Co-designing a Layered Control Architecture

1 Introduction

• Ad hoc, isolated layer design often requires costly redesigns because interactions between layers are checked only after the individual designs are made.Source: §1 · p. 1
• Unlike approaches that design controllers for a given architecture, this formulation makes architectural choices such as sampling ratio and reduced-model order part of the design process.The related approaches analyze layer interactions but leave choices that shape the feasible design space outside their design process.Source: §1 · p. 2
• The framework formulates a multi-objective co-design problem over architecture and tuning variables, with computable conditions for stability and feasibility.It derives closed-loop error dynamics and a computable set-valued constraint map, and proves that iterating constraint sets yields monotonically tightening enclosures.Source: §1 · p. 2
• A simple numerical example searches architectural choices and illustrates the different behaviors of Pareto-optimal LCA designs.Source: §1 · pp. 2–3

2.1 The plant and control architecture

• The plant has state and actuator constraints and is controlled by a higher-level planner and a lower-level tracker.The planner supplies references that the tracker follows over multiple fast-time steps.Source: §2.1 · p. 3

2.2 The planning layer

• Because planning computation takes time, a reference computed during one interval is tracked in the following interval.The planner and tracker operate on separate timescales.Source: §2.1 · p. 3, §2.2 · p. 5

2.1 The plant and control architecture

• The formulation assumes exact discretization so that stability of the discrete-time controller implies stability of the continuous-time plant.The text notes that this assumption can be lifted using methods in related work.Source: §2.1 · p. 3
• The sampling-time ratio Nr is an architectural choice that affects overall performance, and the paper seeks to select it through co-design tradeoffs.The paper notes that practice often uses a rule of thumb, while the best choice depends on the design.Source: §2.1 · p. 4

2.2 The planning layer

• The planner uses a receding-horizon predictive controller with an approximate plant model, whose reduced state and input dimensions are architectural choices.Projection maps convey plant state and input information to the planner.Source: §2.2 · p. 4

2.3 The tracking layer

• The tracker is a static state-feedback controller whose gain is designed for closed-loop stability and whose control computation must fit within the fast sampling interval.The design requires Acl eigenvalues inside the unit disc.Source: §2.3 · p. 5

2.4 Problem statement

• The co-design question is how to formulate an evaluable joint design problem for the planner and tracker given the plant and its constraints.The section aims to quantify tradeoffs in LCA design systematically.Source: §2.4 · pp. 5–6
• The co-design problem jointly selects planner and tracker designs while enforcing LCA stability, feasibility, and compute constraints, enabling tradeoffs among Pareto-optimal solutions.The design variables include architectural choices and tuning parameters.Source: §2.4 · p. 6

3.1 Error evolution on the fast timescale

• The tracking controller assumes the planner reference is an equilibrium; deviations from equilibrium introduce a mismatch that can accumulate between reference updates.Co-design can anticipate the next update to counteract this accumulating error.Source: §3.1 · p. 6

3.2 Error evolution on the slow timescale

• Individually stable planner and controller dynamics can still produce an unstable LCA when the planner model and projection are poorly matched.The lifted dynamics must also be considered when assessing closed-loop stability.Source: §3.2 · p. 7
• For the lifted slow-timescale error, planner-model and projection choices must represent the controller’s Nr-step-ahead closed-loop dynamics.The reference change can counteract compounded mismatch when the model makes the second and third terms in (9b) zero.Source: §3.2 · p. 7
• The framework does not require exact satisfaction of the Nr-step consistency condition; its violation is absorbed by the constraint-tightening map.The condition corresponds to consistency for the Nr-step lifted system.Source: §3.2 · p. 7

4 Constraining the planning layer

• Because the tracker does not enforce state or input constraints, the planner must ensure constraint satisfaction for every provided reference update.The construction uses standard MPC arguments to determine planner constraint sets from the planner and controller design variables.Source: §4 · p. 8

4.1 Construction of Y, R

• The construction uses an RPI set for slow-timescale planning dynamics and an intra-period tube to bound error between reference updates.These sets feed the fixed-point problem for the planner’s constraint sets.Source: §4.1 · p. 8

4.2.1 Existence of (Y⋆, R⋆)

• Under its spectral condition, F is anti-tone, while F◦F is monotone and has a fixed point; F therefore has a fixed point or a two-cycle pair.A fixed point is not required for constraint satisfaction; planner sets can instead be chosen within the relevant bounds.Source: §4.2.1 · p. 9

4.2.2 Computing (Y, R)

• The iteration generates nested bounds that enclose every fixed point of F◦F, with the Hausdorff-distance measure Dk tracking their tightening.The sets can be updated until the change in Dk falls below a threshold.Source: §4.2.2 · p. 9
• Monotonicity or anti-tonicity of F with respect to individual design variables such as Nr, ny, K, and Ap remains open.The paper defers analysis of these variable-by-variable properties to future work.Source: §4.2.2 · p. 10

5.1 LCA design variables

• Architecture variables shape stability and feasibility, tuning variables shape closed-loop behavior, and algorithmic choices affect tractability but are excluded from this co-design procedure.Examples of architectural choices include reduced-model dimension and inter-layer sampling ratio; tuning choices include numerical controller and model parameters.Source: §5.1 · pp. 10–11

5.2 Co-design problem formulation

• The formulation jointly designs the planner and controller subject to stability, LCA feasibility, and real-time computation constraints.The compute constraint requires the planner and controller computations to finish within their respective sampling times.Source: §5.2 · pp. 11–12

5.3 Definition of cost

• The cost function maps design choices to Pareto curves, allowing designers to select among optimal designs according to their preferred objectives.The paper notes that LCA tradeoffs are heterogeneous and application-specific, so there is generally no unified cost definition.Source: §5.3 · p. 12
• The illustrative multi-valued cost combines speed, accuracy, and computational effort rather than imposing one scalarized objective.The paper leaves systematic joint handling or scalarization of these components outside its focus.Source: §5.3 · p. 12

6 Case study

• The simplified study solves the problem for coupled stable linear integrators, fixes several architectural and model choices, and optimizes over integer design variables ny and Nr.It uses exact discretization and solves for K through the discrete-time algebraic Riccati equation.Source: §6 · p. 12
• The study evaluates speed using state change, sampling ratio, and tracking timing; accuracy using tracking error; and compute using controller and planner computation proxies.The compute proxy assumes shared resources for the planner and controller.Source: §6 · p. 13

Numerical results

• The simplified co-design problem is evaluated on a ny × Nr grid spanning [1, 20] × [2, 30], with feasible points and Pareto-optimal solutions plotted.Because K is not optimized in this setup, the grid solutions trace the Pareto curves directly; other feasible solutions with different tuning values may lie north of the plotted curves.Source: Numerical results · pp. 13–14
• The Pareto designs trade speed, accuracy, and compute; the response plots contrast accuracy-, speed-, and compute-focused choices with a diversity-enabled σ-sweet spot.The compute-focused design sacrifices both speed and accuracy for very cheap compute and yields only marginal stability, while the consensus choice is sufficiently accurate and fast within compute limits.Source: Figure 3 · p. 13, Figure 4 · p. 14, Numerical results · p. 14

On solving (18)

• The co-design problem is large, heterogeneous, nonlinear, and non-convex, although it needs to be solved only once offline; practical formalizations should enable polynomial-time navigation of the design space.The formulation is intended to provide stability, constraint-satisfaction, and implementation guarantees for the selected design.Source: §7 · p. 14

On compositionality

• The derivation is not limited conceptually to two layers, but compositional design remains open because new interfaces and an analogous constraint map F are not established.The authors suggest that formalizing compositionality may require tools beyond those presented.Source: §7 · pp. 14–15

LCA co-design problem as a DMP

• The co-design problem resembles a single-layer design method, and clarifying whether it can reduce to that formulation is proposed as a direction for future work.The connection may matter when individual layers, such as a learned world model or advanced motion planner, are expensive to evaluate.Source: §7 · p. 15

On noise and disturbances

• For plants with bounded additive disturbances, the derivations can be generalized by expanding the signal set and constructing a fast-timescale robust positively invariant set.The authors expect this to yield a robust co-design problem with minor modifications.Source: §7 · p. 15

8 Conclusions

• For a linear two-layer LCA, the co-design formulation combines architecture and tuning choices with conditions guaranteeing stability, constraint satisfaction, and online implementability, enabling Pareto tradeoffs in speed, accuracy, and compute.It is presented as an explicit formulation for navigating Pareto-optimal LCA designs.Source: §8 · p. 15

• Some results in this appendix use tools presented in Appendix B.Source: A Proofs and derivations · p. 17

A.1 Derivation of (9a)

• The derivation of (9a) writes the error at k = (n + 1)Nr and concludes by applying interaction relations (4d) and (5).Source: §A.1 · p. 18

A.2 Proof of Lemma 1

• For fixed design variables ϕ and θ, set inclusions imply that Fϕ,θ is order-reversing in (Y, R).The proof propagates inclusions through the intermediate sets and uses Fact 3 with the definition of F in (15).Source: §A.2 · p. 18

• Repeated application of Fϕ,θ composed with itself yields an inclusion chain to which Lemma 3 applies, so the Hausdorff distance Dk is nonincreasing.Source: §A.3 · pp. 18–19

B.1 Facts regarding ⊕, ⊖

• Minkowski sums and linear maps preserve set inclusion, while the Pontryagin difference reverses inclusion in its subtracted set.Source: §B.1 · p. 19

B.2 Hausdorff distances and set inclusions

• Hausdorff distance measures the largest closest-point distance between two sets, and B1 ⊆ B2 ⊆ A2 ⊆ A1 implies dH(A2, B2) ≤ dH(A1, B1).Source: §B.2 · p. 19