Advances in the Theory of Fixed-time Stability with Applications in Constrained Control and Optimization

Driving the state of dynamical systems to a desired point or set is a problem of crucial practical importance. Various constraints are present in real-world applications due to structural and operational requirements. Spatial constraints, i.e., constraints requiring the system trajectories to evolve in some textit{safe} set, while visiting some goal set(s), are typical in safety-critical applications. Furthermore, temporal constraints, i.e., constraints pertaining to the time of convergence, appear in time-critical applications, for instance, when a task must complete within a fixed time due to an internal or an external deadline. Moreover, imperfect knowledge of the operational environment and/or system dynamics, and the presence of external disturbances render offline control policies impractical and make it essential to develop methods for online control synthesis. Thus, from the implementation point-of-view, it is desired to design fast optimization algorithms so that an optimal control input, e.g., min-norm control input, can be computed online. As compared to exponential stability, the notion of fixed-time stability is stronger, with the time of convergence being finite and is bounded for all initial conditions. This dissertation studies the theory of fixed-time stability with applications in multi-agent control design under spatiotemporal and input constraints, and in the field of continuous-time optimization. First, multi-agent control design problems under spatiotemporal constraints are studied. A vector-field-based controller is presented for distributed control of multi-agent systems for a class of agents modeled under double-integrator dynamics. A finite-time controller that utilizes the state estimates obtained from a finite-time state observer is designed to guarantee that each agent reaches its goal location within a finite time while maintaining safety with respect to other agents as well as dynamic obstacles. Next, new conditions for fixed-time stability are developed to use fixed-time stability along with input constraints. It is shown that these new conditions capture the relationship between the time of convergence, the domain of attraction, and the input constraints for fixed-time stability. Additionally, the new conditions establish the robustness of fixed-time stable systems with respect to a class of vanishing and non-vanishing additive disturbances. Utilizing these new fixed-time stability results, a control design method using convex optimization is presented for a general class of systems having nonlinear, control-affine dynamics. Control barrier and control Lyapunov function conditions are used as linear constraints in the optimization problem for set-invariance and goal-reachability requirements. Various practical issues, such as input constraints, additive disturbance, and state-estimation error, are considered. Next, new results on finite-time stability for a class of hybrid and switched systems are proposed using a multiple-Lyapunov-functions framework. The presented framework allows the system to have unstable modes. Finally, novel continuous-time optimization methods are studied with guarantees for fixed-time convergence to an optimal point. Fixed-time stable gradient flows are developed for unconstrained convex optimization problems under conditions such as strict convexity and gradient dominance of the objective function, which is a relaxation of strong convexity. Furthermore, min-max problems are considered and modifications of saddle-point dynamics are proposed with fixed-time stability guarantees under various conditions on the objective function.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168071/1/kgarg_1.pd

Robust prescribed trajectory tracking control of a robot manipulator using adaptive finite-time sliding mode and extreme learning machine method

This study aims to provide a robust trajectory tracking controller which guarantees the prescribed performance of a robot manipulator, both in transient and steady-state modes, experiencing parametric uncertainties. The main core of the controller is designed based on the adaptive finite-time sliding mode control (SMC) and extreme learning machine (ELM) methods to collectively estimate the parametric model uncertainties and enhance the quality of tracking performance. Accordingly, the global estimation with a fast convergence rate is achieved while the tracking error and the impact of chattering on the control input are mitigated significantly. Following the control design, the stability of the overall control system along with the finite-time convergence rate is proved, and the effectiveness of the proposed method is investigated via extensive simulation studies. The results of simulations confirm that the prescribed transient and steady-state performances are obtained with enough accuracy, fast convergence rate, robustness, and smooth control input which are all required for practical implementation and applications

Robust Prescribed Trajectory Tracking Control of a Robot Manipulator Using Adaptive Finite-Time Sliding Mode and Extreme Learning Machine Method

peer reviewedThis study aims to provide a robust trajectory tracking controller which guarantees the prescribed performance of a robot manipulator, both in transient and steady-state modes, experiencing parametric uncertainties. The main core of the controller is designed based on the adaptive finite-time sliding mode control (SMC) and extreme learning machine (ELM) methods to collectively estimate the parametric model uncertainties and enhance the quality of tracking performance. Accordingly, the global estimation with a fast convergence rate is achieved while the tracking error and the impact of chattering on the control input are mitigated significantly. Following the control design, the stability of the overall control system along with the finite-time convergence rate is proved, and the effectiveness of the proposed method is investigated via extensive simulation studies. The results of simulations confirm that the prescribed transient and steady-state performances are obtained with enough accuracy, fast convergence rate, robustness, and smooth control input which are all required for practical implementation and applications

Finite-rate control : stability and performance

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (leaves 88-91).The classical control paradigm addressed problems where communication between one plant and one controller is essentially perfect, and both have either discrete or continuous dynamics. Today, new problems in control over networks are emerging. A complex network involves an interconnection of numerous computational components where the controllers may be decentralized, and the components can have discrete or continuous dynamics. Communication links can be very noisy, induce delays, and have finite-rate constraints. Applications include remote navigation systems over the internet (eg. telesurgery) or in constrained environments (eg. deep sea/Mars exploration). These complexities demand that control be integrated with the protocols of communication to ensure stability and performance. Control over networks is recent and continues to receive growing interest. Initial work has focused on asymptotic stability under finite-rate feedback control, where the only excitation to the system is an unknown (but bounded) finite-dimensional initial condition vector. Such problems reduce to state-estimation under finite-rate constraints.(cont.) More recently, disturbance rejection limitations were derived for the same setting, assuming stochastic exogenous signals entering the system. Although these studies have contributed greatly to our understanding of such systems, input-output stability, performance analysis, and synthesis of coding schemes and controllers under finite-rate constraints remains largely untapped. In this thesis we address how finiterate control impacts input-output stability and performance, and we also construct computable methods for synthesizing controllers and coding schemes to meet control objectives. We first investigate how finite-rate feedback limits input-output stability and closed-loop performance. We assume that exogenous inputs belong to rich deterministic classes of signals, and perform analyses in a worst case setting. Since our results are derived using a robust control perspective, we are able to translate performance demands into optimization problems that can be solved to obtain quantization strategies and controllers in a streamlined manner. We then study how finite-rate feedforward control impacts finite-horizon tracking and navigation.(cont.) We derive performance limitations for each case, and illustrate time and performance tradeoffs. Finally, we investigate feedforward control over noisy discrete channels, and solve a decentralized distributed design problem involving the simultaneous synthesis of a block coding strategy and a single-input single-output linear time-invariant controller. We also illustrate delay versus accuracy tradeoffs.by Sridevi Vedula Sarma.Ph.D

Time-delay systems : stability, sliding mode control and state estimation

University of Technology, Sydney. Faculty of Engineering and Information Technology.Time delays and external disturbances are unavoidable in many practical control systems such as robotic manipulators, aircraft, manufacturing and process control systems and it is often a source of instability or oscillation. This thesis is concerned with the stability, sliding mode control and state estimation problems of time-delay systems. Throughout the thesis, the Lyapunov-Krasovskii (L-K) method, in conjunction with the Linear Matrix Inequality (LMI) techniques is mainly used for analysis and design. Firstly, a brief survey on recent developments of the L-K method for stability analysis, discrete-time sliding mode control design and linear functional observer design of time-delay systems, is presented. Then, the problem of exponential stability is addressed for a class of linear discrete-time systems with interval time-varying delay. Some improved delay-dependent stability conditions of linear discrete-time systems with interval time-varying delay are derived in terms of linear matrix inequalities. Secondly, the problem of reachable set bounding, essential information for the control design, is tackled for linear systems with time-varying delay and bounded disturbances. Indeed, minimisation of the reachable set bound can generally result in a controller with a larger gain to achieve better performance for the uncertain dynamical system under control. Based on the L-K method, combined with the delay decomposition approach, sufficient conditions for the existence of ellipsoid-based bounds of reachable sets of a class of linear systems with interval time-varying delay and bounded disturbances, are derived in terms of matrix inequalities. To obtain a smaller bound, a new idea is proposed to minimise the projection distances of the ellipsoids on axes, with respect to various convergence rates, instead of minimising its radius with a single exponential rate. Therefore, the smallest possible bound can be obtained from the intersection of these ellipsoids. This study also addresses the problem of robust sliding mode control for a class of linear discrete-time systems with time-varying delay and unmatched external disturbances. By using the L-K method, in combination with the delay decomposition technique and the reciprocally convex approach, new LMI-based conditions for the existence of a stable sliding surface are derived. These conditions can deal with the effects of time-varying delay and unmatched external disturbances while guaranteeing that all the state trajectories of the reduced-order system are exponentially convergent to a ball with a minimised radius. Robust discrete-time quasi-sliding mode control scheme is then proposed to drive the state trajectories of the closed-loop system towards the prescribed sliding surface in a finite time and maintain it there after subsequent time. Finally, the state estimation problem is studied for the challenging case when both the system’s output and input are subject to time delays. By using the information of the multiple delayed output and delayed input, a new minimal order observer is first proposed to estimate a linear state functional of the system. The existence conditions for such an observer are given to guarantee that the estimated state converges exponentially within an Є-bound of the original state. Based on the L-K method, sufficient conditions for Є-convergence of the observer error, are derived in terms of matrix inequalities. Design algorithms are introduced to illustrate the merit of the proposed approach. From theoretical as well as practical perspectives, the obtained results in this thesis are beneficial to a broad range of applications in robotic manipulators, airport navigation, manufacturing, process control and in networked systems

Robust Predictive Extended State Observer for a Class of Nonlinear Systems with Time-Varying Input Delay

[EN] This paper deals with asymptotic stabilisation of a class of nonlinear input-delayed systems via dynamic output feedback in the presence of disturbances. The proposed strategy has the structure of an observer-based control law, in which the observer estimates and predicts both the plant state and the external disturbance. A nominal delay value is assumed to be known and stability conditions in terms of linear matrix inequalities are derived for fast-varying delay uncertainties. Asymptotic stability is achieved if the disturbance or the time delay is constant. The controller design problem is also addressed and a numerical example with an unstable system is provided to illustrate the usefulness of the proposed strategy.This work was partially supported by: Ministerio de Economía y Competitividad, Spain (TIN2017-86520-C3-1-R); Universitat Politècnica de València (FPI-UPV 2014 PhD Grant); and Israel Science Foundation (Grant No. 1128/14).Sanz Diaz, R.; García Gil, PJ.; Fridman, E.; Albertos Pérez, P. (2020). Robust Predictive Extended State Observer for a Class of Nonlinear Systems with Time-Varying Input Delay. International Journal of Control. 93(2):217-225. https://doi.org/10.1080/00207179.2018.1562204S217225932Ahmed-Ali, T., Cherrier, E., & Lamnabhi-Lagarrigue, F. (2012). Cascade High Gain Predictors for a Class of Nonlinear Systems. IEEE Transactions on Automatic Control, 57(1), 221-226. doi:10.1109/tac.2011.2161795Artstein, Z. (1982). Linear systems with delayed controls: A reduction. IEEE Transactions on Automatic Control, 27(4), 869-879. doi:10.1109/tac.1982.1103023Basturk, 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.2016.10.006Basturk, 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.2015.05.013Bekiaris-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:10.1115/1.4005278Besançon, G., Georges, D. & Benayache, Z. (2007). Asymptotic state prediction for continuous-time systems with delayed input and application to control. 2007 European control conference (ECC) (pp. 1786–1791).Engelborghs, K., Dambrine, M., & Roose, D. (2001). Limitations of a class of stabilization methods for delay systems. IEEE Transactions on Automatic Control, 46(2), 336-339. doi:10.1109/9.905705Fridman, E. (2001). New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems. Systems & Control Letters, 43(4), 309-319. doi:10.1016/s0167-6911(01)00114-1Fridman, E. (2014). Introduction to Time-Delay Systems. Systems & Control: Foundations & Applications. doi:10.1007/978-3-319-09393-2Fridman, E. (2014). Tutorial on Lyapunov-based methods for time-delay systems. European Journal of Control, 20(6), 271-283. doi:10.1016/j.ejcon.2014.10.001Furtat, I., Fridman, E., & Fradkov, A. (2018). Disturbance Compensation With Finite Spectrum Assignment for Plants With Input Delay. IEEE Transactions on Automatic Control, 63(1), 298-305. doi:10.1109/tac.2017.2732279Germani, A., Manes, C., & Pepe, P. (2002). A new approach to state observation of nonlinear systems with delayed output. IEEE Transactions on Automatic Control, 47(1), 96-101. doi:10.1109/9.981726Guo, 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.978Karafyllis, I., & Krstic, M. (2017). Predictor Feedback for Delay Systems: Implementations and Approximations. Systems & Control: Foundations & Applications. doi:10.1007/978-3-319-42378-4Krstic, 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.2014.11.003Léchappé, V., Moulay, E. & Plestan, F. (2016). Dynamic observation-prediction for LTI systems with a time-varying delay in the input. 2016 IEEE 55th conference on decision and control (CDC) (pp. 2302–2307).Manitius, 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.1979.1102124Mazenc, F. & Malisoff, M. (2016). New prediction approach for stabilizing time-varying systems under time-varying input delay. 2016 IEEE 55th conference on decision and control (CDC) (pp. 3178–3182).Mondie, 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.2003.820147Najafi, 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:10.1080/00207179.2012.751627Najafi, M., Sheikholeslam, F., Hosseinnia, S., & Wang, Q.-G. (2014). Robust H ∞ control of single input-delay systems based on sequential sub-predictors. IET Control Theory & Applications, 8(13), 1175-1184. doi:10.1049/iet-cta.2012.1004Sanz, 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.2016.05.019Sanz, R., García, P., & Albertos, P. (2018). A generalized smith predictor for unstable time-delay SISO systems. ISA Transactions, 72, 197-204. doi:10.1016/j.isatra.2017.09.020Sanz, R., García, P., Fridman, E. & Albertos, P. (2017). A predictive extended state observer for a class of nonlinear systems with input delay subject to external disturbances. 2017 IEEE 56th annual conference on decision and control (CDC) (pp. 4345–4350).Sanz, R., Garcia, P., Fridman, E., & Albertos, 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. doi:10.1002/rnc.4027Shustin, E., & Fridman, E. (2007). On delay-derivative-dependent stability of systems with fast-varying delays. Automatica, 43(9), 1649-1655. doi:10.1016/j.automatica.2007.02.009Suplin, V., Fridman, E., & Shaked, U. (2007). Sampled-data H∞ control and filtering: Nonuniform uncertain sampling. Automatica, 43(6), 1072-1083. doi:10.1016/j.automatica.2006.11.024Yao, J., Jiao, Z., & Ma, D. (2014). RISE-Based Precision Motion Control of DC Motors With Continuous Friction Compensation. IEEE Transactions on Industrial Electronics, 61(12), 7067-7075. doi:10.1109/tie.2014.2321344Zhong, 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.2004.83753

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. 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SMC design for robust H∞ control of uncertain stochastic delay systems

Recently, sliding mode control method has been extended to accommodate stochastic systems. However, the existing results employ an assumption that may be too restrictive for many stochastic systems. This paper aims to remove this assumption and present in terms of LMIs a sliding mode control design method for stochastic systems with state delay. In some cases, the proposed method provides a control scheme for finite-time stabilization of stochastic delay systems

Stabilising Model Predictive Control for Discrete-time Fractional-order Systems

In this paper we propose a model predictive control scheme for constrained fractional-order discrete-time systems. We prove that all constraints are satisfied at all time instants and we prescribe conditions for the origin to be an asymptotically stable equilibrium point of the controlled system. We employ a finite-dimensional approximation of the original infinite-dimensional dynamics for which the approximation error can become arbitrarily small. We use the approximate dynamics to design a tube-based model predictive controller which steers the system state to a neighbourhood of the origin of controlled size. We finally derive stability conditions for the MPC-controlled system which are computationally tractable and account for the infinite dimensional nature of the fractional-order system and the state and input constraints. The proposed control methodology guarantees asymptotic stability of the discrete-time fractional order system, satisfaction of the prescribed constraints and recursive feasibility