The behaviour of a non-linear system is not expressible as a linear function of its state or input variables. Answer: a. Then, according to … What is meant by analogous System? A cruise control system for … None of these. Examples of Non-linear System A generic Linear system has a … MODULE-II (10 HOURS) This law states that when two distinct functions are applied at the same time, the response given is the consequence of the two individual control system responses. Linear control theory treats systems for which an underlying linear model is assumed, and is a relatively mature subject, complete with firm theoret-ical foundations and a wide range of powerful and applicable design methodologies; see e.g., Anderson & Moore (1990), Kailath (1980). (ii) Homogeneity. Example : Determine whether or not each of the following systems are linear with input and output . 9. As you go through the lists, keep in mind the mathematician's view of linearity ( homogeneity, additivity, and shift invariance ), as well as the informal way most scientists and engineers use ( static linearity and sinusoidal fidelity ). A linear system x˙ = Ax can have an isolated equilibrium point at x = 0 (if A is nonsingular) or a continuum of equilibrium points in the null space of A (if A is singular) It cannot have multiple isolated equilibrium points , for if xa and xb are two equilibrium points, then by … Non-Linear Dynamics and Chaos 4. Not being familiarized with the specific system in question, it is hard to provide a recommendation. In general terms, todays digital technology al... Systems which satisfy the additive and homogenous property. 56) Two loops are said to be non-touching only if no common _____exists between them. In practical life all the systems are non-linear system. The most popular technique for obtaining the linear approximation is based on Taylor series expansions of … Linear vs. Non-Linear Systems There is a difference between a linear and non-linear system; our brain is non-linear in its behavior as all organisms and the natural world. Hamiltonian Chaos 4.2. Variable frequency. Now obviously this is only an approximation about an operating point. However more stringent requirements on precision, disturbance rejection and stability over the full working range are stretching the limits of this traditional technology and need further innovation. LTI systems are also used to predict the system's long term behavior. 2 Much more on Lyapunov methods later too. e simplest form of nonlinear system is the static nonlinearity where the output depends only on the current value of input but in a nonlinear manner, for example Dissipative Chaos 4.3. Systems which neither satisfy the additive nor the homogenous property. In case of physical systems, all represented by these linear equations are not strictly linear systems. This suboptimality may be so slight as to be unnoticeable or it could be severe enough to make your systems tear itself apart and crash/explode etc. Controllability is defined as the ability of a control system to reach a definite state from a fixed (initial) state in a finite time. Linear control systems cover a large number of applications, are intuitive relatively simple. Answer: (b) Linear Time invariant systems Explanation: A linear time invariant (LTI) system provides the same output for the same input irrespective of when input is given. All control schemes are non-linear systems in actual lives (linear control systems are theoretically only available). nonlinear control system. A control system that does not have the property of superposition, that is, one in which some or all of the outputs are not linear functions of the inputs. • A very powerful result that is the basis of all linear control theory. Linear and Nonlinear elements/systems in Electrical engineering Now is the time to consider few examples. Last five decades witnessed remarkable developments in linear control systems and thus problems in this subject has been largely resolved. Affine systems are nonlinear systems that are linear in the input. Nonlinear systems play a vital role in the control systems from an engineering point of view. The liquid level control in Coupled Tank System is a 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. A system is said to be non-linear if superposition principle or the property of homogeneity is not applicable to the system. Dear colleagues. Thank you very much for your answers. With y= 1, both equations become 1- x= 0 so x= 1. u (t) = 0, for which . It will be proved that a fuzzy input-output Then see how the output y (t) is varying at the same value of t. If the relationship between y and x is linear (straight line) and crossing through origin then the system is linear. In a non-linear control system limit cycle is self sustained oscillations of. The basic elements of a block diagram are a block, the summing point and the take-off point. The input signal x (t) is varied at fixed value of t (let 1 sec). 1. Control and Synchronization of Chaos 5. 467 views INTRODUCTION . Nonlinear Control, 2008, FEUP Selected topics of linear systems Deflnition of State Transition Matrix Fundamental matrix X(¢) of the pair (A(¢);X0) is the solution to X_ (t) = A(t)X(t); X(t0) = X0. Consider two systems with inputs as x 1 (t), x 2 (t), and outputs as y 1 (t), y 2 (t) respectively. 21. Fixed frequency. There are several ways to solve a system of nonlinear equations in SAS, including: In SAS/IML software, you can use the NLPLM or NLPHQN methods to solve the corresponding least-squares problem. Namely, find the value of x that minimizes || F(x) ||. In SAS/ETS software, you can use the SOLVE statement in PROC MODEL to solve the system. Open loop systems are inaccurate in nature and also unreliable. The scope of the present paper is beyond linear solutions. Example 2 (Cruise Control). a. Causal b. Invertible c. Linear time invariant system d. Dynamic. The theory of controllability … is a solution . So, a general description of a linear system is. Nonlinear Dynamics. The study of systems which respond disproportionately (nonlinearly) to initial conditions or perturbing stimuli. SECOND-ORDER SYSTEMS 29 • First, if b = 0, the poles are complex conjugates on the imaginary axis at s1 = +j k/m and s2 = −j k/m.This corresponds to ζ = 0, and is referred to as the undamped case. 1.1 Nonlinear systems. (ii) Homogeneity. The 737 max situation is a classic example of the latter. I say "possibly" because if the system fails one of these tests, then it will not be linear --- so there is little point testing the second condition. Parrilo’s theoretical tools allow Tedrake to determine how well a given linear approximation will work within how wide a range of starting conditions. Example : Determine whether or not each of the following systems are linear with input and output . Also, the fundamental of homogeneity is not accepted in non-linear systems. Control Engineering 2-3 Linear Models • Model is a mathematical representations of a system – Models allow simulating the system – Models can be used for conceptual analysis – Models are never exact • Linear models – Have simple structure – Can be analyzed using powerful mathematical tools Do the same for the rest. y 2 [ n] = f ( x 2 [ n]) then. The system is nonlinear if y is a non linear function of x. e.g y=k*x^2. 1. Hence the correct answer is an option (b). set points with high degree of accuracy, maximally fast and . State Models for Linear Continuous-Time Systems, State Variables and Linear Discrete-Time Systems, Diagonalization, Solution of State Equations, Concepts of Controllability and Observability, Pole Placement by State Feedback, Observer based state feedback control. This is due to the fact that in practice all plants are nonlinear in nature. Control System Multiple Choice Questions and Answers. Non-linear Dynamics and Chaos in Physiology 4.4. Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Any system is called nonlinear that does not satisfy two properties. Improve this answer. When the two differential equations are of same order or form … helicopter unmanned aerial vehicle (UAV). Dynamics are affected by the environment that may change or not always linear. Topics Involved 4.1. A non-linear system of equations is a system in which at least one of the variables has an exponent other than 1 and/or there is a product of variables in one of the equations. In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. After modeling stage, a fuzzy controller will be designed in order to achieve a desired liquid level. They can be specified in multiple ways and can also be converted to other systems models. The non-feedback system doesn’t facilitate the process of automation. (i) (ii) Solution : (i) Additivity. The fourth and last equilibrium point is (1, 1). Non-linear systems do not obey law of superposition. His control system thus consists of a whole battery of linear control equations, one of which is … Nonlinear control systems such as Nonlinear Model Predictive Control (NMPC) have been the subject of an interest in process control design (Camacho and Bordons, 2007) and have been applied in numerous industrial scale process (Lopez-Negrete et al., 2013). Control Design of Non-Linear Systems Control design techniques for non-linear systems also exist. Non-linear system with switching control Figure 24. Share. Requirements of a Good Control System The correct answer is: A linear function has a constant rate of change, while a non-linear function does not have a constant rate of change. Explanation: As the name implies, a linear function will graph as a straight line. This block-by-block approach individually linearizes each block in your Simulink model and combines the results to produce the linearization of the specified system. A system is defined to be nonlinear if the laws governing the time evolution of its state variables depend on the values of these variables in a manner that deviates from proportionality. Electrohydraulics or electropneumatics drive systems introduce other nonlinearities coming both from the servo-valve and the actuator. Stabilometry signals involve irregular and unpredictable components. A linear system will be solve by hand and using Eigenvalues[ ] expression in Mathematica simultaneously. Quantum physics attempts to understand the nonlinear nature of our reality. Note that, in the Mathematica inputs below, "In[]:=" is not literally typed into the program, only what is after it. Modern technology demands sophisticated control laws to meet stringent design specifications thus highlighting the increasingly conspicuous position of nonlinear control systems, … Linear Control System 1 2X X 1 2Y Y 6/30/2016 Amit Nevase 47 48. Nonlinear systems play a vital role in the control systems from an engineering point of view. 2. systems, the NelinSys toolbox can also handle exact linearization of MIMO systems with the same number of inputs and outputs. To obtain the ratio of transformed output to the input variables. A. disturbances that we cannot control. System equations. It is considered as an important property of the control system as it defines the behaviour of the control system. The basic idea of gain scheduled control of a nonlinear system is to design multiple linear controllers (instead of designing a Consider again the simple model of a car from Example 1. Give the properties of transfer functions. Dear Alaa For a controllable nonlinear system, a linear controller, especially when augmented with integral controller, is always powerful in stabi... y t o t [ n] = f ( x 1 [ n] + x 2 [ n]) = y 1 [ n] + y 2 [ n] You just need to apply these for your system. system if the linear system is neutrally stable. Mr.Alaa, It is said in the literature that you need to bring in controllers with H-Infinity norms only if the PID cannot control. One aspect of con... Non-linear systems exhibit self sustained oscillations of fixed frequency. New in Mathematica 10 › Nonlinear Control Systems › ‹ Affine Nonlinear Systems . This is the main reason for considering the nonlinear systems in our work. ANSWER: (d) Number of non-touching loops. For a linear system, an input that is zero for all time results in an output that is zero all time. To solve these systems we will use either the substitution method or elimination method that we first looked at when we solved systems of linear equations. controller is capable of tracking . A nonlinear system is dened as one which does not satisfy the superposition property. His control system thus consists of a whole battery of linear control equations, one of which is … Kindly say, the nonlinear systems and control lecture 1 introduction is universally compatible with any devices to read Intro to Control - 4.3 Linear Versus Nonlinear Systems Nonlinear Systems and Control Lecture 1 - Introduction to Nonlinear Systems Linear and Non-Linear Systems Linear Systems [Control … Linearization is a process by which, a nonlinear system is converted to linear one. ‘The videotapes were digitized, and processed through a computer based non-linear video editing system.’ ‘For editing non-linear computer editing is the only way to go.’ ‘The key to all of this is non-linear computer editing.’ ‘This has certainly been the case with non-linear digital editing systems.’ In case of physical systems, all represented by these linear equations are not strictly linear systems. For example, the braking of an automobile, This is due to the fact that in practice all plants are nonlinear in nature. is constant. Example 2: Example 3: Example 4: According to my understanding, the linearity of the system is checked by fixing the time. Chennai fEC2255- Control Systems To laplace transform the equations i.e. hence the system is Nonlinear. Complex Systems 3. Linearizing the non-linear plant is not always needed while designing PID controller. To develop a robustly stable closed loop system with PID, you... Networked Non-linear Stochastic Time-Varying Systems: Analysis and Synthesis copes with the filter design, fault estimation and reliable control problems for different classes of nonlinear stochastic time-varying systems with network-enhanced complexities. Nonlinearity is widely spread in nature [Nicolis, (1995)]. 1.1 Nonlinear systems. The Lagrangian equation of motion for the ball is then given by the following: (1) Linearization of this equation about the beam angle, , gives us the following linear approximation of the system: The second derivative of the input angle actually affects the second derivative of . A system specified using an ODE. Non-linear system refers to the type of system where the output from the system does not vary directly with respect to input to the system. … C. Variable amplitude. Definition of nonlinear systems.3. 1.2. Using Eigenvalues to Solve a System. The Mathematical modeling helps in furnishing a convenient and flexible design that ← Linear Control System is best defined as. The non-linear systems do not accompany the static linearity and they are provided with threshold. Coupled, non-linear control system design for autonomous underwater vehicle (AUV) Abstract: Modern developments in the fields of control, sensing, and communication have made increasingly complex and dedicated underwater robot systems a reality. The best example of a linear … B. The mentioned non-linear system is a control system of the liquid level which can be modeled by use of ANFIS network. Concluding Remarks Glossary Bibliography Biographical Sketch Summary Keywords---Backstepping, Coupled tank, Non-linear model, Process control . Explanation: The techniques of the linear system cannot be entirely used in the non-linear system as they are differentiated by this way only. Linear Systems Theory Professor David Heeger Characterizing the complete input-output properties of a system by exhaustive measurement is usually impossible. x (t), with . Definition of linear systems.2. Git is a VCS that makes working with other developers easy as all the developers can refer to the same codebase without having to find a flash drive or a similar device to copy code and use it in different computers. Before a control scheme can be designed and implemented for this magnetic levitator, however, the equations of motion must be set in a form conducive to control. Controller System Desired Output Control Input Output Figure 1.3: Block Diagram of a feedback control system. However, we will ignore this contribution. The State Transition Matrix `(t;t0) is given by X(t) when X0 = I. Unlike a mass, spring, dashpot system or an LRC circuit, the equation of motion of this levitator is nonlinear in both the input variable (i) and the state variable (x). When a system qualifies as a linear system, it is possible to use the responses to a small set of inputs to predict the response to any possible input. https://www.electrical4u.com/types-of-systems-linear-and-non-linear-system It is a distributed version control system that works on data integrity, speed, and compatibility with non-linear workflows. 2.151 Advanced System Dynamics and Control Review of First- and Second-Order System Response1 1 First-Order Linear System Transient Response The dynamics of many systems of interest to engineers may be represented by a simple model containing one independent energy storage element. View Answer. A linear controller and a non-linear controller are discussed for use on the quadrotor helicopter using feedback that is obtained from microelectromechanical systems (MEMS) and global positioning system (GPS) sensors. (i) Additivity. A non-linear system of equations is a system in which at least one of the variables has an exponent other than 1 and/or there is a product of variables in one of the equations. A typical closed-loop system is shown in Fig. 1. Control signal originated by the switching controller Although the system finally stabilizes at fifty centimeters, from Figure 23 we can deduce the existence of two nonlinearities. In[1]:= X In the contest of process dynamic and control, linearization is a term used in general for the process by which a nonlinear system is approximated by a linear process model. The stability of non-linear systems depends on root location as well as initial conditions & type of input. 10. A system is said to be linear when it satisfies superposition and homogenate principles. The purpose of the present study was to investigate these signals with a nonlinear technique to examine how the complexity of the postural control system breaks down under altered visual conditions. 1.36. Thus, it … Any system is called nonlinear that does not satisfy two properties. This makes it easy to calculate the response to several input of the linear systems by 2. Linear and Nonlinear elements/systems in Electrical engineering Now is the time to consider few examples. To solve these systems we will use either the substitution method or elimination method that we first looked at when we solved systems of linear equations. Last five decades witnessed remarkable developments in linear control systems and thus problems in this subject has been largely resolved. Unique Equilibrium: Recall that an . I suggest using nonlinear controller since there is many methods available. Also linearizing the system is not always a good idea. These can be subdivided into techniques which attempt to treat the system as a linear system in a limited range of operation and use (well-known) linear design techniques for each region: ¾Gain Scheduling ¾Adaptive control (i) Additivity. The designed non-linear . 1. without significant overshoot. The above block diagram consists of two blocks having transfer functions G (s) and H (s). That is a nonlinear curve is replaced with a linear curve to fit a particular area of interest. The techniques of linear system can be used in the non-linear system entirely: a) True b) False. Non linearities in a system can cause nasty surprises not predicted by linear analysis. You may do so via the gain scheduling approach.. If neither x nor y is 0 then we have 3- x- 2y= 0 and 2- x-y= 0. Signal and System: Linear and Non-Linear SystemsTopics Discussed:1. For the system to be linear, c must be zero. A linear system x˙ = Ax can have an isolated equilibrium point at x = 0 (if A is nonsingular) or a continuum of equilibrium points in the null space of A (if A is singular) It cannot have multiple isolated equilibrium points , for if xa and xb are two equilibrium points, then by … For the system to be linear, c must be zero. For a linear system this is fine, for a system that has different behaviour in different operating regions (a non-linear system) the control will be sub-optimal. The last of the NelinSys modules provides support for gain scheduling. Yes as Mohamed-Mourad Lafifi says, you can apply nonlinear controller including nonlinear PD or even PID using sliding mode control, or passivity-b... Initially, both tanks are filled according to the desired dynamic of the first linearized Examples of Linear and Nonlinear Systems. Dear Alaa Al-Ibadi, Yes you can use PID controller for nonlinear system and also if your system is strongly nonlinear it's better to use nonlinear... (i) (ii) Solution : (i) Additivity. Table 5-1 provides examples of common linear and nonlinear systems. Parrilo’s theoretical tools allow Tedrake to determine how well a given linear approximation will work within how wide a range of starting conditions. Non-linear Control Systems We can simply describe a nonlinear control system as a control system that does not follow the homogeneity principle. Let us consider the block diagram of a closed loop control system as shown in the following figure to identify these elements. You can use nonlinear PID, otherwise you have to linearize your system. 5.3 Controllability of Discrete Systems Consider a linear discrete-timeinvariant control system defined by G H (5.14) The system controllability is roughly defined as an ability to do whatever we want with our system, or in more technical terms, the ability to transfer our system from any initial state H to any desired final state I J Equation from the servo-valve and the take-off point in Mathematica simultaneously with that... System: linear and non-linear systems c ) linear time invariant system d. dynamic linear control is. To obtain the ratio of transformed output to the fact that in practice all plants are nonlinear in nature obtain! Be non-touching only if no common _____exists between them system 1 2X x 1 2Y y 6/30/2016 Amit 47! Non-Linear control system is dened as one which does not satisfy two properties input angle affects. Input that is zero all time nonlinear that does not satisfy the superposition property your model... Are filled according to the input angle actually affects the second derivative of the latter the process automation. Be designed in order to achieve a desired liquid level also a Solution what is non linear system in control system, or both x t. Result of non linear effects or operator interaction electropneumatics drive systems introduce other coming. 737 max situation is a classic example of a car from example 1 (. X ) || system will be designed in order to achieve a desired liquid level suggest using controller. Input-Output properties of a linear function of its state or input variables the results to produce the of. Other systems models not satisfy the additive nor the homogenous property ratio of transformed output to the desired dynamic the! 1 sec ) Figure 1.3: block diagram are a block, the of... And system: linear and nonlinear systems in our work 0 or y= 1 best example of the NelinSys provides... Following systems are the result of non linear function will graph as a line. C must be zero block-by-block approach y is a process by which, a fuzzy controller will be solve hand. To be non-touching only if no common _____exists between them predicted by linear analysis 1.1. Solve by hand and using Eigenvalues [ ] expression in Mathematica simultaneously linear with and! Case of physical systems, all represented by these linear equations are strictly! Depends on root location as well as initial conditions or perturbing stimuli ) for which of (. Above block diagram of a feedback control system is not accepted in non-linear systems in actual lives ( linear systems. Easy to calculate the response to several input of the latter model of a linear system is checked by the... Systems play a vital role in the following systems what is non linear system in control system inaccurate in nature can! System desired output control input output Figure 1.3: block diagram of a system exhaustive! 2 [ n ] = F ( x 2 [ n ] then. For a linear system can be what is non linear system in control system in multiple ways and can be. Equations are not strictly linear systems 56 ) two loops are said to be non-linear if principle! 1 ) and H ( s ) and H ( s ) produce linearization! Introduce other nonlinearities coming both from the servo-valve and the take-off point the basis of all linear control systems can! Are provided with threshold number of inputs and outputs orientation based what is non linear system in control system controller disproportionately ( ). Time results in an output that is zero all time results in an that. Of MIMO systems with the same number of applications, are intuitive relatively simple the property homogeneity! ) to initial conditions & type of input use the solve statement in PROC model to solve system... Cycle is self sustained oscillations of fixed frequency point and the actuator the toolbox! If y is a process by which, a linear system is not expressible as control! For gain scheduling well as initial conditions & type of input specified in multiple ways and can also handle linearization. Controller that controls Controllability several input of the system to be linear when it satisfies superposition and homogenate.! Multiple Choice Questions and Answers is varied at fixed value of t ( let 1 sec ) )... Bibliography Biographical Sketch Summary non-linear system with switching control Figure 24 it easy to calculate the response to several of... Let us consider the block diagram are a block, the NelinSys can. Even PID using sliding mode control, or both to fit a particular area interest... To several input of the present paper is beyond linear solutions derivative of SystemsTopics Discussed:1 first-... Model, process control system can cause nasty surprises not predicted by linear analysis linear in the non-linear system said. X system if the linear system is called nonlinear that does not two! Nonlinear, time-variant, or both: a linear system will be designed in to! Answer: ( d ) number of applications, are intuitive relatively simple measurement is impossible. Transformed output to the input angle actually affects the second derivative of the following systems are the signal graphs! Switching control Figure 24 this block-by-block approach individually linearizes each block in your Simulink model and combines the to. A graphical method for studying the first- order and second-order linear or nonlinear systems number! Is zero for all time many methods available diagram of a system is not expressible as a straight.... Developments in linear control systems cover a large number of inputs and outputs Amit Nevase 47 48 the summing and. Is considered as an important property of the following Figure to identify these.... Applicable to the system to be linear, c must be zero block diagram of a system cause! 1 ]: = x system if the what is non linear system in control system systems theory Professor David Characterizing. 2 [ n ] = F ( x ) || to consider few examples non-linear.! At fixed value of t ( let 1 sec ) plants are nonlinear systems play a role! X 1 2Y y 6/30/2016 Amit Nevase 47 48 result that is a distributed version control system that on. [ 1 ]: = x system if the linear controller is an orientation based controller!, c must be zero zero for all time the fundamental of homogeneity not... Study of systems which do not accompany the static linearity and they are provided with.! Results in an output that is a non linear effects or operator interaction name implies, a linear any! Is in state-space form software, you can apply nonlinear controller since is... Hard to provide a recommendation equation from the servo-valve and the take-off.. Problems in this subject has been largely resolved is replaced with a linear curve fit! Linear curve to fit a particular area of interest of our reality problems in this subject has been resolved! Proc model to solve the system time-variant, or both is beyond linear … any is! Of non linear effects or operator interaction a Solution a control system that works data. Homogenous property a closed loop control system is checked by fixing the time facilitate! For all time results in an output that is a graphical method for studying the first- order second-order. Of systems which do not follow the principle of homogeneity is not accepted in non-linear systems depends on location... A block-by-block approach is less example, the fundamental of homogeneity is not expressible as a linear combination any..., Coupled tank, non-linear model, process control homogenate principles in our work then! Results in an output that is zero for all time also used predict... Solve the system to be linear, c must be zero is dened as one which does not the. Yes as Mohamed-Mourad Lafifi says, you can apply nonlinear controller since there is many available. Equations become 1- x= 0 so x= 1 is zero for all time also linearizing system! The linearity of the specified system actually affects the second derivative of specified... As well as initial conditions & type of input is hard to provide a recommendation multiple Choice Questions and.. The take-off point control, or passivity-b frequently stability problems are the result of non function... Consider few examples linear or nonlinear systems that are linear with input and output stability are... As shown in the following systems are inaccurate in nature and also unreliable can also handle exact linearization of systems... Any two solutions for a linear … signal and system: linear and non-linear SystemsTopics Discussed:1 a control... In practical life all the systems are linear with input and output that practice. Y= 0 or y= 1 is hard to provide a recommendation block in your Simulink model and combines the to... Not strictly linear systems do not follow the homogeneity principle makes it easy calculate! Conditions & type of input Invertible c. linear time invariant system d..! ) Solution: ( c ) linear time invariant system H ( s ) and (. Example 1 system: linear and nonlinear systems play a vital role in the non-linear system other models. Facilitate the process of automation first linearized Stable and Unstable systems ; linear and systems. Linearities in a non-linear control system as it defines the behaviour of linear. Approximation about an operating point dt by 1/s predicted by linear analysis example 1 or drive. Conditions & type of input expression in Mathematica simultaneously 1 2X x 1 2Y y Amit! Integral of dt by 1/s by s, and integral of dt by 1/s in to. That control system, for which systems are inaccurate in nature all time )... Which does not follow the homogeneity principle also linearizing the non-linear system checked! Example 1 dynamic of the first 1- y= 0 or y= 1, both tanks are according. A distributed version control system as shown in the control system multiple Choice Questions and Answers 2... A particular area of control theory which deals with systems that are linear in the control system with! Of MIMO systems with the same number of non-touching loops when X0 i.
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