Variable Structure Systems: from principles to implementation
2: Department of Postgraduate Study, National Autonomous University of Mexico, Mexico, Mexico
3: Department of Engineering, University of Leicester, Leicester, UK
This book is unique in that it aims to fulfil the fefinite need for an accessible book on variable structure systems and also provdies the very latest results in research on this topic. The book contains many numerical design examples, so that readers can quickly understand the design methodologies and their applications to practical problems.
Inspec keywords: variable structure systems
Other keywords: frequency domain analysis; sliding mode observation; causal nonminimumphase systems; output tracking; diesel generator set; discretetime variable structure control; fuzzy systems; delayed relay control; power electronics; deterministic output noise effects; chaos; deltamodulation; semiglobal stabilisation; second order sliding mode techniques; output feedback control; neural network systems; underwater objects; regulator design; stochastic output noise effects; dynamic systems; robustness; linear uncertain system; automobile applications; motion control; motion control systems; sliding mode control
Subjects: Multivariable control systems
 Book DOI: 10.1049/PBCE066E
 Chapter DOI: 10.1049/PBCE066E
 ISBN: 9780863413506
 eISBN: 9781849190008
 Page count: 427
 Format: PDF

Front Matter
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Part I: Sliding mode control theory
1 Sliding mode control
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The sliding mode control approach is recognised as an efficient tool to design robust controllers for complex highorder nonlinear dynamic plant operating under uncertain conditions. The research in this area was initiated in the former Soviet Union about 40 years ago, and the sliding mode control methodology has subsequently received much more attention from the international control community within the last two decades.
2 Sliding mode regulator design
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The Error Feedback Sliding Mode Regulation Problem has been introduced. Solution conditions are derived for linear systems and different classes of nonlinear systems including systems presented in the Regular and NBCforms. In particular, the combination of VSS and block control techniques allows straightforward solutions to be obtained, specially when compared to the classical solutions of the error feedback regulator problem. Additionally the sliding mode based controller achieves robustness with respect to the uncertainty.
3 Deterministic output noise effects in sliding mode observation
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In this chapter it has been shown that: the modified (with the linear correction term and δregularisation) concept of 'sliding mode observation' does really work, in principle, and provides acceptable quality of the stateestimation process for output noise of a deterministic nature: the averaged stateestimation error norm is shown to be bounded asymptotically; the correct selection of the gainmatrix K in the SMO is related to the corresponding algebraic Riccati equation; and the convergence zone is dependent on the process and observer properties and can be minimised by appropriate selection of the gain matrices.
4 Stochastic output noise effects in sliding mode observation
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The modified (with a linear correction term and δregularisation) concept of 'sliding mode observation' does work and provides good quality state estimates for the case of stochastic output noise: the average of the state estimation error is shown to be bounded asymptotically. Correct selection of the gain matrices Ko and K of the mixed observer is related to a corresponding algebraic Riccati equation. The convergence zone is dependent on the process and observer properties and can be minimised by appropriate selection of the gain matrices.
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Part II: New trends in sliding mode control
5 Discretetime VSS
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This chapter reviews some basic results obtained in the study of discretetime (DT) variable structure control systems (DVSCS) theory during its twentyfive year history. For this purpose, the chapter is organised as follows: in Section 5.1 basic defini tions, assumptions and remarks are introduced that are necessary for the connection with continuoustime (CT) variable structure control systems (CVSCS) and form an introduction of terminologies for DVSCS. Section 5.2 is a brief overview of the more notable works in DVSCS. Section 5.3 gives the definition of a quasisliding mode (QSM) and a DT sliding mode (DSM). In Section 5.4, the Lyapunov stability concept is used to define invariant sets in DVSCS. Section 5.5 gives DSM existence conditions as a new motion phenomenon that is not possible in CVSCS. In Section 5.6, a basic concept of DVSCS, which is founded on the DT equivalent control method and a boundary layer concept for the system with nominal parameters, is presented, while Section 5.7 introduces some disturbance estimation methods. Section 5.8 describes two methods of DVSCS with sliding sectors. In Section 5.9 basic properties of DVSCS are given. Design methods to establish sliding surfaces are summarised in Section 5.10. Section 5.11 gives numerical examples that illustrate the properties of some DVSCS algorithms. Some issues in the practical realisation of DVSCS are given in Section 5.12 and Section 5.13 contains a list of published papers and other work that has been used for the preparation of this chapter.
6 Robustness issues of 2sliding mode control
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The sliding mode control approach is based on keeping exactly a properly chosen constraint by means of high frequency switching of the control. The high frequency control switching leads to the socalled chattering effect which is exhibited by high frequency vibration of the controlled plant and can be dangerous in some applications. It is shown in this section that the arising higherorder sliding mode is never stable, but the instability is local and not crucial if the actuator is fast and stable. The case of a linear autonomous control system is considered. It is shown by the method of description functions that fast stable actuators cause oscillations in a small vicinity of the 2sliding manifold. Correspondent simulation results are presented.
7 Sliding modes, deltamodulation and output feedback control of dynamic systems
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In this chapter, we propose a sliding mode based algorithm for robust differentiation of reference signals with uniformly bounded rates which may also be subject to additive measurement noise. The proposed algorithm is based entirely in the reinterpretation of sliding mode features of Delta Modulation based signal tracking (see Steele [ 1 ] and Norsworthy et al. [2]), in combination with well known properties of the Equivalent Control method (Utkin [3]). We specifically show that an elementary reference signal tracking problem, with control decision inputs restricted to a discrete set, naturally yields a classic delta modulation tracking scheme consisting of a feed forward sign function nonlinearity in feedback connection with a pure integrator. The reference signal is only assumed to be differentiable with an absolutely bounded time deriva tive. The delta modulator output coincides, under ideal sliding conditions, with the equivalent control associated with the tracking problem. This 'equivalent' modulator output signal is just the time derivative of the exogenous reference input signal to the modulator, provided the switched gain is chosen in accordance with the (known) uniform absolute bound of the reference signal rate. Hence, using well known results of the equivalent control method, a first order low pass filtering of the modulator's output asymptotically converges to the time derivative of the input signal.
8 Analysis of sliding modes in the frequency domain
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The considered frequency domain methodology of analysis of SM control systems is based on the notion of the LPRS and an approach that involves substitution of the relay element with the equivalent gain, and analysis of the obtained linearised system. The LPRS comprises both: the oscillatory and the transfer properties of a relay system and succeeds even if the filtering hypothesis fails, and, therefore, can be used as a relatively universal characteristic of a relay system. It is proved that despite the fact that the LPRS is defined via the parameters of the periodic motion in the closedloop system, it is actually a characteristic of the linear part only. Three different formulas of the LPRS for both nonintegrating and integrating linear parts are derived and a methodology of analysis that involves the LPRS is presented. An illustrative example of the frequencydomain analysis of a SM system is considered.
9 Output tracking in causal nonminimumphase systems using sliding modes
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In this chapter an output tracking problem relating to nonminimumphase nonlinear systems has been considered. Nonminimumphase output tracking is a challenging, real life control problem that restricts the use of powerful control techniques such as sliding mode control and feedback linearisation. Taking into consideration uncertain causal systems that have to follow realtime reference profiles only complicates the problem further. In this chapter, the outputtracking problem for causal nonminimum phase systems with uncertainties and disturbances has been tackled by means of a robust nonlinear control technique, sliding mode control.
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Part III: Applications of sliding mode control
10 Sliding mode control and chaos
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In this chapter, we have examined the relationship between SMC and chaos. We have shown that digitising SMC in practice may cause some microlevel 'chaotic' behaviours, such as different periodic behaviours due to different initial conditions, an aspect of sensitivity to initial conditions. An interesting correlation between the periodic trajectories and their symbolic sequences has been explored. We have also discussed two SMCbased chaos control methods: one is the TDFC control and the other is a generalised OGY method. Their effectiveness has also been verified by computer simulations.
11 Sliding modes in fuzzy and neural network systems
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It is a well known fact that sliding mode control (SMC) is a powerful control method ology for both linear and nonlinear systems because of its robustness to parameter changes, external disturbances and unmodelled dynamics. Besides its power, the design of sliding mode controllers needs the information of the system's state, which makes the design relatively austere in some applications where the mathematical modelling of the system is very hard and where the system has a large range of parameter variations together with unexpected and sudden external disturbances. For those applications, a controller that will provide predicted performance even if the model of the system is not very well known, is needed. That controller should also adapt itself to large parameter variations and to unexpected external disturbances. These types of controllers are generally called 'intelligent' controllers, mainly work ing on the principles of fuzzy logic, neural networks, genetic algorithms and other technologies derived from artificial intelligence. The idea of combining these intelli gent control structures with the power of sliding mode control approach has attracted much research. A recent survey on the combination of SMC and intelligent control can be found in Reference 1. In this chapter, the union of sliding mode with neural networks and fuzzy logic is examined with examples from literature, and then a new technique combining neural networks and sliding mode control is presented.
12 SMC applications in power electronics
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Power converters are widely used in applications where it is desired to obtain a totally regulated electric signal from a nonregulated one, keeping optimum energy efficiency in the conversion. These converters can be linear or switched, the latter being the most common due to their better energy efficiency. As will be seen in this chapter, switching converters can be modelled as variable structure systems. They therefore constitute a natural field of application of Sliding Mode Control techniques. The most usual conversion types, namely DCDC, DCAC and ACDC, will be considered here. SMC controllers will be designed and several aspects involving the electronic implementation of the controllers will be discussed.
13 Sliding modes in motion control systems
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In this chapter the sliding mode design method and its application to motion control systems are discussed. The general solution for motion control systems with generalised forces or torques as control inputs is derived and its application to the tuningbelt servosystem as an illustrative example is shown. In this framework the dynamics of the subsystem that generates generalised force is neglected and the force control system is assumed ideal in the sense that it perfectly tracks reference values. The realisation of the control input in both a continuous time and discretetime framework is discussed. IM induction machine motion control and state estimation is discussed with the aim to show the validity of the SMC approach in cases where the dynamics of the torque/force generation is taken into account. It was shown that the same motion dynamics as attained in the previous case could be achieved here too. The design of the IM rotor flux and velocity observer is discussed in the last part of the chapter. The usefulness of the SMC approach is demonstrated in this case too.
14 Sliding mode control for automobile applications
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Automobilerelated control and estimation issues usually involve either highorder, high levels of nonlinearities or limited system state information. Evidently, from the reported results, sliding mode control theory demonstrates its capability of dealing with various problems, either control or estimation ones, with limited information. The results are promising and worthy of further modification or extension.
15 The application of sliding mode control algorithms to a diesel generator set
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This chapter demonstrates the application of both standard, first order sliding mode control and higher order sliding mode control techniques to a specific diesel power generator The specific first order methods considered are sliding mode inte gral tracking (SMIT) control and sliding mode modelfollowing (SMMF) control A second order sliding mode (SOSM) control is employed to demonstrate the higher order sliding mode technique. The SMIT and SMMF control algorithms are modified from the methods in References 1, 2 and 3. respectively. The SOSM control algorithm is modified from References 4 and 5.
16 Motion control of underwater objects by using second order sliding mode techniques
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In this chapter some recent results using a second order sliding mode methodology have been presented. This approach allows almost the same features of the first order sliding mode algorithms to be obtained in terms of simplicity, robustness and decentralisation of control structures, while eliminating the chattering phenomenon. The effectiveness of the relevant algorithm has been demonstrated experimentally on a prototype of an underwater actuation system based on opposite jets. The results appear promising for future generalisation to more complex underwater objects. The evaluation of the advantages of the proposed actuators with respect to traditional thrusters, briefly hinted at in this chapter, requires more systematic experimental as well as theoretical work and will be considered in future work.
17 Semiglobal stabilisation of linear uncertain systemvia delayed relay control
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The time delays that usually occur in relay and sliding mode control systems must be considered in system analysis and design. On the other hand, the presence of time delay does not allow the sliding mode control to be designed in the space of state variables. Even in the simplest onedimensional delayed relay control system only oscillatory solutions can occur. That is why the following directions in relay delayed control require investigation: research into time delay compensation; and control of the amplitude of any oscillations.
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Back Matter
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