Nonlinear Control System Design I-Control of Non Linear System-Lecture Slides, Slides of Nonlinear Control Systems

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Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Nonlinear Control Lecture 7: Nonlinear Control System Design Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 7 1/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Nonlinear Control Problems Stabilization Problems Feedback Control Tracking Problems Tracking Problem in Presence of Disturbance Tracking Problem in Presence of Disturbance Specify the Desired Behavior Some Issues in Nonlinear Control Modeling Nonlinear Systems Feedback and FeedForward Importance of Physical Properties Available Methods for Nonlinear Control Farzaneh Abdollahi Nonlinear Control Lecture 7 2/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Feedback Control I State feedback: for system ẋ = f (t, x , u) I Output feedback for the system ẋ = f (t, x , u) y = h(t, x , u) I The measurement of some states is not available. I an observer may be required Farzaneh Abdollahi Nonlinear Control Lecture 7 5/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Feedback Control I State feedback: for system ẋ = f (t, x , u) I Output feedback for the system ẋ = f (t, x , u) y = h(t, x , u) I The measurement of some states is not available. I an observer may be required I Static control law: I u = γ(t, x) I Dynamic control law: I u = γ(t, x , z) I z is the solution of a dynamical system driven by x : ż = g(t, x , z) I The origin to be stabilize is x = 0, z = 0 Farzaneh Abdollahi Nonlinear Control Lecture 7 5/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available 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 is a.s Farzaneh Abdollahi Nonlinear Control Lecture 7 6/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available 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 is a.s I If RoA is unknown, FB provides local stabilization I If RoA is defined, FB provides regional stabilization I If g.a.s is achieved, FB provides global stabilization Farzaneh Abdollahi Nonlinear Control Lecture 7 6/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available 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 is a.s I If RoA is unknown, FB provides local stabilization I If RoA is defined, FB provides regional stabilization I If g.a.s is achieved, FB provides global stabilization I If FB control does not achieve global stabilization, but can be designed s.t. any given compact set (no matter how large) can be included in the RoA, FB achieves semiglobal stabilization Farzaneh Abdollahi Nonlinear Control Lecture 7 6/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Example I Consider the system ẋ = x2 + u I Linearize at the origin ẋ = u I Stabilize by u = −kx , k > 0 I ∴ the closed loop system ẋ = −kx + x2 I RoA is x < k I It is regionally stabilized I Given any compact set Br = {|x | ≤ r}, we can choose k > r I ∴ FB achieves semiglobal stabilization. I Once k is fixed and the controller is implemented, for x0 < k a.s. is guaranteed I Global stabilization is achieved by FB: u = −x2 − kx Farzaneh Abdollahi Nonlinear Control Lecture 7 7/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Tracking Problems I Asymptotic 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 any initial state in region Ω, the tracking error y(t)− yd(t) goes to zero, while whole state x remain bounded. I A practical point: Sometimes x can just be remained reasonably bounded, i.e., bounded within the range of system model validity. I Perfect tracking: proper initial states imply zero tracking error for all time: y(t) ≡ yd(t) ∀t ≥ 0; in asymptotic/ exponential tracking perfect tracking is achieved asymptotically/ exponentially I Assumption throughout the rest of the lectures: I yd and its derivatives up to a sufficiently high order ( generally equal to the system’s order) are cont. and bounded. I yd and its derivatives available for on-line control computation I yd is planned ahead Farzaneh Abdollahi Nonlinear Control Lecture 7 10/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control I Sometimes derivatives of the desired output are not available. I A reference model is applied to provide the required derivative signals I Example: For tracking control of the antenna of a radar, only the position of the aircraft ya(t) is available at a given time instant (it is too noisy to be differentiated numerically). I desired position, velocity and acceleration to be tracked is obtained by ÿd + k1ẏd + k2yd = k2ya(t) (1) k1 and k2 are pos. constants I ∴ following the aircraft is translated to the problem of tracking the output yd of the reference model I The reference model serves as I providing the desired output of the tracking system in response to the aircraft position I generating the derivatives of the desired output for tracker design. I (1) Should be fast yd to closely approximate ya Farzaneh Abdollahi Nonlinear Control Lecture 7 11/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Tracking Problem I Perfect tracking and asymptotic tracking is not achievable for non-minimum phase systems. I Example: Consider ÿ + 2ẏ + 2y = −u̇ + u. I It is non-minimum phase since it has zero at s = 1. I Assume the perfect tracking is achieved. I ∴ u̇ − u = −(ÿd + 2ẏd + 2yd)⇒ u = − s 2+2s+2 s−1 yd I Perfect tracking is achieved by infinite control input. I ∴ Only bounded-error tracking with small tracking error is achievable for desired traj. I Perfect tracking controller is inverting the plant dynamics Farzaneh Abdollahi Nonlinear Control Lecture 7 12/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control I For T.V disturbance w(t), achieving asymptotic disturbance rejection may not be feasible. look for disturbance attenuation: I achieve u.u.b of the tracking error with a prescribed tolerance: ‖e(t)‖ < , ∀t > T ,,  is a prespecified (small) positive number. I OR consider attenuating the closed-loop input-output map from the disturbance input w to the tracking error e = y − yd I e.g. considering w as an L2 signal, goal is min the L2 gain of the closed-loop I/O map from w to e I For tracking problem one can design: I Static/Dynamic state FB controller I Static/Dynamic output FB controller I Tracking may achieve locally, regionally, semiglobally, or globally: I These phrases refer not only to the size of the initial state, but to the size of the exogenous signals yd ,w I Local tracking means tracking is achieved for sufficiently small initial states and sufficiently small exogenous signals I Global tracking means tracking is achieved for any initial state and any yd ,w Farzaneh Abdollahi Nonlinear Control Lecture 7 14/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Relation between Stabilization and Tracking Problems I Tracking problems are more difficult to solve than stabilization problems I In tracking problems the controller should I not only keep the whole state stabilized I but also drive the system output toward the desired output I However, for tracking problem of the plant: ÿ + f (ẏ , y , u) = 0 I e(t) = y(t)− yd(t) goes to zero I It is equivalent to the asymptotic stabilization of the system ëd + f (ė, e, u, yd , ẏd , ÿd) = 0 (2) with states e and ė I ∴ tracking problem is solved if we can design a stabilizer for the non-autonomous dynamics (2) I On the other hand, stabilization problems can be considered as a special case of tracking problem with desired trajectory being a constant. Farzaneh Abdollahi Nonlinear Control Lecture 7 15/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Specify the Desired Behavior I In Linear control, the desired behavior is specified in I time domain: rise time, overshoot and settling time for responding to a step command I frequency domain: the regions in which the loop transfer function must lie at low and high frequencies I So in linear control the quantitative specifications of the closed-loop system is defined, the a controller is synthesized to meet the specifications I For nonlinear systems the system specification of nonlinear systems is less obvious since I response of the nonlinear system to one command does not reflect the response to an other command I a frequency description is not possible I ∴ In nonlinear control systems some qualitative specifications of the desired behavior is considered. Farzaneh Abdollahi Nonlinear Control Lecture 7 16/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Modeling Nonlinear Systems I Two points in modeling: 1. To obtain tractable yet accurate model, good understanding of system dynamics and control tasks requires. I Note: more accurate models are not always better. They may require unnecessarily complex control design and more computations. I Keep essential effects and discard insignificant effects in operating range of interest. 2. In modeling not only the nominal model for the physical system should be obtained, but also some characterization of the model uncertainties should be provided for using in robust control, adaptive design or simulation. I Model uncertainties: difference between the model and real physical system I parametric uncertainties: uncertainties in parameters I Example: model of controlled mass: mẍ = u I Uncertainty in m is parametric uncertainty I neglected motor dynamics, measurement noise, and sensor dynamics are non-parametric uncertainties. I Parametric uncertainties are easier to characterize; 2 ≤ m ≤ 5 Farzaneh Abdollahi Nonlinear Control Lecture 7 19/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Feedback and FeedForward I Feedback (FB) plays a fundamental role in stabilizing the linear as well as nonlinear control systems I Feedforward (FF) in nonlinear control is much more important than linear control I FF is used to I cancel the effect of known disturbances I provide anticipate actions in tracking tasks I for FF a model of the plant (even not very accurate) is required. I Many tracking controllers can be written in the form: u = FF+ FB I FF: to provide necessary input to follow the specified motion traj and canceling the effect of known disturbances I FB to stabilize the tracking error dynamics. Farzaneh Abdollahi Nonlinear Control Lecture 7 20/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Example I Consider a minimum-phase system A(s)y = B(s)u (3) where A(s) = a0 + a1s + ...+ an−1s n−1 + sn, B(s) = b0 + b1s + ...+ bms m I Objective: make the output y(t) follow a time-varying traj yd(t) 1. To achieve y = yd , input should have a FF term of A(s) B(s) : u = v + A(s) B(s) yd (4) I Substitute (4) to (3): A(s)e = B(s)v , where e(t) = y(t)− yd(t) 2. Use FB to stabilize the system: I v = C(s) D(s) e closed loop system (AC + BD)e = 0. I Choose D and C to poles in desired places I ∴ u = AB yd + C D e I e(t) is zero if initial conditions y (i)(0) = y (i) d (0), i = 1, ..., r ,otherwise exponentially converges to zero Farzaneh Abdollahi Nonlinear Control Lecture 7 21/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Available Methods for Nonlinear Control I There is no general method for designing nonlinear control I Some alternative and complementary techniques to particular classed of control problem are listed below: I Trail-and Error: The idea is using analysis tools such a phase-plane methods, Lyapunov analysis , etc, to guide searching a controller which can be justified by analysis and simulations. I This method fails for complex systems I Feedback Linearization: transforms original system models into equivalent models of simpler form (like fully or partially linear) I Then a powerful linear design technique completes the control design I This method is applicable for input-state linearizable and minimum phase systems I It requires full state measurement I It does not guarantee robustness in presence of parameter uncertainties or disturbances. I It can be used as model-simplifying for robust or adaptive controllers Farzaneh Abdollahi Nonlinear Control Lecture 7 24/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Available Methods for Nonlinear Control I Robust Control is deigned based on consideration of nominal model as well as some characterization of the model uncertainties I An example of robust controls is sliding mode control I They generally require state measurements. I In robust control design tries to meet the control objective for any model in the ”ball of uncertainty.” I Adaptive Control deals with uncertain systems or time-varying systems. I They are mainly applied for systems with known dynamics but unknown constant or slowly-varying parameters. I They parameterizes the uncertainty in terms of certain unknown parameters and use feedback to learn these parameters on-line , during the operation of the system. I In a more elaborate adaptive scheme, the controller might be learning certain unknown nonlinear functions, rather than just learning some unknown parameters. Farzaneh Abdollahi Nonlinear Control Lecture 7 25/26docsity.com Outline Nonlinear Control Problems Specify the Desired Behavior Some Issues in Nonlinear Control Available Methods for Nonlinear Control Available Methods for Nonlinear Control I Gain Scheduling Employs the well developed linear control methodology to the control of nonlinear systems. I A number of operating points which cover the range of the system operation is selected. I Then, at each of these points, the designer makes a linear TV approximation to the plant dynamics and designs a linear controller for each linearized plant. I Between operating points, the parameters of the compensators are interpolated, ( scheduled), resulting in a global compensator. I It is simple and practical for several applications. I The main problems of gain scheduling: I provides limited theoretical guarantees of stability in nonlinear operation I The system should satisfy some conditions: I the scheduling variables should change slowly I The scheduling variables should capture the plant’s nonlinearities”. I Due to the necessity of computing many linear controllers, this method involves lots of computations. Farzaneh Abdollahi Nonlinear Control Lecture 7 26/26docsity.com