LPV model development and control of a solution copolymerization reactor

In this paper, linear parameter-varying (LPV) control is considered for a solution copolymerization reactor, which takes into account the time-varying nature of the parameters of the process. The nonlinear model of the process is first converted to an exact LPV model representation in the state-space form that has a large number of scheduling variables and hence is not appropriate for control design purposes due to the complexity of the LPV control synthesis problem. To reduce such complexity, two approaches are proposed in this paper. First, an approximate LPV representation with only one scheduling variable is obtained by means of a parameter set mapping (PSM). The second approach is based on reformulating the nonlinear model so that it provides an LPV model with a fewer number of scheduling parameters but preserves the same input–output behavior. Moreover, in the implementation of the LPV controllers synthesized with the derived models, the unmeasurable scheduling variables are estimated by an extended Kalman filter. Simulation results using the nonlinear model of the copolymerization reactor are provided in order to illustrate the performance of the proposed controllers in reducing the convergence time and the control effort.     

Gain-scheduled controller synthesis for a nonlinear PDE

Linear parameter-varying (LPV) modelling and control of a nonlinear partial differential equation (PDE) is considered in this article. The one-dimensional viscous Burgers' equation is discretised using a finite difference scheme; the boundary conditions are taken as control inputs and the velocities at two grid points are assumed to be measurable. A nonlinear high-order state space model is generated and proper orthogonal decomposition is used for model order reduction. After assessing the accuracy of the reduced model, a low-order functional observer is designed to estimate the reduced states which are linear combinations of the velocities at all grid points. A discrete-time quasi-LPV model that is affine in scheduling parameters is derived based on the reduced model. A polytopic LPV controller is synthesised based on a generalised plant containing the LPV model and the functional observer. More generally, the proposed method can be used to design an LPV controller for a quasi-LPV system with non-measurable scheduling parameters. Simulation results demonstrate the high tracking performance and disturbance and measurement noise rejection capabilities of the designed LPV controller compared with a linear quadratic Gaussian (LQG) controller based on a linearised model.     

Gain-scheduling control of a floating offshore wind turbine above rated wind speed

This paper presents an application of gain-scheduling(GS) control techniques to a floating offshore wind turbine on a barge platform for above rated wind speed cases. Special emphasis is placed on the dynamics variation of the wind turbine system caused by plant nonlinearity with respect to wind speed. The turbine system with the dynamics variation is represented by a linear parameter-varying(LPV) model, which is derived by interpolating linearized models at various operating wind speeds. To achieve control objectives of regulating power capture and minimizing platform motions, both linear quadratic regulator(LQR) GS and LPV GS controller design techniques are explored. The designed controllers are evaluated in simulations with the NREL 5 MW wind turbine model, and compared with the baseline proportional-integral(PI) GS controller and non-GS controllers. The simulation results demonstrate the performance superiority of LQR GS and LPV GS controllers, as well as the performance trade-off between power regulation and platform movement reduction.     

Systematic Gain‐Scheduling Control Design: A Missile Autopilot Example

The missile autopilot was designed using linear parameter‐varying (LPV) control techniques. The controller provides exponential stability guarantee and performance bound in terms of induced L₂ norm for the missile plant. The systematic gain‐scheduling approach is motivated by the recent development in LPV control theory and provides a well founded and systematic procedure for high performance missile autopilot design problem.     

Output feedback control of linear fractional transformation systems subject to actuator saturation

In this paper, the control problem for a class of linear parameter varying (LPV) plant subject to actuator saturation is investigated. For the saturated LPV plant depending on the scheduling parameters in linear fractional transformation (LFT) fashion, a gain-scheduled output feedback controller in the LFT form is designed to guarantee the stability of the closed-loop LPV system and provide optimised disturbance/error attenuation performance. By using the congruent transformation, the synthesis condition is formulated as a convex optimisation problem in terms of a finite number of LMIs for which efficient optimisation techniques are available. The nonlinear inverted pendulum problem is employed to demonstrate the effectiveness of the proposed approach. Moreover, the comparison between our LPV saturated approach with an existing linear saturated method reveals the advantage of the LPV controller when handling nonlinear plants.     

Probabilistic Robust Linear Parameter-varying Control of a Small Helicopter Using Iterative Scenario Approach

In this paper, we present an iterative scenario approach (ISA) to design robust controllers for complex linear parameter-varying (LPV) systems with uncertainties. The robust controller synthesis problem is transformed to a scenario design problem, with the scenarios generated by identically extracting random samples on both uncertainty parameters and scheduling parameters. An iterative scheme based on the maximum volume ellipsoid cutting-plane method is used to solve the problem. Heuristic logic based on relevance ratio ranking is used to prune the redundant constraints, and thus, to improve the numerical stability of the algorithm. And further, a batching technique is presented to remarkably enhance the computational efficiency. The proposed method is applied to design an output-feedback controller for a small helicopter. Multiple uncertain physical parameters are considered, and simulation studies show that the closed-loop performance is quite good in both aspects of model tracking and dynamic decoupling. For robust LPV control problems, the proposed method is more computationally efficient than the popular stochastic ellipsoid methods.      

Discretization and event triggered digital output feedback control of LPV systems

This paper investigates the problem of discretization and digital output feedback control design for continuous-time linear parameter-varying (LPV) systems subject to a time-varying networked-induced delay. The proposed discretization procedure converts a continuous-time LPV system into an equivalent discrete-time LPV system based on an extension of the Taylor series expansion and using an event-based sampling. The scheduling parameters are continuously measured and modeled as piecewise constant. A new transmission of the measured output to the controller is triggered by significant changes in the parameters, yielding time-varying transmission intervals. The obtained discretized model has matrices with polynomial dependence on the time-varying parameters and an additive norm-bounded term representing the discretization residual error. A two step strategy based on linear matrix inequality conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the LPV model. The conditions can also be used to provide robust (i.e., independent of the scheduling parameter) static output feedback controllers. The viability of the proposed design method is illustrated through numerical examples.     

Dynamic output feedback robust MPC for LPV systems subject to input saturation and bounded disturbance

For linear parameter varying (LPV) systems with unknown scheduling parameters and bounded disturbance, a synthesis approach of dynamic output feedback robust model predictive control (OFRMPC) with input saturation is investigated. By pre-specifying partial controller parameters, a main optimization problem is solved by convex optimization to reduce the on-line computational burden. The main optimization problem guarantees that the estimated state and estimation error converge within the corresponding invariant sets such that recursive feasibility and robust stability are guaranteed. The consideration of input saturation in the main optimization problem improves the control performance. Two numerical examples are given to illustrate the effectiveness of the approach.     

Gain-scheduled control via filtered scheduling parameters

Gain-scheduled control via LPV system models enjoys LMI-based synthesis methods and in particular parameter-dependent Lyapunov matrices have been employed to successfully reduce conservatism. Those controllers derived via parameter-dependent Lyapunov matrices, however, end up with depending on derivatives of scheduling parameters. Though this can be avoided by approximating derivatives or restricting Lyapunov matrices to be partly constant, the former loses guarantee of performance and stability and the latter can cause conservatism. This paper proposes a synthesis method of gain-scheduled controllers that depend on filtered scheduling parameters, instead of derivatives, with a concrete guarantee of a performance level. Moreover, it is shown that the performance level of conventional derivative-dependent gain-scheduled controllers is recovered with arbitrarily small errors.     

Reduced Order Robust Gain‐Scheduling Control Of The Diesel Apu For Series Hybrid Vehicles

This paper presents a reduced order robust gain‐scheduling approach for the control of the diesel auxiliary power unit (APU) for series hybrid vehicles. The nonlinear plant dynamics are converted into a linear parameter‐varying (LPV) form with parametric uncertainties, in which only partial information of the plant states is available. For this type of LPV system, a new reduced order robust gain‐scheduling synthesis method is proposed based on partial state feedback. The parametric uncertainties are considered using multipliers to reduce the conservatism. The reduced order synthesis problem is solved offline by means of linear matrix inequalities (LMIs), and the synthesis result requires much simpler online computation than the explicit controller formulas do. The synthesis method is applied to the diesel APU controller design, and simulation results are given to demonstrate the controller performance.     

Observer‐based synthesis of linear parameter‐varying mixed sensitivity controllers

This article presents a computationally efficient way of synthesizing linear parameter‐varying (LPV) controllers. It reviews the possibility of a separate observer and state feedback synthesis with guaranteed performance and shows that a standard mixed sensitivity problem can be solved in this way. The resultant output feedback controller consists of an LPV observer, augmented with dynamic filters to incorporate integral control and roll‐off properties, and an LPV state feedback gain. It is thus highly structured, which is beneficial for implementation. Moreover, it does not depend on scheduling parameter rates regardless of whether parameter‐dependent Lyapunov matrices are used during synthesis. A representative control design for active flutter suppression on an aeroelastic unmanned aircraft demonstrates the benefits of the proposed method in comparison with state‐of‐the‐art LPV output feedback synthesis.     

风力机的线性变参数主动容错控制

针对风力机具有非线性和参数的不确定性的特征,提出了基于线性变参数(linear parameter varying,LPV)增益调度的风力机主动容错控制方法,降低故障对机组动态特性的影响.基于LPV凸分解方法,将风力机的非线性模型转化为具有凸多面体结构LPV模型,利用线性矩阵不等式(linear matrix inequalities,LMIs)技术对凸多面体各个顶点分别设计满足性能要求的控制器,再利用各顶点设计的反馈控制器得到具有凸多面体结构LPV容错控制器.仿真结果表明,LPV增益调度技术可以成功地应用于风力机系统的容错控制.     

A Survey of Output Feedback Robust MPC for Linear Parameter Varying Systems

For constrained linear parameter varying (LPV) systems, this survey comprehensively reviews the literatures on output feedback robust model predictive control (OFRMPC) over the past two decades from the aspects on motivations, main contributions, and the related techniques. According to the types of state observer systems and scheduling parameters of LPV systems, different kinds of OFRMPC approaches are summarized and compared. The extensions of OFRMPC for LPV systems to other related uncertain systems are also investigated. The methods of dealing with system uncertainties and constraints in different kinds of OFRMPC optimizations are given. Key issues on OFRMPC optimizations for LPV systems are discussed. Furthermore, the future research directions on OFRMPC for LPV systems are suggested.      

Asymptotically optimal orthonormal basis functions for LPV system identification

A global model structure is developed for parametrization and identification of a general class of Linear Parameter-Varying (LPV) systems. By using a fixed orthonormal basis function (OBF) structure, a linearly parametrized model structure follows for which the coefficients are dependent on a scheduling signal. An optimal set of OBFs for this model structure is selected on the basis of local linear dynamic properties of the LPV system (system poles) that occur for different constant scheduling signals. The selected OBF set guarantees in an asymptotic sense the least worst-case modeling error for any local model of the LPV system. Through the fusion of the Kolmogorov n-width theory and Fuzzy c-Means clustering, an approach is developed to solve the OBF-selection problem for discrete-time LPV systems, based on the clustering of observed sample system poles.     

Probabilistic design of LPV control systems

This paper presents an alternative approach to design of linear parameter-varying (LPV) control systems. In contrast to previous methods, which are focused on deterministic algorithms, this paper is based on a probabilistic setting. The proposed randomized algorithm provides a sequence of candidate solutions converging with probability one to a feasible solution in a finite number of steps. The main features of this approach are as follows: (i) The randomized algorithm gives a method for general LPV plants with state space matrices depending on scheduling parameters in a nonlinear manner. That is, the probabilistic setting does not need a gridding of the set of scheduling parameters or approximations such as a linear fractional transformation of the state space matrices. (ii) The proposed algorithm is sequential and, at each iteration, it does not require heavy computational effort such as solving simultaneously a large number of linear matrix inequalities.     

基于回路成形的鲁棒增益调度控制器设计

针对目前基于线性变参数系统的增益调度控制设计中存在的控制结构复杂性问题，提出一种基于回路成形的简单且易实现的增益调度控制结构.在此基础上，提出一个鲁棒增益调度控制设计方法.设计过程首先采用补偿器函数使得被控对象奇异值具有期望的形状，以保证被控对象的性能要求，然后利用小增益定理设计一个鲁棒控制器，得到具有良好性能的、结构简单的鲁棒增益调度控制器.最后针对一个化工过程，说明此方法的有效性.     

A Multi-step Output Feedback Robust MPC Approach for LPV Systems with Bounded Parameter Changes and Disturbance

This paper considers a multi-step output feedback robust model predictive control (OFRMPC) approach for the linear parameter varying (LPV) systems with bounded changes of scheduling parameters and bounded disturbance. Less conservative bounds of future estimation error sets and system parametric uncertain sets are predicted by considering bounded changes of scheduling parameters in LPV systems. In the multi-step OFRMPC approach, an optimization problem is solved to obtain a sequence of controller gains, which considers predictions of future bounds of estimation error sets and system parametric uncertain sets. The optimized sequence of controller gains corresponding to a sequence of Lyaponov matrices have less constraint conditions and also introduce more degree of freedom for the optimization. The proposed multi-step OFRMPC guarantees robust uniform ultimately bounded of the estimation error and robust stability of the observer system. A numerical example is given to demonstrate the effectiveness of the approach.     

Low-complexity linear parameter-varying modeling and control of a robotic manipulator

In this paper, a practical procedure for linear parameter-varying (LPV) modeling and identification of a robotic manipulator is presented, which leads to a successful experimental implementation of an LPV gain-scheduled controller. A nonlinear dynamic model of a two-degrees-of-freedom manipulator containing all important terms is obtained and unknown parameters which are required to construct an LPV model are identified. An important tool for obtaining a model of complexity low enough to be suitable for controller synthesis is the principle-component-analysis-based technique of parameter set mapping. Since the resulting quasi-LPV model has a large number of affine scheduling parameters and a large overbounding, parameter set mapping is used to reduce conservatism and complexity in controller design by finding tighter parameter regions with fewer scheduling parameters. A sufficient a posteriori condition is derived to assess the stability of the resulting closed-loop system. To evaluate the applicability and efficiency of the approximated model, a polytopic LPV gain-scheduled controller is synthesized and implemented experimentally on an industrial robot for a trajectory tracking task. The experimental results illustrate that the designed LPV controller outperforms an independent joint PD controller in terms of tracking performance and achieves a slightly better accuracy than a model-based inverse dynamics controller, while having a simpler structure. Moreover, it is shown that the LPV controller is more robust against dynamic parameter uncertainty.     

LPV gain-scheduled control of SCR aftertreatment systems

Hydrocarbons, carbon monoxide and some of other polluting emissions produced by diesel engines are usually lower than those produced by gasoline engines. While great strides have been made in the exhaust aftertreatment of vehicular pollutants, the elimination of nitrogen oxide (NO _x ) from diesel vehicles is still a challenge. The primary reason is that diesel combustion is a fuel-lean process, and hence there is significant unreacted oxygen in the exhaust. Selective catalytic reduction (SCR) is a well-developed technology for power plants and has been recently employed for reducing NO _x emissions from automotive sources and in particular, heavy-duty diesel engines. In this article, we develop a linear parameter-varying (LPV) feedforward/feedback control design method for the SCR aftertreatment system to decrease NO _x emissions while keeping ammonia slippage to a desired low level downstream the catalyst. The performance of the closed-loop system obtained from the interconnection of the SCR system and the output feedback LPV control strategy is then compared with other control design methods including sliding mode, and observer-based static state-feedback parameter-varying control. To reduce the computational complexity involved in the control design process, the number of LPV parameters in the developed quasi-LPV (qLPV) model is reduced by applying the principal component analysis technique. An LPV feedback/feedforward controller is then designed for the qLPV model with reduced number of scheduling parameters. The designed full-order controller is further simplified to a first-order transfer function with a parameter-varying gain and pole. Finally, simulation results using both a low-order model and a high-fidelity and high-order model of SCR reactions in GT-POWER interfaced with MATLAB/SIMULINK illustrate the high NO _x conversion efficiency of the closed-loop SCR system using the proposed parameter-varying control law.