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Unformatted text preview: Nonlinear & Adaptive Control Module 9Part I Feedback Linearization 1 Topics
1. 2. Differential Geometry concepts Feedback linearization as a design tool. Material in Chapter 13 of Khalil and in attached notes. 2 Special References
1. 2. Nonlinear Control Systems  Isidori 3rd Edition. Nonlinear Dynamical Systems  Nijmeier and van der Schaft The first book in particular is a very well developed discussion of the ideas of feedbacklinearization and differential geometric based designs. 3 Motivating Feedback Linearization Can this idea be generalized?
4 The Need for State Transformation (1/2)
To be feedback linearizable systems need to be of the form : (*) Choose Example: Is it of the form (*) ? Introduce
5 The Need for State Transformation (2/2) 6 Feedback Linearization Pros & Cons Advantages
1. 2. The nonlinear control reduces the system to a linear one (exactly). The nonlinear control results in global asym. stability of the resulting linear behavior. The real system might be different from the nominal system. Thus robustness is needed. Not every system can be treated this way. Disadvantages
1. 2. 7 InputOutput Linearization (1/3) 8 InputOuput Linearization (2/3) 9 InputOutput Linearizability (3/3) 10 Examples 11 The Complete IO system 12 The Complete IO System 13 When Zeros are Present 14 The New System 15 The Brunowsky canonical Form 16 Internal & Zero Dynamics 17 The MinimumPhase Property 18 InputState Linearizability Two PDEs give the necessary & sufficient conditions for feedback linearization. T is however not unique 19 Definition of FeedbackLinearizability 20 Finding T() Also 21 Equivalent State Transformation 22 Brunowsky Canonical Form 23 Brunowsky Canonical Form 24 PDE's 25 PDE's 26 Solving for Statetransformation 27 From Equilibrium to Equilibrium 28 InputState Linearizationrevisited Definition: 29 InputStateLinearizations Question: 30 Feedback Linearizability Theorem 31 Interpretations 32 Example (1/3) 33 Example (2/3) 34 Example (3/3) 35 Control Design Using FL 36 Stabilization 37 What happens if IS Linearized? 38 Block Diagram of FL Closedloop system under feedback(exact) linearization + _
linearization loop poleplacement 39 InputOutput LinearizationTracking
Let us consider the tracking problem where & its derivative up to a sufficiently high order are assumed to be known and bounded. Objective: To make tracks while keeping the whole state bounded. An apparent difficulty with this system is that the output is only indirectly related to the input through the state variable and the nonlinear state equation. One might guess that the difficulty of the tracking control design can be reduced if we can find a direct and simple relation between the system output and the control input
40 Example (1/5) 41 Example (2/5) 42 Example (3/5)
For the above example, the internal dynamics is represented by If this internal dynamics is stable (by which we actually mean stability in BIBO), our tracking control problem has indeed been solved. Otherwise, it will end up with burningup fuses or violent vibration of mechanical members Therefore, the effectiveness of the above control design, based on the reduced order model, hinges upon the stability of the internal dynamics.
43 Example (4/5) 44 Example (5/5)
(perhaps after a transient) 45 Robustness Issue { } 46 Robust Stability Analysis 47 Example (1/2) 48 Example (2/2) 49 Looking Backwards
1. 2. 3. In this module, we learned about differential geometric concepts. Such concepts allowed us to feedback linearize, either completely or partially a special class of nonlinear systems. Various feedback control designs were then obtained using the feedback linearized systems.
50 Looking Forward
1. 2. In the next (and final) module, we will present other design techniques for nonlinear systems. Such techniques apply when feedback linearization dos not, or when it is not even desirable. 51 ...
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This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.
 Spring '09
 ChaoukiT.Abdallah

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