Quadratic Cost Research Articles

Sparse feedback stabilization in high‐order linear systems

AbstractWe consider the column sparsity of the feedback stabilization gain matrix in high‐order linear systems. By means of a special matrix norm and the state transition matrix quadratic cost function (SQF) of the systems, the sparse feedback stabilization controller design problem is formulated as a regularized SQF optimization problem. We further derive the proximal mapping of the special matrix norm, and then based on the gradient descent of the SQF part of the objective function, the proximal gradient method is introduced to develop an algorithm for solving the non‐smooth optimization problem. Numerical examples are given to illustrate the effectiveness of the proposed method.

Indefinite linear quadratic optimal control for discrete time‐varying linear rectangular descriptor systems

AbstractIn this article, the indefinite linear quadratic (LQ) optimal control problem for discrete time‐varying rectangular descriptor systems is investigated, in which the indefinite weight matrices are considered in the quadratic cost function. With a suite of equivalent transformations, the indefinite LQ problem for discrete time‐varying rectangular descriptor systems can be substituted by an equivalent indefinite LQ problem for standard state space systems. Then, the transformed equivalent indefinite LQ problem is solved by establishing a dual Krein space model, and the optimal control and optimal cost value are obtained by invoking the Kalman filter of Krein space. The form of state feedback optimal control corresponding to the original system is given. Next, to guarantee the dynamic part for the resulting optimal closed‐loop system has a unique solution, some necessary and sufficient conditions are proposed. Finally, a numerical example is carried out to demonstrate the availability of established results.

What if We Knew What the Future Brings? Optimal Investment for a Frontrunner with Price Impact

In this paper we study optimal investment when the investor can peek some time units into the future, but cannot fully take advantage of this knowledge because of quadratic transaction costs. In the Bachelier setting with exponential utility, we give an explicit solution to this control problem with intrinsically infinite-dimensional memory. This is made possible by solving the dual problem where we make use of the theory of Gaussian Volterra integral equations.

High-Order Control Barrier Functions

We approach the problem of stabilizing a dynamical system while optimizing a cost and satisfying safety constraints and control limitations. For (nonlinear) affine control systems and quadratic costs, it has been shown that control barrier functions (CBFs) guaranteeing safety and control Lyapunov functions (CLFs) enforcing convergence can be used to (conservatively) reduce the optimal control problem to a sequence of quadratic programs (QPs). Existing works in this category have two main limitations. First, with one exception, they are based on the assumption that the relative degree of the system with respect to a function enforcing a safety constraint is one. Second, the QPs can easily become infeasible, in particular for problems with many safety constraints and tight control limitations. We propose high-order CBFs (HOCBFs), which can accommodate systems of arbitrary relative degrees. For each safety constraint, by using Lyapunov-like conditions, we construct a set of controls that renders the intersection of a set of sets forward invariant, which implies the satisfaction of the original constraint. We formulate optimal control problems with constraints given by HOCBF and CLF, and propose two methods—the penalty method and the parameterization method—to address the feasibility problem. Finally, we show how our methodology can be extended for safe navigation in unknown environments with long-term feasibility. We illustrate the proposed framework on adaptive cruise control and robot control problems.

Homing Problems With Control in the Diffusion Coefficient

In this article, we consider a class of homing problems with controls only in the diffusion-coefficient of a diffusion process in a given closed interval. The controller seeks to minimize an expected cost that accounts for quadratic control costs, and terminal costs until the controlled process hits the end-points of the given interval. Bounds for the value function and the optimal control in question are established under certain conditions. The present results extend and complement an earlier result proved by Lefebvre (2004).

Control-Aware Transmission Scheduling for Industrial Network Systems Over a Shared Communication Medium

In this article, we consider the design of dynamic transmission scheduling policies for the industrial network systems sharing scarce communication resources. Only a few subsystems can obtain channel access for information updates to close their control loops at each time step, which highlights the necessity of designing optimal transmission scheduling schemes to achieve a minimum average linear quadratic cost of the industrial network systems. We first propose a greedy state-error-dependent scheduling (SES) policy based on the one-step expected profit and discuss its stability employing the Lyapunov function method. After formulating the scheduling optimization as a Markov decision process problem and relaxing with a soft constraint, we develop a heuristic near-optimal solution that guarantees the optimality of certainty equivalent controllers, namely, Whittle’s index-inspired error-dependent scheduling (WIES). A stochastic stability condition of WIES is further given based on f-ergodicity. Due to low computational complexity and ease of implementation, the proposed schemes are suitable for large-scale heterogeneous industrial network systems. Finally, simulation results show that the proposed policies outperform the existing round-robin, holding-time-prioritized, and error-aware scheduling schemes.

Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems

We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon T with fixed and known cost matrices Q,R, but unknown and non-stationary dynamics At, Bt. The sequence of dynamics matrices can be arbitrary, but with a total variation, VT, assumed to be o(T) and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all t, we present an algorithm that achieves the optimal dynamic regret of Õ(VT2/5 T3/5). With piecewise constant dynamics, our algorithm achieves the optimal regret of Õ(√ST) where S is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of VT. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary. Our analysis also highlights that the key motif driving the regret is that the LQR problem is in spirit a bandit problem with linear feedback and locally quadratic cost. This motif is more universal than the LQR problem itself, and therefore we believe our results should find wider application.

Observer-Based Quadratic Guaranteed Cost Control for Linear Uncertain Systems with Control Gain Variation

This study proposes a method for designing observer-based quadratic guaranteed cost controllers for linear uncertain systems with control gain variations. In the proposed approach, an observer is designed, and then a feedback controller that ensures the upper bound on the given quadratic cost function is derived. This study shows that sufficient conditions for the existence of the observer-based quadratic guaranteed cost controller are given in terms of linear matrix inequalities. A sub-optimal quadratic guaranteed cost control strategy is also discussed. Finally, the effectiveness of the proposed controller is illustrated by a numerical example. The result shows that the proposed controller is more effective than conventional methods even if system uncertainties and control gain variations exist.

On the Regularity of Optimal Transports Between Degenerate Densities

We study the most common image and informal description of the optimal transport problem for quadratic cost, also known as the second boundary value problem for the Monge–Ampère equation—what is the most efficient way to fill a hole with a given pile of sand?—by proving regularity results for optimal transports between degenerate densities. In particular, our work contains an analysis of the setting in which holes and sandpiles are represented by absolutely continuous measures concentrated on bounded convex domains whose densities behave like nonnegative powers of the distance functions to the boundaries of these domains.

Service scheduling for random requests with fixed waiting costs

We study service scheduling problems in a slotted system in which agents arrive with service requests according to a Bernoulli process and have to leave within two slots after arrival, service costs are quadratic in service rates, and there is also a waiting cost. We consider fixed waiting costs. We frame the problem as an average cost Markov decision process. While the studied system is a linear system with quadratic costs, it has state dependent control. Moreover, it also possesses a non-standard cost function structure rendering the optimization problem complex. Here, we characterize the optimal policy. We also consider a system in which the agents make scheduling decisions for their respective service requests keeping their own cost in view. We frame this scheduling problem as a stochastic game. Here, we provide Nash equilibrium.

On the Extremum Control Problem with Pointwise Observation for a Parabolic Equation

In this paper we consider a control problem with pointwise observation for a one-dimensional parabolic equation which arises in a mathematical model of climate control in industrial greenhouses. We study a general equation with variable diffusion coefficient, convection coefficient, and depletion potential. For the extremum problem of minimizing an integral weighted quadratic cost functional, we establish the existence and uniqueness of a minimizing function. We also study exact controllability and dense controllability of the problem. Necessary conditions for an extremum are obtained, and qualitative properties of the minimizing function are studied.

A novel optimization model for biding in the deregulated power market with pay as a bid settlement mechanism, based on the stochastic market clearing price

In this paper, a novel optimization-based bidding strategy will propose for an electricity generation company in the day-ahead market with a pay-as-bid settlement system. In this proposed model, Market Clearing Price is considered as a stochastic parameter with different distribution functions per hour. Moreover, days are also clustered in three different clusters based on the historical data by using the K-Means Algorithm. In addition, three types of linear, quadratic, and cubic cost functions are taken into account in the proposed model, where the results indicated that the cubic function is the best estimation model for the generation cost. Finally, the League Championship Algorithm is applied in order to solve our proposed model. The proposed model has been applied to real case studies where the obtained results indicated that the proposed model is able to improve the expected revenue by 17% in comparison with the basic model.

Adaptive Control Barrier Functions

It has been shown that optimizing quadratic costs while stabilizing affine control systems to desired (sets of) states subject to state and control constraints can be reduced to a sequence of quadratic programs (QPs) by using control barrier functions (CBFs) and control Lyapunov functions (CLFs). In this article, we introduce adaptive CBFs (aCBFs) that can accommodate time-varying control bounds and noise in the system dynamics while also guaranteeing the feasibility of the QPs if the original quadratic cost optimization problem itself is feasible, which is a challenging problem in current approaches. We propose two different types of aCBFs: parameter-adaptive CBF (PACBF) and relaxation-adaptive CBF (RACBF). Central to aCBFs is the introduction of appropriate time-varying functions to modify the definition of a common CBF. These time-varying functions are treated as high-order CBFs with their own auxiliary dynamics, which are stabilized by CLFs. We demonstrate the advantages of using aCBFs over the existing CBF techniques by applying both the PACBF-based method and the RACBF-based method to a cruise control problem with time-varying road conditions and noise in the system dynamics, and compare their relative performance.

Consensus models with aggregation operators for minimum quadratic cost in group decision making.

In group decision making (GDM), to facilitate an acceptable consensus among the experts from different fields, time and resources are paid for persuading experts to modify their opinions. Thus, consensus costs are important for the GDM process. Notwithstanding, the unit costs in the common linear cost functions are always fixed, yet experts will generally express more resistance if they have to make more compromises. In this study, we use the quadratic cost functions, the marginal costs of which increase with the opinion changes. Aggregation operators are also considered to expand the applications of the consensus methods. Moreover, this paper further analyzes the minimum cost consensus models under the weighted average (WA) operator and the ordered weighted average (OWA) operators, respectively. Corresponding approaches are developed based on strictly convex quadratic programming and some desirable properties are also provided. Finally, some examples and comparative analyses are furnished to illustrate the validity of the proposed models.

Advances in Noise Modeling for Stochastic Systems in Optimal Control

In this paper some noise models for stochastic control systems are described that differ from the well known model of Brownian motion. Some of the processes are Gaussian such as the family of fractional Brownian motions and other processes are non-Gaussian, especially Rosenblatt processes. These processes have a long range dependence property that can be described by a slow decay of the covariance process. A stochastic calculus for these processes exists that is more limited than the calculus for Brownian motion that is inherited from the martingale property but nevertheless is sufficient for addressing some stochastic control problems. In many physical situations the data demonstrates a long range dependence which can justify a choice from these non-Brownian processes. Explicit solutions of stochastic control problems with quadratic cost functionals having driving noise from the Rosenblatt processes are given.

Fragmentation and Strategic Market-Making

AbstractHow does trading in one venue affect the quoting strategies of market makers in other venues? We develop a two-venue imperfect competition model in which market makers face quadratic costs when absorbing shocks. Nonconstant marginal costs imply that absorbing a shock in one venue simultaneously changes marginal costs in all other venues. Moreover, market makers strategically choose which shock(s) to absorb. These two forces may intensify competition, leading to enhanced liquidity. Using Euronext proprietary data, we track individual best bid and ask quotes of intermediaries in each venue. We uncover evidence of strategic cross-venue market-making behavior which is uniquely predicted by our model.

Decentralized control and state estimation of linear time‐periodic systems

AbstractThe main contributions of this article are the design of a decentralized controller and state estimator for linear time‐periodic systems with fixed network topologies. The proposed method to tackle both problems consists of reformulating the linear periodic dynamics as a linear time‐invariant system by applying a time‐lifting technique and designing a discrete‐time decentralized controller and state estimator for the time‐lifted formulation. The problem of designing the decentralized estimator is formulated as a discrete‐time Kalman filter subject to sparsity constraints on the gains. Two different algorithms for the computation of steady‐state observer gains are tested and compared. The control problem is posed as a state feedback gain optimization problem over an infinite‐horizon quadratic cost, subject to a sparsity constraint on the gains. An equivalent formulation that consists in the optimization of the steady‐state solution of a matrix difference equation is presented and an algorithm for the computation of the decentralized gain is detailed. Simulation results for the practical cases of the quadruple‐tank process and an extended 40‐tank process are presented that illustrate the performance of the proposed solutions, complemented with numerical simulations using the Monte Carlo method.

A class of hybrid LQG mean field games with state-invariant switching and stopping strategies

A novel framework is presented that combines Mean Field Game (MFG) theory and Hybrid Optimal Control (HOC) theory to obtain a unique ɛ-Nash equilibrium for a non-cooperative game with switching and stopping times. We consider the case where there exists one major agent with a significant influence on the system together with a large number of minor agents constituting two subpopulations, each agent with individually asymptotically negligible effect on the whole system. Each agent has stochastic linear dynamics with quadratic costs, and the agents are coupled in their dynamics and costs by the average state of minor agents (i.e. the empirical mean field). It is shown that for a class of Hybrid LQG MFGs, the optimal switching and stopping times are state-invariant and only depend on the dynamical parameters of each agent. Accordingly, a hybrid systems formulation of the game is presented via the indexing by discrete events: (i) the switching of the major agent between alternative dynamics or (ii) the termination of the agents’ trajectories in one or both of the subpopulations of minor agents. Optimal switchings and stopping time strategies together with best response control actions for, respectively, the major agent and all minor agents are established with respect to their individual cost criteria by an application of Hybrid LQG MFG theory.

Security control for nonlinear systems under quantization and Round-Robin protocol subject to deception attacks

This paper is concerned with the security control problem for the nonlinear systems with the effects of quantization, communication protocol and deception attacks based on the Takagi–Sugeno (T–S) fuzzy model. The measurement output and control input signals are quantized by dynamic quantizers simultaneously, and the quantized signals are transmitted through the communication channels scheduled by Round-Robin (RR) protocol. Moreover, two Bernoulli processes are utilized to characterize the deception attacks occurring on the different channels respectively. The mode-dependent observer-based controller and dynamic quantizers are designed such that the security in probability of the closed-loop system with the prescribed quadratic cost index can be guaranteed. Then, sufficient design conditions are obtained for the controller gains and quantizers’ parameters based on the linear matrix inequalities (LMIs) approach. In the end, a numerical example is carried out to demonstrate the validity of the proposed method.

Resilient observer-based sliding mode control of connected vehicles with denial-of-service attacks

This paper investigates an observer-based sliding mode control (SMC)) for connected vehicles under denial-of-service attacks. The attacks refer to interrupting communication channels between vehicles. Firstly, a reduced order observer is used to estimate the relative acceleration between neighbor vehicles, and a switching communication topology is introduced to model the attack. Then, an observer based sliding mode controller is proposed to achieve desired stability performance. Moreover, a quadratic cost performance is also defined and the cost upper bound is proved. Some sufficient conditions are provided such that the connected vehicles can achieve robust tracking performance, and input-to-state string stability is guaranteed under zero initial errors. Finally, numerical simulations are given to illustrate the validity of the designed controller.