, incorporated in Ws,t , s = 1, (12) The, so-called, training process consists in a suitable approximation (or estimation) of these values It can be realized off-line (before the beginning of the state estimation) by the ∗ ∗ best selection of the nominal parameters Θ := [A, W1 , W1 ] using some available experimental data (utk , xtk ) |k=1,N and a numerical interpolator algorithm allowing to manage these data as a semi-continuous signals Obviously, the data must be sampled with a fixed supplied frequency to contain enough information to process a special kind of parametric identification [18] including the ”persistent excitation” condition and so on Here we suggests to apply the least-mean square algorithm (see [18]) as well as the integral sliding mode adjustment to attain this aim a) Mean Least Square (MLS) application Eq (5) in its integral form is t Yt := ΦXt + ζ t , Yt := xt -xt−h - Axs ds, s=t−h t ζ t := s=t−h (23) t ˜ fs +ξ 1,s ds, Φ:= ∗ W1 ∗ W2 , Xt := s=t−h σ (xs ) ds ϕ (xs ) us High Order Sliding Mode Neurocontrol 187 The Matrix Least Square estimate Φiden of Φ is given by t t ∗,id W1 Φiden = t ∗,id W2 :=Ψt Γt , Γt−1 := t Xt Xt dτ , Ψt := τ =0 Yt Xt dτ (24) τ =0 or in differential form d iden Φ = Yt Xt Γt − Ψt Γt Xt Xt Γt = Yt − Φiden Xt Xt Γt t dt t d −1 Γt := −Γt Xt Xt Γt , t ≥ t0 =: inf Γt > t dt (25) b) Integral Sliding Mode application Define an auxiliary sliding surface as ¯ sP (t, xt ) := χt + xt (26) where χ (t) is some auxiliary variable and xt is an artificial reconstruction ¯ of the continuous nonlinear dynamics using the classical discrete time techniques for sampling, holding and interpolating By direct differentiation of (26) we get ˙ x (27) sP (t, δ t ) = χt + f (¯t , ut , t) + ξ 1,t , sP (0, x0 ) = ˙ The main objective is to enforce the sliding mode to the surface s (t, δt ) = ˙ ˙ ∀t ≥ via an Integral Sliding Mode controller vt [30] Select χt as χt = −vt , v0 = −x0 and define the relay control vt by ⎧ (aux) st ⎨ kt if st > st vt := (28) ⎩ k (aux) e , e ≤ if s = t t t t Taking into account A1-A3 and (4) for the Lyapunov function candidate V (aux) (δ) := st /2 one has ˙ x ˙ V (aux) (st ) = st st = st f (¯t , ut , t) − vt + ξ 1,t ≤ √ (aux) 2 ¯ st C1 + C2 xt + C3 ut + Υ1 − kt √ (aux) (aux) 2 Selecting kt as kt = C1 + C2 xt + C3 ut + Υ1 + ρ, ρ > we derive ¯ ¯ ¯ ˙ V (aux) (st ) ≤ −¯ st = −¯ 2V (aux) (st ) that implies V (aux) (st ) = for all t ≥ ρ ρ (eq) d tr := 2V (aux) (s0 )/¯ = In view of (27), this means vt = dt xt , ∀t ≥ ε > ρ ¯ or equivalently, (eq) vt ∗ ∗ ˜ = A¯t + W1 σ (¯t ) + W2 ϕ (¯t ) u + ft + ξ 1,t ∀t ≥ ε > x x x (29) Remark It is well-known [29], [3], [4] that for t ≥ there appears the chattering effect in the realization of vt if classical sliding controller is applied One of posible 188 I Chairez, A Poznyak, and T Poznyak techniques to reduce the undesirable chattering performance is to use the output (av) vt of a low-pass filter (av) τ vt ˙ (av) + vt (av) = vt , v0 instead of vt , for small enough τ (In practice τ =0 (30) 0.01) The equation (29), may be rearrange as σ (¯t ) x ϕ (¯t ) ut x ∗ ∗ Yt := ΦXt + ζ t , Φ := W1 W2 , Xt := (eq) Yt := vt − A¯t , x ˜ ζ t := ft + ξ 1,t So, it can be applied the LSM approach discussed above Output Feedback Adaptive Neurocontrol 3.1 Quasi Separation Principle For the given uncertain system (1) the tracking performance index is as follows T Jt = T →∞ T xt − x∗ t lim QC dt (31) t=0 Applying the inequality a + b M ≤ + ε−1 a M + (1 + ε) b M to (31) and minimizing the right-hand side by ε, we get a + b ≤ ( a M + b M )2 , that M leads to Jt ≤ Jt,est + Jt,track T Jt,est := lim T →∞ T T (xt -ˆt ) x QC dt, Jt,track := lim T →∞ T t=0 (ˆt -x∗ ) x t QC dt (32) t=0 As it follows from (32), to minimize the upper bound for Jt (31) it is sufficient to minimize the bounds for two terms in (32): the upper bound (18) for the first term Jt,est is already guaranteed by the DNN-method application (see the subsection 1.2.5) valid for any control satisfying (9); the upper bound for the second term Jt,track in (32) can be minimized by the corresponding design of the control actions ut in (11) providing that this control is from U adm (9) This designing can be realized by the high-order sliding mode approach [13] The relation (32) is referred to as the Quasi Separation Principle 3.2 High-Order Sliding Mode Neurocontrol In general, the trajectory tracking problem may be treated as a special constraint on the tracking error dynamics, i.e., these trajectories must belong to a special High Order Sliding Mode Neurocontrol 189 surface S = where S = S (ˆt − x∗ ) := S (δ t ) This surface is closely related x t with the generalized performance index in the linear quadratic regulator (LQR) problem [22] For nonlinear systems, one of the important structure characteristics is the relative degree [10] According to [15], the desired sliding surface S can reflect this property having also the corresponding relative degree r with respect to the tracking error δ t This property is related to the following system of dierential equations: ă (33) S = S = S = = S (r−1) = The high-order sliding mode control as it is suggested in [15] is able to drive the tracking error δ t to the surfaces (33) in finite time practically without the undesired chattering effect Remark Such controller design requires the exactly knowledge of relative degree r This information is always available in the case of DNN sliding mode control since the dynamics, which we are traying to reach for a given DNN, is always known (we constructed it according to (33)) The controller to be designed is expected to be a discontinuous function of the tracking error if we are going to apply the high-order sliding mode approach that requires the real-time calculation of the S-successive derivatives S (i) , i = r−1 The collection of equation (33) represents a manifold The adaptive version of the high-order sliding mode control is proposed below as a feedback function of ă S, S, , S (r1) [13] and [2] In view of (15) the tracking error dynamics is ˙ x δ t = a (δ t , t) + b (δ t , t) ut , b (δ t , t) := W2,t ϕ (ˆt ) yt − C xt ˆ -φ (t, x∗ ) x x ˆ a (δ t , t) :=Aˆt +W1,t σ (ˆt ) +K1 [yt − C xt ] +K2 t yt − C xt ˆ (34) The task for the desired high-order sliding mode controller is to provide in finite time the convergence to S = keeping (33) By the definition of the relative degree r of S the control action ut firstly appears only in the r-th derivative of ∂ (r) S, i.e., S (r) = h (δ t , t) + g (δ t , t) ut , h (δ t , t) := S (r) u=0 , g (δ t , t) = S = ∂ut In the DNN-case we have g (δ t , t) = b (δ t , t) = W2,t ϕ (ˆt ) , and hence, by the x ¯ properties (21), there exist positive parameters Km , KM and C such that < Km := κ −1 − λ ≤ ∂ (r) ≤ KM := 2κ, S ∂ut S (r) ut =0 ¯ ≤C Following to [16] denote for i = 0, 1, , r − φ0,r φi,r = sign (S) , Ψi,r = |N0,r | |Ni,r | (r−i)/(r−i+1) (r−i)/(r−i+1) = S (i) + β i Ni−1,r Ψi−1,r , Ni,r = S (i) + β i Ni−1,r φ0,r = S φi,r N0,r = |S| Ψo,r = where β i , i = 1, , r − are positive numbers Obviously φi,r = S (i) + (r−i)/(r−i+1) β i Ni−1,r φi−1,r Here, any fixed collection of β i , i = 1, , r − defines 190 I Chairez, A Poznyak, and T Poznyak a high-order sliding mode controller applicable to all nonlinear systems (1) with relative degree equal to r For small relative degrees the controllers are as follows: ut = −αsign (S) ˙ ut = sign S + |S|1/2 sign (S) r=1: r=2: r=3: ă ut = −α S + ˙ S 1/6 + |S| 2/3 ˙ sign S + |S| sign (S) ă ut = S + S + S + |S| r=4: ă ×sign S + ˙ S 1/12 × 1/6 + |S| 3/4 ˙ sign S + 0.5 |S| sign (S) ă Since the controllers given above require the direct measurement of S , S , etc., this can be realized by the direct application of the high-order sliding mode differentiator (see [14] and [15]) given by (j = 1, , n) ˙ ˙ z0 = v0 , zk = vk , k = 1, , n-2, zn−1 = −λ1 Lsign (zn -vn−1 ) ˙ n/n+1 sign (z0 -S) + z1 , z0 = S, zj = S (j) v0 := −λ0 |z0 − S| vk := -λk |zk -vk−1 |(n−k)/(n−k+1) sign (zk -vk−1 ) +zk+1 (35) Remark In the case of external bounded perturbations ξ T ( ξ T ≤ ΞT ) , affecting the tracking error surface S, there exist positive constants μi and si (see [16]) depending exclusively on the differentiator parameters (λk , k = n) such that (i = 0, , n) zi − S (j) ≤ μi ΞT (n−i+1)/(i+1) ΞT (n−i)/(i+1) , vi − S (j+1) ≤ si n n Super-twisting controller The twisting algorithm is one of the simplest and most popular algorithms among the second order sliding mode algorithms There are two ways to use the twisting algorithm [13], [7] and [2]: • for systems with relative degree two; • or for systems with relative degree one introducing an integrator in the loop (twisting-as-a-filter or super-twisting) In this chapter, the latter approach is used The control law ut (36) is the combination of two terms: the first one is defined by its discontinuous time derivative, while the other one (which appears during the reaching phase only) is a continuous function of the available sliding variable The algorithm is defined by High Order Sliding Mode Neurocontrol u1,t = ˙ ut = u1,t + u2,t q -u if |u| >1 -λ |s0 | sign (S) if |S| > |s0 | , u2,t = q -W sign (S) if |u| 0): V := V Δ, x, W , t = Δt ˆ ˜ P1 + xt ˆ −1 −1 ˜ ˜ ˜ ˜ 2−1 k1,t tr W1,t W1,t +2−1 k2,t tr W2,t W2,t +κ P2 + ˜ W2,t ϕ (ˆ) x −1 (43) +λ which time derivative is d −1 ˜ −1 ˙ ˙ ˜ ˙ ˙ ˆ V = Δt P1 Δt + 2ˆt P2 xt + tr k1,t W1,t −2−1 k1,t k1,t W1,t + W1,t x dt −1 ˜ −1 ˙ ˜ ˙ +tr k2,t W2,t −2−1 k2,t k2,t W2,t + W2,t − 2κ ˜ W2,t ϕ (ˆ) x −2 ˜ W2,t ϕ (ˆ) , x +λ (44) d ˜ ˜ ˙ x W2,t ϕ (ˆ) + W2,t ϕ (ˆ) x dt Δ [P1 A + A P1 ] Δt and estimating the rest of the t terms in (42) using the matrix inequality XY + Y X ≤ XΛX + Y Λ−1 Y valid for any X, Y ∈ Rr×s and any < Λ = Λ ∈ Rs×s one gets: Notice that Δt P1 AΔt = ˜ ¯ ˙ 2Δt P1 Δt ≤ Δt δ Λ−1 +δ Λ−1 +2f1 Λf +λmax (Λσ ) lσ Δt + ∗ ∗ ∗ ∗ Δt P1 W1 Λ−1 (W1 ) +W2 Λ−1 (W2 ) +Λ−1 +Λ−1 P1 Δt σ ϕ f ξ1 ˜ ˜ ˆ ˜ ˆ ˜ +2Υ2 +f0 +Υ1 +2f1 xt Λf¯ +f2 x Λ ˜ +et CNδ P1 W1,t σ (ˆt ) + x f ˜ x ˜ x Δt [P1 A+A P1 ] Δt +σ (ˆt ) W1,t P1 Nδ C Λ−1 C+Λ1 Nδ P1 W1,t σ (ˆt ) + ξ2 ˜ ˜ et CNδ P1 W2,t ϕ (ˆ) ut +ut ϕ (ˆ) W2,t P1 Nδ C Λ−1 C+Λ2 Nδ P1 W2,t ϕ (ˆ) ut x x ˜ x ξ2 yt − C xt ˆ + ϕ (x, xt ) u Λϕ -Δt P1 K1 (yt − yt ) -Δt P1 K2 ˜ ˆ ˆ yt − C xt ˆ y By the same reason, it follows 2Δt P1 K1 (yt -ˆt ) ≤ Δt [P1 K1 C+C K1 P1 ] Δt + −1 Δt P1 K1 Λ−1 K1 P1 Δt + Υ2 and, in view of the selection K2 = kP1 C and ξ2 by the inequality (19) in [20], one gets −1 δλmax P1 Δt P1 − 2k n λmax Λξ2 Δt P1 K2 CΔt + ξ 2,t CΔt + ξ 2,t √ Υ2 Additionally, ≥ k √ αP1 − High Order Sliding Mode Neurocontrol 199 yt − C xt ˆ yt − C xt ˆ ˜ ˜ xt − xt P2 W1,t σ (ˆt ) − 2ˆt P2 W2,t ϕ (ˆt ) ut ˆ ˆ x x x 2ˆt P2 Aˆt + W1,t σ (ˆt ) +W2,t ϕ (ˆt ) ut + K1 [yt − C xt ] +K2 x x x ˆ x ≤ xt P2 A + AT P2 ˆ +ˆt P2 K1 CΛK1 C + Λ−1 K1 P2 xt + Δt Λ−1 Δt + Υ2 + ϕ (x, xt ) u x ˆ ˜ ˆ K1 ξ2 2ˆt P2 Λ−1 P2 xt x ˆ K2 ∗ ∗ xt W1 Λ−1 (W1 ) ˆ σ −1 −1 + k λmax CP1 Λ−1 P1 C + K2 ∗ −1 ∗ + W2 Λϕ (W2 ) + λmax (Λσ ) lσ Λϕ + xt ˆ Using the upper bound ut Λu ut ≤ v0 + v1 x Λu (9), after the substitution of ˆ all of these inequalities in (44) one gets the following: Δt ˙ V ≤ Δt [P1 (A + K1 C) + (A + K1 C) P1 ] Δt ˜ ¯ δ Λ−1 + δ Λ−1 + 2f1 Λf + λmax (Λσ ) lσ + Λ−1 + Q0 Δt + K1 ∗ ∗ ∗ ∗ Δt P1 W1 Λ−1 (W1 ) + W2 Λ−1 (W2 ) + Λ−1 + Λ−1 + K1 Λ−1 K1 P1 Δt σ ϕ f ξ1 ξ2 ˆ +ˆt P2 A + AT P2 xt + xt P2 K1 CΛK1 C + Λ−1 K1 + 2Λ−1 P2 xt + x ˆ ˆ K2 ξ2 ˜ ˜ ˆ xt 2f1 λmax Λf + f2 λmax Λf + 2ϕ+ v1 Λu xt + ˆ ¯ ˜ ∗ ∗ ∗ ∗ xt W1 Λ−1 (W1 ) + W2 Λ−1 (W2 ) + λmax (Λσ ) lσ xt ˆ ˆ σ ϕ √ √ −1/2 −1 −1 −1 ˜ +Υ1 +2k nλ 4Υ2 +f0 Υ2 +k λmax CP1 ΛK2 P1 C +2ϕ+ v0 max Λξ √ −1 Δt P1 − Δt Q0 + αP1 − δλmax P1 −k et C + σ (ˆt ) W1,t P1 Nδ C Λ−1 C + Λ1 x ˜ ξ2 ˜ Nδ P1 W1,t σ (ˆt ) + x ˜ et C + ut ϕ (ˆ) W2,t P1 Nδ C Λ−1 C + Λ2 Nδ P1 W2,t ϕ (ˆ) ut − x ˜ x ξ2 ˜ ˜ 2ˆ P2 W1,t σ (ˆt ) − 2ˆ P2 W2,t ϕ (ˆt ) ut − x x x x t t −1 ˜ −1 ˙ ˜ ˙ tr k1,t W1,t 2−1 k1,t k1,t W1,t -W1,t −2κ ˜ W2,t ϕ (ˆ) x −1 ˜ −1 ˙ ˜ ˙ -tr k2,t W2,t 2−1 k2,t k2,t W2,t -W2,t −2 ˜ W2,t ϕ (ˆ) , x +λ d ˜ ˜ ˙ x W2,t ϕ (ˆ) + W2,t ϕ (ˆ) x dt where ϕ+ = λmax [˜ (x, xt ) Λϕ ϕ (x, xt )] Since both Riccati equations (17) adϕ ˆ ˜ ˆ mit positive definite solutions, P1 and P2 , and under the adaptive weights ad√ −1 ˙ Δt P1 αP1 - δλmax P1 justment laws (16), one gets V ≤ χ (k) - k −1/2 αQ Δt P1 Integrating both sides and defining αQ := λmin P1 > 0, we get ⎛ 1⎝ √ lim -k αP1 T →∞ T lim T t=0 T →∞ VT -V0 ≤ χ (k) + T αQ Δt P1 dt+k −1 δλmax P1 −1/2 Q0 P1 ⎞ Δt P1 dt⎠ 200 I Chairez, A Poznyak, and T Poznyak T Defining β t T Δt P := lim Δt P1 dt and applying the Jensens inequality T →∞ T t=0 ⎛ T ⎞2 dt ≥ ⎝ t=0 Δt P dt⎠ the last inequality becomes αQ β + 2k αβ t ˜ t t=0 − χ (k) ≤ 0, α := ˜ βt ≤ √ −2k α + ˜ −1 δλmax P1 αP1 − that leads to [2k α] + 4αQ χ (k) ˜ 2αQ 2χ (k) ≤ 2k α + ˜ [2k α] + 4αQ χ (k) ˜ ... Universal Single-InputSingle-Output (SISO) Sliding- Mode Controllers With Finite-Time Convergence IEEE Transactions on Automatic Control 46, 1447–1451 (2001) 16 Levant, A.: Higher -order sliding modes,... weight-learning procedure for on-line High Order Sliding Mode Neurocontrol 181 identification of a nonlinear plant (a model design) with completely available states DNN observers containing sign-term... designing can be realized by the high- order sliding mode approach [13] The relation (32) is referred to as the Quasi Separation Principle 3.2 High- Order Sliding Mode Neurocontrol In general, the