Polytopic Linear Parameter Varying Research Articles

Multi‐variable and multi‐objective gain‐scheduled control based on Youla‐Kucera parameterization: Application to autonomous vehicles

AbstractThis paper presents a Youla‐Kucera based interpolation between a set of Linear Parameter‐Varying (LPV) controllers, each one being a gain‐scheduled of Linear Time‐Invariant (LTI) controllers designed separately for different operating points. The gain‐scheduling is achieved based on Youla‐Kucera (YK) parameterization. A generalized LPV‐YK control structure is designed to interpolate between various LPV controllers. The closed‐loop system is proved to guarantee the quadratic stability for any continuous/discontinuous interpolating signals in terms of a set of Linear Matrix Inequalities (LMIs). The proposed method can help multi‐variable and multi‐objective systems to achieve high performances at different operating conditions and different critical situations regardless of the interpolation rate. A numerical example is simulated to show the importance of the proposed method to achieve different objectives for lateral control of autonomous vehicles. In addition, the approach has been tested on a real Renault ZOE vehicle to validate its real performance, and compare it with a standard polytopic LPV controller.

Polytopic LPV modeling and gain scheduling [formula omitted] control of ball screw with position- and load-dependent variable dynamics

Ball screw drive (BSD) is a precision transmission mechanism widely used in high-precision positioning or tracking systems. The dynamic behavior of BSD varies with position and load, which causes tracking errors and poor robustness. Therefore, this paper proposes a polytopic linear parameter varying (LPV) model to express the varying dynamic behavior of BSD. The parameters of the LPV model are identified by the closed-loop frequency domain method and Levenberg–Marquardt iterative algorithm. Based on the polytopic LPV model, an LPV gain scheduling (GS) H∞ controller is proposed for the BSD with varying dynamics. Specifically, the controller is designed through polytope-based GS representation and mixed sensitivity synthesis. The most significant part is the proposal of a GS H∞ control algorithm to implement controller parameters that change with changing dynamics. Moreover, the stability of the closed-loop system is achieved by quadratic stabilization with state feedback. Finally, identification experiments and trajectory-tracking comparative experiments are carried out. The experimental results demonstrate that the proposed polytopic LPV modeling and GS H∞ control synthesis are effective in achieving accurate trajectory tracking and excellent robustness.

Robust Switched Velocity-Dependent Path-Following Control for Autonomous Ground Vehicles

This paper proposes a switched velocity-dependent path-following control method for autonomous ground vehicles under uncertain cornering stiffness and time-varying velocity. A switched polytopic linear-parameter-varying (LPV) model combining path-following and vehicle lateral dynamics is established, where the velocity-dependent parameters are divided into several switched regions. In each region, a trapezoidal polytope is adopted to describe the varying parameters. A global <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> performance analysis is carried out for the switched polytopic LPV vehicle system based on the average dwell time approach with multiple Lyapunov functions. Based on the analysis, a switched velocity-dependent controller is synthesized to achieve a desired path-following performance by regulating the steering angle of the front wheels. The proposed switched path-following control specializes in handling large range of varying velocity, and achieves superior path-following performance compared with the conventional gain-scheduling control. Simulations are carried out for typical road maneuvers to illustrate the advantages of the proposed method. The results are also applicable to other LPV analysis and synthesis conditions involving time-varying parameters of large varying ranges.

A robust adaptive linear parameter-varying gain-scheduling controller for aeroengines

This paper presents an on-board model-based adaptive linear parameter-varying (LPV) gain-scheduling controller for aeroengines exhibiting strong nonlinearity, uncertainty, and component performance degradation. A novel hybrid on-board adaptive model is proposed using a dual-loop (fast-slow) structure, which consists of an adaptive LPV model as the fast loop to guarantee the on-board performance, and a component-level model as the slow loop to provide accurate model information at a slow update frequency, without violating the real-time requirement of the fast loop. An innovative mixed adaptive extended Kalman filter is designed to estimate and update the model, by introducing an extended-Kalman-filter-based covariance scaling mechanism and updating the system noise covariance matrix by employing full covariance estimation. The component-level adaptive nonlinear model and mixed adaptive extended Kalman filter increase the accuracy of the on-board model, while the adaptive LPV model enhances its real-time performance. Finally, a robust polytopic adaptive LPV gain-scheduling controller is designed based on the adaptive LPV model by introducing a new scheduling variable, i.e., health parameters estimated by the on-board adaptive model. The proposed adaptive model and controller are implemented to a high fidelity nonlinear aeroengine model. Simulation results demonstrate that the estimation error of health parameters and component degradation level of the proposed method is 0.0763%, which gives an order of magnitude higher health parameter and component degradation level estimation accuracy compared with the other commonly applied models. The proposed model-based adaptive LPV gain scheduling control scheme has a steady-state error of 0.0028%, gives one order of magnitude higher control accuracy than that of three state-of-the-art baseline controllers while demonstrating significantly reduced overshoot and adjustment time. This verifies the robustness and effectiveness of the proposed controller, and verifies that the model can meet the technical requirements of the engine control system and significantly improve the engine control performance.

H∞ Loop Shaping Using Polytopic Weights and Pole Assignment to Missile Autopilot

Modern missiles must deal with stringent performance requirements and ensure robustness across a wide range of operating conditions during flight time. Classic gain-scheduling designs have been successful in practice, but do not provide theoretically ensured bounds on both performance and robustness. Controllers are interpolated at intermediate operating conditions and switching them may cause instability. On the other hand, polytopic linear parameter-varying (LPV) controllers avert this switching and use real-time information about the plant, in order to smooth out the gain scheduling. We hereby propose a novel procedure to yield an output feedback LPV controller that ensures robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{H}_{\infty} $ </tex-math></inline-formula> performance to a missile longitudinal autopilot. Our novel approach considers the four-block loop-shaping <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{H}_{\infty} $ </tex-math></inline-formula> control theory with polytopic LPV weights that use polytopic coordinates of the LPV plant. One is capable of adjusting the singular values of the open-loop plant individually at the polytope vertices, and benefits from a trade-off between linear matrix inequalities (LMI) based optimization tools and the designer experience. This can be construed as a natural extension of the traditional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{H}_{\infty} $ </tex-math></inline-formula> loop-shaping method that uses linear time-invariant (LTI) weights. Assuming the scheduling variables are frozen, we also include LMI conditions for assigning closed-loop poles and hence circumvent controller order reduction. Nonlinear simulations assess the proposed autopilot, and results show an improved robust stability margin, in addition to an improved response to the acceleration command, concerning the LTI-based approach.

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.

Tube-MPC Control via Notch Filter for Flexible Air-Breathing Hypersonic Vehicle with Actuator Fault

In this paper, the tube model predictive control (Tube-MPC) via notch filter is proposed for flexible air-breathing hypersonic vehicle with actuator fault. Firstly, considering the low-order flexible modes, the polytopic linear parameter varying (LPV) model of flexible air-breathing hypersonic vehicle is established using Jacobian linearization and tensor product. Actuator fault and parameter uncertainty are transformed into a lumped disturbance term. The baseline Tube-MPC trajectory tracking controller is developed by designing robust auxiliary feedback control law and nominal control law. In order to weaken the effect of flexibility, a notch filter is introduced to eliminate the frequency response peak of flexible modes. And the effect of notch filter on control performance and stability is analysed. Secondly, an alternative controller design strategy is proposed to ensure stability of the closed-loop system. The filter and the original nonlinear model are combined together such that the new polytopic LPV model is codesigned. On the basis, the adjustable Tube-MPC controller is developed to improve the control performance and ensure the stability of the system. Finally, the effectiveness of the two controllers is verified by simulations.

Development of an autonomous fog computing platform using control-theoretic approach for robot-vision applications

This paper presents the dynamic modelling and linear matrix inequality (LMI) based controller design of a distributed fog computing framework for real-time robot vision applications. A mobile robot vision system acquires the images from an application environment such as a warehouse, where articles are stacked in numerous racks. We characterise the mobile robot vision data (MRVD) using frames per second (FPS) and the image resolution. From the MRVD, object detection is performed by an open-source deep learning (DL) platform for detecting and localising various objects. However, with higher FPS together with high-resolution images, the processing time by the DL algorithm increases significantly. This necessitates the deployment of a distributed computing platform with several computing nodes. In this work, we deploy a distributed fog computing environment (DFCE) for the real-time object detection in an application environment.The processing time required to handle the MRVD is called the service time. However, for efficient auto-scaling performance, the mathematical model of the DFCE, taking into consideration the characteristics of the MRVD is necessary. In this context, we envisage the application of control theory to build the dynamic model of the DFCE. A Linear Parameter Varying (LPV) model is proposed for the DFCE with the service time as the output, the number of fog nodes as the input, and the characteristics of MRVD as the time-varying parameters. At first, an LPV model for the DFCE is derived using system identification, and the model is validated using the real-time test data. The LPV model is converted to a Polytopic LPV (PLPV) model for LMI based controller design. Finally, we develop and validate a Linear Matrix Inequality (LMI) based LPV controller to meet the service time constraints for a given application environment.For localisation and trajectory tracking with obstacle avoidance in the application environment, the mobile robot implements an Extended Kalman Filter (EKF) based simultaneous localisation and mapping (SLAM) and a bug-based path planning algorithm respectively. Finally, we present, detailed controller validation results illustrating the mobile robot navigation together with the auto-scaling control for the fog computing platform to modulate the service time.

Aggressive Maneuver Oriented Robust Actuator Fault Estimation of a 3-DOF Helicopter Prototype Considering Measurement Noises

This article presents a robust actuator fault estimation strategy design for a three-degree of freedom (3-DOF) helicopter prototype which can be adapted to aggressive maneuvers. First, considering large pitch angle condition during flight, nonlinear coupling characteristic of the helicopter system is exploited. As the pitch angle can be measured in real time, a polytopic linear parameter varying (LPV) model is developed for the helicopter system. Furthermore, considering measurement noises in the actual helicopter system, the dynamical model of helicopter system is modified accordingly. Then, based on the modified polytopic LPV model, a robust unknown input observer is developed for the helicopter system to realize actuator fault estimation, in which both measurement noises and large pitch angle are considered. Robust performance of proposed fault estimation approach is guaranteed by using energy-to-energy strategy. And the observer gains are calculated by using linear matrix inequalities. Finally, based on a 3-DOF helicopter prototype, both simulations and experiments are conducted. The effects of measurement noises and large pitch angle on the fault estimation performance are sufficiently demonstrated. And effectiveness as well as advantages of the proposed observer is verified by using comparative analysis.

Tensor product model HOSVD based polytopic LPV controller for suspension anti-vibration system

This paper presents a polytopic linear parameter varying (LPV) controller design methodology for nonlinear anti-vibration suspension based on stochastic road identification. As the main nonlinear component in semi-active suspension, the continuous damping control (CDC) damper value is mapped by damping velocity and control current through a bench experiment. linear parameter varying gain-scheduled method is applied for dealing with the nonlinear deterministic model. As the uncertainties caused by road stochastic excitation to suspension cannot be ignored, it is essential to identify the excitation signal as a measurable variable parameter. Empirical-mode-decomposition (EMD) and support vector machine (SVM) are utilized to identify and classify the road stochastic signal input into standard level, so polytopic LPV controller can be designed with CDC damper velocity, control current, and road level as variable parameters. Meanwhile the exponential increase of high-order plant elements in multi-parameter LPV model makes it too complicated for real-time computing. Hence the tensor product model transformation is utilized to identify the orthogonal basis of system plant so that low-rank approximation of it can be implemented by truncated high-order singular value decomposition (T−HOSVD) method. Real-time simulation shows the performance and feasibility of controller, which can efficaciously identify the road stochastic excitation as measurable variable parameters for nonlinear suspension polytopic LPV control.

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.

Robust Controller Designing for an Air-Breathing Hypersonic Vehicle with an HOSVD-Based LPV Model

This paper focuses on the linear parameter varying (LPV) modeling and controller design for a flexible air-breathing hypersonic vehicle (AHV). Firstly, by selecting the measurable altitude and velocity as gain-scheduled variables, the original longitudinal nonlinear model for AHV is transformed into the LPV model via average gridding division, vertex trimming, Jacobian linearization, and multiple linear regression within the entire flight envelope. Secondly, using the tensor product model transformation method, the obtained LPV model is converted into the polytopic LPV model via high-order singular value decomposition (HOSVD). Third, the validity and applicability of the HOSVD-based LPV model are further demonstrated by designing a robust controller for command tracking control during maneuvering flight over a large envelope.

Finite-time switched LPV control of quadrotors with guaranteed performance

This paper presents a novel finite-time switched linear parameter-varying (LPV) control method for quadrotors subject to large attitude angles and external disturbances. For controller design purpose, both the nonlinear attitude and position dynamics are modeled as a switched LPV system without the idealized assumption of linear approximation. A family of polytopic LPV controllers are designed and each of them is available for a specific parameter subspace of the quadrotor system. The persistent dwell time (PDT) switching logic is adopted to govern the switching behavior among these controllers, which is capable of addressing more general parameter variations than typically used dwell time or average dwell time. By applying an extended cost function, the established switched LPV control method ensures a guaranteed finite-time H∞/LQR performance for the error-tracking system of quadrotors. The advantages of the proposed method are demonstrated via simulations.

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.

Confidence set-membership state estimation for LPV systems with inexact scheduling variables

In this paper, a confidence set-membership state estimator is proposed for a class of polytopic linear parameter varying (LPV) systems with inexact scheduling variables. The set-bounded and Gaussian uncertainties are considered simultaneously in the process disturbances and measurement noises. The purpose of the proposed estimator is to achieve a confidence set of the state with given confidence level. Based on the polytopic LPV uncertain enclosure model, the set-bounded/Gaussian uncertainties of the state are given by using the worst case strategy. The size of the confidence set is minimized to get the optimal gain for the estimator. Meanwhile, the constrained zonotope is adopted to represent set-bounded uncertainties for more accurate results. Finally, a vehicle example is given to illustrate the effectiveness of proposed methods.

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.

Prescribed-time observers of LPV systems: A linear matrix inequality approach

This paper considers prescribed-time observer (PTO) designs for a class of linear parameter-varying (LPV) systems. Firstly, a full-order prescribed-time observer with time-varying gains is developed. The existence conditions are given in terms of linear matrix inequalities (LMIs). In addition, the reduced-order PTO is also considered in this paper. Moreover, it is shown that the existence conditions under which the full-order PTO exists can also guarantee the existence of a corresponding reduced-order PTO. The advantages of the full-order and the reduced-order PTOs over the existing asymptotic convergence observers are that (1) they can achieve exact estimations in almost any prescribed convergence time regardless of what the system initial values are. (2) the proposed time-varying gain PTOs can avoid the conservatism of the unknown input decoupling conditions brought about by the traditional polytopic LPV observer design methods. Finally, two examples are given to illustrate the effectiveness of the proposed methods.