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Lecture: Anti-windup techniques
Automatic Control 2
Anti-windup techniques
Prof. Alberto Bemporad
University of Trento
Academic year 2010-2011
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 1 / 17
Lecture: Anti-windup techniques Saturation effects
Problem
G(z)x(k)
r(k)Kx(k)+Hr(k)
linearcontroller
u(k)saturation
Wednesday, June 2, 2010
Most control systems are designed based on linear theory
A linear controller is simple to implement and performance is good, as longas dynamics remain close to linear
Nonlinear effects require care, such as actuator saturation (always present)
Saturation phenomena, if neglected in the design phase, can lead toclosed-loop instability, especially if the process is open-loop unstable
Main reason: the control loop gets broken if saturation is not taken intoaccount by the controller: u(k) 6= Kx(k) for some k
[1] K.J. Åstrom, L. Rundqwist, “Integrator windup and how to avoid it”, Proc. American ControlConference, Vol. 2, pp. 1693-1698, 1989Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 2 / 17
Lecture: Anti-windup techniques Saturation effects
Example: AFTI-F16 aircraft
Linearized model:
x =�−0.0151 −60.5651 0 −32.174−0.0001 −1.3411 0.9929 00.00018 43.2541 −0.86939 0
0 0 1 0
�
x+� −2.516 −13.136−0.1689 −0.2514−17.251 −1.5766
0 0
�
u
y =� 0 1 0 0
0 0 0 1
�
x
Inputs: elevator and flaperon(=flap+aileron) angles
Outputs: attack and pitch angles
Sampling time: Ts = 0.05 sec (+ ZOH)
Constraints: ±25◦ on both angles
Open-loop response: unstableopen-loop poles:−7.6636,−0.0075± 0.0556j, 5.4530
!"##$%
&'$()*+%,
-').,
/0'$%+1
/0'$%+12+130*"#01)'4)50,
6$%*07)'4)50,
2)*$%)'4)50,
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80*79:);
Wednesday, June 2, 2010
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 3 / 17
Lecture: Anti-windup techniques Saturation effects
Example: AFTI-F16 aircraft
LQR control + actuator saturation ±25◦
The system is unstable !
Actuator saturation cannot be neglected in the design of a good controller
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 4 / 17
Lecture: Anti-windup techniques Saturation effects
Example: AFTI-F16 aircraft
With a controller designed to handle saturation constraints:
There are several techniques to handle input saturation, the most popularones are anti-windup techniques
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 5 / 17
Lecture: Anti-windup techniques Saturation effects
Saturation function
Saturation can be defined as the static nonlinearity
sat(u) =
umin if u< uminu if umin ≤ u≤ umax
umax if u> umax
−2 −1 0 1 2−1
−0.5
0
0.5
1
umin and umax are the minimum and maximum allowed actuation signals(example: ±12 V for a DC motor)If u is a vector with m components, the saturation function is defined as thesaturation of all its components (umin, umax ∈ Rm)
sat(u) =
sat(u1)sat(u2)
...sat(um)
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 6 / 17
Lecture: Anti-windup techniques Anti-windup schemes: Ad hoc methods
The windup problem
reference output
input
To Workspace
integ
Saturation Process
1s
Integrator
1s
Gain
1
Derivative
du/dt
very simple process (=an integrator)controlled by a PID controller (KP = KI = KD = 1)
under saturation −0.1≤ u≤ 0.1
The output takes a long time to go steady-state
The reason is the “windup” of the integrator contained in the PID controller,which keeps integrating the tracking error even if the input is saturating
anti-windup schemes avoid such a windup effect
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 7 / 17
Lecture: Anti-windup techniques Anti-windup schemes: Ad hoc methods
Anti-windup #1: incremental algorithm
y(kT)r(KT)
u(kT)incremental PID
1
1 ! z!1 G(z)!u(kT )
Sunday, May 16, 2010
It only applies to PID control laws implemented in incremental form
u((k+ 1)T) = u(kT) +∆u(kT)
where∆u(kT) = u(kT)− u((k− 1)T)
Integration is stopped if adding a new ∆u(kT) causes a violation of thesaturation bound
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 8 / 17
Lecture: Anti-windup techniques Anti-windup schemes: Ad hoc methods
Anti-windup #2: Back-calculation
G(s)!"e
sTd Kp!
v yr uactuator
!!
!
PID
! !
1
s
1
Tt
!"Kp
Ti
anti-windup
et
Wednesday, June 2, 2010
The anti-windup scheme has no effect when the actuator is not saturating(et(t) = 0)
The time constant Tt determines how quickly the integrator of the PIDcontroller is reset
If the actual output u(t) of the actuator in not measurable, we can use amathematical model of the actuator. Example: et(t) = v(t)− sat(v(t))
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 9 / 17
Lecture: Anti-windup techniques Anti-windup schemes: Ad hoc methods
Anti-windup #2: Back-calculation
reference output
input
To Workspace
integ_aw
Saturation Process
1s
Integrator
1s
Gain
1
Derivative
du/dt
anti-windup scheme for a PID controller (KP = KI = KD = 1, Tt = 1)
Integrator windup is avoided thanks to back-calculation
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 10 / 17
Lecture: Anti-windup techniques Anti-windup schemes: Ad hoc methods
Benefits of the anti-windup scheme
Let’s look at the difference between having and not having an anti-windupscheme
0 10 20 30 40 50 60 70 800
0.5
1
1.5
2
outp
ut y
(t)
0 10 20 30 40 50 60 70 80−0.1
−0.05
0
0.05
0.1
inpu
t u(t
)
0 10 20 30 40 50 60 70 80−5
0
5
inte
gral
term
i(t)
time t
Note that in case of windup we have
strong output oscillations
a longer time to reach thesteady-state
the peaks of control signal
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 11 / 17
Lecture: Anti-windup techniques Anti-windup schemes: Ad hoc methods
Anti-windup #3: Conditional integration
The PID control law is
u(t) = Kp(br(t)− y(t)) + I(t)−KpTddy(t)
dt= Kpbr(t)−Kpyp(t) + I(t)
where yp(t) = y(t) + Tddy(t)
dtis the prediction of the output for time t+ Td
Consider the proportional band [yl(t), yh(t)] for yp(t) in which thecorresponding u is not saturating
yl(t) = br(t) +I(t)− umax
Kp
yh(t) = br(t) +I(t)− umin
Kp
where umin and umax are the saturation limits of the actuatorThe idea of conditional integration is to update the integral term I(t) onlywhen yp is within the proportional band (for PI control simply set yp = y)An hysteresis effect may be included to prevent chattering
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 12 / 17
Lecture: Anti-windup techniques Anti-windup schemes: General methods
Anti-windup #4: Observer approach
The anti-windup method applies todynamic compensators
x(k+ 1) = Ax(k) + Bu(k)+L(y(k)− Cx(k))
u(k) = Kx(k) +Hr(k)
by simply feeding the saturatedinput to the observer
x(k+ 1) = Ax(k) + B sat(v(k)))+L(y(k)− Cx(k))
v(k) = Kx(k) +Hr(k)
u(k) = sat(v(k))
x(k) y(k)A,B
u(k)C
.1$1-
-.1&%$1,+
!"#$%&'$()*+,'-..)/0).$12+$1&,#
r(k)K
x(k)ˆ++H
saturation
v(k)
Wednesday, June 2, 2010
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 13 / 17
Lecture: Anti-windup techniques Anti-windup schemes: General methods
Example
The process to be controlled is obtained by exactly sampling
x =
�
−1.364 0.4693 0.736 1.131−1.08 −1.424 0.1945 −0.71320.0499 0.8704 −0.9675 −0.3388−0.9333 0.8579 −0.5436 −0.9997
�
x+� 0.05574
0−0.04123−1.128
�
u
y = [−1.349 0 0.9535 0.1286 ]x
with T = 0.5sThe input u saturates between ±2Control design by pole placement: poles in e−5T , e−5T , e(−2±j)T
Observer design by pole placement: 4 poles in e−10T
The dynamic compensator is�
x(k+ 1) = Ax(k) + B sat(v(k)) + L(y(k)− Cx(k))v(k) = Kx(k) +Hr(k)
with
K =� 0.4718−1.5344−2.82532.1819
�′
, L=� 1.1821
1.19244.2054−3.6554
�
, H = 1/(C(I− A− BK)−1B) = 8.2668
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 14 / 17
Lecture: Anti-windup techniques Anti-windup schemes: General methods
Example (cont’d)
Compare the results with anti-windup and without anti-windup:
0 5 10 15 20 25 30 35 40 45 50−1
−0.5
0
0.5
1
1.5ou
tput
y(t
)
0 5 10 15 20 25 30 35 40 45 50−2
−1
0
1
2
inpu
t u(t
)
time t
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 15 / 17
Lecture: Anti-windup techniques Anti-windup schemes: General methods
Conclusions
Conditional integration is easy to apply to many controllers, although it maynot be immediate to find the conditions to block integration and to avoidchattering
Back-calculation only requires tuning one parameter, the time constant Tt.But it only applies to PID control
The observer approach is very general and does not require tuning anyadditional parameter. It also applies immediately to MIMO (multi-inputmulti-output) systems
Saturation effects can be included in an optimal control formulation. In thiscase the control action is decided by a constrained optimization algorithm, asin model predictive control (MPC) techniques
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 16 / 17
Lecture: Anti-windup techniques Anti-windup schemes: General methods
English-Italian Vocabulary
anti-windup scheme schema di anti-windupaileron alettonepitch beccheggioroll rollioyaw imbardatamodel predictive control controllo predittivo
Translation is obvious otherwise.
Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011 17 / 17