Polytopic Linear Parameter-varying Systems Research Articles

Stability and Robustness Analysis and Optimization for Gain-Scheduled Control of Aero-Engines

Abstract Gainscheduled control is widely applied in the aerospace domain, yet the selection of design points for gainscheduling controllers to ensure stability and robustness throughout the range of scheduling variables remains theoretically unguided. Therefore, this paper introduces an analysis and optimization method for the stability and robustness of gain-scheduled control, aimed at providing a theoretical framework for design points selection. Initially, the method characterizes the gainscheduled control system as a polytopic Linear Parameter Varying (LPV) system, wherein the design points of the gainscheduled control system correspond to the vertices of the polytopic LPV system. Subsequently, the method utilizes Linear Matrix Inequality techniques to demonstrate the stability of a polytopic LPV system with a corresponding number of vertices, and by assessing the approximation degree between the polytopic LPV system and the gainscheduled control system with an identical number of design points, it evaluates and ensures the stability of the latter, thereby establishing the minimal requirements for the number of design points. After ensuring stability, the method further refines the number of design points within the gainscheduled control system to meet additional robustness and performance considerations. A case study on turbofan engine controls validates the proposed method. New design points, selected via stability and robustness analysis, enhance the system's steadystate phase margin and robustness against model uncertainties. Moreover, compared to a vgap metric-based method, the two methods exhibit similar performance in terms of stability, robustness, and tracking control. However, the proposed method requires fewer design points, resulting in less conservatism.

Adaptive Damping Control Scheme for Wind Grid-Connected Power Systems With Virtual Inertia Control

With the large-scale wind power integration into the power system, the inertia and damping characteristics of the system have been weakened, which is not conducive to fulfill the grid-friendly operation of wind power. Aiming at this, an adaptive control scheme combining virtual inertia control (VIC) and supplementary damping controller (SDC) is proposed in this paper. Firstly, the mathematical model of wind grid-connected power system with VIC is established. Secondly, the polytopic linear parameter varying (LPV) system is built to describe the uncertainty of system operating points, and the gap metric-based nonlinear evaluation method is used for polytope vertex configuration. Next, Youla-Kucera parametrization are extended to polytopic LPV system to design an adaptive damping control, in which VIC and SDC are coordinated to provide damping and virtual inertia simultaneously. Finally, the proposed adaptive control scheme is compared with several existing control techniques to validate its effectiveness and advantages in suppressing system oscillation in the modified IEEE 39-bus system integrated with 2 wind turbines.

Discrete-time ℋ2 and ℋ∞ poly-quadratic filter designs for polytopic LPV systems

This paper addresses novel discrete-time and filter designs for polytopic linear parameter-varying (LPV) systems. A less conservative approach is employed to obtain the synthesis conditions, such that the polytopic LPV system state-space representation may have all its matrices subject to time-varying parameters and be solved by an efficient procedure. Parameter-dependent Lyapunov functions are used to derive both and discrete-time filter design conditions resulting in a poly-quadratic approach in terms of linear matrix inequalities. A multi-objective solution is also provided. The proposed conditions have the advantage of recovering the established poly-quadratic concept as a particular case, which was an unresolved problem until the present paper. Moreover, using the addressed approach, it is possible to obtain improved quadratic and filters. Numerical examples illustrate the proposed design methods applicability as an observer to estimate LPV system states and its performance in comparison with other approaches.

Policy Gradient Reinforcement Learning for Uncertain Polytopic LPV Systems based on MHE-MPC

In this paper, we propose a learning-based Model Predictive Control (MPC) approach for the polytopic Linear Parameter-Varying (LPV) systems with inexact scheduling parameters (as exogenous signals with inexact bounds), where the Linear Time Invariant (LTI) models (vertices) captured by combinations of the scheduling parameters becomes wrong. We first propose to adopt a Moving Horizon Estimation (MHE) scheme to simultaneously estimate the convex combination vector and unmeasured states based on the observations and model matching error. To tackle the wrong LTI models used in both the MPC and MHE schemes, we then adopt a Policy Gradient (PG) Reinforcement Learning (RL) to learn both the estimator (MHE) and controller (MPC) so that the best closed-loop performance is achieved. The effectiveness of the proposed RL-based MHE/MPC design is demonstrated using an illustrative example.

Robust H2 output-feedback bumpless transfer control of polytopic uncertain LPV systems

The problem of Robust Switching Gain-Scheduled Output-Feedback (RSGSOF) H2 controller synthesis with control bumps limitation is tackled in this paper for polytopic uncertain continuous-time linear parameter varying (LPV) systems. The presented robust switching gain-scheduled control design scheme overcomes the limitation of existing methods based on single quadratic Lyapunov function. However, in practice, RSGSOF controller leads to expressive discontinuities at switching instants. Hence, we aim to investigate of H2 output-feedback Bumpless Transfer Controller (BTC) design problem. In this study, not only stability and an upper bound on the H2-norm performance of the closed-loop system can be guaranteed but also solvability conditions are incorporated in the synthesis procedure to reduce control bumps. The proposed BTC composes multiple LPV controllers with the bumps limitation objective and a Lyapunov-based min-switching rule governing the switching among these controllers. Numerical studies demonstrate the effectiveness of the proposed approach.

Gain Scheduled Adaptive Control Scheme for Damping SSOs in PMSG-Integrated Power System Under High Wind Speed Variability

A gain scheduled adaptive control scheme based on a polytopic linear parameter varying (LPV) system is proposed in this paper to damp sub-synchronous oscillations (SSOs) for power system integrated with permanent magnet synchronous generators (PMSGs) and operating under high wind speed variability. Firstly, the linearized state space model of the PMSG-integrated power system is established. Secondly, a polytopic LPV system is developed to account for the stochastic drift behavior and wind speed variability. Next, a gap metric-based nonlinearity measurement method is used for configuring the vertices of the polytope. The adaptive polytopic LPV controller is solved using linear matrix inequality and gain scheduling control theory, while mode identification based on a sliding window fast Fourier transform is employed for adaptively adjusting the polytopic LPV controller. Finally, the proposed polytopic LPV adaptive control scheme is compared with three other control techniques to demonstrate its effectiveness in suppressing SSOs.

Piece‐wise ellipsoidal set‐based model predictive control of linear parameter varying systems with application to a tower crane

AbstractIn this paper, a model predictive control algorithm for polytopic linear parameter varying (LPV) systems is proposed, which is based on an offline calculated sequence of robust one‐step controllable sets. At each sampling instant, a finite‐time optimal control problem is solved subject to a stabilizing constraint on the first state to ensure stability and recursive feasibility guarantees independent of the length of the prediction horizon and the cost function. An algorithm is proposed to compute a sequence of piece‐wise ellipsoidal one‐step controllable sets. The systematic application of the developed algorithm is possible via tensor product model transformation to a class of nonlinear systems. The size and shape of each set depend on the vertices of the polytopic LPV system and influences the control performance. By using the tensor product model transformation, different vertex representations can be obtained automatically. Furthermore, a trade‐off between model complexity and accuracy can be chosen, and the computational effort of the set calculation can be influenced. The developed method is applied to a tower crane system. The tower crane is modeled in a cascaded form with three separate subsystems. The couplings between the subsystems are treated as varying parameters. Each subsystem of the crane is transformed into a polytopic LPV system using the tensor product model transformation. By discarding the nonzero singular values, the complexity of the polytopic LPV model is reduced in terms of the number of vertices.

Discrete‐time state‐feedback control of polytopic LPV systems

AbstractThis paper focuses on a new state‐feedback control synthesis for discrete‐time polytopic linear parameter‐varying (LPV) systems. The synthesis conditions have been derived using an approach less conservative than those currently found in the literature, which allows to deal with polytopic LPV systems subject to time‐varying parameters in all matrices of its state‐space representation. Based on parameter‐dependent Lyapunov functions, a set of linear matrix inequalities‐based conditions is provided to obtain a discrete‐time state‐feedback controller. Another significant advantage of the proposed conditions is the ability to include the poly‐quadratic condition as a particular case, presenting a solution to this so far unresolved problem. In addition, using the same concept, an improved condition for quadratic control synthesis is given. Finally, the effectiveness of the proposed state‐feedback control design method is assessed and compared to that of similar approaches by means of numerical examples.

Inexact induced L2 observer‐based control of polytopic LPV systems: application to clutchless automated manual transmission of pure electric vehicles

A novel method for observer-based control of disturbed polytopic linear parameter varying (LPV) systems with inexactly measured scheduling parameters is investigated in this study. Despite the imperfect scheduling parameters knowledge, the proposed control method ensures the closed-loop induced L 2-norm performance criterion. Unlike the previous methods, the inexact scheduling parameters are not assumed to be proportional to the original scheduling parameters of the LPV system. Employing both cases of parameter-dependent and parameter-independent Lyapunov functions, sufficient design conditions in terms of linear matrix inequalities are provided using Finsler's lemma. Numerical examples are included to demonstrate the efficiency of the proposed method and its superiority over the previous techniques. Moreover, a practical case study, i.e. clutchless automated manual transmission system is considered and controlled by the developed approach numerically.

Gain-scheduled control for morphing aircraft via switching polytopic linear parameter-varying systems

This paper addresses the gain-scheduled output feedback control for morphing aircraft (MA) in full envelope using switching polytopic linear parameter-varying (SPLPV) theory. The SPLPV model of MA is derived by scheduled parameter interval division and convex decomposition. Considering average dwell time (ADT) and mode-dependent average dwell time (MDADT) switching logics, sufficient conditions to ensure that the SPLPV systems are exponential stable with weighted disturbance attenuation indexes are presented based on the multiple parameter-dependent Lyapunov functions (MPDLFs) technique, and the comprehensive algorithm for the controller solution is obtained. To reduce the computational burden and ease controller implementation, a finite number of linear matrix inequalities (LMIs) are developed for solving controller by proposing the intermediate controller variables are depend affinely on the scheduled parameter. In addition, the control matrix of linear parameter-varying (LPV) system is not limited to be constant, which greatly expands the application scope of the proposed method. The simulation validation is carried out on the SPLPV control for MA.

Coprime factorization of polytopic LPV systems based on a homogeneous polynomial approach

AbstractThis paper presents novel linear matrix inequalities (LMI)‐based conditions to address the coprime factorization problem for polytopic linear parameter‐varying (LPV) systems in both the continuous‐ and discrete‐time cases. The proposed conditions rely on the use of homogeneous polynomial parameter‐dependent (HPPD) matrix solutions of arbitrary degree g, which are used to obtain coprime factorization descriptions subject to time‐varying parameters defined inside a polytopic structure with known vertices. For a given degree g, synthesis of right and left coprime factorizations in terms of two special state‐feedback and filtering problems under quadratic performance are provided. Existence conditions are given from linear matrix inequalities relaxations based on an extension of Pólya's theorem and the HPPD approach. The effectiveness of the proposed conditions is evaluated by means of numerical examples.

Event-triggered and self-triggered H ∞ output tracking control for discrete-time linear parameter-varying systems with network-induced delays

In this paper, we address the event-triggered and self-triggered output tracking problem of discrete LPV systems with network-induced delay. Considering the time delay and external disturbance, we formulate the closed-loop event-triggered output tracking problem into a time-delayed discrete polytopic linear parameter-varying system with relative event-triggering mechanism. Then, by constructing the parameter-dependent Lyapunov–Krasovskii functional, we derive a sufficient condition such that the closed-loop system is global asymptotic stable and satisfies the output tracking performance. Further, we develop an approach to design the event-triggering mechanism and output tracking controller. The parameters of event-triggering mechanism and the gains of output tracking controller are obtained by solving linear matrix inequalities. Moreover, we extend the results of the event-triggered control to the self-triggered output tracking control. This not only mitigates the usage of network bandwidth but also avoids the use of additional hardware. Finally, the numerical simulation is included to illustrate the usefulness and effectiveness of the proposed approach.

Robust Controller Synthesis for Airborne Electro-Optical Device Line-of-Sight Stabilization System with Saturating Actuators

A technique of multiobjective robust controller synthesis for an electro-optical device line-of-sight stabilization system with saturating actuators is developed on the basis of polytopic linear parameter-varying system theory. From the computational point of view, the synthesis procedure is reduced to convex optimization under constraints expressed in the form of linear matrix inequalities. The multiobjective synthesis of the robust control law for the stabilization system being considered is performed.

Stability and Stabilization for Polytopic LPV Systems with Parameter-Varying Time Delays

In this paper, we deal with the problem of stability and stabilization for linear parameter-varying (LPV) systems with time-varying time delays. The uncertain parameters are assumed to reside in a polytope with bounded variation rates. Being main difference from the existing achievements, the representation of the time derivative of the time-varying parameter is under a polytopic structure. Based on the new representation, delay-dependent sufficient conditions of stability and stabilization are, respectively, formulated in terms of linear matrix inequalities (LMI). Simulation examples are then provided to confirm the effectiveness of the given approach.

Stability analysis of LPV systems: Scenario approach

This paper discusses a ‘scenario’ approach to prove decay-rate stability of discrete-time polytopic linear parameter-varying systems, dealing with sets of sequences of vertex models of different length. When all sequences have the same length, parameter-trajectory dependent results in earlier literature are obtained as particular cases. The approach in this paper discusses ‘classical’ stability, without the need of probabilistic ingredients present in other scenario-based ideas in literature. A numerical example shows that the proposal achieves a sensible tradeoff between proven performance and computing requirements.

Gain-Scheduling Attitude Control for Complex Spacecraft Based on HOSVD

In this paper, a robust gain-scheduling attitude control scheme for spacecrafts with large rotational appendages is proposed. First, by introducing the higher-order singular value decomposition (HOSVD) method, a polytopic linear parameter varying (LPV) model with a family of weighting coefficients is developed based on the kinetics of a flexible spacecraft. This model eliminates the need of verifying all the gridding points, which is required in conventional controller synthesis process, and reduces the calculation complexity. Second, a generalized plant is derived to guarantee both the system robust stability and the tracking performances. Based on the LPV control theory, a less conservative controller synthesis condition for the polytopic LPV system is deduced. With an online tuning unit, the convex combination of every vertex controller is obtained. For control implementation, the present scheduling parameter is taken as an input for the tuning unit. Numerical results demonstrate the effectiveness and efficiency of the proposed control scheme.

Joint State and Parameter Estimation for Discrete-Time Polytopic Linear Parameter-Varying Systems

Linear parameter-varying systems are very suitable for modelling nonlinear systems, since well-established methods from the linear-systems domain can be applied. Knowledge about the scheduling parameter is an important condition in this modelling framework. In case this parameter is not known, joint state and parameter-estimation methods can be employed, e.g., using interacting multiple-model estimation methods, or using an extended Kalman filter. However, these methods cannot be directly used in case the parameters lie in a polytopic set. Furthermore, these existing methods require tuning in order to have convergence and stability. In this paper, we propose to solve the joint-estimation problem in a two-step, Dual Estimation approach, where we first solve the parameter-estimation problem by solving a constrained optimisation problem in a recursive manner and secondly, employ a robust polytopic observer design for state estimation. Simulations show that our novel method outperforms the existing joint-estimation methods and is a promising first step for further research.

A New Polytopic Modeling with Uncertain Vertices and Robust Control of Robot Manipulators

This paper proposes \(H_{2}\)/\(H_{\infty }\) full state-feedback synthesis for robot manipulators, using uncertain polytopic linear parameter-varying (LPV) system modeling, with pole placement constraints to assign the poles of closed-loop system in a desired linear matrix inequality (LMI) region. The desired state trajectory of the system is used for generating an uncertain polytopic model of the system applying usual Lagrangian equations. The control gain matrix is derived by solving a set of LMIs to design a robust pole placement controller such that a prescribed mixed \(H_{2}\)/\(H_{\infty }\) performance is fulfilled and the response of the manipulator has a proper damping ratio. A sufficient condition is proposed to guarantee the asymptotic stability of the closed-loop uncertain polytopic LPV system against the uncertainties on the vertices. The proposed scheme is applied to controller synthesis of a two-degree-of-freedom manipulator trajectory-tracking problem. The simulation results show the effectiveness of the proposed controller.

H−/H∞ fault detection observer design based on generalized output for polytopic LPV system

This paper proposes an H−/H∞ fault detection observer design method by using generalized output for a class of polytopic linear parameter-varying (LPV) system. First, with the aid of the relative degree of output, a new output vector is generated by gathering the original and its time derivative. The actuator fault is introduced into the measurement equation of the new system. An H−/H∞ observer is designed for the new LPV polytopic system to guarantee the robustness against disturbances and to improve the fault sensitivity, simultaneously. The existence conditions of the H−/H∞ observer are given and solved by a set of linear matrix inequalities (LMIs). Finally simulation results are given to illustrate the effectiveness of the proposed method.

Robust fault reconstruction via learning observers in linear parameter‐varying systems subject to loss of actuator effectiveness

In this study, the problem of robust fault reconstruction in polytopic linear parameter-varying systems, subject to loss of actuator effectiveness and external disturbances, is investigated using a learning observer (LO). A polytopic LO is proposed to achieve system state estimation and actuator fault reconstruction at the same time. The proposed LO is sensitive to incipient faults because no chattering is induced. Both constant and time-varying faults can be accurately reconstructed and detected. Moreover, simultaneous fault reconstruction and detection can be performed using a single LO because the reconstructed fault signal can be employed as a fault alarm. The stability and convergence of polytopic LO and the uniformly ultimately boundedness of the dynamic error system is proved using Lyapunov stability theory and H ∞ techniques. A systematic design of the LO is effectively solved using standard linear-matrix-inequality techniques. At last, simulation results on a satellite system for attitude control verify the effectiveness of the proposed fault-reconstruction approach.