input output equations - 39.8

The matrix can be parameterized,

–ra

–Ka

1

d

i

-------

--------

i

-----

0

v

i

v

a

=

L

a

L

a

a

+

L

a

s

=

A B

a

+

E 0

s

----

dt

ω

r

K

–B

ω

r

–1

T

L

C D

ω

r

0 F

T

L

a

m

0

-----

-----

---------

J

J

J

E =

1

La

= E

-----

La

A =

–ra

ra = –ALa = –AE

-------

La

B =

–Ka

Ka = –BLa = –BE

---------

La

C =

Ka

J =

Ka

–BE

-----

----- =

----------

J

C

C

D =

–Bm

Bm

–BED

---------

= --------------

J

C

39.3BLOCK DIAGRAMS AND UNITS

-each block in a block diagram has an input and output. Making sure that the units are consistent is useful for detecting mistakes and ensuring consitency.

input output equations - 39.9

39.4 SIGNAL FLOW GRAPHS

- these invert block diagrams.

··	1	·	1	x
x	---	x	---
	D		D

39.5ZERO ORDER HOLD

-t.

1			G		( s)	=	1 – e–Ts
				h			------------------
								s
								t

0T

39.6TORSIONAL DAMPERS

-

input output equations - 39.10

39.7 MISC

- controller goals

stabilize - to make the system stable

regulate - to control the system about a central design value follow - to track an input/feedback signal

disturbance rejection - to minimize the effects of noise

-state diagrams can be used with multiple input/multiple output (MIMO) systems

-state based controller designs,

state feedback - in an ideal controller the full state variables can be measured

observer-estimator - determines the system state using limited data.

-Linear Time Invariant (LTI) systems are linear, having constant coefficients. For example, state coefficient matrices are all constant values

-Linear systems allow superposition

-leaf springs are non-linear and have a cubic relationship to displacement

-if a system input (setpoint) is zero, or not used, then the control system is called a regulator.

-stepper motor motion plans should not be approximate, the stepper motor should follow the path, although trapezoidal profiles are more common.

39.8Nyquist Plot

•The method is used to determine system stability using the characteristic equation 1+Gc(s)G(s)H(s)

•If over the range of frequencies the system enters the unstable range of the Nyquists contour shown below.

input output equations - 39.11

• The basic method for producing polar plots is,

1.Write the characteristic equation for a feedback system Gc(s)G(s)H(s)

2.Use phasors to determine a gain and angle.

3.For a large range of frequencies, plot the gain and angle on polar axes.

4.Look for ’encirclements’ of the point at a gain of -1 and -180°.

•If the number of clockwise encirclements is equal to the number of poles in the right hand side of the plane, the system is stable.

•The gain and phase margins can be determined from the polar plot as shown

below,