# (16h) Optimal Control of Dose Delivery in Atmospheric Pressure Plasma Jets

#### AIChE Annual Meeting

#### 2021

#### 2021 Annual Meeting

#### Computing and Systems Technology Division

#### Estimation and Control under uncertainty

#### Sunday, November 7, 2021 - 5:43pm to 6:02pm

This talk will present an approach for the dose delivery problem that does not require solving any optimization problem online and converges to an optimal solution for the true system. The key contribution is the proposed combination of adaptive control and optimization methods that enables effective handling of structural plant-model mismatch in this application to APPJs, for which accurate models are typically unavailable.

The APPJ and its thermal effect are described by a system where the dose in cumulative equivalent minutes (CEM) and the surface temperature are the states, the temperature is the output, and the applied power is the input. It is assumed that a generic first-principles model of the APPJ is known, but a part of the temperature model is subject to uncertainty due to its unknown structure.

One can then formulate the dose delivery problem in terms of a generic optimal control problem (OCP) for this system, which includes a terminal cost, terminal constraints, and input and state path constraints. By using the necessary conditions of optimality given by Pontryagin's maximum principle, we can show that singular arcs are not possible, independently of the model uncertainty. The optimal input is then composed of a sequence of arcs where each arc can be 1) input constraint-seeking, such that it is determined by an input bound, or 2) state constraint-seeking, such that a state constraint remains active [6]. Hence, there is a finite number of arc types from which arc sequences can be formed. It follows that the number of plausible sequences is also finite.

For a given plausible arc sequence, the input is defined by the following decision variables: the switching times to input constraint-seeking arcs and the final time. The switching to state constraint-seeking arcs cannot occur at arbitrary times since it depends on the states. Consequently, the OCP reduces to determining the optimal arc sequence and the optimal decision variables. For a given arc sequence, the OCP can be reformulated in terms of a few decision variables, which is convenient for numerical optimization.

However, the explicit form of the time derivatives of the state constraints is not exactly known due to model uncertainty. Thus, to implement a state-constraint seeking arc, one needs to track a constraint via feedback control. In particular, we consider a single state constraint that consists in an upper bound for the temperature. Moreover, the model in the OCP formulation is not an exact representation of the true APPJ due to plant-model mismatch, thus the solution to that OCP is not the true optimal solution. Hence, we propose a control strategy via feedback linearization and derivative estimation for constraint tracking and an adaptive algorithm based on modifier adaptation to compute the optimal solution to the OCP for the true system despite the presence of plant-model mismatch.

We control the temperature at a setpoint that equals the upper bound minus a back-off parameter by manipulating the applied power, even though a part of the temperature model is unknown and the input and output are subject to disturbances. We use input-output feedback linearization and estimate the time derivative of the temperature to set the rate of temperature variation. For this estimation, we apply a filter that provides deadbeat estimation and minimizes the effect of measurement noise. The proposed control strategy eliminates steady-state error and rejects constant input disturbances without any integral term and ensures fast convergence of the temperature to the setpoint [7].

We also propose an adaptive approach to dealing with the structural mismatch between the true APPJ and its model. To this end, we use modifier adaptation to describe this mismatch using measurements of the cost and constraint functions of the OCP for the true system [8]. This method is used to obtain estimates of the cost and constraint functions for the true system. Hence, at each iteration of the adaptive algorithm, a modified version of the original OCP is solved. If the modified OCPs are such that their solutions are in the basin of attraction of a global optimum of the OCP for the true system, the true system converges to its global optimum.

For simulation studies and experiments, we consider an OCP that consists in the minimization of the final time needed to reach a dose in CEM and a temperature that ensures no dose accumulation after the final time. In addition, the applied power is subject to lower and upper bounds, while the temperature is subject to an upper bound to ensure that the surface is not subject to excessive temperatures. In our studies, the temperature model is subject to structural mismatch and inaccurate model parameters. Also, the true applied power is not known due to input disturbances, and the temperature measurements are noisy.

The optimal arc sequence is composed of three arcs: in the first arc, the maximum power is applied so that the temperature increases as quickly as possible until the path constraint for the temperature becomes active; in the second arc, the temperature is kept constant so that the path constraint remains active; in the third arc, the minimum power is applied so that the temperature decreases as quickly as possible. This sequence is optimal independently of the temperature model. Switching from the first arc to the second arc cannot occur at an arbitrary time since it depends on the temperature. Since the explicit form of the time derivative of the temperature is not exactly known, control via feedback linearization and derivative estimation is used to track the constraint in the second arc. Hence, the OCP reduces to determining the optimal switching time from the second arc to the third arc and the optimal final time. Modifier adaptation is used to perform this task due to the presence of plant-model mismatch.

By using the proposed adaptive algorithm, the cost for the true system approaches its optimal value after few iterations while the constraints become active, which indicates convergence to the optimal solution to the dose delivery problem. For each point evaluated by the algorithm, the proposed control strategy successfully tracks the constraint in the second arc. Hence, despite the existence of disturbances and structural plant-model mismatch, the combination of the adaptive algorithm with a control strategy for constraint tracking enforces convergence to the true optimal solution.

In summary, the proposed approach is a relatively simple way to optimize the dose delivery in APPJs and overcomes the aforementioned challenges faced by MPC strategies. Future work may extend the proposed approach to more complex models, nonthermal effects, and uniform dose delivery on two-dimensional surfaces.

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[3] D. Gidon, B. Curtis, J. A. Paulson, D. B. Graves, and A. Mesbah. Model-based feedback control of a kHz-excited atmospheric pressure plasma jet, *IEEE Trans. Radiat. Plasma Med. Sci.*, 2(2):129â€“137, 2018.

[4] A. D. Bonzanini, J. A. Paulson, D. B. Graves, and A. Mesbah. Toward safe dose delivery in plasma medicine using projected neural network-based fast approximate NMPC. In *Proceedings of the 21st IFAC World Congress*, 5353-5359, Berlin, Germany, 2020.

[5] A. D. Bonzanini, J. A. Paulson, G. Makrygiorgos, and A. Mesbah. Fast approximate learning-based multistage nonlinear model predictive control using Gaussian processes and deep neural networks, *Comput. Chem. Eng.*, 145:107174, 2021.

[6] D. Rodrigues and D. Bonvin. On reducing the number of decision variables for dynamic optimization, *Optim. Control Appl. Meth.*, 41(1):292-311, 2020.

[7] D. Rodrigues and H. Hjalmarsson. Stability and performance analysis of control based on incomplete models, *IFAC-PapersOnLine*, 52(1):874â€“879, 2019.

[8] A. Marchetti, B. Chachuat, and D. Bonvin. Modifier-adaptation methodology for real-time optimization, *Ind. Eng. Chem. Res.*, 48(13):6022â€“6033, 2009.

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