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Concept# Minimum phase

Summary

In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
The most general causal LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system. The system function is then the product of the two parts, and in the time domain the response of the system is the convolution of the two part responses. The difference between a minimum phase and a general transfer function is that a minimum phase system has all of the poles and zeroes of its transfer function in the left half of the s-plane representation (in discrete time, respectively, inside the unit circle of the z-plane). Since inverting a system function leads to poles turning to zeroes and vice versa, and poles on the right side (s-plane imaginary line) or outside (z-plane unit circle) of the complex plane lead to unstable systems, only the class of minimum phase systems is closed under inversion.

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Dominique Bonvin, Kahina Guemghar, Philippe Müllhaupt

Predictive Control of Unstable Nonminimum-phase Systems K. Guemghar, B. Srinivasan, Ph. Mullhaupt, D. Bonvin ´ Institut d’Automatique, Ecole Polytechnique F´d´rale de Lausanne, ee CH-1015 Lausanne, Switzerland. Predictive control is a very eﬀective approach for tackling problems with constraints and nonlinear dynamics, especially when the analytical computation of the control law is diﬃcult. Standard predictive control involves predicting the system behavior over a prediction horizon and calculating the input that minimizes a criterion expressing the system behavior in the future. Only the ﬁrst part of the computed input is applied to the system, and this procedure is repeated with the advent of each new measurement. This methodology is widely used in the process industry where system dynamics are sufﬁciently slow to permit its implementation. In contrast, applications of predictive control to fast unstable dynamic systems are rather limited. Apart from computational considerations, a fundamental limitation arises from the accuracy of that prediction, which can be quite poor due to accumulation of numerical errors if the prediction horizon is large. In this paper, an upper bound on the prediction horizon based on the location of the unstable pole(s) of the linearized system will be provided. The problem becomes more acute when nonminimum-phase systems are considered. Nonminimum phase implies that the system starts in a direction opposite to its reference (inverse response). To control such systems, it is reasonable to predict the maneuvers, thus making predictive control a natural strategy. However, a large prediction horizon is required since it is necessary to look beyond the inverse response. In this context, a lower bound on the prediction horizon based on the location of the unstable zero(s) of the linearized system will be provided. The two bounds mentioned above may lead to the situation where there exist no value of the prediction horizon that can stabilize a given unstable nonmiminum-phase system. For such a case, a combination of tools from diﬀerential geometry and predictive control is proposed in this paper. The envisaged procedure has a cascade structure and is outlined below: 1. Using the input-output linearization technique, the nonlinear system is transformed into a linear subsystem and internal dynamics. However, since the original nonlinear system is nonminimum-phase, the internal dynamics are unstable. 2. The linear subsystem is made arbitrarily fast by using a stabilizing high-gain linear feedback (inner-loop). 3. A predictive control scheme is then used to stabilize the slow internal dynamics (outerloop) by manipulating the reference of the inner loop. The stability of the proposed procedure is analyzed using a singular perturbation approach. The results will be illustrated in simulation on a system consisting of an inverted pendulum on a cart. 1

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This work investigates halide segregation in methylammonium-free wide bandgap perovskites by photoluminescence quantum yield (PLQY) and advanced electron microscopy techniques. Our study reveals how the formation of nano-emitting low-energy domains embedded in a wide bandgap matrix, located at surfaces and grain boundaries, enables a PLQY up to 25%. Intensity-dependent PLQY measurement and PL excitation spectroscopy revealed efficient charge funnelling and the failure of optical reciprocity between absorption and emission, limiting the use of PLQY data to determine the quasi-Fermi level splitting (QFLS) in these layers. Concomitantly, the small spectral overlap between emission and absorption reduces photon re-absorption. We demonstrate that phase segregation and charge funnelling, although harmful for the radiative efficiency of the mixed phase, are essential for achieving high PLQYs, selectively at low energies, otherwise not achievable in non-segregated perovskites with a similar bandgap. This promotes the applicability of this phenomenon in thermally stable high-efficiency emitting devices and color-conversion heterostructures.