sampled data active disturbance rejection output feedback control for systems with mismatched uncertainties

Special Issue Article Sampled-data active disturbance rejection output feedback control for systems with mismatched uncertainties Advances in Mechanical Engineering 2017, Vol 9(1) 1–9 Ó The Author(s) 2017 DOI: 10.1177/1687814016682690 journals.sagepub.com/home/ade Jun You1, Jiankun Sun2, Shuaipeng He3 and Jun Yang2 Abstract This article investigates the sampled-data disturbance rejection control problem for a class of non-integral-chain systems with mismatched uncertainties Aiming to reject the adverse effects caused by general mismatched uncertainties via digital control strategy, a new generalized discrete-time extended state observer is first proposed to estimate the lumped disturbances in the sampling point A disturbance rejection control law is then constructed in a sampled-data form, which will lead to easier implementation in practices By carefully selecting the control gains and a sampling period sufficiently small to restrain the state growth under a zero-order-holder input, the bounded-input bounded-output stability of the hybrid closed-loop system and the disturbance rejection ability are delicately proved even the controller is dormant within two neighbor sampling points Numerical simulation results demonstrate the feasibility and efficacy of the proposed method Keywords Sampled-data control, disturbance rejection, mismatched uncertainty, discrete-time extended state observer Date received: 26 August 2016; accepted: 14 October 2016 Academic Editor: Yongping Pan Introduction In this article, we consider a class of single-input singleoutput (SISO) system with mismatched uncertainties of the form x_ (t) = Ax(t) + Bu u(t) + Bd f ðx(t), v(t), tÞ ym (t) = Cm x(t) ð1Þ y0 (t) = C0 x(t) where x(t) R is the system state vector, u(t) R is the control input, v(t) R is the external disturbance, ym (t) Rr is the measurable outputs, y0 R is the controlled output, and A Rn n , Bu Rn , Bd Rn , Cm Rr n , C0 R1 n are system matrices f (x(t), v(t), t) is an uncertain function representing the lumped disturbance in a general way which possibly includes n external disturbances, unmodeled dynamics, parameter variations, and complex nonlinear dynamics.1–8 In modern control practices, it is a trend that sampled-data controllers with a zero-order holder (ZOH) are being digitally implemented into real-life plants with the rapid development of computational hardware technology It is common that conventional School of Electrical Engineering, Southeast University, Nanjing, China Key Laboratory of Measurement and Control of CSE, Ministry of Education, School of Automation, Southeast University, Nanjing, China College of Automation Engineering, Shanghai University of Electric Power, Shanghai, China Corresponding author: Jun You, School of Electrical Engineering, Southeast University, Nanjing 210096, China Email: youjun@seu.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 control law design is always devoted to continuous-time design for mathematically modeled continuous-time systems due to the convenience of direct stability analysis, typically based on the continuously differentiable Lyapunov function analysis In most practical implementation processes, continuous-time controllers can be discretized directly which exists on the fact that the closed-loop system performance can always be guaranteed while the sampling frequency is fast enough However, in most of the times, this is done without a theoretical support, and the formulated restrictions of the sampling frequency and how it affects the controlled system performance are actually unknown Moreover, only digital sensors are available for some real-life plants, for instance, Global Positioning System (GPS) is a discrete-time sensor and a radar is more naturally represented using a discrete-time model.9 Hence, it is always a challenging but crucial issue to address the problem of designing sampled-data controllers which pursue smaller steady-state error, faster dynamical response, and milder noise susceptibility for system perturbed by matching or mismatched uncertainties In the literature, one of the most popular methods among those existing approaches developed to design sampled-data controllers for nonlinear plants is the discretization method.10–12 It employs a discrete-time approximation model of the plant (typically use Euler approximation) to design discrete-time controllers since it is almost impossible to obtain an exact discrete-time model of the continuous-time nonlinear plant Hence, the results regarding this method are always achieved within a local or semi-global control goal The emulation method, as a global control method, designs a continuous-time controller based on the continuoustime plant, followed by a discretization process to yield sampled-data controllers which will assure the continuous-time nonlinear plant by choosing an appropriate sampling period One can refer to papers13–19 and the references therein In practical engineering, the control performance of modern industrial systems is inevitably affected by various uncertainties including parameter variations, unmodeled dynamics, and external disturbances.20–22 Active anti-disturbance technique is generally required in the controller design to access high-precision control performance.1,23–25 The extended state observer–based control (ESOBC) was originally proposed by Han,26,27 and it is made practical by the tuning method which simplifies its implementation and makes the design transparent to engineers.1 Su et al.28 presented the relationship between time-domain and frequency-domain disturbance observers and its applications for further information However, most of existing disturbance rejection methods, including ESOBC, are concerned with continuous control approaches, which lack sound Advances in Mechanical Engineering justification since most control approaches are digitally implemented in a sampled-data manner To this end, the development of an active disturbance rejection method for system (1) with mismatched uncertainties will be of interest for both theoretical and industrial communities Moreover, the mismatched uncertainties, rather than the so-called matching conditions4,20 are concerned to discover a generalized sampled-data disturbance rejection control law for system (1) Mismatched uncertainties may not act via the same channel with the control input and are regarded as a more general case concerned in uncertainty attenuation problems As an example, the lumped disturbance torques in flight control systems always affect the states directly rather than through the input channels.23,29 This article presents a generalized sampled-data control law design based on a discrete-time extended state observer (ESO), which estimates the unmeasurable states and the lumped disturbance information in the sampling point Explicit formulas to select the control gains and the tunable sampling period based on a detailed stability analysis for the hybrid closed-loop system are presented Numerical simulations are shown to demonstrate the effectiveness of the proposed method The proposed method will be a helpful guideline for direct digital implementation Main results In this section, we present a step-by-step procedure to design a discrete-time ESO-based sampled-data control law to solve the global stabilization problem for system (1) With an added extended variable xn + (t) = d = f ðx(t), v(t), tÞ system (1) can be extended to the following form  x(t) + B  u u(t) + Eh(t) x_ (t) = A  ym (t) = C mx(t) with the denotation of  ÃT x(t) = xT (t), xn + (t) h(t) = df ðx(t), v(t), tÞ dt   A Bd R(n + 1) (n + 1) 01 n   Bu  R(n + 1) Bu = = A E = ½0, , 0, 1ŠT R(n + 1)  m = ½Cm , 0r Š Rr (n + 1) C ð2Þ You et al  C  m ) is Assumption (A, Bu ) is controllable and (A, observable Assumption The lumped disturbances satisfy the following conditions:20  d(t) and h(t) are bounded, that is, there exist  h such that two positive constants d,   jd(t)j d, jh(t)j h, respectively t!‘ In the article, the Assumption is made on the disturbances d(t) that its derivative is close to zero However, this assumption can be relaxed using unknown observer theory.30,31 Construction of discrete-time ESO In what follows, an ESO for system (1) will be built using the sampled-data information ym (tk ) (tk = kT , k = 0, 1, 2, ), where T is the sampling D period The continuous-time state ^x(t) ẳ ẵ^xT , ^xn + T defined in the time region t ½tk , tk + ) ^  u u(t) À Lð^ym (t) À ym (tk )Þ, ^ x_ (t) = A x(t) + B  m^x(t) and L R(n + 1) r is the observer where ^ym (t) = C gain matrix to be assigned later The observer (3) can be rewritten as follows t ½tk , tk + ) ð4Þ Integrating the continuous-time observer (4) from tk to tk + , it can be concluded that ^x(tk + ) can also be generated by the following discrete-time ESO   ^ x(tk + ) = eðAÀLCm ÞT ^x(tk ) + ðT    u u(tk ) + Lym (tk )Þ ẦLCm Þs dsðB D x(tk ) + Gu(tk ) + Nym (tk ) ¼ F^ ð5Þ with the denotation of F =e  C  m ÞT ðẦL ðT G=   u ẦLCm Þs dsB ðT N=   eðAÀLCm Þs dsL The sampled-data disturbance rejection law can be designed as ^ k ), u(t) = u(tk ) = K ^x(tk ) = Kx^x(tk ) + Kd d(t t ½tk , tk + ) 7ị where K = ẵKx , Kd Š Kx is the control gain to be designed Kd is the disturbance compensation gain Motivated by the results,20 Kd is assigned by the following equation h iÀ1 Kd = À C0 ðA + Bu Kx ÞÀ1 Bu C0 A + Bu Kx ị1 Bd t ẵtk , tk + ) ð3Þ ^  u u(tk ) À Lð^ym (t) À ym (tk )Þ ^ x_ (t) = A x(t) + B  À LC  m Þ^x(t) + B  u u(tk ) + Lym (tk ), = ðA  m ðx(t) À x(tk )Þ,  À LC  m Þe(t) À Eh(t) À LC e_ (t) = A 6ị t ẵtk , tk + ) Sampled-data disturbance rejection law design _ = lim h(t) = lim d(t) t!‘ The state and disturbances errors are defined as e(t) = ½eTx (t), ed (t)ŠT where ex (t) = ^x(t) À x(t), ed (t) = ^ À d(t), respectively With equations (2) and (3), the d(t) error dynamics are Remark With equation (7) in mind, the designed discrete-time extended observer can also be presented as ^x(tk + ) = M ^x(tk ) + Nym (tk ) ð8Þ where M = F À GK R(n + 1) (n + 1) and N are two matrices dependent of the sampling period T Hybrid closed-loop system stability analysis Combing equations (1), (6), and (7) together, one can obtain the hybrid closed-loop system as ! ! ! x(t) Bu K x_ (t) A + Bu Kx =  À LC  m e(t) A e_ (t) ! ! ÀBu K Bu Kd + Bd h(t) + + ÀE d(t) ! ^x(t) À ^x(tk ) , t ½tk , tk + ) x(t) À x(tk ) m ÀLC ! ð9Þ If we choose the observer gain L and the feedback  À LC  m are Hurwitz gain Kx such that A + Bu Kx and A matrices, it is easy to verify that the following matrix A + Bu Kx Bu K  À LC m A ! D ¼L is also a Hurwitz one Hence, there exists a positive definite matrix P = PT R(2n + 1) (2n + 1) such that Advances in Mechanical Engineering LT P + PL = ÀaI Lemma There exist proper constants c3 0, c4 0, such that the following inequality holds where a.0 is a constant will be determined later Construct a positive definite and proper Lyapunov where function V (Z(t)) = Z T (t)PZ(t) T T T Z(t) = ½x (t), e (t)Š The derivative of V (Z(t)) along system (9) is given as follows V_ ðZ(t)Þ = Àa k Z(t)k2 + 2Z(t)T ! ! Bu Kd + Bd h(t) P ÀE d(t) # !" ^x(t) À ^x(tk ) ÀBu K T , + 2Z(t) P  m x(t) À x(tk ) ÀLC ð10Þ k Z(t) À Z(tk ) k B u Kd + B d ÀE À Á 2c1 k P k h + d k Z(t) k À Á h + d k Z(t) k = c1  ! h(t)  d(t)  Proof Denote the whole closed-loop system (9) to be _ = C(u(tk ), Z(t)) With the fact that h(t) and d(t) are Z(t) bounded in mind, the following inequality holds for s ½tk , t) c3 ðk Z(s) À Z(tk ) k  + k Z(tk ) k + h + dÞ k Z(t) À Z(tk ) k k C(u(tk ), Z(s)) k ds À Á c3 k Z(tk ) k + h + d (t À kT ) + c3 k Z(s) À Z(tk ) k ds, ð17Þ t ẵtk , tk + ) tk 12ị   ÀBu K  2Z(t)T P  # !" ^ x(t) À ^ x(tk )   m x(t) À x(tk )  ÀLC  !  e(t) À e(tk ) + x(t) À x(tk )   2c2 k Z(t) k Á k P k Á   x(t) À x(t ) Based on equation (17), applying the Gronwall inequality, we have À Á c3 k Z(tk ) k + h + d (t À kT ) ðt À Á 2 2 + c3 k Z(tk ) k + c3 h + c3 d (s À kT )ec3 (tÀs) ds k Z(t) À Z(tk ) k tk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á V (Z(tk )) + h + d ec3 (tÀkT) À , c4 t ½tk , tk + ) ð18Þ where c4 is a constant With Lemma in mind, equation (14) becomes k 4c2 lmax fPg k Z(t) k ðk Z(t) À Z(tk ) k + k d(t) À d(tk ) kÞ c2 k Z(t) kk Z(t) À Z(tk ) k + c2 d k Z(t) k , t ½tk , tk + ) ð13Þ Substituting equations (11) and (13) into equation (10) yields À Á À a k Z(t)k2 + c1 h + c1 d + c2 d k Z(t) k + c2 k Z(t) kk Z(t) À Z(tk ) k , ðt tk ð11Þ where c2 is a constant dependent on the value of K, L Similar to equation (11), there exists a constant c2 0, such that the following estimation holds V_ ðZ(t)Þ ð16Þ where c3 is a constant By equation (16), one can obtain that ðt where c1 and c1 = 2c1 lmax (P) are constants Since K, L are set, one can conclude that  !  ÀBu K    c2   ÀLC m  ð15Þ k Cðu(tk ), Z(s)Þ k Now, we will estimate the items in the right hand side of equation (10) First, with jh(t)j h, k P k = lmax (P), k E k = in mind, we have ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á V (Z(tk )) + h + d ec3 (tÀkT) À , t ½tk , tk + ) t ½tk , tk + )   2Z(t)T P c4 t ½tk , tk + ) ð14Þ Next, we introduce a lemma which plays a key role in the main theorem V_ ðZ(t)Þ À Á À a k Z(t)k2 + c1 h + c1 d + c2 d k Z(t) k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À Á + c2 c4 k Z(t) k V ðZ(tk )Þ + h + d ec3 (tÀkT ) À ! a À pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À c5 V ðZ(t)Þ lmin fPg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀ c (tÀkT ) Á À1 + c6 V ðZ(t)Þ V ðZ(tk )Þ e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À À ÁÁ  V ðZ(t)Þ, + c5 + c6 ec3 (tÀkT ) À (h + d) t ½tk , tk + ) ð19Þ with two positive constants c5 , c6 You et al Theorem Under Assumptions and 2, with the following selection formulas for the parameter a and the allowable sampling period T a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! c5 + 1; T \ ln c3 lmin fPg  1+ c6  ð20Þ and the observer gain L and the feedback control gain  À LC  m are Kx are selected such that A + Bu Kx and A Hurwitz matrices; system (1) under the proposed sampled-data control law (5)–(7) can be rendered bounded stable for any bounded d(t), h(t) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Proof Define j(t) = V (Z(t))= V (Z(tk )) It can be concluded that equation (19) equals to the following inequality ! Á a c6 À c3 (tÀkT ) _ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À c5 j(t) + e À1 j(t) À 2 lmin fPg À À ÁÁ h + d + c5 + c6 ec3 (tÀkT ) À pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , V ðZ(tk )Þ Remark As one can mention, the discrete-time observer (8) will generate the same sampled information ^x(kT ) at the sampling point for the same initial condition with the continuous-time observer (3) Hence, in the design, we could use the continuous-time form (3) to take fully advantages of the continuous-time design method.16,18 As a matter of fact, we not need to consider the signal information for t (kT, (k + 1)T ) for the observer (3), what we need is only the sampled information ^x(kT) which can be generated from the sampled output information y(kT ), k = 0, 1, As one can observe from the proof of Theorem that even the controller is dormant within two neighbor sampling points and the inter-sample information is not analyzed, the fact that the difference Lyapunov function DV (Z) is semi-negative definite can assure the boundedinput bounded-output stability of the hybrid closedloop system Disturbance rejection analysis t ½tk , tk + ) ð21Þ In what follows, we will give a detailed disturbance rejection analysis to show the proposed ESO-based sampled-data disturbance rejection control law will effectively reject the lumped disturbances With equation (20) in mind, the following inequality can be concluded Á c6 À c3 T _ e À1 j(t) À j(t) + 2 À À ÁÁ h + d + c5 + c6 ec3 T À pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , V ðZ(tk )ị t ẵtk , tk + ) 22ị Theorem Under Assumptions and 2, the lumped disturbances from system (1) can be effectively rejected from the output channel in steady-state under the proposed sampled-data control law (7) provided that the observer gain L and the feedback control gain Kx are  À LC  m are Hurwitz selected such that A + Bu Kx , A À1 matrices and C0 (A + Bu Kx ) Bu is invertible Solving the above inequality with j(tk ) = 1, we have j(tk + ) À c6 e Á c3 T  À À À + c5 + c6 e ÀT 1Àe2  ÀT c3 T À1 ÁÁ h + d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ðZ(tk )Þ D + e ẳ r(T ) ! 23ị Proof Substituting the sampled-data controller (7) into system (1) yields x_ (t) = Ax(t) + Bu Kx ex (tk ) + Bu Kx x(tk ) + Bd (d(t) À d(tk )) À Bd ed (tk ), t ½tk , tk + ) ð26Þ which leads to V ðZ(tk + )Þ r(T )V ðZ(tk )Þ ð24Þ According to the condition (20), we have 6¼ c6 (ec3 T À 1) Hence, the difference equation DV (Z) = V (Z(tk + )) À V (Z(tk )) if  V ðZ(tk )Þ !  2 ðc5 + c6 ðec3 T À 1ÞÞ(h + d) À c6 ðec3 T À 1Þ ð25Þ which implies that Z(tk ) decreases for any Z(tk ) satisfies condition (25); hence, the state Z(tk ) is bounded for any bounded d(t) and h(t) if those parameters are properly selected This completes the proof of Theorem By Theorem 1, the following relations hold for the hybrid closed-loop system (1)–(7)–(8) lim x_ (t) = lim x_ (tk ) = 0, lim e(tk ) = t!‘ k!‘ k!‘ ð27Þ Hence, one can conclude from equations (26) and (27) that lim y0 (t) = t!‘ lim y0 (tk ) = C0 ðA + Bu Kx ÞÀ1 x_ (tk ) À C0 ðA + Bu Kx ÞÀ1 k!‘ ðBu Kx ex (tk ) + Bd ðd(tk ) À d(tk )Þ À Bd ed (tk )Þ = ð28Þ Advances in Mechanical Engineering Figure Implementation structure of extended state observer–based sampled-data controller which concludes that the lumped disturbances can be effectively rejected from the output channel in steadystate Remark In this article, we assume some states of the systems are not measurable; hence, the proposed sampled-data control strategy has to be implemented by output feedback of the form of equation (7) If we consider the case that the states are available or can be easily measured, the sampled-data state-feedback control law can be simply designed as ^ k ), t ½tk , tk + ) u(t) = u(tk ) = Kx x(tk ) + Kd d(t  à À1 Kd = À C0 (A + Bu Kx )À1 Bu C0 ðA + Bu Kx ÞÀ1 Bd ð29Þ ^ k ) can be generated from the discrete-time where d(t ESO (8) Remark Figure depicts the implementation structure of practical plant with the proposed ESO-based sampled-data disturbance rejection controller (7) and (8) In practical engineering, the dominated dynamics of a system has been stabilized by feedback control, and the nonlinear character of those uncertainties contained in the system are relatively weak, which means the presence of the uncertainties will not cause much damage to the closed-loop system’s stability Hence, such uncertainties can be regarded as part of the lumped disturbances and can be reasonably handled by sampled-data control law of the form (7) This fact will support the general availability of the proposed control method Example and simulation results Next, we use an example and numerical simulations to show how an ESOBC law is designed and the effectiveness of the proposed method A two-dimensional example Consider a two-dimensional uncertain nonlinear system with mismatching condition of the form x_ (t) = x2 (t) + f ðx(t), v(t), tÞ x_ (t) = À2x1 (t) À x2 (t) + u(t) ð30Þ y(t) = x1 (t) which can also be written as the state-space form x_ (t) = Ax(t) + Bu u(t) + Bd d(t) ð31Þ y(t) = Cx(t) ! ! 0 , with the denotation of A = , Bu = À2 À1 ! , Cm = C0 = C = ½1, 0Š, and d(t) = f (x1 (t), Bd = x2 (t), v(t), t) Clearly, system (30) can be seen as an example of system (1) and can satisfy Assumptions and Let d(t) = x3 (t), one can obtain an extended system as  x(t) + Bu(t)  x_ (t) = A + Eh(t) ð32Þ  x(t) y = C 3 0 1  = ½1, 0, 0Š,  = À2 À1 5, B  u = 5, C where A 0 0 and E = ½0, 0, 1ŠT One can construct a discrete-time You et al ESO-based sampled-data output feedback controller of the following form ^ xðtk + Þ = M ^x(tk ) + Ny(tk ) u(t) = K ^x(tk ), t ½tk , tk + ) ð33Þ ð34Þ  C)T  ^ T , K = ½k1 , k2 , k3 Š, M = e(ẦL x = ½^x1 , ^x2 , dŠ + where ^ Ð T L(AÀL Ð T (AÀL  C)s    C)s  dsBu K, and N = e dsL e Numerical simulations In what follows, we will use numerical simulations for system (30) to demonstrate the effectiveness of the proposed method The disturbance is considered as d(t) = ex1 (t) + v, where v represents the external disturbance assigned as v = acting on the system at the time instant t = 5s, the observer gain vector L is selected as L = ½14, À 66, 125ŠT , and the controller gain K = ½À4, À 4, À 5Š based on the guideline (7) Hence,  À LC  in this case, we can verify that A + Bu Kx and A are both Hurwitz By Theorem 1, we can choose a proper sampling period T = 0:05s The discrete-time observer matrices can be calculated as 0:5286 1:0833 1:0913 M = @ 26:3854 2:8041 3:3161 A, À5:6081 À4:6100 À6:0125 0:5278 N = @ À437:8961 A 500:2623 Figure Trajectories of x1 (t) and ^x1 (tk ) of the closed-loop system (30)–(33)–(34) Figures 2–4 show the response curves of the system and estimated states One can observe from Figures and that the system states converge to the equilibrium quickly in the presence of both internal uncertainties and external disturbances It is illustrated by Figure that the lumped disturbance can be successfully rejected in the output channel And from Figure 4, it is shown that the discrete-time ESO works effectively and leads to high-precision observation of the disturbance d(t) The time history of the sampled-data output feedback control law (34) is shown in Figure Figure Trajectories of x2 (t) and ^x2 (tk ) of the closed-loop system (30)–(33)–(34) Conclusion A novel discrete-time ESO-based sampled-data active disturbance rejection output feedback control law has been proposed in this article to achieve high-precision control performances for a class of systems with mismatched uncertainties With a delicate design procedure, the careful selection of the involved parameters ensures the global stability of the hybrid closed-loop system and disturbance rejection ability Direct digital Figure Trajectories of the disturbance d(t) and estimated disturbance ^d(tk ) from the ESO (33) design strategy will lead to easier implementation Numerical simulations have shown the effectiveness of the proposed method 8 Figure Time history of the sampled-data control input signal u(t) Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Funding The author(s) received no financial support for the research, authorship, and/or publication of this article References Gao Z Active disturbance rejection control: a paradigm shift in feedback control system design In: Proceedings of the American control conference, Minneapolis, MN, 14–16 June 2006, pp.2399–2405 New York: IEEE Li S and Liu Z Adaptive speed control for permanentmagnet synchronous motor system with variations of load inertia IEEE T Ind Electron 2009; 56: 3050–3059 Liu H and Li S Speed control for PMSM servo system using predictive function control and extended state observer IEEE T Ind Electron 2012; 59: 1171–1183 Yang J, Li S, Su J, et al Continuous nonsingular terminal sliding mode control for systems with mismatched disturbances Automatica 2013; 49: 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