Appendix D: Linearization

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# Appendix D: Linearization - PowerPoint PPT Presentation

Appendix D: Linearization. Small-Signal Linearization Linearization by Feedback. Material covered in the P RESENT L ECTURE is shown in yellow. I. DYNAMIC MODELING Deriving a dynamic model for mechanical, electrical, electromechanical, fluid- & heat-flow systems

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## Appendix D: Linearization

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### Appendix D: Linearization

Small-Signal Linearization

Linearization by Feedback

Material covered in the PRESENT LECTURE is shown in yellow

I. DYNAMIC MODELING

• Deriving a dynamic model for mechanical, electrical, electromechanical, fluid- & heat-flow systems
• Linearization the dynamic model if necessary

II. DESIGN OF A CONTROLLER: Several design methods exist

• Classical control or Root Locus Design:

Define the transfer function; Apply root locus, loop shaping,…

• Modern control or State-Space Design:

Convert ODE to state equation; Apply Pole Placement, Robust control, …

• Nonlinear control: Apply Lyapunov’s stability criterion
Introduction
• Real models usually exhibit nonideal and nonlinear characteristics
• Analysis of a system represented by nonlinear partial differential equations, with time varying coefficients is extremely difficult and requires heavy computations
• There is no general analytic method available for solving nonlinear systems
Dealing with Nonlinear Systems

In general, we can take one of the following three options:

• Replace nonlinear elements with “roughly equivalent linear elements”, which often leads to invalid models;
• Develop and solve a nonlinear model, which results in most accurate results, BUT the analysis is too expensive since there is no general analytic methods available for solving.
• Linearize ( = find a linear model that approximates well, or at least fairly well, a nonlinear one) in order to make possible more efficient analysis and control design based on linear models.
Why are linear good?

To summarize: why are linear models useful?

• They are easy to compute, understand and visualize.
• They give predictable outputs, in time and over iterations
• The analysis of linear theory is complete, developed and efficient
• Linear differential equations are easy to solve!
What is a Linear Function?
• Linear functions are the first type of functions one learns in mathematics, yet there is not one single definition of linearity…
• Different answers apply to different contexts, discipline or purposes.

A linear function is:…

• … a function of the general form y = bx + c
• … a function whose derivative is a constant
• … a function in which the output is proportional to the input
• … a straight line? (careful, this only works in 2D representations)

Intuitively, linearity means proportionality of the output with respect to a variable. One variable function are most familiar but functions can be linear in many variables, e.g:

The Real World
• Learning about linear behavior is good, but how useful is it? Is the real word linear at all? The answer is no most of the time.
• Unfortunately, nonlinear dynamics are not fully understood and the best we can do is simulate the real world with linear or low-order approximations.
• To be more precise, linear behavior is simulated locally, at a point or along a small interval in space-time, and then the results are extrapolated about the general domain.
• That means that some degree of prediction is possible, but yet, we do not know everything about nonlinearity.

### Part A: Small-signal Linearization

Approximating the function while considering small disturbances around stable equilibrium points

Applications & Method
• Small signal linearization method is the most widely used
• It is in general done with the help of Taylor series.
• The Taylor expansion of a function f(x) around a point is given by:
Linearization around a point
• If we define ,the Taylor expansion becomes:
• Note that the functions and all the derivatives are evaluated at the linearization point
• If is small (i.e. x is close to ), then we may drop the second and higher-order terms as follows:
Functions of several independent variables
• For a function f of a single variable, x:
• For a function f of two independent variables, x and y:
Around what point is it proper to linearize?
• Intuitively, one would say around zero
• But the general answer is around stable equilibrium points
Example 1: Pendulum
• The dynamic model is given by:
• This equation is nonlinear in the displacement angle θ, because ofthe second term on the left side: f(θ)=(g/l) sin θ
• Equilibrium position ? It satisfies:
• Therefore, at equilibrium:
Example 1: Pendulum (cont’d)
• Linearizing f(θ)about , which is such as with

• Substituting f in the expression above gives:
• Linearizing about a stable equilibrium point requires:
Example 1: Pendulum (cont’d)
• Recall the dynamic model:

where

• At the operating point :

where

• By subtracting the two equations above:
sinx around x = π/2
• The linear approximation of sin x around x = π/2 is a constant
exp (-x) around 0
• The linear approximation of y = exp(-x) around 0 is y = 1-x
• Note that as x goes to infinity, y = exp(-x) converges to 0, while y = 1-x diverges to - infinity
Example 2: Double-pendulum
• See class’ notes.

### Part B: Linearization by Feedback

Subtracting the nonlinear terms out of the equation of motion and adding them to the control.

Pros & Cons

Gives a linear model

• Provided that the computer implementing the control has enough capability to compute the nonlinear terms fast enough
• No matter how large the system variable (θ) becomes.
Example: Pendulum
• Consider a pendulum subjected to a torque T. The dynamic model is then:
• If we compute the Torque T to be:

then the motion is described by:

• The resulting linear control will provided the values of u, based on the measurement of θ