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.

Event‐triggered based finite‐frequency control for vehicle electrohydraulic suspension with force tracking

AbstractThis study investigates the problem of event‐triggered based finite‐frequency control for vehicle electrohydraulic suspensions with force tracking. A novel hierarchical control methodology is proposed by designing event‐triggered finite frequency and active force tracking controllers to improve the suspension performance. In practical application, active suspension system can be constructed as the network control system under consideration of the electrohydraulic actuator's nonlinear dynamics and parametric uncertainties, and in‐vehicle network delay in a unified framework. Then, due to the more sensitiveness of the human body to vertical vibrations in 4–8 Hz and limited in‐vehicle network communication resource, an event‐triggered finite frequency controller is developed based on the generalized KYP lemma and Lyapunov stability theory. The triggered condition is designed for the network control system with in‐vehicle network delay. Other physical constraints including suspension working space, tire dynamic load and actuator saturation are also introduced in this controller design. It can generate the target force to satisfy suspension performance requirements under each triggered instant period. Furthermore, since the nonlinearity and uncertainty always exist in the electrohydraulic actuator, a filter‐based adaptive sliding mode control method is employed to design the active force tracking controller. It can precisely drive electrohydraulic actuator to track the target constant force generated by event‐triggered finite frequency controller. Finally, the results validate the effectiveness and saving communication resource capability of the proposed control methodology.

H∞ observer-based controller synthesis for fractional order systems over finite frequency range

This paper focuses on the H∞ observer-based control problem for fractional order systems (FOSs) over finite frequency range. Firstly, as an extension to the KYP lemma, the structure singular value–μ and its upper bound are analyzed and determined. The necessary and sufficient conditions of the H∞ performance for FOSs over finite frequency range are given through the generalized μ analysis method. This method is different from the traditional μ-analysis method, which is only suitable for integer order systems. Secondly, according to these obtained LMI-based criteria for H∞ performance index and a key projection lemma, we design an H∞ observer-based controller to stabilize the estimation error and decrease the H∞ norm for the FOS simultaneously. With the help of the matrix congruence transformation, the feedback gain matrix is parameterized by a scalar matrix. Thirdly, to deal with the nonlinear terms in matrix inequalities, an iterative linear matrix inequality (ILMI) algorithm is developed. Finally, two numerical examples are provided to explain the correctness of the established results.

Fault Detection and Reliable Controller Design for Fractional-Order Systems Based on Dynamic Observer

In this paper, the problem of the fractional-order linear systems’ simultaneous fault detection and reliable control (SFDC) in the finite frequency domain is investigated. A dynamic observer aimed at detecting faults is designed, which also generates state estimation signals. In particular, it is found that the proposed dynamic-observer-based controller can achieve a better H∞ performance. We derived the detector and controller’s design conditions to achieve H− fault sensitivity performance and H∞ interference attenuation performance in the limited frequency domain based on the generalized KYP lemma, which can achieve a better disturbance attenuation performance and fault sensitivity performance. Finally, the simulation results show the designed method’s effectiveness.

A Structure Exploiting SDP Solver for Robust Controller Synthesis

In this paper, we revisit structure exploiting SDP solvers dedicated to the solution of Kalman-Yakubovic-Popov semi-definite programs (KYP-SDPs). These SDPs inherit their name from the KYP lemma and they play a crucial role in e.g. robustness analysis, robust state feedback synthesis, and robust estimator synthesis for uncertain dynamical systems. Off-the-shelve SDP solvers require O(n6) arithmetic operations per Newton step to solve this class of problems, where n is the state dimension of the dynamical system under consideration. Specialized solvers reduce this complexity to O(n3). However, existing specialized solvers do not include semi-definite constraints on the Lyapunov matrix, which is necessary for controller synthesis. In this paper, we show how to include such constraints in structure exploiting KYP-SDP solvers.

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 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.

LMI-Based Stability Conditions for Continuous Fractional-Order Two-Dimensional Fornasini-Marchesini First Model

In this brief, the structural stability of continuous fractional-order (FO) two-dimensional (2D) Fornasini-Marchesini (FM) first model is investigated. Firstly, the bivariate polynomial-based stability condition is equivalently transformed to be in more tractable form. Then, based on generalized KYP lemma, the transformed conditions are further reduced into the feasibility problem of linear matrix inequalities (LMIs). The main contribution is to establish the LMI-based stability conditions for continuous FO 2D FM first model. The results can be applied in the cases with FO <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha _{i} \in (0,2)$ </tex-math></inline-formula> and be no conservative when the FO in the second dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha _{2} \in [1,2$ </tex-math></inline-formula> ). Lastly, the example is provided to verify the efficiency.

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.

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.

Zonotopic fault detection observer for discrete-time descriptor systems considering − fault sensitivity

This paper addresses a zonotopic fault detection (FD) observer for discrete-time descriptor systems subject to additive actuator faults. The considered descriptor systems are also perturbed by unknown-but-bounded uncertainties including state disturbances and measurement noise. Under a set-based framework, the effects of uncertainties and faults are separated into two state zonotopes. Based on this decomposition, the FD observer gain is designed to be robust against uncertainties and meanwhile sensitive to faults based on a index in a finite-frequency domain. Then, two linear matrix inequality (LMI) conditions are obtained to design an observer that achieves robustness and fault sensitivity at the same time. The generalised KYP lemma is applied to address the fault sensitivity condition. The optimal FD observer gain can be obtained via an offline design procedure. Finally, the proposed fault detection method is applied to a chemical mixing system and the effectiveness is shown.

Robust PD-Type Iterative Learning Control of Discrete Linear Repetitive Processes in the Finite Frequency Domain

This paper studies a robust iterative learning control design for discrete linear repetitive processes in the finite frequency domain. Firstly, the state-space model of the iterative learning process is deduced. Then the dynamic performance condition of the control system in the finite frequency domain is derived by combining it with the stability theory of discrete linear repetitive processes. The system performances in the finite frequency domain are then transformed into the corresponding solutions of the linear matrix inequality by using the generalised KYP lemma. Finally, an integrated state feedback PD-type iterative learning control strategy is proposed. The robust control problem with norm-bounded uncertainty and convex polyhedral uncertainty are also considered in this paper. The simulation of the injection velocity in injection molding verified that the proposed methods in this paper are more effective than the P-type state feedback iterative learning control algorithm.

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.