Steady-state Optimal Control Research Articles

Saturated Trajectory Tracking Controller in the Body-Frame for Quadrotors

This paper introduces a quadrotor trajectory tracking controller comprising a steady-state optimal position controller with a normed input saturation and modular integrative action coupled with a backstepping attitude controller. First, the translational and rotational dynamical models are designed in the body-fixed frame to avoid external rotations and are partitioned into an underactuated position system and a quaternion-based attitude system. Secondly, a controller is designed separately for each subsystem, namely, (i) the position controller synthesis is derived from the Maximum Principle, Lyapunov, and linear quadratic regulator (LQR) theory, ensuring the global exponential stability and steady-state optimality of the controller within the linear region, and global asymptotic stability is guaranteed for the saturation region when coupled with any local exponential stable attitude controller, and (ii) the attitude system, with the quaternion angles and the angular velocity as the controlled variables, is designed in the error space through the backstepping technique, which renders the overall system, position, and attitude, with desirable closed-loop properties that are almost global. The overall stability of the system is achieved through the propagation of the position interconnection term to the attitude system. To enhance the robustness of the tracking system, integrative action is devised for both position and attitude, with emphasis on the modular approach for the integrative action on the position controller. The proposed method is experimentally validated on board an off-the-shelf quadrotor to assess the resulting performance.

Comparative Study of Advanced Modulation and Control Schemes for Dual Three-Phase PMSM Drives With Low Switching Frequencies

In this paper, multiple advanced low-switching-frequency modulation and control schemes are investigated for dual three-phase permanent-magnet synchronous motor (PMSM) drives. The modulation schemes under investigation include multisampling space vector modulation (MS-SVM), selective harmonic elimination pulse width modulation (SHEPWM) and synchronous optimal pulse width modulation (SOPWM) while the control schemes include complex vector control, model predictive control (MPC) and flux trajectory control-based model predictive pulse pattern control (MP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> C). The difference between three-phase and dual three-phase PMSM drives at low switching frequencies is analyzed. Moreover, an optimal pulse width modulation (PWM)-based MP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> C scheme is proposed for the dual three-phase PMSM, where the optimal PWM modulation and MP <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> C control are combined such that the optimal steady-state and dynamic control performance is achieved among the modulation and control schemes investigated. Experimental results are given to compare and validate the proposed control scheme.

Study on the Effect of Exhaust Gas Recirculation Coupled Variable Geometry Turbocharger and Fuel Quantity Control on Transient Performance of Turbocharged Diesel Engine

With increasingly stringent emissions regulations, there are growing demands for the transient performance of diesel engines. This study conducted a transient bench test on a two-stage turbocharged heavy-duty diesel engine to optimize its performance during a load increase (20% to 100% in 1 s) at a constant speed (1200 RPM) transient process. The results showed that the transient control scheme using the low-pressure EGR system resulted in a 42.1% reduction in the peak value of soot emission, a 24.8% decrease in the peak value of NOx emission, a 9.14% decrease in ISFC and a 30.6% increase in maximum IMEP achieved in 1 s, compared to the steady-state optimization control scheme without EGR. Transient control scheme using the high-pressure EGR system resulted in a 24.4% reduction in the peak value of soot emission, a 31.8% reduction in the peak value of NOx emission, a 9.52% reduction in ISFC, and a 31.7% increase in maximum IMEP achieved in 1 s. The comparison of high and low-pressure EGR systems revealed that the low-pressure EGR system produced lower compromising emissions, while alterations in control parameters for the diesel engine with a high-pressure EGR system had a more significant impact on the transient process performance.

Linear-Convex Optimal Steady-State Control

We consider the problem of designing a feedback controller for a multivariable linear time-invariant system, which regulates a system output to the solution of an equality-constrained convex optimization problem despite unknown constant exogenous disturbances; we term this the linear-convex optimal steady-state (OSS) control problem. We introduce the notion of an optimality model, and show that the existence of an optimality model is sufficient to reduce the OSS control problem to a stabilization problem. This yields a design framework for OSS control that unifies and extends existing design methods in the literature. We illustrate the approach via an application to optimal frequency control of power networks, where our methodology recovers centralized and distributed controllers reported in the recent literature.

Steady-State Control of Fuel Cell Based on Boost Mode of a Dual Winding Motor

In recent years, a dual winding motor drive has been proposed in the field of fuel cell vehicles due to its advantages of good performance and high robustness. This new topology and its basic control method have been widely investigated. However, the previous research has not considered the current dynamic property of a fuel cell when studying the power sharing control method, but this is an important research objective for fuel cell durability. Considering the current change principle of a fuel cell, an optimal steady-state control method based on a new dual winding motor architecture boost charging is proposed to optimize the fuel cell life. In addition, in view of the current circulation problem of the fuel cell side winding in the boost mode, a Bang-Bang-PI control algorithm with a relatively constant reference value is proposed to realize the current sharing control. On this basis, the optimized control of the output current ripple of the fuel cell is realized to ensure the steady-state of the proton exchange membrane fuel cell (PEMFC). Finally, the results show that this method can control the stability of the fuel cell efficiently.

Real-Time Loss Minimizing Control of Induction Machines for Dynamic Load Profiles Under Deadbeat-Direct Torque and Flux Control

This article develops a real-time loss minimizing control technique on induction machines for dynamic load profiles. Steady-state loss minimization has been well addressed. However, when the steady-state optimal control is applied to a highly dynamic load profile, more losses are induced during transients due to the slow rotor flux dynamics. Several heuristic methods have been developed for real-time loss reduction, but they are not designed for minimum losses. Loss-minimizing flux trajectories have been generated by dynamic programming but require prior information of load profiles. In this article, a real-time loss minimizing control technique is developed for applications without prior information of load profiles. It resembles the loss-minimizing stator flux trajectory generated by a modified dynamic programming method, which generates the best practical solution for unknown load profiles. Both simulation and experimental results demonstrate the energy saving capability of the proposed method. Significant energy savings have been demonstrated on traction applications.

Quadratic Constrained Periodic Optimization for Bandlimited Linear Systems Via the Fourier-Based Method

AbstractMotivated by engineering applications, we address bounded steady-state optimal control of linear dynamical systems undergoing steady-state bandlimited periodic oscillations. The optimization can be cast as a minimization problem by expressing the state and the input as finite Fourier series expansions, and using the expansions coefficients as parameters to be optimized. With this parametrization, we address linear quadratic problems involving periodic bandlimited dynamics by using quadratic minimization with parametric time-dependent constraints. We hence investigate the implications of a discretization of linear continuous time constraints and propose an algorithm that provides a feasible suboptimal solution whose cost is arbitrarily close to the optimal cost for the original constrained steady-state problem. Finally, we discuss practical case studies that can be effectively tackled with the proposed framework, including optimal control of DC/AC power converters, and optimal energy harvesting from pulsating mechanical energy sources.

Optimal Steady-State Voltage Control Using Gaussian Process Learning

In this article, an optimal steady-state voltage control framework is developed based on a novel linear voltage-power dependence deducted from Gaussian process (GP) learning. Different from other point-based linearization techniques, this GP-based linear relationship is valid over a subspace of operating points and, thus, suitable for a system with uncertainties such as those in power injections due to renewables. The proposed optimal voltage control algorithms, therefore, perform well over a wide range of operating conditions. Both centralized and distributed optimal control schemes are introduced in this framework. The least-squares estimation is employed to provide analytical forms of the optimal control, which offer great computational benefits. Moreover, unlike many existing voltage control approaches deploying fixed voltage references, the proposed control schemes not only minimize the control efforts but also optimize the voltage reference setpoints that lead to the least voltage deviation errors with respect to such setpoints. The control algorithms are also extended to handle uncertain power injections with robust optimal solutions, which guarantee compliance with the voltage regulation standards. As for the distributed control scheme, a new network partition problem is cast, based on the concept of effective voltage control source (EVCS), as an optimization problem which is further solved using convex relaxation. Various simulations on the IEEE 33-bus and 69-bus test feeders are presented to illustrate the performance of the proposed voltage control algorithms and EVCS-based network partition.

Voltage-Based Distributed Optimal Control for Generation Cost Minimization and Bounded Bus Voltage Regulation in DC Microgrids

In conventional DC microgrid control schemes, optimization and real-time control are usually performed at different time scales. It is hard for such control schemes to achieve real-time optimization. Even a small disturbance can result in deviations of bus voltages and output currents from their optimal operating points. In addition, most real-time controllers cannot guarantee the satisfaction of pre-defined constraints on individual bus voltages due to the disconnection between steady-state optimization and real-time control. In this paper, a distributed optimal control scheme is presented for DC microgrid. The objectives are to simultaneously realize generation cost minimization and individual bus voltage regulation. First, the optimal control problem is formulated, based on which a necessary and sufficient optimality condition is derived that characterizes the optimal operating points. The distributed optimal controller is then proposed to dynamically drive the system to operate at the optimality condition. Convergence to the optimal operating points that minimizes generation cost under constraints is guaranteed through rigorous Lyapunov synthesis. Each individual bus voltage can also be maintained within bounds in both transient- and steady-state. The performance of the proposed controller is validated through simulations based on a switch-level microgrid model.

Optimal Setting and Control for Iron Removal Process Based on Adaptive Neural Network Soft-Sensor

Satisfying the process technical requirements while optimizing the control of oxygen and zinc oxide is the main task for optimal operation of the iron removal process. However, due to the complicated mechanism and varied production conditions, the process is difficult to achieve optimal operation with the simple controller. The manual control, as a result, is extensively used in practice. In this paper, we develop an optimal setting and control (OSC) system for the iron removal process to achieve its technical requirements with minimal process consumptions. The OSC system is composed of a neural network soft-sensor, an optimal setting module, an optimal controller, and a fuzzy-logic-based compensator. First, we define the oxygen reaction efficiency (ORE) to measure the difference between the theoretical oxygen amount and its actual amount. Due to the ORE cannot be measured online and it varies with the production conditions, an adaptive weight radial basis function neural network soft-sensor is designed to estimate it. The adaptive weight adjusting method contributes to improve the adaptability of the soft-sensor, and its convergence is discussed. The optimal setting module provides the set-point of outlet ferrous ion concentration for the optimal operation of every reactor. The steady-state optimal control of oxygen and zinc oxide is then established, and the compensator compensates the control inputs utilizing the feedforward and feedback information. Finally, simulations validate that the ORE is important for the optimal control of the process. Furthermore, industrial experiments are presented to verify the effectiveness and potential of the proposed strategy.

Steady-State Predictive Optimal Control of Integrated Building Energy Systems Using a Mixed Economic and Occupant Comfort Focused Objective Function

Control of energy systems in buildings is an area of expanding interest as the importance of energy efficiency, occupant health, and comfort increases. The objective of this study was to demonstrate the effectiveness of a novel predictive steady-state optimal control method in minimizing the economic costs associated with operating a building. Specifically, the cost of utility consumption and the cost of loss productivity due to occupant discomfort were minimized. This optimization was achieved through the use of steady-state predictions and component level economic objective functions. Specific objective functions were developed and linear models were identified from data collected from a building on Texas A&amp;M University’s campus. The building consists of multiple zones and is serviced by a variable air volume, chilled water air handling unit. The proposed control method was then co-simulated with MATLAB and EnergyPlus to capture effects across multiple time-scales. Simulation results show improved comfort performance and decreased economic cost over the currently implemented building control, minimizing productivity loss and utility consumption. The potential for more serious consideration of the economic cost of occupant discomfort in building control design is also discussed.

Linear–quadratic optimal steady state controllers for engineering students and practicing engineers

This paper is an overview of fundamental linear–quadratic optimal control techniques used for linear dynamic systems. The presentation is suitable for undergraduate and graduate students and practicing engineers. The paper can be used by class instructors as supplemental material for undergraduate and graduate control system courses. The paper shows how to find the solution to a dynamic optimization problem: optimize an integral quadratic performance criterion along trajectories of a linear dynamic system over an infinite time period (steady-state linear–quadratic optimal control problem). The solution is obtained by solving a static optimization problem. All derivations done in the paper require only elementary knowledge of linear algebra and state space linear system analysis. The results are presented also for the observer-driven linear–quadratic steady-state optimal controller, output feedback-based linear–quadratic optimal controller, and the Kalman filter-driven linear–quadratic stochastic optimal controller. Having full understanding of derivations of the linear–quadratic optimal controller, observer-driven linear–quadratic optimal controller, optimal linear–quadratic output feedback controller, and optimal linear–quadratic stochastic controller, students and engineers will feel confident to use these controllers in numerous engineering and scientific applications. Several optimal linear–quadratic control case studies involving models of real physical systems, with the corresponding Simulink block diagrams and MATLAB codes, are included in the paper.

Distributed Frequency and Voltage Control for AC Microgrids based on Primal-Dual Gradient Dynamics

With the gradual transformation of power generation towards renewables, distributed energy resources are becoming more and more relevant for grid stabilization. In order to involve all participants in the joint solution of this challenging task, we propose a distributed, model-based and unifying controller for frequency and voltage regulation in AC microgrids, based on steady-state optimal control. It not only unifies frequency and voltage control, but also incorporates the classic hierarchy of primary, secondary and tertiary control layers with each closed-loop equilibrium being a minimizer of a user-defined cost function. By considering the individual voltage limits as additional constraints in the corresponding optimization problem, no superordinate specification of voltage setpoints is required. Since the dynamic model of the microgrid has a port-Hamiltonian structure, stability of the overall system can be assessed using shifted passivity properties. Furthermore, we demonstrate the effectiveness of the controller and its robustness against fluctuations in active and reactive power demand by means of numerical examples.

Efficient Operation of Continuous Reactor Systems using Economic Nonlinear MPC Formulations

A combination of real-time steady-state optimization (RTO) and model predictive control (MPC) used in the process industry for maximizing economic performance can result in sub-optimal operation due to the model mismatch between the RTO and MPC components and delays involved in carrying out RTO step. These problems are alleviated by recently proposed economic nonlinear MPC (ENMPC) that merges the optimization and control layers. For an e¤ective implementation of ENMPC, it is essential that accurate information of states and parameters is available at every sampling instant. The ENMPC schemes proposed in the literature either assume that the model parameters are not time-varying or are measurable/known quantities when they change with time. In practice, variations in these model parameters are seldom measurable online. In this work, two novel ENMPC formulations are proposed to address this problem: (i) Output Feedback ENMPC integrated with state estimator that employs innovation feedback for correcting model predictions and (ii) Adaptive ENMPC integrated with simultaneous state and parameter estimator, in which parameters are estimated simultaneously with the states and are updated in the model used by ENMPC. The efficacies of the proposed approaches are demonstrated by simulating two benchmark reactor control problems. Analysis of the simulation results reveals that the proposed Adaptive ENMPC is able to achieve performance comparable to the ideal case where all states and parameter variations are perfectly known. Also, Adaptive ENMPC as well as Output Feedback ENMPC formulations result in significantly better control when compared with the conventional ENMPC formulation.

Dynamics and Control on a Discrete Multi-Inventory System

In this paper, a discrete two item inventory model for deteriorating items with a linear stock dependent demand rate without regular supply is given first; then, regular supply is taken into account, which may be regarded as feedback control. An objective function is formed to calculate the net profit with respect to possible profits and possible loss. A necessary criterion for the steady state optimal control problem for optimizing the objective function subjected to the constraints given by the difference equations of the inventory is obtained. At last, an improved model is put forward.

Design and control of different pressure thermally coupled reactive distillation for amyl acetate synthesis

The different pressure thermally coupled reactive distillation (DPT-RD) process is an attractive heat-integrated design in chemical engineering development processes. In this study, a rigorous simulation of the DPT-RD process for amyl acetate synthesis in a neat operation mode is performed using the Aspen Plus simulator. The economic feasibility and controllability of the DPT-RD process were evaluated based on the research of steady-state optimization and dynamic control. The optimum DPT-RD process in terms of the total annual cost (TAC) is screened. The results show that 33.74% reduction in energy consumption and 7.96% savings in TAC can be achieved by the DPT-RD process compared to the conventional reactive distillation. The control strategy of the DPT-RD process was developed and the basic control structure with proportional control and the improved control structure with temperature/composition cascade control were presented. The dynamic results indicate that the improved control structure can effectively handle disturbances of ±20% in feed flow rate and of 5% in feed composition. Overall results demonstrate that the DPT-RD process is a promising approach for amyl acetate synthesis.

An energy-efficient operation system for a natural gas liquefaction process: Development and application to a 100 ton-per-day plant

Abstract Production of liquefied natural gas (LNG) is a highly energy-intensive process. The required liquefaction temperature is approximately −160 °C at atmospheric pressure. In this study, we propose a two-level energy-efficient operation system for an LNG process, which consists of real-time steady-state optimization (RTSSO) and real-time control subsystems. The RTSSO follows the typical design to minimize set point energy losses. In contrast, the real-time control subsystem was configured to incorporate the concept of self-optimizing control, which minimizes both implementation energy losses and set point energy losses. Special attention was given to the loss of the evaluation step so that the implementation energy loss is accurately estimated; thus, the designed system can be seamlessly applied to the actual process with guaranteed optimal energy efficiency. The performance of the proposed system was validated in a simulated LNG plant that precisely replicated an actual plant that produces 100 tons of LNG per day.

Fast cooling for a system of stochastic oscillators

We study feedback control of coupled nonlinear stochastic oscillators in a force field. We first consider the problem of asymptotically driving the system to a desired steady state corresponding to reduced thermal noise. Among the feedback controls achieving the desired asymptotic transfer, we find that the most efficient one from an energy point of view is characterized by time-reversibility. We also extend the theory of Schrödinger bridges to this model, thereby steering the system in finite time and with minimum effort to a target steady-state distribution. The system can then be maintained in this state through the optimal steady-state feedback control. The solution, in the finite-horizon case, involves a space-time harmonic function φ, and −logφ plays the role of an artificial, time-varying potential in which the desired evolution occurs. This framework appears extremely general and flexible and can be viewed as a considerable generalization of existing active control strategies such as macromolecular cooling. In the case of a quadratic potential, the results assume a form particularly attractive from the algorithmic viewpoint as the optimal control can be computed via deterministic matricial differential equations. An example involving inertial particles illustrates both transient and steady state optimal feedback control.

Global optimal control of variable air volume air-conditioning system with iterative learning: an experimental case study

The air-conditioning system in a large commercial or high-rise building is a complex multi-variable system influenced by many factors. The energy saving potential from the optimal operation and control of heating, ventilating, and air-conditioning (HVAC) systems can be large, even when they are properly designed. The ultimate goal of optimization is to use the minimum amount of energy needed to improve system efficiency while meeting comfort requirements. In this study, a multi-zone variable air volume (VAV) and variable water volume (VWV) air-conditioning system is developed. The steady state modes and dynamic models of the HVAC subsystems are constructed. Optimal control based on large scale system theory for system-level energy-saving of HVAC is introduced. Control strategies such as proportional-integral-derivative (PID) controller (gearshift integral PID and self-tuning PID) and iterative learning control (ILC) are studied in the platform to improve the dynamic characteristics. The system performance is improved. An 18.2% energy saving is achieved with the integration of ILC and sequential quadratic programming based on a steady-state hierarchical optimization control scheme.

Variable air volume air-conditioning experiment system with advanced controls

A heating, ventilation, and air-conditioning (HVAC) system is a multi-variable strongly coupled large-scale system that is composed of several sub-systems. Considerable research, simulations, and experiments have been conducted on HVAC control. The optimization control of an HVAC system is now the popular issue. The ultimate goal of this paper was to achieve minimum energy consumption and improve system efficiency. Multi-zone variable air volume and variable water volume air-conditioning systems were developed. The dynamic models of HVAC sub-systems were built by the adaptive directional forgetting method. Control strategies such as the gearshift integral proportional-integral-derivative (PID) controller and self-tuning PID controller were studied in the platform to improve the dynamic characteristics of the HVAC system. System performance was improved. The system saved 18.2% of energy with the integration of iterative learning control and sequential quadratic programming based on the steady-state hierarchical optimization control scheme.