Open-loop Poles Research Articles

Input delay margin for consensusability of multi-agent systems

In this paper, we are concerned with the consensus of multi-agent systems with input delay. Among all standard static protocols that achieve the consensus for the multi-agent system under no input delay, we aim to find the maximum input delay such that the system remains consensusable under the same protocols. In the case of continuous-time systems, in view of the continuity of stability with respect to the time delay, the maximum delay margin for consensusability is given for scalar systems and vector systems with a single unstable open-loop pole. For scalar discrete-time systems, we show that the maximum delay margin for consensusability is strictly greater than zero if and only if the open-loop pole of the system is located in a specified interval.

CONTRAPOSITION 2-REGULARIZED NYQUIST STABILITY CRITERIA FOR LINEAR CONTINUOUS-TIME PERIODIC SYSTEMS

By employing the 2-regularized determinant technique about Hilbert-Schmidt operators and creating what we call the contraposition return difference relationship for finite-dimensional linear continuous-time periodic (FDLCP) feedback systems, a contraposition 2-regularized Nyquist stability criterion is developed for asymptotic stability analysis in this study. The stability conditions of the contraposition 2-regularized Nyquist stability criterion are necessary and sufficient, which can be implemented graphically in a way similar to what we do in linear time-invariant cases but without involving open-loop poles distribution and encircling orientation counting of the Nyquist loci. A numeric implementation algorithm for the contraposition 2-regularized Nyquist criterion is also proposed via truncating the harmonic transfer operator of the FDLCP systems.

Pole assignment using state feedback with time delay in friction-induced vibration problems

The dynamic behavior of friction-induced vibration problems is governed by second-order differential equations having asymmetric matrices, due to the coupling of structures and external loads, which are functions of some parameters. Asymmetric systems are prone to unstable vibration (flutter) as a parameter reaches a critical value. Placement of these unstable poles to the left-hand half of the complex pole plane for stabilization can be achieved by active (feedback control) and passive means (structural modification). Moreover, placement of poles is also done to achieve the desirable dynamic response and system performance. However, such active pole assignment control introduces inherent time delays in the feedback control loop. This paper presents a method for assigning complex poles to second-order damped asymmetric systems by using state-feedback control while considering a constant time-delay in the feedback control loop. The control strategy is based on receptances of the symmetric part of the asymmetric open-loop system (without time delay), which can be easily obtained from transfer function measurements. This method does not require the knowledge of mass, damping and stiffness matrices, and hence circumvents the modeling errors (finite element or reduced order). In this research, with numerical examples, it is shown that by means of active state feedback control and by using a relatively small number of available receptances, open-loop poles of the asymmetric system can be assigned precisely. The stability of the closed-loop system is analyzed by computing primary closed-loop poles and the associated critical time delay.

PID controllers for a class of unstable linear time-delay systems: An eigenvalue-loci approach

This paper studies linear time-invariant systems with an input delay and only real open-loop poles. The systems in discussion have one unstable pole but no zeros. To design P, PD, PI and PID-controllers, the closed-loop system eigenvalue-loci are constructed with respect to the controller gains, like the root-locus for non-delay systems. Via the concept of leading locus, stability criteria are derived for the controllers, and performances of output responses are analyzed. Finally the proposed methods are tested with examples to illustrate the application process and demonstrate the effectiveness.

A Field Test to Determine PSS Effectiveness at Multigenerator Power Plants

This paper presents a new field test to adequately assess the oscillation damping effectiveness of the PSSs in a multigenerator power plant. The test allows estimating the location of the dominant open-loop poles (i.e., in the absence of PSSs at the multigenerator plant) from frequency response measurements of the closed-loop multivariable system. This test was first carried out at the Itaipu 60-Hz power station, which was generating 4000 MW at the time. The test validation was made through comparison of obtained frequency response measurements with computer simulations of a detailed dynamic model of the Brazilian interconnected power system.

Optimal pole shifting controller for interconnected power system

Power system stabilizer based on optimal pole shifting is proposed. An approach for shifting the real parts of the open-loop poles to any desired positions while preserving the imaginary parts is presented. In each step of this approach, it is required to solve a first-order or a second-order linear matrix Lyapunov equation for shifting one real pole or two complex conjugate poles, respectively. This presented method yields a solution, which is optimal with respect to a quadratic performance index. The attractive feature of this method is that it enables solutions of the complex problem to be easily found without solving any non-linear algebraic Riccati equation. The present power system stabilizer is based on Riccati equation approach. The control law depends on finding the feedback gain matrix, and then the control signal is synthesized by multiplying the state variables of the power system with determined gain matrix. The gain matrix is calculated one time only, and it works over wide range of operating conditions. To validate the power of the proposed PSS, a linearized model of a simple power system consisted of a single synchronous machine connected to infinite bus bar through transmission line is simulated. The studied power system is subjected to various operating points and power system parameters changes.

Optimal pole assignment for discrete-time systems via Stein equations

This study is concerned with designing a feedback gain to minimise a quadratic performance index with guaranteed pole locations for closed-loop discrete-time linear systems. Firstly, a method that shifts the open-loop poles to desired locations by using a parametric linear Stein equation is presented. Then a recursive approach is proposed to shift every eigenvalue of a discrete-time linear system separately without mode decomposition in each step. By using such method, it is required to solve a linear Stein matrix equation of low order in each step. The presented method yields a solution which is optimal with respect to a quadratic performance index that can be obtained explicitly. The attractive feature of this method comparing with existing results is that it enables solutions to complex problems to be easily found without solving any non-linear algebraic Riccati equations. Moreover, analytical solutions can be obtained which may have advantages in some design problems. Numerical examples illustrate the proposed approach.

Transmission Zeros in Structural Control with Collocated Multi-Input/Multi-Output Pairs

I N ACTIVE vibration control of flexible structures with a singleinput/single-output (SISO) collocated (dual) actuator/sensor pair, the transmission zeros exhibit the well-known property of interlacing, that is, the poles and zeros alternate along the imaginary axis (strictly if the structure is undamped, slightly in the left-half plane for a lightly damped structure). This property is the origin of the guaranteed stability of the so-called low authority control (LAC) strategies for active damping [1,2]. Besides, the transmission zeros coincide with the poles (natural frequencies) of a modified system which depends on the sensor configuration. For a displacement (velocity) sensor, the modified system is the constrained system where the degree of freedom (DOF) alongwhich the control operates is blocked [3,4]. For a force sensor, the modified system is obtained by removing the contribution of the active member to the global stiffness matrix of the structure [2]. As the gain increases, the closedloop poles start from the open-loop poles and those which remain at finite distance move on loops going asymptotically to the transmission zeros. For displacement sensors, the extension of the SISO result to multi-input/multi-output (MIMO) collocated pairs has already been discussed in the literature [5] and, based on the fact that transmission zeros give identically zero output response, it has been inferred that, for MIMO systems, the transmission zeros are the eigenvalues of an associated constrained modes problem. To the authors’ knowledge, however, no formal proof is available. This Note provides such a proof for an undamped structure; the two cases (force actuator, displacement sensor and displacement actuator, force sensor) are discussed separately. II. Force Actuator, Displacement Sensor

On Asymptotic Stabilizability of Linear Systems With Delayed Input

This paper examines the asymptotic stabilizability of linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open-loop system is not exponentially unstable (i.e., all the open-loop poles are on the closed left-half plane). A simple example shows that such results would not be true if the open-loop system is exponentially unstable. It is further shown that such systems, when subject to actuator saturation, are semiglobally asymptotically stabilizable by linear state or output feedback.

On Asymptotic Stabilizability of Discrete-time Linear Systems with Delayed Input

This paper examines the asymptotic stabilizability of discrete-time linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable able and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable (i.e., all the open loop poles are on or inside the unit circle.) It is further shown that such a system, when subject to actuator saturation, is semi-globally asymptotically stabilizable by linear state or output feedback.

EFFECTS OF TIME DELAY ON FEEDBACK STABILISATION OVER SIGNAL-TO-NOISE RATIO CONSTRAINED CHANNELS

The expanding integration of control and communication networks in recent years has generated an increasing interest in control problems with feedback over a communication channel. Significant research activity has concentrated on stabilisation in face of channel effects such as quantisation and data-rate limits. In a recent paper, the authors have studied the problem of feedback stabilisation over a communication channel with a constraint on the admissible signal-to-noise ratio (SNR). It has been shown therein that for a delay-free, linear time-invariant feedback loop, a SNR constraint in the feedback channel imposes fundamental limitations in the ability to achieve closed-loop stability. The present paper extends these results by introducing a time-delay in the loop, and shows that the lowest SNR required for closed-loop stability increases by a factor that may grow exponentially with the time-delay and the unstable open loop poles of the system. This result contributes to the quantification of performance tradeoffs in integrated control and communication environments.

Optimal pole shifting for power system stabilization

A method for shifting the real parts of the complex open-loop poles to any desired positions while preserving the imaginary parts is presented. In each step of this approach, it is required to solve a first-order or a second-order linear matrix Lyapunov equation for shifting one real pole or two complex poles, respectively. This presented method yields a solution, which is optimal with respect to a quadratic performance index. The attractive feature of this method is that it enables solutions to complex problem to be easily found without solving any non-linear algebraic Riccati equation. The gain feedback is calculated one time only and it works over wide range of operating conditions. The digital computation results verify the effectiveness of the proposed PSS for less overshoot and less settling time compared with the open-loop techniques. Moreover, the damping and synchronizing torques of the synchronous machine with and without the proposed robust PSS are calculated.

Improved PI controller with delayed or filtered integral mode

AbstractIntegral action is almost always included in process control systems to eliminate steady‐state offset without uncertain process gain. The open‐loop pole, however, at the origin of the integral term causes some problems such as integral windup. Various methods to solve these problems were studied. For better control performance and robustness, a filter was added to the integral term, which decouples the effective frequency ranges between the integral and proportional terms without degradation of the integral action. It produces a phase lead in a certain frequency range without having a derivative term, enhancing the control performances and stability robustness. Based on the internal model control method or the direct synthesis method, tuning rules for the proposed controller are given.

Arbitrarily fast and robust tracking by feedback

The output of a singe-input-single-output linear feedback system with more than one pole in excess over the zeros in the loop transmission cannot track arbitrarily fast its input (by the root locus). In this work we extend the linear feedback so that some of the open loop poles may depend on the open loop gain; we call this new class quasi-linear feedback systems. We then derive time domain, pole-zero, and frequency domain conditions which ensure arbitrarily fast and robust tracking by quasi-linear feedback, for an arbitrary number of poles in excess over the zeros. We prove that in a particular case these conditions are equivalent, and that the boundedness in frequency of the closed loop transfer function is no longer necessary for achieving arbitrarily fast tracking. The robustness is to external disturbances and initial conditions, and the open loop has to be minimum phase. Some examples are presented which illustrate these results. They also show that this good performance can be obtained with a reduced control effort, and that quasi-linear feedback can alleviate the limitation on performance of non-minimum phase open loops.

Constrained pole placement for linear systems using low-order output feedback controllers

The paper presents a simple approach to the problem of designing low-order output feedback controllers for linear continuous systems. The controller can place all of the closed-loop poles within a circle, C(- f , 1/ g ) , with centre at - f and radius of 1/ g in the left half s-plane. The design method is based on transformation of the original system and then applying the bounded-real-lemma to the transformed system. It is shown that subjected to the solvability of an algebraic Riccati equation (ARE), output feedback controllers can then be systematically derived. Furthermore, the order of the controller is low and equals only the number of the open-loop poles lying outside the circle. A step-by-step design algorithm is given. Numerical examples are given to illustrate the design method.

Tradeoffs in linear time-varying systems: an analogue of Bode's sensitivity integral

A new time-domain interpretation of Bode's integral is presented. This allows for a generalization to the class of time-varying systems which possess an exponential dichotomy. It is shown that the sensitivity function is constrained, on average, by the Lyapunov exponents of the open-loop system which take the place of the magnitude of the open-loop poles.

Semi-global stabilization of linear systems subject to output saturation

It is established that a SISO linear stabilizable and detectable system subject to output saturation can be semi-globally stabilized by linear output feedback if all its invariant zeros are in the closed left-half plane, no matter where the open loop poles are. This result complements a recent result that such systems can always be globally stabilized by discontinuous nonlinear feedback laws, and can be viewed as dual to a well-known result: a linear stabilizable and detectable system subject to input saturation can be semi-globally stabilized by linear output feedback if all its poles are in the open left-half plane, no matter where the invariant zeros are.

Interaction between thermal efficiency and dynamic controllability for heat-integrated reactors

The objective of this work is to quantify the relationship between thermal efficiency and dynamic controllability. For a given feed-effluent heat exchanger (FEHE), relationships between the degree of heat recovery and open-loop poles are derived. Then, similar to the ultimate gain in the control literature, the ultimate heat recovery can be derived. By ultimate heat recovery, we mean the amount of heat recovery such that the open-loop system is at the limit of stability (i.e. having poles at the imaginary axis). Since the controllability is characterized by the pole locations, achievable closed-loop performance of heat integrated systems can also be evaluated. Internal as well as external energy load disturbances are compared. The results clearly quantify the trade-off between the thermal efficiency and controllability. More importantly, the results presented here offer a simple way to synthesize a controllable heat integrated reactor.

Slow stable open-loop poles: to cancel or not to cancel

Stable plant poles can either be shifted by feedback, or they can be approximately cancelled by a controller. While it is known that the cancelling design impacts on the transients due to input disturbances, here we show that the alternative, i.e., the shifting design, impacts on closed-loop robustness. In this paper we consider control loops with integral action and show that shifting slow stable real poles, rather than cancelling them, increases the l 1, H ∞, and the point-wise in frequency norms of the complementary sensitivity. Thus, the choice to either cancel or shift any one stable pole emerges as a design issue that requires a deliberate trade-off between input disturbance response versus output disturbance response and robustness.

IMC Based Robust Pole Placement Procedure for Modal Control of One-Dimensional Heat Conduction Process.

This work details an IMC-based pole placement controller tuning procedure for a multi-loop SISO system in the presence of model uncertainty. For a given uncertainty level, selection of the IMC low-pass filter parameter is done using Kalman filter poles and open-loop poles as respective lower and upper limits. Tuning is facilitated using a convex tuning parameter, which depends on both the structured singular value as a quantitative performance measure and closed-loop behaviour like overshoots and manipulated variable move sizes as qualitative measures. The method is applied to a distributed parameter system using modal control, but can be generally applied for any multi-loop SISO system using a state estimator. Using simulation studies it is shown that the ambiguity encountered during selection of the IMC filter constant is significantly reduced.