Concept of adaptive control.pdf

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1 Concept of adaptive control

1.1 Overview of adaptive controlKey words: Adaptive controller, Adaptive observer, Adaptive identifier

Let us consider a system represented by a state space model (1) and squareevaluation function(2).

x(t) = Ax(t) + Bu(t), x(0) = x0 (1)

y = Cx(t)

x(t) : n × 1, u(t) : r × 1

J =

∞

0

yT (t)Qy(t) + uT (t)Ru(t)

dt (2)

We can obatin u(t) that minimize J by (3),

u(t) = −Kx(t), (3)

where Q and R are weight matrix, and K is feedback matrix (r × n). K is asolution of Riccati equation, K = f (A,B,C,Q,R). We call K as SAS (Stability

Argumentation System) gain.If we can estimate matrix A and B, above K realizes SAS. Of course, wehave to know all sate valiables exactly. However, actual system always containsmodel error. Additionally, we cannot always observe state valiables. In suchcase, we usually apply observer and estimate system status, x(t). To constructa feedback system u(t) = −K x(t), we have to know exact dynamics of a systemin advance.

Adaptive Control System is a system that arranges its controller dynam-ically depends on dynamics of a system to be controlled.

Adaptive Observer is an observer that estimate parameters and statevaliables in a system only by inputs and outputs.Adaptive Identifier is a system that identifies parameters of a system.

1.2 Formally known system

1.2.1 High Gain System

Here, we assume y(s) = yn(s) + yu(s), where

yn(s) = GP (s)1+GC(s)H (s)GP (s)n(s) um(s) = 0yu(s) =

GC(s)GP (s)GM (s)1+GC(s)H (s)GP (s)

um(s) n(s) = 0.

When we choose GC (s) and H (s) so that |GC (s)H (s)| >> 1, yn(s)n(s)

1GC(s)H (s)

→ 0y (s) GM(s) ,





Reference Model Transfer function GM (s).Unknown plant (system) GP (s): Its parameters are unknown.

Disturbance n(s)System output y(s)System input u(s)

Our goal is obtaining GC (s) and H (s) so that y(s) follows output of thereference model even if any disturbance exists.

+ +

+-

um(s)GM(s) GC(s) GP(s)

y(s)

n(s)

H(s)

Figure 3: Assumed system

Here, we introduce typical methods. We will discuss them precisely in fol-lowing sections.

1.3.1 Model Reference Adaptive Control System (MRACS)

A MRACS system arranging controller (Fig. 4) so that output of combined sys-tem (controller and unknown plant) matches that of refenrece model. Monopoliet al. have proposed a method that calculates feasible control parameters onlyby input and output signals of the plant.

Controller PlantInput Output

Combined system

Adaptation mechanism

Reference model

+

- Input

Plant

Identificaion mechanism

Identification model+

-

Output

Output

error

Estimated

output

MARCS control system MARCS identifier

Figure 4: Model Reference Adaptive Control System; MRACS



1.3.2 Self Tuning Regulator (STR)

STR is online-tuning feedback controller. At first, STR assume parameters of unknown plant arbitrarily. Then it construct feedback controller based on givenevaluation function. Next, it identifies parameters according to control inputsand outputs of the plant. After that, it repeats tuning procedure.

Controller PlantControl input

Identifier

Estimated

parametersCalculation ofcontrol parameters

Input Output

Figure 5: Self Tuning Regulator; STR

1.4 Expression of unknown plants

Level of ”unknown”.

White box Structure: known (linear differential equation), parameters andstate valiables: known and observable.

Gray box partially known, partially unknown.

Black box Only inputs and outputs are observable.

Expressions of systems

• Lumped Parameter System

– Linear, time-invaliant

x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

– Linear, time-valiant

x(t) = A(t)x(t) + B(t)u(t)y(t) = C (t)x(t) + D(t)u(t)

– Non-linear, time-invaliant x(t) = f (x(t), u(t))

y(t) = g(x(t), u(t))

– Non-linear, time-valiant

x(t) = f (x(t), u(t), t)y(t) = g(x(t), u(t), t)

• Distributed parameter system

• Lag-time system



2 Time Series model

2.1 ExpressionWe consider a system represented by a state space model as follows.

x(t) = Ax(t) + Bu(t) , x(0) = x0

y(t) = Cx(t) (1)

We can obtain the solution of (1) as (2), and the transfer function of the sys-tem is represented by (3), where s and p means Laplace operator and differentialoperator (= d

dt) respectively.

y(t) = C eAtx0 + C

t0

eA(t−τ )Bu(τ )dτ (2)

Y (s) = G(s)U (s) or y(t) = G( p)u(t) (3)

Here, we assume system noise v(t) and measurement noise w(t). x(t) = Ax(t) + Bu(t) + Dv(t)y(t) = Cx(t) + w(t)

(4)

We assume v(t) and w(t) are independent white noises that have followingproperties.

E [v(t)] = E [w(t)] = 0

E [v(t)wT (t)] = 0

E [v(t)vT (t + τ )] = Qδ (τ )

E [w(t)wT (t + τ )] = Rδ (τ )

E [·] represents an expectation, and δ (t) means Dirac’s delta function. Q andR denote simmetory positive definite matrix and positive semidefinite matrix,respectively.

2.2 Forward and Backward Shift Operators

Here, we consider a discretized system whose sampling time is T .

x(k + 1) = F (T ) x(k) + H (T ) u(k)

y(k) = C (T ) x(k) + D(T ) u(k)

(5)

, where

F (T ) = eAT , H (T ) =

T

0

eAλdλ

B.

If we assume T << 1,



Here, we introduce forwardand backward operators. Forward and backwardoperators q and q −1 are defined as follows.

qy(k) = y(k + 1) (6)

q −1y(k) =

y(k − 1) (k ≥ 1)0 (k = 0)

(7)

q iy(k) = y(k + i) (8)

q −1y(k) =

y(k − i) (k ≥ i)0 (0 ≤ k < i)

(9)

Then we consider stochastic noise-added model represented by following eu-qation.

y(k) = B(q −1)

A(q −1)u(k) + n(k) (10)

A(q −1) = 1 + a1q −1 + · · · + anq −n

B(q −1) = b0 + b1q −1 + · · · + bmq −m

n(k) = H (q −1)m(k) (11)

, where m(k) represents white noise. Then H (q −1) is called as shaping filter.

H (q −1) = D(q −1)

C (q −1) (12)

C (q −1) = 1 + c1q −1 + · · ·+ c pq − p

D(q −1

) = 1 + d1q −1

+ · · · + drq −

r

Therefore, output y (k) is represented by following equation. This system iscalled ”time series model”.

y(k) = B(q −1)

A(q −1)u(k) +

D(q −1)

C (q −1)m(k) (13)

2.3 ARMA model

When D(q −1) = 1, n(k) becomes

n(k) = −c1n(k − 1)− c2n(k − 2) · · · − c pn(k − p) + m(k). (14)

It is called AR model (Auto-Regressive model).

When C (q 1

) = 1, n(k) becomes



It is called MA model (Moving Average model).Generally, n(k) is

n(k) = −c1n(k

−1)

−c2n(k

−2)

· · ·−c pn(k

− p)+m(k)+d1m(k

−1)+

· · ·+drm(k

−r).

(16)It is called ARMA model (Auto-Regressive Moving Average Model).

2.4 Linear Diophantine equation

We consider following system, where A( p) and B ( p) are coprime polynomials.

y(t) = B( p)

A( p)u(t) (17)

A( p) = pn + an−1 pn−1 + · · · + a1 p + a0

B( p) = bm pm + · · · + b1 p + b0

When Q( p) (order n) and D( p) (order n−m) are monic stability polynomial,we can find a uniq pair of R( p) and H ( p) that satisfy (18).

R( p)A( p) + H ( p)B( p) = Q( p)(bmA( p) − D( p)B( p)) (18)

, where

R( p) = rn−1 pn−1 + · · · + r1 p + r0

H ( p) = hn−1 pn−1 + · · · + h1 p + h0.

Equation (18) is called Diophantine equation.By multiplying B−1( p)y(t) to (18) and introducing (17), we obtain (19).

D( p)y(t) = bmu(t) −

R( p)

Q( p)u(t) −

H ( p)

Q( p)y(t) (19)

This equation gives non-minimal realization of the system (17). Figure 1 illus-trates the block diagram of a non-minimal realization.

Arrange equation (18) as

(bmQ( p) − R( p))A( p) = (H ( p) + Q( p)D( p))B( p), (20)

then we set E ( p) and F ( p) as

E ( p)B( p) = bmQ( p) − R( p) (21)

F ( p) = −

H ( p), (22)equation (20 becomes

E ( p)A( p)B( p) = (Q( p)D( p) − F ( p))B( p). (23)

Therefore, we obtain Egardt’s identity (24).

Q(p)D(p) = A(p)E(p) + F (p) (24)



u(t)

y(t)

y(t)

bm

H(p)Q(p)

1

D(p)

R(p)

Q(p)

B(p)

A(p)

+

−−

PLANT

Non-minimal realization

Figure 1: Block diagram of a non-minimal realization



3 Deterministic identifier

3.1 Adaptive Identifier/ControllerAdaptive Control System is a system that arranges its controller dynamicallydepends on dynamics of a system to be controlled.

Adaptive Observer is an observer that estimate parameters and statevaliables in a system only by inputs and outputs.

Adaptive Identifier is a system that identifies parameters of a system.

Controller Plant

Disturbance

Performanceestimation

Adaptivemechanism

Desired performance

System input System outputControl input

+ -

Plant

Rerefencemodel

System output

+ -

Identifier

ArrangeError estimation

Test drive

Observedinput

Parameters

Adaptive controller Adaptive identifier

Figure 1: Schematic view of adaptive control system

3.2 Strictly Positive Real

A rational function f (s) is Positive Real when

• for real s, f (s) is real, and

• for all s : Res > 0, Ref (s) ≥ 0.

Additionally, when a real number λ > 0 that makes f (s − λ) positive realexists, f (s) is Strictly Positive Real (SPR).

Examples

• f (s) = k

s (k > 0) is positive real.

• f (s) = ksa2s

2+a1s+1 (a1, a2, k > 0) is SPR.

3.3 Positive Real System

Let us consider following system;x(t) = Ax(t) + Bu(t)

(1)



, where u(t) and y(t) is m × 1, x(t) is n× 1.The transfer function F (s) of the system is F (s) = C (sI −A)−1B+D. This

system (F (s)) is SPR if positive definite simmetric matrices P,Q,W exist thatsatisfies following conditions.AT P + PA = −Q

BT P = C D +DT = W T W

(2)

If Q is semi-positive definit simmetric, F (s) becomes positive real. D = 0also satisfies the conditions. When D = 0, the conditions becomes simple asfollows.

AT

P + PA = −QBT P = C

(3)

Here, we consider discrete time system denoted as follows.

x(k + 1) = Fx(k) + Hu(k)y(k) = Cx(k) + Du(k)

(4)

The transfer function of the system is G(z) = C (zI − F )−1H + D. Thissystem is SPR if positive definite simmetric matrices P,Q exist that satisfies

following conditions. F T PF − P = −Q

H T PF = C D +DT = H T PH

(5)

If Q is semi-positive definit simmetric, G(z) becomes positive real. Pleasenotice that D = 0 does not satisfies SPR condition in this case because H T PH

must be positive.

3.4 Deterministic identification; error modelAt the beginning, we introduce following error model.

e(t) = Ae(t) + bf (t)e1(t) = cT e(t) + df (t)f (t) = φT (t)ξ (t)

(6)

, where e(t): n dimensional state error vector, f (t): scalar control input, e1(t):observable identification error, φ(t): parameter error.

The parameter error is represented as

φ(t) = θ(t) − θ

, where θ denotes unknown parameters and θ(t) represents adjustable identifca-tion parameters.

By thedeterministic identification, we want to obtain φ(t)(θ(t)) that realizes



ξ(t)

ξ(t)

φT(t)

φT(t)

cT

W(p)e1(t)

e1(t)

f(t)

f(t)

A

d

++ + + b

Figure 2: Error model of the deterministic identification.

The transfer function of the error model and observable error are denotedas follows.

W (s) = cT (sI − A)−1b + d

e1(t) = W ( p)f (t) = W ( p)φT (t)ξ (t) (7)

3.5 Deterministic identifier; continuous time, Narendra’smethod

For the identification, we apply feedback gain of squared ξ (t) (Narendra’s method).In this section, we suppose d = 0 to simplify the problem.

e(t) = Ae(t) + bf (t)e1(t) = cT e(t)f (t) = φT (t)ξ (t) − αξ T (t)Λξ (t)e1(t)

(8)

, where Λ = ΛT

> 0,α > 0.The transfer function and output of the feedback system become

W (s) = cT (sI −A)−1b (9)

e1(t) = W ( p)f (t) = W ( p)φT (t)ξ (t)

1 + αW ( p)ξ T (t)Λξ (t) (10)

Here we assume that W ( p) is SPR, then

φ(t) = ˙θ(t) = −αΓξ (t)e1(t) (11)

realises e1(t) → 0 for t →∞. Notice Γ = ΓT > 0.We can denote (11) in the following form.

θ(t) = −Γ

t

0

ξ (τ )e1(τ )dτ − Λξ (t)e1(t) (12)



ξ(t) e1(t)f(t)

φT(t) W(p)

αξT(t)Λξ(t)

+

-

Figure 3: Narendra’s adaptive model.

3.6 Deterministic identifier; discrete time

Let us consider identification method in discrete time systems. At first, weintroduce Narendra’s method in the discrete time. Let us formulate a systemas follows.

e(k + 1) = F (e(k) + hf (k)e1(k) = cT e(k) + df (k)f (k) = φT (k)ξ (k) − αξ T (k)Λξ (k)e1(k)

(13)

where Λ = ΛT > 0,α > 0.

The transfer function and observable output of the system are

W (z) = cT (zI − F )−1h + d

e1(k) = W (q )f (k) (14)

Here we assume W (q ) is SPR, then (15) realizes deterministic identification.

θ(k + 1) = θ(k)− αΛξ (k)e1(k) (15)



4 System identification; Stochastic identifier

4.1 Least square estimationWe considet following system;

y(k) = B(q −1)

A(q −1)u(k) + n(k) (1)

Here, we assume m=n (dimension of A and B are equal).Then, y(k) is represented by following equations.

y(k) = −a1y(k−1)−a2y(k−2)−· · ·−any(k−n)+b0u(k)+b1u(k−1)+· · ·+bnu(k−n)+w(k)

(2)w(k) = A(q −1)n(k) = n(k) + a1n(k − 1) + · · · + ann(k − n) (3)

Our aim is estimating system parameters ai and bi by observed u(k) andy(k) values (k = 1, 2, · · ·, N ).

At first, we define θ and z(k).

θT = [ −a1 −a2 · · · −an b0 b1 · · · bn ]zT (k) = [ y(k − 1) y(k − 2) · · · y(k − n) u(k) u(k − 1) · · · u(k − n) ]

(4)

Then, equation(2) becomes

y(k) = zT (k)θ + w(k). (5)

We also define vectors y , w and Z as follows.

yT = [ y(1) y(2) · · · y(N ) ]wT = [ w(1) w(2) · · · w(N ) ]Z T = [ z(1) z(2) · · · z(N ) ]

(6)

Then, output y is derived by (7).

y = Zθ + w (7)

Because w(k) represents noise, so we want to minimize J .

J =N k=1

w2(k) = wT w = (y − Zθ)T (y − Zθ) (8)

∂J

∂θ is

∂J

∂θ = −

2Z T

(y − Zθ). (9)

For ∂J ∂θ

= 0, estimated θ (θ) is

θ = (Z T Z )−1Z T y (10)

=

N

z(k)zT (k)

−1

N z(k)y(k).



4.1.1 Consider estimated θ

Let us consider estimation error. When we define θ = θ − θ,

θ = (Z T Z )−

1Z T w (11)

If w(k) is a white noise,

E [w(k)] = 0, E [w(i)wT ( j)] = Qδ ij. (12)

So, if Z is independent from w , E [θ] = 0.

4.1.2 Online identification

We have already observed N data. Now, we obtain N + 1 th data. Let usassume yN +1, Z N +1, P (N ) and q (N ) as follows.

yN +1 =

yN

· · ·

y(N + 1)

, Z N +1 =

Z N

· · ·

zT (N + 1)

(13)

P (N ) = [Z T N Z N ]−1, q (N ) = Z T N yN (14)

Then,

P −1

(N + 1) = P −1

(N ) + z(N + 1)zT

(N + 1), (15)q (N + 1) = q (N ) + z(N + 1)y(N + 1). (16)

By the matrix inversion lemma,

P (N + 1) = P (N ) − P (N ) z(N + 1)zT (N + 1)

1 + zT (N + 1)P (N )z(N + 1)P (N ). (17)

Then, by equation (10),

θ(N + 1) = P (N + 1)q (N + 1) = (18)P (N ) − P (N )

z(N + 1)zT (N + 1)

1 + zT (N + 1)P (N )z(N + 1)P (N )

×[Z T N yN + z(N + 1)y(N + 1)]. (19)

Therefore, we can calculate parameters successively. (N → k)

θ(k + 1) = θ(k) + K (k + 1)[y(k + 1) − zT (k + 1)θ(k)] (20)

K (k + 1) = P (k)z(k + 1)

1 + zT (k + 1)P (k)z(k + 1)

P (k + 1) = [I −K (k + 1)zT (k + 1)]P (k)

Additionally, by K (N + 1) = P (N + 1)z(N + 1),

θ(k + 1) = θ(k) + P (k + 1)z(k + 1)[y(k + 1) − zT (k + 1)θ(k)] (21)

P (k + 1) = P (k) − P (k) z(k + 1)zT (k + 1)

P (k)



4.2 Weighted least square estimation

We can add weight matrix Q to the evaluation function J .

J = wT Qw = (y − Zθ)T Q(y − Zθ) (22)

In this case, estimated θ (θ) becomes

θ = (Z T QZ )−1Z T Qy. (23)

4.3 Extended least square method

When w(k) is not the white noise, former methods cannot cancel the bias er-ror.Here, we introduce extended least square method to estimate not only θ butalso noise properties.

We considet following system;

y(k) = B(q −1)

A(q −1)u(k) + n(k) (24)

Then,

A(q −1)y(k) = B(q −1)u(k) + w(k) (25)w(k) = A(q −1)n(k),

We re-define w(k) by equation(26), where m(k) indicates the white noise,and C (q −1) represents characteristics of the noise.

w(k) = 1

C (q −1)m(k). (26)

C (q −1) = 1 +

p

i=1

ciq −i (27)

Then,

w(k) +

p

i=1

ciw(k − i) = m(k) (28)

y(k) = −

n

i=1

aiy(k − i) +m

i=0

biu(k − i) −

p

i=1

ciw(k − i) + m(k) (29)

Here, we define vectors θ,zm(k),c,w(k) as follows.

θT = [ −a1 −a2 · · · −an b0 b1 · · · bm ]zT m(k) = [ y(k − 1) y(k − 2) · · · y(k − n) u(k) u(k − 1) · · · u(k − m) ]cT = [ −c1 −c2 · · · −c p ]

wT (k) = [ w(k − 1) w(k − 2) · · · w(k − p) ]



Then,y(k) = zT m(k)θ + wT (k)c + m(k). (31)

Additionally, we define Z ,W ,m,y that combiles observed data k = 1, 2,· · ·

, N .

Z T = [ zm(1) zm(2) · · · zm(N ) ]W T = [ w(1) w(2) · · · w(N ) ]mT = [ m(1) m(2) · · · m(N ) ]yT = [ y(1) y(2) · · · y(N ) ]

(32)

Then,y = Zθ + Wc + m. (33)

Finally, we transform equation (33) into

m = y − Ωφ, (34)

where

Ω = [Z ...W ], φ =

θ

· · ·

c

. (35)

We introduce Ω, called extended matrix, so that we can estimate both θ and

c simultaneously.

φ = (ΩT Ω)−1ΩT y (36)

θ(k) = (Z T (k)Z (k))−1[Z T (k)y(k) − Z T (k)W (k)c(k)] (37)

c(k) = (W T (k)W (k))−1W T (k)[y(k) − Z (k)θ(k)] (38)

4.3.1 On-line estimation

Equation (31) gives w(k) = y(k) − zT

m(k)θ(k). Then,

φ(k + 1) = φ(k) + K (k + 1)[y(k + 1) − ΩT (k + 1)φ(k)] (39)

K (k + 1) = P (k)ΩT (k + 1)

I + Ω(k + 1)P (k)Ω(k + 1)

P (k + 1) = [I − K (k + 1)ΩT (k + 1)]P (k).



5 MRACS

5.1 Assumptions and system expressions

In this section, we introduce Model Reference Adaptive Control System (MRACS).To simplify the problem, we only consider a target plant that has followingproverties.

• Time-invaliant linear system.

• Without noise (only unknown parameters).

• Orders of a transfer function are known.

• We know bm is positive or not.

• An invert system is stable.

• Control input is unlimited.

Let us consider a system represented by (1).

y(t) = B( p)

A( p)u(t) (1)

where

A( p) = pnan−1 p

n−1· · ·a1 p + a0

B( p) = bm pmbm−1 p

m−1· · · b1 p + b0

5.2 MRACS in continuous time

As mentioned in section 1, MRACS has an architecture denoted in Fig. 1.

Controller PlantInput Output

Combined system

Adaptation mechanism

Reference model

+-

Figure 1: Model Reference Adaptive Control System; MRACS

Our goal is realizing control system so that a combined system works as thesame as a reference mode. For simplify the problem, we assume single inputsingle output system denoted in (2),

x(t) = APx(t) + bPu(t)

(2)



and we set a model system and tracking error as (3) and (4), respectively.In (3), AM ( p) and BM ( p) are order p and q , respectively.

ym(t) = BM ( p)

AM ( p)um(t) (3)

e1(t) = ym(t) − y(t) (4)

We have to notice following condition;

p − q ≥ n∗ , where n∗ = n − m.

In order to realize an adaptive system without differential signals (= futurevalues of a system), we have to satisfy the condition.

We are going to construct MRACS according to following procedure.

1. Construct non-minimal realization of an unknown plant.

2. Design model-matching controller to zero the output error.

3. Calculate control input.

5.3 non-minimal realization by Diophantine equation

We again consdier Diophantine euqation. For proper A( p) and B( p), by intro-ducing monic stability polinominals Q( p) (order n) and D( p) (order n − m), wecan find polinominals R( p) and H ( p) that satisfy (5).

R( p)A( p) + H ( p)B( p) = Q( p)(bmA( p) − D( p)B( p)) (5)

, where

R( p) = rn−1 p

n−1 + rn−2 pn−2 + · · · + r1 p + r0

H ( p) = hn−1 pn−1 + hn−2 p

n−2 + · · · + h1 p + h0

Equation (6), obtained from (5), represents non-minimal realization of thesystem.

y(t) = 1

D( p)

bmu(t) −

R( p)

Q( p)u(t) −

H ( p)

Q( p)y(t)

(6)

Next, we consider control input that converges the tracking error to zero.Based on (4),

D( p)e1(t) = D( p)ym(t) − D( p)y(t) (7)

= D( p)ym(t) − bmu(t) + R( p)Q( p)

u(t) + H ( p)Q( p)

y(t)

We can D( p)e1(t) = 0 by following u(t).

u(t) = 1

b

D( p)ym(t) +

R( p)

Q(p)u(t) +

H ( p)

Q(p)y(t)

(8)



5.4 Design of adaptive system

As (8), we want to know bm,R( p), and H ( p) in order to zero the tracking error.

So, we denote estimated parameters as bm(t), R( p, t), and H ( p, t). Then,

u(t) = 1

bm(t)

yc(t) +

R( p, t)

Q( p) u(t) +

H ( p, t)

Q( p) y(t)

(9)

, where yc(t) = D( p)BM ( p)AM ( p)

um(t) (model output).

Next, we define w(t) called state valiable filer such as

wT

(t) = 1

Q( p)u(t),

p

Q( p)u(t),· · ·

,

pn−1

Q( p)u(t),

1

Q( p)y(t),

p

Q( p)y(t),· · ·

,

pn−1

Q( p)u(t)

(10), and unknown parameter vector θ such as

θT = [r0, r1, · · · , rn−1, h0, h1, · · · , hn−1] . (11)

Then, we can describe u(t) as

u(t) = 1

bmyc(t) + θT w(t) . (12)

We can represent the non-minimal realization of (12) and the tracking errorby (13) and (14) respectively.

y(t) = 1

D( p)

bmu(t) − θT w(t)

(13)

e1(t) = 1

D( p) −bmu(t) + θT w(t) + yc(t)

(14)

By introducing estimated parameters into (12),

u(t) = 1

bm(t)

yc(t) + θT (t)w(t)

(15)

e1(t) = 1

D( p)

−bm(t)u(t) + θT (t)w(t) + yc(t)

(16)

5.5 Adjustment of control parameters

Here, we define argumented error 1(t) = e1(t) −

e1(t), then

1(t) = W ( p)φT (t)ξ (t)

,

W ( p) = 1D( p) ,

φT (t) = [bm(t) − bm,−(θT (t) − θT )],ξ T (t) = [u(t), wT (t)].

(17)



For example (we assume W ( p) is SPR),

˙bm(t) = −γ 0u(t)1(t) (γ 0 > 0)

˙θ(t) = Γw(t)1(t) (Γ = ΓT > 0)

When we apply this method to actual systems, we have to consider bmbecause u(t) →∞ for bm → 0.

5.6 Example

Let us construct MRACS for following plant and model.

Plant

˙x

(t) =

−a0x

(t) + b0u

(t)y(t) = x(t) (18)

Model

xm(t) = −am0xm(t) + bm0um(t) (am0 > 0)ym(t) = xm(t)

(19)

We can transform (18) as

y(t) = b0

p + q 0u(t).

Then, we obtain A( p) = p + a0 (n = 1)B( p) = b0 (m = 0) ,and

AM ( p) = p + am0

BM ( p) = bm0.

Next, we calculate control input u(t).

u(t) 1

bm0(t)

yc(t) + θT (t)w(t)

(20)

, where yc(t) = D( p) bm0

p+am0um(t).For non-minimal realization, we introduce Q( p) = p+ q 0 and D( p) = p+ d1,

and denotes estimated R( p, t) = r0(t), H ( p, t) = h0(t). Then the state variablefilter becomes

wT (t) =

1

p + q 0u(t),

1

p + q 0y(t)

.

Here, we set unknown vector θ(t) as

θT (t) = [r0(t), h0(t)].

Instead of 1(t), we simply introduce φ(t)

φT (t) = [bm0(t)− bm0, r0 − r0(t), h0 − h0(t)].

Because obviously W ( p) = 1D( p) = 1

p+d1is SPR, we can apply algorithms of

adaptive identifiers. By setting ξ T (t) = [u(t), w(t)], φ(t) = −Γξ (t)e1(t)(Γ =T



6 MRACS on discrete time systems

6.1 Formulation

In this section, we apply MRACS to discrete time systems. At first, we formulateplants and models by (1) and (2).

Plant : A(q −1)y(k) = q −dB(q −1)u(k) (1)

d ≥ 1 : T imedelay(q −dy(k) = y(k − d))A(q −1) = 1 + a1q −1 + · · ·+ anq −n

B(q −1) = b0 + b1q −1 + · · · + bmq −m

Model : AM (q −1)ym(k) = q −dBM (q −1)um(k) (2) AM (q −1) = 1 + am1q −1 + · · · + amnq −n

BM (q −1) = bm0 + bm1q −1 + · · ·+ bmmq −m

, where bm0 > 0, AM (q −1) is stability polynominal.We want e1(k) = ym(k) − y(k) → 0 for k → ∞.

6.2 Diophantine equation

Next, we introduce Diophantine equation (3) and non-minimal realization.

D(q −1) = A(q −1)R(q −1) + q −dH (q −1) (3)

, where D(q −1) = 1+d1q −1 + · · ·+dnq −n is n order monic stability polynominalthat we can set arbitrary. Then R(q −1) and H (q −1) always exist.

R(q −1) = 1 + r1q −1 + · · ·+ rd−1q −(d−1)

H (q −1) = h0 + h1q −1 + · · ·+ hn−1q −(n−1)

By multiplying (3) by y (k),

D(q −1)y(k) = A(q −1)R(q −1)y(k) + q −dH (q −1)y(k)

= q −dB(q −1)R(q −1)u(k) + H (q −1)y(k − d)

= B(q −1)R(q −1)u(k − d) + H (q −1)y(k − d).

Here, we introduce θ, set of unknown parameters, and ξ , input and outputdata

θT = [b0, b0r1 + b1, b0r2 + b1r1 + b2, · · · , bmrd−1, h0, h1, · · · , hn−1]

ξ T = [u(k), u(k − 1), · · · , u(k − (m + d − 1)), y(k), y(k − 1), · · · , y(k − (n− 1))]

, then we obtain (4) that denotes non-minimal realization.



By the way, let us assme that order n = 2, d = 2. The Diohpantine equation(3) becomes

1 + d1q −1

+ d2q −2

= (1 + a1q −1

+ a2q −2

)(1 + r1q −1

) + q −2

(h0 + h1q −1

).

By comparing coeficients, we can derive following equations.

r1 = d1 − a1h0 = d2 − a2 − a2d1 + a21h1 = a2(a1 − d1)

These equations mean that coeficients ri and hj depend on A(q −1) and D(q −1).Let us get back to the system error e1(k). By multiplying e1(k) by D(q −1)q −d,

D(q −1)e1(k + d) = D(q −1)ym(k + d) − B(q −1)R(q −1)u(k) − H (q −1)y(k)

= D(q −1)ym(k + d) − θT ξ (k) (5)

, and we separate b0 and u(k) as θT = [b0, θT ]ξ T (k) = [u(k), ξ T (k)]

.

Then,

D(q

−1

)e1

(k + d) = D(q

−1

)ym(k + d)−

b0

u(k)− ¯

θ

T

ξ (k) (6)For (6) → 0,

u(k) = 1

b0

D(q −1)ym(k + d) − θT ξ (k)

. (7)

6.3 Direct control

Unknown parameters in (6) are θ(b0(k) and ˆθ(k). We denotes y(k) that is cal-culated by means of expected values. Then, we define 1(k):

1(k) = D(q −1)(y(k) − y(k))= φT (k)ξ (k − d) (8)

where φT (k) = θ(k)−θ. This equation represents a case of deterministic identi-fier that has W ( p) = 1. So, we can apply algorithms of deterministic identifiers.

θ(k) = θ(k − 1)− Π(k − 1)ξ (k − d)1(k) (9)

Π(k) = 1

λ1

(k)Π(k − 1)−

λ2(k)Π(k − 1)ξ (k − d)ξ T (k − d)Π(k − 1)

λ1

(k) + λ2

(k)ξ T (k−

d)Π(k−

1)ξ (k−

d) (10)

, where 0 < λ1(k) ≤ 1, 0 ≤ λ2(k) ≤ λ, Π(0) = Π(0)T > 0. By removing

φ(k) (θ(k)) from 1(k),

1(k) = −D(q −1)y(k) + θT (k − 1)ξ (k − d)

1 + ξ T (k − d)Π(k − 1)ξ (k − d) . (11)



7 STR: Self-tuning regulator

7.1 Formulation

In this section, we introduce STR that considers stochastic noize added to plants.At first, we formulate a plant by (12).

Plant : A(q −1)y(k) = q −dB(q −1)u(k) + C (q −1)w(k) (12)

A(q −1) = 1 + a1q −1 + · · ·+ anq −n

B(q −1) = b0 + b1q −1 + · · ·+ bmq −m

C (q −1) = 1 + c1q −1 + · · ·+ cnq −n

where w(k) denotes white noize (average=0, distribution=σ2

). Known parame-ters are m, n, d, and unknown parameters are ai, bj , ck.We assume that B(q −1)and C (q −1) are stability polynominals.

In this section, our goal is minimizing J = E [(ym(k) − y(k))2] (minimizingdistribution).


Next, we consider Diophantine equation and non-minimal realization. Diophan-

tine equation is represented by (13).

C (q −1) = A(q −1)R(q −1) + q −dH (q −1) (13)

R(q −1) = 1 + r1q −1 + · · · rd−1q −(d−1)

H (q −1) = h0 + h1q −1 + · · ·hn−1q −(n−1)

By mutiplying (13) by y(k),

C (q −1)y(k) = A(q −1)R(q −1)y(k) + q −dH (q −1)y(k)

= B(q −1)R(q −1)u(k − d) + H (q −1)y(k − d) + C (q −1)R(q −1)w(k)(14)

Here, we define system error e1(k) = ym(k)− y(k),

C (q −1)e1(k) = C (q −1)ym(k) −C (q −1)y(k)

= C (q −1)ym(k) −B(q −1)R(q −1)u(k − d)−H (q −1)y(k − d)− C (q −1)R(q −1)w(k)(15)

. For (15)→ 0,

u(k) = 1

b0

C (q −1)ym(k + d)−H (q −1)y(k)−BR(q −1)u(k)

(16)

where BR(q −1) = B(q −1)R(q −1)− b0. Then, we obtain

e1(k) = −R(q−1)w(k)



7.3 Control

Same as the case of MRACS, we define θ and ξ as follows.

θT = [b0, b0r1 + b1, b0r2 + b1r1 + b2, · · · , bmrd−1, h0, h1, · · · , hn−1, c1, c2, · · · , cn]

ξ T = [u(k), u(k − 1), · · · , u(k − (m + d − 1)), y(k), y(k − 1), · · · , y(k − (n − 1)),

−ym(k + d − 1), · · · , −ym(k + d − n)]

Then,

u(k) = 1

b0

ym(k + d) −

θT ξ (k)

(17)

ym(k + d) = θT

ξ (k) (18)

where, θT = [b0, θT ], ξ T (k) = [u(k), ξ T (k)].

By φT (k) = θT − θT ,

1(k) = φT (k)ξ (k − d) = ym(k) − y(k)

Then, we can apply algorithms of identifiers.



6 MRACS on discrete time systems

6.1 Formulation

In this section, we apply MRACS to discrete time systems. At first, we formulateplants and models by (1) and (2).

Plant : A(q −1)y(k) = q −dB(q −1)u(k) (1)

d ≥ 1 : T imedelay(q −dy(k) = y(k − d))A(q −1) = 1 + a1q −1 + · · ·+ anq −n

B(q −1) = b0 + b1q −1 + · · · + bmq −m

Model : AM (q −1)ym(k) = q −dBM (q −1)um(k) (2) AM (q −1) = 1 + am1q −1 + · · · + amnq −n

BM (q −1) = bm0 + bm1q −1 + · · ·+ bmmq −m

, where bm0 > 0, AM (q −1) is stability polynominal.We want e1(k) = ym(k) − y(k) → 0 for k → ∞.


Next, we introduce Diophantine equation (3) and non-minimal realization.

D(q −1) = A(q −1)R(q −1) + q −dH (q −1) (3)

, where D(q −1) = 1+d1q −1 + · · ·+dnq −n is n order monic stability polynominalthat we can set arbitrary. Then R(q −1) and H (q −1) always exist.

R(q −1) = 1 + r1q −1 + · · ·+ rd−1q −(d−1)

H (q −1) = h0 + h1q −1 + · · ·+ hn−1q −(n−1)

By multiplying (3) by y (k),

D(q −1)y(k) = A(q −1)R(q −1)y(k) + q −dH (q −1)y(k)

= q −dB(q −1)R(q −1)u(k) + H (q −1)y(k − d)

= B(q −1)R(q −1)u(k − d) + H (q −1)y(k − d).

Here, we introduce θ, set of unknown parameters, and ξ , input and outputdata

θT = [b0, b0r1 + b1, b0r2 + b1r1 + b2, · · · , bmrd−1, h0, h1, · · · , hn−1]

ξ T = [u(k), u(k − 1), · · · , u(k − (m + d − 1)), y(k), y(k − 1), · · · , y(k − (n− 1))]

, then we obtain (4) that denotes non-minimal realization.



By the way, let us assme that order n = 2, d = 2. The Diohpantine equation(3) becomes

1 + d1q −1

+ d2q −2

= (1 + a1q −1

+ a2q −2

)(1 + r1q −1

) + q −2

(h0 + h1q −1

).

By comparing coeficients, we can derive following equations.

r1 = d1 − a1h0 = d2 − a2 − a2d1 + a21h1 = a2(a1 − d1)

These equations mean that coeficients ri and hj depend on A(q −1) and D(q −1).Let us get back to the system error e1(k). By multiplying e1(k) by D(q −1)q −d,

D(q −1)e1(k + d) = D(q −1)ym(k + d) − B(q −1)R(q −1)u(k) − H (q −1)y(k)

= D(q −1)ym(k + d) − θT ξ (k) (5)

, and we separate b0 and u(k) as θT = [b0, θT ]ξ T (k) = [u(k), ξ T (k)]

.

Then,D(q −1)e1(k + d) = D(q −1)y

m(k + d) − b0u(k) − θT ξ (k) (6)

For (6) → 0,

u(k) = 1

b0

D(q −1)ym(k + d) − θT ξ (k)

. (7)

6.3 Direct control

Unknown parameters in (6) are θ(b0(k) and ˆθ(k). We denotes y(k) that is cal-culated by means of expected values. Then, we define 1(k):

1(k) = D(q −1

)(y(k) − y(k))= φT (k)ξ (k − d) (8)

where φT (k) = θ(k)−θ. This equation represents a case of deterministic identi-fier that has W ( p) = 1. So, we can apply algorithms of deterministic identifiers.

θ(k) = θ(k − 1)− Π(k − 1)ξ (k − d)1(k) (9)

Π(k) = 1

λ1(k)

Π(k − 1)−

λ2(k)Π(k − 1)ξ (k − d)ξ T (k − d)Π(k − 1)

λ1(k) + λ2(k)ξ T (k − d)Π(k − 1)ξ (k − d)

(10)

, where 0 < λ1(k) ≤ 1, 0 ≤ λ2(k) ≤ λ, Π(0) = Π(0)T > 0. By removing

φ(k) (θ(k)) from 1(k),

1(k) = −D(q −1)y(k) + θT (k − 1)ξ (k − d)

1 + ξ T (k − d)Π(k − 1)ξ (k − d) . (11)



7 STR: Self-tuning regulator

7.1 Formulation

In this section, we introduce STR that considers stochastic noize added to plants.At first, we formulate a plant by (12).

Plant : A(q −1)y(k) = q −dB(q −1)u(k) + C (q −1)w(k) (12)

A(q −1) = 1 + a1q −1 + · · ·+ anq −n

B(q −1) = b0 + b1q −1 + · · ·+ bmq −m

C (q −1) = 1 + c1q −1 + · · ·+ cnq −n

where w(k) denotes white noize (average=0, distribution=σ2

). Known parame-ters are m, n, d, and unknown parameters are ai, bj , ck.We assume that B(q −1)and C (q −1) are stability polynominals.

In this section, our goal is minimizing J = E [(ym(k) − y(k))2] (minimizingdistribution).


Next, we consider Diophantine equation and non-minimal realization. Diophan-

tine equation is represented by (13).

C (q −1) = A(q −1)R(q −1) + q −dH (q −1) (13)

R(q −1) = 1 + r1q −1 + · · · rd−1q −(d−1)

H (q −1) = h0 + h1q −1 + · · ·hn−1q −(n−1)

By mutiplying (13) by y(k),

C (q −1)y(k) = A(q −1)R(q −1)y(k) + q −dH (q −1)y(k)= B(q −1)R(q −1)u(k − d) + H (q −1)y(k − d) + C (q −1)R(q −1)w(k)(14)

Here, we define system error e1(k) = ym(k)− y(k),

C (q −1)e1(k) = C (q −1)ym(k) −C (q −1)y(k)

= C (q −1)ym(k) −B(q −1)R(q −1)u(k − d)−H (q −1)y(k − d)− C (q −1)R(q −1)w(k)(15)

. For (15)→ 0,

u(k) = 1

b0

C (q −1)ym(k + d)−H (q −1)y(k)−BR(q −1)u(k)

(16)

where BR(q −1) = B(q −1)R(q −1)− b0. Then, we obtain

e1(k) = −R(q−1)w(k)



7.3 Control

Same as the case of MRACS, we define θ and ξ as follows.

θT = [b0, b0r1 + b1, b0r2 + b1r1 + b2, · · · , bmrd−1, h0, h1, · · · , hn−1, c1, c2, · · · , cn]

ξ T = [u(k), u(k − 1), · · · , u(k − (m + d − 1)), y(k), y(k − 1), · · · , y(k − (n − 1)),

−ym(k + d − 1), · · · , −ym(k + d − n)]

Then,

u(k) = 1

b0

ym(k + d) −

θT ξ (k)

(17)

ym(k + d) = θT

ξ (k) (18)

where, θT = [b0, θT ], ξ T (k) = [u(k), ξ T (k)].

By φT (k) = θT − θT ,

1(k) = φT (k)ξ (k − d) = ym(k) − y(k)

Then, we can apply algorithms of identifiers.



Fuzzy theory1. Overview

2. Fuzzy sets

3. Operations for fuzzy sets

4. Fuzzy I/O5. Fuzzy functions

6. Fuzzy rules7. Fuzzy control



１Overview

Fuzzy theory began with a paper on“fuzzy sets”, written by Prof. L.A. Zadehin 1965.

Fuzzy sets are those sets whoseboundary is not clear. Fuzzy logics arecalculation procedures on fuzzy sets.

A technology in which the whole systemcan be roughly defined, that is “fuzzy

theory”

was proposed.



ATTENTIONFuzzy system is deterministic. Neither

stochastic nor ambiguous.



２Fuzzy sets

Ordinal set = Crisp set

Fuzzy set



Crisp set: feeling “HOT”XHOT(T) = 1

0T≧29T < 29

Degree“HOT”

1

Only Yes (degree=1) or NO (degree=0)

HOT : T | T≧ 29

28.9 is not HOT

0 29 T []



Fuzzy set: feeling “HOT”

Degree“HOT”

1

µHOT(T)

Membership function µHOT(T) defines a fuzzy set

28.9≒ 29

0 29 T []



Definition of many fuzzy sets Only by defining each membership function, we can

express many status.

Slightly

HOT HOT Very HOT

Degree

1

0

T []



3 Operations for fuzzy setsµ A(T) ∨ µ B(T) = max µ A(T), µ B(T)

Similar to “OR” logic.

Degree

0

1

T

µ A(T) µ B(T)

µ A(T)∨

µ B(T)

Set of maximum values in µ A(T), µ B(T)



µ A∪ B = min 1, µ A(T) + µ B(T)

Saturated sum ofµ A andµB

Degree

0

1

µ A

(T) µ B

(T)

µ A∪ B

T



µ A(T) ∧ µ B(T) = min µ A(T), µ B(T)

Similar to “ AND” logic.

Degree

1

µ A

(T) µ B

(T)

µ A(T) ∧ µ B(T)0

T



４ Fuzzy Input and Output

Let us input a fuzzy set into a crisp function

A crisp function y = f(x) e.g. y = 2x+1

Let us input a fuzzy set A y=f(x), x∈ A

Output of the function becomes a membership function.

Examples will appear on the next slide.

E l （１）



Examples（１）

Input a fuzzy set ACrisp function y=x

Membership function (output)

Examples（２）



Examples（２）

Crisp inputs Input a fuzzy set

µ (y1) = max µ (x1), µ (x2), µ (x3)

, where y1 = f(x1) = f(x2) = f(x3)



５Fuzzy functions

ｚ(Degree)

y = x(a crisp function)

z=µf (y,x)

Fuzzy set of function

y

x

Fuzzication

A fuzzy function becomes a “cylinder”.

A crisp input to a fuzzy function



A crisp input to a fuzzy function

Output is obtained by an intersection among a fuzzy function andan input.

Input a fuzzy set to a fuzzy function



Input a fuzzy set to a fuzzy function

Projection

Cylindrical extension of an

input fuzzy set.

Intersection among a

fuzzy function and a

fuzzy input



６Fuzzy rules

Rules between fuzzy sets

IF x is R then y is C

Crisp sets

IF x∈

µA then y∈

µBFuzzy sets

Illustration of fuzzy rules



Illustration of fuzzy rulesProjection

Intersection

Input A’ to the fuzzy ruleA fuzzy rule: an intersection amongcylindrical extensions of input and

output fuzzy sets





７Fuzzy control

Control system of a “cooler ”

Input：temperature T []Output：driving voltage V

Designing fuzzy sets

Defining fuzzy rulesDefuzzication

P d f f t l



Procedures of fuzzy control

Input T（crisp input）

↓Output of fuzzy rules（fuzzy）

↓

Evaluate fitness

↓

Defuzzication（crisp output）↓

Driving voltage（crisp）





Fuzzy sets for outputsModerate

Degree

1

0 1

Very lowVery high

HighLow

0.25 0.50 0.75 Output voltage(Max.=1)



Fuzzy rules

Defining relationship between inputs

and outputs

Input (feeling) Output (voltage)

Cold

Cool

Good

Warm

Hot

Very low

Low

Moderate

High

Very high



Operation of fuzzy rules

if xi(µ xi(t)) then yi(µ yi(v))

If xi then yi

“t ” temp., “v” voltage



Calculation of outputFor all rules（relationship between xi and yi）

µ output(v) = (µ yi(v) µ xi(t ))i

in other words,

µ output(v) = max min µ yi(v), µ xi(t ) i

When we input t (crisp), we obtain a membership function of v.



Defuzzication to obtain crisp output



Defuzzication to obtain crisp output

Output v

Degree

Output of fuzzy rules

µ output

Fitness

vc

Choosing the center of the gravity vc

Fitness



Control output for input TDriving voltage

Only setting several fuzzy sets, a fuzzy

controller generates continuous output

function of input T.

Changing membership functions



Changing membership functions

for inputs

Good DegreeCool Warm

Cold Hot

... then, a shape of a control



function is changed.Driving voltage



Conclusion

“Fuzzy” is a tool for designers. It doesnot contribute to the performance of

controller.

We can handle “feelings” by fuzzy sets

without finding “crisp” boundaries.

Only by changing membership functions,

we can arrange a control function.



Artificial neural networks(ANN)

1. Overview

2. Formal neurons3. Perceptron / Multi-layered NN

4. Error back propagation5. Approximate functions

1 Overview



1 Overview

A neural network is a network of

interconnected elements. Theseelements were inspired from studies ofbiological nervous systems.

The function of a neural network is toproduce an output pattern whenpresented with an input pattern.

Biological neuron



axon endings of other neurons

endings

synapse

soma

axon

dendrite

2 Formal neuron



2 Formal neuron

y = f (Σ wi xi - θ )i=1

nInputs

Weights

Fan-out 0 ( x<0)

1 ( x≧0) f(x)=

Threshold

Heaviside function

Calculation of a formal neuron



Calculation of a formal neuron

w1=1

w2=1

θ=1.5

x1

x2

y

x1w1+x2w2=0

0

x1w1+x2w2=1

0

x1w1+x2w2=1

0

x1w1+x2w2=20

0

1

0

0

1

1

1

(0 0)→

0(1 0)→ 0

(0 1)→ 0

(1 1)→ 1

“AND” Logic1Inputs

Output

A formal neuron generates one hyper

plane that divides input space



plane that divides input space

0

x2

1

1 x1

x1w1+x2w2= θ (=1.5)

A hyper plane

(0 0)→ 0

(1 0)→ 0

(0 1)→ 0(1 1)→ 1

“AND” Logic

Other logics by a formal neuron



AND OR NOT

Inputs (x1, x2)w1=1, w2=1

θ=1.5

Inputs (x1, x2)w1=1, w2=1

θ=0.5

Input (x1)w1=-1

θ=-0.5

Only one hyper-plane a formal neuron can generate.

Functions that needs two or

more hyper planes



more hyper planes

0 1

1

x1

x2

Example: XOR

(0,0)→0

(1,0)→1

(0,1)→1(1,1)→0

Combination or network of formal neurons is needed.

A combination for XOR



A combination for XOR

(1, 0)→ 1

y1=f(x1-x2-0.5)

XOR

Inputs

OR

(0, 1)→ 1

y2=f(-x1+x2-0.5)

3 Perceptron;Learning ANN



Learning ANN

“Perceptron” was proposed by Dr. F.

Rosenblatt in 1958. It contains three layers of formal

neurons called the Sensor, Association

and Response.

By changing the weights, a perceptron

can learn correct outputs.

Perceptron: Multi-layered neural network



S layer A layer R layer

x1

x2

z

1 1

(Sensory) (Association) (Response)

A formal neuron

1

Extension of Perceptron;General Multi-layered NN



General Multi layered NN

The original perceptron has a week-

point of learning. It does not guaranteeto reach correct answer in all case.

Prof. Amari and other researchers have

proposed a method called “back

propagation” by introducing sigmoid

function instead of heaviside function.

Introducing sigmoid function



Introducing sigmoid functiony=(1+e-x)-1

Heaviside function Sigmoid function

In order to analyze neural networks, Heavisidefunction is not suitable because its derivative is notcontinuous.

So, we introduce Sigmoid function that hascontinuous derivative.

Calculation in Multi-layered NN



Calculation in Multi layered NN

Notation: Input for ith neuron in lth layer … xli

Output of the neuron … zli

Sensory layer = 0th layer, Response layer = Lth layer

Sensory Layer ：output input signal directly

z0i = xi

Other layers: calculate by Sigmoid function

zl

i= f(xl

i) =

1+e-x i

1

l

Calculation of xli



Ca cu at o oi

xli depends on outputs of (l-1) th layer ×weights

xli = Σ wl

ij zl-1 j

j=0

nl-1

where

wlij : weight from jth neuron in l-1th layer to ith neuronin lth layer.

nl-1

: Number of neurons in l-1th layer

Notation example



(2inputs 1output)

1 1

S layer 0th

A layer 1st

R layer 2nd

x1

x2

zz0

1

z02

z00

w110

w111

w1

12

w120

w121

w122

w210

w211

w212

1

z12

z11 z2

1

z20z1

0

4 Error back propagation



p p g

Main idea

• Changes weights w*** according to output error.

• By comparing with teacher signals, feedback errors to

the neural network.

Output

Teacher signal

Input

Feedback

Output error E



p

For single-output systems,

where t is teacher signal

E = ( t – z )21

2

For multi-output systems,

E = Σ ( t j – zL j )21

2 j=1

nL

Training: changing wlij



according to output error.

wlij = wlij +∆wlij(new) (current)

∂E

∂wl

ij

∆wlij = - η

η: Learning parameter（0<η<1）

Calculate ∆wl



ij

∆wlij = - η ∂E

∂wlij

∂E

∂wlij

∂xli

∂xli

= - η





Calculate ∆wl (Cont.)



ijAt the Lth layer (R layer),

E = ( t – z )2 = ( t – f(xL1) )212 12

The derivative of Sigmoid function f(x) becomes

d f(x)

1+e-x

e-x

=dx ( )

2

1+e-x

1

1+e-x

1

1 -( )= = f(x)(1-f(x))

Therefore, at the Lth layer,



δL1 = - ∂E

∂xL1

= (t-z)z(1-z)

( Notice：f(xLi)=z because of R layer )

In the case of other layers,

δl j = zl

j(1- zl j) Σ wl+1

ijδl+1

i

nl+1

i=1

This result indicates δ of lth layer depends on δ in l+1th layer

Conclusion



ANN = Network of formal neurons.

Formal neurons with Heaviside functionrealizes logics.

Back propagation method for neurons

with Sigmoid function realizes learning

ability and approximation of continuous

functions.



Overview of SOM



The process of formation of a topologically orderedmapping from the signal space onto the neural

network is defined by the Self-organizing map (SOM)algorithm.

The “feature maps” thereby realized can ofteneffectively be used for the preprocessing of patterns

for their recognition, or, if the neural network is aregular two-dimensional array, to project andvisualize high-dimensional signal space on such a

two-dimensional display. As a theoretical scheme, on the other hand, the

adaptive SOM processes, in a general way, mayexplain the organizations found in various brain

structures.

Demonstrations



Mapping “colors” onto 2D plane.

A color is a vector (R, G, B).Map size: 10x10 units

Initial state: randomized

SO M can categorize the colors order 3 vectors)onto a 2D m ap without any teacher signal.

Each trial generates different m ap.

SOM array: a “map”



Input vector x(t) (order n)

SOM array

Each unit

mi(t)is n×１vector.

m×m units are allocated

Training SOM = Changing mi(t)

Schematic view of SOM



An input (vector)

By iterative training, near units

has near vectors.

Winner node: an unit that has the

smallest norm among the input.

According to the winner node,

we can categorize input signals.

To train a SOM



• We have to feed many data.

• We have to fix the learning time (step) T.• We do not need any teacher signal.

Procedure of training



New input

(ex. a color)

Find a“

winner node”

thathas smallest difference

between the new input.




The winner node affects

values of nodes located

around it.






Training procedure of SOM



(1) Finding a winner node mc(t)

mc(t) | |x(t)-mc(t)| = min |x(t)-mi(t)|

(2) Renew unit mi(t) to mi(t+1)mi(t+1) = mi(t) + hci(t) (x(t)-mi(t))

hci(t) = α(t)

hci(t) = 0

（i∈ Nc）

（other cases）

α(t) = α0(1-t/T) Nc(t) = Nc(0)(1-t/T)

※Blue parametersare given in advance.

T: Training periodα: learning coefficient (0<a<1) Nodes that is located within N

cfrom the winner node mc(t) Nc becomes smaller

Using trained SOM



Unknown data

Finding a “winner node”,

we can understand the

nearest approximation for

the input.

Locations of nodes indicate

categories of known data.

Demonstration



SOM categorize vision data from a camera onthe top of a mobile robot.

The robot estimate its situation according tothe trained result of SOM.

Automatic boundary generation

(Learning Vector Quantization: LVQ)



(Learning Vector Quantization: LVQ)

• LVQ generates boundaries of classified categories on a SOM.

In this method, the location of a unit will be changed.

• On LVQ procedure, a map itself represents a space of input

vectors. So, each point in a map represents an input signal, and

position of each node represents its status.

• Training procedures of LVQ need teacher signal. (SOM does

not need any teacher)

• Not contents of a cell but its position will be changed during

training procedure.

Category RED Category Blue

Example: we want to find the boundary.



• Input: a point on the map. In this case, a map has large dimension.

• We can distinguish which category a point belongs to by a winner node (nearest node).

On LVQ procedure, a map itself represents a space of inputvectors. So, each point in a map represents an input signal,

and position of each node represents its status.



New input ＝Blue (teacher)Winner (nearest) = RED: categorizing failed

Get away

from the

input

If a winner node (nearest to a input) belongs to same

category as a teacher, current categorizing is correct.

Correct ! Coming near to the input



New input (teacher=RED)







Training procedure of LVQ



Finding winner mc

to input x

Here, we denote that x and mc belongs to S r and S s respectively.

Then, we renew the value (position) of the winner as following.

α(t ) (0<α(t )<1) is a coefficient of learning.

Demonstration:

Solving TSP by LVQ



Solving TSP by LVQTSP: Traveling Salesman Problem

• Let us assume a salesman who starting from his home city, is to visitexactly once each city on a given list and then return home.

• A TSP problem is a problem such that he selects the order in which he visits

the cities so that the total of the distances traveled in his tour is minimum.

• Assume that he knows, for each pair of cities, the distance from one to the

other. Then he has all the data necessary to find the minimum, but it is by

no means obvious how to use these data in order to get the answer.

• So, TPS is difficult problem.

Overview of TSP solver

Input: locations (x,y) of cities to be visited.We construct a “ring” of LVQ nodes.



We construct a ring of LVQ nodes.

Winner

n : distance

from a winner

on the ring.

LVQ node

Moved LVQ node

Winner node

A focused city

By following rules, LVQ nodes covers cities successively.





When a LVQ node is pulled by many cities, we generatetwo LVQ nodes both side of the node.



If a LVQ node is not pulled by any city, it disappears.



By the procedures, the ring of LVQ nodes becomes expanded.





Finally, the ring of LVQ nodes covers all cities.

Initialize：Put one node at (0,0)

Focus on one of cities at random

Fi d i

Start



Find a winner

The winner moves to the city

Increase LVQ node

Decrease LVQ node

Reduce gain G: G← αG

Are all cities covered by LVQ nodes?

Focused on all cities？

Finish

Yes

No

Yes

No

Conclusion

Self-organizing Map（ＳＯＭ）



Self-organizing Map（ＳＯＭ）Categorize unknown data without teacher

signal. A result is represented as a map on a 2D

plane.

Example: state recognition of a robot.

Learning Vector Quantizaton (LVQ)

Generating boundaries according toteacher signals.

Example: TSP solver.



Genetic algorithm (GA)

Intelligent control part II

Overview of GA

GA = Genetic algorithm



GA Genetic algorithm

GA is a category of algorithms for

optimization, mainly inspired from biologicalevolution procedure such as “naturalselection”

GA is suitable for large or complexoptimization problem that other deterministicalgorithms need too much time.

GA contains many heuristic operations.

Outputs of GA strongly depends on its initialstate.

Schematic view of GA (1)①Coding: define “genes” that represent candidates of a solution



1111010

10110111011010

11110111011011

0011010

1000111

1000111 10110000011010

Population (max. number of genes) must be defined.

Schematic view of GA (2)② Selection and reproduction based on evaluations.



0011010

1000111

0011010

1111010

1011010

1011000

1111011

1011011

1000111

1011011

55

7030

2590

43

66

83

17

72

A fitness function (defined by a user) evaluates genes.

3

Schematic view of GA (2)② Selectionand reproduction based on evaluations.



1000111

1011010

1111011

1011011

1011011

70

90

66

83

72


3

Schematic view of GA (2)② Selectionandreproductionbased on evaluations.



100011170

1011010

90

1111011

66

1011011

83

1011011

72

101101090

1011010

90

1011011

83

1011011

72

100011170




4

Schematic view of GA (3)③ Crossover: generate “children”.



1011010

11110101011010 1011011

10110111011011

1011011

10110111000111

1000111

Select a pair of “parents”..., and exchange parts of their genes.

4

Schematic view of GA (3)③ Crossover: generate “children”.



1011010

11110101011010 1011011

100001110110111011011

10110111000111

1011111

Select a pair of “parents”..., and exchange parts of their genes.



6

One generation in GA

①



①Defining a coding, the population, and a fitness function

②Selection and reproduction

③Crossover Continue

④Mutation



... that is all about GA.

From now on, let us consider

algorithms of selection,

reproduction, crossover, and

mutation, respectively.

7

Selection and reproduction

• Procedure to keep the population and to

select (possibly) good genes



select (possibly) good genes.

• A fitness function evaluates genes.

• “Selection” and “reproduction” relates

each other.

There are two ideologies;

A gene have possibility to live according to its fitness.

Genes that have low fitness must die.

8

Roulette selection A gene that has high fitness has high possibility to duplicate it.

Agene that has lowfitnessmay live.



A gene that has low fitness may live.

• Selection by “a roulette”.• A gene that has high fitness occupies large region.

• Iterate selections population times.

• Suitable for large population cases.Probability to

be reproduce Fitness of gene i

Sum of fitness values

9

Expected-value selectionGenes that has low fitness must die.



• An expected value = fitness / population• Determine number of reproductions according to the value

• Suitable for small population cases.

Fitness

Expected

Reproduction

Example: population = 10

10

Ranking selectionGenes that has low fitness must die.



• Determine a ranking according to the fitness.

• Reproduce a gene based on its rank.

Number of reproductionRank

12

3

4

.

.

.

106

4

3

.

.

.

109

8

7

.

.

.

Gene 5Gene 2

Gene 8

Gene 1

.

.

.

(Linear) (Non-linear)

Crossover

A crossover exchanges parts of parent genes.



“Children” hopefully succeed “good

characteristics” of parents.

Crossover procedures must consider the

coding in order to avoid mortal genes. Mortal gene = inadmissible answer.

Here I would like to introduce general methods

for crossover.

11

Simple crossover

(1) Simple crossover (One-point crossover)



Parents Children

A crossover point is determined at random.

12（２）Multipoint Crossover Parents Children

Two-points crossover



Three-points crossover

（３）Uniform Crossover

Parents

Using a mask pattern

Children

13

Mutation

• Change a locus of a gene at random.

So often mutation results a random search



• So often mutation results a random search.

• A mutation also consider the coding in orderto avoid mortal genes.

<1011101000>→<1001101000>



15

Coding

In order to keep the condition (error < 10-5),



we express x by 22bit code.

s1=<1000101110110101000111>

Boundary condition:<0000000000000000000000>=-1.0

<1111111111111111111111>= 2.0.

• In this case, no mortal gene exists.

• At the beginning, we generate genes at random.

16

Fitness function

In this case, we apply f(x) itself as a fitness function.



s1=<1000101110110101000111>

s2=<0000000111000000010000>s3=<1110000000111111000101>

f(s1) = 2.586345

f(s2) = 1.078878f(s

3) = 3.250650 BEST

17 A result of optimization



TRUE

1.850542

Population = 50, Prob. mutation = 0.01

Simple crossover, Prob. crossover = 0.25

Roulette selection



Calculation cost

19

(number of cities –1)!

2number of routes =



2

Too many candidates to search

#city #route #city #route

20

Coding and crossover Let us use a list of visiting cities as a gene ....

1 2



2

35

4 67

98

s1=<12345|6789>

s2=<19283|7465>

s’1=<12345|7465>

s’2=<19283|6789>

（Mortal gene!）

We cannot apply simple crossover to this coding.

We have to change the crossover procedure

21

Crossovers for TSPResearchers on the field of GA often use TSP as a

benchmark. So, there are many proposals about

crossover procedures for TSP



crossover procedures for TSP.

• Partially Matched Crossover, PMX

• Ordered Crossover, OX

• Cycle crossover ,CX

22（１）Partially Matched Crossover (PMX)

s1=<123| 4567| 89>s2=<452| 1876| 93>(i) Parents



s’ 1=<***| 1876| ** >

s’ 2=<***| 4567| ** > Corresponding pairs1-4, 8-5, 7-6, 6-7

(ii) Exchanging

(iii) Insertion s’

1=<*23| 1876| *9>

s’ 2=<**2| 4567| 93> Additional pairs3-2, 9-3

(iv) Completion s’ 1=<423| 1876| 59>s’ 2=<182| 4567| 93>

This crossover loses orders of visiting cities in parent genes.

23（２）Ordered Crossover (OX)

s1=<123| 4567| 89>s2=<452| 1876| 93>(i) Parents

Order after 2nd



s’ 1=<218| 4567| 93>s’ 2=<345| 1876| 92>

(iii) Insertion of remained

genes according to their

original orders.

s’ 1=<***| 4567| ** >s’ 2=<***| 1876| ** >

934521876

crossover point

93218

(ii) Copy

This crossover loses correspondence between locus and a city.

24

（３）Cycle crossover (CX)

s1=<123456789>

s2=<412876935>(i) Parents



(ii) Find a cycle

s1=< 1 2 3 4 5 6 7 8 9 >

s2=< 4 1 2 8 7 6 9 3 5 >

s’1=< 1 2 3 4 * * * 8 * >

(iii) Exchange remained genes

s’1=< 1 2 3 4 7 6 9 8 5 >

(s’

2 is applied the same completion)

25

Demonstration

Cities：１０



Population：１０Cycle crossover and ranking selection

Ratio of mutation: 10%Fast, but not global optimum.

Variation of genes will be lost.

26

Conclusion

GA is a category of optimization algorithms

that are inspired from natural selection.



p

Fast, but no guarantee of global optimum.

We have to consider a procedure of

crossover depends of the coding.

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