Quadratic Cost Function Research Articles

Mixed H∞ and passivity guaranteed cost proportional‐integral control for interval‐valued type‐2 fuzzy systems with input delay

AbstractThe paper focuses on the problem of mixed and passivity guaranteed cost control of an interval‐valued type‐2 (IT‐2) fuzzy system with time‐varying input delay and exogenous disturbance. For this, an IT‐2 fuzzy‐model‐based system with a mismatched membership grade function is formulated. The main objective is to design a proportional‐integral controller that stimulates our system to attain its equilibrium state even with the occupancy of exogenous disturbance and input delay in the system. In addition, a quadratic cost function is constructed such that the cost function of the closed‐loop IT‐2 fuzzy system is lower than the upper‐bound cost index. Mixed and passivity performance indices are then calculated to address the exogenous disturbance in the system. A set of new criteria ensuring the asymptotic stability of the closed‐loop IT‐2 fuzzy system with input delay are derived in terms of linear matrix inequalities with the aid of Lyapunov stability theory and the employment of a Wirtinger‐based inequality. Finally, two example simulations are conducted to validate our theoretical results.

Guaranteed Cost Impulsive Synchronization of Uncertain Multiplex Networks

This brief mainly deals with the issue of guaranteed cost impulsive synchronization of uncertain multi-agent systems from the multiplex network prospective, where the uncertain time-varying system parameters are supposed to be norm-bounded. By means of a group of linear matrix inequalities, the effective guaranteed cost impulsive control is designed to not only ensure that the trajectories of all followers synchronize with the leader, but also guarantee that the control cost is no higher than an upper bound of the quadratic guaranteed cost function. Numerical example demonstrates the feasibility of theoretical results.

Security tracking control for discrete-time stochastic systems subject to cyber attacks

This paper is concerned with the security tracking problem under the quadratic cost criterion for a class of discrete-time stochastic linear networked control systems (NCSs) exposed to cyber attacks, covering false data injection attacks as well as a class of DoS attacks. Taking into account factors such as network congestion and the defensive role of intrusion detection systems, successful attack events are modeled as a Bernoulli random sequence. To describe the transient trajectory of an NCS under the impact of a random attack, a probabilistic definition of secure trackability is taken. Therefore, an observer-based dynamic output feedback controller is designed in order to achieve the specified probabilistic secure trackability. Specifically, the probabilistic safety output tracking problem is transformed into an input-to-state stability problem in the probabilistic sense for closed-loop systems with some new sufficient conditions, provided that an augmented incremental model is utilized Then, the controller parameters and the upper bound of the quadratic cost function are determined by solving matrix inequalities, while easy-solution forms of the matrix inequalities to be solved are presented by the Schur complementary lemma. Both simulation studies and practical experiments demonstrate the effectiveness of the proposed control scheme.

Optimal reset law design based on guaranteed cost control method for Lipschitz nonlinear systems

AbstractThis article proposes a systematic approach for optimal reset law design of a class of nonlinear systems. By using the guaranteed cost control method, sufficient conditions for the design of optimal reset law are derived in terms of linear matrix inequalities. In an offline design procedure, the reset law is computed that minimizes the upper bound of a quadratic cost function. The proposed method can be implemented for real‐time applications even with small sampling time. The simulation results verify the efficacy and effectiveness of the proposed theoretical results.

Stochastic event-based LQG control: An analysis on strict consistency

This paper studies the discrete-time linear–quadratic-Gaussian problem under partial-information-based sensors and event-based transmissions. By constructing a special form of stochastic event-based protocols, the optimal control policy is obtained, which minimizes an average quadratic cost function under a fixed transmission protocol. Then, analytical expressions of the corresponding transmission and control performance are derived. To evaluate the efficiency of event-based protocols, the concept of strict consistency is introduced, which requires that the control performance of event-based protocols be strictly better than that of the periodic protocol under the same transmission rate. Based on the obtained analytical expressions, the strict consistency is proved by showing that the periodic protocol corresponds to a local maximum point in the design space of parameters. To further improve the control performance of stochastic event-based protocols, an iterative parameter design algorithm (with finite iterations) is proposed for ensuring a strict reduction in cost functions after each iteration. Finally, numerical simulations are provided to illustrate the theoretical results.

Solving the Economic Growth Acceleration Model with Memory Effects: An Application of Combined Theorem of Adomian Decomposition Methods and Kashuri–Fundo Transformation Methods

The primary purpose of this study is to solve the economic growth acceleration model with memory effects for the quadratic cost function (Riccati fractional differential equation), using Combined Theorem of Adomian Polynomial Decomposition and Kashuri–Fundo Transformation methods. The economic growth model (EGM) with memory effects for the quadratic cost function is analysed by modifying the linear fractional differential equation. The study’s significant contribution is to develop a linear cost function in the EGM for a quadratic non-linear cost function and determine the specific conditions of the Riccati fractional differential equation (RFDEs) in the EGM with memory effects. The study results showed that RFDEs in the EGM involving the memory effect have a solution and singularity. Additionally, this study presents a comparison of exact solutions using Lie symmetry, Combined Theorem of Adomian Polynomial Decomposition, and Kashuri–Fundo Transformation methods. The results showed that the three methods have the same solution. Furthermore, this study provides a numerical solution to the RFDEs on the EGM with memory effects. The numerical simulation results showed that the output value of Y(t) for the quadratic cost function in the economic growth model is significantly affected by the memory effect.

EFFECT OF RISK-SENSITIVE STOCHASTIC OPTIMAL CONTROL WITH TRACKING IN AN EVAPORATOR

This work presents an application of the Risk-Sensitive (R-S) control with tracking applied to a stochastic nonlinear system which models the operation of an electronic expansion valve (EEV) in a conventional evaporator. A novel dynamical stochastic equation represents the mathematical model of the evaporator system. The R-S stochastic optimal problem consists of the design of an optimal control u(t) such that the state reaches setpoint values (SP) and minimizes the exponential quadratic cost function. The presence of disturbances and errors in the sensor measurements is represented by Gauss white noise in the state equation, with the coefficient v(e/(2?^2 )) . One novel characteristic in this proposal is that the coefficient of the control into the state equation contains the state term. The error and exponential quadratic cost function show that the R-S control has a better performance versus the classical PID (Proportional, Integral Derivative) control. Key Words: Optimal Risk-Sensitive control with tracking, modelling of the evaporator.

Dissipativity in Infinite-Horizon Optimal Control: Willems’ 1971 Paper Revisited

Jan Willems introduced the system-theoretic notion of dissipativity in his foundational two-part paper which appeared in the Archive of Rational Mechanics and Analysis in 1972. Even earlier, in a likewise pivotal 1971 IEEE Transactions on Automatic Control paper, he investigated infinite-horizon least-squares optimal control and the algebraic Riccati equation from a dissipativity point of view. This note revisits infinite-horizon optimal control leveraging strict dissipativity. We discuss the interplay between dissipativity and stability properties in continuous-time infinite-horizon problems without assuming linear dynamics or quadratic cost functions. Finally, we compare our recent results from Faulwasser and Kellett (2021) to the original findings of Willems (1971).

Solution of Combined Heat and Power Economic Dispatch Problem Using Genetic Algorithm

This research proposes a synergistic meta-heuristic algorithm for solving the extreme operational complications of combined heat and power economic dispatch problem towards the advantageous economic outcomes on the cost of generation. The combined heat and power (CHP) is a system that provides electricity and thermal energy concurrently. For its extraordinary efficiency and significant emission reduction, it is considered a promising energy prospect. The broad application of combined heat and power units requires the joint dispatch of power and heating systems, in which the modelling of combined heat and power units plays a vital role. The present research employs the genetic optimization algorithm to evaluate the cost function, heat and power dispatch values encountered in a system with simple cycle cogeneration unit and quadratic cost function. The system was first modeled to determine the various parameters of combined heat and power units towards solving its economic dispatch problem directly. In order for modelling to be done, a general structure of combined heat and power must be defined. The test system considered consists of four units: two conventional power units, one combined heat and power unit and one heat-only unit. The algorithm was applied to test system while taking into account the power and heat units, bounds of the units and feasible operation region of cogeneration unit. Output decision variables of 4-unit test systems plus cost function from Genetic Algorithm (GA), was determined using appropriate codes. The proposed algorithm produced a well spread and diverse optimal solution and also converged reasonably to the actual optimal solution in 51 iterations. The result obtained compared favourably with that obtained with the direct solution algorithm discussed in a previous paper. We conclude that the genetic algorithm is quite efficient in dealing with non-convex and constrained combined heat and power economic dispatch problem.

A mean-field stochastic linear-quadratic optimal control problem with jumps under partial information

In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson stochastic martingale measure, and the quadratic cost function contains cross terms. In addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. This is the so-called optimal control problem of mean-field type. Firstly, the existence and uniqueness of the optimal control is proved. Secondly, the adjoint processes of the state equation is introduced, and by using the duality technique, the optimal control is characterized by the stochastic Hamiltonian system. Thirdly, by applying a decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information. Fourthly, the existence and uniqueness of the solutions of two Riccati equations are proved. Finally, we discuss a special case, and establish the corresponding feedback representation of the optimal control by means of filtering technique.

Optimal control of SDEs with expected path constraints and related constrained FBSDEs

In this paper, we consider optimal control of stochastic differential equations subject to an expected path constraint. The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs. In particular, the compensated process in our adjoint equation is deterministic, which seems to be new in the literature. For the typical case of linear stochastic systems and quadratic cost functionals (i.e., the so-called LQ optimal stochastic control), a verification theorem is established, and the existence and uniqueness of the constrained reflected FBSDEs are also given.

Efficient Implementation and Sampling Period Analysis of MPC for Water Distribution Networks*

In this paper we consider the design and implementation of Model Predictive Control (MPC) for water distribution networks (WDNs). First, we define the nonlinear differential algebraic equations that model a WDN, using both classical methods and simplified pump implementations. Then the cost function used in the MPC algorithm is formulated, where extra steps are taken to build a quadratic and convex cost function. The entire control problem is solved using a tailored sequential quadratic programming (SQP) method. The resulting control algorithm for WDNs is tested on a small distribution network to both illustrate the effectiveness and to perform additional tests on the effects of reducing the sampling period. The simulation results of this experiment indicate that the presented SQP–MPC can be implemented at faster rates than 1 hour and that this results in improved economic benefits.

Stable Reinforcement Learning for Optimal Frequency Control: A Distributed Averaging-Based Integral Approach

Frequency control plays a pivotal role in reliable power system operations. It is conventionally performed in a hierarchical way that first rapidly stabilizes the frequency deviations and then slowly recovers the nominal frequency. However, as the generation mix shifts from synchronous generators to renewable resources, power systems experience larger and faster frequency fluctuations due to the loss of inertia, which adversely impacts the frequency stability. This has motivated active research in algorithms that jointly address frequency degradation and economic efficiency in a fast timescale, among which the distributed averaging-based integral (DAI) control is a notable one that sets controllable power injections directly proportional to the integrals of frequency deviation and economic inefficiency signals. Nevertheless, DAI does not typically consider the transient performance of the system following power disturbances and has been restricted to quadratic operational cost functions. This paper aims to leverage nonlinear optimal controllers to simultaneously achieve optimal transient frequency control and find the most economic power dispatch for frequency restoration. To this end, we integrate reinforcement learning (RL) to the classic DAI, which results in RL-DAI control. Specifically, we use RL to learn a neural network-based control policy mapping from the integral variables of DAI to the controllable power injections which provides optimal transient frequency control, while DAI inherently ensures the frequency restoration and optimal economic dispatch. Compared to existing methods, we provide provable guarantees on the stability of the learned controllers and extend the set of allowable cost functions to a much larger class. Simulations on the 39-bus New England system illustrate our results.

Selective LQ wide area damping of power networks based on the spectral decomposition of Gramians

Inter-area oscillations in large power network are poorly controlled via local decentralized control. One promising strategy for wide-area damping of these oscillations is to minimize the quadratic state and control cost function for a linearized system model. In this paper, we improve this linear-quadratic method by introducing novel selective approach. Namely, instead of using the full state we minimize in the cost function only the selected spectral part of the state. The proposed selective LQ optimal problem is reformulated via the selected spectral part of the closed-loop observability Gramian. This spectral part is a solution to the modal Lyapunov equation specially designed for a chosen spectral range. Novel approach allows to take into account in global optimization only the selected set of oscillations and therefore to improve flexibility and accuracy of controllers setting. Numerical experiments with the IEEE New England test system confirm that the selective approach improves the performance of the conventional LQ method for suppressing low frequency inter-area oscillations.

Gradient Descent for Resource Allocation with Packet Loss*

This paper studies the effect of packet loss during the application of the weighted gradient descent to solve a resource allocation problem with piecewise quadratic cost functions in a multi-agent system. We define two performance metrics that measure, respectively, the deviation from the constraint and the error on the expected cost function. We derive upper bounds on both metrics: both bounds are proportional to the difference between the initial cost function and the cost function evaluated at the minimizer. Then, we extend the analysis of the constraint violation to open multi-agent systems where agents are replaced: based on a preliminary result and simulations we show that the combination of replacements and losses makes the constraint violation error diverge with time.

Hetero-Functional Network Minimum Cost Flow Optimization: A Hydrogen-Natural Gas Network Example

Over the past decades, engineering systems have developed as networks of systems that deliver multiple services across multiple domains. This work aims to develop an optimization program for a dynamic, hetero-functional graph theory-based model of an engineering system. The manuscript first introduces a general approach to define a dynamic system model by integrating the device models in the hetero-functional graph theory structural model. To this end, the work leverages Petri net dynamics and the hetero-functional incidence tensor. The respective Petri net-based models are translated into the quadratic program canonical form to finalize the optimization program. The optimization program is demonstrated through the application of the program to a hydrogen-natural gas infrastructure test case. Four distinct scenarios are optimized to demonstrate potential synergies or cascading network effects of policy across infrastructures. This work develops the first hetero-functional graph theory-based optimization program and demonstrates that the program can be used to optimize flows across a multi-operand network, transform the operands in the network, store operands over time, analyze the behavior for a quadratic cost function, and implement it for a generic, continuous, large flexible engineering systems of arbitrary topology.

Integral Reinforcement Learning for Tracking in a Class of Partially Unknown Linear Systems With Output Constraints and External Disturbances

In this paper, the H∞ tracking control problem of partially unknown linear systems with output constraints and disturbance is studied by the reinforcement learning (RL) method. Firstly, an augmented system is established based on the reference trajectory dynamics and target system dynamics, and a special cost function is established to realize asymptotic tracking. In addition, the barrier function (BF) is used to transform the augmented system, and the output constraints is realized simultaneously by minimizing the quadratic cost function of the transformed system. Using only the obtained data and part of the system dynamics, the optimal control strategy and the worst disturbance strategy are obtained by using the integral reinforcement learning (IRL). Rigorous stability analysis shows that the proposed method can make the trajectory of system states converge, and the output of the control strategy can make the tracking error asymptotically stable. Finally, a simulation example is conducted to verify the effectiveness of the proposed algorithm.

Optimal LQG Control of Networked Systems Under Traffic-Correlated Delay and Dropout

Transmission delay and packet dropout are inevitable network-induced phenomena that severely compromise the control performance of network control systems. The real-time network traffic is a major dynamic parameter that directly influences delay and reliability of transmission channels, and thus, acts as an unavoidable source of induced coupling among all network sharing systems. In this letter, we analyze the effects of traffic-induced delay and dropout on the finite-horizon quality-of-control of an individual stochastic linear time-invariant system, where quality-of-control is measured by an expected quadratic cost function. We model delay and dropout of the channel as generic stochastic processes that are correlated with the real-time network traffic induced by the rest of network users. This approach provides a pathway to determine the required networking capabilities to achieve a guaranteed quality-of-control for systems operating over a shared-traffic network. Numerical evaluations are performed using realistic stochastic models for delay and dropout. As a special case, we consider exponential distribution for delay with its rate parameter being traffic-correlated, and traffic-correlated Markov-based packet drop model.

Robust MPC for Systems With Model Uncertainties and Measurement Outliers

This paper considers the observer-based output feedback robust model predictive control (RMPC) problem for systems with model uncertainties and possible measurement outliers. For the sake of alleviating the effects from possible abnormal measurements, we design a set of observer-based output feedback RMPC controllers with the saturation constraint where the saturation level is adaptive according to the estimation errors. The purpose of the addressed problem is to design a set of desired RMPC controllers so as to guarantee the robustness and the asymptotical stability of the closed-loop system. Sufficient stability conditions are obtained by solving a time-varying terminal constraint set of an auxiliary optimization problem, and the corresponding control law and the upper bound of the quadratic cost function are derived. In addition, an algorithm including both off-line and on-line parts is provided to find a sub-optimal solution. Finally, two simulation examples are employed to illustrate the effectiveness of the proposed RMPC approach.

Explicit stochastic model predictive control for anti-pitching a high-speed multihull

Heave and pitch motion control of a high-speed multihull is important for restraining its large heave and pitch motions and improving its seaworthiness. However, stochastic ocean wave disturbances and strict input constraints on the T-foil and the flap highly increase difficulties in heave and pitch motion control. This paper proposes a explicit stochastic model predictive control (ESMPC) approach for anti-pitching the multihull without the violation of the control input constraints. A quadratic cost function is designed for the stochastic receding-horizon optimization of MPC under control input constraints, by considering the stochastic control model for the multihull as the predictive model. Then, the stochastic optimization is converted into deterministic optimization based on a Kalman-filter-based estimator and state mean prediction, and the ESMPC law is obtained by a combination of offline design and online search, which reduces the complexity of online optimization and enhances real-time control performance. The mean-square boundedness of stochastic system state is theoretically analyzed using the Lyapunov theory. The control effectiveness is illustrated by simulations and experiments, from which the pitch motion and the heave motion are reduced by 30% and 40%, respectively.