Abstract
We add the σmodification and the lowfrequency 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 \(\sigma\)modification (RMRAC), and the RMRAC with lowfrequency learning (MRMRAC) schemes on tracking error convergence, steadystate tracking error, disturbance rejection and the presence of overshoot have been studied. The numerical experiments suggest that \(2 \le \gamma \le 20,\) \(10 \le \sigma \le 100,\) and \(20 \le \lambda \le 80\) for the MRMRAC provide fast adaptation, no overshoot, and low tracking error in the controlled response. The parameter \(\lambda > 0\) suppresses highfrequency oscillations in the closedloop control system, and \(\gamma\) serves to tune the controller gains. The control scheme has been tested using the software, LAVAP, that simulates well an APSP.
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1 Introduction
An atmospheric plasma spray process (APSP) is a versatile tool to produce a variety of coatings such as thermal barrier, functionally graded and wear/corrosion resistant that have applications in aerospace, automotive, agriculture, and biomedical fields. Like in any manufacturing process, reproducibility and repeatability of the coating quality are of prime importance. As depicted in Fig. 1, in a typical APSP, a mixture of argon (Ar) and hydrogen (H_{2}) gases injected into a gun passes over an electric arc and produces a plasma that exits at high velocity and temperature.
Powder particles injected with or without a carrier gas into the plasma through a powder port traverse with the plasma toward the substrate to be coated.
It is generally believed in the coating industry that the mean of particles’ temperature and axial velocity (collectively called mean particles’ states, MPSs) just before impacting the substrate determine the coating quality. Disturbances in process variables such as the injection velocity and the arc voltage due to the nozzle wear, the powder injector wear, and pulsing/clogging of powder particles significantly affect the MPSs and hence the coating quality.
To make an APSP highly efficient and to minimize the variability in the coating quality, it is important to implement an efficient and robust adaptive controller capable of fast adaptation to disturbances in process variables and produce the desired MPSs with minimal deviations. Even though a few researchers have implemented online diagnoses and control for various thermal spray processes, aspects of fast adaptation, stability, and robustness to external disturbances have not been addressed. Fincke et al. [1] experimentally demonstrated the application of realtime diagnostics and control to the thermal spray process by monitoring the velocity and the temperature of particles as well as the shape and the trajectory of the spray pattern by varying the arc current and the flow rates of the primary and the secondary gases. However, the control system was designed by integrating several oneinput and oneoutput proportionalintegralderivative (PID) controllers, which is not an efficient control strategy. Li et al. [2,3,4] developed a model based on the process estimation and control of the mean particles’ velocity and particles’ melting in a highvelocity oxygen fuel (HVOF) spray process. A closedloop proportionalintegral (PI) controller was coupled to the estimation model to regulate the volumebased average of particles’ velocity and their degree of melting by controlling the inlet gas flow rates and combustion pressure. Through numerical simulations, the authors showed the effectiveness of the feedback controller and its robustness for various disturbances introduced in the HVOF spray process. Srinivasan et al. [5] employed an active sensor to monitor the system with a feedback controller to control the MPSs by varying the primary gas flow rate and the current.
Dykhuizen and Neiser [6] implemented a PI controller in a wire plasma spray process and showed that it kept MPSs uniform in a real production environment. Sampath et al. [7] presented an integrated approach by identifying the process maps between particles’ states and coating properties and introduced a feedback controller for maintaining MPSs within the prescribed range. Kanta et al. [8] used a fuzzy logic (FL) controller to keep the inflight particle characteristics within prescribed bounds. Kanta et al. [9] implemented an artificial intelligent (AI) system using the artificial neural network (ANN) and the FL controllers for online controlling of the APSP. The FL controller incorporates the knowledge, the understanding of the process, and the expertise of the operator to establish rules that are regularly updated. The conventional PI and PID controllers are fixed gain controllers that have poor adaptability and control performance for nonlinear plants like an APSP in the presence of external disturbances. Planche et al. [10] developed an automatic system using an ANN, FL controllers and an emulator that replicated the dynamic behavior of the APSP.
The fast adaptation and the stability of a closedloop system with minimal tracking error and disturbance rejection have not been addressed in the abovecited papers. Kim [11] designed and tested a simple linear controller using the AutoRegressive with eXogenous (ARX) input model and the input/output data for inductively coupled plasma torches to regulate the temperature and the axial velocity of the plasma jet by varying the input power. The feedback control simulation exhibited stable performance with minimal tracking error and disturbance rejection.
The model reference adaptive controller (MRAC) described in our previous work [12] could adaptively adjust the current, and the argon (Ar) and the hydrogen (H_{2}) flow rates in response to variations in the MPSs detected via sensors mounted on the observation window located just before the substrate. The integration of the MRAC into the APSP achieved faster convergence of the tracking error of the MPSs to their desired values and the stability of the closedloop system. However, it is not robust in the presence of model uncertainties and can become unstable due to parameter drifts, high gains, fast adaptation, and highfrequency noise. Some of these can be mitigated by using a robust MRAC (RMRAC) that modifies relations used in the MRAC. The RMRAC with fixed \(\sigma\)modification in the MRAC [13] is quite robust without explicitly knowing the plant dynamics and bounds on the external disturbances. It provides bounded closedloop signals but loses asymptotic convergence of the tracking error in the absence of disturbances. Furthermore, the fixed \(\sigma\)modification can introduce a steadystate error or bursting phenomenon [13]. Yucelen and Haddad [14] have proposed a lowfrequency learning to the MRAC by filtering out highfrequency oscillations in the controller response that preserves the asymptotic convergence of the tracking error to zero and achieves faster adaptation using high adaptive gains in the absence of external disturbances.
Here we report on the implementation of the modified RMRAC (MRMRAC) scheme by incorporating lowfrequency learning into the RMRAC for an APSP that in the presence of bounded external disturbances is stable and adaptively adjusts input parameters to achieve the desired MPSs within small bounds. The steadystate error associated with the \(\sigma\)modification is avoided, and the fast adaptation using high gain is achieved by using lowfrequency learning with lowpass filters for estimating gains. The controller design is based on a linearized model of the MPSs with unknown external bounded disturbances. Thus, the MRMRAC gives bounded responses of the closedloop control and convergence of the tracking error to a small number. The ranges of control parameters of the MRAC, the RMRAC and the MRMRAC schemes for the APSP to achieve low steadystate tracking error of the MPSs, disturbance rejection of average injection velocity of particles and arc velocity, and no overshoot of MPSs are presented using numerical simulations. It is found that the MRMRAC performs well when \(2 \le \gamma \le 20,\) \(10 \le \sigma \le 100,\) and \(20 \le \lambda \le 80\). These ranges of values have been found through numerical experiments. Parameters \(\lambda\), \(\gamma\) and \(\sigma\), respectively, are used to suppress highfrequency oscillations in the closedloop control system, adjust controller gains and damp out highfrequency oscillations.
2 Methodology
2.1 Numerical simulations of the APSP
As shown in Fig. 1 an APSP involves several interrelated parameters of which a few are listed in the figure. The APSP is numerically analyzed by using the software, LAVAP [15], that simulates the formation of the plasma from the flow of a mixture of Ar and H_{2} gases over an electric arc and of the plasma exiting the gas gun. Through a powder port it injects particles into the plasma at a random velocity and computes their acceleration, melting, resolidification, evaporation, and trajectories to the substrate to be coated. Due to the complexity of the plasma dynamics inside the torch, we follow many other published works and do not simulate several electromechanical interactions occurring within the gun. The governing equations and the mathematical formulation used in LAVAP are lucidly presented in [15] and [16]. The software LAVAP numerically solves these equations by the finite volume method from over the simulation region exhibited in Fig. 2. The assumptions, boundary conditions, and values of parameters used to simulate the APSP using LAVAP are listed in our previous work [12]. The assumptions are summarized below for a reader interested in improving upon this work.

(1)
The plasma jet is an axisymmetric, unsteady, compressible, Newtonian, turbulent, and chemically reacting multicomponent mixture with only temperaturedependent thermodynamic and transport properties. We assume that the plasma is in local thermodynamic equilibrium (LTE), is optically thin, and neglects (i) effects of the gravitational and the buoyancy forces as compared to those of the viscous drag force, and (ii) of the carrier gas flowing through the powder port at ~ 5 slm (standard liters per minute).

(2)
We consider the mean value of the arc voltage (voltage fluctuations caused by changes in the current are neglected) and chemical reactions among different species including ionization, dissociation, and recombination. However, we ignore turbulence modulation due to the injection transverse to the jet axis of the carrier gas and the powder particles.

(3)
Powder particles are rigid spheres, randomly vary in diameter, do not interact with each other, exchange heat with the plasma, can melt due to temperature rise, and the internal convection within a molten particle has a negligible effect on the heat transfer. The temperature distribution in a particle is taken to be axisymmetric.
The linearization of the nonlinear dynamics of particles’ states around a known steadystate solution called an equilibrium point gives the following multiinput and multioutput (MIMO) state space (SS) model for the mean axial velocity \(v\) and the mean temperature \(T\) [12].
Here, \(v\left( t \right)\) is the mean axial velocity, \(T\left( t \right)\) the mean temperature, \(P\left( t \right)\) the Ar flow rate, \(Q\left( t \right)\) the H_{2} flow rate, \(I\left( t \right)\) the current, and constants \(a_{v} ,a_{T} ,b_{11} , \ldots ,b_{23}\) depend on the equilibrium point. Variables P, Q and I are collectively denoted below by \(u\left( t \right)\).
The 30–100μmdiameter zirconia (ZrO_{2}) powder particles are injected at random velocities within a specified range through the powder port located 6 mm away from the nozzle exit and 8 mm above the jet axis. The average values of particles’ velocities and temperatures are computed in the 1 cm wide window located at 9.5 cm from the nozzle exit, as shown in Fig. 2 at t = 9.1, 9.2 …, 9.9, 10 ms. We note that the particle characteristics in the observation window reach a steady state at t = 9 ms in the absence of disturbances.
In the APSP, noise parameters such as powder pulsation, powder clogging, and powderport wear change the average injection velocity of particles. The nozzle wear and erosion influence fluctuations in the voltage. The effect of noise parameters is modeled by varying the average injection velocity of powder particles, and that of the nozzle wear and the cathode/anode erosion by introducing fluctuations in the voltage. The effect of simultaneously varying the two on the MPSs is depicted in Fig. 3. The values of other process parameters such as the current, the argon flow rate, the H_{2} flow rate, the mass flow rate of powder particles, and the particle size, respectively, are 500 A, 40 slm, 10 slm, 20 g/min, and 30–100 μm. The MPSs are output after every 0.01 ms.
2.2 Design of \({\varvec{\sigma}}\)modified RMRAC with lowfrequency learning for the APSP
We modify the MRAC to get a robust adaptive controller for the APSP that minimizes errors between the desired MPSs, \(y_{{{\text{des}}}} \left( T \right)\), and the measured MPSs, \(y\left( t \right)\), due to bounded changes in the noise variables. To achieve the desired MPSs in less than 50 ms, the controller adjusts inputs such as the Ar and the H_{2} flow rates and the current within the following prescribed ranges, \(20 {\text{ slm}} \le P \le 60 {\text{ slm}},\) \(0 \le Q \le 20 {\text{ slm}},\) and \(300 A \le I \le 600 A\).
Setting \(y\left( t \right) = \left\{ {v\left( t \right),T\left( t \right)} \right\}^{T}\) we rewrite Eq. (1) as
where \(y_{0}\) is the MPSs at time = 0 when the system is in a steady state and an unknown smooth disturbance \(d\left( t \right)\) satisfying \(\left\ {d\left( t \right)} \right\_{2} \le d_{{{\text{max}}}} ,\left\ {\dot{d}\left( t \right)} \right\_{2} \le \dot{d}_{{{\text{max}}}}\) with positive bounds \(d_{max} {\text{ and }} \dot{d}_{max}\) is introduced. The elements of matrices
depend upon conditions at time = 0.
For minimizing the errors between \(y_{{{\text{des}}}} \left( T \right)\) and \(y\left( t \right)\), the inputs \(u\left( t \right)\) are varied according to the following control law [12]:
where \(K\left( t \right) \in {\mathbb{R}}^{3 \times 2} {\text{ and }}L\left( t \right) \in {\mathbb{R}}^{3 \times 3}\) are the controller gain matrices at time \(t\). We choose the following reference model to meet the design criteria:
Here \(y_{{\text{m}}} \left( t \right) \in {\mathbb{R}}^{2}\) is a reference output vector and the piecewise bounded output vector \(r\left( t \right) \in {\mathbb{R}}^{3}\) contains the \(y_{des} \left( t \right)\). That is, \(r\left( t \right) = \left\{ {v_{{{\text{des}}}} \left( t \right), T_{{{\text{des}}}} \left( t \right), 0} \right\}^{T}\). To achieve \(y_{{{\text{des}}}} \left( t \right)\) within 50 ms, we take
For \(d\left( t \right) = 0\) in Eq. (2), the asymptotic convergence of the tracking error \(e\left( t \right) \equiv y\left( t \right)  y_{m} \left( t \right)\) is achieved by using the control law listed as Eq. (3) and the following adaptive law of the MRAC scheme [12].
Here \({\Lambda } = {\Lambda }^{{\text{T}}} \in {\mathbb{R}}^{3 \times 3}\) and \(\tilde{P} = \tilde{P}^{T} \in {\mathbb{R}}^{2 \times 2}\) are positive definite matrices. Note that we have placed ‘\(\widetilde{{}}\)’on the matrix \(P\) in Eq. (23) of [12] to rule out confusion with the Ar flow rate \(P\left( t \right)\) used herein.
The MRAC scheme using the adaptive law of Eq. (5) may suffer from instabilities such as the parameter drift, the high gain instability or the instability due to fast adaptation in the presence of certain external disturbances [17]. The RMRAC by incorporating the \(\sigma\)modification in Eq. (5) proposed by Ioannou and Kokotovic [18] can avoid these instabilities. Thus, the adaptive law of Eq. (5) is modified to that given below as Eq. (6).
Here \(\sigma\) is a positive constant. The second term on the righthand side of Eq. (6) acts as damper that bridges the tracking performance with the robustness achieved by using a fixed \(\sigma\)modification. However, the asymptotic convergence of the tracking error in the absence of external disturbances is not guaranteed. The tracking error is guaranteed to be of the order of the disturbance and \(\sigma\) [13]. A larger value of \(\sigma\) increases the robustness to uncertainties and disturbances but may lead to poor tracking of the steadystate error. A small value of \(\sigma\) decreases the norm of the tracking error but may create the ‘bursting’ phenomenon [13].
To prevent the steadystate error due to the \(\sigma\)modification and the highfrequency oscillations and to enable fast adaptation in the presence of bounded external disturbances, we use in the RMRAC scheme lowfrequency learning with lowpass filters \(K_{{\text{f}}} \left( t \right) \in {\mathbb{R}}^{3 \times 2}\) and \(L_{{\text{f}}} \left( t \right) \in {\mathbb{R}}^{3 \times 3}\) described in [14] with their rates of evolutions given by
In Eq. (7,) the design parameter \(\lambda > 0\) serves as the cutoff parameter to suppress highfrequency oscillations in the closedloop control system. The lowfrequency learning is incorporated by enforcing a distance between \(K\left( t \right) \,{\text{and }} L\left( t \right)\) and the estimated filter gains \(K_{{\text{f}}} \left( t \right), L_{{\text{f}}} \left( t \right)\) [14]. The adaptive laws for the modified RMRAC (MRMRAC) scheme for estimating gains matrices \(K\left( t \right)\) and \(L\left( t \right)\) in terms of the tracking error, \(e\left( t \right)\) are given below as Eq. (8).
The theorem included in the Supplementary Material guarantees the stability of the closedloop system and establishes properties of the controller performance. In the absence of an external disturbance, the asymptotic property of the tracking error is guaranteed by the \(\sigma\)—modified adaptive laws using lowfrequency learning, and the MRMRAC and the MRAC schemes have identical performance. The incorporation of the lowfrequency learning in the adaptive laws of the MRAC converts a pure integral type MRAC to a proportionalintegral type MRAC [14]. The MRMRAC enables fast learning and improves the robustness. The equations for the three controllers, the MRAC, the RMRAC and the MRMRAC schemes, are summarized in Table 1.
The architecture of the MRMRAC and its implementation in the APSP are, respectively, illustrated in Figs. 4 and 5. The adaptive gain matrix used is
where the constant \(\gamma\) serves as a tuning parameter to achieve the desired MPSs. We choose \(\tilde{P} = I_{2 \times 2}\) and \(l = 1\), and the following initial gains for the desired output array \(r\left( t \right) = \left\{ {90\frac{m}{s}, 2850 K, 0} \right\}^{T}\) [12]:
\(\begin{array}{*{20}c} {K_{0} = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ \end{array} } \right];} & {L_{0} = \left[ {\begin{array}{*{20}c} {0.0004} & {0.0123} & 0 \\ {0.0001} & {0.0031} & 0 \\ {0.0045} & {0.1430} & 0 \\ \end{array} } \right]} \\ \end{array}\).
3 Results
3.1 Comparison of performances of MRAC, RMRAC, and MRMRAC
To achieve the desired MPSs of \(v_{{{\text{des}}}} \left( t \right) =\) 90 m/s and \(T_{{{\text{des}}}} \left( t \right) =\) 2850 K within 50 ms of introducing a disturbance, we explore the effect of the value of \(\gamma\) on the MRAC performance. For a step variation in the average injection velocity and \(\gamma\) arbitrarily set = 1, 10 and 20, the corresponding responses are presented in the first column of Fig. 6. The variations in the average injection velocity of particles for \(\gamma = 10\) are attenuated within 50 ms. The value \(\gamma\) = 1 provides a low convergence rate of the plant responses thereby necessitating more time to reach the desired MPSs. Highfrequency oscillations occur in the control responses for \(\gamma = 20\).
In the second column of Fig. 6, we have illustrated the effect of the damping parameter \(\sigma\) = 0.1, 1, 10 and 100 while keeping \(\gamma = 20\) for fast adaptation of the controller. Whereas an increase in \(\sigma\) from 0.1 to 10 reduces the frequency of oscillations in the responses, it increases the steadystate error between the desired and the computed MPSs. Both \(\sigma = 10\) and \(\sigma = 100\) significantly damp the response.
To investigate the effect of the filter constant \(\lambda\) having values 1, 10 and 50, we use the MRMRAC for fast adaptation with \(\gamma = 20\) and a high damping coefficient \(\sigma = 100\). The fast adaptation using a highgain learning rate is achieved for \(\lambda =\) 50. Results displayed in the third column of Fig. 6 reveal that the highfrequency oscillations have been diminished and the effect of disturbances has been attenuated within 20 ms. For \(\gamma = 20\), \(\sigma = 100\) and \(\lambda = 50,\) the smoothly varying MPSs and the controller provided inputs illustrate that the designed MRMRAC scheme achieves fast and robust adaptation without producing both highfrequency oscillations and steadystate errors. Thus, the performance of the MRMRAC is superior to that of the MRAC.
3.2 Effect of disturbance variations on performance of the MRMRAC
For two sets of simultaneous disturbances in the arc voltage and the injection velocity shown in rows 1 and 2 of Fig. 7, time histories of the MPSs and of the three input variables provided by the MRMRAC scheme are exhibited in the remaining rows of Fig. 7 for \(\gamma = 20\), \(\sigma = 100\) and \(\lambda = 50\). This elucidates the effectiveness of the designed controller in mitigating effects of the disturbances. Note that for large disturbances using a large value of \(\gamma\) provides inputs that are outside their limiting values. Of course, in practice disturbances are not limited to those stipulated here. Nevertheless, these results evince that the MRMRAC can warn the operator that the coating process must be stopped since the disturbances are too large to be controllable.
3.3 Discussion of results
While a large value of \(\gamma\) in the MRAC scheme increases the adaptation rate for each input considered herein, the response exhibits highfrequency oscillations, and the controller performance can become unstable. The range \(2 \le \gamma \le 10\) for the MRAC scheme provides a low tracking error of the MPSs and no overshoot both in the values of the MPSs and the inputs.
The RMRAC scheme with \(\sigma \ge 1\) helps to reduce highfrequency oscillations associated with the fast adaptation but it has steadystate errors in the MPSs. Small values of \(\sigma\) in the RMRAC can create a bursting phenomenon as seen at time of 20–50 ms for \(\sigma\) = 0.1 in Fig. 6 in the controller provided inputs. Large values of \(\sigma\) give a larger steadystate error.
Including a lowpass filter in the RMRAC minimizes the steadystate error and suppresses these highfrequency oscillations. An increase in the filter constant \(\lambda\) of the MRMRAC minimizes the steadystate error due to damping.
The numerical experiments suggest that \(2 \le \gamma \le 20,\) \(10 \le \sigma \le 100,\) and \(20 \le \lambda \le 80\) for the MRMRAC provide adequate performance that has low tracking error in the controlled responses, fast adaptation, and no overshoot.
4 Conclusions
We have implemented a modified robust model reference adaptive controller (MRMRAC) in an atmospheric plasma spray process (ASPS) for automatically adjusting the three input parameters, namely the current, the argon flow rate and the hydrogen flow rate in the gas gun to maintain desired values of the mean particles’ axial velocity and temperature just before they arrive at the substrate to be coated. The controller is based on the sigmamodified adaptive laws with a lowpass weighted filter. We have established the robustness of the MRMRAC in the presence of bounded simultaneous disturbances in the powder injection velocity and the arc voltage. The MRMRAC scheme performs better than the standard MRAC scheme under the same external disturbances, provides smooth variations in the inputs and the outputs without creating highfrequency oscillations and steadystate errors, and signals that the process be stopped when inputs needed fall outside the prescribed limits.
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Guduri, B., Batra, R.C. Robust model reference adaptive controller for atmospheric plasma spray process. SN Appl. Sci. 4, 128 (2022). https://doi.org/10.1007/s42452022050120
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DOI: https://doi.org/10.1007/s42452022050120