Christian H. Mayr

PID controller design for nonlinear systems represented by discrete-time local model networks

This paper deals with proportional–integral–derivative (PID) controller design for nonlinear systems represented by local model networks. The proposed method is based on the concept of parallel distributed compensators where the scheduling of the local model network is adopted for the PID parameters. The proposed design method for nonlinear PID controllers considers closed-loop stability by means of a Lyapunov stability criterion as well as closed-loop performance. All PID parameters are determined by a multi-objective genetic algorithm (multiGA), which handles the trade-off between stability and performance. A simulation example demonstrates the effectiveness of the proposed method.

Piecewise Quadratic stability analysis for local model networks

This paper deals with the problem of stability analysis of dynamic local model networks. Established methods in this context are mainly based on Lyapunov stability theory and are targeted to be as little conservative as possible. In previous works the so called Piecewise Quadratic Lyapunov approach was developed. For discrete time systems the state space is partitioned into local subspaces, which are defined by the validity functions of the local models. Because of the overlapping validity functions, so-called uncertainty terms exist which describe the influence of the dynamics of other local models. In this respect, it is necessary to pay attention to the determination of these uncertainty terms. This paper presents and discusses a method to determine the upper bounds for the uncertainty terms of the local models. The method is based on quadratic optimization to achieve a stability criterion where the conservatism is not additionally increased. The effectiveness of the proposed method is shown by a simulation example in connection with the Piecewise Quadratic Lyapunov approach as a stability criterion.

Hybrid powertrain control using optimization and cycle based predictive control algorithms

In this paper a predictive control scheme for hybrid electric powertrains is introduced which makes use of cyclic operation such as in city buses or dockside cranes. The predictive controller is formulated as quadratic programming problem and based on cyclic operation of the vehicle. To take the nonlinearities of the electrical circuit into account a surrogate model is used. For the prediction of the future torque a wavelet approach is used for the optimization criterion. The effectiveness of the proposed method is shown by a simulation example. This approach can be easily adapted to a large family of hybrid powertrain configurations.

Dynamic NOx emission modelling using local model networks

More and more stringent emission regulations and the desire to reduce fuel consumption lead to an increasing demand for efficient and reliable modelling tools in the automotive industry. When conventional physical modelling is not possible due to the lack of precise, formal knowledge about the system, black-box- and grey-box-oriented nonlinear system identification procedures are a widely used concept to create models based on measured input and output data of the process. In this context, local model networks are an established approach for nonlinear dynamic system identification as they provide not only accurate but also interpretable models and therefore allow a better understanding of the true system than pure black-box models. As a consequence, local model networks provide a basis for the development of systematic approaches to stability analysis and nonlinear controller design. In this article, local model network–based dynamic NOx emission modelling is presented. A robust and efficient local model network training algorithm is described, and the proposed concepts are validated using real measurement data. An important advantage of the architecture of local model networks is their good interpretability which is an important advantage for the design of controllers or observers. Additionally, stability analysis of both the nonlinear open- and closed-loop system is possible based on Lyapunov stability theory.

Stability analysis of data-driven local model networks

This article discusses stability analysis of data-driven dynamic local model networks. In contrast to traditional fuzzy modelling, the structure and complexity of such model architectures is not unique when only observed input- and output data are available for their parametrization. The present article complements the well-known trade-off between accuracy and complexity by the notion of stability. For this purpose, existing Lyapunov stability criteria for local model networks are extended by a decay rate which represents a scalar and quantitative stability measure. It allows to compare models with different degrees of complexity also in view of their stability. For some of the commonly available Lyapunov stability criteria, the individual local model transitions are crucial. Therefore, in this article, an approach is introduced to determine the actually occurring model transitions by means of the identification data. The methods presented in the article are illustrated and discussed by means of a simulation example. It is shown how model complexity and the related approximation quality can have an adverse impact on the stability and how the outcome of different Lyapunov criteria is affected by the proper determination of local model transitions.

Nonlinear stable PID controller design using local model networks

This paper addresses PID controller design using local model networks. The proposed method uses a common quadratic Lyapunov function to guarantee stability of the closed loop. To solve the resulting bilinear matrix inequalities (BMI) an iterative procedure is introduced which is based on state of the art linear matrix inequalities solvers (iLMI). Due to the fact that the Lyapunov approach requires a state-space model a suitable closed-loop state-space system is introduced. An example demonstrates the effectiveness of the proposed method.

Relaxed fuzzy Lyapunov approach for dynamic local model networks

This paper deals with the problem of stability analysis of dynamic local model networks. Established methods in this context are mainly based on Lyapunov stability theory and are targeted to be as little conservative as possible. In this respect it is essential to take into account the transitions between the different local models. For that purpose this paper presents and discusses a method to determine possible model transitions of such dynamic local model networks utilizing identification or simulation data. The effectiveness of the proposed method is shown by a simulation example in connection with the fuzzy Lyapunov approach as a stability criterion. The example demonstrates how identification data can be used to reduce the conservatism compared to standard approaches.

Automatic tuning of PID controllers in engine control units by means of local model networks and evolutionary optimization

In this work a new approach for a fully automated calibration of nonlinear PID controllers and feedforward maps is introduced. Controller design poses a particularly challenging task in the application to internal combustion engines due to the nonlinear controller structure, which is usually prescribed by the manufacturer of the engine control unit (ECU). A dynamic local model network is used to represent the actual physical process as its architecture can beneficially be adopted for scheduling of the nonlinear controller parameters. The presented calibration technique uses a genetic algorithm to calibrate the nonlinear PID controller and a static model inversion to determine the feedforward map. Finally, an example demonstrates the effectiveness of the proposed method.

Stability Analysis of PID Controlled Local Model Networks

Abstract This paper addresses closed-loop stability analysis of PID controlled local model networks. The proposed method allows to investigate exponential or asymptotic stability of the closed-loop system. For this purpose a common quadratic Lyapunov function is used as stability criterion. Due to the fact that the Lyapunov approach requires a state-space model a suitable closed-loop state-space system with integration of the controller parameters is introduced. An example demonstrates the effectiveness of the proposed method.