Infinite Horizon Optimization Problem Research Articles

Design of reduced-order observer based steady state optimal linear-quadratic feedback controllers

In this paper we formulate and solve a problem encountered in engineering applications when a linear-quadratic (LQ) optimal feedback controller uses state estimates obtained via a reduced-order observer. Due to the use of state estimates instead of the actual state variables, the optimal quadratic performance is degraded in a pretty complex manner. In the paper, we show how to find the exact formula for the optimal performance degradation (optimal performance loss) in linear time invariant systems for the steady state case (infinite horizon optimization problem). The optimal performance loss is obtained in terms of solution of a reduced-order algebraic Lyapunov equation whose dimension is equal to the dimension of the reduced-order observer. The quantities that impact the performance criterion loss are identified. Practical examples (an inverted pendulum on a cart and an aircraft) show that the optimal performance loss can be very significant in some applications, and even very high in the presented linear-quadratic optimal control of an inverted pendulum problem. We have shown, using the derived formula, how the optimal performance loss can be considerably reduced by properly choosing the reduced-order observer initial conditions via the least square method.

Using computational techniques of fixed point theory for studying the stationary infinite horizon problem from the financial field

<abstract><p>In this article, we will solve optimization problems from the financial and economic field with constants, infinite-horizon iterative techniques and elements from fixed point theory. We will resort to Ćirić contractions in Banach space and the main result consists of the existence of a fixed point to solve an important class of infinite-horizon iterative schemes for optimization problems with state constraints for which the value function is merely lower semi-continuous. The developed tools allowed us to solve the stationary infinite-horizon optimization problems, especially for the maximization of the utility of households. We present some fixed point results that are fundamental for the development of our contributions: Notably, existence, monotonicity, attainability and results in the Ćirić contribution. We show the convergence in norm with probability for an iterative procedure defined for our problem under the stated assumptions. By using the Ćirić operator and the Reich-Rus type $ \psi F $-contraction, we prove the existence of the results of the optional cost function of an infinite horizon problem in a complete metric space. For a particular case, we realize a numerical simulation in C++. The conclusions are that the convergence, the existence and the uniqueness results of an optimal cost function of an infinite horizon problem in a Banach space can be treated by resorting to the Ćirić operator.</p></abstract>

Stabilization of Discrete-Time Multiplicative-Noise System Under Decentralized Controllers

In this article, we study the mean-square stabilization of discrete-time multiplicative-noise stochastic systems under decentralized controllers. Two controllers are involved in the system where each controller has access to different information and one information set is contained by another one. The adopted information structure is adapted open-loop. The main contribution of the article is twofold. On one hand, we give the unique explicit solution of the finite-horizon optimization problem in terms of difference Riccati equations. On the other hand, we characterize the equivalent condition for the mean-square stabilization under the decentralized controllers via the corresponding algebraic Riccati equations. Moreover, we derive the optimal and stabilizing solution for the infinite-horizon optimization problem.

A Promised Value Approach to Optimal Monetary Policy

This paper characterizes optimal commitment policy in the New Keynesian model using a novel recursive formulation of the central bank's infinite horizon optimization problem. In our recursive formulation motivated by Kydland and Prescott (1980), promised inflation and output gap---as opposed to lagged Lagrange multipliers---act as pseudo-state variables. Using three well known variants of the model---one featuring inflation bias, one featuring stabilization bias, and one featuring a lower bound constraint on nominal interest rates---we show that the proposed formulation sheds new light on the nature of the intertemporal trade-off facing the central bank.

Improved predictive control approach to networked control systems based on quantization dependent Lyapunov function

This paper considers model predictive control (MPC) for the linear discrete-time systems in the presence of packet loss, quantization and actuator saturation. Compared with the previous work ([45]), this paper presents an improved networked MPC approach for networked control systems (NCSs) by applying the quantization dependent Lyapunov function (QDLF) method which leads to less conservative results. The additional improvement is made by placing the heavier weighting on the system corresponding to the actual linear feedback law and choosing the relative weighting on the actual and auxiliary feedback laws which further improves the control performance over the existing method. It is shown that the closed-loop stability is guaranteed and a quantized state-feedback controller is derived by solving the infinite horizon optimization problem. Moreover, this method is further extended to multiple-input case. A numerical example is given to illustrate the effectiveness of the proposed approach.

Two-Scale Stochastic Control for Integrated Multipoint Communication Systems With Renewables

Increasing threats of global warming and climate changes call for an energy-efficient and sustainable design of future wireless communication systems. To this end, a novel two-scale stochastic control framework is put forth for smart-grid powered coordinated multi-point (CoMP) systems. Taking into account renewable energy sources, dynamic pricing, two-way energy trading facilities, and imperfect energy storage devices, the energy management task is formulated as an infinite-horizon optimization problem minimizing the time-averaged energy transaction cost, subject to the users’ quality of service requirements. Leveraging the Lyapunov optimization approach as well as the stochastic subgradient method, a two-scale online control (TS-OC) approach is developed for the resultant smart-grid powered CoMP systems. Using only historical data, the proposed TS-OC makes online control decisions at two timescales, and features a provably feasible and asymptotically near-optimal solution. Numerical tests further corroborate the theoretical analysis, and demonstrate the merits of the proposed approach.

A Finite Convergence Criterion for the Discounted Optimal Control of Stochastic Logical Networks

Stochastic logical networks (SLNs) are discrete-time stochastic dynamical systems with Boolean (or multivalued) logical state variables. The discounted infinite horizon optimal control problem for SLN is addressed in this paper. By resorting to the equivalent Markov decision process description, the infinite horizon optimization problem is presented in algebraic form. Then using the increasing-dimension technique, an improved finite convergence criterion, which can find the optimal stationary policy, is derived for value iteration approach. To demonstrate the theoretical value of this approach, it is applied to the optimization problems of the human–machine game and the p53-Mdm2 gene network.

Infinite-Horizon Problems Under Periodicity Constraint

We study soue infinite-horizon optimization problems on spaces of periodic functions for non periodic Lagrangians. The main strategy relies on the reduction to finite horizon thanks in the introduction of an avering operator.We then provide existence results and necessary optimality conditions in which the corresponding averaged Lagrangian appears.

Convergence of trajectories in infinite horizon optimization

In this paper, we investigate the convergence of a sequence of minimizing trajectories in infinite horizon optimization problems. The convergence is considered in the sense of ideals and their particular case called the statistical convergence. The optimality is defined as a total cost over the infinite horizon.

A generalization of Fatou\u2019s lemma for extended real-valued functions on \u03c3-finite measure spaces: with an application to infinite-horizon optimization in discrete time

Given a sequence {f_{n}}_{n in mathbb {N}} of measurable functions on a σ-finite measure space such that the integral of each f_{n} as well as that of limsup_{n uparrowinfty} f_{n} exists in overline{mathbb {R}}, we provide a sufficient condition for the following inequality to hold: lim supn↑∞∫fndμ≤∫lim supn↑∞fndμ.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\limsup_{n \\uparrow\\infty} \\int f_{n} \\,d\\mu\\leq \\int\\limsup_{n \\uparrow\\infty} f_{n} \\,d\\mu. $$\\end{document} Our condition is considerably weaker than sufficient conditions known in the literature such as uniform integrability (in the case of a finite measure) and equi-integrability. As an application, we obtain a new result on the existence of an optimal path for deterministic infinite-horizon optimization problems in discrete time.

A Bellman Approach to Periodic Optimization Problems

We consider an infinite horizon optimization problem with arbitrary but finite periodicity in discrete time. The problem can be formulated as a fix-point problem for a contraction operator, and we provide a solution scheme for this class of problems. Our approach is an extension of the classical Bellman problem to the special case of non-autonomy that periodicity represents. Solving such problems paves the way for consistent and rigorous treatment of, for example, seasonality in discrete dynamic optimization. In an illustrative example, we consider the decision problem in a fishery with seasonal fluctuations. The example demonstrates that rigorous treatment of periodicity has profound influence on the optimal policy dynamics compared to the case where seasonality is abstracted from by considering average effects only.

Dynamic Energy Management for Smart-Grid-Powered Coordinated Multipoint Systems

Due to increasing threats of global warming and climate change concerns, green wireless communications have recently drawn intense attention toward reducing carbon emissions. Aligned with this goal, the present paper deals with dynamic energy management for smart-grid powered coordinated multipoint (CoMP) transmissions. To address the intrinsic variability of renewable energy sources, a novel energy transaction mechanism is introduced for grid-connected base stations that are also equipped with an energy storage unit. Aiming to minimize the expected energy transaction cost while guaranteeing the worst-case users’ quality of service, an infinite-horizon optimization problem is formulated to obtain the optimal downlink transmit beamformers that are robust to channel uncertainties. Capitalizing on the virtual-queue-based relaxation technique and the stochastic dual-subgradient method, an efficient online algorithm is developed yielding a feasible and asymptotically optimal solution. Numerical tests with synthetic and real data corroborate the analytical performance claims and highlight the merits of the novel approach.

Solving infinite horizon optimization problems through analysis of a one-dimensional global optimization problem

Infinite horizon optimization (IHO) problems present a number of challenges for their solution, most notably, the inclusion of an infinite data set. This hurdle is often circumvented by approximating its solution by solving increasingly longer finite horizon truncations of the original infinite horizon problem. In this paper, we adopt a novel transformation that reduces the infinite dimensional IHO problem into an equivalent one dimensional optimization problem, i.e., minimizing a Holder continuous objective function with known parameters over a closed and bounded interval of the real line. We exploit the characteristics of the transformed problem in one dimension and introduce an algorithm with a graphical implementation for solving the underlying infinite dimensional optimization problem.

Endogenous Time Preference and Optimal Capital Deepening: A Rational Reconstruction of F. A. Hayek's Capital Theory

This paper argues that F. A. Hayek anticipated the notion of `recursive utility' and analytically reconstructs his informal exposition of the \textit{optimal saving process}. The scope of analysis is restricted to Hayek's largely unrecognized contribution in \textit{Utility Analysis and Interest} in 1936, restated as chapters seventeen and eigthteen in \textit{The Pure Theory of Capital}, first published in 1941. It is shown that Hayek characterized efficient dynamic choice as an infinite series of two-period optimality conditions by transforming an infinite-horizon optimization problem into a perpetual confrontation of current and prospective utility, that he hinted at the axiomatic base of stationary and weakly separable dynamic preferences, and that he endogenized the subjective discount rate to substantiate his claim that the \textit{interest-rate path in a perfect-foresight equilibrium is unidirectionally determined by the marginal productivity of investment} (and not by thrift). Hayek's vision of dynamic social efficiency and dynamic equilibrium is completely characterized.

Robust Consumption-Investment Problem on Infinite Horizon

In our paper we consider an infinite horizon consumption-investment problem under a model misspecification in a general stochastic factor model. We formulate the problem as a stochastic game and finally characterize the saddle point and the value function of that game using an ODE of semilinear type, for which we provide a proof of an existence and uniqueness theorem for its solution. Such equation is interested on its own right, since it generalizes many other equations arising in various infinite horizon optimization problems.

NECESSARY AND SUFFICIENT CONDITIONS FOR DYNAMIC OPTIMIZATION

We characterize the necessary and sufficient conditions for optimality in discrete-time, infinite-horizon optimization problems with a state space of finite or infinite dimension. It is well known that the challenging task in this problem is to prove the necessity of the transversality condition. To do this, we follow a duality approach in an abstract linear space. Our proof resembles that of Kamihigashi (2003), but does not explicitly use results from real analysis. As an application, we formalize Sims's argument that the no-Ponzi constraint on the government budget follows from the necessity of the tranversality condition for optimal consumption.

A Matrix Expression of Infinite Horizon Optimal Control Problem for Stochastic Logical Dynamical Systems

The stochastic logical control dynamical system with finite state is considered. After giving two equivalent descriptions of stochastic logical dynamical system: in term of discrete-time evolution equation and in term of Markov process, the infinite horizon optimization problem is presented in an algebraic form. Based on semi-tensor product of matrix and the increasing dimensional technique, we establish a succinct matrix expression of dynamic programming and Bellman's equation for the optimal control problem.

Stochastic Dynamic Programming based Energy Management of HEV's: an Experimental Validation

This paper addresses an experimental validation of an energy management strategy on a parallel Hybrid Electric Vehicle (HEV). The strategy under consideration is based on Stochastic Dynamic Programming. The control law (determining the torque split between the engine and the motor) is computed off-line by solving an infinite horizon optimization problem. It results in a time-invariant state feedback controller function of vehicle acceleration and velocity, battery state of charge and engine state. This controller is first validated in simulation and then implemented in the vehicle electronic control unit. Experimental results highlight the good behavior of the control strategy. During a 35 km urban route, the strategy succeeds in regulating the battery state of charge and judiciously uses the powertrain.

Dynamical Methods for Linear Hamiltonian Systems with Applications to Control Processes

The nonautonomous version of the Yakubovich Frequency Theorem characterizes the solvability of an infinite horizon optimization problem in terms of the validity of the Frequency and Nonoscillation Conditions for a linear Hamiltonian system, which is defined from the coefficients of the quadratic functional to be minimized. This paper describes those nonautonomous linear Hamiltonian systems satisfying the required properties. Two groups appear, depending on whether they are uniformly weakly disconjugate or not. It also contains a previous analysis of the long-term behavior of the Grassmannian and Lagrangian flows under the presence of exponential dichotomy, which is required for the classification and has interest by itself.

A reduction of topological infinite-horizon optimization to periodic optimization in a class of compact 2-manifolds

We consider a general class of infinite-horizon optimization problems, where the phase space is a two dimensional manifold satisfying the Jordan curve theorem, and the set of feasible curves satisfies general conditions similar to, but weaker than the solution set of a control system. We verify the existence of an optimal periodic solution for this class of problems. Applying this result we obtain two new results: we generalize a previous result for discounted infinite-horizon control problems, and provide a Poincaré–Bendixson type result for continuous generalized semi-flows.