Minimal Actuator Selection Problem

• Minimal actuator selection is the process of choosing the smallest subset of actuators to meet controllability and performance requirements in dynamic systems.
• The approach leverages structural controllability through graph-theoretic methods and matching algorithms, reducing the problem to NP-hard set cover formulations.
• Advanced techniques, including greedy, convex optimization, and reinforcement learning methods, are developed to balance control energy, robustness, and cost efficiency.

The minimal actuator selection problem encompasses the design and analysis of methodologies for selecting the smallest subset or schedule of actuators in dynamic systems, typically linear time-invariant (LTI) or more generally nonlinear, so as to ensure properties such as controllability, optimal control performance, energy or disturbance attenuation, robustness to actuator failures, and satisfaction of structural constraints. This problem arises fundamentally in control theory, network science, cyber-physical systems, robotics, and industrial applications where resource constraints, hardware costs, or reliability issues drive the need for sparse actuation patterns.

1. Structural Controllability and Graph-Theoretic Formulations

In classical LTI systems, minimal actuator selection is tightly linked to the concept of structural controllability. For a system $\dot{x} = A x + B u$, the structure of $A$ and the dedicated input pattern of $B$ (each input directly affecting one state) induces a directed graph $D(A) = (X,E)$, where $X$ is the set of state nodes and edges correspond to nonzero $A_{ij}$ entries (Pequito et al., 2012). Actuator selection is reframed as finding input placements so that the augmented graph $D(A,B)$ is spanned by a disjoint union of input cacti (collections of input-rooted paths and cycles), guaranteeing structural controllability almost surely for arbitrary weighting of nonzero elements.

This is formalized via bipartite matching in the associated graph $B(D(A))$, where the minimal number of dedicated inputs required is exactly the number of right-unmatched vertices in a maximum matching. All possible minimal input configurations correspond to all possible maximum matchings, and the minimal dedicated-input placement can be constructed in polynomial time $O(\sqrt{n}|E|)$ using the Hopcroft–Karp algorithm (Pequito et al., 2012).

2. Complexity: NP-Hardness and Set-Cover Reductions

General formulations where actuator placement is subject to additional constraints—such as controllability with respect to a given input matrix $B$, bounds on control energy, or robustness against actuator failures—lead to NP-hard problems. The minimal actuator selection problem reduces to set cover or set multicover under the PBH controllability criterion: for given $A, B$, the minimal selection problem is equivalent to a binary integer linear program where the rows correspond to eigenvalue coverage needs and the columns to actuator channels (Ballotta et al., 13 Jan 2026). If $A$ has all simple eigenvalues, the problem reduces to set cover; for higher geometric multiplicities, it becomes a set multicover problem. Robust selection against up to $f$ actuator failures increases the coverage demand from $g_i$ to $g_i+f$ per eigenspace.

Approximation algorithms for set cover or multicover offer $O(\ln p)$-factor guarantees, where $p$ is the number of elements (modes), and no polynomial-time algorithm may achieve substantially better performance unless P=NP (Ballotta et al., 13 Jan 2026).

3. Control-Energy and Performance-Constrained Selection

Beyond mere controllability, selection typically optimizes system-theoretic metrics such as minimum required energy, controllability Gramian volume, average or worst-case effort ($$\mathrm{tr}(W^{-1}), \log\det(W^{-1}), 1/\lambda_{\min}(W)$$), or closed-loop performance (Tzoumas et al., 2015, Tzoumas et al., 2014, Chen et al., 2016). For given $A$ and time horizon $T$, the selection of actuators $S$ yields a Gramian $W_S$, and the minimum-energy to realize a state transfer is $x_f^\top W_S^{-1} x_f$. Placing a bound $J(S)\leq E_{\max}$ makes the problem a submodular cover, allowing greedy selection algorithms with provable $O(\log n)$ approximability (Tzoumas et al., 2015, Tzoumas et al., 2014).

Supermodularity arises since $-J(S)$ is submodular—the marginal gain in energy decreases as $S$ grows. This structure allows efficient algorithms for operating under cardinality or energetic constraints (Tzoumas et al., 2015, Tzoumas et al., 2014). In the worst-case control energy setting, exact solutions are available for symmetric positive-definite $A$ via closed-form expressions involving the inverse of a Cauchy matrix; in the general case, the problem is highly nonconvex (Chen et al., 2016).

4. Algorithmic Approaches and Approximation Guarantees

Prominent algorithmic approaches include:

• Graph-theoretic & Matching: Polynomial-time enumeration for dedicated-input systems under purely structural constraints (Pequito et al., 2012).
• ILP/Set Cover: General selection cast as binary ILP or set (multi)cover, solved exactly (exponential in worst case) or approximately (greedy $H(p)$-approximation) (Ballotta et al., 13 Jan 2026).
• Greedy and Matroid-based: Forward & reverse greedy algorithms exploit submodularity, matroid structure, and cardinality constraints for energy metrics, providing performance bounds parameterized by submodularity ratio $\gamma$ and curvature $\alpha$ (Guo et al., 2019, Guo et al., 2021).
• Convex Optimization: Semidefinite programming and relaxations, often employing group $\ell_2$ sparsity penalties for actuator selection in optimal feedback problems, with proximal gradient methods scaling to large $n$ (Zare et al., 2018).
• Mixed-Integer SDPs/MIBMIs: Output-feedback and robust control designs recast as MI-SDPs or MIBMIs, solved via branch-and-bound or successive convex approximations, sometimes with slicing for binary recovery (Nugroho et al., 2018, Taha et al., 2017, Ebrahimi et al., 2018).
• Reinforcement Learning & Data-Driven Policy: Sequential actuator selection for nonlinear or hybrid applications, formulated as submodular MDPs (e.g., Dueling Double-DQN for assembly, PPO-CMA in flow control), yielding near-greedy performance with competitive complexity (Ye et al., 24 Apr 2025, Paris et al., 2022).

Approach	Complexity	Approximation
Bipartite matching	Polynomial (structural)	Exact (unstructured)
ILP/Set cover	Exponential/poly	$H(p)$-approx.
Greedy (submodular)	Polynomial	$O(\log n)$
Convex optimization	Polynomial	Relaxed sparsity
MI-SDP/MIBMIs	Exponential	Global opt. if feasible
RL methods	Empirical polynomial	Near-greedy, data-adaptive

Approximation ratios follow from submodular set cover theory: greedy selection achieves $(1-\frac{1}{e})$ for monotone submodular functions, and worst-case bounds are tight for NP-hard coverage (Tzoumas et al., 2015, Guo et al., 2019, Ye et al., 24 Apr 2025).

5. Extensions: Time-Varying, Nonlinear, Robust, Output-Feedback Selection

Time-varying actuator schedules, joint sensor-actuator selection, output-feedback stabilizability, and systems with nonlinear dynamics significantly generalize the problem (Taha et al., 2017, Nugroho et al., 2018, Nugroho et al., 2018, Nugroho et al., 2020). Schedules can be constructed so as to activate only a constant average number of actuators per time step while ensuring systemic controllability metrics stay within guaranteed bounds; spectral sparsification and leverage score sampling underlie efficient deterministic and randomized constructions (Siami et al., 2018).

Robust actuator selection, aiming to maintain controllability under actuator failures, raises coverage constraints equivalent to the robust set multicover problem and receives similar algorithmic treatment (Ballotta et al., 13 Jan 2026, Guo et al., 2021). In nonlinear dynamic systems, the placement problem is embedded in Lyapunov-theoretic matrix inequalities with selection variables, solved via customized MISDP branch-and-bound routines with structure exploitation for scalability (Nugroho et al., 2020).

6. Numerical Case Studies and Practical Implementations

Empirical results confirm theoretical bounds and demonstrate computational viability. Exact polynomial-time matching algorithms solve structural problems on digraphs of hundreds of nodes (Pequito et al., 2012). Greedy and matroid-based algorithms scale to real-world networks and power systems of 100+ states, providing significant actuator savings versus random selection (Guo et al., 2019, Guo et al., 2021). Convex optimization and proximal gradient methods handle large PDE discretizations (e.g., $n=256$) with near-linear convergence (Zare et al., 2018). RL-based frameworks and sequential placement policies match or outperform model-free baselines in precision assembly and flow control, achieving substantial resource reduction with near-optimal performance (Ye et al., 24 Apr 2025, Paris et al., 2022).

7. Open Directions and Research Challenges

Research continues on:

• Dropping strict controllability in favor of "approximate" reachability or singular Gramian designs (Tzoumas et al., 2015).
• Time-varying, stochastic, and distributed actuator selection algorithms with robustness to failures and partial information (Siami et al., 2018, Guo et al., 2021).
• Extensions to nonlinear, hybrid, or uncertain systems, especially under operational constraints and feedback limitations (Nugroho et al., 2020, Taha et al., 2017, Nugroho et al., 2018).
• Integrating RL and data-driven approaches for online, adaptive placement in complex environments (Ye et al., 24 Apr 2025, Paris et al., 2022).
• Joint sensor-actuator selection, output-feedback configurations, and large-scale system implementations with scalable optimization routines (Nugroho et al., 2018, Nugroho et al., 2018).

A plausible implication is that future research will focus on generalized actuator selection frameworks that combine combinatorial optimization, convex programming, graph theoretical insights, and data-driven methods, all tailored for robustness, scalability, and system-theoretic guarantees.