Abstract: A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modelled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions. PubDate: 2020-11-18
Abstract: Achieving a simplified model is a major issue in the field of fractional-order nonlinear systems, especially large-scale systems. So that in addition to simplifying the model, the outstanding features of the fractional-order modeling, such as memory feature, are preserved. This paper presented the homotopy singular perturbation method (HSPM) to reduce the complexity of the model and use the advantages of both models of the fractional order and the integer order. This method is a combination of the fractional-order singular perturbation method (FOSPM) and the homotopy perturbation method (HPM). Firstly, the FOSPM is developed for fractional-order nonlinear systems; then, a modification of the HPM is proposed. Finally, the HSPM is presented using a combination of these two methods. fractional-order nonlinear systems can be divided into two lower-order subsystems such as nonlinear or linear integer-order subsystem and linear fractional-order subsystem using this hybrid method. Convergence analysis of tracking error to zero is theoretically presented, and the effectiveness of the proposed method is also evaluated with two examples. Next, the number and location of equilibrium points are compared between the original system and the subsystems obtained from the proposed method. In the end, we show that the stability of fractional-order nonlinear system can be determined by investigating the stability of the subsystems using Theorem 3 and Lemma 2. PubDate: 2020-11-13
Abstract: This paper addresses the mean stability analysis and \(L_1\) performance of continuous-time Markov jump linear systems (MJLSs) driven by a two time-scale Markov chain, in the scenario in which the temporal scale parameter \(\epsilon \) tends to zero. The jump process considered here is bivariate, with slow and fast components. Our approach relies on a convergence analysis involving the semigroup that generates the first-moment dynamics of the MJLS when the switching frequency of the fast part of the Markov chain tends to infinity. In this setup, we introduce a new definition of stability in a limit case, and connect it with the mean stability of an averaged MJLS. In the particular case where the averaged MJLS is positive, we also derive suitable criteria for assessing mean stability and \(L_1\) performance. These criteria are expressed in terms of the Hurwitz stability of a matrix (whose dimension is independent of the cardinality of the state space of the fast switching component), of linear programming, and of the 1-norm of a certain transfer matrix, which makes them suitable for computational purposes. We also establish comparisons between our (two-time-scale) approach and existing one-time-scale approaches from the literature, and show that our criteria are based on matrices of relatively smaller dimensions, which do not depend on the scale parameter \(\epsilon \) . The effectiveness of the main results is discussed through numerical examples of epidemiological and compartmental models. PubDate: 2020-11-11
Abstract: In this paper, we consider the input-to-state stabilization of an ODE-wave feedback-connection system with Neumann boundary control, where the left end displacement of the wave equation enters the ODE, while the output of the ODE is fluxed into boundary of the wave equation. The disturbance is appeared as a nonhomogeneous term in the ODE. Based on the backstepping approach, a state feedback control law is designed to guarantee the exponential input-to-state stability of the closed-loop system. The resulting closed-loop system has been shown to be well-posed by the semigroup approach. Moreover, we construct an exponentially convergent state observer based on which an output feedback control law is obtained, and the closed-loop system is proved to be input-to-state stable. PubDate: 2020-09-22
Abstract: This paper studies a notion of topological entropy for switched systems, formulated in terms of the minimal number of trajectories needed to approximate all trajectories with a finite precision. For general switched linear systems, we prove that the topological entropy is independent of the set of initial states. We construct an upper bound for the topological entropy in terms of an average of the measures of system matrices of individual modes, weighted by their corresponding active times, and a lower bound in terms of an active-time-weighted average of their traces. For switched linear systems with scalar-valued state and those with pairwise commuting matrices, we establish formulae for the topological entropy in terms of active-time-weighted averages of the eigenvalues of system matrices of individual modes. For the more general case with simultaneously triangularizable matrices, we construct upper bounds for the topological entropy that only depend on the eigenvalues, their order in a simultaneous triangularization, and the active times. In each case above, we also establish upper bounds that are more conservative but require less information on the system matrices or on the switching, with their relations illustrated by numerical examples. Stability conditions inspired by the upper bounds for the topological entropy are presented as well. PubDate: 2020-09-09
Abstract: We investigate input-to-state stability (ISS) of infinite-dimensional collocated control systems subject to saturated feedback. Here, the unsaturated closed loop is dissipative and uniformly globally asymptotically stable. Under an additional assumption on the linear system, we show ISS for the saturated one. We discuss the sharpness of the conditions in light of existing results in the literature. PubDate: 2020-09-07
Abstract: We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion. PubDate: 2020-08-10
Abstract: Linear differential systems, in Willems’ behavioral system theory, are defined to be the solution sets to systems of linear constant coefficient PDEs, and they are naturally parameterized in a bijective way by means of polynomial modules. In this article, introducing appropriate topologies, this parametrization is made continuous in both directions. Moreover, the space of linear differential systems with a given complexity polynomial is embedded into a Grassmannian. PubDate: 2020-08-09
Abstract: This paper deals with the minimization of \(H_\infty \) output feedback control. This minimization can be formulated as a linear matrix inequality (LMI) problem via a result of Iwasaki and Skelton 1994. The strict feasibility of the dual problem of such an LMI problem is a valuable property to guarantee the existence of an optimal solution of the LMI problem. If this property fails, then the LMI problem may not have any optimal solutions. Even if one can compute parameters of controllers from a computed solution of the LMI problem, then the computed \(H_\infty \) norm may be very sensitive to a small change of parameters in the controller. In other words, the non-strict feasibility of the dual tells us that the considered design problem may be poorly formulated. We reveal that the strict feasibility of the dual is closely related to invariant zeros of the given generalized plant. The facial reduction is useful in analyzing the relationship. The facial reduction is an iterative algorithm to convert a non-strictly feasible problem into a strictly feasible one. We also show that facial reduction spends only one iteration for so-called regular \(H_\infty \) output feedback control. In particular, we can obtain a strictly feasible problem by using null vectors associated with some invariant zeros. This reduction is more straightforward than the direct application of facial reduction. PubDate: 2020-07-12
Abstract: Motivated by the ubiquitous sampled-data setup in applied control, we examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a solvable matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: (1) its Lie series expansion enjoys a type of strong convergence; (2) the origin is an equilibrium; (3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of the dynamics at the origin, in particular, global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability. PubDate: 2020-06-17
Abstract: In this paper, we study the hierarchic control problem for a linear system of a population dynamics model with an unknown birth rate. Using the notion of low-regret control and an adapted observability inequality of Carleman type, we show that there exist two controls such that, the first control called follower solves an optimal control problem which consists in bringing the state of the linear system to the desired state, and the second one named leader is supposed to lead the population to extinction at final time. PubDate: 2020-06-08
Abstract: In this paper, we introduce a weak maximum principle-based approach to input-to-state stability (ISS) analysis for certain nonlinear partial differential equations (PDEs) with certain boundary disturbances. Based on the weak maximum principle, a classical result on the maximum estimate of solutions to linear parabolic PDEs has been extended, which enables the ISS analysis for certain nonlinear parabolic PDEs with certain boundary disturbances. To illustrate the application of this method, we establish ISS estimates for a linear reaction–diffusion PDE and a generalized Ginzburg–Landau equation with mixed boundary disturbances. Compared to some existing methods, the scheme proposed in this paper involves less intensive computations and can be applied to the ISS analysis for a wide class of nonlinear PDEs with boundary disturbances. PubDate: 2020-05-27
Abstract: Model order reduction (MOR) techniques are often used to reduce the order of spatially discretized (stochastic) partial differential equations and hence reduce computational complexity. A particular class of MOR techniques is balancing related methods which rely on simultaneously diagonalizing the system Gramians. This has been extensively studied for deterministic linear systems. The balancing procedure has already been extended to bilinear equations, an important subclass of nonlinear systems. The choice of Gramians in Al-Baiyat and Bettayeb (In: Proceedings of the 32nd IEEE conference on decision and control, 1993) is the most frequently used approach. A balancing related MOR scheme for bilinear systems called singular perturbation approximation (SPA) has been described that relies on this choice of Gramians. However, no error bound for this method could be proved. In this paper, we extend SPA to stochastic systems with bilinear drift and linear diffusion term. However, we propose a slightly modified reduced order model in comparison to previous work and choose a different reachability Gramian. Based on this new approach, an \(L^2\)-error bound is proved for SPA which is the main result of this paper. This bound is new even for deterministic bilinear systems. PubDate: 2020-05-25
Abstract: We establish the local input-to-state stability of a large class of disturbed nonlinear reaction–diffusion equations w.r.t. the global attractor of the respective undisturbed system. PubDate: 2020-05-06
Abstract: Classical discrete-time adaptive controllers provide asymptotic stabilization and tracking; neither exponential stabilization nor a bounded noise gain is typically proven. In our recent work, it is shown, in both the pole placement stability setting and the first-order one-step-ahead tracking setting, that if the original, ideal, projection algorithm is used (subject to the common assumption that the plant parameters lie in a convex, compact set and that the parameter estimates are restricted to that set) as part of the adaptive controller, then a linear-like convolution bound on the closed-loop behaviour can be proven; this immediately confers exponential stability and a bounded noise gain, and it can be leveraged to provide tolerance to unmodelled dynamics and plant parameter variation. In this paper, we solve the much harder problem of adaptive tracking; under classical assumptions on the set of unmodelled parameters, including the requirement that the plant be minimum phase, we are able to prove not only the linear-like properties discussed above, but also very desirable bounds on the tracking performance. We achieve this by using a modified version of the ideal projection algorithm, termed as vigilant estimator: it is equally alert when the plant state is large or small and is turned off when it is clear that the disturbance is overwhelming the estimation process. PubDate: 2020-04-06
Abstract: This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative by exploiting state constraints that are known to hold for all trajectories of interest (e.g., path constraints that describe feasible trajectories in the context of dynamic optimization, or constraints that explicitly describe invariant sets containing all system trajectories). However, effective bounding algorithms of this type are currently only available for problems with linear constraints. Moreover, the theoretical results underlying these algorithms do not apply to constraints that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear constraints. This article contributes a new differential inequalities theorem that permits the use of a very general class of nonlinear state constraints. Moreover, a new algorithm is presented for efficiently exploiting nonlinear constraints to achieve tighter bounds. The proposed approach is shown to produce very sharp bounds for two challenging case studies. PubDate: 2020-02-26
Abstract: This article considers the problem of optimally recovering stable linear time-invariant systems observed via linear measurements made on their transfer functions. A common modeling assumption is replaced here by the related assumption that the transfer functions belong to a model set described by approximation capabilities. Capitalizing on recent optimal recovery results relative to such approximability models, we construct some optimal algorithms and characterize the optimal performance for the identification and evaluation of transfer functions in the framework of the Hardy Hilbert space and of the disk algebra. In particular, we determine explicitly the optimal recovery performance for frequency measurements taken at equispaced points on an inner circle or on the torus. PubDate: 2020-02-13
Abstract: We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance rejection for infinite-dimensional discrete-time systems and propose a design method of finite-dimensional controllers by using a solution of the Nevanlinna–Pick interpolation problem with both interior and boundary conditions. We apply our results to systems with state and output delays. PubDate: 2019-12-12
Abstract: The purpose of this work is to obtain restrictions on the asymptotic structure of two-dimensional hybrid dynamical systems. Previous results have been achieved by the authors concerning hybrid dynamical systems with a single impact surface and a single state space. Here, this work is extended to hybrid dynamical systems defined on a directed graph; each vertex corresponds to a state space and each directed edge corresponds to an impact. PubDate: 2019-11-16
Abstract: In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately, their result is only concerned with fixed sampling times, and thus, it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature. Precisely, we establish a new version of the PMP that can handle free sampling times. As in the aforementioned work by Bourdin and Trélat, we obtain a first-order necessary optimality condition written as a nonpositive averaged Hamiltonian gradient condition. Furthermore, from the freedom of choosing sampling times, we get a new and additional necessary optimality condition which happens to coincide with the continuity of the Hamiltonian function. In an autonomous context, even the constancy of the Hamiltonian function can be derived. Our proof is based on the Ekeland variational principle. Finally, a linear–quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of fixed sampling times and highlighting its continuity in the instance of optimal sampling times. PubDate: 2019-09-17
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