This paper considers two different control problems for deterministic systems with stochastic initial conditions where, in addition to the usual asymptotic behavior requirement, we are interested in the transient behavior of the state distribution evolution. For the first one, we study control design such that the state trajectories enter target set at a given transient time with a prescribed minimum cumulative distribution. For the second one, we propose control design where the distribution of state variable at transient time is close to the target distribution. We illustrate the efficacy of the proposed solutions to the aforementioned control problems through numerical examples.
In this paper, we introduce a modeling framework for free molecular flow (FMF) processes (such as, deposition processes under an ultra-high vacuum condition) that is suitable for model-based control design. The generic dynamical model is comprised of four important elements in such processes: (i) particle transfer, which is modeled based on the wellknown Knudsen cosine law; (ii) particle leakage; (iii) adsorption and desorption described by a (nonlinear) sticking function; and (iv) control input particle flux. As a starting point for obtaining accurate control on the deposition process in FMF regime, we propose a control design method for stabilization with guaranteed transient behavior for fluxes. It is based on a point-wise min-norm control approach, employing both control Lyapunov and control barrier functions. Lastly, we validate our model, applied to a cylindrical geometry, with existing results in literature and evaluate the effectiveness of the proposed control method for controlling the fluxes.
Ultra-high vacuum chemical vapor deposition is a thin film deposition process that features excellent film purity, but is sensitive to the processing variations (such as, the precursors and their dispensers, the reactor's initial condition, etc.). In this paper, we present the design of a ultra-high vacuum chemical vapor deposition reactor with in-situ partial pressure atomic absorption spectroscopy measurement that improves reproducibility and observability of such a process. Our main contributions are: (i). a conceptual control systems design of ultra-high vacuum chemical vapor deposition; (ii). atomic absorption spectroscopy based sensor design for the real-time in-situ partial pressure measurements; (iii). a flux dynamical model; (iv). experimental reactor design; and (v). experimental validation of model components and the atomic absorption spectroscopy measurement technique. Our results show that the proposed sensor systems are able to provide real-time measurements of the partial pressure inside the reactor and our proposed flux dynamical model agrees with the measured partial pressure. The latter allows us to use it in the design of model-based output feedback control of the partial pressure.
We study a control design problem for nonlinear affine systems whose initial condition is a random variable with known distribution. In this control problem, the control objectives are two-folds: (i). the closed-loop system attains a minimum cumulative distribution over a prescribed containment set at the end of transient time; and (ii) it converges exponentially to a desired trajectory. A solution to the control problem is obtained by imposing desired contraction properties on the closed-loop system such that both control objectives can be met. The efficacy of the proposed controller design is shown on the control of robotic manipulator.
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