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

Control of an indoor autonomous mobile communications relay via antenna diversity

Presented in this thesis is a motion planning scheme for enabling a quadrotor unmanned aerial vehicle (UAV) to serve as an autonomous communications relay in indoor or GPS-denied environments. The goal of the algorithm is to maximize the throughput of the end-to-end communications channel. An extremum-seeking controller steers the quadrotor while collision avoidance is provided by artificial potential fields. Extremum-seeking is model-free adaptive control method; it\u27s applicable in situations where there is a nonlinearity in the control problem and the nonlinearity has a local minimum or maximum. The extremum-seeking controller presented here is driven by antenna diversity and attempts to optimize the inputs to an unknown, time-varying cost function characterized by the RF environment. Each of the multiple antennas onboard the quadrotor receives the same incoming packets and provides associated signal strength measurements. The extremum-seeking controller then uses these measurements to autonomously fly the quadrotor communications relay to an optimal location so as to maximize throughput, all without positioning data. This work is motivated by the need to extend the operating ranges of robots in complex urban and indoor environments. The algorithm and necessary technical background are presented in detail. Simulations results verify the validity of the proposed extremum-seeking approach. Experiments demonstrate the feasability of implementing the extremum-seeking controller with tangible hardware

Extremum Seeking With Enhanced Convergence Speed for Optimization of Time-Varying Steady-State Behavior of Industrial Motion Stages

Recently, an extremum-seeking control (ESC) approach has been developed for optimization of generically time-varying steady-state responses of nonlinear systems. A generic filter structure was introduced, the so-called dynamic cost function, which has been instrumental in facilitating the use of ESC in the more generic, time-varying context. However, the dynamic cost function must operate sufficiently slow compared to the time-varying nature of the system responses, thereby compromising the convergence speed of the ESC scheme. In this work, a modified ESC approach is proposed that incorporates explicit knowledge about the user-defined dynamic cost function, able to enhance the convergence speed of the ESC scheme. Moreover, we provide a stability analysis for this extended approach. The main contribution of this work is the experimental demonstration of both ESC approaches for the performance optimal tuning of a variable-gain control (VGC) strategy employed on a high-accuracy industrial motion stage setup, exhibiting generically time-varying steady-state responses. VGC is able to enhance the system performance by balancing the typical linear control tradeoff between low-frequency disturbance suppression properties and sensitivity to high-frequency disturbances in a more desirable manner. We experimentally show that, for the unknown disturbance situation at hand, the variable-gain controller can be automatically tuned using both ESC approaches to achieve the optimal system performance. In addition, enhanced convergence speed with the modified ESC approach is evidenced experimentally.acceptedVersio

Extremum Seeking-based Iterative Learning Linear MPC

In this work we study the problem of adaptive MPC for linear time-invariant uncertain models. We assume linear models with parametric uncertainties, and propose an iterative multi-variable extremum seeking (MES)-based learning MPC algorithm to learn on-line the uncertain parameters and update the MPC model. We show the effectiveness of this algorithm on a DC servo motor control example.Comment: To appear at the IEEE MSC 201

Multi-Parametric Extremum Seeking-based Auto-Tuning for Robust Input-Output Linearization Control

We study in this paper the problem of iterative feedback gains tuning for a class of nonlinear systems. We consider Input-Output linearizable nonlinear systems with additive uncertainties. We first design a nominal Input-Output linearization-based controller that ensures global uniform boundedness of the output tracking error dynamics. Then, we complement the robust controller with a model-free multi-parametric extremum seeking (MES) control to iteratively auto-tune the feedback gains. We analyze the stability of the whole controller, i.e. robust nonlinear controller plus model-free learning algorithm. We use numerical tests to demonstrate the performance of this method on a mechatronics example.Comment: To appear at the IEEE CDC 201

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