Near-Optimal Static State-Feedback Law

• The paper presents a near-optimal static state-feedback law that minimizes a quadratic cost while meeting structural and informational constraints.
• It details methodologies including deadbeat and hybrid strategies, achieving competitive ratios and ensuring undominated performance in distributed control systems.
• The study quantifies trade-offs between available model information and performance, offering guidelines for effective design in networked, large-scale systems.

A near-optimal static state-feedback law is a control strategy for deterministic or stochastic dynamical systems in which the control input at each time depends only on the instantaneous system state via a static gain matrix, i.e., $u = Kx$. The “near-optimal” qualifier indicates that the law is constructed to minimize a specified performance cost reasonably close to the theoretical optimum, often under given structural, informational, or stability constraints. This concept is of central importance in modern control design, particularly for large-scale, distributed, and information-constrained systems, as it balances closed-loop performance against architectural, computational, and implementational limitations.

1. Structural Constraints and Limited Information in Static State Feedback

In practical applications, static state-feedback laws are often required to be structured according to the physical interconnection or measurement architecture of the system. The canonical problem is to design $u = Kx$ with $K$ subject to specific sparsity or block-diagonal constraints, where $K$ may be block-diagonal, banded, or otherwise structured according to a “control graph” that encodes which state variables are accessible to each actuator or decision maker (Farokhi et al., 2011).

A key development is the paper of limited model information control design methods. These are maps $\Gamma: \mathcal{P} \rightarrow \mathcal{K}$ from the set of admissible plant models, structured via a “plant graph,” to the set of admissible static control gains, structured via a (possibly sparser) “design graph.” In the extreme “communication-less” case (i.e., totally disconnected design graph with self-loops), each subcontroller can access only its own subsystem’s local parameters and state, yielding diagonal $K$ by construction. This restriction induces a hierarchy of controllers according to the breadth of accessible model information (Farokhi et al., 2011).

This setting gives rise to the notion of optimal structured static state-feedback control as the solution to:

$\min_{K \in \mathcal{K}_{\text{struct}}} J_P(K)$

where $J_P(K)$ is a separable quadratic cost, and $\mathcal{K}_{\text{struct}}$ is the set of controllers adhering to a strict structure.

2. Performance Metrics: Competitive Ratio and Domination

Given structured controllers designed under information constraints, the performance must be benchmarked against the achievable optimum. Two performance metrics are paramount (Farokhi et al., 2011):

• Competitive ratio:

$$r_P(\Gamma) = \sup_{P} \frac{J_P(\Gamma(P))}{J_P(K^*(P))}$$

where $K^*(P)$ denotes the fully centralized (unconstrained) optimal feedback. The competitive ratio quantifies the worst-case multiplicative performance loss incurred by the suboptimally designed controller. For static state feedback, this competitive ratio often yields a lower bound on the best possible undominated design under specified informational restrictions.

• Domination: A controller synthesis method $\Gamma$ is said to “dominate” another $\Gamma'$ if for all $P$, $J_P(\Gamma(P)) \leq J_P(\Gamma'(P))$ and strictly less for at least one $P$. An “undominated” design is not beaten by any other in the worst-case sense, within the family of considered information structures.

By comparing controllers via their competitive ratios and the notion of domination, systematic trade-offs between decentralization, locality, and achievable performance are revealed.

3. Optimal Strategies under Structural Constraints

For fully-actuated systems partitioned into $q$ subsystems (i.e., each block $B_{ii}$ invertible), canonical synthesis methods arise:

• Deadbeat strategy: For communication-less designs and no sinks in the plant graph, the optimal local gain for subsystem $i$ is

$K_{ii}^{\Delta} = -B_{ii}^{-1} A_{ii}$

leading to

$\Gamma^{\Delta}(A,B) = -B^{-1}A$

which drives the system to the origin in a single time step (“deadbeat” action). This achieves the minimal competitive ratio among all communication-less designs:

$r_P(\Gamma^{\Delta}) = 1 + 1/\epsilon^2$

where $\epsilon$ is a uniform lower bound on the minimal singular values of all $B_{ii}$ (Farokhi et al., 2011).

• Hybrid strategy for systems with sinks: If the plant graph has sinks (subsystems not affecting others), the deadbeat prescription is suboptimal. In this case, the deadbeat law is applied to non-sinks, while for each sink, the optimal subsystem LQR gain (via local Riccati equation) is used. This hybrid controller is shown to be undominated within the admissible family.

Such strategies are not only theoretically optimal under imposed constraints but are implementable using only locally available plant parameters and access to interconnection topologies.

4. Trade-off between Model Information and Performance

A central result is the quantification of the trade-off between available model information (as expressed by the design graph) and the best achievable performance (Farokhi et al., 2011). If local designers have access solely to their own subsystem’s model, the deadbeat competitive ratio cannot be improved. However, enriching the model information—for example, by providing subsystems with (portions of) the global interconnection graph—can sometimes strictly improve the achievable competitive ratio.

Specifically, if the design graph is at least as dense as the plant graph (i.e., each subsystem designer can access models of all “affecting” subsystems), then it is possible to “mix” local and neighboring information to synthesize gains that get strictly closer to true centralized optimality. The performance penalty for restricting communication or designer information is thus explicitly characterized and bounded.

Design graph (information)	Competitive ratio bound	Achievable performance
Totally disconnected (local only)	$1 + 1 / \epsilon^2$	Deadbeat, undominated
Supergraph of plant graph	$< 1 + 1 / \epsilon^2$ possible	Closer to centralized optimal

This result provides a quantitative guideline for design decisions in large-scale, privacy-sensitive, or otherwise communication-constrained networks: increasing the available model information directly reduces worst-case inefficiency.

5. Synthesis Procedures and Implementation

The synthesis of near-optimal static state-feedback laws under structural or informational constraints proceeds as follows:

1. Graph specification: Define the plant interconnection (plant graph) and the maximal information subsets accessible to each subcontroller (design graph).
2. Gain computation: For each subsystem $i$:
   • If fully local (communication-less), compute $K_{ii}$ using only $A_{ii}$, $B_{ii}$.
   • If sinks present, solve the decoupled LQR Riccati equation for those.
   • If additional neighbor models are available, augment local models accordingly.
3. Controller assembly: Stack local gains to form the overall $K$ matrix with block structure dictated by the design graph.
4. Performance certification: Calculate the competitive ratio (and/or verify via dominating property) to ensure that the design meets application-specific performance thresholds.

Typical resource requirements involve local matrix factorizations or Riccati solves and rudimentary combinatorial searches over the design and plant graph entries, making the procedures scalable and suitable for distributed implementation.

6. Applications, Limitations, and Extensions

This framework has immediate applicability to distributed, decentralized, or privacy-constrained control—networked control systems, power grids, and multi-agent robotics, where controller design must respect strict locality conditions. It generalizes to settings involving time-varying or stochastic interconnections by suitable modification of the structural constraints.

However, strong limitations include: (i) applicability largely to fully-actuated systems (each $B_{ii}$ invertible); (ii) possible conservatism for poorly connected networks or highly asymmetric interconnections; (iii) suboptimality relative to the centralized LQR optimum is inherent unless information restrictions are relaxed.

Extensions in literature consider robustness to plant uncertainties, privacy-preserving synthesis, dynamic (as opposed to static) feedback, or explicit consideration of communication delays, but the outlined “limited model information” paradigm remains foundational for near-optimal structured static state-feedback law design.

7. Summary

Near-optimal static state-feedback laws under structural and informational constraints are characterized by their synthesis via locally-available model data, certified worst-case performance via competitive ratio, and systematic trade-offs between locality and achievable optimality. The core theoretical and practical results establish that, for fully-actuated, discrete-time systems, one may efficiently synthesize undominated, computationally tractable feedback laws whose suboptimality relative to the fully centralized solution is explicitly and quantitatively bounded by the amount of shared model information (Farokhi et al., 2011). This paradigm underlies much of the modern theory and application of distributed optimal control in large-scale complex systems.