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
Fall 2010
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Nonlinear Control System Design-Non Linear Control Systems-Lecture Slides, Slides of Nonlinear Control Systems
Dr. Javed Iftikhar delivered this lecture at A.P. University of Law for Non Linear Control Systems course. It includes: Stabilization, Problems, Feedback, Control, Tracking, Disturbance, Desired, Behavior, Modeling, Stable
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2011/2012
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Nonlinear Control
Lecture 7: Nonlinear Control System Design
Fall 2010
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
Stabilization Problems
I (^) Asymptotic Stabilization Problem: Given a nonlinear dynamic system:
x ˙ = f (x, u, t)
find a control law, u, s.t. starting from anywhere in region Ω x → 0 as t → ∞. I (^) If the objective is to drive the state to some nonzero set-point xd , it can be simply transformed into a zero-point regulation problem x − xd as the state. I (^) Static control law: the control law depends on the measurement signal directly, such as proportional controller. I (^) Dynamic control law: the control law depends on the measurement through a differential Eq, such as lag-lead controller
Feedback Control
I (^) State feedback: for system ˙x = f (t, x, u) I (^) Output feedback for the system
x˙ = 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 (^) 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 (^) 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 (^) 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 (^) 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
Example: Stabilization of a Pendulum
I (^) Consider the dynamics of the pendulum: J θ¨ − mgl sin θ = τ
I (^) Objective: take the pendulum from a large initial angel (θ = 60o^ ) to the vertical up position I (^) A choice of stabilizer: a feedback part for stability (PD)+ a feedforward part for gravity compensation: τ = −kd θ˙ − kp θ−mgl sin θ
kd and kp are pos. constants. I (^) ∴ globally stable closed-loop dynamics:prove it J θ¨ + kd θ˙ + kp θ = 0 I (^) In this example feedback (FB) and feedforward (FF) control actions modify the plant into desirable form.
Example: Stabilization of a Inverted Pendulum with Cart
I (^) Consider the dynamics of the inverted pendulum shown in Fig.: (M + m)¨x + ml cos θ θ¨ − ml sin θ θ˙^2 = u m¨x cos θ + ml θ¨ − ml x˙ θ˙ sin θ + mg sin θ = 0
mass of the cart is not negligible I (^) Objective: Bring the inverted pendulum from vertical-down at the middle of the lateral track to the vertical-up at the same lateral point. I (^) It is not simply possible since degree of freedom is two, # inputs is one (under actuated).
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 y¨d + k 1 y˙d + k 2 yd = k 2 ya(t) (1)
k 1 and k 2 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
Tracking Problem
I (^) Perfect tracking and asymptotic tracking is not achievable for non-minimum phase systems. I (^) Example: Consider ¨y + 2 ˙y + 2y = − u˙ + u. I (^) It is non-minimum phase since it has zero at s = 1. I (^) Assume the perfect tracking is achieved. I (^) ∴ u˙ − u = −(¨yd + 2 ˙yd + 2yd ) ⇒ u = − s^2 +2 s−s 1 +2 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
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 L 2 signal, goal is min the L 2 gain of the closed-loop I/O map from w to e
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 L 2 signal, goal is min the L 2 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