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## Über dieses Buch

This edited book aims at presenting current research activities in the field of robust variable-structure systems. The scope equally comprises highlighting novel methodological aspects as well as presenting the use of variable-structure techniques in industrial applications including their efficient implementation on hardware for real-time control. The target audience primarily comprises research experts in the field of control theory and nonlinear dynamics but the book may also be beneficial for graduate students.

## Inhaltsverzeichnis

### Comparison of Backstepping-Based Sliding Mode and Adaptive Backstepping for a Robust Control of a Twin Rotor Helicopter

Abstract
In this contribution, two robust MIMO backstepping control approaches for a twin rotor aerodynamic system (TRAS) test-rig are considered. The TRAS represents a nonlinear system with significant couplings. A nonlinear multibody model of the TRAS with lumped unknown disturbance torques is derived using Lagrange’s equations. Herewith, both a backstepping-based sliding mode control and an adaptive backstepping control are designed to track desired trajectories for the azimuth angle and the pitch angle. An explicit expression is derived for the reaching time in the case of the backstepping-based sliding mode control. In order to estimate immeasurable angular velocities and unknown disturbance torques for the backstepping-based sliding mode control, a discrete-time extended Kalman filter (EKF) is employed. For the adaptive backstepping, a robust sliding mode differentiator is used instead to estimate the angular velocities. Moreover, in the adaptive backstepping control approach, the disturbance compensation is realised with the help of additional adaptive control parts driven by the tracking errors of the controlled variables. The overall stability of the proposed controllers in combination with the corresponding estimator is investigated thoroughly by simulations. Furthermore, in order to validate the proposed control schemes, experiments are performed on the dedicated test-rig and a comparison of the two proposed control structures is provided as well.
Saif Siddique Butt, Hao Sun, Harald Aschemann

### Robust Congestion Controller for a Single Virtual Circuit in Connection-Oriented Communication Networks

Abstract
In this contribution, we consider the problem of data flow control for a single virtual connection in communication networks. The connection is described by the maximum link capacity, the non-negligible propagation delay and an unknown, time-varying data loss rate. We propose a discrete-time sliding mode controller, which generates non-negative and upper bounded transmission rates. In addition, it ensures that the queue length in the bottleneck link buffer is always limited. Moreover, with a sufficiently large memory buffer in the bottleneck node, it guarantees full utilization of the available bandwidth. The controller uses a dead-beat sliding hyperplane in order to ensure fast response to unknown changes of the link capacity and to an unpredictable data loss rate. However, if the straightforward dead-beat paradigm was used, unacceptably large transmission rates would be generated. Therefore, we use the reaching law approach in this chapter to decrease excessive magnitudes of the control signal at the start of the control process.
Piotr Leśniewski, Andrzej Bartoszewicz

### Interval Methods for Robust Sliding Mode Control Synthesis of High-Temperature Fuel Cells with State and Input Constraints

Abstract
Fuel cell systems provide a way to produce electric energy in future decentralized power supply grids. In the case of using high-temperature fuel cells, it becomes possible to exploit not only the provided electric power but also the process heat in order to maximize the overall system efficiency. However, the efficiency maximization goes along with a high flexibility with respect to temporal variations of the electric power that is demanded by corresponding consumers. Such power variations impose restrictions on intelligent fuel cell control systems. Such control strategies do not only have to make sure that the supplied fuel gas (typically hydrogen and mixtures with methane or carbon monoxide) is stoichiometrically balanced with the demanded electric power. It is also inevitable to control the fuel cell itself from a thermodynamic point of view. This control has to make sure that sufficiently smooth temperature trajectories can be tracked during the heating phase of the system and that a priori unknown but bounded disturbances are robustly compensated at high-temperature operating points. For this purpose, interval-based sliding mode control procedures can be implemented. This contribution gives an overview of how interval methods can be combined with the fundamental sliding mode methodology in a variable-structure control synthesis. The efficiency of the presented methods is highlighted for the control of solid oxide fuel cells in various simulations.
Andreas Rauh, Luise Senkel

### Experimental and Numerical Validation of a Reliable Sliding Mode Control Strategy Considering Uncertainty with Interval Arithmetic

Abstract
Real applications are often affected by uncertainty caused by, for example, unknown parameters, sensor inaccuracies, and noise processes. These effects influence control procedures in a significant way and have to be taken into consideration in simulations and experiments to ensure stability of a real system. Often, the dynamics of a considered system can be described by nonlinear equations. To control such systems, sliding mode techniques are advantageous in compensating not explicitly modeled disturbances that influence a system. In this contribution, common sliding mode controllers are extended and combined with interval arithmetic to enhance their performance. This can be achieved by an adaptive calculation of the state-dependent gain stabilizing the variable-structure part of the system—the so-called switching amplitude. Therefore, an exact knowledge of the system parameters is not necessary because their true values are assumed to be located in specified range bounds. Moreover, stochastic uncertainty is taken into consideration representing process and measurement noise that affect practically every real system.
Luise Senkel, Andreas Rauh, Harald Aschemann

### A Sliding Mode Control with a Bang–Bang Observer for Detection of Particle Pollution

Abstract
This chapter presents a single-input single-output (SISO) adaptive sliding mode control combined with an adaptive bang–bang observer to improve a metal–polymer composite sensor system. The proposed techniques improve the disturbance rejection of a sensor system and thus their reliability in an industrial environment. The industrial application is based on the workplace particulate pollution of welding fumes. Breathing welding fumes is extremely detrimental to human health and exposes the lungs to great hazards, therefore an effective ventilation system is essential. Typically, sliding mode control is applied in actuator control. In this sense, the proposed application is an innovative one. It seeks to improve the performance of sensors in terms of robustness with respect to parametric uncertainties and in terms of insensibility with respect to disturbances. In particular, a sufficient condition to obtain an asymptotic robustness of the estimation of the proposed bang–bang observer is designed and substantiated. The whole control scheme is designed using the well-known Lyapunov approach. A particular sliding surface is defined to obtain the inductive voltage as a controlled output. The adaptation is performed using scalar factors of the input–output data with the assistance of an output error model. A general identification technique is obtained through scaling data. To obtain this data, recursive least squares (RLS) methods are used to estimate the parameters of a linear model using input–output scaling factors. In order to estimate the parametric values in the small-scale range, the input signal requires a high frequency and thus a high sampling rate is needed. Through this proposed technique, a broader sampling rate and input signal with low frequency can be used to identify the small-scale parameters that characterise the linear model. The results indicate that the proposed algorithm is practical and robust.
Manuel Schimmack, Paolo Mercorelli

### Sliding Mode Control for a Hydrostatic Transmission in Combination with a Sliding Mode Observer

Abstract
Hydrostatic transmissions are continuously variable hydraulic power converters, which provide lots of advantages and represent a characteristic drive train component in, e.g. all types of working machines, city vehicles and renewable energy plants. In high-performance motion control systems, however, hydrostatic transmissions are less frequently used than electrical and mechanical drives due to their nonlinear behaviour, the impact of unknown disturbances like leakage volume flows as well as disturbance torques, and model uncertainty. In this contribution, a sliding mode approach is applied to the tracking control of a hydrostatic transmission. Moreover—in order to robustly reconstruct the immeasurable system states and the unknown disturbances—a gain-scheduled modified Utkin sliding mode observer is proposed that is based on extended linearisation techniques. This observer-based control structure is compared with an alternative approach, where a flatness-based tracking control is combined with a nonlinear reduced-order observer. The efficiency and the performance of the proposed control structure are highlighted by both simulations and meaningful experimental results.
Hao Sun, Harald Aschemann

### Sliding Mode Observation with Iterative Parameter Adaption for Fast-Switching Solenoid Valves

Abstract
Control of the armature motion of fast-switching solenoid valves is highly desired to reduce noise emission and wear of material. For feedback control, information of the current position and velocity of the armature are necessary. In mass production applications, however, position sensors are unavailable due to cost and fabrication reasons. Thus, position estimation by measuring merely electrical quantities is a key enabler for advanced control, and, hence, for efficient and robust operation of digital valves in advanced hydraulic applications. The work presented here addresses the problem of state estimation, i.e., position and velocity of the armature, by sole use of electrical measurements. The considered devices typically exhibit nonlinear and very fast dynamics, which makes observer design a challenging task. In view of the presence of parameter uncertainty and possible modeling inaccuracy, the robustness properties of sliding mode observation techniques are deployed here. The focus is on error convergence in the presence of several sources for modeling uncertainty and inaccuracy. Furthermore, the cyclic operation of switching solenoids is exploited to iteratively correct a critical parameter by taking into account the norm of the observation error of past switching cycles of the process. A thorough discussion on real-world experimental results highlights the usefulness of the proposed state observation approach.
Tristan Braun, Johannes Reuter

### Sliding Mode Observer for Fault Diagnosis: LPV and Takagi–Sugeno Model Approaches

Abstract
This chapter investigates recently proposed fault reconstruction methods by sliding mode observers defined by two different model classes: linear parameter varying and Takagi–Sugeno models. Both model classes are used to design the sliding mode observers. They may be considered as a polytopic extension of the canonical form restricted to uncertain linear time-invariant systems originally introduced by Edwards and Spurgeon. This approach is best suited for plants which can be thought of as predominantly linear in the characteristics or for nonlinear plants which can be modelled well (at least locally) by linear approximations. For highly nonlinear plants which are operated in a large operating range, a structure restricted to uncertain linear time-invariant systems is not ideal, as the sliding term would then have to capture both: the nonlinear plant dynamics and the influence of the faults. The chapter describes the observer design for linear parameter varying and Takagi–Sugeno models, which are illustrated by the means of the inverted pendulum and the wind turbine benchmark from the literature. Simulation results are shown to demonstrate the capability of the designed observers.
Horst Schulte, Florian Pöschke

### Sliding Mode State and Fault Estimation for Decentralized Systems

Abstract
The interconnection of dynamical systems gives rise to interesting challenges for control in terms of stability, robustness and the overall performance of the global interconnected systems as well as the fault tolerance of the individual subsystems. Interconnected systems can be developed either from a standpoint of centrality of control based on the construction and design of a global system that satisfies the above requirements. Alternatively, the interconnected system can be decentralized which means that the stability, performance, etc., requirements are achieved at the local (subsystem) levels. To develop a good “fault-tolerant control” strategy for decentralized systems it is necessary to take account of various faults or uncertainties that may occur throughout all local levels of the system. A powerful way to achieve this is to use robust state and fault estimation methods accounting for the model–reality mismatch that is inevitable when (a) systems are linearized and (b) when faults occur in subsystem components such as actuators, sensors, etc. The chapter develops a strategy for decentralized state and fault estimation based on the Walcott–Żak form of sliding mode observer (SMO) with linear matrix inequality (LMI) formulation. This strategy is shown to be advantageous when considering the estimation problem for a large number of interconnected subsystems. After developing the design procedure a tutorial example of two interconnected linear systems with nonlinear interconnection functions shows that the states as well as actuator and sensor faults can be robustly estimated. Finally, an application-oriented example of a three-machine power system is given which has actuator faults as well as nonlinear machine interconnections.
Zheng Huang, Ron J. Patton, Jianglin Lan

### Fault Diagnosis of Nonlinear Differential-Algebraic Systems Using Hybrid Estimation

Abstract
Modern technical systems often contain components capable for extensive autonomous actions. Thus, an integrated system supervision is essential in enabling an adequate reaction for compensation of unpredictable substantial variations. This is addressed by fault detection, isolation and identification techniques discussed in this article. Therefore, an overview is given about modelling of systems subject to faults, continuous state estimation utilizing an unscented Kalman filter and hybrid state estimation by the interacting multiple model approach. These methods are generalized for application to nonlinear differential-algebraic equations, i.e. DAE systems. DAE systems arise in such fields as discretization of partial differential equations or optimization problems. However, the appearance of DAE systems most often results from an object-oriented modelling (OOM) approach. Since OOM is probably the most relevant approach for modelling complex systems, the generalization and adaptation of supervision methods to DAE is the principal subject of this contribution. Finally, the proposed fault identification approach is applied to a hydraulic system, and the related results are discussed in detail.
Dirk Weidemann, Ilja Alkov

### Towards Robust Fault-Tolerant Model Predictive Control with Constraints for Takagi–Sugeno Systems

Abstract
This chapter deals with the problem of a robust fault-tolerant model predictive control (RFT-MPC) for discrete-time nonlinear systems described by Takagi-Sugeno models. The RFT-MPC is a mixture of the $$\mathscr {H_\infty }$$-based parallel distributed controllers and the fast model Predictive Control. The approach proposed in the paper is based on a series of offline and online computations. For the given Takagi-Sugeno system, PDC is designed without considering input and state constraints. Moreover, the idea of robust invariant sets is employed to deal with both the input and state constraints. This also provides an efficient way to introduce the MPC algorithm. Therefore, enhancing the invariant set enlarges the domain of attraction. As the robustness is achieved offline, the MPC is not employed until large enough faults occur. Otherwise, it serves as a fault-tolerant control distributing any compensation actions between actuators to avoid their saturation if possible. Finally, an illustrative example is provided, proving the efficiency and quality of the proposed multi-stage RFT-MPC.
Piotr Witczak, Marcin Witczak

### Constrained Model Predictive Control of Processes with Uncertain Structure Modeled by Jump Markov Linear Systems

Abstract
Linear systems with abrupt changes in its structure, e.g. caused by component failures of a production system, can be modelled by the use of jump Markov linear systems (JMLS). This chapter proposes a finite horizon model predictive control (MPC) approach for discrete-time JMLS considering input constraints as well as constraints for the expectancy of the state trajectory. For the expected value of the state as well as a quadratic cost criterion, recursive prediction schemes are formulated, which consider dependencies on the input trajectory explicitly. Due to the proposed prediction scheme, the MPC problem can be formulated as a quadratic program (QP) exhibiting low computational effort compared to existing approaches. The resulting properties concerning stability as well as computational complexity are investigated and demonstrated by illustrative simulation studies.
Jens Tonne, Olaf Stursberg
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