Nonlinear Control Systems 1. - Introduction To Nonlinear .
Nonlinear Control Systems1. - Introduction to Nonlinear SystemsDept. of Electrical EngineeringDepartment of Electrical EngineeringUniversity of Notre Dame, USAEE60580-01Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-011 / 54
Course OverviewThe course has three modulesAdvanced Tools for ODEsStability ConceptsNonlinear Controller SynthesisText - lecture notes that are transcribed from a variety of texts andmonographsHomework (once a week) 1-3 problems - We’ll arrange recitationsessions for HW groups (2-3 students). HW due the day afterrecitation. Recitations are optional after the first week.Two exams - Midterm/Final - closed book.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-012 / 54
Course OverviewNovel features of this course1Emphasis of relationships between tools and concepts, rather thandetailed discussion of proofs (proofs are in notes).2Topics on bifurcation and structural stability3Use more modern approach of treating Lyapunov functions as stabilitycertificates. Examines computational methods for certifying systemstability.4Emphasis on three stabilization methods; constructive methods basedon CLF, older geometric methods use in feedback linearization, andpassivity-based methods.5Introduces relatively recent ideas regarding Model-free controlschemes that provide a ”hybrid systems” approach to the regulationof nonlinear processes.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-013 / 54
Dynamical SystemsRather than starting with an ODE as the system model, let usconsider a more general formulation in which the process is seen as amathematical system.Σ (X , G , U, φ) where X is a topological space (state space), G is astrongly ordered group of time instants, U is a linear space of inputsignals defined on G , and φ : G X U X is a transition map.We assume the transition map is continuous andφ(0, x, u) x(identity)φ(s t, x) φ(t, φ(s, x, u[0,s] ), u[s,s t] )We call x : G X a state trajectory or orbit. The value x(t) denotesthe system ”state” at time instant t G .Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-014 / 54
Dynamical SystemsIf there is no input (i.e. u 0), then this is an unforced system(X , G , φ) and with the given assumptions we know that the orbits donot intersect and partition the phase space into equivalence classes.This justifies the notion of ”state”.Principle of Superpositionφ(t; x, αu1 βu2 ) αφ(t; x, u1 ) βφ(t; x, u2 )means the system is linear. You studied this in linear systems theory(EE550) when G with either Z (discrete-time) or R(continuous-time).We focus on systems that do not satisfy the principle ofsuperposition. These systems exhibit a wide range of interestingbehaviors that do not appear in linear systems.The objective of the first couple lectures is to present examplesintroducing these interesting behaviors. These behaviors will providethe motivation for our subsequent study of nonlinear control systems.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-015 / 54
Discrete-time System based on Logistic MapLet us consider a discrete-time system Σ (X , G , φ) whereX [0, 1], G Z, and φ : [0, 1] [0, 1] is given byφ(x) ax(1 x)(1)with a being a positive real constant. This is called the logistic map.Orbits are generated by the recursive equationxk 1 axk (1 xk ),x0where a 0 and {xk } k 0 is the orbit.Models population of organism in a box of fixed size. Whenpopulation is low, then there is abundant food/space and xk isincreasing. When population is high, there is a shortage offood/space and xk is decreasing.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-016 / 54
Logistic Mapcobweb plot10.90.9a 30.30.20.20.10.10population state 01214161820generation (k)Figure: Cobweb Graph and State Trajectory for Logistic System with a 2 andx0 0.2Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-017 / 54
Logistic MapOne can computationally examine what happens if the food supply, a,is increased without expanding the logistic map’s positive support (i.e.the living space). This gives rise to a number of scenariosExtinction: (0 a 1) - In this case the graph of Φ(x) never crossesthe 45 degree line formed by y x. x[k] decreases with eachgeneration and asymptotically approaches 0. In other words, for thisrange of a the population goes extinct due to a lack of food.Stable Population: (1 a 3) - In this case the 45 degree line fory x intersects the graph y Φ(x) at two points xu 0 andxs 1 1a . One can readily verify that the point xs 1 1a is anasymptotically stable equilibrium in the sense that x[k] xs ask . This case, therefore, corresponds to a scenario in which thepopulation achieves a “stable” fixed level.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-018 / 54
Logistic Map Limit Cycling Population: (3 a 1 6 3.449) - In this casethe population goes through a “boom and bust” cycle or what will belater referred to as a limit cycle. Fig. 2 shows the cobweb plot andstate trajectory for the case when a 3.4. In this figure thepopulation bounces between two different sizes on each generation.cobweb plot1h 0.30.30.20.20.10population state .9102468101214161820generation (k)Figure: Cobweb Graph and State Trajectory for Logistic System with h 3.4and x0 0.2Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-019 / 54
Logistic MapRoad to Chaos: (3.449 a 3.570) - For a just greater than3.449 1 6, the period 2 limit cycle begins doubling untila 3.570 at which point the population’s state trajectory becomeschaotic as shown in Fig. 3. The term ”chaos” formally means thatthe future states vary in a discontinuous manner with the initialcondition x0 .cobweb plot1h 0.30.30.20.20.10population state .9101002003004005006007008009001000generation (k)Figure: Cobweb Graph and State Trajectory for Logistic System with h 3.9and x0 0.2Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0110 / 54
Nonlinear Continuous-time OscillatorA nonlinear dynamical system, S (X , G , U, φ), is said to becontinuous-time if the index set G is the set of reals, R.Continuous-time systems can also exhibit a range of behaviors thatinclude convergence to fixed points, limit cycles, and chaos.The Duffing oscillator satisfies a second order differential equation ofthe formẍ γ ẋ ω 2 x x 3 Γ cos(Ωt)(2)where ω, γ, , and Γ are all positive real constants.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0111 / 54
Duffing OscillatorThe Duffing equation (2) maybe viewed as a forced oscillatorwhose spring provides arestoring force that is a cubicfunction of the system state.When 0 then the spring isa ”hardening” spring and when 0, then this spring is”soft”.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0112 / 54
Duffing OscillatorNumerically integrating equation (2) forward in time shows a periodic limitcycle when γ 0.1, 0.25, ω 1, Γ 0.5 and Ω 2.time seriesphase space0.40.40.30.30.20.20.10.1Poincaré section (GAM .5-1-0.500.511.52Figure: State trajectory (left), phase plane trajectory (middle), and Poincarésection (right) for forced Duffing oscillator (2) with γ 0.1, 0.25, ω 1,Γ 0.5, and Ω 2.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0113 / 54
Poincare SectionPoincaré section provides aconvenient way of viewing thebehavior of periodic statetrajectories. The method fixesa cross section, Σ, of the phasespace and determines a map Fthat maps a state x Σ ontothe state trajectory’s firstreturn, F (x), to the crosssection.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0114 / 54
Duffing Oscillator ChaosWhen we change Γ 1.5 in the Duffing oscillator, the Poincaré sectionshows a much more complex structure. This structure is sometimes calleda chaotic or strange attractor for the system.time seriesphase spacePoincaré section (GAM 0120140160180-3-2-10123-3-2-1012Figure: State trajectory (left), phase plane trajectory (middle), and Poincarésection (right) for forced Duffing oscillator (2) with γ 0.1, 0.25, ω 1,Γ 1.5, and Ω 2.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0115 / 54
Nonlinear Regulation ProblemsConsider an input-output system whose state trajectory, x : R Rn ,system output y : R Rm , and control input u : R Rp satisfy thefollowing set of equations,ẋ(t) f (x, u)y h(x, u)u k(x)We assume that the functions f : Rn Rp Rn andh : Rn Rp Rm are known.Given a desired output signal, yd : R Rm , the objective is to findthe state feedback controller k : Rn Rp such that the output signaly (t) yd (t) as t .Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0116 / 54
Regulation through Local LinearizationConsider a system ẋ f (x, u) where u is fixed and let φ denote thesystem’s flow.A point x Rn is called an equilibrium point if f (x , u) 0 for thegiven u Rm . This equilibrium point for the unforced system (i.e.u 0) is said to be hyperbolic if the eigenvalues of the Jacobian f(x , 0) , evaluated at x has no eigenvalues with zero realmatrix , xparts.The Hartman-Großman theorem establishes that the flow φ in asuitably small neighborhood of a hyperbolic equilibrium istopologically equivalent to the flows of the linearized system thatsatisfies f f ẋ (x , 0) (x x ) (x , 0) u(3) x u A(x x ) BuDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0117 / 54
Regulation through Local LinearizationTopological equivalence means there is a smooth invertiblestate-space transformation between the flows of the nonlinear systemand its linearization that preserves the direction of time.This suggests that if one were to design a state feedback controllermatrix, K , such that the control signal u K (x x ) asymptoticallystabilizes the linearized system about this equilibrium point, then weshould achieve adequate regulation of the nonlinear system.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0118 / 54
Nonholonomic Control of Two-wheeled RobotLet us see how well this ”linearized” control works for a two-wheeledrobotic vehicleThe “plant” is a two-wheeled robotic vehicle. Let F denote the forceapplied along the body’s x-axis and T be the torque about thevehicle’s center of mass.The control vector is u(t) [F (t), T (t)]T . The state variables arethe plant’s center of mass, x and y , the body angle, θ, the velocity,vx , and the angular rate, ω.yxDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0119 / 54
Regulation through Local LinearizationThe first step in developing such a state feedback control is to findthe linearization in equation (3) for our two-wheeled cart.We start by introducing the new tracking variables z1 x xd ,z2 y yd , z3 θ θd , z4 vx vd , and z5 ω where(xd (t), yd (t)) is the trajectory we want our vehicle to track in theplane, θd (t) is the direction of the desired trajectory’s velocity vector,and vd (t) is the magnitude of that desired velocity vector.With this change of variables our system equations in Fig. (z4 vd ) cos(z3 θd ) ẋdż10 0 ż2 (z4 vd ) sin(z3 θd ) ẏd 0 0 ż3 0 0z5 θ̇d ż4 1 0 v̇dż50 10? become u1 u2 F (x) G (x)uDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0120 / 54
Regulation through Local LinearizationComputing the Jacobian matrix for F and Gu yields the followinglinearized system equation z10 0 vd sin(θd ) cos(θd ) 0ż1 0 0 vd cos(θd ) sin(θd ) 0 z2 ż2 ż3 0 0001 z3 0 0 ż4 000 z4 z50 0000ż5 0 0 0 0 u1 0 0 1 0 u20 1 Az BuDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0121 / 54
Regulation through Local LinearizationWe can then use a number of methods that design stabilizingcontrollers for this system. In particular, we’ll compute the linearquadratic regulator (LQR) that find the state gains, K , such that thecontroller u Kz minimizes the cost functionalZ J[u] (z T z u T u)dτ0This controller was simulated in the following MATLAB script withthe desired trajectory 2πtẋd (t) 50 sin, xd (0) 050 4πtẏd (t) 50 cos, yd (0) 050The LQR control was recomputed at each time instant using thedesired reference trajectory states. We indeed obtain tracking of thedesired reference trajectory, though the vehicle’s initial transientshows some significant oscillation while it is picking up speed.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0122 / 54
Regulation through Local Linearization250desired trajvehicle traj200initial positionfinal position150direction of timevehicle tracking100direction of 700800xFigure: (left) MATLAB script - (right) trajectories for linearized control with(x0 , y0 ) (50, 0)Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0123 / 54
Regulation through Local LinearizationDept. of Electrical Engineering (ND)200desired trajvehicle traj150100direction oftime50yA limitation of the precedinglinearization approach is thatthe topological equivalence isonly local (i.e. in aneighborhood of theequilibrium point). Thissuggests that if we were tostart the vehicle further awayfrom the desired referencetrajectory then our controlstrategy might fail. This indeedis the case for our system.0-50vehicle is lost-100-150-200-400-2000200400600800xNonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0124 / 54
Regulation through Global LinearizationOne way of overcoming this limitation is to adopt a moresophisticated feedback linearization method that uses feedback toforce the system to appear as a linear dynamical system.This feedback linearization is a state transformation that transformsthe original system states onto a state vector consisting of the desiredtracking outputs and their derivatives.The advantage of this approach is that if that state transformation is“global”, then the controls we develop for this “feedback” linearizedsystem are also global and can thereby ensure asymptotic tracking ofthe reference trajectory for any initial vehicle condition.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0125 / 54
Regulation through Global LinearizationIn the feedback linearization approach we will find it convenient tointroduce a change of control variables in whichZ tu1 (t) F (t) v1 (s)ds0u2 (t) T (t) v2 (t)The original control, u1 F , is then treated as another system state,thereby extending the state vector of the original system.With this change of control variable we obtain the following stateequations for our cart, xvx cos θ0 0 y vx sin θ 0 0 0 0 v1d ω θ 0 0 v2Fdt vx ω 0 1 0F01 0Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0126 / 54
Regulation through Global LinearizationWe now introduce a state transformation which is obtained by takingthe derivatives of the tracking error.z1 x xd z2 ẋ ẋd , z3 ẍ ẍdz4 y yd , z5 ẏ ẏd , z6 ÿ ÿdThe differential equations for these components are then readilycomputed asż1 vx cos θ ẋd z2ż2 F cos θ vx ω sin θ ẍd z3ż3 .v1 cos θ vx v2 sin θ (2F ω sin θ vx ω 2 cos θ) x dż4 vx sin θ ẏd z5ż5 F sin θ vx ω cos θ ÿd z6ż6 .v1 sin θ vx v2 cos θ (2F ω cos θ vx ω 2 sin θ) y dDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0127 / 54
Regulation through Global LinearizationThese equations have the form of two chains of integrators driven bythe inputs into states z3 and z6 . This means we can rewrite the abovedifferential equations in the following form, ż1ż2ż3ż4ż5ż6 ż 000000 100000cos θsin θ010000000000000100 vx sin θvx cos θ000010 v1v2z1z2z3z4z5z6 001000000001 . (2F ω sin θ vx ω 2 cos θ) x d . 2F ω cos θ vx ω 2 sin θ y d Az E (α ρv )where A is the linear matrix representing the chain of integrators, αand ρ are matrices whose components are functions of the originalsystem states, F , ω, θ, and vx .Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0128 / 54
Regulation Through Global LinearizationNote that if we select the control v to have the form . xd 1v ρ α . Kzy(4)dThen the resulting state equation is given byż (A EK )z(5)whereT K E 0 0 1 0 0 00 0 0 0 0 1 k11 k12 k13 k14 k15 k16k21 k22 k23 k24 k25 k26 The important thing to note here is that equation (5) is a lineardifferential equation and so if we can select K so that A EK is aHurwitz matrix, then we would have globally stabilized our vehicularsystem using the control in equation (4).Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0129 / 54
Regulation through Global Linearization300250desired trajx0 (50,0)x0 (250,250)direction oftime200150initial positionfinal positiondirection of timeperfect trackingy10050direction gure: Feedback Linearized Controller (left) script - (right) trajectoriesDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0130 / 54
Bursting in Adaptive SystemsAdaptive control systems are systems that adapt the controller’s gainsin response to how well the controlled system is actually behaving.Even when this approach is used to adapt a linear dynamical system,the overall system will be highly nonlinear since the control signal is aproduct of the gain and the system state; both of which in turn aretime-varying.This nonlinearity makes can make it difficult to ensure the adaptionrule is well behaved (i.e. stable) and it sometimes leads to unexpectedbehaviors such as bursting; a phenomenon that occurs when anadaptive system suddenly exhibits a burst of oscillatory behaviorbefore returning back to normal.Bursting is a qualitative behavior found in biological systems andturbulent fluid flow. The objective of this section is to illustratebursting in a simple adaptive control problem and show how thenonlinearities in that system give rise to the bursting phenomena.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0131 / 54
Bursting in adaptive linear systemsAs a simple example, let us consider a linear plant whose statex : R R satisfiesẋ ax bu(6)in which a and b are unknown system constants but in which weassume we can measure the system state x(t).The objective is to select a control signal u : R R so that x tracksa reference model whose state xm : R R satisfiesẋm am xm bm r(7)where am 0, bm 0, r (t), and xm (t) are all known.The control input, u, is generated by the following equationu(t) k1 x(t) k2 r (t)(8)where k1 and k2 are “control gains”.The objective is to select these gains so that the model tracking error,e x xm , asymptotically goes to zero as time goes to infinity.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0132 / 54
Bursting in adaptive linear systemsLet us first note that the tracking error satisfies the differentialequationė ẋ ẋm ax b(k1 x k2 r ) am x bm rmNote that if we select k1 a aand k2 bbmb ,thenė ax b(k1 x k2 r ) am x bm r a ambm ax b x r am x b m rbb am (x xm ) am e(9)Since am 0, this would imply that for k1 k1 and k2 k2 thate(t) asymptotically goes to 0 as t .Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0133 / 54
Bursting in adaptive linear systemsThis particular choice for the control gains, however, requires that oneknow what a and b are ahead of time. Since these system parametersare unknown, our actual control law uses a k1 and k2 that are notequal to the ”optimal” values.For this non-optimal choice, the modeling error, e, satisfiesė ax b(k1 x k2 r ) am x bm r ax b((k1 k1 )x (k2 k2 )r ) bk1 x bk2 r am bm rWe can then use equation (9) to reduce the above equation toė am e b(k1 k1 )x b(k2 k2 )r(10)which more clearly shows how deviations in k1 and k2 from theiroptimal values impact the model tracking error.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0134 / 54
Bursting in Adaptive Linear SystemsIn model reference adaptive control (MRAC), one seeks an adaptationrule that adjusts the controller parameters k1 and k2 to ensure modeltracking.This adaptation rule may be expressed as a differential equationk̇1 φ1 (k1 , x, r , e),k̇2 φ2 (k2 , x, r , e)(11)where φ1 and φ2 are the functions we need to determine.The adaptation rule in equation (11) takes as input the current plantoutput, x, the reference input r and tracking error e to adjust thegains k1 and k2 .Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0135 / 54
Bursting in Adaptive Linear SystemsTo see what a plausible form for these adaptation functions shouldbe, let us consider a cost functionbb1(k1 k1 )2 (k2 k2 )2V (e, k1 , k2 ) e 2 22γ2γwhere γ 0 is a constant.Note that V 0 and let us consider the time derivative of this cost,V̇ , as e, k1 , and k2 follow the error dynamics in equation (10) underthe adaptation rule (11).In particular, if one can show that V̇ 0, then since V is boundedbelow by zero, then V (t) must be a monotone decreasing function oftime that converges to a limit point.So let us compute V̇ V V VV̇ ė k̇1 k̇2 e k1 k2bb e ė (k1 k1 )k̇1 (k2 k2 )k̇2(12)γγDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0136 / 54
Bursting in Adaptive Linear SystemsInserting equation (10) into (12) yields,V̇ am e 2 b(k1 k1 )(xe 11k̇1 ) b(k2 k2 )(xr k̇2 )γγNote that if we selectk̇1 φ1 (k1 , x, e, r ) γxek̇2 φ2 (k2 , x, e, r ) γxr(13)then we can ensure V̇ 0 for all e 6 0 and this would suggest V (t)is a decreasing function of time.The adaptation rule in equation (13) is sometimes called the MIT ruleand it represents one of the earliest adaptive control laws proposed foraerodynamic systems in the 1960’s.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0137 / 54
Bursting in Adaptive Linear SystemsVarious forms of the MIT rule have been used, but one well knownvariation takes the formxk̇1 γ θ x2ek̇2 γxr(14)This is sometimes called the normalized MIT rule since the size of thek1 update is normalized by the size of the current system state.Let us now look at simulations of the normalized MIT rule and seewhat happens. In this simulation, we let the plant and referencemodel beẋ 1.8x 2uẋm 3x 3rDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0138 / 54
Bursting in Adaptive Linear SystemsLet us first simulate this with γ 1 and θ 0.1 using the normalized MITrule from equation (14). The top plot shows e(t) 0 as time goes toinfinity. The lower plot shows k1 and k2 and the optimal gains (k1 and k2 )for this example.3x, plantxm, 0500timeFigure: Simulation of Normalized MIT rule for ẋ 1.8x 2u and ẋm 3x 3rwith no noise in the measured plant state.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0139 / 54
Bursting in Adaptive Linear SystemAn important thing to note about the simulation is that there is nomodeling error in the linear plant and there is no measurement noise.To see how our proposed adaptive control works under theseconditions, let us assume that the plant state, x, satisfies the ODE,ẋ 1.8x 0.1x 2 2uy x 0.1νwhere ν is a zero mean white noise process with unit variance.We now use the measured output y , rather than the true state in ourfeedback controller and adaptation rules,ê y xmu k1 y k2 ryk̇1 γ ê0.1 y 2k̇2 γ êrDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0140 / 54
Bursting in Adaptive Linear SystemsThe simulation results show t there is a burst in which the systembecomes highly oscillatory and then recovers tracking again.86x, plantxm, 200250300350400450500timeThe lower plot shows that k1 and k2 have a linear drift and bursting occurswhen these gains change sign.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0141 / 54
Bursting in Nonlinear SystemsThere are numerous other systems where the bursting phenomenonplays an important functional role.This occurs in the dynamics of cell membranes where the ”burst”corresponds to an impulse representing the transmission ofinformation down a nerve cell’s axon.A simplified version of the Fitzhugh-Nagumo model for nerveconduction.v̇ẇv3 w (y I )3 φ(v a bw ) v The variable v represents the cell membrane’s potential, w representsa ”gating” variable, and I represents the stimulating current(assumed to be constant).For a suitable constant I , the membrane potential exhibits a sustainedoscillation as shown in the top plot of Fig. 9.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0142 / 54
Bursting in Nonlinear SystemsBursting occurs when a small modulation in the driving currrent I y isintroduced. The burst is triggered when the current perturbation, y ,exceeds a given threshold.Voltage Potential with Constant Current, 50500Voltage Potential with Slow variation in I3bursting210-1-2-3-4050100150200250300Figure: Membrane Potential v , (top) constant applied current I - (bottom)modulated current I yDept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0143 / 54
Reachability in Nonlinear Control SystemsConsider a linear time-invariant systemẋ(t) Ax(t) Bu(t),x(0) 0(15)We’re interested in finding a control input, u that transitions thesystem state from x(0) 0 to the state x(T ) x1 where T 0.The state x1 is said to be reachable if there exists a finite time T 0and a control input u : [0, T ] Rm such thatZ Tx1 e A(T τ ) Bu(τ )dτ0The system is said to be reachable if every state in Rn is reachablefrom the origin.The necessary and sufficient condition for the reachability of thelinear system in equation (15) is that rank A AB A2 B · · · An 1 B n(16)Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0144 / 54
Reachability in Nonlinear Control SystemsIn particular, one usually examines reachability with respect tononlinear systems that satisfy the following state equationsẋ(t) A(x) B(x)uwhere A(x) is an n n matrix of functions, aij : Rn Rn , in whichA(0) 0 and B(x) is an n m matrix of functions, bij : Rn Rm .Since we know the flows about a hyperbolic equilibrium of thenonlinear system are topologically equivalent to flows about thelinearized system’s equilibirum, this suggests that one might be ableto use the reachability condition in equation (16) to determinewhether or not the nonlinear system is reachable.This is, however, not as simple as it seems on the surface. A simpleexample can be used to illustrate how nonlinearity complicates thisquestion.Dept. of Electrical Engineering (ND)Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-0145 / 54
Reachability in Nonlinear Control SystemsLet us consider the rigid body dynamics for a satellite controlled bytwo
Dept. of Electrical Engineering (ND) Nonlinear Control Systems 1. - Introduction to Nonlinear SystemsEE60580-01 13 / 54. Poincare Section Poincar e section provides a convenient way of viewing the behavior of periodic state tra
Outline Nonlinear Control ProblemsSpecify the Desired Behavior Some Issues in Nonlinear ControlAvailable Methods for Nonlinear Control I For linear systems I When is stabilized by FB, the origin of closed loop system is g.a.s I For nonlinear systems I When is stabilized via linearization the origin of closed loop system isa.s I If RoA is unknown, FB provideslocal stabilization
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I Applied Nonlinear Control, J. J. E. Slotine, and W. Li, Prentice-Hall, 1991 I Nonlinear System Analysis, M. Vidyasagar, 2nd edition, Prentice-Hall, 1993 I Nonlinear Control Systems, A. Isidori, 3rd edition Springer-Verlag, 1995 Farzaneh Abd
Nonlinear oscillations of viscoelastic microcantilever beam based on modi ed strain gradient theory . nonlinear curvature e ect, and nonlinear inertia terms are also taken into account. In the present study, the generalized derived formulation allows modeling any nonlinear . Introduction Microstructures have considerably drawn researchers' .
Nonlinear Space Plasma Physics (I) [SS-8041] Chapter 1 by Ling-Hsiao Lyu 2005 Spring 1-4 Probability Approach Chaos, fractal, and turbulence are popular ways to describe different stages of nonlinear phenomena. Nonlinear wave solutions obtained analytically by pseudo-potential method can be considered as a chaos type of nonlinear phenomena.
Calicut University P.O, Malappuram, Kerala, India 673 635 506A. School of Distance Education Spectrum: Literature & Contemporary Issues Page 2 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION STUDY MATERIAL FOURTH SEMESTER BA/BSc (2017 ADMISSION ONWARDS) COMMON COURSE : ENG4A06 : SPECTRUM: LITERATURE AND CONTEMPORARY ISSUES Prepared by: Smt. Smitha.N, Assistant Professor on Contract, School .
