Feedback Linearization Presented by : Shubham Bhat (ECES-817)

Feedback Linearization- Single Input case

Feedback Linearization- Contd.

Problem Statement

Example- Controlling a fluid level in a tank

Example – Contd.

Input State Linearization

Input State Linearization-Contd.

pole-placement loop Linearization Loop - x 0 Input State Linearization-Contd.

Our objective is to make the output y(t) track a desired trajectory y d (t) while keeping the whole state bounded, where y d (t) and its time derivatives up to a sufficiently high order are assumed to be known and bounded. Input Output Linearization

To generate a direct relationship between the output y and the input u, differentiate the output y Input Output Linearization-Contd.

Note : The control law is defined everywhere, except at the singularity points such that x2= -1. Input Output Linearization-Contd.

Internal Dynamics If we need to differentiate the output r times to generate an explicit relationship between output y and input u, the system is said to have a relative degree r. The system order is n. If r<= n, there is an part of the system dynamics which has been rendered “unobservable”. This part is called the internal dynamics, because it cannot be seen from the external input-output relationship. If the internal dynamics is stable, our tracking control design has been solved. Otherwise the tracking controller is meaningless. Therefore, the effectiveness of this control design, based on reduced-order model, hinges upon the stability of the internal dynamics.

Assume that the control objective is to make y track yd(t). Differentiating y leads to the first state equation. Choosing control law Internal Dynamics

Internal Dynamics- Contd

Internal Dynamics in Linear Systems

Extending the notion of zeros to nonlinear systems is not a trivial proposition. For nonlinear systems, the stability of the internal dynamics may depend on the specific control input. The zero-dynamics is defined to be the internal dynamics of the system when the system output is kept at zero by the input. A nonlinear system whose zero dynamics is asymptotically stable is an asymptotically minimum phase system. Zero-Dynamics is an intrinsic feature of a nonlinear system, which does not depend on the choice of control law or the desired trajectories. Extension of Internal Dynamics to Zero Dynamics

Lie derivative and Lie bracket Diffeomorphism Frobenius Theorem Input-State Linearization Examples The zero dynamics with examples Input-Output Linearization with examples Opto-Mechanical System Example Mathematical Tools

Lie Derivatives

Lie Brackets

Example - Lie Brackets

Properties of Lie Brackets

Diffeomorphisms and State transformations

Example

Frobenius Theorem- Completely Integrable

Frobenius Theorem- Involutivity

Frobenius theorem

Frobenius theorem- example

Input-State Linearization

Conditions for Input-State Linearization

How to perform input-state Linearization

Consider a mechanism given by the dynamics which represents a single link flexible joint robot. Its equations of motion is derived as Because nonlinearities ( due to gravitational torques) appear in the first equation, While the control input u enters only in the second equation, there is no easy way to design a large range controller. Example system

Checking controllability and involuvity conditions. It has rank 4 for k>0 and IJ> infinity. Furthermore, since the above vector fields are constant, they form an involutive set. Therefore the system is input-state linearizable. Example system- Contd.

Let us find out the state-transformation z = z(x) and the input transformation so that input-state linearization is achieved. Example system - Contd.

Accordingly, the input transformation is Example system- Contd.

Finally, note that The above input-state linearization is actually global, because the diffeomorphism z(x) and the input transformation are well defined everywhere. Specifically, the inverse of the state transformation is Example system- Contd.

Input-Output Linearization of SISO systems

Generating a linear input-output relation

Normal Forms

Zero Dynamics

Zero Dynamics- Contd.

Local Asymptotic Stabilization

Example System

Global Asymptotic Stability Zero Dynamics only guarantees local stability of a control system based on input-output linearization. Most practically important problems are of global stabilization problems. An approach to global asymptotic stabilization based on partial feedback linearization is to simply consider the control problem as a standard lyapunov controller problem, but simplified by the fact that putting the systems in normal form makes part of the dynamics linear. The basic idea, after putting the system in normal form, is to view as the “input” to the internal dynamics, and as the “output”.

Steps for Global Asymptotic Stability The first step is to find a “ control law” which stabilizes the internal dynamics. An associated Lyapunov function demonstrating the stabilizing property. To get back to the original global control problem. Define a Lyapunov function candidate V appropriately as a modified version of Choose control input v so that V be a Lyapunov function for the whole closed-loop dynamics.

Local Tracking Control

Tracking Control

Inverse Dynamics

Inverse Dynamics- Contd.

Application of Feedback Linearization to Opto-Mechanics For the double slit aperture, the irradiance at any point in space is given as: = wavelength = 630 nm k = wave number associated with the wavelength a = center-to-center separation = 32 um b = width of the slit = 18 um z = distance of propagation =1000 um

Plant Model - + Motor DynamicsPlant Model U X2X2 Y= X 1 Plant Model

Input-State Linearization

Pole-Placement loop Plant Model - + Motor DynamicsPlant Model U(x,v) X2X2 Y - 0 Input-State Linearization- Block diagram

Input-Output Linearization

Zero Dynamics

Conclusion Control design based on input-output linearization can be made in 3 steps: Differentiate the output y until the input u appears Choose u to cancel the nonlinearities and guarantee tracking convergence Study the stability of the internal dynamics If the relative degree associated with the input-output linearization is the same as the order of the system, the nonlinear system is fully linearized. If the relative degree is smaller than the system order, then the nonlinear system is partially linearized and stability of internal dynamics has to be checked.

Homework Problems