Predictive time optimal algorithm for a third-order dynamical system with delay

Both linear PID controllers and fuzzy controllers are designed and implemented for a boost converter. Comparison between the two controllers is made in the aspect of design, implementation and experimental results. Design of fuzzy controllers is based on heuristic knowledge of the converter and tuned using trial and error, while the design of linear PID and PI controllers is based on the frequency response of the boost converter. Implementation of linear controllers is quite straightforward, while implementation of fuzzy controllers has its unique issues. A comparison of experimental results indicates that the performance of the fuzzy controller is superior to that of the linear PID and PI controllers. The fuzzy controller is able to achieve faster transient response, has more stable steady-state response, and is more robust under different operating points.

This paper investigates the performance of adaptive particle swarm optimization (APSO) algorithm to find the best optimal gain parameters (Kp, Ki, Kd) in linear PID (L-PID) and nonlinear PID (NL-PID) controllers for the trajectory control of a nonlinear dynamical system (NLDS). Mathematical modeling, stability and performance analysis of a NLDS are being carried out with L-PID and NL-PID controllers. It is found that the APSO based tuning of NL-PID controller provides better tracking of NLDS than L-PID controller in respect of its search domain, premature convergence, and computational efficiency. Simulation analysis of various variables of controllers and system response such as error, cost function, and response of NLDS and their comparison to that of APSO tuned L-PID controller is described.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

. Recently, digital technologies and software packages as MATLAB and MATCAD are widely used for modeling and research of control systems. When analyzing and synthesizing the control system, ”manual” operations do not allow one to cover fully the systematic approach both from the constructive and computational points of view. The most widely used in practice and at the same time simplest control device is the linear PID controller. However, computer modeling and digital studies of this controller are not at the required level. This paper reveals some features of the functioning of a linear digital PID controller with three settings on a 2nd order object. Considering these assumptions, in the paper, the object modeled in the terms of output and state variables and the known theoretical parameters occurring in the ACS with a PID controller are more adequately confirmed. For the experiments, the Matlab/Simulink system with the ”Simulink Response Optimization/Signal Constraint” package was used. In order to facilitate access to the package, some important features of the algorithm for solving the control problem are revealed using a specific example.

An analytical design method for a conventional proportional-integral (PI) controller is developed for the optimal control of the liquid level loop in order to explicitly handle important control specifications. The constrained optimal control problem for the liquid level loop is formulated and converted into an unconstrained optimization problem to find an optimal PI controller tuning rule. The proposed PI controller minimizes the rate of change of the outlet flow and the deviation of the level subject to two important constraints in the level loop: the maximum allowable rate of change in the outlet flow rate and the maximum allowable deviation of the level. The simulation results show that the constrained optimal control of the liquid level loop can be successfully implemented using a simple PI controller with the proposed design method.

In this study, the problem of finding an optimal controller for nonlinear systems with one input and a reference tracking signal is approached. With the problem's formulation, any desired signal can be tracked instantly with a closed-loop controller without the need for integral terms. Presentation lies at the heart of optimal control. This study, however, does not consider the integral term, allowing tracking and stability to occur naturally. It has a broad scope with a wide range of applications, namely when dealing with affine nonlinear systems, which provide geometric control unification with asymptotic stability in some cases. A common scenario that comes from optimal control involves the minimization of integral cost functionals. Issues like asymptotic stability or even tracking to the desired reference signal have always been the main limitations. In this study, the main theorem allows the solution of optimal control problems with no-integral terms, in other words tracking problems with input/state constraints, providing closed-loop controllers. A DC motor with a pendulum in upright position is an example of an application for which singular optimal control is tested in this study. The results confirm both asymptotic stability and optimal tracking with an accuracy of 95%. The main contributions of this study include an optimal closed-loop controller with no mixed initial/final conditions, input/state constraints, asymptotic stability guarantee, a strong connection with geometric tools and finally the possibility to generalize to systems with multiple inputs. As a conclusion, general nonlinear control systems can be included in the optimal control methodology presented in this study including input/state constraints. Due to the lack of integral terms, the problem can be solved in closed form by using an optimal closed-loop controller.

The centralized paradigm of a single controller and a single plant upon which modern control theory is built is no longer applicable to modern cyber-physical systems of interest, such as the power-grid, software defined networks or automated highways systems, as these are all large-scale and spatially distributed. Both the scale and the distributed nature of these systems has motivated the decentralization of control schemes into local sub-controllers that measure, exchange and act on locally available subsets of the globally available system information. This decentralization of control logic leads to different decision makers acting on asymmetric information sets, introduces the need for coordination between them, and perhaps not surprisingly makes the resulting optimal control problem much harder to solve. In fact, shortly after such questions were posed, it was realized that seemingly simple decentralized optimal control problems are computationally intractable to solve, with the Wistenhausen counterexample being a famous instance of this phenomenon. Spurred on by this perhaps discouraging result, a concerted 40 year effort to identify tractable classes of distributed optimal control problems culminated in the notion of quadratic invariance, which loosely states that if sub-controllers can exchange information with each other at least as quickly as the effect of their control actions propagates through the plant, then the resulting distributed optimal control problem admits a convex formulation. The identification of quadratic invariance as an appropriate means of convexifying distributed optimal control problems led to a renewed enthusiasm in the controller synthesis community, resulting in a rich set of results over the past decade. The contributions of this thesis can be seen as being a part of this broader family of results, with a particular focus on closing the gap between theory and practice by relaxing or removing assumptions made in the traditional distributed optimal control framework. Our contributions are to the foundational theory of distributed optimal control, and fall under three broad categories, namely controller synthesis, architecture design and system identification. We begin by providing two novel controller synthesis algorithms. The first is a solution to the distributed H∞ optimal control problem subject to delay constraints, and provides the only known exact characterization of delay-constrained distributed controllers satisfying an H∞ norm bound. The second is an explicit dynamic programming solution to a two player LQR state-feedback problem with varying delays. Accommodating varying delays represents an important first step in combining distributed optimal control theory with the area of Networked Control Systems that considers lossy channels in the feedback loop. Our next set of results are concerned with controller architecture design. When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given -- indeed the task of designing this architecture is now as important as the design of the control laws themselves. To address this task, we formulate the Regularization for Design (RFD) framework, which is a unifying computationally tractable approach, based on the model matching framework and atomic norm regularization, for the simultaneous co-design of a structured optimal controller and the architecture needed to implement it. Our final result is a contribution to distributed system identification. Traditional system identification techniques such as subspace identification are not computationally scalable, and destroy rather than leverage any a priori information about the system's interconnection structure. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to estimate local dynamics within a large interconnected system. To that end we propose a promising heuristic for identifying the dynamics of a subsystem that is still connected to a large system. We exploit the fact that the transfer function of the local dynamics is low-order, but full-rank, while the transfer function of the global dynamics is high-order, but low-rank, to formulate this separation task as a nuclear norm minimization problem. Finally, we conclude with a brief discussion of future research directions, with a particular emphasis on how to incorporate the results of this thesis, and those of optimal control theory in general, into a broader theory of dynamics, control and optimization in layered architectures.

Model-based synthesis of nonlinear PI and PID controllers

The design of rate allocation and queue length control in computer networks is treated as a stochastic optimal control problem. The performance index is chosen to achieve the twin objectives of minimising queue length fluctuations and fully utilising the available bandwidth. Simple, practically realisable optimal control schemes are obtained for both LANs and WANs. An adaptive scheme is proposed where the auto-regressive parameters of the traffic, needed for gain calculations, are estimated by an LMS algorithm. Discrete event simulations are carried out to verify the fluid-flow models used in developing the controllers, to compare their performance against PI controllers proposed previously, and to study the effect of self-similar traffic. Two key results are obtained. First, queue-length fluctuations, and hence potentially packet losses, are much smaller for the optimal feedforward controller than for the PI controller. Second, in contrast to uncontrolled queues, the queue length variance decreases with increasing Hurst parameter for self-similar traffic.

Recently, the Gravitational-Search Algorithm (GSA) has been presented as a promising physics-inspired stochastic global optimization technique. It takes its derivation and features from laws of gravitation. This paper applies the GSA to design optimal controllers of a nonlinear system consisting of a doubly-fed induction generator (DFIG) driven by a wind turbine. Both the active and the reactive power are controlled and processed through a back-to-back converter. The active power control loop consists of two cascaded proportional integral (PI) controllers. Another PI controller is used to set the q-component of the rotor voltage by compensating the generated reactive power. The GSA is used to simultaneously tune the parameters of the three PI controllers. A time-weighted absolute error (ITAE) is used in the objective function to stabilize the system and increase its damping when subjected to different disturbances. Simulation results will demonstrate that the optimal GSA-based coordinated controllers can efficiently damp system oscillations under severe disturbances. Moreover, simulation results will show that the designed optimal controllers obtained using the GSA perform better than the optimal controllers obtained using two commonly used global optimization techniques, which are the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO).

The authors propose a method by which to design an optimum vibration control system composed of continuous body structures using modal analysis and sensitivity analysis. The mechanical structures are modeled first by FEM. This spatial model is of such a high degree-of-freedom (DOF) that a control system cannot be designed. This spatial model is transformed into a modal model in order to reduce DOF. The optimum control system is designed by applying the optimum control theory to this reduced model. However, stability of the control system is not always guaranteed because the modal model ignores high-order modes. An optimization method of the control system which guarantees stability of the actual system is proposed in this paper. We propose also an experimental optimization of the control system based on experimental modal analysis. The optimization is performed using sensitivity analysis and nonlinear programming. The effectiveness of the proposed method is verified by both simulation and experiment.

Studies by simulation on the model of a multistage flash (MSF) desalination plant have shown certain features of the model developed, on the basis of physical laws and correlations. The most important of these is nonlinearity which relates to the changing of the process characteristics with the operating conditions. It is shown that a fixed PID controller cannot be optimal, if the operating point changes from the one at which the optimal controller was designed. In this paper, the parameter variations in the linearised plant model have been modelled over the operating region of interest, and these are mapped into the parameter space of the optimal PID controller. The resulting algorithm is a parameter scheduling scheme for the MSF plant for optimal PID control with a choice among four criteria (ISE, IAE, ISTE, and IATE). The scheme illustrated is with reference to the top brine temperature control of an actual 18 stage MSF desalination plant.

Batch crystallizers are predominantly used in chemical industries like pharmaceuticals, food industries and specialty chemicals. The nonlinear nature of the batch process leads to difficulties when the objective is to obtain a uniform Crystal Size Distribution (CSD). In this study, a linear PI controller is designed using classical controller tuning methods for controlling the crystallizer outlet temperature by manipulating the inlet jacket temperature; however, the response is not satisfactory. A simple PID controller cannot guarantee a satisfactory response that is why an optimal controller is designed to keep the concentration and temperature in a range that suits our needs. Any typical process operation has constraints on states, inputs and outputs. So, a nonlinear process needs to be operated satisfying the constraints. Hence, a nonlinear controller like Generic Model Controller (GMC) which is similar in structure to the PI controller is implemented. It minimizes the derivative of the squared error, thus improving the output response of the process. Minimization of crystal size variation is considered as an objective function in this study. Model predictive control is also designed that uses advanced optimization algorithm to minimize the error while linearizing the process. Constraints are fed into the MPC toolbox in MATLAB and Prediction, Control horizons and Performance weights are tuned using Sridhar and Cooper Method. Performances of all the three controllers (PID, GMC and MPC) are compared and it is found that MPC is the most superior one in terms of settling time and percentage overshoot.

The research aims to design robust controllers that achieve the stability of a single-wheeled robot under the presence of friction factors and to track different parameters to verify robust stability. This paper presents a new study of the unicycle robot system that is controlled using advanced control methodologies. The paper aims to improve the work of the unicycle robot system, due to its effective impact on improving the performance of driving the robot, which is reflected in the smoothness of the vehicle speed change, ensuring the stability of the robot and the safety of the investor in the uncertain work environment. The main goal is to achieve high dynamic performance for the unicycle robot system. The studied system is non-linear and is subject to the restrictions of the friction factor change with the speed change of the unicycle robot. What increases the difficulty of controlling this type of control system is the uncertainty of some parameters of the control system, such as friction factors. In this paper, two advanced control methodologies were proposed: the optimal controller and the optimal parametric controller. The research results showed that both the optimal and optimal parametric controllers succeeded in achieving stability despite the uncertainty of the parameters and multiple friction factors, but with a relative superiority of the optimal parametric controller. Previous research has discussed many controllers such as classical and advanced controllers such as sliding control and fuzzy control, but it has not previously dealt with the optimal parametric controller that will be discussed in this research.

Robotic manipulator control is a challenging task due to its nonlinear, interacting multi-input–multi-output dynamics. In this paper, we proposed an optimal robust fractional-order fuzzy PID controller based on multi-objective particle swarm optimization (MOPSO) algorithm for a two-link robotic manipulator. In order to minimize position and trajectory-tracking error, MOPSO finds the optimal parameters of fuzzy membership functions and order of the fractional operators. To show the effectiveness of the proposed controller, the software-in-the-loop real-time simulation is executed, and the results are compared to conventional fuzzy PID, fractional-order PID, and linear PID controllers.