Nonlinear Control Lecture 8: Nonlinear Control System Design
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinNonlinear ControlLecture 8: Nonlinear Control System DesignFarzaneh AbdollahiFarzaneh AbdollahiDepartment of Electrical EngineeringAmirkabir University of TechnologyFall 2011Nonlinear ControlLecture 81/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinNonlinear Control ProblemsStabilization ProblemsFeedback ControlTracking ProblemsTracking Problem in Presence of DisturbanceTracking Problem in Presence of DisturbanceSpecify the Desired BehaviorSome Issues in Nonlinear ControlModeling Nonlinear SystemsFeedback and FeedForwardImportance of Physical PropertiesAvailable Methods for Nonlinear ControlFarzaneh AbdollahiNonlinear ControlLecture 82/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinNonlinear Control ProblemsIIObjective of Control design: given a physical system to be controlledand specifications of its desired behavior, construct a feedback controllaw to make the closed-loop system display the desired behavior.Control problems:1. Stabilization (regulation): stabilizing the state of the closed-loop systemaround an Equ. point, like: temperature control, altitude control ofaircraft, position control of robot manipulator.2. Tracking (Servo): makes the system output tracks a given time-varyingtrajectory such as aircraft fly along a specified path, a robot manipulatordraw straight lines.3. Disturbance rejection or attenuation: rejected undesired signals such asnoise4. Various combination of the three aboveFarzaneh AbdollahiNonlinear ControlLecture 83/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinStabilization ProblemsIAsymptotic Stabilization Problem: Given a nonlinear dynamic system:ẋ f (x, u, t)Ifind a control law, u, s.t. starting from anywhere in region Ω x 0 ast .If the objective is to drive the state to some nonzero set-point xd , it canbe simply transformed into a zero-point regulation problem x xd as thestate.IIStatic control law: the control law depends on the measurement signaldirectly, such as proportional controller.Dynamic control law: the control law depends on the measurement througha differential Eq, such as lag-lead controllerFarzaneh AbdollahiNonlinear ControlLecture 84/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinFeedback ControlIState feedback: for system ẋ f (t, x, u)IOutput feedback for the systemIIẋ f (t, x, u)y h(t, x, u)The measurement of some states is not available.an observer may be requiredFarzaneh AbdollahiNonlinear ControlLecture 85/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinFeedback ControlIState feedback: for system ẋ f (t, x, u)IOutput feedback for the systemIII f (t, x, u)y h(t, x, u)The measurement of some states is not available.an observer may be requiredStatic control law:IIẋu γ(t, x)Dynamic control law:IIIu γ(t, x, z)z is the solution of a dynamical system driven by x: ż g (t, x, z)The origin to be stabilize is x 0, z 0Farzaneh AbdollahiNonlinear ControlLecture 85/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinIFor linear systemsIIWhen is stabilized by FB, the origin of closed loop system is g.a.sFor nonlinear systemsIFarzaneh AbdollahiWhen is stabilized via linearization the origin of closed loop system isa.sNonlinear ControlLecture 86/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinIFor linear systemsIIWhen is stabilized by FB, the origin of closed loop system is g.a.sFor nonlinear systemsIIFarzaneh AbdollahiWhen is stabilized via linearization the origin of closed loop system is a.sIf RoA is unknown, FB provides local stabilizationNonlinear ControlLecture 86/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinIFor linear systemsIIWhen is stabilized by FB, the origin of closed loop system is g.a.sFor nonlinear systemsIIIFarzaneh AbdollahiWhen is stabilized via linearization the origin of closed loop system is a.sIf RoA is unknown, FB provides local stabilizationIf RoA is defined, FB provides regional stabilizationNonlinear ControlLecture 86/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinIFor linear systemsIIWhen is stabilized by FB, the origin of closed loop system is g.a.sFor nonlinear systemsIIIIFarzaneh AbdollahiWhen is stabilized via linearization the origin of closed loop system is a.sIf RoA is unknown, FB provides local stabilizationIf RoA is defined, FB provides regional stabilizationIf g.a.s is achieved, FB provides global stabilizationNonlinear ControlLecture 86/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinIFor linear systemsIIWhen is stabilized by FB, the origin of closed loop system is g.a.sFor nonlinear systemsIIIIIFarzaneh AbdollahiWhen is stabilized via linearization the origin of closed loop system is a.sIf RoA is unknown, FB provides local stabilizationIf RoA is defined, FB provides regional stabilizationIf g.a.s is achieved, FB provides global stabilizationIf FB control does not achieve global stabilization, but can be designeds.t. any given compact set (no matter how large) can be included in theRoA, FB achieves semiglobal stabilizationNonlinear ControlLecture 86/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinExampleIConsider the system ẋ x 2 uILinearize at the originIStabilize by u kx, k 0I the closed loop system ẋ kx x 2IRoA is x kIIt is regionally stabilizedIGiven any compact set Br { x r }, we can choose k r FB achieves semiglobal stabilization.IIIẋ uOnce k is fixed and the controller is implemented, for x0 k a.s. isguaranteedGlobal stabilization is achieved by FB: u x 2 kxFarzaneh AbdollahiNonlinear ControlLecture 87/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinExample: Stabilization of a PendulumIConsider the dynamics of the pendulum:J θ̈ mgl sin θ τIObjective: take the pendulum from a large initialangel (θ 60o ) to the vertical up positionIA choice of stabilizer:a feedback part for stability (PD) a feedforward part for gravity compensation:τ kd θ̇ kp θ mgl sin θkd and kp are pos. constants.I globally stable closed-loop dynamics:prove itJ θ̈ kd θ̇ kp θ 0IIn this example feedback (FB) and feedforward (FF)control actions modify the plant into desirable form.Farzaneh AbdollahiNonlinear ControlLecture 88/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinExample: Stabilization of an Inverted Pendulum with CartIConsider the dynamics of the inverted pendulumshown in Fig.:(M m)ẍ ml cos θθ̈ ml sin θθ̇2 umẍ cos θ ml θ̈ ml ẋ θ̇ sin θ mg sin θ 0mass of the cart is not negligibleIObjective: Bring the inverted pendulum fromvertical-down at the middle of the lateral trackto the vertical-up at the same lateral point.IIt is not simply possible since degree of freedomis two, # inputs is one (under actuated).Farzaneh AbdollahiNonlinear ControlLecture 89/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinTracking ProblemsIAsymptotic Tracking Problem: Given a nonlinear dynamics:ẋ f (x, u, t)y h(x, u, t)and a desired output, yd , find a control law for the input u s.t. starting from anyinitial state in region Ω, the tracking error y (t) yd (t) goes to zero, while wholestate x remain bounded.IA practical point: Sometimes x can just be remained reasonably bounded, i.e.,bounded within the range of system model validity.IPerfect tracking: proper initial states imply zero tracking error for all time:y (t) yd (t) t 0; in asymptotic/ exponential tracking perfect tracking isachieved asymptotically/ exponentiallyIAssumption throughout the rest of the lectures:I y and its derivatives up to a sufficiently high order ( generally equal to thedsystem’s order) are cont. and bounded.I y and its derivatives available for on-line control computationdI y is planned aheaddFarzaneh AbdollahiNonlinear ControlLecture 810/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinISometimes derivatives of the desired output are not available.IA reference model is applied to provide the required derivative signalsIExample: For tracking control of the antenna of a radar, only the position ofthe aircraft ya (t) is available at a given time instant (it is too noisy to bedifferentiated numerically).Idesired position, velocity and acceleration to be tracked is obtained byÿd k1 ẏd k2 yd k2 ya (t)(1)k1 and k2 are pos. constantsI following the aircraft is translated to the problem of tracking the output yd ofthe reference modelIThe reference model serves asI providing the desired output of the tracking system in response to theaircraft positionI generating the derivatives of the desired output for tracker design.I(1) Should be fast for yd to closely approximate yaFarzaneh AbdollahiNonlinear ControlLecture 811/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinTracking ProblemIPerfect tracking and asymptotic tracking is not achievable fornon-minimum phase systems.IExample: Consider ÿ 2ẏ 2y u̇ u.IIt is non-minimum phase since it has zero at s 1.IAssume the perfect tracking is achieved.I u̇ u (ÿd 2ẏd 2yd ) u sIPerfect tracking is achieved by infinite control input.I Only bounded-error tracking with small tracking error is achievable fordesired traj.IPerfect tracking controller is inverting the plant dynamicsFarzaneh AbdollahiNonlinear Control2 2s 2s 1Lecture 8yd12/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinTracking Problem in Presence of DisturbanceIAsymptotic disturbance rejection:Given a nonlinear dynamics:ẋ f (x, u, w , t)y h(x, u, w , t)and a desired output, yd , find a control law for the input u s.t. startingfrom any initial state in region Ω, the tracking error y (t) yd (t) goes tozero, while whole state x remain bounded.IWhen the exogenous signals yd and w are generated by a known model,asymptotic output tracking and disturbance rejection can be achieved byincluding such model in the FB controller.Farzaneh AbdollahiNonlinear ControlLecture 813/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinIFor T.V disturbance w (t), achieving asymptotic disturbance rejectionmay not be feasible.look for disturbance attenuation:IIachieve u.u.b of the tracking error with a prescribed tolerance:ke(t)k , t T , is a prespecified (small) positive number.OR consider attenuating the closed-loop input-output map from thedisturbance input w to the tracking error e y ydFarzaneh AbdollahiIe.g. considering w as an L2 signal, goal is min the L2 gain of theclosed-loop I/O map from w to eNonlinear ControlLecture 814/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinIFor T.V disturbance w (t), achieving asymptotic disturbance rejectionmay not be feasible.look for disturbance attenuation:IIachieve u.u.b of the tracking error with a prescribed tolerance:ke(t)k , t T , is a prespecified (small) positive number.OR consider attenuating the closed-loop input-output map from thedisturbance input w to the tracking error e y ydIIFor tracking problem one can design:IIIe.g. considering w as an L2 signal, goal is min the L2 gain of theclosed-loop I/O map from w to eStatic/Dynamic state FB controllerStatic/Dynamic output FB controllerTracking may achieve locally, regionally, semiglobally, or globally:IIIThese phrases refer not only to the size of the initial state, but to the sizeof the exogenous signals yd , wLocal tracking means tracking is achieved for sufficiently small initial statesand sufficiently small exogenous signalsGlobal tracking means tracking is achieved for any initial state and anyyd , wFarzaneh AbdollahiNonlinear ControlLecture 814/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinRelation between Stabilization and Tracking ProblemsIITracking problems are more difficult to solve than stabilization problemsIn tracking problems the controller shouldIIInot only keep the whole state stabilizedbut also drive the system output toward the desired outputHowever, for tracking problem of the plant:ÿ f (ẏ , y , u) 0IIe(t) y (t) yd (t) goes to zeroIt is equivalent to the asymptotic stabilization of the systemë f (ė, e, u, yd , ẏd , ÿd ) 0(2)with states e and ėI tracking problem is solved if we can design a stabilizer for thenon-autonomous dynamics (2)IOn the other hand, stabilization problems can be considered as a specialcase of tracking problem with desired trajectory being a constant.Farzaneh AbdollahiNonlinear ControlLecture 815/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinSpecify the Desired BehaviorIIn Linear control, the desired behavior is specified inIIIISo in linear control the quantitative specifications of the closed-loopsystem is defined, the a controller is synthesized to meet the specificationsFor nonlinear systems the system specification of nonlinear systems is lessobvious sinceIIItime domain: rise time, overshoot and settling time for responding to astep commandfrequency domain: the regions in which the loop transfer function must lieat low and high frequenciesresponse of the nonlinear system to one command does not reflect theresponse to an other commanda frequency description is not possible In nonlinear control systems some qualitative specifications of thedesired behavior is considered.Farzaneh AbdollahiNonlinear ControlLecture 816/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinISome desired qualitative specifications of nonlinear system:IStability must be guaranteed for the nominal model, either in local orglobal sense. In local sense, the region of stability and convergence are ofinterest.IIIIstability of nonlinear systems depends on initial conditions and onlytemporary disturbances may be translated as initial conditionsRobustness is the sensitivity effect which are not considered in the designlike persistent disturbance, measurement noise, unmodeled dynamics, etc.Accuracy and Speed of response for some typical motion trajectories in theregion of operation. For instance, sometimes appropriate control is desiredto guarantee consistent tracking accuracy independent of the desired traj.Cost of a control which is determined by # and type of actuators, sensors,design complexity.IThe mentioned qualitative specifications are not achievable in a unifieddesign.IA good control can be obtained based on effective trade-offs of them.Farzaneh AbdollahiNonlinear ControlLecture 817/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinNonlinear Control ProblemsIA Procedure of designing control1.2.3.4.5.Specify the desired behavior and select actuators and sensorsmodel the physical plant by a set of differential Eqsdesign a control lawanalyze and simulate the resulting control systemimplement the control system in hardwareIExperience and creativity are important factor in designing the controlISometimes, addition or relocation of actuators and sensors may makecontrol of the system easier.IModeling Nonlinear SystemsIModeling is constructing a mathematical description (usually as a set ofdifferential Eqs.) for the physical system to be controlled.Farzaneh AbdollahiNonlinear ControlLecture 818/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinModeling Nonlinear SystemsITwo points in modeling:1. To obtain tractable yet accurate model, good understanding of systemdynamics and control tasks requires.INote: more accurate models are not always better. They may requireunnecessarily complex control design and more computations.Keep essential effects and discard insignificant effects in operatingrange of interest.2. In modeling not only the nominal model for the physical system should beobtained, but also some characterization of the model uncertainties shouldbe provided for using in robust control, adaptive design or simulation.IIModel uncertainties: difference between the model and real physical systemI parametric uncertainties: uncertainties in parametersIExample: model of controlled mass: mẍ uI Uncertainty in m is parametric uncertaintyI neglected motor dynamics, measurement noise, and sensor dynamics arenon-parametric uncertainties.I Parametric uncertainties are easier to characterize; 2 m 5Farzaneh AbdollahiNonlinear ControlLecture 819/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinFeedback and FeedForwardIFeedback (FB) plays a fundamental role in stabilizing the linear as well asnonlinear control systemsIFeedforward (FF) in nonlinear control is much more important than linearcontrolFF is used toIIIIIcancel the effect of known disturbancesprovide anticipate actions in tracking tasksfor FF a model of the plant (even not very accurate) is required.Many tracking controllers can be written in the form: u FF FBIIFF: to provide necessary input to follow the specified motion traj andcanceling the effect of known disturbancesFB to stabilize the tracking error dynamics.Farzaneh AbdollahiNonlinear ControlLecture 820/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinExampleIConsider a minimum-phase systemA(s)y B(s)u(3)where A(s) a0 a1 s . an 1 s n 1 s n , B(s) b0 b1 s . bm s mIObjective: make the output y (t) follow a time-varying traj yd (t)1. To achieve y yd , input should have a FF term ofA(s)u v ydB(s)IA(s)B(s) :(4)Substitute (4) to (3): A(s)e B(s)v , where e(t) y (t) yd (t)2. Use FB to stabilize the system:IIC (s)v D(s)e closed loop system (AD BC )e 0.Choose D and C to poles in desired placesAB ydCDeI u Ie(t) is zero if initial conditions y (i) (0) yd (0), i 1, ., r ,otherwiseexponentially converges to zeroFarzaneh Abdollahi (i)Nonlinear ControlLecture 821/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinExample Cont’dIIIf some derivatives of yd are notavailable, one can simply omitthem from FFonly boundedtracking error is guaranteed,This method is not applicable fornon-min phase systems.IIIFarzaneh Abdollahilow freq. components of desiredtraj in FF, good tracking infreq lower than the LHP zeros ofplantBy defining FF term asDAe(t) AD BC[ BB1 1]AydB 1 ydIf B1 eliminates the RHP zeros ofB good tracking for desiredtraj with frequencies lower thanthe RHP zeroes (but we may nothave internal stability)Nonlinear ControlLecture 822/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinImportance of Physical PropertiesIIn nonlinear control design, explanation of the physical properties maymake the control of complex nonlinear plants simple;IExample: Adaptive control of robot manipulator was long recognized tobe far of reach.IBecause robot’s dynamics is highly nonlinear and has multiple inputsUsing the two physical facts:IIIpos. def. of inertia matrixpossibility of linearly parameterizing robot dynamicsyields adaptive control with global stability and desirable tackingconvergence.Farzaneh AbdollahiNonlinear ControlLecture 823/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinAvailable Methods for Nonlinear ControlIIThere is no general method for designing nonlinear controlSome alternative and complementary techniques to particular classed ofcontrol problem are listed below:ITrail-and Error: The idea is using analysis tools such as phase-planemethods, Lyapunov analysis , etc, to guide searching a controller which canbe justified by analysis and simulations.IIThis method fails for complex systemsFeedback Linearization: transforms original system models intoequivalent models of simpler form (like fully or partially linear)Farzaneh AbdollahiIIIIIThen a powerful linear design technique completes the control designThis method is applicable for input-state linearizable and minimum phasesystemsIt requires full state measurementIt does not guarantee robustness in presence of parameter uncertainties ordisturbances.It can be used as model-simplifying for robust or adaptive controllersNonlinear ControlLecture 824/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinAvailable Methods for Nonlinear ControlIRobust Control is deigned based on consideration of nominal model aswell as some characterization of the model uncertaintiesIIIIAn example of robust controls is sliding mode controlThey generally require state measurements.In robust control design tries to meet the control objective for any model inthe ”ball of uncertainty.”Adaptive Control deals with uncertain systems or time-varying systems.IIIThey are mainly applied for systems with known dynamics but unknownconstant or slowly-varying parameters.They parameterizes the uncertainty in terms of certain unknownparameters and use feedback to learn these parameters on-line , during theoperation of the system.In a more elaborate adaptive scheme, the controller might be learningcertain unknown nonlinear functions, rather than just learning someunknown parameters.Farzaneh AbdollahiNonlinear ControlLecture 825/26
Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for NonlinAvailable Methods for Nonlinear ControlIGain Scheduling Employs the well developed linear control methodologyto the control of nonlinear systems.IIIIIA number of operating points which cover the range of the systemoperation is selected.Then, at each of these points, the designer makes a linear TVapproximation to the plant dynamics and designs a linear controller foreach linearized plant.Between operating points, the parameters of the compensators areinterpolated, ( scheduled), resulting in a global compensator.It is simple and practical for several applications.The main problems of gain scheduling:IIprovides limited theoretical guarantees of stability in nonlinear operationThe system should satisfy some conditions:IIIthe scheduling variables should change slowlyThe scheduling variables should capture the plant’s nonlinearities”.Due to the necessity of computing many linear controllers, this methodinvolves lots of computations.Farzaneh AbdollahiNonlinear ControlLecture 826/26
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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