Control Co-Design (CCD) Synthesis Overview

• Control Co-Design (CCD) is a framework that jointly optimizes physical system design and controller synthesis to enhance performance, robustness, and efficiency.
• It leverages diverse mathematical methods—such as convex optimization, bilevel programming, and game theory—to tackle interconnected design challenges.
• CCD embeds safety, performance, and resource constraints directly into the design of dynamic, cyber-physical systems.

Control Co-Design (CCD) Synthesis Problem

Control Co-Design (CCD) refers to the rigorous, simultaneous synthesis of both the physical design (plant or system parameters) and the controller for dynamic, typically cyber-physical, systems. Unlike traditional sequential approaches where plant and controller are designed separately, CCD optimizes the coupled system-end-to-end, exploiting and explicitly constraining plant–controller interactions to achieve performance, robustness, and efficiency objectives, often under resource, safety, and structural constraints. CCD synthesis spans both finite- and infinite-dimensional models, may incorporate networked or distributed architectures, and leverages optimization, convexity, game theory, and system-theoretic safe set and Lyapunov-based conditions.

1. Mathematical Foundations and Problem Structure

CCD problems are formulated as joint optimization (or equilibrium) problems over both physical (plant) and control (often policy) parameters. The general form is:

$\min_{x,\,K}\ J(x, K)$

subject to

$$\begin{aligned} &\text{System dynamics and constraints (ODE, PDE, hybrid, discrete):} \ &\qquad x_{k+1} = f(x_k,\,K(\cdot),\,w_k,\,\cdots), \ &\text{Physical and plant constraints:}\quad h(x)\leq 0,\ &\text{Control constraints:}\quad g(\cdot, K)\leq 0,\ &\text{Additional constraints: stochastic safety, communication, or energy use.} \end{aligned}$$

Decision variables may include continuous plant parameters (geometry, mass, inertia, actuator/sensor placement), discrete architecture variables (communication links, layouts), and parametric or functional controllers (static or dynamic, centralized or distributed policies, feedback gains, or even deep neural network parameters in reinforcement learning contexts).

The interplay between plant and controller is encoded by coupled system models: plant parameters appear explicitly in the system matrices or PDE coefficients, controller design impacts closed-loop poles, reachable sets, and energy consumption, and additional couplings may arise from state- or configuration-dependent communication or failure models (Hu et al., 2018, Najafirad et al., 9 Nov 2025, Yadav et al., 29 Dec 2025).

2. Principal CCD Synthesis Methodologies

CCD synthesis employs diverse methodologies contingent on model structure, control law class, and constraint type:

• Convex programs (LP/QP/SDP/LMI-based approaches): When system models are linear or can be convexified through intelligent parametrization (e.g., under Lyapunov or structural controllability constraints), CCD reduces to tractable LP, QP, or SDP (Hu et al., 2018, Najafirad et al., 9 Nov 2025).
• Gradient-based or coordinate-optimization: For nonconvex, often nonlinear closed-loop conditions involving Lipschitz nonlinearities, static quadratic matrix equations, or Lyapunov constraints, coordinate/gradient-descent—possibly with alternating or Armijo line search—is used (Chanekar et al., 2022, Yadav et al., 29 Dec 2025).
• Bilevel or nested optimization: When inner- and outer-loops correspond respectively to plant and controller (or vice versa), and the control law is designed repeatedly for trial plant parameters, bilevel optimization or nested loops are employed (Wu et al., 2022, Bayat et al., 2023, Nash et al., 2021).
• Stochastic or game-theoretic formulations: CCD under uncertainty, stochastic loads/demands, or adversarial communication scenarios is addressed using Markov chain models, Nash equilibria, or min–max (robust) control frameworks (Wang et al., 2023, Aggarwal et al., 1 Mar 2025, Nash et al., 2021).
• Surrogate/data-driven and evolutionary algorithms: For large-scale, nonconvex domains (e.g., layout+control of WEC farms), fast surrogate models (ANNs, many-body expansions) combined with evolutionary search (GA, CMA-ES) are used to render the CCD tractable (Azad et al., 2024, Bayat et al., 2023).
• Set-based and reachability-based synthesis: For robust safety under bounded uncertainty, CCD is framed as a set-inclusion problem, propagating reachable sets and ensuring robust positive invariance via support function constraints, often leading to efficient one-level optimization (Bird et al., 2023).
• Atomic norm and structural methods: When sparse architectures or minimal communication are required (e.g., distributed $\mathcal{H}_2$ optimization), atomic norm minimization and combinatorial assignment algorithms formally balance architecture cost against closed-loop performance (Matni, 2014, Pequito et al., 2015).

3. Safety, Robustness, and Performance Constraints

Modern CCD formulations embed stringent safety and performance constraints, leveraging system-theoretic analysis:

• Lyapunov and multiple-Lyapunov function conditions: Stability, stochastic safety (ASE, ASAS, PSP), and practical safety are guaranteed via Lyapunov conditions, yielding algebraic constraints on plant and controller parameters; for switched or randomly switched systems, multiple Lyapunov functions and state-dependent constraints are used (Hu et al., 2018, Bird et al., 2023).
• Reachable set and robust invariance: Set-based CCD requires that all future closed-loop states, under all allowable disturbances from a convex initial set, reside within a safe invariant polytope; this is encoded via support function inequalities that are convex in the design variables (Bird et al., 2023).
• Stochastic safety: For systems with Markovian, fading, or packet-drop models, safety is enforced in expectation or with high probability, yielding convex or quadratic constraints on policy variables and state-occupancy distributions (Hu et al., 2018, Wang et al., 2023).
• Compositional dissipativity: For distributed systems (e.g., AC microgrids), subsystem-level dissipativity and global network compositional stability are enforced via structured LMIs, ensuring $L_2$-gain, voltage/frequency regulation, and proportional power sharing under arbitrary load disturbances (Najafirad et al., 9 Nov 2025).
• Actuator/sensor and communication limitations: Input/output constraints, actuator saturations, quantized communication, delays, and network-induced random losses are systematically embedded within the system model and optimization structure (Matni, 2014, Hu et al., 2018, Pequito et al., 2015).

4. Structural and Architectural CCD

CCD can integrate the simultaneous selection of sensors, actuators, and control/communication architecture:

• Structural system and assignment approaches: The minimal-cost co-design of input, output, and feedback configuration for arbitrary pole-placement or structural controllability/observability is formulated as a combinatorial/graph-theoretic optimization. Under state-irreducibility, this reduces to weighted assignment problems solvable in polynomial time via the Hungarian algorithm, unifying I/O and feedback co-design (Pequito et al., 2015).
• Atomic norm and regularization for distributed control: The Regularization for Design (RFD) framework amplifies this by structural sparsity regularization using atomic norms, leading to jointly optimal co-design of communication topology and distributed controller via convex SOCP (Matni, 2014).

5. Networked, Stochastic, and Multi-Disciplinary Extensions

CCD has broad application across networked, stochastic, and multi-disciplinary systems:

• Networked control with state-dependent communication: In industrial wireless NCS, plant states directly modulate transmission reliability (e.g., shadow fading), necessitating joint optimization of motion/operation policy and communication power to maintain stochastic safety (Hu et al., 2018).
• Stochastic co-design: For infrastructure systems (e.g., water networks), CCD is cast as a stochastic expectation-minimization problem, where infrastructure sizing (tank volume) and control policy (price threshold, pumping schedules) are co-designed under exogenous noise, modeled using finite-state Markov chains; long-run empirical cost converges almost surely to expected cost (Wang et al., 2023).
• Game-theoretic communication–control: In adversarial or multi-agent settings, scheduler–controller interactions are modeled as non-cooperative games, with equilibrium computed via solution of generalized Lyapunov–Sylvester equations and projected gradient/response dynamics (Aggarwal et al., 1 Mar 2025).
• High-dimensional/multi-physics systems: For PDE-governed systems (parabolic dynamics), CCD involves discretization to ODEs, joint optimization of boundary feedback and PDE coefficients, Lyapunov-based stability certificates, and gradient-based optimization (Yadav et al., 29 Dec 2025). Multi-disciplinary settings (e.g., wind/wave energy or autonomous vehicles) utilize modular, monotone co-design frameworks capturing component-level monotonicity and Pareto optimality across control, sensing, and computation (Zardini et al., 2022, Zardini et al., 2020, Azad et al., 2024).

6. Computational Algorithms and Representative Examples

The following table summarizes representative CCD synthesis formulations and solution approaches, illustrating breadth across domains:

Application Domain	CCD Methodology	Key Mathematical Framework	arXiv Reference
State-dependent NCS	Convex LP/QP, Lyapunov constraints	Multiple Lyapunov, state-occupancy, safety LP	(Hu et al., 2018)
Structural architecture	Assignment, atomic norm, SOCP	Graph theory, QI subspaces, atomic norms	(Pequito et al., 2015, Matni, 2014)
Hydrokinetic turbine	Direct transcription, feedback law	Collocation, OLOC vs. feedback, BEMT	(Amini et al., 2023, Jiang et al., 2022)
Precision motion stage	Bilevel nested, H∞ synthesis	Bi-level loop, mixed-sensitivity, SVD-IFT	(Wu et al., 2022)
Nonlinear systems	Static matrix eq., Lyapunov constraint	Quadratic matrix, gradient-based descent	(Chanekar et al., 2022)
PDE (parabolic) dynamics	Discrete Lyapunov, gradient descent	Finite-difference, matrix Lyapunov equation	(Yadav et al., 29 Dec 2025)
Water distribution network	Markov chain, expectation minimization	Stationary distribution, empirical SLLN	(Wang et al., 2023)
AC Microgrid (distributed)	LMI, dissipativity, passivity	IF/OFP passivity indices, networked LMI	(Najafirad et al., 9 Nov 2025)
Active suspension	Set-based reachability, tube invariance	Support function constraints, zonotopes	(Bird et al., 2023)
Multi-physics, multi-agent	Modular monotone, Pareto front	MDPI, task-driven feasibility	(Zardini et al., 2022, Zardini et al., 2020)

Simulation studies demonstrate the practical impact of CCD synthesis: for example, in precision motion systems, weight reduction by 42% and bandwidth increase by 28% are achieved relative to sequential design (Wu et al., 2022); in industrial NCS, joint policy outperforms separation-based design under fading (Hu et al., 2018); and in distributed AC microgrids, optimal sparse topologies with guaranteed voltage/frequency regulation are found (Najafirad et al., 9 Nov 2025).

7. Emerging Directions and Practical Implications

Contemporary CCD research addresses increasingly complex, data-driven, and uncertain environments:

• Digital Twin and reinforcement learning integration: Multi-generation frameworks incorporate data from successive system iterations, online quantile-regression uncertainty modeling, and Deep RL-based joint synthesis of plant and policy, yielding significant improvements in performance and robustness in, e.g., automotive suspension (Tsai et al., 12 Oct 2025).
• Surrogate modeling and hybrid optimization: For large-scale, multi-layout optimization (WEC farms), surrogate models (ANNs) are coupled with evolutionary algorithms, drastically reducing optimization time and enabling site-dependent layouts and controls (Azad et al., 2024).
• Formal modularity and monotonicity: By recasting control and estimation modules as monotone feasibility relations, scalable, modular CCD becomes tractable for complex robotic platforms and autonomy stacks, supporting formal Pareto-front computation and system-level trade-off analysis (Zardini et al., 2022, Zardini et al., 2020).

The Control Co-Design Synthesis Problem thus represents a unifying, extensible paradigm for optimal, robust, and efficient integration of plant and controller (and, increasingly, architecture, communication, and resource allocation) in modern dynamical systems engineering, and continues to evolve through developments in optimization, learning, and formal system theory.