Difference between linear and nonlinear control system pdf
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- Nonlinear control systems
- An Algebraic Approach to Linear and Nonlinear Control
- Analysis and Design of Nonlinear Control Systems
- Nonlinear control
Last five decades witnessed remarkable developments in linear control systems and thus problems in this subject has been largely resolved.
Nonlinear control systems are those control systems where nonlinearity plays a significant role, either in the controlled process plant or in the controller itself. Nonlinear plants arise naturally in numerous engineering and natural systems, including mechanical and biological systems, aerospace and automotive control, industrial process control, and many others. Nonlinear control theory is concerned with the analysis and design of nonlinear control systems. It is closely related to nonlinear systems System — nonlinear theory in general, which provides its basic analysis tools.
Nonlinear control systems
Last five decades witnessed remarkable developments in linear control systems and thus problems in this subject has been largely resolved.
The scope of the present paper is beyond linear solutions. Modern technology demands sophisticated control laws to meet stringent design specifications thus highlighting the increasingly conspicuous position of nonlinear control systems, which is the topic of this paper. 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. 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. 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. Robustness comparison of linear and nonlinear control strategies [ 7 ] a Disturbance b Tracking performance.
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.
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. 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. Considering a general architecture [ 7 ] with a nonlinear system assumed to be in canonical form i.
To achieve the objective of directing the states to zero even in the presence of disturbances, we define a sliding surface as. The control law for SMC consists of a nominal feedback control term and an additional part to deal with uncertainties. The overall control law can be written as. This term can be designed based on a Lyapunov function of the form.
Differentiating w. Using norm bounded assumption, 5 reduces to. 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. 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. Block diagram of SMC [ 46 ]. It also significantly reduces the chattering phenomena. 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. ISMC scheme for humanoid robotic arm [ 47 ]. 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. Performance of nonlinear controller - Results show that there is no overdose of anesthesia [ 52 ]. 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.
An Algebraic Approach to Linear and Nonlinear Control
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. This document was translated from BibT E X by bibtex2html. Publications about 'nonlinear control'. Articles in journal or book chapters. We consider a nonlinear SISO system that is a cascade of a scalar "bottleneck entrance" with a stable positive linear system.
Analysis and Design of Nonlinear Control Systems
Essays on Control pp Cite as. The analysis and design of control systems has been greatly influenced by the mathematical tools being used. Kalman brought forward state space analysis around For nonlinear systems, differential geometric concepts have been of great value recently. We will argue here that algebraic methods can be very useful for both linear and nonlinear systems.
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear , time-variant , or both.
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analytically described by equations such as algebraic, difference, differential and so on A non-linear controller handles the nonlinearities in large range of operation part of a control system so that the model uncertainties may be tolerated.