Linear Parameter-varying State-space Research Articles

Machine Learning-Based Modeling and Predictive Control of Combustion Phasing and Load in a Dual-Fuel Low-Temperature Combustion Engine

<div>Reactivity-controlled compression ignition (RCCI) engine is an innovative dual-fuel strategy, which uses two fuels with different reactivity and physical properties to achieve low-temperature combustion, resulting in reduced emissions of oxides of nitrogen (NO<sub>x</sub>), particulate matter, and improved fuel efficiency at part-load engine operating conditions compared to conventional diesel engines. However, RCCI operation at high loads poses challenges due to the premixed nature of RCCI combustion. Furthermore, precise controls of indicated mean effective pressure (IMEP) and CA50 combustion phasing (crank angle corresponding to 50% of cumulative heat release) are crucial for drivability, fuel conversion efficiency, and combustion stability of an RCCI engine. Real-time manipulation of fuel injection timing and premix ratio (PR) can maintain optimal combustion conditions to track the desired load and combustion phasing while keeping maximum pressure rise rate (MPRR) within acceptable limits.</div> <div>In this study, a model-based controller was developed to track CA50 and IMEP accurately while limiting MPRR below a specified threshold in an RCCI engine. The research workflow involved development of an imitative dynamic RCCI engine model using a data-driven approach, which provided reliable measured state feedback during closed-loop simulations. The model exhibited high prediction accuracy, with an <i>R</i><sup>2</sup> score exceeding 0.91 for all the features of interest. A linear parameter-varying state space (LPV-SS) model based on least squares support vector machines (LS-SVM) was developed and integrated into the model predictive controller (MPC). The controller parameters were optimized using genetic algorithm and closed-loop simulations were performed to assess the MPC’s performance. The results demonstrated the controller’s effectiveness in tracking CA50 and IMEP, with mean average errors (MAE) of 0.89 crank angle degree (CAD) and 46 kPa and Mean absolute percentage error (MAPE) of 9.7% and 7.1%, respectively, while effectively limiting MPRR below of 10 bar/CAD. This comprehensive evaluation showcased the efficacy of the model-based control approach in tracking CA50 and IMEP while constraining MPRR in the dual-fuel engine.</div>

Offline output feedback robust anti-windup MPC-LPV using relaxed LMI optimization

This paper addresses a new technique of constrained output feedback robust model predictive control (RMPC) with anti-windup (AW) synthesis adopting linear parameter varying state-space (LPV-SS) systems via relaxed linear matrix inequalities (LMI) procedure. We proposed an output feedback control design with a conservatism reduction, including a robust AW-LPV synthesis to solve the robust stability under saturated signal control mode. Lyapunov stability functions through the LPV emphasis are considered. To evaluate the efficiency of the proposed method, we presented a time response analysis, closed-loop z-plane, worst-case upper bound, and we compared the performance of the proposed method with some MPC benchmarks using a numerical example.

Integrated Comfort-Adaptive Cruise and Semi-Active Suspension Control for an Autonomous Vehicle: An LPV Approach

This paper presents an integrated linear parameter-varying (LPV) control approach of an autonomous vehicle with an objective to guarantee driving comfort, consisting of cruise and semi-active suspension control. First, the vehicle longitudinal and vertical dynamics (equipped with a semi-active suspension system) are presented and written into LPV state-space representations. The reference speed is calculated online from the estimated road type and the desired comfort level (characterized by the frequency weighted vertical acceleration defined in the ISO 2631 norm) using precomputed polynomial functions. Then, concerning cruise control, an LPV H2 controller using a linear matrix inequality (LMI) based polytopic approach combined with the compensation of the estimated disturbance forces is developed to track the comfort-oriented reference speed. To further enhance passengers’ comfort, a decentralized LPV H2 controller for the semi-active suspension system is proposed, minimizing the effect of the road profile variations. The interaction with cruise control is achieved by the vehicle’s actual speed being a scheduling parameter for suspension control. To assess the strategy’s performance, simulations are conducted using a realistic nonlinear vehicle model validated from experimental data. The simulation results demonstrate the proposed approach’s capability to improve driving comfort.

Linear parameter-varying subspace identification: A unified framework

In this paper, we establish a unified framework for subspace identification (SID) of linear parameter-varying (LPV) systems to estimate LPV state–space (SS) models in innovation form. This framework enables us to derive novel LPV SID schemes that are extensions of existing linear time-invariant (LTI) methods. More specifically, we derive the open-loop, closed-loop, and predictor-based data-equations (input–output surrogate forms of the SS representation) by systematically establishing an LPV subspace identification theory. We also show the additional challenges of the LPV setting compared to the LTI case. Based on the data-equations, several methods are proposed to estimate LPV-SS models based on a maximum-likelihood or a realization based argument. Furthermore, the established theoretical framework for the LPV subspace identification problem allows us to lower the number of to-be-estimated parameters and to overcome dimensionality problems of the involved matrices, leading to a decrease in the computational complexity of LPV SIDs in general. To the authors’ knowledge, this paper is the first in-depth examination of the LPV subspace identification problem. The effectiveness of the proposed subspace identification methods are demonstrated and compared with existing methods in a Monte Carlo study of identifying a benchmark MIMO LPV system.

Robust Predictive Control Algorithm Based on Parameter Variation Rate Information of Functional-Coefficient ARX Model

Considering the conservativeness caused by the polytopic linear parameter varying (LPV) model, which is constructed using only the upper and lower bound information of the state-dependent auto-regressive model with eXogenous input and radial basis function network type coefficients (RBF-ARX model). In this paper, a robust predictive control (RPC) algorithm based on the parameter variation rate information of the RBF-ARX model is proposed. By using the information, the size of the convex polytopic sets used to wrap the system's polytopic LPV model is compressed greatly. Thus, it improves greatly the control performance and reduces the conservativeness of the subsequent RPC algorithm. The conversion from the RBF-ARX model to the polytopic LPV state space model just uses the RBF-ARX model itself, and the derived LPV model is a special quasi-LPV autoregressive model. So, it is not necessary to assume that the time varying parameters and/or the bounds of the parameter variation rate in the polytopic LPV model must be known or measured. An example of a widely used continuous stirred-tank reactor process control is studied to illustrate the effectiveness of the proposed approach in terms of using parameter variation rate information of the RBF-ARX model to improve step response control performance and anti-jamming performance.

Vehicle Active Damping Control with Considering Time Delay

A model predictive control (MPC) based active damping controller for automotive driveline oscillations with time-delay consideration is proposed. A simplified driveline model considering time delay is modeled and converted to a linear parameter varying state space equation. Based on the model and model predictive control theory, an active damping controller is designed for drivability and comfortability improvement. In order to verify the designed controller, a driveline with engine is modeled to simulate the tip-in/out driving operation. An MPC active damping controller without considering time delay is simulated together with the proposed controller. The simulation results show that, by adopting the new MPC active damping controller, the vibration of the vehicle is reduced and the drivability and comfortability are improved.

Continuous-time identification of periodically parameter-varying state space models

This paper presents a new frequency domain identification technique to estimate multivariate Linear Parameter-Varying (LPV) continuous-time state space models, where a periodic variation of the parameters is assumed or imposed. The main goal is to obtain an LPV state space model suitable for control, from a single parameter-varying experiment. Although most LPV controller synthesis tools require continuous time state space models, the identification of such models is new. The proposed identification method designs a periodic input signal, taking the periodicity of the parameter variation into account. We show that when an integer number of periods is observed for both the input and the scheduling, the state space model representation has a specific, sparse structure in the frequency domain, which is exploited to speed up the estimation procedure. A weighted non-linear least squares algorithm then minimizes the output error. Two initialization methods are explored to generate starting values. The first approach uses a Linear Time-Invariant (LTI) approximation. The second estimates a Linear Time-Variant (LTV) input–output differential equation, from which a corresponding state space realization is computed.

Embedding of Nonlinear Systems in a Linear Parameter-Varying Representation

This paper introduces a systematic approach to synthesize linear parameter-varying (LPV) representations of nonlinear (NL) systems which are originally defined by control affine state-space representations. The conversion approach results in LPV state-space representations in the observable canonical form. Based on the relative degree concept of NL systems, the states of a given NL representation are transformed to new coordinates that provide its normal form. In the SISO case, all nonlinearities of the original system are embedded in one NL function which is factorized to construct the LPV form. An algorithms is proposed for this purpose. The resulting transformation yields an LPV model where the scheduling parameter depends on the derivatives of the inputs and outputs of the system. In addition, if the states of the NL model can be measured or estimated, then the procedure can be modified to provide LPV models scheduled by these states. Examples are included for illustration.

Crucial Aspects of Zero-Order Hold LPV State-Space System Discretization

In the framework of Linear Parameter-Varying (LPV) systems, controllers are commonly designed in continuous-time, but implemented on digital hardware. Additionally, LPV system identification is formulated exclusively in discrete-time, needing structural information about the plant, which is often provided by first principle continuous-time models. These imply that LPV system discretization is an important issue for both system identification and controller implementation. Discretization approaches of LPV state-space systems are introduced and analyzed in terms of approximation error, considering ideal zero-order hold actuation and sampling of the input-output signals and the scheduling parameter of the system. Criteria to choose appropriate sampling times with the investigated methods are also presented.

LPV Identification of a Diesel Engine Torque Model

Torque estimation has an increasing importance in the field of automotive control, as most engine controllers rely on torque estimation today. In this paper a new approach based on a linear parameter varying (LPV) model is proposed. It relies on a non linear subspace identification method and exploits the natural dependency of many engine processes on the rotational speed, but does not use a physical model, which is very time consuming and difficult to obtain. To this end, first a LPV state space method is introduced and then experimental results are presented which show that the model exhibits high accuracy.