Linear Time-invariant System Subject Research Articles

Hyperstability of Linear Feed-Forward Time-Invariant Systems Subject to Internal and External Point Delays and Impulsive Nonlinear Time-Varying Feedback Controls

This paper investigates the asymptotic hyperstability of a single-input–single-output closed-loop system whose controlled plant is time-invariant and possesses a strongly strictly positive real transfer function that is subject to internal and external point delays. There are, in general, two controls involved, namely, the internal one that stabilizes the system with linear state feedback independent of the delay sizes and the external one that belongs to an hyperstable class and satisfies a Popov’s-type time integral inequality. Such a class of hyperstable controllers under consideration combines, in general, a regular impulse-free part with an impulsive part.

Decentralized Voltage Stabilization and Robust Performance Satisfaction of Islanded Inverter-Interfaced Microgrids

In this article, an improved robust control strategy for voltage stabilization and desired performance satisfaction of islanded inverter-interfaced microgrids consisting of several distributed generations (DGs) with general topology is presented. The main advantages of the proposed control manner are as follows: 1) its structure is fully decentralized, 2) the design process is scalable, 3) it does not impose any limitations on the microgrid parameters and line dynamics, 4) it provides stability and the desired performance of nominal microgrid system, 5) it maintains robust stability as well as robust performance of the closed-loop system against microgrid topology changes, plug-and-play (PnP) operation of DG units, and load unmodeled dynamics, and 6) each local controller is obtained from a unique convex optimization problem which results in optimal performance of the system and also robustness to several sequential changes. To achieve these objectives, first, each DG subsystem is modeled as a linear time-invariant (LTI) system affected by disturbances caused by the local load current and the load voltages of its neighboring DGs. Next, the PnP functionality of DGs and the microgrid topology changes are modeled as a new polytopic-type uncertainty. Thereafter, the design problem is transformed into a dynamic output feedback controller for an LTI system subject to polytopic-type uncertainty with <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</b> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance criteria. Finally, a convex linear-matrix-inequality-based optimization problem with a noniterative direct synthesis process is proposed to solve the controller design problem. The performance of the presented controller is appraised via several simulation studies accomplished in MATLAB/SimPowerSystems Toolbox.

Delay Margin of Low-Order Systems Achievable by PID Controllers

This paper concerns the delay margin achievable using proportional-integral-derivative (PID) controllers for linear time-invariant (LTI) systems subject to variable, unknown time delays. The basic issue under investigation addresses the question: What is the largest range of time delay so that there exists a single PID controller to stabilize the delay plants within the entire range? Delay margin is a fundamental measure of robust stabilization against uncertain time delays and poses a fundamental, longstanding problem that remains open except in simple, isolated cases. In this paper, we develop explicit expressions of the exact delay margin and its upper bounds achievable by a PID controller for low-order delay systems, notably the first- and second-order unstable systems with unknown constant and possibly time-varying delays. The effect of nonminimum phase zeros is also examined. PID controllers have been extensively used to control and regulate industrial processes that are typically modeled by first- and second-order dynamics. While furnishing the fundamental limits of delay within which a PID controller may robustly stabilize a delay process, our results should also provide useful guidelines in tuning PID parameters and in the analytical design of PID controllers.

Robust Decomposition and Structured Control of an Islanded Multi-DG Microgrid

This paper presents a new robust control strategy for islanded operation of a microgrid consisting of several distributed generation (DG) units with arbitrary topology. Each DG unit is connected to its local load through a voltage-source converter, a series RL filter and a step-up transformer. The multi-DG microgrid with arbitrary connectivity graph is modeled as a linear time-invariant system subject to a parametric, norm-bounded uncertain block. By utilizing the developed model for multi-DG microgrids, a systematic approach for the design of structured dynamic output feedback control is proposed. A convex optimization problem is developed which yields a low-order dynamic controller of arbitrary structure such that the microgrid channels are all decoupled at the presence of load parameter uncertainties. Simulation case studies performed in MATLAB/SimPowerSystems Toolbox verify the performance of the proposed strategy.

Auxiliary Fault Tolerant Control With Actuator Amplitude Saturation and Limited Rate

In this paper, the problem of fault tolerant tracking control for a linear time-invariant system subject to actuator faults and saturations is addressed. An auxiliary system is developed to ensure actuators behave within amplitude and rate limits under the influence of partial loss of control effectiveness. Based on the auxiliary system, a fault tolerant compensation controller is constructed to guarantee tracking errors to converge to a small region. Some relationships among tracking errors, command signals, actuator faults, amplitude and rate limits as well as controller parameters are comprehensively studied and explicitly illustrated with formulas. An example of rudder-roll damping control for a cruise keeping ship is included to illustrate the proposed procedures and their effectiveness.

Lower Bounds on Delay Margin of Second-Order Unstable Systems

This paper examines the delay margin achievable by using PID controllers for linear time-invariant (LTI) systems subject to variable, unknown time delays. We derive explicit lower bounds on the delay margin of second-order unstable delay systems achievable by PID control, which provide a priori the ranges of delay over which a second-order delay plant is guaranteed to be stabilizable by a PID controller, and more generally, by a finite-dimensional LTI controller. Analysis shows that our results are less conservative in most cases than lower bounds obtained elsewhere using more sophisticated and general LTI controllers.

Control With Minimal Cost-Per-Symbol Encoding and Quasi-Optimality of Event-Based Encoders

We consider the problem of stabilizing a continuous-time linear time-invariant system subject to communication constraints. A noiseless finite-capacity communication channel connects the process sensors to the controller/actuator. The sensor's state measurements are encoded into symbols from a finite alphabet, transmitted through the channel, and decoded at the controller/actuator. We suppose that the transmission of each symbol costs one unit of communication resources, except for one special symbol in the alphabet that is “free” and effectively signals the absence of transmission. We explore the relationship between the encoder's average bit-rate, its average consumption of communication resources, and the ability of the controller and encoder/decoder pair to stabilize the process. We present a necessary and sufficient condition for the existence of a stabilizing controller and encoder/decoder pair, which depends on the encoder's average bit-rate, its average resource consumption, and the unstable eigenvalues of the process. Moreover, if this condition is satisfied, a stabilizing encoding scheme can be constructed that consumes resources at an arbitrarily small rate, provided the encoder has access to a sufficiently precise clock or large memory. The paper concludes with the analysis of a simple emulation-based controller and event-based encoder/decoder pair that are easy to implement, stabilize the process, and have average bit-rate and resource consumption within a constant factor of the optimal bound.

Plug-and-Play Voltage Stabilization in Inverter-Interfaced Microgrids via a Robust Control Strategy

This paper proposes a decentralized control strategy for the voltage regulation of islanded inverter-interfaced microgrids. We show that an inverter-interfaced microgrid under plug-and-play (PnP) functionality of distributed generations (DGs) can be cast as a linear time-invariant system subject to polytopic-type uncertainty. Then, by virtue of this novel description and use of the results from theory of robust control, the microgrid control system guarantees stability and a desired performance even in the case of PnP operation of DGs. The robust controller is a solution of a convex optimization problem. The main properties of the proposed controller are that: 1) it is fully decentralized and local controllers of DGs that use only local measurements; 2) the controller guarantees the stability of the overall system; 3) the controller allows PnP functionality of DGs in microgrids; and 4) the controller is robust against microgrid topology change. Various case studies, based on time-domain simulations in MATLAB/SimPowerSystems Toolbox, are carried out to evaluate the performance of the proposed control strategy in terms of voltage tracking, microgrid topology change, PnP capability features, and load changes.

An analysis and design framework for robust control of a multi-axis active vibration isolation system with unknown payload

This paper presents a framework for model-based analysis of robust stability and performance for a multi-axis active vibration isolation system with constant but unknown payload and subject to modelling errors associated with structural flexibility. The theoretical treatment involves a linear time-invariant system subject to real parameter uncertainty associated with the unknown payload. A set of performance indices are formulated based on generalized () and measures. A method for stability/performance verification is then developed using a parameter-dependent Lyapunov function that incorporates the kinetic energy of the uncertain payload mass. This allows nonconservative bounds on the performance indices to be established via numerical solution of a corresponding set of matrix inequalities. The approach is especially suitable, and computationally efficient, for multi-degree-of-freedom systems as the overall (symmetric positive-definite) properties of the system mass matrix are accounted for without involving information for each scalar parameter. The associated LMIs can therefore be solved in polynomial time with respect to the number of unknown parameters. Numerical examples for the case of sky-hook damping control and multi-objective control are provided that demonstrate the effectiveness of the method as a tool for model-based controller evaluation and multi-objective optimization.

Coding scheme based on spherical polar coordinate for control over packet erasure channel

SUMMARY Like feedback strategy, coding scheme is an important part of networked control system design as well. On the basis of spherical polar coordinate, a novel coding scheme is proposed for stabilization problem of discrete linear time invariant system subject to packet erasure channel with feedback. The coding scheme uses encoder without access to control inputs. In the case that a decoder does not use control inputs, a definite relation between the quantized data and the corresponding quantization error is established, which helps to analyze the stability of system, and a selective quantization method is adopted, by which finite data rate is obtained. In the case that a decoder uses control inputs, instead of quantizing the system state at each time step as usual, the encoder quantizes the initial state all the time by updating quantizer. Sufficient conditions guaranteeing the system stable are presented for two cases, respectively, and the corresponding design methods for coding schemes are given. Copyright © 2012 John Wiley & Sons, Ltd.

Characterization of Fixed Modes in a Structurally Constrained Control System with Input and Output Delays

AbstractThe problem of controlling a linear time-invariant (LTI) system subject to input and output delays by means of structurally constrained controllers is considered in this paper. Necessary and sufficient conditions are given to determine if a distinct mode of the system is fixed with respect to a decentralized or overlapping information flow structure. In general, a fixed mode of a time-delay system may not be fixed if the length of the delay varies. However, it is shown that a specific type of fixed modes remain fixed for any value of delay. Conditions are also provided to characterize this type of delay-independent fixed modes. The importance of the results obtained is illustrated by a numerical example.

Spatially Periodic Disturbance Rejection With Spatially Sampled Robust Repetitive Control

Repetitive controllers have been shown to be effective for tracking periodic reference commands or for rejecting periodic disturbances. Typical repetitive controllers are synthesized in temporal domain where the periods of the reference or disturbance signals are assumed to be known and stationary. For periodic references and disturbances with varying periods, researchers usually resort to adaptive and robust control approaches. For rotational motion systems where the disturbances or reference signals are spatially periodic (i.e., periodic with respect to angular displacement), the temporal period of the disturbance and reference signals will be inversely proportional to the rotational speed and vary accordingly. Motivating by reducing halftone banding for laser printers, we propose a design framework for synthesizing spatially sampled repetitive controller by reformulating a linear time-invariant system subject to spatially periodic disturbances using angular displacement as the independent variable. The resulting nonlinear system can be represented as a quasi-linear parameter-varying (quasi-LPV) system with the angular velocity as one of the varying state-dependent parameters. An LPV self-gain–scheduling controller that includes a spatially sampled repetitive control can be designed to take into consideration bounded model uncertainty and input nonlinearity, such as actuator saturation. Using the signal from an optical encoder pulse as a triggering interrupt, experimental results verified the effectiveness of the proposed approach in rejecting spatially periodic disturbances that cannot be compensated with fixed period temporal repetitive controllers.

Synthesis of Stabilizing Antiwindup Controllers Using Piecewise Quadratic Lyapunov Functions

We consider the problem of antiwindup controller synthesis based on the general antiwindup framework presented in Kothare (Automatica, vol. 30, no. 12, pp. 1869-1883, 1994) applicable to linear time-invariant systems (LTI) subject to a saturating actuator. Our synthesis approach takes advantage of the fact that the antiwindup system is a piecewise affine system and thus, we can utilize piecewise quadratic Lyapunov function theory Johansson and Rantzer (IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 555-559, Apr. 1998), Rantzer and Johansson (IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 629-637, Apr. 2000), and Johansson (Proc. 14th World Congr., Beijing, China, 1999, pp. 521-5260) to determine a stabilizing antiwindup control law. The synthesis problem is expressed in terms of bilinear matrix inequalities (BMIs) and is solved using an iterative approach as well as using commercial software. The performance of the system is optimized by minimizing an upper bound on the induced gain of the system. The proposed approach is demonstrated using examples.

STABILITY OF LINEAR TIME‐INVARIANT OPEN‐LOOP UNSTABLE SYSTEMS WITH INPUT SATURATION

ABSTRACTThis paper analyzes the stability of a linear time‐invariant open‐loop unstable system subject to input saturation. First, we extend the idea of approximating the locally asymptotically stable region (controllable set) of the system for the case where the control is small enough to be unsaturated (inside the linear region), to the case when the control is allowed to saturate. It is shown that, when the Lyapunov descent criterion and the Kuhn‐Tucker Theorem is applied, a superior locally asymptotically stable region is found. A technique for approximating the locally asymptotically stable region is presented.

LPV Control of Uncertain Systems by Observer with Pole Placement and H2 Constraints

Linear parameter varying (LPV) systems are linear plants whose dynamics depend on a vector of time varying exogenous parameters. They can be viewed as a representation of a linear time invariant (LTI) system subject to time varying uncertainties. Thus LTI techniques such as H2 control with pole placement can be applied to derive a controller for LPV systems. This problem is addressed in this paper. Furthermore, when state is not available for measure an observer is designed to allow a feedback control.

ON THE STABILITY OF SYSTEMS WITH MIXED TIME‐VARYING PARAMETERS

The well-known scaled small gain condition guarantees stability for a linear time invariant system subject to bounded complex nonlinear and/or time-varying perturbations. A polynomial time computable condition is derived that can be substantially less conservative for gain scheduled and other multivariable systems with repeated real time-varying parameters. The proof is a generalization of the purely-complex case given in Andrew Packard's thesis. © 1997 by John Wiley & Sons, Ltd.

ON THE STABILITY OF SYSTEMS WITH MIXED TIME-VARYING PARAMETERS

The well-known scaled small gain condition guarantees stability for a linear time invariant system subject to bounded complex nonlinear and/or time-varying perturbations. A polynomial time computable condition is derived that can be substantially less conservative for gain scheduled and other multivariable systems with repeated real time-varying parameters. The proof is a generalization of the purely-complex case given in Andrew Packard’s thesis.

Output regulation for linear systems subject to input saturation

The purpose of this paper is to examine the problem of controlling a linear time-invariant system subject to input saturation in order to have its output track (or reject) a family of reference (or disturbance) signals produced by some external generator. It is shown that a semiglobal framework for this problem is a natural one. Within this framework, a set of solvability conditions are given and feedback laws that solve the problem are constructed. The theory developed in this paper incorporates earlier results established for linear systems without input saturation.

Stability robustness of linear discrete-time systems under time-varying perturbations

Abstract The stability robustness of an asymptotically stable linear time-invariant discrete-time nominal system subject to linear time-varying perturbations is analysed. Based on the stability theory of linear time-varying continuous-time systems and the matrix norm theory, both the norm bounds and the element bounds on linear time-varying perturbations are obtained for the nominal system. The new bounds are simple to implement and they have less conservative results when the condition number of the transformation matrix is small.

Matrices computation of an unknown-input observer

An original method for the design of the Luenberger observer is developed for a linear time-invariant system subject to unknown inputs. The paper mainly discusses the problem of determining the matrices of the observer