WP4 - Description

WP4: Analysis.

One of the main goals of PADECOT is to introduce analysis tools in order to studying the robustness and performance properties of the developed feedback control laws, and hence, in order to also deriving performance and robustness guarantees of the developed algorithms in the control of traffic. The activities of WP4 are organized in the following tasks.

Task 4.1: Stability analysis of the Moskowitz model under the proposed controller with the construction of a Lyapunov function.

A1. For the class of systems considered in Task 2.1.A1 of WP2, local asymptotic stability of the closed-loop system, under the developed feedback law, in the C1 norm of the PDE state is established as follows. Due to the potential formation of shock waves in the solutions of quasilinear, first-order hyperbolic PDEs (which is related to the fundamental restriction for systems with time-varying delays that the delay rate is bounded by unity), one should limit herself/himself to a certain feasibility region around the origin. It is then shown that the PDE predictor-feedback law achieves asymptotic stability of the closed-loop system and an estimate of its region of attraction is provided. The analysis combines Lyapunov-like arguments and input-to-state stability (ISS) estimates.

N. Bekiaris-Liberis and M. Krstic, Compensation of actuator dynamics governed by quasilinear hyperbolic PDEs,,Automatica, vol. 92, pp. 29--40, 2018.

N. Bekiaris-Liberis and M. Krstic, Control of nonlinear systems with actuator dynamics governed by quasilinear first-order hyperbolic PDEs, European Control Conference, 2018.

A2. For the class of systems considered in Task 2.1.A2 of WP2, the closed-loop system, under the developed predictor-feedback control law, is shown to be locally asymptotically stable, in the C1 norm of the PDE state, via the employment of a Lyapunov-like argument and the introduction of a backstepping transformation. The stability result is local due to an inherent limitation of the class of transport PDEs under consideration, which ensures the well-posedness of the given transport PDE. More specifically, this restriction guarantees that, in an equivalent formulation of the transport PDE that employs a constant PDE domain and a transport speed that depends on the boundary valuesof the PDE state as well as its first-order spatial derivative, the transport speed remains always strictly positive as well as uniformly bounded from above and below by finite constants.

N. Bekiaris-Liberis and M. Krstic, Compensation of transport actuator dynamics with input-dependent moving controlled boundary, IEEE Transactions on Automatic Control, vol. 63, pp. 3889--3896, 2018.

N. Bekiaris-Liberis and M. Krstic, Compensation of transport actuator dynamics with input-dependent moving controlled boundary, European Control Conference, 2018.

A3. For the model introduced in Task 2.2.A we study its observability properties. Specifically, we provide sufficient and necessary conditions for the structural observability as well as the strong structural observability of the model, adopting a graph-theoretic approach; it should be noted that observability properties are rarely studied in the literature on traffic estimation. As a result, we characterize the fixed detectors configuration that guarantee the proper operation of the developed traffic state estimation scheme. The stability properties of the estimator are also discussed.

N. Bekiaris-Liberis, C. Roncoli, and M. Papageorgiou, Highway traffic state estimation per lane in the presence of connected vehicles, Transportation Research Part B, vol. 106, pp. 1--28, 2017.

I. Papamichail, N. Bekiaris-Liberis, A. I. Delis, D. Manolis, K.-S. Mountakis, I. K. Nikolos, C. Roncoli, & M. Papageorgiou, Motorway traffic flow modelling, estimation and control with vehicle automation and communication systems, Annual Reviews in Control, to appear, 2019.

A4. All of the designs in Task 2.3.A are explicit since they are constructed interlacing a feedback linearizing transformation (which is introduced and which is inspired by the so-called Hopf-Cole transformation) with PDE backstepping. Due to the fact that the linearizing transformation is locally invertible, only regional stability results are established, which are, nevertheless, accompanied with region of attraction estimates. Our stability proofs are based on the utilization of the linearizing transformation together with the employment of backstepping transformations, suitably formulated to handle the case of bilateral actuation and sensing.

Specifically, we first establish the well-posedness of the feedforward controllers for the original nonlinear PDE system, for reference outputs that belong to Gevrey class (of certain order) with sufficiently small magnitude, via the employment of the feedback linearizing transformation, which allows us to convert the original nonlinear problem to a motion planning problem for a linear heat equation. Second, we establish local asymptotic stability of the closed-loop system (via the utilization of a modified version of the feedback linearizing transformation), under the full-state feedback laws, in H1 norm, employing a Lyapunov functional and we provide an estimate of the region of attraction of the controllers. Our stability result is local in H1 norm due to the fact that the linearizing transformation is invertible only locally and, in particularly, the size of the supremum norm of the transformed PDE state should be appropriately restricted. Finally, we show that the bilateral, observer-based output-feedback controller achieves local asymptotic stabilization of the reference trajectory in H1 norm.

N. Bekiaris-Liberis and R. Vazquez, Nonlinear bilateraloutput-feedback control for a class of viscous Hamilton-Jacobi PDEs, Automatica, vol. 101, pp. 223--231, 2019.

N. Bekiaris-Liberis and R. Vazquez, Nonlinear bilateral full-state feedback trajectory tracking for a class of viscous Hamilton-Jacobi PDEs, IEEE Conference on Decision and Control, 2018.

A5. For the predictor-feedback control design presented in Task 2.1.A3 for multi-input linear systems with distinct input delays, input-to-state stability, with respect to additive plant disturbances, as well as robustness to constant multiplicative uncertainties affecting the inputs are established. Furthermore, it is shown that the exact predictor-feedback controller is inverse optimal with respect to a meaningful differential game problem. The proofs capitalize on the availability of a backstepping transformation, which is formulated appropriately in a recursive manner.

X. Cai, N. Bekiaris-Liberis, and M. Krstic, Input-to-state stability and inverse optimality of predictor feedbackfor multi-input linear systems, Automatica, vol. 103, pp. 549--557, 2019.

Task 4.2: Stability analysis of the ARZ model by constructing a Lyapunov function.

A1. For the model developed in Task 3.2.A1 of WP3, it is shown that for all physically meaningful initial conditions this non-standard model admits a globally defined, unique, classical solution that remains positive and bounded for all times. Moreover, it is shown that the feedback law developed in Task 3.2.A1 of WP3 achieves global stabilization in the supremum norm of the logarithmic deviation of the state from its equilibrium. The proofs of these results are based on: i) The introduction of an appropriate nonlinear transformation of the PDE state, ii) the construction of a solution, and iii) the derivation of estimates on the solutions of the system.

I. Karafyllis, N. Bekiaris-Liberis, and M. Papageorgiou, Feedback control of nonlinear hyperbolic PDE systems inspired by traffic flow models, IEEE Transactions on Automatic Control, vol. 64, pp. 3647--3662, 2019.

I. Karafyllis, N. Bekiaris-Liberis, and M. Papageorgiou, Traffic flow inspired analysis and boundary control for a class of 2x2 hyperbolic systems, European Control Conference, 2018.

A2. Employing an input-output approach, we prove that the predictor-based ACC law with integral action developed in Task 3.2.A2 guarantees all four typical requirements of ACC designs, namely, (1) stability of each individual vehicular system, (2) zero steady-state spacing error between the actual and the desired inter-vehicle spacing, (3) string stability of homogenous platoons of vehicular systems, and (4) non-negative impulse response of each individual vehicular system, for any delay value smaller than the desired time-headway, which constitutes a physically intuitive limitation. We also establish string stability robustness of the predictor-based ACC law to delay mismatch between the real delay and the delay value available to the designer.

N. Bekiaris-Liberis, C. Roncoli, and M. Papageorgiou, Predictor-based adaptive cruise control design with integral action, IFAC Symposium on Control in Transportation Systems, 2018.

A3. The control strategy in Task 3.2.A3 is developed for a linearized version of the ARZ-type traffic flow model with ACC around a uniform, congested equilibrium profile, which is proved to be open-loop unstable. The closed-loop system under the proposed controller is shown to be exponentially stable (in C1 norm), constructing a Lyapunov functional. Furthermore, employing an input-output approach, an additional stability property of the closed-loop system is established, namely convective stability, which is important from a traffic control point of view as it guarantees the non-amplification of speed perturbations, as these propagate upstream.

N. Bekiaris-Liberis and A. Delis, PDE-based feedback control of freeway traffic flow via time-gap manipulation of ACC-equipped vehicles, IEEE Transactions on Control Systems Technology, provisionally accepted, 2019.

N. Bekiaris-Liberis and A. Delis, Feedback control of freeway traffic flow via time-gap manipulation of ACC-equipped vehicles: A PDE-based approach, IFAC Workshop on Control of Transportation Systems, 2019.