Finite-time stability and stabilization of time-delay systems

International audienceFinite-time stability and stabilization of retarded-type functional differential equations are developed. First, a theoretical result on finite-time stability inspired by the theory of differential equations, using Lyapunov functionals, is given. As it may appear not easily usable in practice, we show how to obtain finite-time stabilization of linear systems with delays in the input by using an extension of Artstein's model reduction to nonlinear feedback. With this approach, we give an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input

Finite-Time Boundedness and Stabilization of Networked Control Systems with Time Delay

The finite-time control problem of a class of networked control systems (NCSs) with time delay is investigated. The main results provided in the paper are sufficient conditions for finite-time stability via state feedback. An augmentation approach is proposed to model NCSs with time delay as linear systems. Based on finite time stability theory, the sufficient conditions for finite-time boundedness and stabilization of the underlying systems are derived via linear matrix inequalities (LMIs) formulation. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed results

An Improved Finite-Time Stability and Stabilization of Linear System with Constant Delay

Practical systems in engineering fields often require that values of state variables, during the finite-time interval, must not exceed a certain value when the initial values of state are given. This leads us to investigate the finite-time stability and stabilization of a linear system with a constant time-delay. Sufficient conditions to guarantee the finite-time stability and stabilization are derived by using a new form of Lyapunov-Krasovskii functional and a desired state-feedback controller. These conditions are in the form of LMIs and inequalities. Two numerical examples are given to show the effectiveness of the proposed criteria. Results show that our proposed criteria are less conservative than previous works in terms of versatility of minimum bounds and larger bounds of time-delay

On finite-time stabilization of a class of nonlinear time-delay systems: Implicit Lyapunov-Razumikhin approach

International audienceTheorems on Implicit Lyapunov-Razumikhin functions (ILRF) for asymptotic, exponential, finite-time and nearly fixed-time stability analysis of nonlinear time-delay systems are presented. Based on these results, finite-time stabilization of a special class of such systems is addressed. These systems are represented by a chain of integrators with a time-delay term multiplied by a function of instantaneous state vector. Possible explicit restriction on nonlinear time-delay terms is discussed. Simple procedure of control parameters calculation is given in terms of linear matrix inequalities (LMIs). Some aspects of digital implementations of the presented nonlinear control law are touched upon. Theoretical results are illustrated by numerical simulations

Delay-Scheduled State-Feedback Design for Time-Delay Systems with Time-Varying Delays - A LPV Approach

International audienceThis paper is concerned with the synthesis of delay-scheduled state-feedback controllers which stabilize linear systems with time-varying delays. In this framework, it is assumed that the delay is approximately known in real-time and used in the controller in a scheduling fashion. First, a new model transformation turning a time-delay system into an uncertain LPV system is introduced. Using this transformation, a new delay-dependent stability test based on the so-called full block S-procedure is developed and from this result, a new delay-dependent stabilization result is derived. Since the resulting LMI conditions depend polynomially on the parameters, a relaxation result is then applied in order to obtain a tractable finite set of finite-dimensional LMIs. The interests of the approach resides in 1) the synthesis of a new type of controllers scheduled by the delay value which has a lower memory consumption than controllers with memory (since it is not necessary to store past values of the state), and 2) an easy consideration of uncertainties on the delay knowledge

Robustness analysis of discrete predictor-based controllers for input-delay systems

In this article, robustness to model uncertainties are analysed in the context of discrete predictor-based state-feedback controllers for discrete-time input-delay systems with time-varying delay, in an LMI framework. The goal is comparing robustness of predictor-based strategies with respect to other (sub)optimal state feedback ones. A numerical example illustrates that improvements in tolerance to modelling errors can be achieved by using the predictor framework.The authors are grateful for grant nos. DPI2008-06737-C02-01, DPI2008-06731-C02-01, DPI2011-27845-C02-01 and PROMETEO/2008/088 from the Spanish and Valencian governments.González Sorribes, A.; Sala, A.; García Gil, PJ.; Albertos Pérez, P. (2013). Robustness analysis of discrete predictor-based controllers for input-delay systems. International Journal of Systems Science. 44(2):232-239. https://doi.org/10.1080/00207721.2011.600469S232239442Boukas, E.-K. (2006). Discrete-time systems with time-varying time delay: Stability and stabilizability. Mathematical Problems in Engineering, 2006, 1-10. doi:10.1155/mpe/2006/42489Du, D., Jiang, B., & Zhou, S. (2008). Delay-dependent robust stabilisation of uncertain discrete-time switched systems with time-varying state delay. International Journal of Systems Science, 39(3), 305-313. doi:10.1080/00207720701805982El Ghaoui, L., Oustry, F., & AitRami, M. (1997). A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 42(8), 1171-1176. doi:10.1109/9.618250Gao, H., & Chen, T. (2007). New Results on Stability of Discrete-Time Systems With Time-Varying State Delay. IEEE Transactions on Automatic Control, 52(2), 328-334. doi:10.1109/tac.2006.890320Gao, H., Wang, C., Lam, J., & Wang, Y. (2004). Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay. IEE Proceedings - Control Theory and Applications, 151(6), 691-698. doi:10.1049/ip-cta:20040822Gao, H., Chen, T., & Lam, J. (2008). A new delay system approach to network-based control. Automatica, 44(1), 39-52. doi:10.1016/j.automatica.2007.04.020Garcia , P , Castillo , P , Lozano , R and Albertos , P . 2006 . Robustness with Respect to Delay Uncertainties of a Predictor Observer Based Discrete-time Controller . Proceeding of the 45th IEEE Conference on Decision and Control . 2006 . pp. 199 – 204 .Guo , Y and Li , S . 2009 . New Stability Criterion for Discrete-time Systems with Interval Time-varying State Delay . Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference . 2009 . pp. 1342 – 1347 .Hägglund, T. (1996). An industrial dead-time compensating PI controller. Control Engineering Practice, 4(6), 749-756. doi:10.1016/0967-0661(96)00065-2V.J.S. Leite, and Miranda, M.F. (2008), ‘Robust Stabilization of Discrete-time Systems with Time-varying Delay: An LMI Approach’,Mathematical Problems in Engineering, 2008, 15 pages (doi:10.1155/2008/875609)Liu, X. G., Tang, M. L., Martin, R. R., & Wu, M. (2006). Delay-dependent robust stabilisation of discrete-time systems with time-varying delay. IEE Proceedings - Control Theory and Applications, 153(6), 689-702. doi:10.1049/ip-cta:20050223Lozano, R., Castillo, P., Garcia, P., & Dzul, A. (2004). Robust prediction-based control for unstable delay systems: Application to the yaw control of a mini-helicopter. Automatica, 40(4), 603-612. doi:10.1016/j.automatica.2003.10.007Manitius, A., & Olbrot, A. (1979). Finite spectrum assignment problem for systems with delays. IEEE Transactions on Automatic Control, 24(4), 541-552. doi:10.1109/tac.1979.1102124Michiels, W., & Niculescu, S.-I. (2003). On the delay sensitivity of Smith Predictors. International Journal of Systems Science, 34(8-9), 543-551. doi:10.1080/00207720310001609057Palmor, Z.J. (1996), ‘Time-delay Compensation – Smith Predictor and Its Modifications’, inThe Control Handbook, ed. W.S. Levine, Boca Raton: CRC Press, pp. 224–237Pan, Y.-J., Marquez, H. J., & Chen, T. (2006). Stabilization of remote control systems with unknown time varying delays by LMI techniques. International Journal of Control, 79(7), 752-763. doi:10.1080/00207170600654554Richard, J.-P. (2003). Time-delay systems: an overview of some recent advances and open problems. Automatica, 39(10), 1667-1694. doi:10.1016/s0005-1098(03)00167-5Wang, Q.-G., Lee, T. H., & Tan, K. K. (1999). Finite-Spectrum Assignment for Time-Delay Systems. Lecture Notes in Control and Information Sciences. doi:10.1007/978-1-84628-531-8He, Y., Wu, M., Han, Q.-L., & She, J.-H. (2008). Delay-dependentH∞control of linear discrete-time systems with an interval-like time-varying delay. International Journal of Systems Science, 39(4), 427-436. doi:10.1080/00207720701832531Yue, D., & Han, Q.-L. (2005). Delayed feedback control of uncertain systems with time-varying input delay. Automatica, 41(2), 233-240. doi:10.1016/j.automatica.2004.09.006Zhang, B., Xu, S., & Zou, Y. (2008). Improved stability criterion and its applications in delayed controller design for discrete-time systems. Automatica, 44(11), 2963-2967. doi:10.1016/j.automatica.2008.04.01

Stability and stabilization of fractional order time delay systems

U ovom radu predstavljeni su neki osnovni rezultati koji se odnose na kriterijume stabilnosti sistema necelobrojnog reda sa kašnjenjem kao i za sisteme necelobrojnog reda bez kašnjenja.Takođe, dobijeni su i predstavljeni dovoljni uslovi za konačnom vremenskom stabilnost i stabilizacija za (ne)linearne (ne)homogene kao i za perturbovane sisteme necelobrojnog reda sa vremenskim kašnjenjem. Nekoliko kriterijuma stabilnosti za ovu klasu sistema necelobrojnog reda je predloženo korišćenjem nedavno dobijene generalizovane Gronval nejednakosti, kao i 'klasične' Belman-Gronval nejednakosti. Neki zaključci koji se odnose na stabilnost sistema necelobrojnog reda su slični onima koji se odnose na klasične sisteme celobrojnog reda. Na kraju, numerički primer je dat u cilju ilustracije značaja predloženog postupka.In this paper, some basic results of the stability criteria of fractional order system with time delay as well as free delay are presented. Also, we obtained and presented sufficient conditions for finite time stability and stabilization for (non)linear (non)homogeneous as well as perturbed fractional order time delay systems. Several stability criteria for this class of fractional order systems are proposed using a recently suggested generalized Gronwall inequality as well as 'classical' Bellman-Gronwall inequality. Some conclusions for stability are similar to those of classical integerorder differential equations. Finally, a numerical example is given to illustrate the validity of the proposed procedure

New Results on Delay-Dependent Stability Analysis and Stabilization of Time-Delay Systems

The interconnection between physical systems is accomplished by flow of information, energy and material, alternatively known as transport or propagation. As such flows may take a finite amount of time, the reaction of real world systems to exogenous or feedback control signals, from automatic control perspective, are not instantaneous. This results time-delays in systems connected by real-world physical media. Indeed, examples of time-delay systems span biology, ecology, economy, and of course, engineering. To this end, it is known that an arbitrary small delay may destabilize a stable system whereas, a delay in the controller may be used to stabilize a system that is otherwise not stabilizable by using a delay-free controller. In general, the presence of time-delay in a system makes the system dynamics infinite-dimensional, and analysis of such systems is complex.This thesis investigates stability analysis and stabilization of time-delay systems. It proposes a delay-decomposition approach for stability analysis of systems with single delay that leads to a simple LMI condition using a Lyapunov-Krasovskii functional. Moreover, a static state feedback controller is designed for systems with state and input-delay using this delay-decomposition approach. Numerical comparison of the present results vis-`a-vis the existing ones for the systems with constant delay considered shows that the present ones are superior. Next, a PI-type controller is implemented for systems with input-delay to improve the tolerable delay bound. Other problems considered is to analyze the stability of systems with two delays. As the number of delays incorporated in the system dynamics increases, it becomes further complex for analysis

Rejection of mismatched disturbances for systems with input delay via a predictive extended state observer

[EN] The problem of output stabilization and disturbance rejection for input-delayed systems is tackled in this work. First, a suitable transformation is introduced to translate mismatched disturbances into an equivalent input disturbance. Then, an extended state observer is combined with a predictive observer structure to obtain a future estimation of both the state and the disturbance. A disturbance model is assumed to be known but attenuation of unmodeled components is also considered. The stabilization is proved via Lyapunov-Krasovskii functionals, leading to sufficient conditions in terms of linear matrix inequalities for the closed-loop analysis and parameter tuning. The proposed strategy is illustrated through a numerical example.PROMETEOII/2013/004; Conselleria d'Educacio; Generalitat Valenciana, Grant/Award Number: TIN2014-56158-C4-4-P-AR; Ministerio de Economia y Competitividad, Grant/Award Number: FPI-UPV 2014; Universitat Politecnica de ValenciaSanz Diaz, R.; García Gil, PJ.; Fridman, E.; Albertos Pérez, P. (2018). Rejection of mismatched disturbances for systems with input delay via a predictive extended state observer. International Journal of Robust and Nonlinear Control. 28(6):2457-2467. https://doi.org/10.1002/rnc.4027S24572467286Stability and Stabilization of Systems with Time Delay. (2011). IEEE Control Systems, 31(1), 38-65. doi:10.1109/mcs.2010.939135Fridman, E. (2014). Introduction to Time-Delay Systems. Systems & Control: Foundations & Applications. doi:10.1007/978-3-319-09393-2Watanabe, K., & Ito, M. (1981). A process-model control for linear systems with delay. IEEE Transactions on Automatic Control, 26(6), 1261-1269. doi:10.1109/tac.1981.1102802Astrom, K. J., Hang, C. C., & Lim, B. C. (1994). A new Smith predictor for controlling a process with an integrator and long dead-time. IEEE Transactions on Automatic Control, 39(2), 343-345. doi:10.1109/9.272329Matausek, M. R., & Micic, A. D. (1996). A modified Smith predictor for controlling a process with an integrator and long dead-time. IEEE Transactions on Automatic Control, 41(8), 1199-1203. doi:10.1109/9.533684García, P., & Albertos, P. (2008). A new dead-time compensator to control stable and integrating processes with long dead-time. Automatica, 44(4), 1062-1071. doi:10.1016/j.automatica.2007.08.022Normey-Rico, J. E., & Camacho, E. F. (2009). Unified approach for robust dead-time compensator design. Journal of Process Control, 19(1), 38-47. doi:10.1016/j.jprocont.2008.02.003Manitius, A., & Olbrot, A. (1979). Finite spectrum assignment problem for systems with delays. IEEE Transactions on Automatic Control, 24(4), 541-552. doi:10.1109/tac.1979.1102124Artstein, Z. (1982). Linear systems with delayed controls: A reduction. IEEE Transactions on Automatic Control, 27(4), 869-879. doi:10.1109/tac.1982.1103023Krstic, M. (2008). Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch. Automatica, 44(11), 2930-2935. doi:10.1016/j.automatica.2008.04.010Léchappé, V., Moulay, E., Plestan, F., Glumineau, A., & Chriette, A. (2015). New predictive scheme for the control of LTI systems with input delay and unknown disturbances. Automatica, 52, 179-184. doi:10.1016/j.automatica.2014.11.003Sanz, R., Garcia, P., & Albertos, P. (2016). Enhanced disturbance rejection for a predictor-based control of LTI systems with input delay. Automatica, 72, 205-208. doi:10.1016/j.automatica.2016.05.019Basturk, H. I., & Krstic, M. (2015). Adaptive sinusoidal disturbance cancellation for unknown LTI systems despite input delay. Automatica, 58, 131-138. doi:10.1016/j.automatica.2015.05.013Basturk, H. I. (2017). Cancellation of unmatched biased sinusoidal disturbances for unknown LTI systems in the presence of state delay. Automatica, 76, 169-176. doi:10.1016/j.automatica.2016.10.006Sanz, R., Garcia, P., Albertos, P., & Zhong, Q.-C. (2016). Robust controller design for input-delayed systems using predictive feedback and an uncertainty estimator. International Journal of Robust and Nonlinear Control, 27(10), 1826-1840. doi:10.1002/rnc.3639Mondie, S., & Michiels, W. (2003). Finite spectrum assignment of unstable time-delay systems with a safe implementation. IEEE Transactions on Automatic Control, 48(12), 2207-2212. doi:10.1109/tac.2003.820147Zhong, Q.-C. (2004). On Distributed Delay in Linear Control Laws—Part I: Discrete-Delay Implementations. IEEE Transactions on Automatic Control, 49(11), 2074-2080. doi:10.1109/tac.2004.837531Zhou, B., Lin, Z., & Duan, G.-R. (2012). Truncated predictor feedback for linear systems with long time-varying input delays. Automatica, 48(10), 2387-2399. doi:10.1016/j.automatica.2012.06.032Zhou, B., Li, Z.-Y., & Lin, Z. (2013). On higher-order truncated predictor feedback for linear systems with input delay. International Journal of Robust and Nonlinear Control, 24(17), 2609-2627. doi:10.1002/rnc.3012Besançon G Georges D Benayache Z Asymptotic state prediction for continuous-time systems with delayed input and application to control IEEE 2007 Kos, GreeceNajafi, M., Hosseinnia, S., Sheikholeslam, F., & Karimadini, M. (2013). Closed-loop control of dead time systems via sequential sub-predictors. International Journal of Control, 86(4), 599-609. doi:10.1080/00207179.2012.751627Léchappé V Moulay E Plestan F Dynamic observation-prediction for LTI systems with a time-varying delay in the input IEEE 2016 Las Vegas, NVCacace, F., Conte, F., Germani, A., & Pepe, P. (2016). Stabilization of strict-feedback nonlinear systems with input delay using closed-loop predictors. International Journal of Robust and Nonlinear Control, 26(16), 3524-3540. doi:10.1002/rnc.3517Mazenc, F., & Malisoff, M. (2017). Stabilization of Nonlinear Time-Varying Systems Through a New Prediction Based Approach. IEEE Transactions on Automatic Control, 62(6), 2908-2915. doi:10.1109/tac.2016.2600500Guo, L., & Chen, W.-H. (2005). Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach. International Journal of Robust and Nonlinear Control, 15(3), 109-125. doi:10.1002/rnc.978Fridman, E. (2003). Output regulation of nonlinear systems with delay. Systems & Control Letters, 50(2), 81-93. doi:10.1016/s0167-6911(03)00131-2Isidori, A., & Byrnes, C. I. (1990). Output regulation of nonlinear systems. IEEE Transactions on Automatic Control, 35(2), 131-140. doi:10.1109/9.45168Ding, Z. (2003). Global stabilization and disturbance suppression of a class of nonlinear systems with uncertain internal model. Automatica, 39(3), 471-479. doi:10.1016/s0005-1098(02)00251-0Chen, W.-H., Yang, J., Guo, L., & Li, S. (2016). Disturbance-Observer-Based Control and Related Methods—An Overview. IEEE Transactions on Industrial Electronics, 63(2), 1083-1095. doi:10.1109/tie.2015.2478397Fridman, E., & Shaked, U. (2002). An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control, 47(11), 1931-1937. doi:10.1109/tac.2002.804462Fridman, E., & Orlov, Y. (2009). Exponential stability of linear distributed parameter systems with time-varying delays. Automatica, 45(1), 194-201. doi:10.1016/j.automatica.2008.06.00

Robust controller design for input-delayed systems using predictive feedback and an uncertainty estimator

[EN] This paper deals with the problem of stabilizing a class of input-delayed systems with (possibly) nonlinear uncertainties by using explicit delay compensation. It is well known that plain predictive schemes lack robustness with respect to uncertain model parameters. In this work, an uncertainty estimator is derived for input-delay systems and combined with a modified state predictor, which uses current available information of the estimated uncertainties. Furthermore, based on Lyapunov-Krasovskii functionals, a computable criterion to check robust stability of the closed-loop is developed and cast into a minimization problem constrained to an LMI. Additionally, for a given input delay, an iterative-LMI algorithm is proposed to design stabilizing tuning parameters. The main results are illustrated and validated using a numerical example with a second-order dynamic system.This work was partially supported by projects PROMETEOII/2013/004, Conselleria d Educació, Generalitat Valenciana, and TIN2014-56158-C4-4-P-AR, Ministerio de Economía y Competitividad, Spain.Sanz Diaz, R.; García Gil, PJ.; Albertos Pérez, P.; Zhong, Q. (2017). Robust controller design for input-delayed systems using predictive feedback and an uncertainty estimator. International Journal of Robust and Nonlinear Control. 27(10):1826-1840. https://doi.org/10.1002/rnc.3639S182618402710Stability and Stabilization of Systems with Time Delay. (2011). IEEE Control Systems, 31(1), 38-65. doi:10.1109/mcs.2010.939135Normey-Rico, J. E., Bordons, C., & Camacho, E. F. (1997). Improving the robustness of dead-time compensating PI controllers. Control Engineering Practice, 5(6), 801-810. doi:10.1016/s0967-0661(97)00064-6Michiels, W., & Niculescu, S.-I. (2003). On the delay sensitivity of Smith Predictors. International Journal of Systems Science, 34(8-9), 543-551. doi:10.1080/00207720310001609057Normey-Rico, J. E., & Camacho, E. F. (2008). Dead-time compensators: A survey. Control Engineering Practice, 16(4), 407-428. doi:10.1016/j.conengprac.2007.05.006Guzmán, J. L., García, P., Hägglund, T., Dormido, S., Albertos, P., & Berenguel, M. (2008). Interactive tool for analysis of time-delay systems with dead-time compensators. Control Engineering Practice, 16(7), 824-835. doi:10.1016/j.conengprac.2007.09.002Manitius, A., & Olbrot, A. (1979). Finite spectrum assignment problem for systems with delays. IEEE Transactions on Automatic Control, 24(4), 541-552. doi:10.1109/tac.1979.1102124Artstein, Z. (1982). Linear systems with delayed controls: A reduction. IEEE Transactions on Automatic Control, 27(4), 869-879. doi:10.1109/tac.1982.1103023Moon, Y. S., Park, P., & Kwon, W. H. (2001). Robust stabilization of uncertain input-delayed systems using reduction method. Automatica, 37(2), 307-312. doi:10.1016/s0005-1098(00)00145-xYue, D. (2004). Robust stabilization of uncertain systems with unknown input delay. Automatica, 40(2), 331-336. doi:10.1016/j.automatica.2003.10.005Yue, D., & Han, Q.-L. (2005). Delayed feedback control of uncertain systems with time-varying input delay. Automatica, 41(2), 233-240. doi:10.1016/j.automatica.2004.09.006Lozano, R., Castillo, P., Garcia, P., & Dzul, A. (2004). Robust prediction-based control for unstable delay systems: Application to the yaw control of a mini-helicopter. Automatica, 40(4), 603-612. doi:10.1016/j.automatica.2003.10.007Gonzalez, A., Garcia, P., Albertos, P., Castillo, P., & Lozano, R. (2012). Robustness of a discrete-time predictor-based controller for time-varying measurement delay. Control Engineering Practice, 20(2), 102-110. doi:10.1016/j.conengprac.2011.09.001Karafyllis, I., & Krstic, M. (2013). Robust predictor feedback for discrete-time systems with input delays. International Journal of Control, 86(9), 1652-1663. doi:10.1080/00207179.2013.792005Krstic, M. (2010). Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems. IEEE Transactions on Automatic Control, 55(2), 287-303. doi:10.1109/tac.2009.2034923Bekiaris-Liberis, N., & Krstic, M. (2011). Compensation of Time-Varying Input and State Delays for Nonlinear Systems. Journal of Dynamic Systems, Measurement, and Control, 134(1). doi:10.1115/1.4005278Karafyllis, I., Malisoff, M., Mazenc, F., & Pepe, P. (Eds.). (2016). Recent Results on Nonlinear Delay Control Systems. Advances in Delays and Dynamics. doi:10.1007/978-3-319-18072-4Cacace, F., Conte, F., Germani, A., & Pepe, P. (2016). Stabilization of strict-feedback nonlinear systems with input delay using closed-loop predictors. International Journal of Robust and Nonlinear Control, 26(16), 3524-3540. doi:10.1002/rnc.3517Fridman, E., & Shaked, U. (2002). An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control, 47(11), 1931-1937. doi:10.1109/tac.2002.804462Fridman, E., & Shaked, U. (2002). A descriptor system approach to H/sub ∞/ control of linear time-delay systems. IEEE Transactions on Automatic Control, 47(2), 253-270. doi:10.1109/9.983353Chen, W.-H., & Zheng, W. X. (2006). On improved robust stabilization of uncertain systems with unknown input delay. Automatica, 42(6), 1067-1072. doi:10.1016/j.automatica.2006.02.015Krstic, M. (2008). Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch. Automatica, 44(11), 2930-2935. doi:10.1016/j.automatica.2008.04.010Léchappé, V., Moulay, E., Plestan, F., Glumineau, A., & Chriette, A. (2015). 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