A Two-Dimensional Systems StabilityAnalysis of Vehicle Platoons

The main contributions of this dissertation are in the field of stability analysis of linear and nonlinear two-dimensional systems. The study of stability of such systems is motivated by the “string stability” or “platooning” problem: In order to achieve tighter spacing between vehicles travelling one after the other in one direction, i. e. in a string or platoon, the driver is replaced by an automatic controller designed to keep a specified distance towards the preceding vehicle. It is shown how such a vehicle platoon can be modelled as a two-dimensional system. Here, two-dimensional refers to the fact that the system depends on two independent vari- ables such as time t and position within the string k. However, two-dimensional systems describing a vehicle string generically exhibit a singularity at the stability boundary. The existence of this singularity at the stability boundary prevents application of most stabil- ity criteria known in the literature, since this marginal case is almost always explicitly or implicitly excluded. Bounded-input bounded-output stability of linear continuous-discrete two-dimensional systems is studied in the frequency domain paying particular attention to systems with nonessential singularities of the second kind at the stability boundary. A two-dimensional version of Parseval’s Theorem and the corresponding induced operator norm are derived. The results are applied to a string of vehicles and sufficient conditions for string stability are deduced. Sufficient conditions for different forms of stability of linear two-dimensional systems in the time domain are developed using a two-dimensional quadratic Lyapunov function and linear matrix inequalities. It is shown that systems permitting a two-dimensional Lyapunov function with a negative definite divergence are exponentially stable. It is proven, however, that two-dimensional systems with a singularity at the stability boundary (such as two-dimensional descriptions of vehicle strings) cannot satisfy the re- quired conditions for exponential stability as the divergence of the Lyapunov function can never be sign definite. If the divergence is only negative semidefinite, stability of the system can be guaranteed. Provided additional conditions on the Lyapunov function and the initial conditions are satisfied, asymptotic stability of systems whose Lyapunov functions have a negative semidefinite divergence can be shown. Extending the results mentioned above, sufficient conditions for stability, exponential stability and asymptotic stability of nonlinear two-dimensional systems are deduced. Sim- ilar to the results on linear two-dimensional systems, exponential stability can be guar- anteed if the divergence of the Lyapunov function is strictly negative. For systems with merely nonpositive divergence stability is also shown. Asymptotic stability of nonlinear two-dimensional systems can be proven if not only the initial conditions but also the Lya- punov function itself and the state space equations satisfy additional smoothness conditions. Instead of a quadratic Lyapunov function, a wider class of Lyapunov functions is allowed in the proofs of stability of nonlinear two-dimensional systems. The notion and theory of (integral) input to state stability is used instead of linear matrix inequalities to derive the results. All proofs and results for the stability of linear and nonlinear two-dimensional systems in the time domain are given in a unified notation, studying systems with continuous and discrete independent variables simultaneously. The theoretical results on linear two-dimensional systems are used to analyse the (string) stability of a linear unidirectional homogenous string with different time headways and com- munication range 1 and 2. The stability results for nonlinear two-dimensional systems are applied to rigorously prove string stability of a nonlinear string with variable time headway.

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