This paper addresses the problem of hybrid control for a class of switched uncertain systems. The switched system under consideration is subject to structured uncertain dynamics in a linear fractional transformation (LFT) form and time-varying input delays. A novel hybrid controller is proposed, which consists of three major components: the integral quadratic constraint (IQC) dynamics, the continuous dynamics, and the jump dynamics. The IQC dynamics are developed by leveraging methodologies from robust control theory and are utilised to address the effects of time-varying input delays. The continuous dynamics are structured by feeding back not only measurement outputs but also some system's internal signals. The jump dynamics enforce a jump (update/reset) at every switching time instant for the states of both IQC dynamics and continuous dynamics. Based on this, robust stability of the overall hybrid closed-loop system is established under the average dwell time framework with multiple Lyapunov functions. Moreover, the associated control synthesis conditions are fully characterised as linear matrix inequalities, which can be solved efficiently. An application example on regulation of a nonlinear switched electronic circuit system has been used to demonstrate effectiveness and usefulness of the proposed approach.
more »
« less
ISS and integral-ISS of switched systems with nonlinear supply functions
The problem of input-to-state stability (ISS) and its integral version (iISS) are considered for switched nonlinear systems with inputs, resets and possibly unstable subsystems. For the dissipation inequalities associated with the Lyapunov function of each subsystem, it is assumed that the supply functions, which characterize the decay rate and ISS/iISS gains of the subsystems, are nonlinear. The change in the value of Lyapunov functions at switching instants is described by a sum of growth and gain functions, which are also nonlinear. Using the notion of average dwell-time (ADT) to limit the number of switching instants on an interval, and the notion of average activation time (AAT) to limit the activation time for unstable systems, a formula relating ADT and AAT is derived to guarantee ISS/iISS of the switched system. Case studies of switched systems with saturating dynamics and switched bilinear systems are included for illustration of the results.
more » « less- NSF-PAR ID:
- 10307533
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematics of Control, Signals, and Systems
- ISSN:
- 0932-4194
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
Abstract Periodic event‐triggered control (PETC) evaluates triggering conditions only at periodic sampling times, based on which it is decided whether the controller needs to be updated. This article investigates the global stabilization of nonlinear systems that are affected by external disturbances under PETC mechanisms. Sufficient conditions are provided to ensure the resulting closed‐loop system is input‐to‐state stable (ISS) for the state feedback and the observer‐based output feedback configurations separately. The sampling period and the triggering functions are chosen such that the ISS‐Lyapunov function of continuous dynamics is also the ISS‐Lyapunov function of the overall system. Based on that, sufficient conditions in the form of linear matrix inequalities are provided for the PETC design of incrementally quadratic nonlinear systems. Two simulation examples are provided to illustrate the effectiveness of the proposed method.
-
more » « less
This article focuses on state‐of‐charge (SOC) estimation in a lithium‐ion battery, using measurements of terminal voltage and bulk force. A nonlinear observer designed using Lyapunov analysis relying on lower and upper bounds of the Jacobian of the nonlinear output function is utilized. Rigorous analysis shows that the proposed observer has feasible design solutions only in each piecewise monotonic region of the output functions and has no constant stabilizing observer gain when the entire SOC range is considered. The nonmonotonicity challenge is then addressed by designing a hybrid nonlinear observer that switches between several constant observer gains. The global stability of the switched system is guaranteed by ensuring overlap between regions and an adequate dwell time between switches. The performance of the observer is evaluated first through simulations using a high‐fidelity battery model and then through experiments. The performance of the nonlinear observer is compared with that of an extended Kalman Filter. Simulation results show that with no model uncertainty the nonlinear observer provides estimates with an RMS error of 1.1%, while the EKF performs better, providing an RMS error less than 1%. However, when model error is introduced into this nonmonotonic system, the EKF becomes unstable for even very small model errors in the output curves. The nonlinear observer, on the other hand, continues to perform very well, providing accurate estimates and never becoming unstable. The experimental results verify the observations from the simulation and the experimental EKF is found to become unstable due to model errors, while the hybrid nonlinear observer continues to work reliably.
-
This paper generalizes the LaSalle–Yoshizawa Theorem to switched nonsmooth systems. The Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A common candidate Lyapunov function that has a negative semidefinite generalized time derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle–Yoshizawa-like results for the switched system. Of independent interest, are the results on approximate continuity and Filippov regularization of set-valued maps, reduction of differential inclusions using Lipschitz continuous regular functions, and comparative remarks on different generalizations of the time derivative along the trajectories of a nonsmooth system.more » « less
-
Bartocci, Ezio ; Putot, Sylvie (Ed.)Switched systems are known to exhibit subtle (in)stability behaviors requiring system designers to carefully analyze the stability of closed-loop systems that arise from their proposed switching control laws. This paper presents a formal approach for verifying switched system stability that blends classical ideas from the controls and verification literature using differential dynamic logic (dL), a logic for deductive verification of hybrid systems. From controls, we use standard stability notions for various classes of switching mechanisms and their corresponding Lyapunov function-based analysis techniques. From verification, we use dL's ability to verify quantified properties of hybrid systems and dL models of switched systems as looping hybrid programs whose stability can be formally specified and proven by finding appropriate loop invariants, i.e., properties that are preserved across each loop iteration. This blend of ideas enables a trustworthy implementation of switched system stability verification in the KeYmaera X prover based on dL. For standard classes of switching mechanisms, the implementation provides fully automated stability proofs, including searching for suitable Lyapunov functions. Moreover, the generality of the deductive approach also enables verification of switching control laws that require non-standard stability arguments through the design of loop invariants that suitably express specific intuitions behind those control laws. This flexibility is demonstrated on three case studies: a model for longitudinal flight control by Branicky, an automatic cruise controller, and Brockett's nonholonomic integrator.more » « less
