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 (H2) gases injected into a gun passes over an electric arc and produces a plasma that exits at high velocity and temperature.

Fig. 1
figure 1

Parameters and variables in a typical APSP

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 real-time 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 one-input and one-output proportional-integral-derivative (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 high-velocity oxygen fuel (HVOF) spray process. A closed-loop proportional-integral (PI) controller was coupled to the estimation model to regulate the volume-based 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 closed-loop system with minimal tracking error and disturbance rejection have not been addressed in the above-cited 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 (H2) 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 closed-loop 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 high-frequency noise. Some of these can be mitigated by using a robust MRAC (R-MRAC) that modifies relations used in the MRAC. The R-MRAC 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 closed-loop signals but loses asymptotic convergence of the tracking error in the absence of disturbances. Furthermore, the fixed \(\sigma\)-modification can introduce a steady-state error or bursting phenomenon [13]. Yucelen and Haddad [14] have proposed a low-frequency learning to the MRAC by filtering out high-frequency 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 R-MRAC (MR-MRAC) scheme by incorporating low-frequency learning into the R-MRAC 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 steady-state error associated with the \(\sigma\)-modification is avoided, and the fast adaptation using high gain is achieved by using low-frequency learning with low-pass filters for estimating gains. The controller design is based on a linearized model of the MPSs with unknown external bounded disturbances. Thus, the MR-MRAC gives bounded responses of the closed-loop control and convergence of the tracking error to a small number. The ranges of control parameters of the MRAC, the R-MRAC and the MR-MRAC schemes for the APSP to achieve low steady-state 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 MR-MRAC 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 high-frequency oscillations in the closed-loop control system, adjust controller gains and damp out high-frequency 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, LAVA-P [15], that simulates the formation of the plasma from the flow of a mixture of Ar and H2 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, re-solidification, 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 LAVA-P are lucidly presented in [15] and [16]. The software LAVA-P 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 LAVA-P are listed in our previous work [12]. The assumptions are summarized below for a reader interested in improving upon this work.

  1. (1)

    The plasma jet is an axisymmetric, unsteady, compressible, Newtonian, turbulent, and chemically reacting multi-component mixture with only temperature-dependent 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. (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. (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.

Fig. 2
figure 2

The 1-cm-wide observation window located 9.5 cm from the nozzle exit is used to measure the mean particles’ axial velocity and temperature. For a powder mass flow rate of 20 g/min, at any time nearly 2,767 and 142 particles, respectively, are in the simulation region and the observation window [12]

The linearization of the nonlinear dynamics of particles’ states around a known steady-state solution called an equilibrium point gives the following multi-input and multi-output (MIMO) state space (SS) model for the mean axial velocity \(v\) and the mean temperature \(T\) [12].

$$\begin{array}{*{20}c} {\begin{array}{*{20}c} {\dot{v}\left( t \right) = a_{v} v\left( t \right) + b_{11} P\left( t \right) + b_{12} Q\left( t \right) + b_{13} I\left( t \right),} & {v\left( 0 \right) = v_{0} ,} & {t \ge 0} \\ \end{array} } \\ {\begin{array}{*{20}c} {\dot{T}\left( t \right) = a_{T} T\left( t \right) + b_{21} P\left( t \right) + b_{22} Q\left( t \right) + b_{23} I\left( t \right),} & {T\left( 0 \right) = T_{0} ,} & {t \ge 0} \\ \end{array} } \\ \end{array}$$

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 H2 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 30–100-μm-diameter zirconia (ZrO2) 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 powder-port 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 H2 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.

Fig. 3
figure 3

Effect of combined variations in the average injection velocity of ZrO2 particles and the arc voltage on the MPSs just before they arrive at the substrate

2.2 Design of \({\varvec{\sigma}}\)-modified R-MRAC with low-frequency 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 H2 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

$$\dot{y}\left( t \right) = Ay\left( t \right) + Bu\left( t \right) + d\left( t \right), y\left( 0 \right) = y_{0}$$

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

$$A = \left[ {\begin{array}{*{20}c} {a_{v} } & 0 \\ 0 & {a_{T} } \\ \end{array} } \right];B = \left[ {\begin{array}{*{20}c} {b_{11} } & {b_{12} } & {b_{13} } \\ {b_{21} } & {b_{22} } & {b_{23} } \\ \end{array} } \right]$$

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]:

$$u\left( t \right) = - K\left( t \right)y\left( t \right) + L\left( t \right)r\left( t \right)$$

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:

$$\dot{y}_{{\text{m}}} \left( t \right) = A_{{\text{m}}} y_{{\text{m}}} \left( t \right) + B_{{\text{m} }} r\left( t \right), y\left( 0 \right) = y_{m0} .$$

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

$$A_{{\text{m}}} = \left[ {\begin{array}{*{20}c} { - 0.5} & 0 \\ 0 & { - 0.5} \\ \end{array} } \right];B = \left[ {\begin{array}{*{20}c} {0.5} & 0 & {0.5} \\ 0 & {0.5} & {0.5} \\ \end{array} } \right].$$

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].

$$\begin{array}{*{20}c} {\dot{K}\left( t \right) = {\Lambda }B_{m}^{T} \tilde{P}e\left( t \right)y^{T} \left( t \right)sgn\left( l \right),} & {K\left( 0 \right) = K_{0} } \\ {\dot{L}\left( t \right) = - {\Lambda }B_{m}^{T} \tilde{P}e\left( t \right)r^{T} \left( t \right)sgn\left( l \right),} & {L\left( 0 \right) = L_{0} } \\ \end{array}$$

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 R-MRAC 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).

$$\begin{array}{*{20}c} {\dot{K}\left( t \right) = {\Lambda }B_{m}^{T} \tilde{P}e\left( t \right)y^{T} \left( t \right)sgn\left( l \right) - \sigma K\left( t \right),} & {K\left( 0 \right) = K_{0} } \\ {\dot{L}\left( t \right) = - {\Lambda }B_{m}^{T} \tilde{P}e\left( t \right)r^{T} \left( t \right)sgn\left( l \right) - \sigma L\left( t \right),} & {L\left( 0 \right) = L_{0} } \\ \end{array}$$

Here \(\sigma\) is a positive constant. The second term on the right-hand 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 steady-state error. A small value of \(\sigma\) decreases the norm of the tracking error but may create the ‘bursting’ phenomenon [13].

To prevent the steady-state error due to the \(\sigma\)-modification and the high-frequency oscillations and to enable fast adaptation in the presence of bounded external disturbances, we use in the R-MRAC scheme low-frequency learning with low-pass 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

$$\begin{array}{*{20}c} {\dot{K}_{{\text{f}}} \left( t \right) = \lambda \left( {K\left( t \right) - K_{{\text{f}}} \left( t \right)} \right),} & {K_{{\text{f}}} \left( 0 \right) = K_{0} } \\ {\dot{L}_{{\text{f}}} \left( t \right) = \lambda \left( {L\left( t \right) - L_{{\text{f}}} \left( t \right)} \right),} & {L_{{\text{f}}} \left( 0 \right) = L_{0} } \\ \end{array}$$

In Eq. (7,) the design parameter \(\lambda > 0\) serves as the cut-off parameter to suppress high-frequency oscillations in the closed-loop control system. The low-frequency 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 R-MRAC (MR-MRAC) 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).

$$\begin{aligned} \dot{K}\left( t \right) = & \Lambda B_{m}^{T} \tilde{P}e\left( t \right)y^{T} \left( t \right)sgn\left( l \right) - \sigma \left( {K\left( t \right) - K_{{\text{f}}} \left( t \right)} \right),K\left( 0 \right) = K_{0} \\ \dot{L}\left( t \right) = & - \Lambda B_{m}^{T} \tilde{P}e\left( t \right)r^{T} \left( t \right)sgn\left( l \right) - \sigma \left( {L\left( t \right) - L_{{\text{f}}} \left( t \right)} \right),L\left( 0 \right) = L_{0} \\ \dot{K}_{{\text{f}}} \left( t \right) = & \lambda \left( {K\left( t \right) - K_{{\text{f}}} \left( t \right)} \right),K_{{\text{f}}} \left( 0 \right) = K_{0} \\ \dot{L}_{{\text{f}}} \left( t \right) = & \lambda \left( {L\left( t \right) - L_{{\text{f}}} \left( t \right)} \right),L_{{\text{f}}} \left( 0 \right) = L_{0} \\ \end{aligned}$$

The theorem included in the Supplementary Material guarantees the stability of the closed-loop 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 low-frequency learning, and the MR-MRAC and the MRAC schemes have identical performance. The incorporation of the low-frequency learning in the adaptive laws of the MRAC converts a pure integral type MRAC to a proportional-integral type MRAC [14]. The MR-MRAC enables fast learning and improves the robustness. The equations for the three controllers, the MRAC, the R-MRAC and the MR-MRAC schemes, are summarized in Table 1.

Table 1 Equations of the MRAC, the R-MRAC and the MR-MRAC schemes

The architecture of the MR-MRAC and its implementation in the APSP are, respectively, illustrated in Figs. 4 and  5. The adaptive gain matrix used is

$${\Lambda } = \gamma \times \left[ {\begin{array}{*{20}c} {10^{ - 8} } & 0 & 0 \\ 0 & {10^{ - 10} } & 0 \\ 0 & 0 & {10^{ - 9} } \\ \end{array} } \right]$$

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]:

Fig. 4
figure 4

Schematic of the MR-MRAC architecture

Fig. 5
figure 5

Schematic of the implementation of the MR-MRAC scheme into an APSP

\(\begin{array}{*{20}c} {K_{0} = \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ \end{array} } \right];} & {L_{0} = \left[ {\begin{array}{*{20}c} {0.0004} & {0.0123} & 0 \\ {0.0001} & {0.0031} & 0 \\ {0.0045} & {0.1430} & 0 \\ \end{array} } \right]} \\ \end{array}\).

3 Results

3.1 Comparison of performances of MRAC, R-MRAC, and MR-MRAC

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. High-frequency oscillations occur in the control responses for \(\gamma = 20\).

Fig. 6
figure 6

Results in the first column illustrate the effect of the value of \(\gamma\) in the MRAC scheme on the controller provided inputs to the process and on the outputs from the APSP, those in the second column are with the damping constant \(\sigma\) in the R-MRAC scheme, and results in the third column are with the filter constant \(\lambda\) = 1, 10 and 50 in the MR-MRAC (\(\gamma = 20\) and \(\sigma = 100\)) scheme

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 steady-state 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 MR-MRAC for fast adaptation with \(\gamma = 20\) and a high damping coefficient \(\sigma = 100\). The fast adaptation using a high-gain learning rate is achieved for \(\lambda =\) 50. Results displayed in the third column of Fig. 6 reveal that the high-frequency 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 MR-MRAC scheme achieves fast and robust adaptation without producing both high-frequency oscillations and steady-state errors. Thus, the performance of the MR-MRAC is superior to that of the MRAC.

3.2 Effect of disturbance variations on performance of the MR-MRAC

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 MR-MRAC 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 MR-MRAC can warn the operator that the coating process must be stopped since the disturbances are too large to be controllable.

Fig. 7
figure 7

The closed-loop responses of the MR-MRAC system in the presence of combined disturbances in the average powder injection velocity and the arc voltage

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 high-frequency 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 R-MRAC scheme with \(\sigma \ge 1\) helps to reduce high-frequency oscillations associated with the fast adaptation but it has steady-state errors in the MPSs. Small values of \(\sigma\) in the R-MRAC 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 steady-state error.

Including a low-pass filter in the R-MRAC minimizes the steady-state error and suppresses these high-frequency oscillations. An increase in the filter constant \(\lambda\) of the MR-MRAC minimizes the steady-state 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 MR-MRAC 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 (MR-MRAC) 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 sigma-modified adaptive laws with a low-pass weighted filter. We have established the robustness of the MR-MRAC in the presence of bounded simultaneous disturbances in the powder injection velocity and the arc voltage. The MR-MRAC scheme performs better than the standard MRAC scheme under the same external disturbances, provides smooth variations in the inputs and the outputs without creating high-frequency oscillations and steady-state errors, and signals that the process be stopped when inputs needed fall outside the prescribed limits.