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Sampled‐data design for robust control of open two‐level quantum systems with operator errors

2016; Institution of Engineering and Technology; Volume: 10; Issue: 18 Linguagem: Inglês

10.1049/iet-cta.2016.0368

1751-8652

Daoyi Dong, Ian R. Petersen, Yuanlong Wang, X. X. Yi, Herschel Rabitz,

Spectroscopy and Quantum Chemical Studies

IET Control Theory & ApplicationsVolume 10, Issue 18 p. 2415-2421 Research ArticleFree Access Sampled-data design for robust control of open two-level quantum systems with operator errors Daoyi Dong, School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, 2600 AustraliaSearch for more papers by this authorIan R. Petersen, School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, 2600 AustraliaSearch for more papers by this authorYuanlong Wang, Corresponding Author yuanlong.wang.qc@gmail.com School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, 2600 Australia Institute of Cyber-Systems and Control, State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorXuexi Yi, Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun, 130024 People's Republic of ChinaSearch for more papers by this authorHerschel Rabitz, Department of Chemistry, Princeton University, Princeton, New Jersey, 08544 USASearch for more papers by this author Daoyi Dong, School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, 2600 AustraliaSearch for more papers by this authorIan R. Petersen, School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, 2600 AustraliaSearch for more papers by this authorYuanlong Wang, Corresponding Author yuanlong.wang.qc@gmail.com School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, 2600 Australia Institute of Cyber-Systems and Control, State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorXuexi Yi, Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun, 130024 People's Republic of ChinaSearch for more papers by this authorHerschel Rabitz, Department of Chemistry, Princeton University, Princeton, New Jersey, 08544 USASearch for more papers by this author First published: 01 December 2016 https://doi.org/10.1049/iet-cta.2016.0368Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract This study proposes a sampled-data design method for robust control of open two-level quantum systems with operator errors. The required control performance is characterised using the concept of a sliding mode domain related to fidelity, coherence or purity. The authors have designed a control law offline and then utilise it online for a two-level system subject to decoherence with operator errors in the system model. They analyse three cases of approximate amplitude damping decoherence, approximate phase damping decoherence and approximate depolarising decoherence. They design specific sampling periods for these cases that can guarantee the required control performance. 1 Introduction Manipulating and controlling quantum systems is a key task in diverse research areas including quantum physics, molecular chemistry and quantum information [1–4]. Several analysis tools and control methods in the area of classical control theory have been applied to investigating control problems in quantum systems. For example, optimal control theory has been used to assist in control design for several classes of quantum mechanical problems [5–7]. Feedback control has been developed for the manipulation of quantum states, the preparation of quantum entanglement and the implementation of quantum error-correction [2, 8–15]. Amongst the research on quantum control, robust control design of quantum systems is significant for the development of practical quantum technology since the existence of uncertainties and errors is unavoidable. Several robust control methods, such as control [15], risk sensitive control [16], sliding mode control [17, 18], noise filtering [19], and sampling-based learning control [20, 21], have been investigated in the area of quantum control. In developing practical quantum information technology, decoherence, which occurs when a quantum system interacts with its environment [22], has been a key issue that needs to be carefully addressed. Decoherence control in quantum systems has become one of the major objectives in quantum control. The methods of error-avoiding codes [23], error-correction codes [24] and dynamical decoupling [25] have been developed for decoherence control. However, there exist few results that investigated the robustness problem for open quantum systems with decoherence when uncertainties or errors exist in the system model [26, 27]. In [28–30], a sampled-data control design approach was presented for the robust control design of a qubit system with decoherence having imprecise parameters or uncertainties in the system–environment interaction or the system Hamiltonian. Here, we propose a robust control design approach for an open quantum system subject to decoherence where operator errors exist in the system model. The operator errors can arise from modelling errors or external disturbances. For example, a phase damping decoherence term for a two-level quantum system can be characterised by a functional of [e.g. ]. However, practical decoherence might correspond to a functional of [i.e. ] where we refer to as the operator error. For an open quantum system with operator errors, we employ the method of sampled-data control to design a control law and characterise the control performance using the sliding mode domain (SMD) [18, 31, 32]. Several sampled-data control results have been presented for quantum systems where discrete-time measurement outputs are used to design feedback controllers [16, 33]. From the measurement collapse postulate (see, e.g. [4]), we know that the sampling process in quantum control unavoidably changes the system's state. One well known example of this is the quantum Zeno effect, which is the inhibition of transitions between quantum states by frequent measurement of the state (see, e.g. [34, 35]). However, in practice it is usually difficult to implement frequent measurements for quantum systems. Similar to [18], we assume that a smaller measurement period is associated with a higher cost for accomplishing the periodic measurements. We will use the sampling process (that is realised using projective measurement in this paper) as a control tool and aim to design sampling periods as large as possible to guarantee the required control performance for open two-level quantum systems with operator errors. Since the sampling process serves as a control tool as well as the role of acquiring information, the sampled-data design method in this paper is different from the sampled-data approach in classical control theory and it may be looked as a partial feedback control scheme for quantum systems. The paper is constructed as follows. Section 2 formulates the quantum control problem and defines the required control performance. The main methods of sampled-data design and results for robust control design of open quantum systems are presented in Section 3. The detailed proofs of the main results are provided in Section 4. Section 5 gives the conclusions. 2 Problem formulation of quantum robust control The state of an open quantum system can be described by a density operator (a positive Hermitian density matrix) with . This paper considers a two-level quantum system with Markovian dynamics and assumes that the evolution of the system is described by the following Lindblad equation [22, 36]: (1)where the commutation operator is defined as , is set, characterises the strength of decoherence and For such a two-level quantum system, we assume that the Hamiltonian can be divided into two parts , where is the free Hamiltonian and () is the control Hamiltonian with In particular, three typical classes of decoherence will be considered [29, 32]: amplitude damping decoherence (ADD), phase damping decoherence (PDD) and depolarising decoherence (see, e.g. [4]). An open two-level quantum system with these three classes of decoherence may be described by a Lindblad equation using Pauli operators [28]. Since the Lindblad equation is an approximate model for the open two-level system, the system model may not be exactly described in a given form. There may exist operator errors due to inaccurate modelling as well as time-varying coupling between the system and its environment. We assume that for the practical system, the operators , and in their Lindblad equation should be replaced by the following , and : where . Usually . The practical models under consideration in this paper are listed as follows: (a) Approximate ADD (2)where and . This case is related to ADD where the population of the two-level system can change (e.g. through loss of energy by spontaneous emission). (b) Approximate PDD (3)This case is related to PDD where a loss of quantum coherence can occur without loss of energy in the two-level system. (c) Approximate depolarising decoherence (4)This case is related to depolarising decoherence that maps pure states into mixed states. For a quantum state , its purity can be defined as . A pure state may also be represented by a unit complex vector in a Hilbert space, where , and refers to the adjoint of . The fidelity between and may be defined as . A projective measurement with on a two-level system in state will make its state collapse into with probability or into with probability , where and are two eigenstates of with corresponding eigenvalues and 1, respectively. Such a process is also referred as the measurement collapse postulate. Moreover, the coherence is another useful quantity that may be defined as , where and (see, e.g. [37, 38]). This paper aims to design control laws for two-level quantum systems guaranteeing required control performance when operator errors exist in the system model. We use the concept of a SMD to define the required control performance for the three cases above as follows (see [28] for more detailed explanations). Definition 1.The SMD for an open two-level quantum system with approximate ADD is defined as , where is a given constant. Definition 2.The SMD for an open two-level quantum system with approximate PDD is defined as , where is a given constant. Definition 3.The SMD for an open two-level quantum system with approximate depolarising decoherence is defined as , where is a given constant. An SMD provides a unified concept to define the required performance. We aim to drive and then maintain the state of a two-level quantum system in a corresponding SMD. However, the decoherence may take the system's state away from the SMD. The sampling process (projective measurement) unavoidably changes the sampled system's state. Thus, we expect that the control law can guarantee that the system's state remains in the domain, except that the measurement operation may take it away from the domain with a small probability. 3 Robust control design methods and results In this paper, we present a sampled-data design approach for robust control of open two-level systems with operator errors, where the required control performance is defined using the concept of an SMD and a sampling period as large as possible is designed to guarantee the performance. The sampling process can be taken as a control tool for modifying the dynamics non-unitarily. For the case with approximate ADD, we also need to design a control law to steer the system's state back to the relevant SMD when the system's state collapses out of the SMD due to the sampling process. It is possible to design an appropriate control Hamiltonian to achieve this task (we call it Hamiltonian control in this paper). For each case, we first assume that no operator errors exist in the free Hamiltonian, i.e. . For example, it is possible to easily obtain the exact free Hamiltonian, but only approximate decoherence dynamics can be obtained. Then we consider the existence of operator errors in the free Hamiltonian, i.e. . For simplicity, we assume that there are the same operator errors for in and the decoherence term. In the sequel, we will provide the main design methods and theoretical results. When the assumption does not hold, it is straightforward to extend the proposed approach to obtain similar results. 3.1 Approximate ADD The objective is to present a control strategy to guarantee the required control performance when operator errors exist in the system model. According to Definition 1, we specify the required control performance as follows: (a) maintain the system's state in the SMD in which the system's state has a high fidelity () with the sliding mode state and (b) once the system's state collapses out of upon making a measurement (sampling), steer it back to within a short time period and maintain the state in for the following time period (where and is the sampling period). We use to characterise the fraction of the total sampling time interval over which the Hamiltonian control is applied. We usually choose to satisfy so that the system's state is kept in for most time [28]. The specific value of can be given considering practical applications. To guarantee the required performance, we design a control strategy based on sampled-data measurements as follows: For any sampling time (), (i) if the measurement result on the system corresponds to , let it evolve with zero control for and sample again at ; (ii) otherwise, apply a Hamiltonian control to drive the system's state back into a subset of from to , then sample again at . We switch the control between (i) and (ii) based on the result of sampled data. In (ii), the Hamiltonian control should steer the system's state into to guarantee the required performance when . We may define as . The sampling period and the Hamiltonian control may be designed offline in advance. The sequel will outline how to design the sampling period and establish a relationship between and aiming at guaranteeing the required performance. For two-level systems with approximate ADD, if its initial state is the excited state , the decoherence will make the system evolve toward the ground state . To guarantee the control performance defined by , we have the following results for designing a sampling period. Theorem 1.For a two-level quantum system with initial state at , the system evolves to subject to (2) where . If with (5)where , the state will remain in (where ). When a projective measurement with the operator is implemented at , the probability of failure satisfies . Hence, we can use as the sampling period to guarantee the required control performance. When the sampled data corresponds to , a Hamiltonian control is needed to steer the state back to a subset of . If we define as , when , the system's state will remain in for a time period after we drive it back to using a Hamiltonian control within the time period [28]. Now we consider that there exist operator errors in the free Hamiltonian, i.e. . The following results can be used to determine the required sampling period. Theorem 2.For a two-level quantum system with initial state at , the system evolves to subject to (2) where . If with (6)where , the state will remain in (where ). When a projective measurement with the operator is made at , the probability of failure satisfies . 3.2 Approximate PDD For a two-level quantum system with approximate PDD, the coherence is defined as where and . The PDD will reduce the coherence of the system. We aim to guarantee that the state has coherence by periodic sampling when approximate PDD exists. The following results can be used to determine the required sampling period. Theorem 3.For a two-level quantum system with initial state satisfying at , the system evolves to subject to (3) where . If with (7)the state will remain in . When a periodic projective measurement with the operator is made on the system, the sampling period can guarantee that the state remains in . Now we consider that there exist operator errors in the free Hamiltonian, i.e. . We have the following results to determine the sampling period. Theorem 4.For a two-level quantum system with initial state satisfying at , the system evolves to subject to (3) where . If with (8)the state will remain in . When a periodic projective measurement with the operator is made on the system, the sampling period can guarantee that the state remains in . 3.3 Approximate depolarising decoherence For a two-level quantum system, depolarising decoherence may reduce the state purity . We aim at guaranteeing that the purity is not less than by periodic sampling when there exists approximate depolarising decoherence. The following results can be used to determine the required sampling period. Theorem 5.For a two-level quantum system with initial state satisfying at , the system evolves to subject to (4) where . If with (9)where and , the state will remain in . If periodic projective measurements with the operator are made, the sampling period can guarantee that the state remains in . When we consider that there exist operator errors in the free Hamiltonian, i.e. , we have the same sampling period in Theorem 5 to guarantee the required robustness. Remark 1.We consider three types of approximate decoherence. As a special case, when there exists no decoherence (i.e. for a closed quantum system), possible operator errors can be treated as uncertainties in the system Hamiltonian. It is straightforward to use the conclusion in [28] to design a sampling period for guaranteeing a similar performance defined by . Moreover, it is also straightforward to generalise the proposed method for two-level quantum systems when different types of approximate decoherence coexist. Example 1.Here we give an example. We assume , (10% errors), (1% probability of failure), and . Then the corresponding sampling periods are , , , and . Considering the example in [28], these sampling periods should be in the order of a nanosecond. Example 2.We consider the case of approximate depolarising decoherence and use the parameter settings and the designed sampling period in Example 1. We assume that and there is no operator error in the free Hamiltonian. The operator error coefficients () are assumed to be six independent white noise processes uniformly distributed in . A step length of is used to discretise (4) for numerical simulation: (10)We then recursively calculate . When , and , a projective measurement using is performed. The simulation result is shown in Fig. 1, where the vertical axis is the purity and the horizontal axis is the time . From Fig. 1, it is clear that the control law can indeed keep the system's state in the sliding mode domain since the purity is greater than 0.96. Fig. 1Open in figure viewerPowerPoint Purity versus evolution time for approximate depolarising decoherence 4 Proof of the main results 4.1 Proof of Theorem 1 Proof.For a two-level quantum system, its state can be represented in terms of the Bloch vector : (11)Considering the open two-level quantum system subject to (2), when and , using (11), we have (12)Denoting (13)where , we have (14)Let . We have the solution . Hence, (15) (16)When where satisfies (5), we have (17)Hence, if we make a measurement with on the system, the probability of failure satisfies . □ 4.2 Proof of Theorem 2 Proof.For the open two-level quantum system subject to (2), when and , using (11), we have (18)Denoting (19)we have (20)Let . We have the optimal solution Hence (21) (22)When where satisfies (6), we have (23)Hence, if we make a measurement with on the system, the probability of failure satisfies . □ 4.3 Proof of Theorem 3 Proof.For a two-level quantum system subject to (3), when , using (11), we have (24) (25)Let . We have (26)Using the Cauchy–Schwarz inequality, we know (27) (28)Hence, we have (29)Since , we know (30)If where satisfies (7), we have . □ 4.4 Proof of Theorem 4 Proof.For a two-level quantum system subject to (3), when , using (11), we have (31) (32)Let . We have (33)Using the Cauchy–Schwarz inequality, we know (34)Together with (28), we have (35)Since , we know (36)If where satisfies (8), we have . □ 4.5 Proof of Theorem 5 Proof.For a two-level quantum system subject to (4), when , and , using (11), we have (37)Let and . It is clear that By integration, we have (38)If where satisfies (9), we have □ 5 Conclusions In this paper, we investigated a robust control design problem for an open two-level quantum system with operator errors in the model. Sampled-data control approaches are employed to design control laws for three cases including approximate ADD, approximate PDD and approximate depolarising decoherence. We present several sufficient conditions to guarantee the required control performance. The control laws may be designed offline in advance and then they can be used online to open two-level systems with operator errors. Although we only consider two-level quantum systems in this paper, the proposed approach can also be generalised to multilevel quantum systems with some simple dynamics forms. However, for general open multilevel systems, we need to develop new robust control methods when operator errors exist in the system model. 6 Acknowledgments This work was supported by the Australian Research Council (DP130101658, FL110100020) and National Natural Science Foundation of China (no. 61374092). H. Rabitz acknowledges support from the US ARO. 7 References 1Dong D., and Petersen I.R.: ' Quantum control theory and applications: a survey', IET Control Theory Appl., 2010, 4, pp. 2651– 2671CrossrefWeb of Science®Google Scholar 2Wiseman H.M., and Milburn G.J.: ' Quantum measurement and control' ( Cambridge University Press, Cambridge, England, 2010) Google Scholar 3Altafini C., and Ticozzi F.: ' Modeling and control of quantum systems: an introduction', IEEE Trans. Autom. Control, 2012, 57, pp. 1898– 1917CrossrefWeb of Science®Google Scholar 4Nielsen M.A., and Chuang I.L.: ' Quantum Computation and Quantum Information' ( Cambridge University Press, Cambridge, England, 2000) Google Scholar 5D'Alessandro D.: ' Introduction to quantum control and dynamics' ( Chapman & Hall/CRC, 2007) CrossrefGoogle Scholar 6Boscain U., and Mason P.: ' Time minimal trajectories for a spin 1/2 particle in a magnetic field', J. Math. Phys., 2006, 47, p. 062101CrossrefWeb of Science®Google Scholar 7Fu S., and Chen M.Z.Q.: ' Optimal control of single spin- quantum systems', IET Control Theory Appl., 2014, 8, pp. 86– 93Wiley Online LibraryWeb of Science®Google Scholar 8Doherty A.C. Habib S., and Jacobs K. et al.: ' Quantum feedback control and classical control theory', Phys. Rev. A, 2000, 62, p. 012105CrossrefWeb of Science®Google Scholar 9van Handel R. Stockton J.K., and Mabuchi H.: ' Feedback control of quantum state reduction', IEEE Trans. Autom. Control, 2005, 50, (6), pp. 768– 780CrossrefWeb of Science®Google Scholar 10Shabani A., and Jacobs K.: ' Locally optimal control of quantum systems with strong feedback', Phys. Rev. Lett., 2008, 101, p. 230403CrossrefPubMedWeb of Science®Google Scholar 11Qi B. Pan H., and Guo L.: ' Further results on stabilizing control of quantum system', IEEE Trans. Autom. Control, 2013, 58, (5), pp. 1349– 1354CrossrefWeb of Science®Google Scholar 12Sayrin C. Dotsenko I., and Zhou X. et al.: ' Real-time quantum feedback prepares and stabilizes photon number states', Nature, 2011, 477, pp. 73– 77CrossrefCASPubMedWeb of Science®Google Scholar 13Mirrahimi M., and van Handel R.: ' Stabilizing feedback controls for quantum systems', SIAM Optim. Control, 2007, 46, (2), pp. 445– 467CrossrefWeb of Science®Google Scholar 14Qi B., and Guo L.: ' Is measurement-based feedback still better for quantum control systems?', Syst. Control Lett., 2010, 59, pp. 333– 339CrossrefWeb of Science®Google Scholar 15James M.R. Nurdin H.I., and Petersen I.R.: ' control of linear quantum stochastic systems', IEEE Trans. Autom. Control, 2008, 53, pp. 1787– 1803CrossrefWeb of Science®Google Scholar 16James M.R.: ' Risk-sensitive optimal control of quantum systems', Phys. Rev. A, 2004, 69, p. 032108CrossrefCASWeb of Science®Google Scholar 17Dong D., and Petersen I.R.: ' Sliding mode control of quantum systems', New J. Phys., 2009, 11, p. 105033CrossrefWeb of Science®Google Scholar 18Dong D., and Petersen I.R.: ' Sliding mode control of two-level quantum systems', Automatica, 2012, 48, pp. 725– 735CrossrefWeb of Science®Google Scholar 19Soare A. Ball H., and Hayes D. et al.: ' Experimental noise filtering by quantum control', Nat. Phys., 2014, 10, pp. 825– 829CrossrefCASWeb of Science®Google Scholar 20Chen C. Dong D., and Long R. et al.: ' Sampling-based learning control of inhomogeneous quantum ensembles', Phys. Rev. A, 2014, 89, p. 023402CrossrefWeb of Science®Google Scholar 21Dong D. Chen C., and Qi B. et al.: ' Robust manipulation of superconducting qubits in the presence of fluctuations', Sci. Rep., 2015, 5, p. 7873CrossrefCASPubMedWeb of Science®Google Scholar 22Breuer H.-P., and Petruccione F.: ' The theory of open quantum systems' ( Oxford University Press, Oxford, UK, 2002, 1st edn.) Google Scholar 23Lidar D.A. Chuang I.L., and Whaley K.B.: ' Decoherence-free subspaces for quantum computation', Phys. Rev. Lett., 1999, 81, pp. 2594– 2597CrossrefWeb of Science®Google Scholar 24Knill E. Laflamme R., and Viola L.: ' Theory of quantum error correction for general noise', Phys. Rev. Lett., 2000, 84, pp. 2525– 2528CrossrefCASPubMedWeb of Science®Google Scholar 25Viola L. Knill E., and Lloyd S.: ' Dynamical decoupling of open quantum systems', Phys. Rev. Lett., 1999, 82, pp. 2417– 2421CrossrefCASWeb of Science®Google Scholar 26Qi B.: ' A two-step strategy for stabilizing control of quantum systems with uncertainties', Automatica, 2013, 49, pp. 834– 839CrossrefWeb of Science®Google Scholar 27Ticozzi F., and Viola L.: ' Quantum Markovian subsystems: invariance', attractivity, and control, IEEE Trans. Autom. Control, 2008, 53, pp. 2048– 2063CrossrefWeb of Science®Google Scholar 28Dong D. Petersen I.R., and Rabitz H.: ' Sampled-data design for robust control of a single qubit', IEEE Trans. Autom. Control, 2013, 58, pp. 2654– 2659CrossrefWeb of Science®Google Scholar 29Dong D. Petersen I.R., and Rabitz H.: ' Sampled-data design for robust decoherence control of a single qubit'. Proc. of 51st IEEE Conf. on Decision and Control, Maui, USA, 10–13 December 2012, pp. 1668– 1673CrossrefGoogle Scholar 30Wang Y. Chen C., and Dong D.: ' Further results on sampled-data design for robust control of a single qubit', Int. J. Control, 2014, 87, pp. 2056– 2064Web of Science®Google Scholar 31Dong D., and Petersen I.R.: ' Notes on sliding mode control of two-level quantum systems', Automatica, 2012, 48, pp. 3089– 3097CrossrefWeb of Science®Google Scholar 32Dong D. Petersen I.R., and Yi X.X. et al.: ' Sampled-data design for decoherence control of a single qubit with operator errors'. Proc. of 2012 Australian Control Conf., Sydney, Australia, 15–16 November 2012, pp. 13– 18Google Scholar 33Maalouf A.I., and Petersen I.R.: ' Sampled-data LQG control for a class of linear quantum systems', Syst. Control Lett., 2012, 61, pp. 369– 374CrossrefWeb of Science®Google Scholar 34Itano W.M. Heinzen D.J., and Bollinger J.J. et al.: ' Quantum Zeno effect', Phys. Rev. A, 1990, 41, pp. 2295– 2300CrossrefCASPubMedWeb of Science®Google Scholar 35Beige A. Braun D., and Tregenna B. et al.: ' Quantum computing using dissipation to remain in a decoherence-free subspace', Phys. Rev. Lett., 2000, 85, pp. 1762– 1765CrossrefCASPubMedWeb of Science®Google Scholar 36Lindblad G.: ' On the generators of quantum dynamical semigroups', Commun. Math. Phys., 1976, 48, pp. 119– 130CrossrefWeb of Science®Google Scholar 37Lidar D.A., and Schneider S.: ' Stabilizing qubit coherence via tracking-control', Quantum Inf. Comput., 2005, 5, pp. 350– 363Web of Science®Google Scholar 38Zhang J. Wu R.B., and Li C.W. et al.: ' Protecting coherence and entanglement by quantum feedback controls', IEEE Trans. Autom. Control, 2010, 55, (3), pp. 619– 633CrossrefWeb of Science®Google Scholar Citing Literature Volume10, Issue18December 2016Pages 2415-2421 FiguresReferencesRelatedInformation