Unstable Poles Research Articles

On Feedback Stabilization of Neutral Time Delay Systems with Infinitely Many Unstable Poles

It is shown that strictly proper neutral time delay systems having at least one asymptotic pole chain converging to a vertical line Re(s) = σ > 0 cannot be stabilized by a proper controller. A special coprime factorization can be obtained for such systems with infinitely many unstable poles, provided they are bi-proper (i.e. proper, but not strictly proper). From this factorization, stabilizing feedback controllers are obtained. Necessarily, all stabilizing controllers for this type of plants are also bi-proper.

Stable and Robust Controller Synthesis for Unstable Time Delay Systems via Interpolation and Approximation

In this paper, we study the robust stabilization of a class of single input single output (SISO) unstable time delay systems by stable and finite dimensional controllers through finite dimensional approximation of infinite dimensional parts of the plant. The plant of interest is assumed to have finitely many non-minimum phase zeros but is allowed to have infinitely many unstable poles in the open right half plane. Conservatism of the proposed methods is illustrated by numerical examples for which infinite dimensional strongly stabilizing controllers are derived in the literature.

Modified PI controller for the stabilization of high-order unstable delayed systems with complex conjugate poles and a minimum phase zero

This work deals with the stabilization problem of a particular class of high-order unstable linear systems with time delay. In particular, systems with one unstable pole, q — 1 complex conjugate stable poles and one minimum phase zero. To solve this problem a modified version for the traditional PI controller, called in this work the Proporcional Integral Filtered PIf, is proposed. This new scheme includes a low-pass first order filter and allows improving the existing results on controlling high-order systems with time delay. Necessary and sufficient conditions for the existence of the PIf controller are expressed in terms of the parameters of the system and the maximum allowable time-delay magnitude. The proposed control scheme is illustrated by a numerical example applied to the temperature control of an unstable Continuously Stirred Tank Reactor (CSTR) linear model focused on the production of propylene glycol.

Performance analysis of networked control systems over AWGN fading channels

Abstract In this paper, the optimal performance of multi-input and multi-output (MIMO) linear time-invariant (LTI) discrete-time networked control systems (NCSs) are investigated based on state-space representation. In communication channels, channel fading, additive white Gaussian noise (AWGN) and scaling parameter matrix are considered. An average performance index is constructed. Moreover, the control energy and the output energy of the system are considered simultaneously in the output regulation performance. The scaling parameter matrix is introduced, which may be considered as an encoder or modification on the channel. The optimal regulation performances of the system are obtained, and the relationship is given between the optimal regulation performances and the steady-state solution of the corresponding algebraic Riccati equation (ARE). Results indicate that the optimal performance is influenced by the non-minimum phase zeros, unstable poles, the scaling parameter matrix, and the characteristics of the communication noise and fading. In addition, the channel power constraint is considered in Theorem 2 . We introduce a novel power weighting factor, which can weigh the relative importance of system output and channel input energy. The obtained results will give some guidance for the design of NCSs. Finally, a simulation example is given to demonstrate effectiveness of the theoretical results.

On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays

The paper contributes to stabilization problems of linear systems subject to time-varying delays. Drawing upon small gain criteria and robust analysis techniques, upper and lower bounds on the largest allowable time-varying delay are developed by using bilinear transformation and rational approximates. The results achieved are not only computationally efficient but also conceptually appealing. Furthermore, analytical expressions of the upper and lower bounds are derived for specific situations that demonstrate the dependence of those bounds on the unstable poles and nonminumum phase zeros of systems.

PD–PID controller for delayed systems with two unstable poles: a frequency domain approach

ABSTRACTThis paper deals with the stability problem of linear delayed systems containing two unstable real poles by means of PD controllers. The analysis presented is based on frequency domain techniques. Necessary and sufficient conditions for the existence of stabilising controllers are given in terms of the parameters of the system and the time delay size. The main result is extended to delayed systems with two unstable poles and n stable real poles. PID controllers are also considered in order to control the studied systems, obtaining similar stability conditions. Numerical examples are presented in order to illustrate the control performance.

MAS Consensus and Delay Limits Under Delayed Output Feedback

In this technical note, we study the consensus problem for discrete-time multi-agent systems over an undirected, fixed network communication graph, focusing on the robustness of consensus with respect to communication delay. We assume that the agents' input is subject to a constant albeit possibly unknown time delay, and is generated by a distributed dynamic output feedback control protocol. Drawing upon concepts and techniques from robust control, notably those concerning gain margin optimization and analytic interpolation, we derive explicit, closed-form conditions for general linear agents to achieve consensus. Our results display an explicit dependence of the consensus condition on the delay value as well as on the agent's unstable poles and non-minimum phase zeros, showing that delayed communication between agents will generally hinder consensus and impose restrictions on the network topology. We also show that a lower bound on the maximal delay allowable for consensus can be computed by a simple line search method.

Active control of transonic buffet flow

Transonic buffet is a phenomenon of aerodynamic instability with shock wave motions which occurs at certain combinations of Mach number and mean angle of attack, and which limits the aircraft flight envelope. The objective of this study is to develop a modelling method for unstable flow with oscillating shock waves and moving boundaries, and to perform model-based feedback control of the two-dimensional buffet flow by means of trailing-edge flap oscillations. System identification based on the ARX algorithm is first used to derive a linear model of the input–output dynamics between the flap rotation (the control input) and the lift and pitching moment coefficients (system outputs). The model features a pair of unstable complex-conjugate poles at the characteristic buffet frequency. An appropriate reduced-order model (ROM) with a lower dimension is further obtained by a balanced truncation method that keeps the pair of unstable poles in the unstable subspace but truncates the dynamics in the stable subspace. Based on this balanced ROM, two kinds of feedback control are designed by pole assignment and linear quadratic methods respectively. These independent designs, however, result in similar suboptimal static output feedback control laws. When introduced in numerical simulations, they are both able to completely suppress the buffet instability. Furthermore, the resulting controllers are even able to stabilize buffet flows with nonlinear disturbances and in off-design flow conditions, thus implying their robustness. The analysis of the feedback control laws indicates that parameters (frequency and phase) corresponding to the ‘anti-resonance’ of the linear input–output model are vital for optimal control. The best performance is obtained when the control operates close to the ‘anti-resonance’, which is supported by the optimal frequency and the phase of the open-loop control as well as by the optimal phase of the closed-loop control.

Performance limitations for LTI discrete-time systems in the presence of stochastic disturbance

ABSTRACTThis paper concerns the fundamental limitations of disturbance rejection for linear time-invariant discrete-time systems under the assumption that the sensor noise is ignorable. The environmental disturbance is modelled as a linear combination of a step signal, a stationary stochastic signal and several sinusoidal signals with different frequencies. Disturbance rejection is measured by the steady-state variance of the residual input of the plant. In studying this problem, it is shown that for minimum phase systems, perfect disturbance rejection can be achieved by minimising the closed-loop sensitivity function. Unlike the minimum phase systems, the performance limitations for non-minimum phase systems are lower bounded by its non-minimum phase zeros, unstable poles and characteristic quantities of the disturbance, where the typical and integral-type control are respectively addressed. Numerical examples are provided to illustrate the utility of the results by analysing the achievable performance.

Effects of Delays, Poles, and Zeros on Time Domain Waterbed Tradeoffs and Oscillations

This letter aims for a simple and accessible explanation as to why oscillations naturally arise due to tradeoffs in feedback systems, and how these can be aggravated by delays and unstable poles and zeros. Such results have been standard for decades using frequency domain methods, which yield a rich variety of familiar “waterbed” tradeoffs. While almost trivial for control experts, frequency domain methods are less familiar to many scientists and engineers who could benefit from the insights such tradeoffs can provide. So here we present an entirely time domain model using discrete time dynamics and l 1 norm performance. A simple waterbed effect is that imposing zero steady state response to a step naturally create oscillations that double the response to periodic disturbances. We also show how this tradeoff is further aggravated not only by unstable poles and zeros, but also delays, in a way clearer than in the frequency domain versions.

Preactuated multirate feedforward for a high-precision stage with continuous time unstable zeros

A plant with unstable zeros is known to be difficult to control because of initial undershoot of step response and unstable poles in its inversion system. There are two reasons why a plant has unstable zeros in the discrete time domain: 1) non-collocation of actuators and sensors, 2) discretization by zero-order-hold. Problem 2) has been solved by the perfect tracking control (PTC) method based on multirate feedforward control proposed by our research group. However, the conventional PTC method cannot achieve the perfect tracking for a plant with continuous time unstable zeros generated by 1) because of the divergence of the desired state trajectories. Preactuation Perfect Tracking Control (PPTC) method proposed by our group solves problem 1) by the state trajectory generation based on a time axis reversal. This paper verifies the effectiveness of PPTC by simulations and experiments in comparison with several single rate feedforward control methods.

Input Shaper Optimization with a Constraint on the Spectrum Distribution

This paper presents a procedure to parametrize input shapers with piecewise equally distributed time delays. The procedure seeks to minimize residual vibrations around an undesirable frequency but at the same time avoids introducing zeros in the right half plane. This ensures that the introduction of the inverse of the input shaper in a closed feedback loop does not introduce unstable poles in the system. The requirement of stable spectra for the input shaper appears as an additional constraint in a constrained optimization problem that includes minimizing the response time and residual vibrations while preserving required properties of the shaper. The novel introduction of a spectral constraint makes the optimization problem nonsmooth and nonconvex, necessitating special optimization algorithms for its reliable solution. The framework is outlined in detail and the results of the optimization are presented indicating the success of the procedure.

Consensus problems for discrete-time agents with communication delay

In this paper, consensus problems for first-order multi-agent systems with unstable pole are investigated over undirected network, where the network is affected by communication delay. By jointly utilizing both the delayed distributed state information and agent’s own historical input information in the protocol, sufficient condition related to agent dynamic, network topology and communication delay is obtained for consensus, which is also proven to be necessary in case of one step delay. Furthermore, the consensus problem is studied on condition that the delay is time-varying. Finally, numerical simulations are provided to demonstrate the effectiveness of the proposed theoretical results.

An alternative method for local surveillance of coolant flow in BWRs fuel channels

A new method to perform “local surveillance” of fuel channels is proposed, it is based on the reduced model of March-Leuba for BWR dynamics along with the measurements of the characteristic frequency of the neutronic signal in steady state. The steam voids are one of the main characteristics in BWR reactors and their speed is a good indicator to verify that there is not an abnormal restriction blocking the coolant flow through the fuel channel. Previous analysis using the March-Leuba reduced model have been performed by using the transit time of steam voids through the fuel assemblies, in those analysis the whole length of the fuel assembly was considered. In this study, a “local surveillance” of the coolant flow is proposed by assessing the measurements of different monitors allocated at different levels of the fuel assemblies. As a new characteristic for the proposed “local surveillance”, the Prony’s method is applied to obtain the frequency from the more unstable pole, which is used to obtain the transit time of steam voids. In addition, a rational approximation is used because the original Padé (2,2) approximation considered in the March-Leuba original reduced model does not follow the asymptotic behavior of the exponential in the range of local monitors. To verify the proposed method LPRM signals are analyzed from an actual power plant and the results are compared against the results obtained by using the method of first minimum of the transfer function between two detectors of neutronic flux, which are placed at different levels. Results show agreement between the first minimum of the transfer function and the proposed local surveillance method (using the rational approximation). However, the proposed local surveillance method shows transit time differences smaller than 5% for different APRMs inasmuch the method of the first minimum of the transfer function shows transit time differences up to 22% for different LPRMs. It is here proposed to use both methods for the reactor characterization and to verify if there is or is not an abnormal restriction blocking the coolant flow through the fuel channel.

Three-axis gap clearance I-PD controller design based on coefficient diagram method for 4-pole hybrid electromagnet

ABSTRACT4-pole hybrid electromagnetic systems have a potential usage in many industrial areas, such as clean room design, transportation, semi-conductor manufacturing due to providing mechanical contact-free operation with considerably low energy consumption. However, the main problem of magnetic levitation process: it has highly nonlinear nature and even if it can be linearized, it has unstable pole(s), which makes the system vulnerable in terms of stability. In this paper, to overcome the instability issue and track the desired references for each degree of freedom, a modified PID controller (so called I-PD) design technique based on coefficient diagram method (CDM) has been proposed. CDM is an algebraic design applied to polynomial structure of the system on the parameter space, where a specific diagram is used to present and interpret the essential data. It is quite simple to apply with a visual support, requires basic mathematical computations for field engineers, and offers a good equilibrium in terms of simplicity, stability, minimum overshoot and robustness, which are crucial specifications for maglev applications. The effectiveness and feasibility of CDM-based I-PD controller have been compared with CDM-based classical PID controller over an experimental set-up.

Stable Simulation of Multiport Passive Distributed Networks Using Time Marching Method

In this paper, we present a stable method for simulating passive distributed networks connected to linear and nonlinear devices using the time marching (TM) method. The method is based on the use of the Z -transform for analyzing and obtaining the TM equations, and the application of cepstrum for preventing the apparition of unstable poles in the Z -domain counterparts of the TM equations. The efficiency of the proposed method is demonstrated by comparing its results with simulations obtained using the following: 1) a full-wave FDTD approach, and 2) CST Cable Studio.

A uniform LMI formulation for tuning PID, multi-term fractional-order PID, and Tilt-Integral-Derivative (TID) for integer and fractional-order processes

In this paper first the Multi-term Fractional-Order PID (MFOPID) whose transfer function is equal to ∑j=1Nkjsαj, where kj and αj are unknown and known real parameters respectively, is introduced. Without any loss of generality, a special form of MFOPID with transfer function kp+ki/s+kd1s+kd2sμ where kp, ki, kd1, and kd2 are unknown real and μ is a known positive real parameter, is considered. Similar to PID and TID, MFOPID is also linear in its parameters which makes it possible to study all of them in a same framework. Tuning the parameters of PID, TID, and MFOPID based on loop shaping using Linear Matrix Inequalities (LMIs) is discussed. For this purpose separate LMIs for closed-loop stability (of sufficient type) and adjusting different aspects of the open-loop frequency response are developed. The proposed LMIs for stability are obtained based on the Nyquist stability theorem and can be applied to both integer and fractional-order (not necessarily commensurate) processes which are either stable or have one unstable pole. Numerical simulations show that the performance of the four-variable MFOPID can compete the trivial five-variable FOPID and often excels PID and TID.

Mean square detectability of LTI systems over finite-state digital block-fading channels

This paper studies the mean square quadratic (MSQ) detectability for multi-output networked systems over finite-state digital block-fading channels. The packet-loss rate of each digital fading channel depends on the channel power gain, as well as packet length and power level used for transmission. A finite-state random process is introduced to model time-varying fading channels, which characterizes various configurations of physical communication environment and/or different channel fading amplitudes. Necessary and sufficient conditions for MSQ detectability over finite-state Markov digital block-fading channels are given in the form of algebraic Riccati equations or linear matrix inequalities (LMIs). The estimation gain is given as a function of estimated/observed channel state. In addition, explicit conditions on network for MSQ detectability over finite-state independent identically distributed (i.i.d.) digital block-fading channels are presented in terms of the unstable poles of the multi-output plant. Finally, an application to Gilbert-Elliott channels (GECs) is provided to demonstrate the derived results.

Fundamental Limits on Uncertain Delays: When Is a Delay System Stabilizable by LTI Controllers?

This paper concerns the stabilization of linear time-invariant (LTI) systems subject to uncertain, possibly time-varying delays. The fundamental issue under investigation, referred to as the delay margin problem, addresses the question: What is the largest range of delay such that there exists a single LTI feedback controller capable of stabilizing all the plants for delays within that range? Drawing upon analytic interpolation and rational approximation techniques, we derive fundamental bounds on the delay margin, within which the delay plant is guaranteed to be stabilizable by a certain LTI output feedback controller. Our contribution is threefold. First, for single-input single-output (SISO) systems with an arbitrary number of plant unstable poles and nonminimum phase zeros, we provide an explicit, computationally efficient bound on the delay margin, which requires computing only the largest real eigenvalue of a constant matrix. Second, for multi-input multi-output (MIMO) systems, we show that estimates on the variation ranges of multiple delays can be obtained by solving LMI problems, and further, by finding bounds on the radius of delay variations. Third, we show that these bounds and estimates can be extended to systems subject to time-varying delays. When specialized to more specific cases, e.g., to plants with one unstable pole but possibly multiple nonminimum phase zeros, our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and zeros, thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized by a LTI controller.

Control Under Stochastic Multiplicative Uncertainties: Part I, Fundamental Conditions of Stabilizability

In this two-part paper we study stabilization and optimal control of linear time-invariant systems with stochastic multiplicative uncertainties. We consider structured multiplicative perturbations, which, unlike in robust control theory, consist of static, zero-mean stochastic processes, and we assess the stability and performance of such systems using mean-square measures. While Part 2 of this paper tackles and solves optimal control problems under the mean-square criterion, Part 1 is devoted to the stabilizability problem. We develop fundamental conditions of mean-square stabilizability which ensure that an open-loop unstable system can be stabilized by output feedback in the mean-square sense. For single-input single-output systems, a general, explicit stabilizability condition is obtained. This condition, both necessary and sufficient, provides a fundamental limit imposed by the system's unstable poles, nonminimum phase zeros and time delay. For multi-input multi-output systems, we provide a complete, computationally efficient solution for minimum phase systems possibly containing time delays, in the form of a generalized eigenvalue problem readily solvable by means of linear matrix inequality optimization. Limiting cases and nonminimum phase plants are analyzed in depth for conceptual insights, revealing, among other things, how the directions of unstable poles and nonminimum phase zeros may affect mean-square stabilizability in MIMO systems. Other than their independent interest, stochastic multiplicative uncertainties have found utilities in modeling networked control systems pertaining to, e.g., packet drops, network delays, and fading. Our results herein lend solutions applicable to networked control problems addressing these issues.