SIMPLE DERIVATION OF A STATE OBSERVER OF LINEAR

TIME-VARYING DISCRETE SYSTEMS

Yasuhiko Mutoh

Department of Applied Science and Engineering, Sophia University, 7-1, Kioicho, Chiyoda-ku, Tokyo, Japan

Keywords:

Pole Placement, State Observer, Linear Time-Varying System, Discrete System.

Abstract:

In this paper, a simple calculation method to derive the Luenberger observer for linear time-varying discrete

systems is presented. For this purpose, the simple design method of the pole placement for linear time-varying

discrete systems is proposed. It is shown that the pole placement controller can be derived simply by ﬁnding

some particular ”output signal” such that the relative degree from the input to this new output is equal to the

order of the system. Using this fact, the feedback gain vector can be calculated directly from plant parameters

without transforming the system into any standard form.

Then, this method is applied to the design of the observer, i.e., because of the duality of linear time-varying

discrete system, the state observer can be derived by simple calculations.

1 INTRODUCTION

The design of the state observer for linear time-

varying discrete systems is well established. As for

the continuous case, the condition for a system to be

a state observer is very simple. However, different

from the time-invariant case, calculation procedure to

obtain the observer gain is not straightforward. This

paper gives a simple calculation method to design

the state observer for linear time-varying discrete sys-

tems.

Since the design of the observer is based on

the pole placement technique, simpliﬁed calculation

method to derive the pole placement feedback gain

vector for linear time-varying discrete systems is con-

sidered ﬁrst. We deﬁne the pole placement of lin-

ear time-varying discrete systems as follows. The

problem is to ﬁnd a time-varying state feedback gain

for linear discrete time-varying discrete system, so

that the closed loop system is equivalent to the time-

invariant system with desired poles.

Usually, the pole placement design procedure

needs the change of variable to the Flobenius stan-

dard form, and hence, is very complicated. To sim-

plify this procedure, it will be shown that the pole

placement controller can be derived simply by ﬁnd-

ing some particular ”output signals” such that the rel-

ative degree from the input to this output is equal to

the order of the system [4]. Using this fact, the feed-

back gain vector can be calculated directly from plant

parameters without transforming the system into any

standard form.

Because of the duality of the linear discrete time-

varying system, the simpliﬁed pole placement tech-

nique can be applied to thedesign of the state observer

for linear discrete time-varying discrete systems.

In the sequel, the simple pole placement technique

is proposed in Section 2, and then, this method is used

to the observer design problem in Section 3.

2 POLE PLACEMENT OF

LINEAR DISCRETE

TIME-VARYING SYSTEMS

Consider the following linear time-varying discrete

system with a single input.

x(k+ 1) = A(k)x(k) + b(k)u(k) (1)

Here, x ∈ R

n

and u ∈ R

1

are the state variable and

the input signal respectively. A(k) ∈ R

n×n

and b(k) ∈

R

n

are time-varying parameter matrices. The problem

is to ﬁnd the state feedback

u = h

T

(k)x(k) (2)

which makes the closed loop system equivalent to

the time invariant linear system with arbitrarily stable

poles.

30

Mutoh Y. (2010).

SIMPLE DERIVATION OF A STATE OBSERVER OF LINEAR TIME-VARYING DISCRETE SYSTEMS.

In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 30-34

DOI: 10.5220/0002945800300034

Copyright

c

SciTePress

Deﬁnition 1. The system (1) is called completely

reachable in step n from the origin, if for any x

1

∈ R

n

,

there exists a ﬁnite input u(m) (m = k, ··· , k + n − 1)

such that x(k) = 0 and x(k+ n) = x

1

.

Lemma 1. The system (1) is completely reachable in

step n from the origin, if and only if

rank [ b(k+ n− 1),Φ(k+ n, k+n−1)b(k+ n− 2),

··· ,Φ(k+n, k + 1)b(k) ]

= rank U

R

(k) = n,

∀

k (3)

where Φ(i, j) is the transition matrix from k = j to k = i,

i.e.,

Φ(i, j) = A(i− 1)A(i− 2)··· A( j) i > j (4)

∇∇

Now, consider the problem of ﬁnding a new output

signal y(k) such that the relative degree from u(k) to

y(k) is n. Here, y(k) has the following form.

y(k) = c

T

(k)x(k) (5)

Then, the problem is to ﬁnd a vector c(k) ∈ R

n

that

satisﬁes this condition.

Lemma 2. The relative degree from u to y deﬁned by

(5) is n, if and only if

c

T

(k+ 1)b(k) = 0

c

T

(k+ 2)Φ(k+ 2, k + 1)b(k) = 0

.

.

. (6)

c

T

(k+ n− 1)Φ(k+ n − 1, k+ 1)b(k) = 0

c

T

(k+ n)Φ(k+ n, k + 1)b(k) = 1

(Here, c

T

(k + n)Φ(k+ n, k+ 1)b(k) = 1 without loss

of generality.) ∇∇

Proof : This is obvious by checking y(k + 1), ···,

y(k+ n).

If the system (1) is completely reachable in step n,

there exists a vector c(k) such that the relative degree

from u(k) to y(k) = c

T

(k)x(k) is n. And, from (6),

such a vector, c(k), is obtained by

c

T

(k) = [0, · · · , 0, 1] [ b(k − 1), Φ(k, k − 1)b(k−2),

··· , Φ(k, k + 1− n)b(k− n) ]

−1

= [0, 0, · ·· , 1]U

−1

R

(k− n) (7)

The next step is to derive the state feedback for the

arbitrary pole placement.

The new output, y(k) = c

T

(k)x(k), with c(k) ob-

tained by (7), satisﬁes the following equations.

y(k) = c

T

(k)x(k)

y(k+ 1) = c

T

(k+ 1)Φ(k+1, k)x(k)

.

.

. (8)

y(k+ n− 1) = c

T

(k+ n− 1)Φ(k+n−2, k)x(k)

y(k+ n) = c

T

(k+ n)Φ(k+n−1, k)x(k) +u(k)

Let q(z) be a desired stable polynomial of z-

operator, i.e.,

q(z) = z

n

+ α

n−1

z

n−1

+ ··· + α

0

(9)

By multiplying y(k + i) by α

i

(i = 0, ··· ,n − 1) and

then summing them up, the following equation is ob-

tained from (8).

q(p)y(k) = d

T

(k)x(k) + u(k) (10)

where d(k) ∈ R

n

is deﬁned by the following.

d

T

(k) = [ α

0

, α

1

, ··· , α

n−1

, 1 ]

×

c

T

(k)

c

T

(k+ 1)Φ(k+ 1, k)

.

.

.

c

T

(k+ n)Φ(k+ n, k)

(11)

Hence, the state feedback,

u = −d

T

(k)x(k) + r(k) (12)

makes the closed loop system as follows.

q(z)y(k) = r(k) (13)

where r(k) is an external input signal.

This control system can be summarized as fol-

lows. The given system is

x(k+ 1) = A(k)x(k) + b(k)u(k) (14)

and, using (4), (9), and (11) the state feedback for the

pole placement is given by

u(k) = −d

T

(k)x(k). (15)

Then, the closed loop system becomes

x(k+ 1) = (A(k) − b(k)d

T

(k))x(k). (16)

Let T(k) be the time varying matrix deﬁned by

T(k) =

c

T

(k)

c

T

(k+ 1)Φ(k+ 1, k)

.

.

.

c

T

(k+ n− 1)Φ(k+ n − 1, k)

(17)

and deﬁne the new state variable w(k) by the follow-

ing equations.

x(k) = T(k)w(k), w =

y(k)

y(k+ 1)

.

.

.

y(k+ n− 1)

(18)

SIMPLE DERIVATION OF A STATE OBSERVER OF LINEAR TIME-VARYING DISCRETE SYSTEMS

31

Using the above, (16) is transformed into

w(k+ 1)

= T

−1

(k+ 1)(A(k) − b(k)d

T

(k))T(k)w(k)

=

0 1 ··· 0

.

.

.

.

.

.

.

.

.

.

.

. 1

−α

0

··· ··· −α

n−1

w(k)

= A

∗

w(k) (19)

This implies that the closed loop system is equiv-

alent to the time invariant linear system which has the

desired closed loop poles (det(zI − A

∗

) = q(z)).

Theorem 2. If the system (1) is completely reachable

in step n, then, the matrix for the change of variable,

T(k), given by (17) is nonsingular for all k. ∇∇

Example 1.

Consider the following unstable system.

x(k+ 1) = A(k)x(k) + b(k)u(k) (20)

where

A(k) =

1 2+ cos0.1k

2+ sin0.2k 2

b(k) =

0

1

(21)

From (7), c

T

(k) is obtained as follows.

c

T

(k) = [ 0, 1 ][ b(k− 1), A(k− 1)b(k− 2) ]

−1

=

1

2+ cos0.1(k− 1)

0

(22)

The purpose is to design the state feedback so that the

closed loop system is equivalent to the linear time in-

variant system with λ

1

= 0.4 and λ

2

= 0.5 as its closed

loop poles. This implies that the desired closed loop

characteristic polynomial is

q(z) = z

2

+ 0.9z+ 0.2.

From (11),

d

T

(k) = [ 0.2, 0.9, 1 ]

×

c

T

(k)

c

T

(k+ 1)A(k)

c

T

(k+ 2)A(k+ 1)A(k)

=

d

1

(k) d

2

(k)

(23)

In the above, d

1

(k) and d

2

(k) are given by

x

x

1

2

0

10

20

-10

-20

Figure 1: Responce of the state variable (x) of the system.

d

1

(k) =

0.2

γ(k− 1)

+

0.9

γ(k)

+

1

γ(k+ 1)

+ 2+ sin0.2k

d

2

(k) = 0.9+

γ(k)

γ(k+ 1)

+ 2

where

γ(k) = 2+ cos0.1k

Fig.1 shows the simulation results.

3 STATE OBSERVER

In this section, we consider the design of the observer

for the following linear time-varying system.

x(k+ 1) = A(k)x(k) + b(k)u(k)

y(k) = g

T

(k)x(k) (24)

Here, y(k) ∈ R is the output signal of this system.

The problem is to design the full order state observer

for (24). Consider the following system as a candidate

of the observer.

ˆx(k+ 1) = F(k) ˆx(k) + b(k)u(k) + h(k)y(k)

= F(k) ˆx(k) + b(k)u(k) + h(k)g

T

(k)x(k)

(25)

where F(k) ∈ R

n×n

, and h(k) ∈ R

n

. Deﬁne the state

error e(k) ∈ R

n

by

e = x(k) − ˆx(k) (26)

Then, e(k) satisﬁes the following error equation.

e(k+ 1) = F(k)e(k) + (A(k) − F(k)

−h(k)g

T

(k))x(k) (27)

Hence, (25) is a state observer of (24) if F(k) and

h(k) satisfy the following condition.

F(k) = A(k) − h(k)g

T

(k) (28)

F(k) : arbitrarily stable matrix

Then, the problem is to ﬁnd h(k) such that F(k)

is equevalent to a constant matrix F

∗

with arbitrar-

ily stable poles. Consider the pole placement control

problem of the following system.

w(k+ 1) = A

T

(−k)w(k) + g(−k)v(k) (29)

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

32

where w(k) ∈ R

n

and v(k) ∈ R

1

are the state variable

and an input signal.

Let Ψ(i, j) be the state transient matrix of the sys-

tem (29). Then, we have the following relation.

Φ

T

(i, j) = Ψ(− j, −i) (30)

Deﬁnition 2. The system (24) is called completely

obsermable in step n, if from y(k), y(k+ 1), ···, y(k+

n− 1), the state, x(k), can be determined uniquely for

any k.

Lemma 3. The system (24) is completely observable

in step n, if and only if

rank

g

T

(k)

g

T

(k+ 1)Φ(k+ 1, k)

.

.

.

g

T

(k+ n− 1)Φ(k+ n − 1, k)

= rank U

o

(k) = n, ∀k (31)

From the property of the duality of the time vary-

ing discrete system, if the pair (A(k), g

T

(k)) is com-

pletely observable in step n, the pair (A

T

(−k), g(−k))

is completely reachable in step n. Then, if the pair

(A(k), g

T

(k)) is completely observable in step n, the

system (29) has a state feedback

v(k) = h

T

(−k)w(k) (32)

such that the closed loop system is equevalent to the

linear time invariant system with arbitrarily stable

poles.

This implies that for some state transformation

matrix, P(−k) ∈ R

n

,

P

−1

((k+ 1))(A

T

(−k) − g(−k)h

T

(−k))P(k)

= F

∗T

(33)

where, F

∗T

is a constant matrix with arbitrarily stable

poles. From this and the duality, we have the follow-

ing equation.

P

−1

(−k)(A(k) − h(k)g

T

(k))P(−(k+ 1))

= F

∗

(34)

Hence, using this h(k), the state observer for the

system (24) is obtained.

Example 2.

Consider the following system.

x(k+ 1) = A(k)x(k) + b(k)u(k)

y(k) = g

T

(k)x(k) (35)

where

A(k) =

0 1

−0.7 −(1.2 + 0.5cos0.4k)

b(k) =

1

1

, g

T

(k) = [ 2, 1 ] (36)

The dual system matrices are as follows.

A

T

(−k) =

0 −0.7

1 −(1.2+ 0.5cos0.4(−k))

g(−k) =

2

1

(37)

From (7), c

T

(−k) for the new output matrix is ob-

tained as

c

T

(−k) = [ 0, 1 ][ g(−(k− 1)),

A

T

(−(k− 1))g(−(k− 2)) ]

−1

=

1

γ(−(k− 1))

−1 2

(38)

where,

γ(−k) = 4.7− 2λ(k)

λ(k) = 1.2+ 0.5cos0.4k. (39)

The purpose is to design the state feedback so that

the closed loop system is equivalent to the linear time

invariant system with λ

1

= 0.3 and λ

2

= 0.4 as its

closed loop poles. This implies that the desired closed

loop characteristic polynomial is

q(z) = z

2

+ 0.7z+ 0.12.

From (11), d

T

(−k) is calculated as follows.

d

T

(−k) = [ 0.12, 0.7, 1 ]

×

c

T

(−k)

c

T

(−(k+ 1))A(−k)

c

T

(−(k+ 2))A(−(k + 1))A(−k)

=

d

1

(−k) d

2

(−k)

(40)

Here, d

1

(k) and d

2

(k) are

d

1

(k) = −

0.12

γ(k− 1)

+

0.7

γ(k)

+

1

γ(k+ 1)

(0.2− λ(k+ 1))

d

2

(k) =

0.12

γ(k− 1)

+

0.7

γ(k)

(0.2− λ(k))

+

1

γ(k+ 1)

{−0.2− (0.2+ λ(k+ 1))λ(k)}

SIMPLE DERIVATION OF A STATE OBSERVER OF LINEAR TIME-VARYING DISCRETE SYSTEMS

33

x

x

1

1

^

0

2

4

6

-2

-4

-6

Figure 2: Responce of x

1

(k) and ˆx

1

(k).

x

x

^

2

2

1

2

3

-1

-2

-3

0

Figure 3: Responce of x

2

(k) and ˆx

2

(k).

e

e

1

2

0

2

3

-1

-2

-3

1

Figure 4: Responce of the estimation error (e

1

(k) = x

1

(k)−

ˆx

1

(k), e

2

(k) = x

2

(k) − ˆx

2

(k)).

Hence, the observer gain vector, h(k), is obtained as

h(k) = −d(k) (41)

and, using this h(k), the observer is

ˆx(k+ 1) = {A(k) − h(k)g

T

(k)} ˆx(k)

+b(k)u(k) + h(k)y(k) (42)

Fig.2 ∼ 4 show the simulation results with u(k) =

2cos(0.9k). The initial condition of the plant is

x

1

(1) = x

2

(1) = 1.

4 CONCLUSIONS

In this paper, a simple design method for the state

observer for linear time-varying discrete systems is

proposed. We ﬁrst proposed the simple derivation

method of the pole placement state feedback gain for

liner time-varying discrete system. Feedback gain can

be calculated directly from the plant parameters with-

out the transformation of the system into any standard

form, which makes the design procedure very sim-

ple. This technique is applied to the observer design

procedure using the duality of the linear time-varying

system. The author appreciates the helpful comments

of the anonymous reviewers.

REFERENCES

Kwakaernaak H. C., Linear Optimal Control Systems.

Wiley-Interscience, 1972

Chi-Tsong Chen C., Linear System Theory and Design

(Third edition). Oxford University Press, 1999

T. Kailath C., Linear Systems. Prentice-Hall, 1980

Tse E., and Athans M., Optimal Minimal-Order Observer-

Estimators for Discrete Linear Time-Varying System.

IEEE, Transaction on AC, AC-15, 4, 416–426, 1970

Mutoh Y, Simple Design of the State Observer for Linear

Time-Varying Systems. 6-th ICINCO, 2009

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