Generalized Kalman Yakubovich Popov Lemma Research Articles

Finite frequency H ∞ control for singular systems with application to a 2-machine 6-bus power network

This paper is concerned with the H ∞ control problem for singular systems over finite frequency ranges. The disturbance is assumed to be within low-/middle-/high-frequency ranges. Both the state feedback and static output feedback controllers are designed. First, by combining the Lyapunov function method and the generalized Kalman–Yakubovich–Popov lemma, a new finite frequency bounded real lemma (BRL) is proposed, which ensures the resulting closed-loop system to be regular, impulse free and stable with given finite frequency H ∞ performance bound. It is shown by theoretical analysis and numerical simulation that the proposed BRL has less conservatism than the existing ones in the literature. With the aid of the Finsler's Lemma, two improved finite frequency BRLs are further presented for the design of controllers. Then, by utilizing convexification techniques, sufficient conditions for solving the desired state feedback and static output feedback gains are presented in terms of linear matrix inequalities, respectively. Finally, the applicability and effectiveness of the proposed control methods are verified by a 6-bus power network system.

Multi-Objective Finite-Frequency H ∞/GH 2 Static Output-Feedback Control Synthesis for Full-Vehicle Active Suspension Systems: A Metaheuristic Optimization Approach

In this paper, the problem of multi-objective control for active suspension systems with polytopic uncertainty is addressed via <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GH</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> static output feedback with a limited-frequency characteristic. For the overall analysis of the performance demanding both vehicle-ride comfort related to vertical- and transversal-directional dynamics and the time-domain constraints related to the driving maneuverability, a seven-degree-of-freedom full-vehicle model with an active suspension system is investigated. The robust static output-feedback control strategy is adopted because some state variables may not be directly measured in a realistic implementation. In designing this control, the finite-frequency H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance using the generalized Kalman—Yakubovich—Popov lemma is optimized to improve the passenger’s ride comfort, while the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GH</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance is optimized to guarantee the constraints concerning the suspension deflection limitation, road-holding ability, and actuator saturation problem. This control synthesis problem is formulated as nonconvex bilinear matrix inequalities and requires simultaneous consideration of different finite-frequency domain ranges for vertical and transversal motions for evaluating the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance. These design difficulties are overcome by the proposed multi-objective quantum-behaved particle swarm optimizer, which efficiently explores the relevant trade-offs between the considered multiple performance objectives and eventually provides the desired Pareto-optimal control set. Further, the numerical simulation cases of a full-vehicle active suspension system are presented to illustrate the effectiveness of the proposed control synthesis methodology in frequency and time domains.

Improving Vibration Performance of Electric Vehicles Based on In-Wheel Motor-Active Suspension System via Robust Finite Frequency Control

This paper presents a robust finite frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> control strategy for improving vibration performance and ride comfort of electric vehicles through in-wheel motor-active suspension system(IWM-ASS). Since the human body is much sensitive to the vertical vibration of 4 -8 Hz, the main objective is dedicated to deal with the vibration challenge that matches the characteristics of the human body by applying the finite-frequency technique. Firstly, the uncertain quarter-vehicle active suspension model with dynamic damping in-wheel motor driven system is established, in which in-wheel motor is suspended as dynamic vibration absorber(DVA) to isolate the force transmitted to motor bearing in IWM-ASS. Based on the framework of generalized Kalman–Yakubovich–Popov lemma and stability theory, then the performance index of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> norm from external disturbance to controlled output for IWM-ASS is attenuated within the concerned frequency range while other system requirements such as parameter uncertainty, suspension deflection constraint and actuator saturation are also guaranteed in controller design. The resulting robust finite frequency state feedback <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> controller is finally designed utilizing two new theorems, and solved via a set of linear matrix inequalities. Simulations for frequency-domain and time-domain responses are implemented and compared with the entire frequency control method to evaluate the effectiveness of the proposed strategy. It can be concluded from the results that the developed control strategy can effectively attenuate the negative vibration and enhance ride comfort and road-holding ability for electric vehicles of IWM-ASS.

Finite-frequency fixed-order dynamic output-feedback control via a homogeneous polynomially parameter-dependent technique

This paper investigates the problem of fixed-order dynamic output-feedback (DOF) control for linear polytopic systems over finite-frequency ranges. Firstly, based on the generalized Kalman–Yakubovich-Popov lemma, we formulate the necessary and sufficient conditions for the finite-frequency disturbance-attenuation performance as bilinear matrix inequalities (BMIs), which are known to be NP-hard. In light of the homogeneous polynomially parameter-dependent technique, we construct relaxed synthesis conditions by employing higher-order decision variables dependent on the uncertainty parameter. To address the BMI problem, we develop an iterative procedure, under which feasible solutions to the original non-convex programming are achieved by vicariously solving a sequence of tractable convex approximations. Finally, we verify the efficacy of the theoretical results by an active suspension system.

H − / L ∞ UIO‐based actuator and sensor fault diagnosis design in the finite frequency domain for discrete‐time Lipschitz nonlinear systems

This study investigates the problem of fault diagnosis observer design in the finite frequency domain for discrete-time Lipschitz nonlinear systems simultaneously subjected to actuator and sensor faults and unknown input disturbances. An augmented descriptor system is developed by constructing an augmented state consisting of system states and sensor faults. A novel H−/L∞ unknown input observer (UIO) is proposed in the finite frequency domain using the generalized Kalman–Yakubovich–Popov lemma to render the residual sensitive to actuator faults and robust against disturbances, which can achieve a better fault diagnosis result compared with existing observers in the full frequency domain. In the fault diagnosis scheme, a time-varying threshold is presented based on the L∞ performance index to detect actuator faults, and the sensor fault estimation is simultaneously achieved through an asymptotical estimation of the augmented state. Design conditions for the proposed H−/L∞ UIO are derived and converted into linear matrix inequalities to solve its design matrices together. The effectiveness of the developed H−/L∞ UIO and the fault diagnosis scheme are validated via simulations of an example of a single-link flexible joint robot.

Robust feedback feed-forward PD-type iterative learning control for uncertain discrete systems over finite frequency ranges

This paper proposes a robust feedback feed-forward proportional-derivative type (PD-type) iterative learning control (ILC) scheme for a class of linear discrete systems with polytopic uncertainty over finite frequency ranges. First, the ILC process is transformed into an equivalent discrete linear repetitive process. Then, with the help of the generalized Kalman–Yakubovich–Popov lemma, the issue of a robust control law design algorithm in the process model is converted into the problem of solutions to the corresponding linear matrix inequality conditions. This procedure not only meets the robust performance specifications along the trial, but also guarantees the monotonic converge of the trial-to-trial error dynamics over finite frequency ranges. Finally, the simulations for a direct current servo motor system are adopted to show the effectiveness and superiority of the proposed method. Compared with previously established works, the new algorithm is more effective in delivering higher performance and achieving better tracking effect.

Finite-frequency self-triggered model predictive control for Markov jump systems subject to actuator saturation

In this paper, a self-triggered saturated model predictive control (SMPC) for Markov jump system (MJS) is studied in finite-frequency domain. First, the MJS with multi-modes is transformed into a single mode system with embedded transition probability to prepare for the use of generalized Kalman–Yakubovich–Popov (GKYP) lemma. Second, a dynamic self-triggered mechanism with time-varying threshold is developed to predict the next triggering time according to the current time, states, and jumping mode. Third, for disturbances that only possess finite-frequency band, a co-design of triggering condition and SMPC is realized by properly employing GKYP lemma. Compared with existing works, the main motivation of this paper is to improve the anti-interference ability within a specified frequency range. Moreover, we also aim to shorten the gap between the finite band control theory and its applications to practical engineering problems. Finally, the effectiveness of the proposed approach is demonstrated by both the numerical and practical simulations.

Model reduction for discrete-time switched linear time-delayed systems with finite-frequency specifications

The paper focusses on the model reduction of discrete-time switched linear systems with finite-frequency specifications. First, we propose a performance index that can estimate systematic performance over finite frequency. And then, we use GKYP (generalized Kalman–Yakubovich–Popov lemma) and introduce some relaxation matrices with variable parameters to analyse the problem in different situations. Based on the above conditions, sufficient conditions can be obtained for meeting the index of finite frequency. Finally, we illustrate two numerical examples. The results show that the proposed method can improve approximation performance.

Robust finite frequency static output feedback control for uncertain fuzzy systems via a sum‐of‐squares approach

AbstractThis article deals with the problem of robust static output feedback control in finite frequency (FF) domain for polynomial Takagi–Sugeno fuzzy systems. By using generalized Kalman–Yakubovich–Popov lemma and polynomial Lyapunov functions, sufficient conditions are proposed for controllers design in terms of sum of squares. These conditions include neither transformation matrices nor equality constraints, which simplifies the numerical solution. The proposed method is designed in the FF domain to reduce the conservativeness generated by those designed in the full frequency domain. Some numerical examples are provided to demonstrate the applicability of the proposed approach.

A Universal Framework of the Generalized Kalman–Yakubovich–Popov Lemma for Singular Fractional-Order Systems

The well-known generalized Kalman–Yakubovich–Popov lemma is widely used in system analysis and synthesis. However, the corresponding theory for singular systems, especially singular fractional-order systems (SFOSs), is lacking. Therefore, many control problems for this type of systems cannot be optimized in limited frequency ranges. In this article, a universal framework of the finite frequency band generalized Kalman–Yakubovich–Popov lemma for SFOSs is established, the bounded real lemma in the sense of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L_{\infty }}$ </tex-math></inline-formula> -norm is derived for different frequency ranges, and the corresponding controller is designed to improve the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L_{\infty } }$ </tex-math></inline-formula> performance index of SFOSs. Three illustrative examples are given to demonstrate the correctness and effectiveness of the theoretical results.

Generalised KYP lemma with its application in finite frequency H ∞ distributed filter design for nonideally interconnected networked control systems

This paper addresses the generalised Kalman-Yakubovich-Popov (GKYP) lemma and its application in finite frequency (FF) distributed filter design for networked control systems (NCSs). The network communication channels between subsystems are considered to be affected by uncertainties. Firstly, based on the proposed linear matrix inequality characterisation for an FF region and S-procedure, a GKYP lemma is presented for NCSs. After that, the FF bounded realness and FF positive realness in terms of individual subsystems are studied as special cases of the proposed GKYP lemma. Moreover, as an application of the developed GKYP lemma, the FF distributed filtering problem for nonideally interconnected NCS is studied. Finally, illustrative examples are given to show that the developed EF methods have less conservation than the common method over the entire frequency region.

Hybrid Finite-Frequency Attitude Control Motivated by Volterra Series

An analytical solution of Euler’s equation is exhibited using the Volterra series theory in the frequency domain. Based on the Volterra series, the nonlinear output frequency response functions of Euler’s equation are formulated by a numerical algorithm to reveal an energy-transfer phenomenon. The output responses of Euler’s equation have some higher frequency parts than those of the inputs. It provides motivation to design a finite-frequency controller for Euler’s equation to accommodate the high-frequency parts of the outputs. A hybrid passive/finite gain control scheme fused with the generalized Kalman–Yakubovich–Popov lemma is used to generate a controller that is effective for stabilizing the angular velocities of Euler’s equation. Additionally, quaternions are considered in the proposed hybrid finite-frequency controller to stabilize the attitude of spacecraft. Simulation results are demonstrated to validate the effectiveness of the proposed control schemes.

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.

Improved finite frequency &lt;i&gt;H&lt;/i&gt;&lt;SUB align="right"&gt;∞ 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.

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.

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.

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.

Actuator and sensor faults estimation for discrete-time descriptor linear parameter-varying systems in finite frequency domain

ABSTRACTThis paper investigates the problem of fault estimation for discrete-time descriptor linear parameter-varying systems with actuator faults, sensor faults and external disturbances. First, in order to estimate the actuator and sensor faults simultaneously, an augmented system is constructed, where the auxiliary state vector consists of the original state vector and the sensor fault vector. Then, considering the fact that many faults usually occur in finite frequency ranges, a novel fault estimation observer with finite frequency specifications is designed, where the conservatism can be reduced compared with the traditional fault estimation design methods in full frequency domain. By employing the generalised Kalman–Yakubovich–Popov lemma and two finite frequency H∞ performance indices, the sufficient conditions of the proposed observer are given in terms of linear matrix inequalities. Furthermore, by using slack variables, improved results on the fault estimation observer design are obtained, which are convenient to calculate fault estimation observer parameters for different frequency ranges. An example is presented to illustrate the effectiveness of the proposed method.