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Institutional Subscription. 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. 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.
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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. The system dynamics can be altered by varying K and C. 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.
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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.
Nonlinear Control Systems: An Introduction
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 ].
Further control approaches based on physical principles e. Another trend which is gaining popularity since last two decades is to introduce adaptability and optimality in traditional nonlinear control strategies. The most effective and advanced nonlinear control schemes are the adaptive controllers. The basic idea of an adaptive controller is to estimate uncertain parameters and to tune the controller online to adopt to the dynamic situation where the variations in the system parameters or environment are significant.
Adaptive control got more popularity in s in the aerospace control community. However, due to lack of well-understood stability tools and proper hardware, it lost its charm. Another particular reason for this loss of interest was incident of the plane crash which was controlled by an adaptive law [ 84 ]. Nevertheless, interest in adaptive control has revived especially in social robotic because of its suitability for a highly nonlinear, unstructured and dynamic human environment as compared to the controlled and well-ordered industrial work cells.
In general, there are two main types of adaptive control schemes i. In MRAC, the adaptation mechanism estimates suitable parameters, so that the system behave the same as the reference model. Self tuning adaptive control [ 85 ]. Adaptive controllers may employ dynamic model and estimate the uncertain parameters [ 86 ] or may be dynamic model free. Model based adaptive controllers can be easily tuned and their performance is usually better as observed be Khan et al. However, dynamic model becomes very cumbersome as the DOF increases. This may results in computational and communication issues thus making the implementation of dynamic model based adaptive schemes more challenging.
Therefore, dynamic model free adaptive controllers can be the best choice for a high DOF plant, eliminating the need for a dynamic model. The most effective and advanced nonlinear control adaptive controllers are very powerful in terms of tracking accuracy and fast response. However, adaptive control action usually involves high energy and aggressiveness. This may be a major disadvantage in application where optimal usage of resources is critical e. On the other hand, traditional linear optimal control techniques e.
Reinforcement learning is one of the key methods to introduce optimality into the adaptive control schemes. Sutton et al. Reinforcement learning based optimal adaptive controllers solve nonlinear algebraic Riccati equation online. It usually consists of an Actor-Critic structure as shown in Fig. Critic evaluates the action taken by the actor i. Subsequently, the control policy is updated and finally developing an optimal control policy over time.
As an educational example, an inverted pendulum mounted on a moving cart is presented in this section. The system is driven by a horizontal force u applied at the cart. Assume viscous friction of coefficient b. The nonlinearity arises from the trigonometric functions, products, and squares of the outputs.
The other two functions f 2 and f 4 could be worked out algebraically. The nonlinear state-space model can be linearized around a suitable equilibrium, say the origin of the 4-dimensional state space.
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It should turn out that some of the eigenvalues have positive real parts. That corresponds to an unstable system. Such a relocation is useful not only to stablize a system but also to achieve desired closed-loop behavior such as good damping and a fast response. Nature is nonlinear and the best way will be to deal with it with nonlinear approaches.
Although, linear control has been successfully employed for decades. The problem with linear system is that it may not be able to cater for the modern and advanced technology such as multi-degree of freedom robots with under actuation or constraints , UAVS, submarines and missile control in which more complex system operations are involved and greater application ranges are required. The decision to choose linear or nonlinear control for a particular application may be a difficult one.
Linear control is well tested and the people in industry have confidence in it. There are plenty of excellent analysis tools available for linear systems such as Bode plot, root locus, Nyquist stability criteria, Laplace transform, Z-transform and Fourier transform etc. In contrast, nonlinear system needs complex mathematical analysis e. Lypunov stability criterion, Popov criterion and singular perturbation methods.
Mathematical modelling may also be cumbersome for nonlinear systems. Introduction to passivity. Bruce Francis bruce. A good undergraduate course on linear systems and a liking of mathematics. A course on state-space theory would be advantageous but is not strictly necessary. Here are course notes you may need to get up to speed:.
The most popular textbook for this subject is Nonlinear Systems , 3rd ed. Other very nice books are Nonlinear Systems Analysis , 2nd ed. A more elementary, easy to read text for Chapter 2 of the course is Differential Equations with Applications and Historical Notes , by Simmons. Lecture times. Approximately every week, homework will be assigned for the next week.
Random Processes in Nonlinear Control Systems by A A Pervozvanskii, Volume 15
You must work on the homework and hand it in. You may work in groups, but what you hand in must be your own write-up. I will read your solutions, but they won't count toward your course grade. Repeat: To receive a course grade, each student taking the course for credit must hand in homework; I will read it to see how you're doing, but it's not counted towards your grade.
Midterm test. October 20, 9 am - 11 am. The test is "limited open book. You will be provided with a booklet in which to write your solutions. Some students don't like them. If you are one of those students, you may bring your own paper. The test covers only Chapter 2. There are four problems:. There are three small parts, the first requiring a formal proof in logic notation. Linear and nonlinear functions; derivatives.
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Motion in the plane; reference Problem The Picard-Lindelof theorem. Final exam.
Dec 9, 9 am - noon, BA The exam is "limited open book.