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2.004 Dynamics and Control II Spring 2008

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H ( s ) H ( s )1 2H ( s ) H ( s )1 2U ( s ) Y ( s ) U ( s ) Y ( s )X ( s )

H ( s )1

2

+

+H ( s ) + H ( s )1 2Y ( s ) Y ( s )U ( s )

U ( s )

H ( s )

Massachusetts Institute of Technology

Department of Mechanical Engineering

2.004 Dynamics and Control II Spring Term 2008

Lecture 41

Reading:

Nise: Secs. 5.15.3

1 Block Diagram Algebra (Interconnection Rules)

a)Series (Cascade) Connection:

Since the output of the rst block is X(s) = H1(s)U(s),

Y (s) = H2(s)X(s) = H1(s)H2(s)U(s)

Note: This is only true if the connection of Hs(s) to H1(s) does not alter the output of H1(s) known as the non-loading condition.

b)Parallel Connection In this case the input U(s) is applied to both inputs and the outputs are summed:

Y (s) = H1(s)U(s) + H2(s)U(s) = (H1(s) + H2(s))U(s)

Example 1

Express 6

H(s) = s2 + 5s + 6

1copyright c D.Rowell 2008

41

U ( s ) Y ( s ) 3s + 3 2

s + 2

o r d e r o f b l o c k s i s a r b i t r a r y

-

+Y ( s )

U ( s )

6

s + 2

6

s + 3

+Y ( s )

U ( s )1

U ( s )2

H ( s )-

+Y ( s )

U ( s )1

U ( s )2

H ( s )

H ( s )

+

U ( s ) Y ( s )H ( s )1 H ( s )2 U ( s ) Y ( s )H ( s )2 H ( s )1

as (a) a series connection, and (b) a parallel connection of rst-order blocks

a) Series: 6 3 2

H(s) = = s2 + 5s + 6 s + 3

s + 2

b) Parallel: Using partial fractions we nd

6 6 H(s) =

s + 2 s + 3

Notes:

a) These two systems are equivalent.

b) A partial fraction expansion is eectively a parallel implementation.

c) A factored representation of H(s) is eectively a series implementation.

c)Associative Rule:

Y (s) = (U1(s) + U2(s))H(s) U1(s)H(s) + U2(s)H(s) d)Commutative Rule:

The order does not matter in a series connection.

42

G ( s )c G ( s )pE ( s )R ( s ) C ( s )

c o n t r o l l e r

p l a n t

e r r o r

+

-

f e e d b a c k p a t h

r e f e r e n c e

i n p u t

c o n t r o l l e d

o u t p u t

G ( s )cG ( s )p

E ( s )V ( s )

c o n t r o l l e r

c a r d y n a m i c s

e r r o r

+

-

f e e d b a c k p a t h

d e s i r e d

s p e e d

a c t u a l

s p e e d

V ( s )dq ( s )

2 The Closed-Loop Transfer Function

a) Unity feedback

Notes:

(a) The term unity feedback means that the actual output value is used to generate the error signal (the feedback gain is 1).

(b) In control theory transfer functions in the forward path are often designated by G(s) (see below).

(c) It is common to use R(s) to designate the reference (desired) input, and C(s) to designate the controlled (output) variable.

From the block diagram: C(s) = (Gp(s)Gc(s))E(s)

and E(s) = R(s) C(s)

or C(s) = Gp(s)Gc(s)(R(s) C(s)

Rearranging:

C(s) Gc(s)Gps Gcl(s) = =

R(s) 1 + Gc(s)Gps

is the unity feedback closed-loop transfer function.

Example 2

Find the closed-loop transfer function for the automobile cruise control example:

43

For the car mv + Bv = Fp = Ke so that

V (s) KsGp(s) = =

(s) ms + B

For the controller (s) = KcE(s)

(s)Gc(s) = = Kc

E(s)

Then from above V (s) Gc(s)Gp(s)

Gcl(s) = = Vd(s) 1 + Gc(s)Gp(s)

KcKe ms+B KcKeGcl(s) = = ,

1 + KcKe ms + (B + KcKe)ms+B

and by inspection the closed-loop dierential equation is

mv + (B + KcKe)v = KcKevd.

Aside: Use the Laplace transform nal value theorem to nd the steady state velocity to a step input vd(t) = vd For the step input

vd vd(s) =

s

and in the Laplace domain

KcKe vd v(s) = Gc1(s)Vd(s) =

ms + (B + KcKe) s

The F.V. theorem states limtf(t) = lims 0sF (s) so that

KcKe vd vss = limtv(t) = lims0s

ms + (B + KcKe) s

KcKe vss =

B + KcKe

which is same as we obtained before.

3 Closed-Loop Transfer Function With Sensor Dynamics:

Until now we have assumed that the output variable y(t) is measured instantaneously, and without error. Frequently the sensor has its own dynamics - for example the sensor might be temperature measuring device modeled as a rst-order system:

44

RCf r o m s e n s o r

t o c o n t r o l l e r

Vs e n s o r

Vo u t

f i l t e r

1

t s + 1s

s y s t e m

o u t p u t

s e n s o r

f e e d b a c k

s i g n a l

t t

where s is the sensor time constant. The closed-loop block diagram is

G ( s )c G ( s )pE ( s )R ( s ) C ( s )

c o n t r o l l e r

p l a n t

e r r o r

+

-r e f e r e n c ei n p u t

a c t u a l

o u t p u t

H ( s )

s e n s o r

i n d i c a t e d o u t p u t

where H(s) is the transfer function of the sensor. In this case:

C(s) = (Gc(s)Gp(s))E(s)

but now E(s) is the indicated error (as opposed to the actual error):

E(s) = R(s) H(s)C(s) so

C(s) = Gc(s)Gp(s)(R(s) H(s)C(s)) or

C(s)(1 + Ge(s)Gp(s)H(s)) = Ge(s)Gp(s)H(s).

C(s) Gc(s)Gp(s)Gcl(s) = =

R(s) 1 + Gc(s)Gp(s)H(s)

is the modied closed-loop transfer function.

Example 3

Suppose that velocity sensor in the cruise control is noisy, and a simple electrical lter is used to smooth the output. Find the eect of the lter on the closed-loop dynamics.

45

cE ( s )V ( s )

c o n t r o l l e r

p l a n t

e r r o r

+

-r e f e r e n c ei n p u t

a c t u a l

s p e e d

f i l t e r

i n d i c a t e d s p e e d

K Km s + Be

1

R C s + 1

d V ( s )

Using Kirchos Voltage Law (KVL) we nd

RC vout + vout = vsensor

so that Vout(s) 1

H(s) = = Vsensor(s) RCs + 1

Then the closed-loop transfer function is

KcKeV (s) Gc(s)Gp(s) ms+B = = =Gcl(s) Va(s) 1 + Gc(s)Gp(s)H(s) 1 + KcKe (ms+B)(RCs+1)

KcKe(RCs + 1) =

(ms + B)(RSs + 1) + KcKe

KcKe(RCs + 1) Gcl(s) =

mRCs2 + (BRC + m)s + (B + KcKe)

and the dierential equation relating the speed of the car to the desired speed command is now

mRCv + (BRC + m)v + (B + KcKe)v = KcKeRC vd + KcKevd

and we note:

1) we now have a second-order system - the dynamics may change signicantly,

2) we have derivative action on the RHS of the dierential equation.

46