Closed-loop Characteristic Equation Research Articles

Generalized Control Coupling Effect of Spinning Guided Projectiles

This article focuses on the analysis of generalized control coupling effect of spinning guided projectiles. A concept of so-called equivalent dynamics is first proposed in this article to avoid nonlinearity and facilitate understanding of control coupling for spinning guided projectiles. First, the coupling induced by an arbitrary-order dynamic is formulated. Two useful theorems showing the property of the concept are presented standby. Second, considering the main elements involved in the control loop, i.e., autopilot, actuator, and inertial measurement unit, a comprehensive model in the spinning frame is established for spinning guided projectiles. With aid of the second theorem, the equivalent dynamics of the multiple elements are introduced into the model, in which the closed-loop characteristic equation is described with the complex summation method. Then, the closed-loop complex steady gain is analyzed and an integrated decoupling is applied to eliminate the coupling effects completely. This can be implemented readily via presetting a leading angle in engineering. Furthermore, it is noted that for this high-order system, numerical method solving the characteristic equation is the only way to obtain dynamic stability conditions. Finally, a case study is conducted to demonstrate the effectiveness and robustness of the decoupling approach. In addition, the stability problem is figured out with numerical method.

The Effect of Replacing Si-MOSFETs with WBG Transistors on the Control Loop of Voltage Source Inverters

The operation of a voltage source inverter (VSI) depends on its output LC filter and the PWM modulator delay. The VSI model includes serial equivalent resistance based on the resistances of the active inverter bridge transistors, the filter coil winding, and additional PCB elements, such as traces and connectors, in addition to the large equivalent resistances that result from power losses in the coil core and switches. These dynamic power losses depend on the switching frequency and the inverter load. This paper investigates the change in equivalent serial resistance that occurs if the standard Si-MOSFET switches are replaced with wide bandgap (WBG) transistors with correspondingly lower equivalent serial resistance. The paper further investigates how such a change influences the design of the controller, given that replacement of the switches shifts the roots of the closed-loop characteristic equation. Theoretical analyses of the influence of equivalent serial resistance for a multi-input single-output passivity-based controller and a single-input single-output coefficient diagram method are also presented. These analyses are applied to both types of switches. Two methods are used to measure the serial equivalent resistance for a given VSI using Si and WBG switches. The possibility of replacing switches with WBG technology in existing inverters was assessed, and the corresponding controller adjustment that would be required. The theoretical analysis is verified via the use of an experimental VSI.

DS-Based 2-DOF PID Controller of a Regasification System for LNG-Fueled Marine Engine

As international regulations on greenhouse gas emissions are becoming stricter, the development of eco-friendly main engines is required for ocean-going ships. To use cryogenic LNG as a fuel for marine engines, it has to be converted to gas. This processing device is called an LNG regasification system. This paper presents a DS-based PID controller for glycol temperature control of a regasification system for LNG-fueled marine engines. In controller design, linear PID controllers have something of a catch-22 relationship: Fast response requires a large gain, which results in a large overshoot. To solve this problem, a DS-based 2-DOF PID controller is considered, which consists of a PID controller to reject disturbances in regulatory response and a set-point filter to reduce overshoot in servo response. The controller design focuses on improving disturbance rejection performance. The DS method is based on comparing the desired closed-loop characteristic equation with the closed-loop characteristic equation consisting of a control object and PID controller. The proposed controller is applied to the glycol temperature control of the regasification system for LNG-fuel marine engines, and its performance is compared with existing methods to show its effectiveness and applicability.

Design of Linear PID Controller for Pure Integrating Systems with Time Delay Using Direct Synthesis Method

The pure integrating process model with dead-time is inherently an unstable open-loop system. This paper presents a 2-DOF (2-degree of freedom) PID controller, based on the direct synthesis (DS) method, to improve the set-point tracking and load disturbance rejection performance for the pure integrating processes with time delays (PIPTD) models. The 2-DOF PID controller consists of a linear PID controller for rejecting the load disturbance in regulatory response and a set-point filter for relieving the overshoot in servo response. The DS method is based on comparing the desired closed-loop characteristic equation with the closed-loop characteristic equation, consisting of plant and PID controller. The adjustment parameter is only one desired time constant, with the roots of the desired characteristic equation placed at the same location, and the desired time constant was expressed as a function of the maximum magnitude of the sensitivity function (MS), for providing convenience in the selection of the desired time constant. The proposed controller is simulated by applying it to three other processes and comparing it with conventional methods, to demonstrate its effectiveness and applicability.

Desired Characteristic Equation Based PID Controller Tuning for Lag-Dominating Processes With Real-Time Realization on Level Control System

Objective of this study is to design proportional-integral-derivative (PID) controller with cascaded lag filter based on desired characteristic equation for lag-dominating processes. Here, attempt is made to formulate the closed-loop characteristic equation which comprises of multiple poles placed at the desired locations. Moreover, a guideline is also provided to choose the parameters of the closed-loop characteristic equation based on optimal values for performance and robustness indices. Efficacy of the proposed scheme is verified through simulation study with first-order plus dead time (FOPDT), second-order plus dead time (SOPDT), and pure integrating plus dead time (IPDT) processes in comparison with recently reported PID tuning relations. Robustness of the proposed controller is also tested in presence of perturbation for process model parameters. In addition, Kharitonov's interval polynomial is used for robust stability analysis. Real-time experimentation is also performed on a level control loop.

Stabilizing and Robust Fractional PID Controller Synthesis for Uncertain First-Order plus Time-Delay Systems

In this paper, by using the noninteger P I λ D μ controllers, we conduct an investigation into the subject of robust stability area of time-delay interval process. Our method is based on setting up of the noninteger interval closed-loop characteristic equation using the inferior and superior bounds of uncertain parameters into several vertexes. We have combined the composition of the value set of vertex with the zero exclusion principle to analyse the stability of the uncertain process. A generalized version of the Hermite–Biehler theorem, applicable to fractional quasipolynomials, is exploited to determine the stability region of each vertex. The robust stability region of the noninteger regulator can be given by the crossing of the stability area of all the vertex characteristic noninteger quasipolynomials. By using the value set method and zero exclusion theory, the effectiveness of the stability region can be tested. Also, we propose a suitable procedure to determine the whole of stabilizing parameters for an interval process. An explicative example is given to point out the advantage and reliability of the approach.

Coupling analysis and dynamic stability boundary of spinning missiles considering actuator dynamics and autopilot

This paper focuses on coupling property and dynamic stability of spinning missiles equipped with a two-loop autopilot. The closed-loop characteristic equation is established in the complex form, on which we investigated the dominant factors of the cross coupling, i.e., actuator dynamics, airframe configuration and autopilot, to provide physical understandings of coupling property. By coupling analysis, it is found that coupling can be mitigated by the autopilot. However, the proposed static decoupling approach is necessary to eliminate steady coupling completely. Furthermore, by solving the forth-order characteristic equations with Cardan formula, the dynamic stability boundary is derived numerically. As actuator dynamics is extremely significant to stability, the proposed stability boundary is derived with consideration of it, and thus, it is more practical than results in previous studies. The effectiveness of decoupling method and the correctness of dynamic stability boundary is verified by mathematical simulations. Theoretical and numerical results also reveal that high actuator bandwidth and less time delay is beneficial to dynamic stability.

Inclusion of Current Limiter Nonlinearity in the Characteristic Analysis of Sustained Subsynchronous Oscillations in Grid-Connected PMSGs

The interactions between permanent magnet synchronous generators (PMSGs) and weak grids may cause negatively damped or growing subsynchronous oscillation (SSO). The growing-SSO would evolve into a sustained-SSO when it is predominantly governed by the nonlinearities in the PMSG's converter control, such as current limiter. Most of the existing frequency-domain modeling and stability analysis methods are in the small-signal sense and thus unable to consider the impact of such nonlinearities. Besides, these methods can hardly give vital information to characterize the sustained-SSO, for instance, the magnitude. This paper presents a detailed characteristic analysis of the sustained-SSO while accounting for the effect of current limiter nonlinearity. First, a single-input-single-output (SISO) frequency-domain model of the PMSG is developed. This SISO model comprises the transfer functions of outer-/inner-loop controls and phase-locked loop (PLL). The nonlinear part, i.e., current limiter, is incorporated into the model using the describing function (DF) method. Then, the equivalent transformation on the system's closed-loop characteristic equation is used to separate the SISO model into independent linear and nonlinear parts. This separation enables the use of the DF-based Nyquist criterion for evaluating the system stability and determining oscillation attributes, such as the magnitude and frequency. Both the analytic results and time-domain simulations reveal that the current limiter has a significant impact on the characteristics of the sustained-SSO.

Approximate and exact stability analysis of 1-DOF and 2-DOF single-actuator real-time hybrid substructuring

This paper presents approximate and exact stability analysis to determine the critical time delays of one and two degree of freedom (DOF) single-actuator real-time hybrid substructuring (RTHS). The approximate stability analysis is based on applying the Routh-Hurwitz stability criterion using a first-order Taylor expansion of the actuator dynamics represented by a constant gain and pure time delay. The exact stability analysis involves solving the critical frequencies first by canceling the exponential delay term of the closed-loop characteristic equation through multiplication by its complex conjugate. The exact critical time delays are then found using the phase of the closed-loop characteristic equation evaluated at the critical frequencies. These stability analysis techniques are applied to several 1-DOF and 2-DOF mass-spring configurations of single-actuator RTHS. Results show that the critical time delays are highly dependent on the mass, stiffness, and damping partitioning of the physical and numerical substructures as well as the actuator gain. This information provides useful insight into the stability behavior of a particular substructure partitioning configuration to evaluate the feasibility of conducting RTHS tests of these systems.

Further Results on Dominant Pole Placement via Stability Mapping Approach

Dominant roots of the closed loop characteristic equation play a crucial role in terms of the performance of Linear Time Invariant (LTI) systems. Within the scope of this study, a dominant pole placement approach which has two main phases is proposed for PI/PID type controllers. In the first phase, characteristic equation is partitioned into its dominant and non-dominant polynomial pairs and dominant poles are placed to predetermined locations. In the second phase, it is required to determine how far the non-dominant poles can be placed. In the current study, this requirement is transformed into a stability problem and Lyapunov Equation-based stability mapping approach is used. This combined approach creates a more flexible design environment compared to the currently existing approaches in literature. In order to demonstrate this flexibility, two benchmark case studies are included with different definitions of dominant pole placement problem.

A robust fractional-order PID controller design based on active queue management for TCP network

ABSTRACTIn this paper, a robust fractional-order controller is designed to control the congestion in transmission control protocol (TCP) networks with time-varying parameters. Fractional controllers can increase the stability and robustness. Regardless of advantages of fractional controllers, they are still not common in congestion control in TCP networks. The network parameters are time-varying, so the robust stability is important in congestion controller design. Therefore, we focused on the robust controller design. The fractional PID controller is developed based on active queue management (AQM). D-partition technique is used. The most important property of designed controller is the robustness to the time-varying parameters of the TCP network. The vertex quasi-polynomials of the closed-loop characteristic equation are obtained, and the stability boundaries are calculated for each vertex quasi-polynomial. The intersection of all stability regions is insensitive to network parameter variations, and results in robust stability of TCP/AQM system. NS-2 simulations show that the proposed algorithm provides a stable queue length. Moreover, simulations show smaller oscillations of the queue length and less packet drop probability for FPID compared to PI and PID controllers. We can conclude from NS-2 simulations that the average packet loss probability variations are negligible when the network parameters change.

New Results on Eigenvalue Distribution and Controller Design for Time Delay Systems

This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with commensurate time delays and its application to proportional-integral-derivative (PID) controller design for a delay plant via dominant eigenvalue assignment. A new result on the root distribution of a quasi-polynomial is first produced by applying part of Pontryagin's conclusions. This result which gives a necessary and sufficient condition can be directly used to judge the number of the right-half plane eigenvalues of the characteristic equation of a time delay system. Based on the proposed result, necessary and sufficient conditions on dominant eigenvalue assignment for PID control of time delay systems are presented and an algorithm is then provided to determine the PID controller gains. The proposed approaches can assign some (one or two) eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones, which enables us to design the controller according to the desired performance indexes for a standard first-order system or a standard second-order system in addition to stability. The method is effective for the closed-loop characteristic equation being retarded type or neutral type. The controller gains to achieve the control objective can be characterized by a straightforward computation. Further, a result on degradation to proportional-integral (PI) control of time delay systems is given.

Research on Fuzzy Immune Self-Adaptive PID Algorithm Based on New Smith Predictor for Networked Control System

We first analyze the effect of network-induced delay on the stability of networked control systems (NCSs). Then, aiming at stochastic characteristics of the time delay, we introduce a new Smith predictor to remove the exponential function with the time delay in the closed-loop characteristic equation of the NCS. Furthermore, we combine the fuzzy PID algorithm with the fuzzy immune control algorithm and present a fuzzy immune self-adaptive PID algorithm to compensate the influence of the model deviation of the controlled object. At last, a kind of fuzzy immune self-adaptive PID algorithm based on new Smith predictor is presented to apply to the NCS. The simulation research on a DC motor is given to show the effectiveness of the proposed algorithm.

Robust fractional controller for interval first order system with time delay

In this paper, a new method is proposed to define the parameters of fractional order PIλ controller that stabilise an uncertain first order system with time delay. In fact, based on an extension version of Hermite-Biehler theorem and Pontryagin theorem the complete set stabilising PIλ parameters (KP, Ki, λ) is determined. The closed-loop characteristic equation of the system is a fractional order quasi-polynomial with interval uncertainty structure which is decomposed into several vertex. In the complex plane the value set of the vertex is a convex polygon. Simulation results are given to show the effectiveness of the proposed method.

A PID controller set of guaranteeing stability and gain and phase margins for time-delay systems

The stabilizing parameter sets and the guaranteed gain-margin (GM) and phase-margin (PM) regions of proportional-integral-derivative (PID) controllers for a class of processes with time-delay are discussed in the paper. The admissible range of stabilizing proportional-gain is first derived by a version of Hermite–Biehler Theorem and the evaluation of some properties of the functions involved in the closed-loop characteristic equation. Then, the stabilizing region in integral-derivative plane, for a fixed proportional-gain, is drawn and identified directly in terms of a graphical stability criterion applicable to time-delay systems. Further, in the stabilizing region, the gain-margin and phase-margin specifications are considered using the same strategy as drawing the stability boundary lines, based on the technique of gain-phase margin tester (GPMT). Illustrating examples are followed in each design step to show the effectiveness of the method.

Robust PID Force Control of Ionic Polymer Actuators

This paper presents a simple robust PID tuning method for force control of ionic polymer actuators. The model is represented by a linear-time invariant system which consists of electrical and electro-mechanical models. The uncertainty is represented by an interval polynomial set of the closed-loop characteristic equation. Using Kharitonov's theorem, we show that just one Kharitonov polynomial stability critevion is necessary and sufficient for satisfying the robust stability of the system. The PID gain is determined by pole placement based on the derived Kharitonov polynomial. Experimental results show the effectiveness of the PID force control by the proposed method.

PID Stabilization of Linear Neutral Time-Delay Systems in a Numerical Approach

In this paper, the stabilization of neutral time-delay systems is investigated. An efficient numerical approach is presented in an algorithm to establish results so that stability of such systems is achieved and stabilizing PID parameters are determined directly. It is based on determining the rightmost characteristic roots and Nyquist plot. The Newton-Raphson’s iterative method based on Lambert W function is used for the calculation of these stabilizing roots directly from the closed-loop characteristic equation of the neutral time-delay system and then stability is checked by Nyquist plot and step response of closed-loop system. Two numerical examples are included to illustrate the effectiveness of the proposed approach.

Design of decoupling Smith control for multivariable system with time delays

In order to solve the decoupling control problem of multivariable system with time delays, a new decoupling Smith control method for multivariable system with time delays was proposed. Firstly, the decoupler based on the adjoint matrix of the multivariable system model with time delays was introduced, and the decoupled models were reduced to first-order plus time delay models by analyzing the amplitude-frequency and phase-frequency characteristics. Secondly, according to the closed-loop characteristic equation of Smith predictor structure, proportion integration (PI) controllers were designed following the principle of pole assignment for Butterworth filter. Finally, using small-gain theorem and Nyquist stability criterion, sufficient and necessary conditions for robust stability were analyzed with multiplicative uncertainties, which could be encountered frequently in practice. The result shows that the method proposed has superiority for response speed and load disturbance rejection performance.

Stabilization of a Class of Linear Systems With Input Delay and the Zero Distribution of Their Characteristic Equations

This paper is concerned with stabilization of linear systems with arbitrarily large but bounded time-varying delay in the input. A pole assignment based low gain feedback design is adopted to solve the problem. Both delay-dependent and delay-independent results are presented and a series of sufficient conditions for guaranteeing the stability of the closed-loop system are established. By using properties of this class of low gain feedback laws and the properties of certain transcendental equations, distribution of the zeros of the closed-loop characteristic equation is described. As a result, a necessary and sufficient condition is identified that guarantees the stability of the closed-loop system. A numerical example is given to illustrate the effectiveness of the proposed approach.

Decoupling Smith Control for Multivariable System with Time Delays

AbstractIn order to solve the decoupling control problem of multivariable system with time delays, a new decoupling Smith control method for multivariable system with time delays was proposed. Firstly, the decoupler based on the adjoint matrix of the multivariable system model with time delays was introduced. And the decoupled models were reduced to first-order plus time delay models by analyzing the amplitude-frequency and phase-frequency characteristics. Secondly, according to the closed-loop characteristic equation of Smith predictor structure, PI controllers were designed following the principle of pole assignment for Butterworth filter. Finally, using small-gain theorem and Nyquist stability criterion, sufficient and necessary conditions for robust stability were analyzed with multiplicative uncertainties which encountered frequently in practice. The illustrative example was given to show the superiority of response speed and load disturbance rejection performance.