Kalman-Yakubovich-Popov Lemma Research Articles

Observer–controller synthesis approach in low frequency for T–S fuzzy systems

For the output feedback control problem of continuous-time T–S fuzzy systems with unknown premise variables, an H ∞ observer–controller design method in the low-frequency domain is proposed. First, an observer–controller structure is given, the unknown premise variables are limited by Lipschitz conditions. Then, the system stability conditions are obtained by the negativeness of eigenvalues' real parts. To achieve better control performance of the system in low frequency, the H ∞ index for attenuating the unknown low-frequency disturbance is guaranteed by generalised Kalman–Yakubovich–Popov lemma. Then, the stability and robustness conditions are converted into linear matrix inequality forms, which can be solved directly by a convex optimisation technique. Finally, several simulation examples carried out to show the effectiveness of the proposed method.

Fault estimation for fractional-order linear systems with polytopic uncertainties in the finite frequency domain

This paper investigates the problem of fault estimation for fractional-order systems with polytopic uncertainties. The faults and disturbances are considered to belong to finite frequency domains, and a fractional-order robust fault estimation filter is proposed for fault estimation with the known frequency ranges. As applications of the generalized Kalman–Yakubovich–Popov lemma, the fault estimation filter design problem is transformed into a multi-objective optimisation problem. By using the projection lemma and a linearising change of variables, sufficient conditions for the existence of the robust fault estimator are derived on the basis of a set of linear matrix inequalities. Numerical examples are used to demonstrate the validity of the presented method.

Robust over finite frequency ranges H∞ model reduction for uncertain 2D continuous systems

This paper deals with the problem of robust H∞ model reduction for two-dimensional (2D) continuous systems described by Roesser model with polytopic uncertainties, over finite frequency (FF) domain. The problem to be solved in the paper is to find a reduced-order model such that the approximation error system is asymptotically stable, which is able to approximate the original continuous systems system with comparatively small and minimised H∞ performance when frequency ranges of noises are known beforehand. Via the use of the generalised Kalman Yakubovich Popov (gKYP) lemma, homogeneous polynomially parameter-dependent matrices, Finsler's lemma and we introduce many slack matrices, sufficient conditions for the existence of H∞ model reduction for different FF ranges are proposed and then unified in terms of solving a set of linear matrix inequalities (LMIs). To highlight the effectiveness of the proposed H∞ model reduction design, two illustrative examples are given.

Sparse Representation of Feedback Filters in Delta-Sigma Modulators

In this paper, we propose sparse representation of FIR (Finite Impulse Response) feedback filters in delta-sigma modulators. The filter has a sparse structure, that is, only a few coefficients are non-zero, that stabilizes the feedback modulator, and minimizes the maximum magnitude of the noise transfer function at low frequencies. The optimization is described as an ℓ1 minimization with linear matrix inequalities (LMIs), based on the generalized KYP (Kalman-Yakubovich-Popov) lemma. A design example is shown to illustrate the effectiveness of the proposed method.

Improved finite frequency <i>H</i><SUB align="right">∞ filtering for Takagi-Sugeno fuzzy systems

This paper investigates the design problem of H∞ filtering for discrete nonlinear systems in the Takagi-Sugeno (T-S) form. Our aim is to design a new filter guaranteeing an H∞ performance level in specific finite frequency (FF) ranges. Using the well-known generalised Kalman Yakubovich Popov lemma, Finsler's lemma, sufficient conditions for the existence of H∞ filters for different FF ranges are proposed and then unified in terms of solving a set of linear matrix inequalities (LMIs). Two examples are given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods.

ON COERCIVITY AND THE FREQUENCY DOMAIN CONDITION IN INDEFINITE LQ-CONTROL

We introduce a coercivity condition as a time domain analogue of the frequency criterion provided by the famous Kalman-YakubovichPopov lemma. For a simple stochastic linear quadratic control problem we show how the coercivity condition characterizes the solvability of Riccati equations.

Fault Detection for 2-D Continuous-Discrete State-Delayed Systems in Finite Frequency Domains

This paper investigates the fault detection problem for two dimensional (2-D) continuous discrete state-delay Roesser systems in finite frequency domains. Two performance indexes H∞ and H- are used to measure the fault sensitivity and the disturbance robustness in finite frequency. Based on this, the fault detection problem is converted into a filtering problem by designing a filter to generate a residual signal. By the generalized KYP lemma, convex design conditions are obtained, which are expressed in terms of linear matrix inequalities (LMIs). An example is provided to demonstrate the feasibility and effectiveness of the proposed method.

Robust Semipassivity and Practical Stability for Uncertain Switched Nonlinear Systems

This paper addresses several issues including robust semipassivity and practical stability analysis for a switched nonlinear system with structural uncertainties. First, robust semipassivity is firstly defined to describe the overall semipassivity property of switched nonlinear systems without requiring the conventional semipassivity property of each active subsystem. Then, the robust semipassive system is shown to be practically stable. Second, a state-dependent switching law is designed to render the switched system robust semipassive. This switching law is a generalization of KYP lemma. Third, robust semipassification for a switched nonlinear system is achieved by the design of a state-dependent switching law and a set of feedback controllers. Finally, a composite state-dependent switching law is designed to render the feedback interconnection of switched nonlinear systems robust semipassive. The effectiveness of the obtained resultsis verified by two numerical examples.

Fault detection for discrete time-delay networked systems with round-robin protocol in finite-frequency domain

ABSTRACTThe paper considers the fault detection problem over finite-frequency domain for a class of discrete time-delayed networked systems subject to round-robin protocol. This protocol is a periodic one to schedule the information transmitted via a shared communication network with an intent to prevent the data from collisions. In this scenario, the error dynamics can be regarded as a periodic protocol-induced system. The aim of the problem addressed is to design a fault detection filter that (1) the sensitivity of the residual to the fault is enhanced by means of a maximised index and (2) the effect from the exogenous disturbances on the residual is decreased with considering a minimised index. By resorting to a Lyapunov function, an improved version of generalised Kalman–Yakubovich–Popov lemma (GKYP) is first developed for protocol-induced periodic system. Then, in light of such a lemma, sufficient conditions are derived to guarantee both the fault sensitivity and the disturbance attenuation capability by solving a set of matrix inequalities. Finally, the usefulness of the presented fault detection algorithm is utilised via a numerical simulation.

Fault detection for fractional‐order linear systems in finite frequency domains

This study is concerned with the fault detection problem for fractional-order linear systems with disturbances. The H − and H ∞ performance indices of fractional-order systems in finite frequency domains are applied to measure the fault sensitivity and the disturbance robustness. A fractional-order fault detection observer is designed to satisfy the two finite frequency performance indices simultaneously. Based on the generalised Kalman–Yakubovich–Popov lemma and the projection lemma, the design conditions are obtained in terms of linear matrix inequalities. Compared with the existing full frequency approaches, the proposed finite frequency one can get better results when the frequency ranges of faults and disturbances are known. The effectiveness of the proposed method is validated by numerical examples.

An improved H∞ filtering for Markovian jump system with time-varying delay based on finite frequency domain

In traditional analysis of system performance, Markovian jump system is often considered in the full frequency domain. However, some unknown factors, such as noise interference, often act on a specific frequency domain, so there is a big defect in putting the system analysis in the full frequency domain. In this paper, the finite frequency domain is introduced into the Markovian jump system. By using generalized KYP lemma, the system is divided into three frequency bands: low, medium and high. The disturbance in different frequency domain is analyzed and the H∞ filter is designed. In addition, in dealing with inequality relations, the improved Wirtinger inequality method is used, and the Projection lemma is used in dealing with filter coupling terms, and time-varying delay problem in finite frequency domain is also considered. Finally, some examples are given to illustrate the effectiveness of our methods and the specific parameters and convergence graph of the finite frequency domain filter are obtained.

An oracle for the discrete-time integral quadratic constraint problem

In the theory of integral quadratic constraints (IQCs), the condition for robust performance is given in terms of a frequency-domain inequality. This inequality can be transformed into an equivalent finite-dimensional linear matrix inequality by the application of the Kalman–Yakubovich–Popov (KYP) lemma and the introduction of an auxiliary matrix variable. Thus, the IQC problem can be expressed as a semi-definite program (SDP) and solved using convex optimization tools such as general-purpose SDP solvers based on interior point methods. However, by introducing the auxiliary variable, the size of the IQC problem significantly increases, and the aforementioned solvers may fail due to the high computational cost of the SDP. To address this issue, specialized SDP solvers have been developed to efficiently solve the SDP by exploiting its structure. An alternative approach consists of using cutting plane methods to directly solve the frequency-domain inequality without resorting to the KYP lemma. Cutting plane methods require an oracle that confirms if a given candidate solution satisfies the frequency-domain inequality. If the candidate solution is not feasible, the oracle generates a hyperplane that separates the candidate solution from the set of feasible solutions. In this paper, the analytic center cutting plane method is applied to the discrete-time IQC problem. The oracle for the discrete-time IQC problem is based on the spectrum of an extended symplectic matrix pencil and does not require an invertibility assumption that holds in continuous-time but not necessarily in discrete-time. An illustrative example is provided that shows the efficacy of the proposed method and points out directions for future research.

Finite frequency range iterative learning fault-tolerant control for discrete time-delay uncertain systems with actuator faults

The subject area considered is discrete linear time delay systems operating repetitively on a finite time interval with actuator faults, where the system resets at the end of each operation. Regulation of the dynamics is by iterative learning control and performance goals imposed over finite frequency intervals for the case of uncertainty in the dynamic model. To derive the results, the generalized Kalman–Yakubovich–Popov lemma is used. A simulation based case study is also given to demonstrate the applicability of the new results.

Strong delay-independent stability of linear delay systems

This paper presents a new necessary and sufficient condition for testing the strong delay-independent stability of linear systems subject to a single delay. The proposed method follows from the use of matrix polynomials constraints and the Kalman–Yakubovich–Popov lemma. The resulting condition can be checked exactly by solving a feasibility problem in terms of a linear matrix inequality (LMI). Simple numerical examples are given to show the effectiveness of the proposed method.

Pole Path Assignment of Constrained SISO Affine Nonlinear Systems

This paper proposes a feedback linearization control scheme for affine nonlinear systems with a constrained control signal where a fast response is required. Instead of placing the linearized system poles at exact locations, ray-like paths in the left hand plane are selected where the poles can freely vary to efficiently exploit the admissible input signal range to increase the speed of response. A stability test for the proposed method is derived via the multivariable circle criterion and the Kalman–Yakubovich–Popov lemma. Simulation results demonstrate how the method significantly increases the speed of response in spite of the input constraints compared to fixed pole placement.

Robust fault estimation in the finite-frequency domain for multi-agent systems

In this paper, a distributed fault estimation observer (DFEO) is designed for multi-agent systems in the finite-frequency domain. The DFEO consists of a network of local fault estimation observers corresponding to each agent. Here communication topology is described as a graph with loops. By using the generalized Kalman–Yakubovich–Popov lemma, the proposed DFEO is designed to avoid the overdesign problem generated by the entire-frequency domain. Furthermore, the design conditions for DFEO are formulated as linear matrix inequalities. Finally, compared with the [Formula: see text] approach in the entire-frequency domain, the proposed DFEO design approach shows better performance in numeral simulations of satellites attitude synchronization.

Hybrid frequency domain control for large flexible structures

A hybrid finite frequency controller is proposed for the vibration suppression of a large flexible structure mounted with collocated sensors and actuators. The controller has passive characteristics at low frequencies and small gain characteristics at high frequencies. Compared with a strictly positive real controller based on the standard Kalman–Yakubovich–Popov lemma, the hybrid finite frequency controller has less energy consumption but can obtain approximately identical performance. Furthermore, when the plant passivity is violated at high frequencies by noncollocation of sensors and actuators, the strictly positive real controller based on the Kalman–Yakubovich–Popov lemma is no longer able to attenuate the vibration of the large flexible structures, while the hybrid finite frequency controller is effective in suppressing the vibration and avoiding spillover instability. Simulation results are presented to validate the effectiveness of the hybrid finite frequency controller.

Zonotopic fault detection observer design for Takagi–Sugeno fuzzy systems

ABSTRACTThis paper considers zonotopic fault detection observer design in the finite-frequency domain for discrete-time Takagi–Sugeno fuzzy systems with unknown but bounded disturbances and measurement noise. We present a novel fault detection observer structure, which is more general than the commonly used Luenberger form. To make the generated residual sensitive to faults and robust against disturbances, we develop a finite-frequency fault detection observer based on generalised Kalman–Yakubovich–Popov lemma and P-radius criterion. The design conditions are expressed in terms of linear matrix inequalities. The major merit of the proposed method is that residual evaluation can be easily implemented via zonotopic approach. Numerical examples are conducted to demonstrate the proposed method.

Interval frequency negative imaginary systems

ABSTRACTThis paper studies interval frequency negative imaginary (IFNI) systems based on minimal state-space realisations. Firstly, the concept of the IFNI transfer functions is introduced, and the relationship between the IFNI transfer functions and the interval frequency positive real transfer functions is established. Secondly, based on the generalised KYP lemma, a necessary and sufficient condition is derived for IFNI transfer functions with minimal state-space realisation in terms of linear matrix inequalities. Our results coincide with the existing negative imaginary lemma when the interval frequency set becomes the positive frequency set. Also, a time domain interpretation of the IFNI property is provided in terms of the system input, output and state. Finally, two examples are used to illustrate the developed theory.

Closed-form solutions of singular KYP lemma: strongly passive systems, and fast lossless trajectories

In this paper, we deal with a special class of passive systems, which possess the characteristic property of having no finite spectral zeros. We call these systems strongly passive. It is well known that, for these systems, storage functions, i.e. solutions to the linear matrix inequality (LMI) arising from the Kalman–Yakubovich–Popov (KYP) lemma, cannot be obtained by the conventional approach of algebraic Riccati equations (AREs) and Hamiltonian matrices. In this paper, we first show that a strongly passive system always admits a unique storage function. We then provide a closed-form expression for this unique storage function. Using the closed-form formula of the unique storage function we characterise the ‘lossless’ trajectories of strongly passive systems and show that such systems admit impulsive lossless trajectories on the half-line; we call them fast lossless trajectories. This adds to the existing notion that such systems do not admit any ‘slow’ lossless trajectories.