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Register a new userTime-Varying Optimization of Networked Systems with Human Preferences
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Added by
Ana Ospina
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper considers a time-varying optimization problem associated with a network of systems, with each of the systems shared by (and affecting) a number of individuals. The objective is to minimize cost functions associated with the individuals preferences, which are unknown, subject to time-varying constraints that capture physical or operational limits of the network. To this end, the paper develops a distributed online optimization algorithm with concurrent learning of the cost functions. The cost functions are learned on-the-fly based on the users feedback (provided at irregular intervals) by leveraging tools from shape-constrained Gaussian Processes. The online algorithm is based on a primal-dual method, and acts effectively in a closed-loop fashion where: i) users feedback is utilized to estimate the cost, and ii) measurements from the network are utilized in the algorithmic steps to bypass the need for sensing of (unknown) exogenous inputs of the network. The performance of the algorithm is analyzed in terms of dynamic network regret and constraint violation. Numerical examples are presented in the context of real-time optimization of distributed energy resources.
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We consider optimization problems for (networked) systems, where we minimize a cost that includes a known time-varying function associated with the systems outputs and an unknown function of the inputs. We focus on a data-based online projected gradient algorithm where: i) the input-output map of the system is replaced by measurements of the output whenever available (thus leading to a closed-loop setup); and ii) the unknown function is learned based on functional evaluations that may occur infrequently. Accordingly, the feedback-based online algorithm operates in a regime with inexact gradient knowledge and with random updates. We show that the online algorithm generates points that are within a bounded error from the optimal solution of the problem; in particular, we provide error bounds in expectation and in high-probability, where the latter is given when the gradient error follows a sub-Weibull distribution and when missing measurements are modeled as Bernoulli random variables. We also provide results in terms of input-to-state stability in expectation and in probability. Numerical results are presented in the context of a demand response task in power systems.
We present a method to over-approximate reachable tubes over compact time-intervals, for linear continuous-time, time-varying control systems whose initial states and inputs are subject to compact convex uncertainty. The method uses numerical approximations of transition matrices, is convergent of first order, and assumes the ability to compute with compact convex sets in finite dimension. We also present a variant that applies to the case of zonotopic uncertainties, uses only linear algebraic operations, and yields zonotopic over-approximations. The performance of the latter variant is demonstrated on an example.
A major limitation of online algorithms that track the optimizers of time-varying nonconvex optimization problems is that they focus on a specific local minimum trajectory, which may lead to poor spurious local solutions. In this paper, we show that the natural temporal variation may help simple online tracking methods find and track time-varying global minima. To this end, we investigate the properties of a time-varying projected gradient flow system with inertia, which can be regarded as the continuous-time limit of (1) the optimality conditions for a discretized sequential optimization problem with a proximal regularization and (2) the online tracking scheme. We introduce the notion of the dominant trajectory and show that the inherent temporal variation could re-shape the landscape of the Lagrange functional and help a proximal algorithm escape the spurious local minimum trajectories if the global minimum trajectory is dominant. For a problem with twice continuously differentiable objective function and constraints, sufficient conditions are derived to guarantee that no matter how a local search method is initialized, it will track a time-varying global solution after some time. The results are illustrated on a benchmark example with many local minima.
There is an increasing interest in designing differentiators, which converge exactly before a prespecified time regardless of the initial conditions, i.e., which are fixed-time convergent with a predefined Upper Bound of their Settling Time (UBST), due to their ability to solve estimation and control problems with time constraints. However, for the class of signals with a known bound of their $(n+1)$-th time derivative, the existing design methodologies are either only available for first-order differentiators, yielding a very conservative UBST, or result in gains that tend to infinity at the convergence time. Here, we introduce a new methodology based on time-varying gains to design arbitrary-order exact differentiators with a predefined UBST. This UBST is a priori set as one parameter of the algorithm. Our approach guarantees that the UBST can be set arbitrarily tight, and we also provide sufficient conditions to obtain exact convergence while maintaining bounded time-varying gains. Additionally, we provide necessary and sufficient conditions such that our approach yields error dynamics with a uniformly Lyapunov stable equilibrium. Our results show how time-varying gains offer a general and flexible methodology to design algorithms with a predefined UBST.
We study predictive control in a setting where the dynamics are time-varying and linear, and the costs are time-varying and well-conditioned. At each time step, the controller receives the exact predictions of costs, dynamics, and disturbances for the future $k$ time steps. We show that when the prediction window $k$ is sufficiently large, predictive control is input-to-state stable and achieves a dynamic regret of $O(lambda^k T)$, where $lambda < 1$ is a positive constant. This is the first dynamic regret bound on the predictive control of linear time-varying systems. Under more assumptions on the terminal costs, we also show that predictive control obtains the first competitive bound for the control of linear time-varying systems: $1 + O(lambda^k)$. Our results are derived using a novel proof framework based on a perturbation bound that characterizes how a small change to the system parameters impacts the optimal trajectory.
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