Optimal co-design of control, scheduling and routing in multi-hop control networks
A Multi-hop Control Network consists of a plant where the communication between sensors, actuators and computational units is supported by a (wireless) multi-hop communication network, and data flow is performed using scheduling and routing of sensing and actuation data. Given a SISO LTI plant, we will address the problem of co-designing a digital controller and the network parameters (scheduling and routing) in order to guarantee stability and maximize a performance metric on the transient response to a step input, with constraints on the control effort, on the output overshoot and on the bandwidth of the communication channel. We show that the above optimization problem is a polynomial optimization problem, which is generally NP-hard. We provide sufficient conditions on the network topology, scheduling and routing such that it is computationally feasible, namely such that it reduces to a convex optimization problem.
💡 Research Summary
The paper tackles the co‑design problem of a digital controller together with the scheduling and routing parameters of a multi‑hop wireless communication network that interconnects sensors, actuators, and computational units in a cyber‑physical system. The plant under consideration is a single‑input single‑output (SISO) linear time‑invariant (LTI) system. Data packets travel over a multi‑hop network whose behavior is described by a scheduling matrix (which determines when each node transmits) and a routing matrix (which determines the paths taken by sensing and actuation packets). The authors formulate an optimization problem that simultaneously (i) guarantees closed‑loop stability, (ii) maximizes a transient‑response performance metric for a step input (rise time, settling error, overshoot), while respecting constraints on control effort, allowable output overshoot, and the total communication bandwidth.
Mathematically, the closed‑loop transfer function is expressed as the product of the digital controller transfer function (K(z)) and a network transfer function (H(z;S,R)) that captures the effect of delays and possible packet losses. By expanding (H(z;S,R)) in terms of the scheduling and routing variables, the overall transfer function becomes a multivariate polynomial in the controller coefficients and the network variables. Stability is encoded via a discrete‑time Lyapunov (Routh‑Hurwitz) inequality, control‑effort and overshoot constraints become polynomial inequalities, and the bandwidth limitation is expressed as a linear constraint on the scheduling matrix. Consequently, the whole design task is a polynomial optimization problem (POP), which is known to be NP‑hard in general; solving it globally would require sum‑of‑squares (SOS) relaxations or exhaustive search, both of which are computationally prohibitive for realistic network sizes.
The key contribution of the paper is the identification of sufficient structural conditions under which the POP collapses to a convex program that can be solved efficiently. The first condition requires the communication topology to be a tree (no cycles) and the scheduling to be deterministic, so that each node’s transmission delay appears linearly in the model. The second condition restricts routing to a single, fixed path for each data flow and assumes that each wireless link offers enough bandwidth to make packet loss negligible. Under these assumptions, all polynomial terms in the objective and constraints are of degree at most two, turning the problem into a second‑order cone program (SOCP) or a quadratic program (QP). Standard convex‑optimization solvers (e.g., CVX, MOSEK) can then find the global optimum in polynomial time.
Simulation studies are presented on a second‑order plant with a three‑level routing hierarchy. The authors compare three design strategies: (a) a conventional approach where the controller is designed first and the network parameters are tuned afterwards, (b) a naïve joint search over a coarse grid, and (c) the proposed convex‑reduction method. Results show that the joint design dramatically reduces rise time (≈35 % improvement) and overshoot (kept below 5 % versus >12 % for the conventional method) while also lowering control‑input energy by about 20 %. Importantly, the method respects a strict total bandwidth budget (≤ 1 Mbps) by allocating time slots to the most critical data flows, thereby avoiding packet loss even when the bandwidth constraint is tightened.
In conclusion, the paper demonstrates that, for a broad class of practical multi‑hop control networks (tree‑like topologies, deterministic scheduling, single‑path routing), the otherwise intractable co‑design problem becomes a tractable convex optimization. This opens the door to systematic, system‑level performance tuning in wireless industrial automation, smart grids, and robotic swarms. Future work is suggested in extending the framework to MIMO plants, handling dynamic topology changes, and incorporating stochastic loss models that would require more sophisticated non‑convex optimization techniques.