Popov Lemma Research Articles

N-D representation and generalised Kalman–Yakubivich–Popov lemma of spatially interconnected systems with interconnected chains

ABSTRACTThis paper presents a multidimensional (N-D) hybrid Roesser model for a class of spatially interconnected systems (SISs) with interconnected chains. A generalised Kalman–Yakubivich–Popov (KYP) lemma for N-D hybrid Roesser model is developed, which plays a significant role in the system control and synthesis. As the application of the developed generalised KYP lemma, robust finite-frequency H∞ filter problem for SISs with interconnected chains is studied. Moreover, it is shown that the developed finite-frequency filter design methods are generally less conservative than the entire frequency ones. Finally, a practical example is provided to validate the effectiveness of the proposed method.

Actuator fault estimation for discrete-time switched systems with finite-frequency

This paper deals with the problem of actuator fault estimation observer design for discrete-time switched systems with finite-frequency. The frequencies of the fault and the unknown input disturbance are assumed to be in a finite range. A finite frequency fault estimation observer is proposed to estimate the actuator fault in the entire frequency domain. Based on the generalized Kalman–Yakubovic–Popov (KYP) lemma and the switched Lyapunov function method, the efficient conditions are obtained to achieve the fault estimation observer design. Finally, an example is given to illustrate the proposed technique.

Finite frequency H∞ filter design for T-S fuzzy systems: New approach

This paper studies the problem of H∞ filter analysis and design for discrete-time Takagi–Sugeno (T-S) fuzzy systems, in a finite frequency (FF) domain. Based on a fuzzy basis dependent Lyapunov function and time domain interpretations of Generalized Kalman Yakubovich Popov (GKYP) lemma, combined with generalised S-procedure, sufficient conditions are developed, guaranteeing that the filtering error system is stable and has a prescribed H∞ attenuation level, over a specified FF domain of the external disturbances. Furthermore, several additional slack variables are introduced to the derived conditions by applying Finsler’s lemma twice, which leads to performance improvement and less conservatism in the solution. Thus, new filter design conditions are proposed in the formulation of Linear Matrix Inequality (LMI) terms in three different frequency ranges, (low, middle and high frequencies). A numerical example is given to demonstrate the feasibility, the effectiveness and the less conservatism of the proposed approach in comparison with another solution that appears in the literature.

Formula omitted] fault detection observer in finite frequency domain for linear parameter-varying descriptor systems

This paper considers H−∕H∞ fault detection observer design in the finite frequency domain for a class of linear parameter-varying (LPV) descriptor systems. We propose a novel fault detection observer with a non-singular structure. To make the residual sensitive to faults and robust against disturbances, we develop a finite frequency H−∕H∞ design method based on a generalized Kalman–Yakubovich–Popov lemma for LPV descriptor systems. Design conditions for the proposed fault detection observer are derived and converted as linear matrix inequalities. Simulation studies are conducted to demonstrate the performance of the proposed method.

Stability and Robust Stabilization of 2-D Continuous Systems in Roesser Model Based on GKYP Lemma

This paper is concerned with the stability and Robust stabilization problem for 2-D continuous systems in Roesser model, based on Generalized Kalman$-$Yakubovich$-$Popov lemma in combination with frequency-partitioning approach. Sufficient conditions of stability of the systems are formulated via linear matrix inequality technique. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

Output Feedback Control of Multirate Sampled-Data Systems With Frequency Specifications

This paper addresses the problem of $H_{\infty }$ output feedback controller design of multirate sampled-data systems with frequency specifications. The general multirate situation is considered, where not only the inputs and outputs but also different channels of the inputs or the outputs have different sampling rates. The aim of this paper is to design an output feedback controller for multirate sampled-data systems in order to ensure an $H_{\infty }$ performance level over a restricted frequency, which includes the standard $H_{\infty }$ control problem as a special case and can better reflect the requirement of attenuating disturbances over restricted frequency ranges. The problem is solved via reformulating the multirate sampled-data system with an output feedback controller and utilizing the generalized Kalman–Yakubovic–Popov lemma to deal with the restricted frequency specifications. For a given restricted frequency range, convex conditions are derived for the existence of a required controller and cases for low/high/entire frequency ranges are also addressed specifically. Finally, illustrative examples are presented to show the effectiveness and advantage of the proposed controller design methods.

A finite frequency domain-based approach for active fault tolerant control of linear time-delay systems with residual feedback

This paper presents a finite frequency-based active fault tolerant control approach for the compensation of unknown failures in linear time-delay plants. An integration of fault detection filter based on observer technology and [Formula: see text] controller in residual feedback form is considered in the active fault tolerant control system. Different from the traditional schemes, exact fault estimation is not necessary in the proposed active fault tolerant control. To achieve the desired system performance when a fault occurs, the residual is directly embedded in the control loop as a feedback term to compensate the influence of fault. By employing the Generalized Kalman–Yakubovich–Popov lemma, we derive the sufficient conditions of the existence of such an active fault tolerant control plant, and iterative algorithms are applied to obtain the solutions to the fault detection filter and controller parameter matrices. Finally, simulation results are proposed to demonstrate the effectiveness of the developed scheme.

Robust $$H_{\infty }$$ H ∞ Filters for Uncertain Systems with Finite Frequency Specifications

This paper deals with $$H_{\infty }$$ filtering problem of linear discrete-time uncertain systems with finite frequency input signals. The uncertain parameters are supposed to reside in a polytope. By applying the generalized Kalman–Yakubovich–Popov lemma, polynomially parameter-dependent Lyapunov function and some key matrices to eliminate the product terms between the filter parameters and the Lyapunov matrices, an improved condition is obtained for analyzing the $$H_{\infty }$$ performance of the filtering error system. Then sufficient condition in terms of linear matrix inequality is established for designing filters with a guaranteed $$H_{\infty }$$ filtering performance level. Finally, a numerical examples are used to demonstrate the effectiveness of the proposed method.

Finite-frequency fault detection for two-dimensional Fornasini–Marchesini dynamical systems

ABSTRACTThis paper is concerned with the fault detection problem for two-dimensional (2-D) discrete-time systems described by the Fornasini–Marchesini local state-space model. The goal of the paper is to design a fault detection filter to detect the occurrence of faults in finite-frequency domain. To this end, a finite-frequency H− index is used to describe fault sensitivity performance, and a finite-frequency H∞ index is used to describe disturbance attenuation performance. In light of the generalised Kalman–Yakubovich–Popov lemma for 2-D systems and matrix inequality techniques, convex conditions are derived for this fault detection problem. Based on these conditions, a numerical algorithm is put forward to construct a desired fault detection filter. Finally, a numerical example and an industrial example are given to illustrate the effectiveness of the proposed algorithm.

Iterative learning fault-tolerant control for differential time-delay batch processes in finite frequency domains

This paper develops a fault-tolerant iterative learning control law for a class of differential time-delay batch processes with actuator faults using the repetitive process setting. Once the dynamics are expressed in this setting, stability analysis and control law design makes use of the generalized Kalman–Yakubovich–Popov (KYP) lemma in the form of the corresponding linear matrix inequalities (LMIs). In particular, sufficient conditions for the existence of a fault-tolerant control law are developed together with design algorithms for the associated matrices. Under the action of this control law the ILC dynamics have a monotonicity property in terms of an error sequence formed from the difference between the supplied reference trajectory and the outputs produced. An extension to robust control against structured time-varying uncertainties is also developed. Finally, a simulation based case study on the model of a two-stage chemical reactor with delayed recycle is given to demonstrate the feasibility and effectiveness of the new designs.

Semidefinite Representations of Gauge Functions for Structured Low-Rank Matrix Decomposition

This paper presents generalizations of semidefinite programming formulations of 1-norm optimization problems over infinite dictionaries of vectors of complex exponentials, which were recently proposed for superresolution, gridless compressed sensing, and other applications in signal processing. Results related to the generalized Kalman--Yakubovich--Popov lemma in linear system theory provide simple, constructive proofs of the semidefinite representations of the penalty functions used in these applications. The connection leads to several extensions to gauge functions and atomic norms for sets of vectors parameterized via the nullspace of matrix pencils. The techniques are illustrated with examples of low-rank matrix approximation problems arising in spectral estimation and array processing.

On Finite frequency H∞ performance for discrete linear time delay systems

ABSTRACTThis paper is concerned with the finite frequency H∞ performance of discrete linear systems with time delay. The disturbance of the system is considered to be restricted in low/middle/high frequency. A novel summation inequality is proposed by including more free weighed matrices. Then by using generalised Kalman–Yakubovich–Popov Lemma and a Lyapunov–Krasovskii functional like method, a new bounded real lemma is presented for the finite frequency H∞ performance analysis. The advantages and reduced conservatism of the new criterion are illustrated by a classical example from the literature.

H∞filter design for linear time-invariant systems with polytopic uncertainties in finite frequency domain

SUMMARY This paper deals with the problem of finite frequency H∞ full-order filter design for discrete-time and continuous-time linear systems, with polytopic uncertainties. Based on the generalized Kalman–Yakubovich–Popov lemma and a parameter-dependent Lyapunov function, a set of sufficient conditions are established in terms of matrix inequalities, ensuring that the filtering error system is stable and the H∞ attenuation level, from disturbance to the estimation error, is smaller than a given value over a prescribed finite frequency domain of the external disturbances. Then, in order to linearize and relax the obtained matrix inequalities, we introduce a large number of slack variables by applying Finsler's lemma twice, which provides extra degrees of freedom in optimizing the guaranteed H∞ performance. This leads to performance improvement and reduction of conservatism in the solution. It is shown later that the robust filter gains can be obtained by solving a set of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods. Copyright © 2016 John Wiley & Sons, Ltd.

Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.

Experimentally verified generalized KYP Lemma based iterative learning control design

This paper considers iterative learning control law design for plants modeled by discrete linear dynamics using repetitive process stability theory. The resulting one step linear matrix inequality based design produces a stabilizing feedback controller in the time domain and a feedforward controller that guarantees convergence in the trial-to-trial domain. Additionally, application of the generalized Kalman–Yakubovich–Popov (KYP) lemma allows a direct treatment of differing finite frequency range performance specifications. The results are also extended to plants with relative degree greater than unity. To support the algorithm development, the results from an experimental implementation are given, where the performance requirements include specifications over various finite frequency ranges.

Delay-independent stability analysis of linear time-delay systems based on frequency discretization

This paper studies strong delay-independent stability of linear time-invariant systems. It is known that delay-independent stability of time-delay systems is equivalent to some frequency-dependent linear matrix inequalities. To reduce or eliminate conservatism of stability criteria, the frequency domain is discretized into several sub-intervals, and piecewise constant Lyapunov matrices are employed to analyze the frequency-dependent stability condition. Applying the generalized Kalman–Yakubovich–Popov lemma, new necessary and sufficient criteria are then obtained for strong delay-independent stability of systems with a single delay. The effectiveness of the proposed method is illustrated by a numerical example.

Fault Detection for Two-Dimensional Roesser Systems With Sensor Faults

This paper investigates the problem of fault detection for 2-D systems with lock-in-place sensor faults. Our attention is focused on detecting sensor faults in the presence of disturbances. To this end, a fault detection filter is designed, through which a residual signal is generated for both the fault-free and faulty cases. In light of the generalized Kalman–Yakubovich–Popov lemma for 2-D systems and matrix inequality techniques, convex filter design conditions are derived. Based on these conditions, an algorithm is proposed to calculate the parameters of a desired fault detection filter. Moreover, a residual evaluation function and a threshold are proposed for 2-D systems. Finally, an example is employed to illustrate the effectiveness of the proposed fault detection method.

Finite-Frequency Fault Detection for Two-Dimensional Roesser Systems

This paper focuses on the fault detection problem of 2-D systems described by the Roesser model. To detect faults effectively in the presence of disturbances, a fault detection filter is designed to satisfy a finite-frequency $H_{-}$ index and a finite-frequency $H_\infty $ index simultaneously. The corresponding finite-frequency performance analysis conditions are obtained by the aid of the generalized Kalman–Yakubovich–Popov lemma. Then, convex filter design conditions are derived by constructing a hyperplane tangent combined with linear matrix inequality techniques. An algorithm is proposed to construct a desired fault detection filter. Finally, a numerical example is given to show the effectiveness of the proposed method.

Fault detection for singular switched linear systems with multiple time-varying delay in finite frequency domain

ABSTRACTThis paper deals with the problem of the fault detection (FD) for continuous-time singular switched linear systems with multiple time-varying delay. In this paper, the actuator fault is considered. Besides, the systems faults and unknown disturbances are assumed in known frequency domains. Some finite frequency performance indices are initially introduced to design the switched FD filters which ensure that the filtering augmented systems under switching signal with average dwell time are exponentially admissible and guarantee the fault input sensitivity and disturbance robustness. By developing generalised Kalman–Yakubovic–Popov lemma and using Parseval's theorem and Fourier transform, finite frequency delay-dependent sufficient conditions for the existence of such a filter which can guarantee the finite-frequency H− and H∞ performance are derived and formulated in terms of linear matrix inequalities. Four examples are provided to illustrate the effectiveness of the proposed finite frequency method.

Integrated Servo-Mechanical Design of Robust Mechatronics Based on Ambiguous Chance Constraints

In this brief, we propose a convex separable parametrization technique for finite frequency reshaping of the mechanical plant and low-order controller in high-performance mechatronic systems simultaneously. Using the generalized Kalman–Yakubovich–Popov lemma with low-order moments and support for parametric uncertainties, the individual chance-constrained robust stability criterion is formulated and solved via several convex linear matrix inequalities while ensuring stable zeros with a positive realness condition. Our simulation results on the Pb–Zr–Ti active suspension in a commercial $3.5''$ dual-stage hard disk drive achieve a high-bandwidth control system and desired disturbance attenuation capabilities, with a stability of 98.2% in proportion of perturbed systems as compared with standard methods in the current literature.