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.

Reliable finite-frequency H ∞ control for switched systems with actuator faults and mode-dependent average dwell time

This paper deals with the reliable finite-frequency H ∞ control problem for switched systems with actuator faults and mode-dependent average dwell time (MDADT). The objective of designing reliable controllers is to guarantee the stability and robustness of the closed-loop system in the case of actuator faults. The actuator fault is solved using the convex combination method. A finite-frequency H ∞ performance index is first defined in the frequency domain, and then a finite-frequency performance condition is derived by the generalised Kalman-Yakubovi c ˘ -Popov (GKYP) lemma. Combined with the MDADT approach and linear matrix inequality technology, a sufficient condition for the existence of a set of reliable finite-frequency H ∞ controllers is derived. As a comparison, a sufficient condition is also given for the existence of a set of standard finite-frequency H ∞ controllers. Finally, the effectiveness and feasibility of the proposed method are verified using a tilt-rotor aircraft. In addition, simulations and comparisons show that the designed reliable controller can stabilise the system even if actuator faults occur.

Finite-frequency fuzzy fault-tolerant static output feedback H∞ control for Diesel engine air-path system

This paper focuses on the design of H∞ finite-frequency (FF) fault-tolerant static output feedback (FTSOFC) problem of Diesel engine air-path system with consideration of external disturbances and actuator faults. Initially, a Diesel engine air-path nonlinear model is described by Takagi-Sugeno (T-S) fuzzy model based descriptor approach. The aim is to regulate intake and exhaust manifold pressures to the desired reference pressures by controlling the Geometry Turbine (VGT) and Exhaust Gas Recirculation (EGR) valves. Then, the robust integrator-based control strategy is developed to track the desired reference signals despite the presence of disturbances and actuator faults. By using the extended Generalized Kalman Yakubovich Popov (GKYP) lemma, Lyapunov functions and independent slack matrices, sufficient conditions are established to ensure both the good tracking of reference pressures and the prescribed H∞ performance with FF domain of the fuel flow variation and actuator faults. Finally, simulation results are given to demonstrate the effectiveness of the proposed approach.

Finite Frequency H∞ Control of Fractional-Order Continuous–Discrete 2-D Roesser Models

The H∞ control problem of fractional-order continuous-discrete 2D Roesser models in the finite frequency is investigated in this paper. Firstly, based on the frequency-partitioning method and generalized Kalman–Yakubovič–Popov lemma, the finite frequency bounded real lemmas in form of linear matrix inequalities (LMIs) are proposed, which can be solved by LMI solvers immediately. Then, the H∞ state feedback controller is designed, and the existence conditions are established in form of bilinear matrix inequalities (BMIs). Moreover, by the provided iterative algorithm, these BMI-based conditions can be solved based on LMIs. Finally, the numerical examples are provided verify the validity of our results.

H ∞ negative imaginary static output feedback controller for low frequency networked control systems

In this paper, an H ∞ negative imaginary static output feedback (SOF) controller is designed for a low frequency (LF) networked control system (NCS) with time-delays. A difference operator is introduced to deal with the time-delays in a feedback configuration via integral quadratic constraints (IQCs). Both H ∞ performance and negative imaginariness are analysed via LF IQCs and a generalised Kalman–Yakubovich–Popov (GKYP) lemma. A two-stage algorithm is presented to obtain the H ∞ negative imaginary SOF controller for the LF NCS with time-delays. Simulation results are given to show the effectiveness and superiority of the H ∞ negative imaginary SOF controller.

Finite Frequency H∞ Control for Doubly Fed Induction Generators with Input Delay and Gain Disturbance

Due to the rapid development of wind power, the stable operation of doubly fed induction generators (DFIGs) has attracted much attention. This paper focuses on the finite frequency (FF) H∞ control for the DFIG with input delay, aiming to reduce the effects of current harmonic interferences and gain disturbances on the DFIG and improve the stability of the system. First, a DFIG state–space model with input delay under current harmonics was constructed. Second, based on the DFIG state–space model, an FF H∞ state-feedback controller was designed from the frequency domain perspective, which makes the DFIG stable and robust against harmonic interferences and gain disturbances. Third, via the generalized Kalman–Yakubovich–Popov (GKYP) lemma and the Lyapunov theory, the FF H∞ performance was evaluated in the form of linear matrix inequalities (LMIs), and then the state feedback FF H∞ controller was designed. Finally, the simulation results showed the efficiency of the proposed approach.

Finite‐frequency output feedback MPC for Markov jump systems

AbstractThis paper investigates a general design framework of dynamic output feedback model predictive control (DOFMPC) for Markov jump systems within both time and frequency domain. Such a design with guaranteed H∞ and quadratic performance is formulated by a standard semi‐definite programming (SDP), and it is achieved by employing a special congruence transformation. The SDP condition greatly reduces the computational effort by eliminating bilinear matrix inequalities or equation constraints reported in existing references. Specifically, the H∞ norm of the transfer function is optimized within three types of frequency ranges on account of generalized Kalman–Yakubovic–Popov (GKYP) lemma. The quadratic index is optimized online via SDP. Finally, we verify the feasibility and effectiveness of the proposed method from both theoretical and practical point of view.

Fault detection in wind turbine transmission systems via finite frequency observers

AbstractThis paper addresses fault detection for wind turbine transmission system (WTTS) subject to actuator stuck faults and wind disturbances. Taking into account the frequency domain characteristics of such faults and disturbances, a modified observer is designed in terms of finite frequency (FF) performance indices that have detection performances superior to some formerly proposed ones. With the generalized Kalman–Yakubovich–Popov (GKYP) lemma and the Lyapunov theory, the FF performance indices are related to linear matrix inequalities (LMIs), and parametrization of the corresponding FF modified observer is completed by convex optimization under LMI constraints. One typical advantage of the modified observer is that fault and disturbance signals in distinctive FF domains can be dealt with simultaneously. This makes the residual system sensitive to actuator stuck faults and robust against external wind disturbances. Simulations are included to show that the design method achieves fault detection performances better than those of the entire frequency observers.

Vehicle Actuator Fault Detection With Finite-Frequency Specifications via Takagi-Sugeno Fuzzy Observers: Theory and Experiments

This article presents a new nonlinear observer-based method to detect the faults of both steering and torque actuators of autonomous ground vehicles. To this end, the nonlinear vehicle system is reformulated in a Takagi-Sugeno (TS) fuzzy model with both measured and unmeasured nonlinear consequents, called N-TS fuzzy model. Differently from the classical TS fuzzy technique with linear consequents, this N-TS fuzzy reformulation enables an effective use of differential mean value theorem to deal with unmeasured nonlinearities, which is known as a major challenge in TS fuzzy observer design. Moreover, the N-TS fuzzy form also allows reducing not only the design conservatism but also the numerical complexity of the design conditions as well as the observer structure, which is crucial for real-time vehicle application. To minimize the disturbance effect with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_\infty$</tex-math></inline-formula> performance and maximize the fault sensibility with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_-$</tex-math></inline-formula> performance, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> information on the disturbance/fault frequency ranges is taken into account in the N-TS fuzzy observer design via the generalized Kalman–Yakubovich–Popov (KYP) lemma. Based on Lyapunov stability theory, the design of the proposed multiobjective <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_-/\mathscr {H}_\infty$</tex-math></inline-formula> N-TS fuzzy fault detector is recast as an optimization problem with strict linear matrix inequality (LMI) constraints, which can be effectively solved with numerical solvers. Both numerical and experiments are performed under realistic driving conditions to demonstrate the theoretical and practical interests of the new finite-frequency fuzzy fault detection method.

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.

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.

Finite-Frequency $\mathcal {H}_{-}/\mathcal {H}_{\infty }$ Memory Fault Detection Filtering Design for Uncertain Takagi–Sugeno Fuzzy Affine Systems

This article is concerned with the finite-frequency <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{-}/\mathcal {H}_{\infty }$</tex-math></inline-formula> memory fault detection filtering problem for discrete-time Takagi&#x2013;Sugeno fuzzy affine systems with norm-bounded uncertainties. The objective is to design a piecewise affine memory filter by using system historical information such that the resulting closed-loop filtering error system is asymptotically stable with the prescribed finite-frequency <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{-}/\mathcal {H}_{\infty }$</tex-math></inline-formula> performance. Based on the generalized Kalman&#x2013;Yakubovi&#x010D;&#x2013;Popov lemma combined with the celebrated <inline-formula><tex-math notation="LaTeX">$\mathcal {S}$</tex-math></inline-formula>-procedure, new sufficient conditions for the fuzzy affine filtering error system to have the finite-frequency <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{-}/\mathcal {H}_{\infty }$</tex-math></inline-formula> performance are given at first. By further using piecewise fuzzy quadratic Lyapunov functions and Projection lemma, the filtering analysis results for the filtering error system to be asymptotically stable with the prescribed finite-frequency <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{-}/\mathcal {H}_{\infty }$</tex-math></inline-formula> performance are obtained. Then, the filtering synthesis is carried out with the aid of matrix inequality convexification techniques, and the synthesis results are described in terms of linear matrix inequalities. It is further shown that a better filtering performance can be achieved by using more system historical information. Finally, simulation is provided to verify the effectiveness of the proposed approach.

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.

LMI-Based Stability Analysis of Continuous-Discrete Fractional-Order 2D Roesser Model

The manuscript investigates the problem of structural stability of continuous-discrete fractional-order 2D Roesser model. This model includes one continuous fractional-order dimension with fractional-order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,2)$ </tex-math></inline-formula> and one discrete dimension. By the equivalent transform and generalized Kalman–Yakubovič–Popov lemma, the necessary and sufficient stability conditions for structural stability of continuous-discrete fractional-order 2D Roesser model are established. Our results are all in the form of linear matrix inequalities. And compared with the existing results, our results have no conservativeness and can be applied in the cases with fractional-order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,2)$ </tex-math></inline-formula> in the continuous fractional-order dimension. Illustrated examples are provided to verify the effectiveness of our results.

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.

A New Passivity Preserving Model Order Reduction Method: Conic Positive Real Balanced Truncation Method

This article is dedicated to model order reduction of linear time-invariant passive systems, i.e., positive real (PR) systems, based on the balanced truncation (BT) concept. The main feature of this article is that we have used the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , in model order reduction for the first time in the proposed method, which results in more accurate reduced-order models. It is worth noting that this hypothesis has been never made before in any other model order reduction techniques. Furthermore, new gramians are calculated corresponding to their associated new Riccati equations and Lur’e equations, which are in turn achieved by using Kalman–Yakubovic–Popov (KYP) lemma and linear matrix inequalities (LMI). Thereafter, a novel passivity preserving model order reduction algorithm, which takes the phase angle of the transfer function, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G(s)$ </tex-math></inline-formula> , into account, is presented. It is demonstrated that the proposed method is a generalization of the positive real balanced truncation (PRBT) method, and it is perfectly capable of providing more accurate approximation error compared to PRBT. Finally, numerical examples are included to figure out the effectiveness of the presented method.

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.

Improved filtering H∞ finite frequency of Takagi-Sugeno fuzzy systems

The work treats the filter H∞ finite frequency (FF) in Takagi-Sugeno (T-S) two dimensional (2-D) systems described by Fornasini-Marchesini local state-space (FM LSS)models. The goal of this work is to find an FF H∞ T-S fuzzy filter model design in such a way that the error system is stable and has a reduced FF H∞ performance over FF area swith noise is established as aprerequisite. Via the use of the generalized Kalman Yakubovich Popov (gKYP) lemma, Lyapunov functions approach, Finsler’s lemma, and parameterize slack matrices, new design conditions guaranteeing the FF H∞ T-S fuzzy filter method of FM LSS models are developed by solving linear matrix inequalities (LMIs). At last, the simulation results are provided to show the effectiveness and the validity of the proposed FF T-S fuzzy of FM LSS models strategy by a practical application has been made.

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.