Temperature control by a state-delay nash strategy: theory and experiments

Abstract: This work expands the Nash equilibrium algorithm to those systems that have a state delay within the deterministic case, a Linear-Quadratic (LQ) type performance index with generalized cross terms is used, following the constructive approach on optimal control: first, propose the strategies form, close the loop and find the Cauchy solution; secondly, express the cost function according to those expressions; and finally use the Bellman equation as restrictions to find the strategies parameters. The stability of the closed loop is demonstrated. The algorithm is applied to a thermal prototype with an ESP32 micro-controller and its performance is compared with optimal PI controls inside two commercial/industrial PID controllers REX-C100.

• The paper introduces an explicit Nash strategy for LQ systems with state delays, using coupled Riccati equations to define control actions.
• It demonstrates improved transient performance in temperature control by outperforming conventional PI controllers in experimental tests.
• The study integrates analytic solution techniques with Lyapunov–Krasovskii based stability proofs for robust real-time control applications.

State-Delay Nash Strategies for Temperature Control: Theory and Experiments

Analytical Framework for State-Delay Nash Equilibria

This paper advances the theory of differential games by constructing explicit Nash equilibrium strategies for deterministic Linear-Quadratic (LQ) systems with state delays. It rigorously addresses the analytic characterization of Nash feedback strategies when the performance index encompasses generalized cross terms, extending previous work that mainly relied on implicit or numerical solutions for similar delay systems. The control synthesis follows the constructive approach that first postulates the structure of admissible state-dependent strategies, closes the loop to obtain the Cauchy solution, and then formulates the cost function for each player in terms of this solution. Optimization proceeds via constraints derived from the Bellman equation, avoiding i-smooth calculus and instead using classical functional derivatives for time-delay systems.

The resulting control strategy for player $i$ is given as:

$$u_i(t) = - (R_{i,i})^{-1} B_i^T \Pi_{i,0} x(t) - (R_{i,i})^{-1} B_i^T \int_{-h}^0 \Pi_{i,1}(\theta) x(t+\theta) d\theta$$

where $\Pi_{i,0}, \Pi_{i,1}(\cdot)$ are matrices determined via a system of coupled partial differential Riccati-type equations. The solution method incorporates decoupling techniques inspired by [9], yielding explicit formulas for real-time control implementation.

Stability Analysis and Functional Criteria

Closed-loop stability is proven by constructing Lyapunov–Krasovskii functionals appropriate for delay differential dynamics. The paper provides sufficient conditions for asymptotic stability by deriving an algebraic Riccati equation (ARE) involving the closed-loop matrices, delay integrals, and performance weights. If a positive definite solution to the ARE exists, stability in the sense of Lyapunov is guaranteed. This approach is noteworthy for rigorously bridging the gap between Bellman functional optimization and stochastic stability criteria typically employed in the literature.

Experimental Validation on Thermal Systems

Empirical evaluation is performed on a physical prototype simulating a food dehydration process controlled by two resistive heaters, with an ESP32 microcontroller executing the Nash strategy and commercial REX-C100 PID controllers serving as benchmarks. Plant identification uses recursive least squares (RLS) with a measured transport delay of $h = 2$ s and sampling time $T = 0.5$ s, yielding quantified system parameters. Nash strategies are computed offline in MATLAB and deployed for real-time control.

Comparative results between Nash equilibrium control and optimal PI controllers reveal pronounced improvements in transient metrics:

• Integral Squared Error (ISE): Nash strategy achieves $2.40 \times 10^3$, substantially lower than PI's $6.68 \times 10^3$.
• Integral Absolute Error (IAE): Nash strategy attains $46.3$, outperforming PI's $52.1$.
• Integral Time-w. Squared Error (ITSE): Nash strategy yields $4.83 \times 10^7$, better than PI's $$u_i(t) = - (R_{i,i})^{-1} B_i^T \Pi_{i,0} x(t) - (R_{i,i})^{-1} B_i^T \int_{-h}^0 \Pi_{i,1}(\theta) x(t+\theta) d\theta$$0.

Eigenvalue analysis of the performance index matrices confirms positive definiteness and hence admissibility of the Nash strategies. Closed-loop stability is corroborated by numerical ARE solutions with $$u_i(t) = - (R_{i,i})^{-1} B_i^T \Pi_{i,0} x(t) - (R_{i,i})^{-1} B_i^T \int_{-h}^0 \Pi_{i,1}(\theta) x(t+\theta) d\theta$$1.

Theoretical and Practical Implications

The explicit Nash strategy forms derived here substantially enhance the theoretical tractability of deterministic LQ differential games with state delays. By allowing analytical controller synthesis and facilitating practical real-time implementations on hardware, the results overcome the limitations of earlier stochastic, mean-field, or hierarchical formulations that relied heavily on computationally expensive numerical solvers.

Practically, this framework is applicable to a broad set of industrial systems involving transport delays, such as process heating, chemical reactors, and multi-agent networked control. The constructive decoupling approach for Riccati-type equations enables robust Nash equilibrium computation without resorting to advanced variational calculus, streamlining both theoretical extension and hardware deployment.

Future Directions

This research lays the groundwork for several strategic extensions. The framework could be generalized to $$u_i(t) = - (R_{i,i})^{-1} B_i^T \Pi_{i,0} x(t) - (R_{i,i})^{-1} B_i^T \int_{-h}^0 \Pi_{i,1}(\theta) x(t+\theta) d\theta$$2-player differential games, as indicated in the appendix, accommodating complex multi-agent scenarios with significant inter-player coupling. Further investigation is warranted into stochastic, adaptive, and learning-based controller variants for uncertain environments and nonstationary delay characteristics. Theoretical development towards hierarchical and mean-field Nash equilibria in time-delay domains is also anticipated.

Conclusion

This paper delivers an authoritative mathematical and experimental account of Nash equilibrium synthesis for deterministic LQ systems with state delays, providing explicit strategic formulas and validating them on a realistic thermal system. The approach demonstrates improved transient and integral performance relative to optimal PI controllers and supplies rigorous stability guarantees. The theoretical advancements and practical feasibility presented promise broad applicability to both cooperative and competitive control in delayed systems (2606.08344).