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We add the σ-modification and the low-frequency learning to the model reference adaptive controller (MRAC) (Guduri et al. in SN Appl Sci 3:1–21, 2021) to make it robust in the presence of two simultaneous bounded disturbances and maintain consistent mean particles’ temperature and velocity collectively called mean particles’ states (MPSs) when they impact the substrate to be coated. The MPSs affect the coating quality. Even though results are applicable to several coating processes, we consider an atmospheric plasma spray process (APSP). It is shown that the proposed controller can quickly adopt to disturbances in the average injection velocity of powder particles and in the arc voltage to change the input current, and the argon and the hydrogen flow rates to maintain constant values of the MPSs. The effects of the parameter values in the MRAC, the MRAC with $$\sigma$$ σ -modification (R-MRAC), and the R-MRAC with low-frequency learning (MR-MRAC) schemes on tracking error convergence, steady-state tracking error, disturbance rejection and the presence of overshoot have been studied. The numerical experiments suggest that $$2 \le \gamma \le 20,$$ 2 ≤ γ ≤ 20 , $$10 \le \sigma \le 100,$$ 10 ≤ σ ≤ 100 , and $$20 \le \lambda \le 80$$ 20 ≤ λ ≤ 80 for the MR-MRAC provide fast adaptation, no overshoot, and low tracking error in the controlled response. The parameter $$\lambda > 0$$ λ > 0 suppresses high-frequency oscillations in the closed-loop control system, and $$\gamma$$ γ serves to tune the controller gains. The control scheme has been tested using the software, LAVA-P, that simulates well an APSP.

The coatings produced by an atmospheric plasma spray process (APSP) must be of uniform quality. However, the complexity of the process and the random introduction of noise variables such as fluctuations in the powder injection rate and the arc voltage make it difficult to control the coating quality that has been shown to depend upon mean values of powder particles’ temperature and speed, collectively called mean particles’ states (MPSs), just before they impact the substrate. Here, we use a science-based methodology to develop a stable and adaptive controller for achieving consistent MPSs and thereby decrease the manufacturing cost. We first identify inputs into the APSP that significantly affect the MPSs and then formulate a relationship between these two quantities. When the MPSs deviate from their desired values, the adaptive controller is shown to successfully adjust the input parameters to correct them. The performance of the controller is tested via numerical experiments using the software, LAVA-P, that has been shown to well simulate the APSP.

Functionally graded coatings (FGCs) have a material composition continuously varying through the thickness but uniform in the surface parallel to the coated substrate. When used as a thermal barrier on a metallic substrate, the coating composition varies from an almost pure metal near the substrate to a pure ceramic adjacent to the outer surface exposed to a hot environment. Challenging issues in producing high quality FGCs in the presence of external disturbances with an atmospheric plasma spray process (APSP) include controlling the mean temperature, the mean axial velocity, and the positions of the constituent material particles when they arrive at the substrate to be coated. The unavoidable disturbances include fluctuations in the arc voltage and clogging of the powder in the delivery system. For a two-constituent coating, this work proposes using three modified robust model reference adaptive controllers based on the σ-modified laws and low frequency learning. One controller adjusts the current and flow rates of argon and hydrogen into the torch. The other two controllers adjust the distance of the two powder injector ports from the plasma jet axis and the average injection velocity of each powder. It is shown through numerical experiments that the three controllers implemented in an APSP consistently produce high-quality FGCs.

This chapter presents the σ-modification and the low-frequency learning to the model reference adaptive control (MRAC) to improve the transient performance of a closed-loop control system under uncertainties and external disturbances. The σ-modification improved the robustness to external bounded disturbances, and the low-frequency learning enabled fast adaption without creating high-frequency oscillations. The feasibility of the resulting robust adaptive control architecture is tested for the atmospheric plasma spray process (APSP) to achieve consistency in ceramic coatings through numerical simulations. The robustness and adaptability of the modified and the standard MRAC architecture are investigated under artificially induced disturbances. The proposed architecture shows better performance than the standard MRAC under external disturbances. The results depict the improved transient performance with faster convergence and smooth variations in closed-loop control signals without creating high-frequency oscillations or steady-state errors.