Robust and Optimal PID Controller Synthesis for Linear Time Invariant Systems

We dealt with new approaches to the design of Proportional-Integral-Derivative (PID) controllers and solved three important open problems: 1) Optimal design of H∞ continuous time controllers 2) Optimal design of H∞ discrete time controllers and 3) Design of PID controllers for prescribed settling time. We also deal with optimal Dynamic Compensator design for controllable and observable systems. The main result of the first problem is a constructive determination of the set Sγ of stabilizing continuous PI and PID controllers achieving an H∞ norm bound of γ on the error transfer function. This result utilizes the computation of the complete stabilizing set S. We also point out connections between this H∞ design and Gain and Phase Margin designs. The main result of the second problem is a constructive characterization of the set Sγ of stabilizing digital controllers achieving a prescribed bound γ on the error transfer function. This is accomplished by utilizing the computation of S, the set of all PID stabilizing controllers. The minimum achievable γ, denoted γ∗ is also determined. The main result of the third problem is a constructive determination of the set S(σ) of stabilizing PI and PID controllers with closed loop poles having real parts less than −σ. The signature method is applied to obtain the set S(σ) in the controller parameter space. The maximum achievable σ for a given plant is also determined. The main result of the last problem is a new approach to design an optimal dynamic compensator. The system is augmented with a proper number of integrators and the state feedback of the augmented system is considered with a design parameter. The dynamic compensator is then designed such that the eigenvalues of the augmented system is identical to the closed loop specboundtrum of the implemented system with the compensator. By sweeping over the design parameter, multiple design specifications are compared within achievable boundary of performances

Robust and Optimal PID Controller Synthesis for Linear Time Invariant Systems

We dealt with new approaches to the design of Proportional-Integral-Derivative (PID) controllers and solved three important open problems: 1) Optimal design of H∞ continuous time controllers 2) Optimal design of H∞ discrete time controllers and 3) Design of PID controllers for prescribed settling time. We also deal with optimal Dynamic Compensator design for controllable and observable systems. The main result of the first problem is a constructive determination of the set Sγ of stabilizing continuous PI and PID controllers achieving an H∞ norm bound of γ on the error transfer function. This result utilizes the computation of the complete stabilizing set S. We also point out connections between this H∞ design and Gain and Phase Margin designs. The main result of the second problem is a constructive characterization of the set Sγ of stabilizing digital controllers achieving a prescribed bound γ on the error transfer function. This is accomplished by utilizing the computation of S, the set of all PID stabilizing controllers. The minimum achievable γ, denoted γ∗ is also determined. The main result of the third problem is a constructive determination of the set S(σ) of stabilizing PI and PID controllers with closed loop poles having real parts less than −σ. The signature method is applied to obtain the set S(σ) in the controller parameter space. The maximum achievable σ for a given plant is also determined. The main result of the last problem is a new approach to design an optimal dynamic compensator. The system is augmented with a proper number of integrators and the state feedback of the augmented system is considered with a design parameter. The dynamic compensator is then designed such that the eigenvalues of the augmented system is identical to the closed loop specboundtrum of the implemented system with the compensator. By sweeping over the design parameter, multiple design specifications are compared within achievable boundary of performances

Modern Design of Classical Controllers

Classical controller design emphasizes simple low-order controllers. These classical controllers include Proportional-Integral (PI), Proportional-Integral-Derivative (PID), and First Order. In modern control theory, it is customary to design high-order controllers based on models, even for simple plants. However, it was shown that such controllers are invariably fragile, and this led to a renewal of interest in classical design methods. In the present research, a modern approach to the design of classical controllers (by introducing a complete stabilizing set in the space of the design parameters) is described. When classical specifications such as gain margin, phase margin, bandwidth, and time-delay tolerance are imposed, the achievable performance can be easily determined graphically. The objective of this research is to determine the controller gains, contained in the stabilizing set, which satisfy desired performance specifications such as crossover frequency and closed-loop stability margins. The design procedure starts with the calculation of the stabilizing set using recent methods. Then, a simple parametrization produces ellipses and straight lines (for PI controller design) and cylinders and planes (for PID controller design) in the space of controller gains. Each set of ellipses/cylinders and straight lines/planes represents constant magnitude and constant phase loci for the controller. The main result is that the crossing points, which are the intersection of ellipses/cylinders and straight lines/planes, are selected such that they are contained in the stabilizing set of controllers. They provide the controller gains that we need to satisfy our desired robust performance, seen as desired gain margin, phase margin, gain crossover frequency, and time-delay tolerance in our system. Then, using these crossing points contained in the stabilizing set, a new plot with information about the achievable performance in terms of gain margin, phase margin, and gain crossover frequency is constructed. Each point of this achievable performance can be used to retrieve the controller’s gains, which are contained in the stabilizing set. This result provides the possibility to analyze the system’s achievable performance by exploring the stabilizing set and considering different desired configurations in the performance capabilities for the system using a PI or PID controller. This expands our possibilities when designing controllers by considering different classical controller’s configurations. This research considers the discrete-time and continuous-time linear time invariant systems and cases including First Order with time-delay in the system, and the extension to the controller design for multivariable systems. Finally, the design procedure is illustrated with different examples and real applications for all such cases

Synthesis and design of PID controllers

controllers for discrete-time systems and time-delayed systems. By using bilinear transformation and orthogonal transformation, earlier research results obtained in the continuous-time case are extended to discrete-time situation. The complete set of stabilizing PID controllers for the discrete-time systems is thus obtained. Moreover, this set remains to be a union of convex sets when one particular parameter is fixed. Thus a method to design robust and non-fragile digital PID controllers is proposed by following a similar design procedure for the continuous-time systems. In order to find the stabilizing controller set for systems with time-delays, the relationship between the Nyquist Criterion and Pontryagins theory is investigated. The conditions under which one can correctly apply the Nyquist Criterion to time-delayed systems are derived. Then, the complete set of stabilizing PID controllers for arbitrary order LTI systems with time-delay up to a given value is obtained. Furthermore, the stability issue of a system with fixed-delay is also studied and a formula which provides complete knowledge of the distribution of the closed-loop poles is presented. Based on this formula, stabilizing P and PI controller sets for the system with fixed-delay can be computed

Computer aided synthesis and design of PID controllers

This thesis aims to cover some aspects of synthesis and design of Proportional- Integral-Derivative (PID) controllers. The topics include computer aided design of discrete time controllers, data-based design of discrete PID controllers and data- robust design of PID controllers. These topics are of paramount in control systems literature where a lot of stress is laid upon identification of plant and robust design. The computer aided design of discrete time controllers introduces a Graphical User Interface (GUI) based software. The controllers are: Proportional (P), Proportional-Derivative (PD),Proportional-Integral (PI) and Proportional-Integral- Derivative (PID) controllers. Different performance based design methods with these controllers have been introduced. The user can either explore the performance by interactively choosing controllers one by one from the entire set and visualizing its performance or specify some performance constraints and obtaining the resulting set. In data-based design, the thesis presents a way of designing PID controllers based on input-output data. Thus, the intermediate step of identification of model from data is removed, saving considerable effort. Moreover, the data required is step response data which is easier to obtain in case of discrete time system than frequency response data. Further, a GUI developed for interactive design is also described. In data-robust design, the problem of uncertainty in data is explored. The design method developed finds the stabilizing set which can robustly stabilize the plant with uncertainty. It has been put forward as an application to interval linear programming. The main results of this research include a new way of designing discrete time PID controllers directly from the data. The simulations further confirm the results. Robust design of PID controllers with data uncertainty has also been established. Additionally, as a part of this research, a GUI based software has been developed which is expected to be very beneficial to the designers in manufacturing, aerospace and petrochemical industries. PID controllers are widely used in the industry. Any progress in this field is well acknowledged both in the industry and the academia alike. This thesis attempts a small step further in this direction

A Data-driven Approach to Robust Control of Multivariable Systems by Convex Optimization

The frequency-domain data of a multivariable system in different operating points is used to design a robust controller with respect to the measurement noise and multimodel uncertainty. The controller is fully parametrized in terms of matrix polynomial functions and can be formulated as a centralized, decentralized or distributed controller. All standard performance specifications like $H_2$, $H_\infty$ and loop shaping are considered in a unified framework for continuous- and discrete-time systems. The control problem is formulated as a convex-concave optimization problem and then convexified by linearization of the concave part around an initial controller. The performance criterion converges monotonically to a local optimal solution in an iterative algorithm. The effectiveness of the method is compared with fixed-structure controllers using non-smooth optimization and with full-order optimal controllers via simulation examples. Finally, the experimental data of a gyroscope is used to design a data-driven controller that is successfully applied on the real system

Theoretical analysis and experimental validation of a simplified fractional order controller for a magnetic levitation system

Fractional order (FO) controllers are among the emerging solutions for increasing closed-loop performance and robustness. However, they have been applied mostly to stable processes. When applied to unstable systems, the tuning technique uses the well-known frequency-domain procedures or complex genetic algorithms. This brief proposes a special type of an FO controller, as well as a novel tuning procedure, which is simple and does not involve any optimization routines. The controller parameters may be determined directly using overshoot requirements and the study of the stability of FO systems. The tuning procedure is given for the general case of a class of unstable systems with pole multiplicity. The advantage of the proposed FO controller consists in the simplicity of the tuning approach. The case study considered in this brief consists in a magnetic levitation system. The experimental results provided show that the designed controller can indeed stabilize the magnetic levitation system, as well as provide robustness to modeling uncertainties and supplementary loading conditions. For comparison purposes, a simple PID controller is also designed to point out the advantages of using the proposed FO controller