Dominant Pole Assignment Research Articles

Parametric Dynamic Distributed Containment Control of Continuous-Time Linear Multi-Agent Systems with Specified Convergence Speed

This paper focuses on the distributed containment control of continuous-time linear multi-agent systems (MASs) with multiple leaders over fixed topology. A parametric dynamic compensated distributed control protocol is proposed in which both the information from the observer in the virtual layer and actual adjacent agents are employed. The necessary and sufficient conditions of the distributed containment control are derived based on the standard linear quadratic regulator (LQR). On this basis, the dominant poles are configured by using the modified linear quadratic regulator (MLQR) optimal control and Geršgorin's circle criterion, hence the containment control with specified convergence speed of the MAS is achieved. Another main advantage of the proposed design is, in the case of virtual layer failure, by adjusting parameters the dynamic control protocol reduces to static, and the convergence speed can still be specified through the dominant pole assignment method combined with inverse optimal control. Finally, typical numerical examples are presented to demonstrate the effectiveness of theoretical results.

Design of decentralized proportional–integral proportional–retarded controllers in discrete-time domain for two-input two-output processes

In this paper, a decentralized proportional–integral proportional–retarded (PI-PR) controller design method is proposed for two-input two-output (TITO) systems in discrete-time domain. The well-known dominant pole assignment (DPA) approach is used as the basis of the proposed approach. The controller design starts with the decoupling of a given TITO system into two sub-systems and continues with the design of proportional–integral–retarded (PIR) controllers for each sub-system, respectively. The feasible discrete PIR controller parameter set is obtained through the Nyquist stability criterion by considering the desired closed-loop performance specifications. The obtained PIR controllers are then implemented using a PI-PR control structure to avoid poor performance of the closed-loop system (CLS) transient response, which can be caused by the controller zeros. Moreover, a case study is presented to show the performance of the PI-PR controller in a simulation environment. It is shown that the proposed control structure provides a satisfactory performance when compared with the other proportional–integral–derivative (PID) control methods from the literature.

Analytical design of discrete PI–PR controllers via dominant pole assignment

In this study, a new discrete proportional integral–proportional retarded (D-PI–PR) controller design method is proposed. The proposed controller involves replacing the PD controller in the classical PI–PD control structure with a PR controller in discrete time and is implemented using dominant pole assignment, which is carried out by help of the modified Nyquist plot approach. The method starts with the determination of the controller parameters Ki and Kr in terms of the parameter Kp for a chosen delay parameter h after the computation of the desired dominant poles depending on the required closed-loop system performance criteria. The next step is to find the feasible Kp interval, in which the dominant pole assignment is guaranteed. To eliminate the unwanted impact of the controller zeros on the transient response of the system, the D-PIR structure is then transformed into the D-PI–PR structure. For the demonstration of the success of the proposed algorithm, first, second and third-order systems with time delays are used and the simulation results are compared with some other PID-type controllers design methods from the literature.

Robust PID controller design via dominant pole assignment for systems with parametric uncertainties

AbstractIn this paper, a novel robust PID controller design technique is proposed for the parametric uncertain systems via the dominant pole assignment approach. In the closed‐loop, it is aimed that two poles are placed in the desired (dominant) region, and it is guaranteed that the remaining (unassigned) poles are located far away from the dominant pole region under all possible perturbations. The robust PID controller design technique is firstly given for the interval type characteristic polynomials with the help of vertex results. After that, the proposed method is generalized to cover the affine‐linear type characteristic polynomials. The method is based on the well‐known robust stability theorems and the generalized Nyquist theorem. The success of the proposed design technique is demonstrated on the control systems through simulation studies for both the interval and affine‐linear cases and compared with the other robust PID controllers from the literature. It is shown that the proposed robust PID controllers guarantee the desired pole configuration in the closed‐loop, and the closed‐loop performance specifications are satisfied even in the worst case.

Neutral PID Control Loop Investigated in Terms of Similarity Theory

A study to which extent of parameter setting the PID controller can be designed by means of dominant pole assignment for non-minimum-phase plant with delay is dealt with in the paper. Two options of the control loop are investigated, namely with and without the measurement filtering where the latter leads to a neutral time delay system which therefore has to satisfy the strong stability condition. The control loop is described in dimensionless quantities obtained by dimensional analysis. The influence of filtering on the loop dynamics is investigated particularly the effect of filter absence at all. This effect is demonstrated by the position of neutral root chain, i.e. how far from the origin lie the roots on a vertical line parallel to the imaginary axis of complex plane. To show boundaries of the dominant three-pole placement possibility in the control loop the trio of poles is placed repeatedly as long as the neutral root chain crosses the imaginary axis and the boundaries are mapped in the space of similarity numbers.

Dominant pole and eigenstructure assignment for positive systems with state feedback

In this paper, the dominant pole assignment problem, the dominant eigenstructure assignment problem and the robust dominant pole assignment problem for linear time-invariant positive systems with state feedback are considered. The dominant pole assignment problem is formulated as a linear programming problem, and the dominant eigenstructure problem is formulated as a quasiconvex optimisation problem with linear constraints. The robust dominant pole assignment problem is formulated as a non-convex optimisation problem with non-linear constraints which is solved using particle swarm optimisation (PSO) with an efficient scheme which employs the dominant eigenstructure assignment technique to accelerate the convergence of the PSO procedure. Each of the three problems can be further constrained by requiring that the controller has a pre-specified structure, or the gain matrix have both elementwise upper and lower bounds. These constraints can be incorporated into the proposed scheme without increasing the complexity of the algorithms. Both the continuous-time case and the discrete-time case are treated in the paper.

A Matrix Technique for Dominant Pole Placement of Discrete-Time Time-Delayed Systems

A matrix method is introduced for determination of robust-stability zones of the general linear time invariant discrete-time dynamics with large delays against parametric uncertainties. The technique employs Kronecker Product and unique properties of palindrome polynomials. These polynomials are subset of self-inversive polynomials which exert advantageous tools for examination of the distribution of its zeros. The main motivation in this paper is to develop a practical tool for determination of robust stability zones against parametric uncertainties and dominant pole assignment in discrete-time domain. A sufficient condition for robust stability and dominant pole assignment is presented. The procedure for the solution is demonstrated via some example case studies.

ULTIMATE-FREQUENCY BASED THREE-POLE DOMINANT PLACEMENT IN DELAYED PID CONTROL LOOP

Besides its original use in the state space based design the pole assignment is also applied in tuning the PID controllers where only an approximate model is considered. However, the common assumption of time delay in this model imposes the limitation that only such pole prescription which results in the dominant pole assignment can be effective in tuning the control loop with time delay. The three parameter PID controller is capable to assign just three poles and a dominance guarantee of their prescription is crucial in the pole placement. A novel method of selecting a trio of numbers p1,2,3 to make them the dominant poles of the control loop is dealt with in the paper with an additional minimizing the absolute error integral. The dominance of each of the placement trials is checked by an argument increment criterion and an optimum of relative damping of the response is assessed to minimize the control error integral. The disturbance rejection response as the critical point of view in time delay process control is taken as criterion in the controller design.

Designing control systems using exact and symbolic manipulations of formulae

The potential of symbolic algebra for the design of control systems is illustrated through several examples. It has been shown in particular, that (a) the exact calculations provided by symbolic algebra can be used to utilize some of the direct (but numerically error prone) methods efficiently in control system design, and (b) symbolic manipulations by a computer can help control engineers at several stages of the design. The block diagram reduction, calculation of all stabilizing controllers, dominant pole assignment and robust pole assignment are taken as case studies.

Optimizing fuzzy neural networks for tuning PID controllers using an orthogonal simulated annealing algorithm OSA

In this paper, we formulate an optimization problem of establishing a fuzzy neural network model (FNNM) for efficiently tuning proportional-integral-derivative (PID) controllers of various test plants with under-damped responses using a large number P of training plants such that the mean tracking error J of the obtained P control systems is minimized. The FNNM consists of four fuzzy neural networks (FNNs) where each FNN models one of controller parameters (K, T/sub i/, T/sub d/, and b) of PID controllers. An existing indirect, two-stage approach used a dominant pole assignment method with P=198 to find the corresponding PID controllers. Consequently, an adaptive neuro-fuzzy inference system (ANFIS) is used to independently train the four individual FNNs using input the selected 176 of the 198 PID controllers that 22 controllers with parameters having large variation are abandoned. The innovation of the proposed approach is to directly and simultaneously optimize the four FNNs by using a novel orthogonal simulated annealing algorithm (OSA). High performance of the OSA-based approach arises from that OSA can effectively optimize lots of parameters of the FNNM to minimize J. It is shown that the OSA-based FNNM with P=176 can improve the ANFIS-based FNNM in averagely decreasing 13.08% error J and 88.07% tracking error of the 22 test plants by refining the solution of the ANFIS-based method. Furthermore, the OSA-based FNNMs using P=198 and 396 from an extensive tuning domain have similar good performance with that using P=176 in terms of J.

Calculation of PID controller parameters by using a fuzzy neural network

Abstract In this paper, we use the fuzzy neural network (FNN) to develop a formula for designing the proportional-integral-derivative (PID) controller. This PID controller satisfies the criteria of minimum integrated absolute error (IAE) and maximum of sensitivity (M s). The FNN system is used to identify the relationship between plant model and controller parameters based on IAE and M s. To derive the tuning rule, the dominant pole assignment method is applied to simplify our optimization processes. Therefore, the FNN system is used to automatically tune the PID controller for different system parameters so that neither theoretical methods nor numerical methods need be used. Moreover, the FNN-based formula can modify the controller to meet our specification when the system model changes. A simulation result for applying to the motor position control problem is given to demonstrate the effectiveness of our approach.

Fuzzy neural networks for tuning PID controller for plants with underdamped responses

In this paper, the fuzzy neural network (FNN) for tuning proportional-integral-derivative (PID) controller for plants with underdamped step responses is proposed. The underdamped systems are modeled by second-order-plus-dead-time transfer functions. For deriving the FNN, the dominant pole assignment method is applied to design the PID controllers for a batch of test plant models that represent the plants with underdamped responses. Then, a fuzzy neural modeling method is utilized to identify the relationship between the parameters that characterize the plant dynamics and the controller parameters. We then utilize the obtained FNN to tune the PID controller for plants with underdamped responses. Several simulation examples are given to demonstrate the effectiveness and robustness of the FNN obtained.

State feedback control of time delay system: Conformal mapping aided design

State feedback based on state variables resulting from functional description of time delay system is dealt with while the modelling approach promotes possibility to select state variables primarily as available outputs. Due to this model concept the resulting state feedback is endowed with better implementation feasibility than in case of equivalent conventional state model. The state feedback design is a specific combination of dominant pole assignment with conformai mapping investigation of characteristic quasi-polynomial of the system. The design procedure is demonstrated on an application to a real heating system control.

NEW TUNING METHOD FOR PID CONTROL OF A PLANT WITH UNDER‐DAMPED RESPONSE

ABSTRACTIn this paper, a tuning formula is derived for PID control of plants with under‐damped step responses. The under‐damped systems are modeled by second‐order plus dead time transfer functions. To derive the tuning rule, the dominant pole assignment method was applied to design the PID controllers for a variety of plant models. Then, the correlation between the controller parameters and the parameters of the models was found and the tuning formula derived. Several simulation examples are given to demonstrate the effectiveness and robustness of this formula.

On robust eigenvalue configuration

This study focuses on a class of disconnected subsets of the complex plane, of interest in the context of dominant pole assignment and filter design. It is first observed that the robust stability conditions originally put forth are in fact necessary and sufficient for the number of eigenvalues (matrices) or zeros (polynomials) in any given connected component to be the same for all the members of the given family. Polynomic semiguardian maps are then identified for a class of disconnected regions of interest. These maps are in fact essentially guarding with respect to one-parameter families.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Polytopes of polynomials with zeros in a prescribed set

In the publication by A.C. Bartlett, C.V. Holot, and H. Lin (Proc. Amer. Contr. Conf., Minneapolis, MN, 1987) a fundamental result is established on the zero locations of a family of polynomials. It is shown that the zeros of a polytope P of nth-order real polynomials are contained in a simply connected set D if and only if the zeros of all polynomials along the edges of P are contained in D. The present authors are motivated by the fact that the requirement of simple connectedness of D may be too restrictive and applications such as dominant pole assignment and filter design where the separation of zeros is required. They extend the edge criterion of Bartlett et al. to handle any set D whose complement D/sup c/ has the following property: every point D in D/sup c/ lies on some continuous path which remains within D/sup c/ and is unbounded. This requirement is typically verified by inspection and allows for a large class of disconnected sets. >

One extending hypersphere method to handle dominant pole assignment

It is pointed out that the stability hypersphere approach presented by C.B. Soh et al. (see ibid vol.AC-32, p.239-244, Mar. 1987), can be extended to handle the case where the region is not simply connected. The extension is useful in application such as dominant pole assignment and filter design where the separation of zeros is required. >

主要極配置によるI-PD型補償器の設計法

主要極配置によるI-PD型補償器の設計法