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EE 3CL4, §31 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
EE3CL4:Introduction to Linear Control Systems
Section 3: Fundamentals of Feedback
Tim Davidson
McMaster University
Winter 2014
EE 3CL4, §32 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Outline
1 Transfer Function (review)
2 Open loop controlStability & Performance
3 Closed loop controlStability & Performance
4 Open loop control vs closed loop control
5 Step resp.: 2nd-order G(s)A taste of pole-placement designExtensions
6 Steady-state error
7 Summary and plan
EE 3CL4, §34 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Transfer function
• Y (s) = G(s)U(s)
• Stability (more details later):
the output y(t) is bounded for all bounded inputs u(t)if and only ifthe poles of G(s) are in the open left half plane
• Note that in this setting u(t) is an arbitrary input, notnecessarily the unit step function
EE 3CL4, §36 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Open loop controlSomewhat of an oxymoron, but this is conventional terminology
• Denote input to process by U(s)
• Y (s) = G(s)U(s) = G(s)Gc(s)R(s)
• If G(s) = nG(s)dG(s) and Gc(s) = nC (s)
dC (s) ,
Y (s)
R(s)= Gc(s)G(s) =
nC(s)nG(s)
dC(s)dG(s)
• Key quantity, error: E(s) = R(s)− Y (s)
EE 3CL4, §37 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Stability & Performance
• E(s) =(
1−Gc(s)G(s))
R(s) + G(s)Td (s)
• For good tracking want Gc(s) ≈ 1G(s)
• One design strategy: with G(s) = nG(s)dG(s) , Gc(s) = nC(s)
dC(s)
• Put poles of Gc(s) near zeros of G(s), andput zeros of Gc(s) near poles of G(s)
• Known as “pole-zero cancellation”• That strategy very sensitive to knowledge of G(s)• Problematic if G(s) has zeros in the right half plane
EE 3CL4, §38 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Stability & Performance
• E(s) =(
1−Gc(s)G(s))
R(s) + G(s)Td (s)
• Controller cannot mitigate effect of Td (s)
• No capability for disturbance rejection
• If G(s) and Gc(s) are stable, so is the “open loop”.What if G(s) is not stable?
• In theory, controller can stabilize Y (s)R(s) by pole-zero
cancellation• but difficult to implement to sufficient accuracy• cannot stabilize Y (s)
Td (s)
EE 3CL4, §39 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Stability & Performance
• In practice, we rarely model G(s) accurately, and it maychange with age, temperature and many other effects
• What if G(s) not perfectly known?
• Let T (s) = Y (s)R(s) (when Td (s) = 0)
• Let actual process transfer function be G(s) + ∆G(s)
• What is the ratio of relative error in T (s) due to relativeerror in G(s) for small errors?
• That is, what is lim∆G(s)→0∆T (s)/T (s)∆G(s)/G(s)
• Can show (prove for yourself) that this “sensitivity” = 1• That means that for small errors the relative error in
T (s) is the same as the relative error in G(s);we can’t design Gc(s) to make it smaller
EE 3CL4, §311 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Closed loop control
• Error: E(s) = R(s)− Y (s)
• Measured error: Ea(s) = R(s)− H(s)(
Y (s) + N(s))
.
• In the general case, Ea(s) 6= E(s).
• When H(s) = 1 and N(s) = 0, Ea(s) = E(s).
EE 3CL4, §312 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
The output signal
What is the output Y (s)? (Calculate yourself for practice)
Y (s) =Gc(s)G(s)
1 + H(s)Gc(s)G(s)R(s)
+G(s)
1 + H(s)Gc(s)G(s)Td (s)
− H(s)Gc(s)G(s)
1 + H(s)Gc(s)G(s)N(s)
EE 3CL4, §313 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
The error signal, H(s) = 1
What is the error E(s) = R(s)− Y (s)?To simplify things, consider the case where H(s) = 1
E(s) =1
1 + Gc(s)G(s)R(s)
− G(s)
1 + Gc(s)G(s)Td (s)
+Gc(s)G(s)
1 + Gc(s)G(s)N(s)
Recall, Ea(s) = E(s) only if H(s) = 1 and N(s) = 0.
EE 3CL4, §314 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Loop gain, H(s) = 1
Define loop gain: L(s) = Gc(s)G(s)
E(s) =1
1 + L(s)R(s)− G(s)
1 + L(s)Td (s) +
L(s)
1 + L(s)N(s)
What do we want?
EE 3CL4, §315 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Stability, H(s) = 1
E(s) =1
1 + L(s)R(s)− G(s)
1 + L(s)Td (s) +
L(s)
1 + L(s)N(s)
• Stability: bounded inputs lead to bounded errors
• For simplicity, let Td (s) = 0, N(s) = 0
• G(s) = nG(s)dG(s) ; Gc(s) = nC (s)
dC (s) ; L(s) = nC (s)dC (s)
nG(s)dG(s)
• Hence,1
1 + L(s)=
dC(s)dG(s)
dC(s)dG(s) + nC(s)nG(s)
• =⇒ closed loop poles are roots of dC(s)dG(s) + nC(s)nG(s)
• These can be in left half plane even if G(s) is unstable,but they can also be in the right half plane if G(s) is stable
EE 3CL4, §316 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Performance: s-domain,H(s) = 1
E(s) =1
1 + L(s)R(s)− G(s)
1 + L(s)Td (s) +
L(s)
1 + L(s)N(s)
What else do we want, in addition to stability?• Good tracking: E(s) does depend only weakly on R(s)
=⇒ L(s) large where R(s) large
• Good disturbance rejection:=⇒ L(s) large where Td (s) large
• Good noise suppression:=⇒ L(s) small where N(s) large
EE 3CL4, §317 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
A taste of loop shaping,H(s) = 1
Possibly easier to understand in pure freq. domain, s = jω
Recall that L(s) = Gc(s)G(s),G(s): fixed; Gc(s): controller to be designed
• Good tracking: =⇒ L(s) large where R(s) large|L(jω)| large in the important frequency bands of r(t)
• Good dist. rejection: =⇒ L(s) large where Td (s) large|L(jω)| large in the important frequency bands of td (t)
• Good noise suppr.: =⇒ L(s) small where N(s) large|L(jω)| small in the important frequency bands of n(t)
Typically, L(jω) is a low-pass function. How big should |L(jω)| be?
Any constraints? Stability! Any others?
EE 3CL4, §318 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Inherent constraints, H(s) = 1
Define sensitivity: S(s) =1
1 + L(s)
Define complementary sensitivity: C(s) =L(s)
1 + L(s)
E(s) = S(s)R(s)− S(s)G(s)Td (s) + C(s)N(s)
Note that S(s) + C(s) = 1.Trading S(s) against C(s), with stability,
is a key part of the art of control design
EE 3CL4, §319 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Performance: time-domain
• Difficult for arbitrary inputs
• In classical control techniques, typically assessed via
• nature of transient component of step response• how fast does system respond?• how long does it take to settle to new operating point
• steady-state error for constant changes in position, orvelocity or acceleration; that is steady-state error for
• step input; ramp input, parabolic input
EE 3CL4, §320 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Trade-off exampleDisk drive system
Y (s) =5000Ka
s3 + 1020s2 + 20000s + 5000KaR(s)
+s + 1000
s3 + 1020s2 + 20000s + 5000KaTd (s)
Coarsely design Ka to balance properties of step responseand response to step disturbance
EE 3CL4, §321 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Responses for Ka = 10
Disturbance step response and step response
Low gain:• steady-state disturbance might not be negligible• slow transient response for step input
EE 3CL4, §322 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Responses for Ka = 10,100
Disturbance step response and step response
Medium gain:• steady-state disturbance much reduced• faster transient response for step input,
but now some overshoot
EE 3CL4, §323 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Responses forKa = 10,100,1000
Disturbance step response and step response
High gain:• steady-state disturbance almost completely rejected• fast transient response for step input,
but now significant overshoot• Actually can show by Routh Hurwitz technique (later)
that loop is unstable for Ka ≥ 4080
EE 3CL4, §324 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Robustness
• What if G(s) is not perfectly known?
• How does T (s) = Y (s)R(s) change as G(s) changes?
• As in open loop case, look at ratio of relative error in T (s)due to a relative error in G(s), for small errors,lim∆G(s)→0
∆T (s)/T (s)∆G(s)/G(s)
• For an open loop system this sensitivity is 1
• For the closed loop system, with H(s) = 1,this sensitivity is S(s) = 1
1+Gc(s)G(s)
• Now can design controller to manage sensitivity
EE 3CL4, §326 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Open loop vs closed loop
Error: E(s) = R(s)− Y (s); Loop gain: L(s) = Gc(s)G(s),
Open loop, with process input Gc(s)R(s) + Td (s)
Eol(s) =(
1− L(s))
R(s)−G(s)Td (s)
Closed loop, with H(s) = 1,
Ecl(s) =1
1 + L(s)R(s)− G(s)
1 + L(s)Td (s) +
L(s)
1 + L(s)N(s)
EE 3CL4, §327 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Advantages of feedback
• Can mitigate disturbances
• Can stabilize (most) unstable processes
• Can mitigate errors in the model of process
EE 3CL4, §328 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Price of feedback
• more components than open loop
• introduces noise, but can “shape” the response
• less gain: Gc(s)G(s)1+Gc(s)G(s) instead of Gc(s)G(s)
• potential for instability even when G(s) is stable
More complicated to analyze and design,but often advantages are worth the price
EE 3CL4, §330 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
A second-order systemNow examine, in detail, a particular class of second ordersystems
Y (s) =G(s)
1 + G(s)R(s) =
ω2n
s2 + 2ζωns + ω2n
R(s)
EE 3CL4, §331 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Step response• What is the step response?
• Set R(s) = 1/s; take inverse Laplace transform of Y (s)
• Y (s) =ω2
n
s(
s2+2ζωns+ω2n
)• For the case of 0 < ζ < 1,
y(t) = 1− 1β
e−ζωnt sin(ωnβt + θ)
where β =√
1− ζ2 and θ = cos−1 ζ.
• Recall pole positions of G(s)1+G(s) (ignore the zero):
EE 3CL4, §332 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Typical step responses, fixed ωn
EE 3CL4, §333 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Typical step responses, fixed ζ
EE 3CL4, §334 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Key parameters of(under-damped) step response
With β =√
1− ζ2 and θ = cos−1 ζ,
y(t) = 1− 1β
e−ζωnt sin(ωnβt + θ)
EE 3CL4, §335 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Peak time and peak value
y(t) = 1− 1β
e−ζωnt sin(ωnβt + θ)
• Peak time: first time dy(t)/dt = 0• Can show that this corresponds to ωnβTp = π
• Hence, Tp =π
ωn√
1− ζ2
• Hence, peak value, Mpt = 1 + e−(ζπ/√
1−ζ2)
EE 3CL4, §336 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Percentage overshoot
Let fv denote the final value of the step response.
Percentage overshoot defined as:
P.O. = 100Mpt − fv
fv
In our example, fv = 1, and hence
P.O. = 100 e−(ζπ/√
1−ζ2)
EE 3CL4, §337 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Overshoot vs Peak TimeThis is one of the classic trade-offs in control
EE 3CL4, §338 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Steady-state error, ess
In general this is not zero. (See “Steady-state error” section)
However, for our second-order system,
y(t) = 1− 1β
e−ζωnt sin(ωnβt + θ)
Hence ess = 0
EE 3CL4, §339 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Settling time
y(t) = 1− 1β
e−ζωnt sin(ωnβt + θ)
• How long does it take to get within ±x% of final value?• Approx. when e−ζωnt < x/100• When x = 2, that corresponds to ζωnTs ≈ 4;
that is, 4 time constants
• In that case, Ts =4ζωn
EE 3CL4, §340 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Rise time (under-damped)
y(t) = 1− 1β
e−ζωnt sin(ωnβt + θ)
• How long to get to the target (for first time)?• Tr , the smallest t such that y(t) = 1
EE 3CL4, §341 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
10%–90% Rise time
• What is Tr in over-damped case? ∞• Hence, typically use Tr1, the 10%–90% rise time
EE 3CL4, §342 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
10%–90% Rise time
• Difficult to get an accurate formula• Linear approx. for 0.3 ≤ ζ ≤ 0.8 (under-damped),
EE 3CL4, §343 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Design problem
For what values of K and p is• the settling time ≤ 4 secs, and• the percentage overshoot ≤ 4.3%?
T (s) =G(s)
1 + G(s)=
Ks2 + ps + K
=ω2
n
s2 + 2ζωns + ω2n,
where ωn =√
K and ζ = p/(2√
K )
EE 3CL4, §344 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Pole positions
Ts ≈4ζωn
P.O. = 100 e−(ζπ/√
1−ζ2)
• For Ts ≤ 4, ζωn ≥ 1• For P.O. ≤ 4.3%, ζ ≥ 1/
√2
Where should we put the poles of T (s)?
EE 3CL4, §345 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Pole positions
ζωn ≥ 1 ζ ≥ 1/√
2
s1, s2 = −ζωn ± jωn√
1− ζ2 = −ωn cos(θ)± jωn sin(θ)
where θ = cos−1(ζ).
EE 3CL4, §346 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Final design constraints
ζωn ≥ 1 ζ ≥ 1/√
2
p ≥ 2 p ≥√
2K
EE 3CL4, §347 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Caveat
• Our work on transient response has been for systemswith
T (s) =ω2
n
s2 + 2ζωns + ω2n
• What about other systems?
EE 3CL4, §348 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Poles, zeros and transientresponse
• Consider a general transfer function T (s) = Y (s)R(s)
• Step response: Y (s) = T (s)1s
• Consider case with DC gain = 1; no repeated poles• Partial fraction expansion
Y (s) =1s
+∑
i
Ai
s + σi+∑
k
Bks + Ck
s2 + 2αks + (α2k + ω2
k )
• Step response
y(t) = 1 +∑
i
Aie−σi t +∑
k
Dke−αk t sin(ωk t + θk )
EE 3CL4, §349 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
EE 3CL4, §351 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Steady-state error
E(s) = R(s)− Y (s) =1
1 + Gc(s)G(s)R(s)
If the the conditions are satisfied, the final value theoremgives steady-state tracking error:
ess = limt→∞
e(t) = lims→0
s1
1 + Gc(s)G(s)R(s)
One of the fundamental reasons for using feedback, despitethe cost of the extra components, is to reduce this error.
We will examine this error for the step, ramp and parabolicinputs
EE 3CL4, §352 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Step, ramp, parabolic
EE 3CL4, §353 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Step input
ess = limt→∞
e(t) = lims→0
s1
1 + Gc(s)G(s)R(s)
• Step input: R(s) = As
• ess = lims→0sA/s
1+Gc(s)G(s) = A1+lims→0 Gc(s)G(s)
• Now let’s examine Gc(s)G(s). Factorize num., den.
Gc(s)G(s) =K∏M
i=1(s + zi)
sN∏Q
k=1(s + pk )
where zi 6= 0 and pk 6= 0.• Limit as s → 0 depends strongly on N.• If N > 0, lims→0 Gc(s)G(s)→∞ and ess = 0• If N = 0,
ess =A
1 + Gc(0)G(0)
EE 3CL4, §354 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
System types
• Since N plays such a key role,it has been given a name
• It is called the type number
• Hence, for systems of type N ≥ 1,ess for a step input is zero
• For systems of type 0, ess = A1+Gc(0)G(0)
EE 3CL4, §355 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Position error constant
• For type-0 systems, ess = A1+Gc(0)G(0)
• Sometimes written as ess = A1+Kp
where Kp is the position error constant
• Recall Gc(s)G(s) =K
QMi=1(s+zi )
sNQQ
k=1(s+pk )
• Therefore, for a type-0 system
Kp = lims→0
Gc(s)G(s) =K∏M
i=1(zi)∏Qk=1(pk )
• Note that this can be computed from positions of thenon-zero poles and zeros
EE 3CL4, §356 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Ramp input
• The ramp input, which represents a step change invelocity is r(t) = At .
• Therefore R(s) = As2
• Assuming conditions of final value theorem aresatisfied,
ess = lims→0
s(A/s2)
1 + Gc(s)G(s)= lim
s→0
As + sGc(s)G(s)
= lims→0
AsGc(s)G(s)
• Again, type number will play a key role.
EE 3CL4, §357 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Velocity error constant
• For a ramp input ess = lims→0A
sGc(s)G(s)
• Recall Gc(s)G(s) =K
QMi=1(s+zi )
sNQQ
k=1(s+pk )
• For type-0 systems, Gc(s)G(s) has no poles at origin.Hence, ess →∞
• For type-1 systems, Gc(s)G(s) has one pole at the origin.Hence, ess = A
Kv, where Kv =
KQ
i ziQk pk
• Note Kv can be computed from non-zero poles and zeros
• Suggests formal definition of velocity error constant
Kv = lims→0
sGc(s)G(s)
• For type-N systems with N ≥ 2, for a ramp input ess = 0
EE 3CL4, §358 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Parabolic input
• The parabolic input, which represents a step change inacceleration is r(t) = At2/2.
• Therefore R(s) = As3
• Assuming conditions of final value theorem aresatisfied,
ess = lims→0
s(A/s3)
1 + Gc(s)G(s)= lim
s→0
As2Gc(s)G(s)
• Again, type number will play a key role.
EE 3CL4, §359 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Acceleration error constant
• For a parabolic input ess = lims→0A
s2Gc(s)G(s)
• Recall Gc(s)G(s) =K
QMi=1(s+zi )
sNQQ
k=1(s+pk )
• For type-0 and type-1 systems, Gc(s)G(s) has at most onepole at origin. Hence, ess →∞
• For type-2 systems, Gc(s)G(s) has two poles at the origin.Hence, ess = A
Ka, where Ka =
KQ
i ziQk pk
• Again, Ka can be computed from non-zero poles and zeros
• Suggests formal definition of acceleration error constant
Ka = lims→0
s2Gc(s)G(s)
• For type-N systems with N ≥ 3, for a parabolic input ess = 0
EE 3CL4, §360 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Summary of steady-state errors
EE 3CL4, §361 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Robot steering system,P control
Let’s examine a proportional controller:
Gc(s) = K1
• Gc(s)G(s) = K1K/(τs + 1)
• Hence, Gc(s)G(s) is a type-0 system.• Hence, for a step input,
ess =A
1 + Kp
where Kp = K1K .
EE 3CL4, §362 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Robot steering system,P control example
• Let G(s) = 1s+2 = 0.5
0.5s+1 .
• Proportional control, Gc(s) = K1. Choose K1 = 18.
• Since Gc(s)G(s) is type-0:• finite steady-state error for a step,• unbounded steady-state error for a ramp
• In this example, Kp = KK1 = 10
• The steady-state error for a step input will be1
1+Kp= 10% of the height of the step.
• For a unit step the steady-state error will be 0.1.
EE 3CL4, §363 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Robot steering system,P control example
EE 3CL4, §364 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Robot steering system,PI control
Let’s examine a proportional-plus-integral controller:
Gc(s) = K1 +K2
s=
K1s + K2
s
• When K2 6= 0, Gc(s)G(s) = K (K1s+K2)s(τs+1)
• Hence, Gc(s)G(s) is a type-1 system.• Hence, for a step input, ess = 0• For ramp input,
ess =AKv,
where Kv = lims→0 sGc(s)G(s) = KK2
EE 3CL4, §365 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Robot steering system,PI control example
• Same system: G(s) = 1s+2 = 0.5
0.5s+1 .
• Prop. + Int. control, Gc(s) = K1 + Kss = K1s+K2
s .Choose K1 = 18 and K2 = 20.
• Now since Gc(s)G(s) is type-1:• zero steady-state error for a step• finite-steady state error for a ramp
• In this example Kv = KK2 = 10
• The steady-state error for a ramp input will be1
Kv= 10% of the slope of the ramp.
• For a unit ramp the steady-state error will be 0.1.
EE 3CL4, §366 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Robot steering system,PI control example
EE 3CL4, §368 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Summary: Desirable properties
With H(s) = 1, E(s) = R(s)− Y (s), L(s) = Gc(s)G(s),
E(s) =1
1 + L(s)R(s)− G(s)
1 + L(s)Td (s) +
L(s)
1 + L(s)N(s)
• Stability
• Good tracking in the steady state
• Good tracking in the transient
• Good disturbance rejection (good regulation)
• Good noise suppression
• Robustness to model mismatch
EE 3CL4, §369 / 69
Tim Davidson
Transferfunctions
Open loopStability &Performance
Closed loopStability &Performance
Open loop vsclosed loop
Step resp.:2nd-orderG(s)A taste ofpole-placementdesign
Extensions
Steady-stateerror
Summary andplan
Plan: Analysis and designtechniques
Rest of course: about developing analysis and designtechniques to address these goals
• Routh-Hurwitz:• Enables us to determine stability without having to find
the poles of the denominator of a transfer function
• Root locus• Enables us to show how the poles move as a single
design parameter (such as an amplifier gain) changes
• Bode diagrams• There is often enough information in the Bode diagram
of the plant/process to construct a highly effectivedesign technique
• Nyquist diagram• More advanced analysis of the frequency response that
enables stability to be assessed even for complicatedsystems