Scientists and engineers engaged in the field of Nonlinear Control Systems will find it an extremely useful handy reference book. Springer Professional. Back to the search result list. Table of Contents Frontmatter Chapter 1. Introduction Abstract. In this chapter we give an introduction to control theory and nonlinear control systems. In Section 1. Section 1. A few typical nonlinear control systems are presented in Section 1. The purpose of this chapter is to present some basic topological concepts of point sets. What we discuss here is very elementary, thus should not be considered as a comprehensive introduction to topology.

But it suffices for our goal-providing a foundation for further discussion, particularly for the introduction of differential manifold and the geometrical framework for nonlinear control systems. Many standard text books such as [1, 2, 3] can serve as further references. This chapter provides an outline of Differential Geometry.

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First we describe the fundamental structure of a differentiable manifold and some related basic concepts, including mappings between manifolds, smooth functions, sub-manifolds. The concept of fiber bundle is also introduced. Then vector fields, their integral curves, Lie derivatives, distributions are discussed intensively. The dual concepts, namely, covector fields, their Lie derivatives with respect to a vector field, co-distributions and the relations with the prime ones are also discussed.

This chapter provides a fundamental tool for the analysis of nonlinear control systems. Geometry, algebra, and analysis are usually called the three main branches of mathematics. This chapter introduces some fundamental results in algebra that are mostly useful in systems and control. In section 4. Ring and algebra are introduced briefly in section 4. As a tool, homotopy is investigated in section 4. Sections 4. In sections 4. Section 4. In section 5. The materials are mainly based on [11, 10].

We also refer to [5, 9] for related results, to [4] for later developments. Section 5. Access provided by: anon Sign Out. Software toolbox for analysis and design of nonlinear control systems and its application to multi-AUV path-following control Abstract: In the paper, we present a software toolbox for rigorous analysis and design of nonlinear continuous and discrete-continuous digital control systems based on the reduction method and sublinear vector Lyapunov functions. For these systems, the toolbox provides solution of the following problems: verification of dynamic properties of dissipativity, asymptotic and practical stability; computation of estimates of basic direct indicators of dynamic quality accuracy, domains of attraction and dissipativity, settling time and others ; synthesis of parameters that ensure desirable or optimal quality of the system.

In addition, we show how the toolbox could be applied to solve the path-following control problem for a single autonomous underwater vehicle AUV as well as for multi-AUV formations. In our statements of the path-following control problem, we take into account together uncertainties of AUVs model, inaccuracy of measurements, and constraints on control actions. Historical role of analytical concepts in analysis and design of nonlinear control systems is briefly outlined.

Recent advancements in these systems from applications perspective are examined with critical comments on associated challenges. It is anticipated that wider dissemination of this comprehensive review will stimulate more collaborations among the research community and contribute to further developments.

Keywords: Nonlinear control ; Modern control ; Challenges in control ; Technological applications. Many physical processes are represented by nonlinear models. Examples include; coulomb friction, gravitational and electrostatic attraction, voltage-current characteristics of most electronic systems and drag on a vehicle in motion. Recently, many researchers from such broad areas like process control, biomedical engineering, robotics, aircraft and spacecraft control have shown an active interest in the design and analysis of nonlinear control strategies.

Thus, most of the real problems necessitate invariably bumping into nonlinearities [ 1 ]. Primary reasons behind growing interest in nonlinear control include [ 2 ]; improvement of linear control systems, analysis of hard nonlinearities, need to deal with model uncertainties and design simplicity. Nonlinear strategies improve trivial approaches by taking into accounts the dynamic forces like centripetal and Coriolis forces which vary in proportion to the square of the speed.

So, the linear control laws put a serious constraint on speed of motion to achieve a specified accuracy. However, simple nonlinear controller can reasonably compensate the nonlinear forces thus achieving high speed in an ample workspace. Also, hard nonlinearities like dead-zones, hysteresis, Coulomb friction, stiction, backlash and saturation do not permit linear approximation of real-world systems [ 3 ]. After predicting these nonlinearities, nonlinear approaches properly compensate these to achieve unmatched performance. Moreover, real systems often exhibit uncertainties in the model parameters primarily due to sudden or slow change in the values of these parameters.

A nonlinear controller through robustness or adaptability can handle the consequences due to model uncertainties [ 4 ]. Modern technology such as high accuracy high speed robots require fulfilling strict design requirements.

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The positioning of such robots is a nonlinear problem since it involves coordinate transformation matrices having sine and cosine terms. This approximation leads to non-uniform damping throughout the work-envelope and results other undesirable effects [ 5 ]. Considering a 6 Degree Of Freedom DOF manipulator, the over performance of a nonlinear strategy particularly in the presence of a disturbance is evident from Fig. Non applicability of superposition and homogeneity in case of nonlinear systems results in major implications on the analysis and design of the control systems [ 6 ].

The straight forward relationship between transfer function zero s and pole s locations and time response does not hold valid in general. An unforced nonlinear system can possess limit cycles not speculated by linear theory. The controllability and observability cannot be determined simply based on rank tests. Owing to the pertinent importance of nonlinear control in wide range of recent applications, this paper presents a brief comment on the subject topic. Interested readers are encouraged to refer to the original literature cited for more specific details.

The remaining paper is structured as follows: Section 2 briefly presents historical perspective of nonlinear control systems while recent advances and challenges are discussed in Section 3 and 4 respectively. Finally Section 5 comments on conclusion. However, the governor was made to work without concrete analytical concepts [ 8 ]. In , A.

Lyapunov, a Russian mathematician, presented two methods in order to determine the stability of dynamic systems described by Ordinary Differential Equations ODE. The second method called as direct method of Lyapunov, can determine stability without actually solving the ODE and thus finds potential in stability analysis of nonlinear control systems [ 9 ]. Lyapunov showed that if the linear approximation of a system is stable near an equilibrium point, then the truly nonlinear system will be stable for some neighborhood of that point.

Based on the proposed nonlinear second order equations, approaches were developed to predict various phenomena of nonlinear systems, which include subharmonic oscillations, limit cycles, jump phenomena and frequency entrainment. As a subject, control engineering was in its infancy till late s when scientific community started to face the problem of servomechanisms control [ 10 ]. The event of Second World War boosted research in nonlinear control of servomechanisms due to the functional requirements imposed by fire-control systems and control of guided vehicles.

During , three main analytical approaches used for analyzing nonlinear systems include; the describing function, the phase plane method and various methods involving relay systems. In the classical era, most problems involved single input, single output, linear, finite dimensional and time-invariant systems. Year is considered as start of modern era for nonlinear control [ 8 ]. The two key application drivers during this time were defense and space race.

Other industrial avenues where nonlinear control were applied include automobiles, ships, steel, paper, minerals etc. The nonlinear, time varying, highly dimensional, poorly modelled and multivariable nature of the encountered real systems were outside the bounds of classical control theory.

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The digital computer was first introduced as a design tool and later as a component of a control system. In early s, scientists investigated that the notions of energy and dissipation are linked with Lyapunov theory. So, dynamic systems can be viewed as energy transformation mechanisms. Based on this concept, Willems Jan proposed a theory for dissipative systems [ 11 ]. In s, Sontag and Wang proposed theory of input-to-state stability for nonlinear control systems [ 12 ], which can analyze stability of complex structures based on behavior of elementary subsystems and has been successfully applied to biological and chemical processes [ 13 ].

In , Isidori presented geometric control theory by introducing the concept of zero dynamics [ 14 ]. With the ability to analyze controllability and observability, differential geometry finds enormous potential in the domain of nonlinear control systems. A critical review of early history of nonlinear control shows that concepts related with optimality, stability and uncertainty were descriptive rather than constructive.

Table 1 summarizes prominent historical advances which directly or indirectly enriched the domain of nonlinear control systems. Table 1 Historical overview of advances in nonlinear control.

Developments in pure and applied Mathematics and to some extent in Physics have a great role in evolving nonlinear control strategies. Application of Differential algebra and multivariable calculus for understanding, formulation and conceptual solutions to the problems in automatic control resulted in various nonlinear control strategies. Detailed reviews of these strategies are reported in [ 5 , 40 ]. As an educational example, a variable structure control technique, SMC is selected here to be examined from mathematical perspective due to its robustness feature and long history of theoretical and practical developments.

This control technique has now become a de-facto solution to handle modeling and parametric uncertainties of a nonlinear system. Its other distinguishing features are reduced-order compensated dynamics and finite-time convergence. The core idea behind SMC is to drive the nonlinear dynamics of the plant onto the selected sliding surface reaching phase. The dynamics is then maintained at this surface for all subsequent time irrespective of nonlinearities. Figure 2 conceptualizes this concept.

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## Nonlinear control - Wikipedia

Considering a general architecture [ 7 ] with a nonlinear system assumed to be in canonical form i. The choice of elements of C ensures that s becomes Hurwitz monic polynomial. This ultimately makes the feedback system insensitive to the matched disturbances. The control law for SMC consists of a nominal feedback control term and an additional part to deal with uncertainties. Differentiating w. The inequality in 7 establishes that that sliding mode takes place in finite time even in the presence of uncertainties.

Choice of positive values of the constants c i in 2 ensures that the poles of the feedback system are in Left Half Plane LHP. An ideal SMC may require infinitely fast switching in an attempt to accurately track the reference trajectory. However, practical switched controllers have imperfections limiting the switching frequency. Thus the representative point may oscillate around the selected sliding surface leading to an undesirable phenomenon termed as chattering.

Figure 3 illustrates this concept. Solutions to this problem are discussed in [ 41 ]. In the last two decades, the advancements in nonlinear control systems have been in two folds; advances in theoretical approaches and more importantly application driven developments. In theory, major breakthrough has been seen in the areas of sliding control, feedback linearization and nonlinear adaptive strategies.

Recently, nonlinear control systems have gained high popularity primarily due to the extensive application of theoretical concepts to solve real world problems in various domains like electrical, mechanical, medical, avionics, space etc. Moreover, the advances in computer hardware and information technology have greatly resolved the computational constraints on analysis and design of nonlinear control systems.

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Robotic manipulators have reshaped the industrial automation and are now an integral part of most of the modern plants. Although linear control strategies like PID [ 43 ] have been the main workhorse in industry since decades, however, the trend to employ nonlinear approaches is gradually increasing [ 44 ]. A typical example of implementation of a nonlinear approach i.

SMC on a custom developed pseudo-industrial platform [ 45 ] is presented in [ 42 ]. The control objective was to ensure tracking of desired trajectory q d. The system dynamics can be altered by varying K and C. It also significantly reduces the chattering phenomena.

## Nonlinear Control Systems

They have also demonstrated compliance control via this scheme employing joint torque sensors [ 47 ]. The scheme is illustrated in Fig. Recently, the book by Speirs et al. They have addressed the associated actuator saturation issues by introducing anti-windup compensators. In medical domain, recent applications of nonlinear control includes anesthesia administration and control of devices for rehabilitation and prosthetics [ 49 50 51 ].

Bispectral Index BIS. DOH level of represents awake state while the level of refers to moderate hypnotic state and is considered as safe range to execute surgery. As shown in the figure, all the patients achieved the desired level of hypnosis. The last decade has seen the emergence of the systems biology approach to understand biological systems in a holistic manner [ 53 ]. Rather than enumerating individual components molecules, proteins the systems biology approach focuses on the interactions between subsystems in order to understand the emergent dynamic behavior of the living system.

Both the structural and functional organizations are important for characterizing the so-called symbiotic state of the biological system under study [ 54 ]. The structural organization involves network topologies including gene regulatory networks and biochemical reaction networks, and physical layers including molecules within organelles, organelles within cells, cells within tissues, tissues with organs and so forth. The functional organization involves cell functions including cell growth, cell division, cell differentiation specialization and cell death apoptosis.

The essence of systems biology is an understanding of the nonlinear dynamics of the biological system, which in turn requires the developments of computational models. Construction and analysis of such models of different subsystems modules of a system allow us to identify feedback loops in the system.

Computer simulations could be used to test a drug or therapy before expensive clinical trials. In control-theoretic terms, a disease could be represented by some region of the state space of a living system. Viewed as a control system, the reference or desired state of the system could represent the healthy state. The drug or therapy could be represented by a controller in the loop. An application of nonlinear control can be found in [ 55 ] where the authors apply feedback linearization and optimal control strategies to a nonlinear state-space model of HIV infection.

Other applications of optimal control of biological systems can be found in [ 56 ]. The curious reader might wonder why nonlinearity is ubiquitous in biological systems. This question can be answered fully in a brief account like this text but a justification will be provided by a few examples. Under appropriate assumptions, the mass action kinetics and mass action-like kinetics.

Mass action-like kinetics are obeyed by intracellular biochemical reactions, cell-cell interactions and inter-species interactions including the epidemiological and predator-prey interactions. The state variables in mass action-like kinetics multiply which gives rise to nonlinear terms in the differential equations [ 53 ].

Computing such a probability involves multiplication of the abundances of the reactant species. Other sources of nonlinearity include the feedback mechanisms in which a product of a reaction cascade could enhance or inhibit the reactants. Simplifications resulting from specific assumptions e.

The rich variety of such rate-equation models has been outlined in Fig. In chemical industry, the control of basic variables like pressure, levels, temperatures, flows and some quality variables is usually achieved by PID based law enhanced with advanced structures such as feedforward control, cascade control, ratio control, dead-time compensators etc. Common process characteristics in a chemical industry include [ 57 ]; unmeasured state variables and disturbances, multivariable interactions between variables, high order and distributed processes, uncertain and time-varying parameters and dead-time on inputs and measurements.

Complex problems arising due to constraints, delays, lags and model uncertainty can better be resolved using nonlinear control approaches. Owing to these reasons, Model Predictive Control MPC , explicitly based on nonlinear dynamic representation of the chemical process, was developed. MPC is based on mathematical optimization and is now de-facto standard in nonlinear control of process industries. Other related concepts include Dynamic Programming DP and iterative learning control for batch processes. Nonlinear control techniques are gradually replacing their linear counterparts in refineries and petrochemical plants to handle chemicals and polymers [ 59 ].

A detailed review presenting challenges and progress of control systems in chemical industry is reported in [ 60 ]. Power systems e. The major contributions of nonlinear control in power systems is to; regulate frequency and voltage, adequately damp the oscillations and preserve synchronisation in the presence of disturbances.

In power electronics, commonly encountered circuits exhibit nonlinear dynamics primarily due to the consequences of cyclic switching [ 68 ]. Existence of chaos, bifurcations and limit cycles in the circuits highlights the governing role of nonlinear control. In this context, MPC and SMC are common in power electronics owing to its ability to handle system constraints, multi-variable case and nonlinearities. Applications of MPC with prominent works in power electronics has been systematically reviewed in [ 69 ]. Noticeable recent advancements include; SMC [ 70 ] and its variants [ 71 , 72 ], Observer-based control [ 73 ] and reinforcement learning based nonlinear control [ 74 ].