On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields

Partial Stability Concept in Extremum Seeking Problems

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov's direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system with oscillating inputs are derived with the use of Lie bracket approximation techniques. These conditions are exploited to describe a broad class of extremum-seeking controllers ensuring the partial stability of the set of minima of a cost function. The obtained theoretical results are illustrated by the Brockett integrator and rotating rigid body.Comment: This is the author's version of the manuscript accepted for publication in the Proceedings of the Joint 8th IFAC Symposium on Mechatronic Systems and 11th IFAC Symposium on Nonlinear Control Systems (MECHATRONICS & NOLCOS 2019

Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).Peer ReviewedPostprint (author's final draft

Realtime Optimization of MPC Setpoints using Time-Varying Extremum Seeking Control for Vapor Compression Machines

Recently, model predictive control (MPC) has received increased attention in the HVAC community, largely due to its ability to systematically manage constraints while optimally regulating signals of interest to setpoints. For example, in a common formulation of an MPC control problem for variable compressor speed vapor compression machines, the setpoints often include the zone temperature and the evaporator superheat temperature. However, the energy consumption of vapor compression systems has been shown to be sensitive to these setpoints. Further, while superheat temperature is often preferred because it can be easily correlated to heat exchanger efficiency (and therefore cycle efficiency), direct measurement of superheat is not always available. Therefore, identifying alternate signals in the control of vapor compression machines that correlate to efficiency is desired. Conventionally, methods for maximizing the energy efficiency rely on the use of mathematical models of the physics of vapor compression systems. These model-based approaches attempt to describe the influence of commanded inputs on the thermodynamic behavior of the system and the consumed electrical energy, and they are used to predict the combination of inputs that both meet the heat load requirements and minimize energy consumption. However, these models of vapor compression systems rely on simplifying assumptions in order to produce a mathematically tractable representation. Further, they are difficult to derive and calibrate, and often do not describe variations over long time scales, such as those due to refrigerant losses or accumulation of debris on the heat exchangers. In this paper, we consider a model-free extremum seeking algorithm that adjusts setpoints provided to a model predictive controller. While perturbation-based extremum seeking methods have been known for some time, they suffer from slow convergence rates---a problem emphasized by the long time constants associated with thermal systems. Our method uses a new algorithm (time-varying extremum seeking), which has dramatically faster and more reliable convergence properties. In particular, we regulate the compressor discharge temperature using an MPC controller with setpoints selected from a model-free time-varying extremum seeking algorithm. We show that the relationship between compressor discharge temperature and power consumption is convex (a requirement for this class of realtime optimization), and use time-varying extremum seeking to drive these setpoints to values that minimize power. The results are compared to the traditional perturbation-based extremum seeking approach. Further, because the required cooling capacity (and therefore compressor speed) is a function of measured and unmeasured disturbances, the optimal compressor discharge temperature setpoint must vary according to these conditions. We show that the energy optimal discharge temperature is tracked with the time-varying extremum seeking algorithm in the presence of disturbances

Proportional-Integral Extremum Seeking for Optimizing Power of Vapor Compression Systems

Conventionally, online methods for minimizing power consumption of vapor compression systems rely on the use of physical models. These model-based approaches attempt to describe the influence of commanded inputs, disturbances and setpoints on the thermodynamic behavior of the system and the resultant consumed electrical power. These models are then used online to predict the combination of inputs for a measured set of thermodynamic conditions that both meets the heat load and minimizes power consumption. However, these models of vapor compression systems must contain nonlinear terms of sufficient complexity in order to accurately describe the region near the optimum operating point(s), but also must rely on simplifying assumptions in order to produce a mathematically tractable representation. For these reasons, model-based online optimization of vapor compression machines have not gained traction in application, and have created an opportunity for model-free techniques such as extremum seeking control, which is gradient descent optimization implemented as a feedback controller. While traditional perturbation-based extremum seeking controllers for vapor compression systems have proven effective at minimizing power without requiring a process model, the algorithm\u27s requirement for multiple distinct timescales has limited the applicability of this method to laboratory tests where boundary conditions can be carefully controlled, or simulation studies with unrealistic convergence times. Perturbation-based extremum seeking requires that the control input be manipulated with a time constant approximately two orders of magnitude slower than the slowest vapor compression system dynamics, otherwise instabilities in the closed loop system occur. As a result, convergence to the optimum for slow processes such as thermal systems is restrictive due to inefficient estimation of the gradient, and slow (integral-action dominated) adaptation in the extremum seeking control law. In order to address this timescale separation issue, we have previously developed an algorithm called ``time-varying extremum seeking that more efficiently estimates the gradient of the performance metric and applied this algorithm to the problem of setpoint optimization for compressor temperatures. That algorithm improved the convergence rate to one timescale slower than the vapor compression machine dynamics. In this paper, we optimize power consumption through the application of a newly-developed proportional--integral extremum seeking controller (PI-ESC) that converges at the same timescale as the process. This method uses the improved gradient estimation routines of time-varying extremum seeking but also modifies the control law to include terms proportional to the estimated gradient. This modification of the control law, in turn, requires a revision to the gradient estimator in order to avoid bias. PI-ESC is applied to the problem of compressor discharge temperature selection for a vapor compression system so that power consumption is minimized. Because of the improved convergence properties of PI-ESC, we show that optimum values of discharge temperature can be tracked in the presence of realistic disturbances such as variation in the outdoor air temperature---enabling application of extremum seeking control to vapor compression systems in environments where previous methods have failed. The method is demonstrated experimentally on a 2.8 kW split ductless room air conditioner and in simulation using a custom-developed Modelica model