CONTROL SYSTEM ANALYIS

Control System Analysis

Assignment number 01

The non linear systems are often linearized by making the use of approximation technique. The non linear equation of motion can be linearized as following.

The original non linear equation which is given as:

u? = Xuu + Xww - gcos(?0?) + XdEdE

To linearized this equation we will use single state equation approximation .The method of approximation is illustrated below.

x ? (t) = f (x , t) ----------- (1)

Let's predict x(t + ?T) from x(t) in which ?T is a time step. To make the prediction, a Taylor series expansion of x(t) is performed.

x(t + ?T) = x(t) + x ? (t)?T + x(t) ??? ? + . . . . . . . .

The first term on the right side is called a 0th -order term, the next term is a 1st -order term, the next is a 2nd-order term, etc. As a simple approximation, we retain the first two terms on the right side and neglect the others. This is called a 1st-order approximation. By neglecting the 2nd and higher order terms, we get

?x = f . ?T

x (t + ?T) = x(t) + ?x -------------- (2)

It is the 1st - order numerical integrator for equation (1).It is also known as Euler Algortihm. Equation (2) is used in a computer program in a loop that predicts x(t) at times T, 2T, 3T, etc. Figure (1) shows the error in the prediction over one time step. As you might expect, the error decreases as the time step decreases.

Figure () f versus t

A System of State Equations

Let's now extend the 1st -order numerical integrator to a set on n state equations. The set of n state equations is

?1(t) = f1(x1 , x2 , x3, . . . , xn , t )

?2(t) = f2(x1 , x2 , x3, . . . , xn , t )

?3(t) = f3(x1 , x2 , x3, . . . , xn , t )

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?n(t) = fn(x1 , x2 , x3, . . . , xn , t ) ---------------- (3)

To predict the states x1(t + ?T), x2(t + ?T), … , xn(t+ ?T) from the states x1(t), x2(t), … , xn(t), we perform Taylor series expansions of each of the state variables.

xr (t + ?T) = xr(t) + xr?(t)?T + xr ?? ?(t) + . . . --------------- (4)

Where,

(r = 1,2,3,4 , . . . . . . . , n)

Neglecting the 2nd -order and higher-order terms

?xr = fr . ?T (r = 1,2,3,4 , . . . . . . . , n)

xr( t + ?T) = xr(t) + ?xr (r = 1,2,3,4 , . . . . . . . , n) ------------ (5)

Equation (5) is the 1st -order numerical integrator for Eq. (3). Figure 1 applies to each of the state variables showing the errors in the state variables over one time step.

Linearization

To linearize the equations of motion following steps are needed to be taken.

u(t) = u0 + ?u(t)

For steady ?ight, ...