FLEA

1
FLEA

• Prof. dr. ir. J. Hellendoorn

2
Contents

• Fuzzy modeling
• Fuzzy control
• Fuzzy modeling and fuzzy control
• Takagi-Sugeno control
• Heuristic fuzzy control
• Sliding mode control

3
Basic notions in control theory

• The open-loop system (process, plant)
• x f(x,u,t) state equation
• y g(x,u,t) output equation
• where
• x (x1, x2 , , xn) is a state vector
• y (y1, y2 , , yn) is an output vector
• u (u1, u2 , , un) is an input vector
• f, g are nonlinear vector-valued functions

4
Linear time invariant systems

• x Ax Bu state equation
• y Cx output equationwhere
• A(nn), B(nm), C(rn) are system matrices
• For the discrete case
• x(k 1) Ax(k) Bu(k)
• y(k) Cx(k)

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A linear system
D

B
?
C

x(t)
x(t)
u(t)

y(t)

A
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Linearized nonlinear system I

• x f(x,u,t)can be linearized around an
  equilibrium point xd and its corresponding ud.
• x Ad x Bd uwhere

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Linearized nonlinear system II

• Then the behavior of the nonlinear system in the
  neighborhood of (xd, ud) can be analyzed by
  studying the behavior of the linear system
• In particular, if the linear system is
  asymptotically stable in a region ? around (xd,
  ud), so is the nonlinear system.

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Stabilization nonlinear I

• Find a control law u h(x, t)such that the
  closed-loop system x f(x, g(x, t), t)is
  asymptotically stable in a region ? around some
  desired (setpoint) xd.
• That is, starting anywhere in ? we will have x
  xd ? 0 as t ? ?

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Stabilization nonlinear II

• If ? is small then the stabilization problem can
  be solved by linearization of the nonlinear
  system around xd.
• If ? is large then linearization will not work.

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Stabilization linear I

• Find a control law u Kxsuch that the
  closed-loop system x Ax BKxis
  asymptotically stable in a region ? around some
  desired (setpoint) xd.
• That is, starting anywhere in ? we will have x
  xd ? 0 as t ? ?

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Stabilization linear II

• For linear systems, if the system is
  asymptotically stable in ?, then it is
  asymtotically stable in the large.

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P, PD-controllers

• P-controllers
• u KP e, where e xd x
• find a KP such that e ? 0 as t ? ?
• PD-controllers
• u KP e KD e
• find a Kp and KD such that e ? 0 as t ? ?

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PID-controllers

• PID-controller
• u KP e KD e KI
• find a Kp, KD and KI such that e ? 0 as t ? ?

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Fuzzy models

• Pure fuzzy models
• Takagi-Sugeno models
• Fuzzy approximation of conventional nonlinear
  models

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Fuzzy controllers

• Model based
• On pure fuzzy models
• On TS-models
• On fuzzy approximations
• On nonlinear models
• Heuristic controllers
• Fuzzy PID
• Fuzzy controllers for ill-structured systems

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Types of nonlinearities I

• Nonlinearities due to simple analytic functions
  such as powers, sinusoids, exponentials of a
  single state variable or products of state
  variables
• Functions that possess convergent
  Taylor-expansions at all points and thus can be
  approximated by linear expressions in the
  neighborhood of any desired xd.

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Types of nonlinearities II

• Piecewise linear approximations which consist of
  a stet of linear functions describing the overall
  system in different regions of the state space.
• The dynamical equations become linear in any
  particular region and the controllers for
  different regions can be joined at the boundaries.

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Saturation, dead zone, relay
y
saturation
u
dead zone
y
ideal relay
u
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Relay with hysteresis
y
u
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Pure fuzzy models
e
xd
e
p

• x is the water level
• A pump p
• A valve v with a disturbance so that water can
  flow out of tank
• xd desired water level
• Error e x xd
• Input u (the amount of water pumped in or out

v
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Control problem

• Keep error as close to zero at all times, without
  knowing the disturbance.
• I.e., find a u such that e(k) ? 0, as k ? ?, for
  each initial state.

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The linguistic model I

• Linguistic values for e
• e is NBe x xd lt 0 and big
• e is NSe x xd lt 0 and small
• e is ZEe x xd 0
• e is PSe x xd gt 0 and small
• e is PBe x xd gt 0 and big

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The linguistic model II

• Linguistic values for u
• u is NBu big amount pumped out
• u is NSu small amount pumped out
• u is ZEu no action
• u is PSu small amount pumped in
• u is PBu big amount pumped in

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The linguistic model III
e(k1)
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Rules

• Each matrix element is a linguistic rule of the
  form
• If e(k) is NBe and u(k) is PBu then e(k1) is ZEe

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General rules

• If e(k) is LEi and u(k) is LUi then e(k1) is LEi
  where LEi ? NBe, NSe, ZEe, PSe, PBeand LUi ?
  NBu, NSu, ZEu, PSu, PBu

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The linguistic model

• The fuzzy sets representing the linguistic values
  of e and u

NB
NS
ZE
PS
PB
1
0
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A linguistic rule

• A linguistic relation on e ? u ? e
• The set of all linguistic rules is the union of
  the fuzzy relations Ri presented above.

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Computing with a fuzzy model I

• Given input values for e and u, say LEin and LUin
  at k the fuzzy input is
• Given the fuzzy relation R corresponding to all
  rules and LEin?LUin , the output at k1 is

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Computing with a fuzzy model II

• Where LEout is the value of error at k1.
• Thus the open-loop system is

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The fuzzy controller

• The fuzzy controller is of the formwhere K is
  a fuzzy relation on e ? u.
• The closed-loop system then is

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The design problem

• For a desired LEdes(k1) find a K such
  thatfor each initial LE

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Example

• For our example suppose LEdes(k1) ZEe
• ZEe (k1) (NBe(k) ? ? ) o R
• ZEe (k1) (NSe(k) ? ? ) o R
• ZEe (k1) (ZEe(k) ? ? ) o R
• ZEe (k1) (PSe(k) ? ? ) o R
• ZEe (k1) (PBe(k) ? ? ) o R

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The fuzzy model for ZEe (k1)
e(k1)
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The question marks

• Thus from
• ZEe (k1) (NBe(k) ? PBu(k) ) o R
• ZEe (k1) (NSe(k) ? PSu(k) ) o R
• ZEe (k1) (ZEe(k) ? ZEu(k) ) o R
• ZEe (k1) (PSe(k) ? NSu(k) ) o R
• ZEe (k1) (PBe(k) ? NBu(k) ) o R

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Conditions on K

• Therefore, K must be such that for each K
• NBe(k) o K PBu(k)
• NSe(k) o K PSu(k)
• ZEe (k) o K ZEu(k)
• PSe (k) o K NSu(k)
• PBe (k) o K NBu(k)

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The controller I

• We have to find ? in ZEe(k1) (NBe(k) ? ? ) o
  R
• But from the model we have that ZEe(k1)
  (NBe(k) ? PBu(k)) o R
• Since ? LE(k) o K PBu(k)
• We have that K must be such that LE(k) o K
  PBu(k)

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The controller II

• Thus, the fuzzy controller is the fuzzy relation
  K on e ? u which corresponds to the set of rules
• If e(k) is NBe then u(k) is PBu
• If e(k) is NSe then u(k) is PSu
• If e(k) is ZEe then u(k) is ZEu
• If e(k) is PSe then u(k) is NSu
• If e(k) is PBe then u(k) is NBu

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Stability of closed-loop systems I

• The closed-loop system
• can be rewritten as
• where

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Stability of closed-loop systems II

• Let LEdes be the desired error
• Theorem the closed-loop system is stable for any
  initial state iff for any n ? N

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Idea of the proof

• Hence
• Hence
• If and are normal then
  and
• Therefore

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Takagi-Sugeno fuzzy models

• A single rule Ri Ri if x1 is A1i and and
  xn is Ani then x Ai x Bi u where A1i is
  the linguistic value of the state variable x1 in
  the i-the rule.

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Computing a single rule I

• For a given input (x10, x20, , xn0) and (u10,
  u20, , un0)
• Step 1 Compute

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Computing a single rule II

• Step 2 Compute
• Step 3 Compute

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Computing a all rules

• For a given input (x10, x20, , xn0) and (u10,
  u20, , un0)
• Repeat Step 1 - 3 for each Ri
• Step 4 Compute the global output

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The global open-loop system

• The global open-loop system is
• Problem 1 How to build such a model?
• Identification
• fuzzy approximation of existing nonlinear model
• Problem 2 Asymptotically stable?

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Asymptotically stable

• Theorem The global open-loop system is
  asymptotically stable in the large if there
  exists a common positive definite matrix P such
  that

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The fuzzy controller for T-S

• The (local) controller for each rule is uiKix
• Since Ai x Bi u is a linear system, Ki can be
  designed via a conventional linear control
  technique (e.g., pole placement)
• Thus, the controller for each rule is Ri if
  x1 is A1i and and xn is Ani then uj Kj x

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The global controller

• The global controller is then given as
• where mj(x) is computed in exactly the same
  matter as for the open-loop system
• TS-control is a weighted sum of (local affine)
  controllers

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The global closed-loop system
(GC-L)
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Stability I

• When is (GC-L) asymptotically stable in the
  large?
• Let Aij Ai Bi Kj, i 1,2, j1,2,
• Then

(GC-L)
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Stability II

• Theorem (GC-L) is asymptotically stable in the
  large iff there exists a common positive definite
  P such that

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Controller design

• Find such Kjs (j 1, 2, ) such that there
  exists a common definite P such that

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Fuzzy approximation

• Fuzzy approximation of a nonlinear model
• Consider a nonlinear system
• where f is a vector-valued nonlinear function

(1)
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Local linearization

• When (1) is linearized around xd we obtain
• where xd ?f/?x at x xd and x f is an
  approximation error.

(2)
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Controller for linearized system

• Since (2) is linear, the controller becomes
• where Kd is a control matrix stabilizing (2)
  around xd f (xd) is an estimate of f(xd)

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Closed-loop system

• The closed-loop system is then given as
• where
• Observe here that xd is an equilibrium point when

(3)
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Asymptotically stable

• (3) is asymptotically stable around xd if Ad
  Kd is a Hurwitz matrix(all eigenvalues of Ad
  Kd have a negative real part, Relilt0).
• The original nonlinear system is stable around xd
  if the linearized one is.

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Fuzzy approximation

• The fuzzy approximation around xd
• Ri if x1d is A1i and and xnd is Ani then
  x fj Aj(x - xd) u
• The global open-loop system is

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The fuzzy controller

• RiC if x1d is A1j and and xnd is Anj then
  uj Kj (x - xd) - f (xd)
• The closed-loop system (Ri, RiC) then is
• D(x, xd ) compensation for errors of Ai and
  KiDi f (xd) - f (xd)

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Local stability (around xd)

• Take into account the robustness ofAi and
  Kiwith respect to D(x, xd ) and Di.

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Global closed-loop system

• Global closed-loop system (far from xd)
• Ri if x1d is A1i and and xnd is Ani then
  x fj Aj(x - xd) u
• RiC if x1d is A1j and and xnd is Anj then
  uj Kj (x - xd) - f (xd)
• The global closed-loop system

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Heuristic fuzzy controllers
x
e
xd
e
De
De
De
De
t
0
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Time map of output x
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The domains of e and De

• How to choose the domains of e and De?
• Changing the domain of e affects the overshoot
  and undershoot characteristics of the controller
• Changing the domain of De affects rise time and
  settling time
• Changing the domain of e and De affects the
  system trajectory

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Domain of De is too small
Demax
e
emax
emin
Demin
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Domain of e is too small
Demax
e
emax
emin
Demin
68
Domain of De and e are OK
Demax
e
emax
emin
Demin
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Membership distribution
u
e
Equally spaced terms
Wide terms around zero
Narrow terms around zero
Little overshoot slow approach
Much overshoot fast approach
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Undershoot overshoot
x
xd
t
0
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Overlap of membership functions
e
50 overlap, Normal width
66 overlap, Double width
No overlap, Half width
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Undershoot overshoot
x
xd
t
0
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Identifying rules
Improve rise time
Improve overshoot
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Sliding mode fuzzy control

• Given a nonlinear system model including
• Model uncertainties
• Parameter fluctuations
• Disturbances
• Find a robust control law that makes the system
  asymptotically stable.

75
Figure switching line
switching line
De
K
K
e
sgt0
slt0
slt0
s e le
s0
sgt0
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Sliding mode introduction

• Drawback of sliding mode control
• Leads to chattering of the control variables
• High stress of mechanical and electrical elements
• Therefore, introduction of a boundary layer near
  the switching line

77
De
boundary layer
K
K
sgt0
e
slt0
slt0
s e le
s0
sgt0
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Boundary layer and fuzzy
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Fuzzy sliding mode

• Fuzzy sliding mode control is an extension of
  sliding mode with boundary layer
• Control law