Decentralized string-stability analysis for heterogeneous cascades subject to load-matching requirements

Abstract

A heterogeneous cascade of stable linear time-invariant subsystems is studied in terms of the spatial and temporal propagation of boundary conditions. The particular context requires constant spatial boundary conditions to be asymptotically matched by the interconnection signals along the string (e.g., to match supply to demand in steady state). Furthermore, the transient response associated with a step change in the spatial boundary condition must remain bounded across space in a string-length independent fashion. With this in mind, an infinite cascade abstraction is considered. A corresponding decentralized string-stability certificate for the desired behaviour is established in terms of the subsystem H ∞ norms, via Lyapunov-type analysis of a two-dimensional model in Roesser form. Verification of the certificate implies uniformly bounded interconnection signals in response to the following system inputs: (i) a square-summable (across space) sequence of initial conditions; and (ii) a uniformly-bounded (across time) finite-energy input applied as the spatial boundary condition (e.g., finite duration on-off pulse). The decentralized nature of the certificate facilitates subsystem-by-subsystem design of local controllers that achieve string-stable behaviour overall. This application of the analysis is explored within the context of a scalable approach to the design of distributed distant-downstream controllers for the sections of an automated irrigation channel.

Heterogeneous string stability of unidirectionally interconnected MIMO LTI systems

“…The leader is represented by a signal q 0 (t) ∈ R nq , which can be considered as a spatial boundary condition to the string. In some applications q 0 is interpreted as a reference signal to be asymptotically followed by the other components (e.g., reference position in longitudinal platooning problems (Ioannou and Chien, 1993), water flow-rate in automated irrigation channels (Soltanian and Cantoni, 2015)). The components of the string are denoted by Σ i , i ∈ N, and are described by the following LTI state-space equations…”

“…The choice of space ∞ corresponds to spatially uniform boundedness of signals, a sensible assumption and requirement in most of the applications. The spatial boundary (reference) function is usually assumed to be q 0 ∈ L 2 or q 0 ∈ L ∞ or both (Soltanian and Cantoni, 2015). Variables of interest, z i , are often required to belong to L ∞ (e.g., spacing errors in platooning problems to avoid collisions).…”

“…Specifically, spatial and temporal evolution of signals and the notion of string stability are examined for interconnections which are subject to nonzero initial conditions, disturbances and reference inputs, and where the flow of information between the components is unidirectional along the string. Such systems arise in many fields of application, for instance, automated irrigation channels (Soltanian and Cantoni, 2015), supply chains (Huang et al, 2007), harmonic oscillators Yu et al (2015), lateral (McAree and Veres, 2016) and longitudinal (Ioannou and Chien, 1993) control of vehicle platoons.…”

“…(Time,Space)-domain approaches are based on infinite dimensional operators (D'Andrea and Dullerud, 2003); (vector) Lyapunov functions (Swaroop and Hedrick, 1996); and the theory of 2-D systems (Knorn, 2013;Soltanian and Cantoni, 2015). Soltanian and Cantoni derived a distributed sufficient condition for heterogeneous string stability (HSS) demanding uniform boundedness in both time and space (Soltanian and Cantoni, 2015). A great advantage of their results is that HSS of the string can be achieved by satisfying a set of independent local conditions.…”

Structured computation of optimal controls for constrained cascade systems

“…This paper is about LQ optimal control problems with system dynamics arising from the cascade interconnection of N subsystems subject to constraints over a finite time horizon. Such systems can be found in the study of irrigation and drainage networks (Li, Cantoni, & Weyer, 2005;Puig et al, 2009;Soltanian & Cantoni, 2015), mutli-reservoir and hydro-power systems (Labadie, 2004;Şahin & Morari, 2010), vehicle platoons (Seiler, Pant, & Hedrick, 2004), and supply chain management (Disney & Towill, 2003). Attention is drawn to exploiting structure in the spatial dimension of the dynamics.…”