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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

Appendix D: Linearization

Small-Signal Linearization

Linearization by Feedback

material covered in the p resent l ecture is shown in yellow
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
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
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
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
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.
possible answers
Possible Answers…

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
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

Part A: Small-signal Linearization

Approximating the function while considering small disturbances around stable equilibrium points

applications method
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
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
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
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
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
Example 1: Pendulum (cont’d)
  • Linearizing f(θ)about , which is such as with

small leads to:

  • Substituting f in the expression above gives:
  • Linearizing about a stable equilibrium point requires:
example 1 pendulum cont d17
Example 1: Pendulum (cont’d)
  • Recall the dynamic model:

where

  • At the operating point :

where

  • By subtracting the two equations above:
sin x around x 2
sinx around x = π/2
  • The linear approximation of sin x around x = π/2 is a constant
exp x around 0
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
Example 2: Double-pendulum
  • See class’ notes.
part b linearization by feedback

Part B: Linearization by Feedback

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

pros cons
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
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