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Linear Lyapunov Cone-Systems

183

5.3 Reachability

Definition 19.

The fractional Lyapunov (P,Q,V) -cone-system (34) is called reachable if for any given

X ∈ there exists ,0qZq

∈

> and an input sequence ,0,1,,1Q

Uq q

∈

=−… that

steers the state of the system from

, i.e.

X= .

Theorem 12.

The fractional Lyapunov

(P,Q,V) -cone-system (34) is reachable:

a) For

A satisfying the condition

XAXA

, i.e. RaaIA

∈

, if and only if the

matrix:

,( )

R PBQ A I N PBQ

−−

⎡

⎤

⎣

⎦

(45)

contains n linearly independent monomial columns,

00 1

PA P A

−

+ .

b) For

AaIaR≠∈, if and only if the matrix

PBQ

−

contains n linearly independent

monomial columns.

Proof:

From (38),(39),(40) and from the definitions 2 and 19, we have that the fractional discrete-

time Lyapunov

(P,Q,V)

-cone-system (34) is reachable if and only if the fractional positive

discrete-time Lyapunov system, with the matrices

ˆˆ ˆ

ˆˆ

,,,,

AABCD, is reachable – so from

the theorem for the reachability of positive discrete-time Lyapunov systems (Przyborowski

P.,2008a). follows the hypothesis of the theorem 12. □

5.4 Controllability to zero

Definition 20.

The fractional Lyapunov (P,Q,V) -cone-system (34) is called controllable to zero if for any

given nonzero

∈

there exist ,0qZq

∈

> and an input sequence

,0,1,,1Q

Uq q∈= −… that steers the state of the system from

to 0

= .

Theorem 13.

The fractional Lyapunov

(P,Q,V) -cone-system (34) is controllable to zero if and only if

2q = and:

() 0

nn n n

PA P I I A I N I

−

⊗

+⊗ + ⊗= (46)

Proof:

From (38),(39),(40) and from the definitions 2 and 19, we have that the fractional discrete-

time Lyapunov

(P,Q,V) -cone-system (34) is controllable to zero if and only if the

fractional positive discrete-time Lyapunov system, with the matrices

ˆˆ ˆ

ˆˆ

,,,,

AABCD, is

controllable – so from the theorem for the controllability of positive discrete-time Lyapunov

systems (Przyborowski P., 2008a). follows the hypothesis of the theorem 13. □

Automation and Robotics

184

5.5 Dual fractional Lyapunov cone-systems

Definition 21.

The fractional Lyapunov system described by the equations:

10 1

NT TT

iii i

AX XA CU

Δ=++

(47a)

iii

YBXDU=+

(47b)

is called the dual system respect to the system (34). The matrices

,,,,,AABCD

iii

UY are the same as in the system (34).

5.6 Observability

Definition 22.

The fractional Lyapunov

(P,Q,V) -cone-system (34) is called observable in q -steps, if

can

be uniquely determined from the knowledge of the output

Y and 0,

UiZ

=∈for

[0, ]iq∈

Definition 23.

The fractional Lyapunov

(P,Q,V) -cone-system (34) is called observable, if there exists a

natural number

1q ≥ , such that the system (34) is observable in q -steps.

Theorem 14.

The fractional Lyapunov (P,Q,V) -cone-system (34) is observable:

a) For

A satisfying the condition

AAX= , i.e.

aI a R

∈ , if and only if the matrix:

()

VCP

VCP A I N

−

⎡

⎤

⎢

⎥

⎣

⎦

(48)

contains n linearly independent monomial rows,

00 1

APAP A

−

+ .

b) For

AaIaR≠∈, if and only if the matrix

VCP

−

contains

linearly independent

monomial rows.

Proof:

From (38),(39),(40) and from the definitions 2 and 20, we have that the fractional discrete-

time Lyapunov

(P,Q,V) -cone-system (34) is controllable to zero if and only if the

fractional positive discrete-time Lyapunov system, with the matrices

ˆˆ ˆ

ˆˆ

,,,,AABCD

, is

observable – so from the theorem for the observability of positive discrete-time Lyapunov

systems (Przyborowski P., 2008a). follows the hypothesis of the theorem 14. □

6. Examples

Consider the state, input and output cones generated by the matrices

12 10 10

1 1 01 01

PQV

−

===

⎡

⎤⎡⎤⎡⎤

⎢

⎥⎢⎥⎢⎥

⎣

⎦⎣⎦⎣⎦

(49)

Linear Lyapunov Cone-Systems

185

6.1 Example 1

Consider the continuous-time Lyapunov system (7) with the matrices

74 40 12

,,,

25 0 1 11

24 00

11 00

AAB

−− − −

===

−− −

−

⎡

⎤⎡ ⎤ ⎡⎤

⎢

⎥⎢ ⎥ ⎢⎥

⎣

⎦⎣ ⎦ ⎣⎦

⎡⎤⎡⎤

⎢⎥⎢⎥

⎣⎦⎣⎦

(50)

This system is

(P,Q,V) -cone-system with ,PQ and V defined by (49) since:

12 7 4 12 1 0

11 2 511 0 3

−

−−−− −

⎡

⎤⎡ ⎤⎡ ⎤ ⎡ ⎤

⎢

⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

−

−−

⎣

⎦⎣ ⎦⎣ ⎦ ⎣ ⎦

−

⎡⎤

⎢⎥

−

⎣⎦

are the Metzler matrices and

12 12 10 10

11 1101 01

10 24 12 20

01 1 1 1 1 01

100010 00

01 0001 00

−

−−

⎡

⎤⎡ ⎤⎡ ⎤ ⎡ ⎤

⎢

⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣

⎦⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡

⎤⎡ ⎤⎡ ⎤ ⎡ ⎤

⎢

⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣

⎦⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡⎤⎡⎤⎡⎤⎡⎤

⎢⎥⎢⎥⎢⎥⎢⎥

⎣⎦⎣⎦⎣⎦⎣⎦

are matrices with nonnegative entries.

has the eigenvalues:

1, 3

−=− and

has the eigenvalues:

=−

=− therefore the system is asymptotically stable, since all the sums of the

eigenvalues:

11 1 2 21 2 2

()5,()2,()7,()4

μλμλμλμ

+=− +=− +=− +=−

have negative real parts.

For this system the reachability matrix

()

3( )

04e

−+

⎡

⎤

⎢

⎥

⎢

⎥

⎣

⎦

∫

Automation and Robotics

186

and the observability matrix

()

3( )

4e 0

−+

⎡

⎤

⎢

⎥

⎢

⎥

⎣

⎦

∫

are the monomial matrices for every

t > . Therefore, the system is reachable and

observable.

6.2 Example 2

Consider the discrete-time Lyapunov system (22) with the matrices

0.3 0.2 0.2 0 1 2

,,,

0.1 0.2 0 0.5 1 1

24 00

11 00

AAB

−

⎡

⎤⎡ ⎤ ⎡⎤

===

⎢

⎥⎢ ⎥ ⎢⎥

⎣

⎦⎣ ⎦ ⎣⎦

−

⎡⎤⎡⎤

⎢⎥⎢⎥

⎣⎦⎣⎦

(50)

This system is

(P,Q,V) -cone-system with ,PQ and V defined by (49) since:

0 1

1 2 0.3 0.2 1 2 0.1 0 0.2 0

ˆˆ

1 1 0.1 0.2 1 1 0 0.4 0 0.5

−

−−

⎡

⎤⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤

===

⎢

⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣

⎦⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

are the Metzler matrices satisfying conditions:

01 0 1

11 11 22 11

01 0 1

11 22 22 22

ˆˆ ˆˆ

0.3 0.2 0.5 0, 0.2 0.2 0.4 0

ˆˆ ˆ ˆ

0.3 0.5 0.8 0, 0.2 0.5 0.7 0

aa aa

aa a a

+=+= > += + = >

+=+=> +=+=>

and

12 1210 10 10 24 12 20

1 1 1 1 01 01 01 1 1 1 1 01

100010 00

010001 00

−−

−

−− −−

⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤

====

⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥

⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦

⎡⎤⎡⎤⎡⎤⎡⎤

⎢⎥⎢⎥⎢⎥⎢⎥

⎣⎦⎣⎦⎣⎦⎣⎦

are matrices with nonnegative entries.

has the eigenvalues:

0.1, 0.4

= and

has the eigenvalues:

0.2,

0.5

= therefore the system is asymptotically stable, since all the eigenvalues:

11 1 2 21 2 2

( ) 0.3, ( ) 0.6, ( ) 0.6, ( ) 0.9

μλμλμλμ

+= += += +=

have moduli less than one.

Linear Lyapunov Cone-Systems

187

The system is reachable and observable because the matrix

PBQ

−

has 2n

monomial

columns, and the matrix

VCP

−

has 2n

monomial rows.

The system is not controllable to zero in finite number of steps since the matrix

0.3 0 0 0

00.60 0

()

000.60

0000.9

PA P I I A

−

⊗+⊗ =

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

is not a the nilpotent matrix, but the system is controllable to zero in the infinite number of

steps since it is asymptotically stable.

6.3 Example 3

Consider the discrete-time Lyapunov system (34) with

and the matrices

0.5 0.8 0.17 0

,,,

0.6 0.9 1 0.4 1 1

24 00

,(2)

11 00

AAB

CD n

−

−−

===

−

===

⎡

⎤

⎢

⎥

⎡⎤⎡⎤

⎢

⎥

⎢⎥⎢⎥

⎣⎦⎣⎦

⎢

⎥

⎢

⎥

⎣

⎦

⎡⎤⎡⎤

⎢⎥⎢⎥

⎣⎦⎣⎦

(51)

This system is (P,Q,V) -cone-system with ,PQ and V defined by (49) since:

0.3 2 0.17 0 1 0 2 0 0 0

ˆˆ ˆ

ˆˆ

,,,,

00.1 1 0.4 01 01 00

AA BCD

⎡ ⎤ ⎡ ⎤ ⎡⎤ ⎡⎤ ⎡⎤

== ===

⎢ ⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢⎥

⎣ ⎦ ⎣ ⎦ ⎣⎦ ⎣⎦ ⎣⎦

The system is (P,Q,V) -cone-system because:

01 0 1

11 11 22 11

01 0 1

11 22 22 22

ˆˆ ˆˆ

0.3 0.17 0.25 0.72 0, 0.1 0.17 0.25 0.52 0

ˆˆ ˆˆ

0.3 0.4 0.25 0.95 0, 0.1 0.4 0.25 0.75 0

aaN aaN

aa N aa N

++= + + = > ++=+ + = >

++=++ = > ++=++ = >

and the matrices

ˆˆ

CD have nonnegative entries.

For the instant 100i

we have

(1) 0.8268 1

AIN I

+− = <

⎛⎞

⎜⎟

⎝⎠

∑

so the system is stable in the meaning of the the definition 18.

Automation and Robotics

188

The system is reachable and observable because the matrix

PBQ

−

has 2n

monomial

columns, and the matrix

VCP

−

has 2n

monomial rows.

The system is not controllable to zero in finite number of steps since the matrix

0.72 0 0 0

00.520 0

(())

000.950

0 0 0 0.75

nn n n

PA P I I A I N I

−

⊗+⊗ + ⊗ =

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

is not a zero matrix.

7. Conclusions

In this paper three types of systems have been considered. For the continuous-time linear

Lyapunov cone-systems, the necessary and sufficient conditions for being the cone-system,

the asymptotic stability and sufficient conditions for the reachability and observability have

been be established. For the discrete-time linear Lyapunov cone-systems, the necessary and

sufficient conditions for being the cone-system, the asymptotic stability, reachability,

observability and controllability to zero have been established. For the fractional discrete-

time linear Lyapunov cone-systems, the necessary and sufficient conditions for being the

cone-system, the reachability, observability and controllability to zero and sufficient

conditions for the stability have been established. The considerations have been illustrated

on the numerical examples.

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239.

Pneumatic Fuzzy Controller Simulation vs

Practical Results for Flexible Manipulator

Juan Manuel Ramos-Arreguin

, Jesus Carlos Pedraza-Ortega

Efren Gorrostieta-Hurtado

, Rene de Jesus Romero-Troncoso

Jose Emilio Vargas-Soto

and Francisco Hernandez-Hernandez

Universidad Tecnológica de San Juan del Río,

Universidad Autónoma de Querétaro,

Universidad de Guanajuato,

Universidad Anáhuac del Sur,

México

1. Introduction

The flexible manipulators have many industrial applications; but most of the reported

works are using electrical or hydraulic actuators. These actuators have a linear behaviour

and the control is easier than pneumatic actuators; but the pneumatic position control is a

highly non linear problem, due to the air compressibility behaviour and internal friction

(Moore & Pu, 1993). Because of these conditions, there are certain difficulties in pneumatics

cylinder control design. The main disadvantage of electrical actuators is the low power-

weight rate, the high current related with its load and it is heavy. The hydraulic actuators

are not ecological, needs hydraulic oil and return lines to the pump is needed. By the other

hand, the pneumatic actuators are clean, economy, light, faster, have a great power-weight

rate and return lines are not needed. However, pneumatic actuators are not used into

flexible manipulators developed due to their highly non linear behaviour. It is important to

note that research lines; such as pneumatic control, embedded systems and flexible

manipulators, are used in separate way.

As a matter of fact, most of manipulators robots use electric or hydraulic actuators; however,

the pneumatic actuators are being used in recent years (Ramos at al, 2006a; Ramos et al,

2006b) to control a flexible manipulator arm. This is the beginning of a project which

involves the use of a pneumatic cylinder to control a flexible manipulator robot. Our first

approach is to use one degree of freedom, but the main goal is to have a two degree of

freedom flexible manipulator.

Several pneumatic controllers had been developed; for example, the Model Reference

Adaptive Control, MRAC (Suarez & Luis, 2005); however, the pneumatic model used for

the control design, have the next considerations; a lineal actuator, a lineal valve, without

damping systems at the sides, ideal gas, adiabatic changes and constant viscous friction.

Other works have been focused in friction parameter identification techniques of cylinder

pneumatic (Wang & Wang, 2004), dynamic modelling and simulation (Jozsef & Claude,

Automation and Robotics

192

2003), analytic and experimental research (Henri & Hollerbach, 1998) and the development

of robotic hands using cylinder pneumatics.

Flexible manipulators can be used only under two conditions: a) when the robot weight

must be minimized, and b) when the collisions in the work space need to be avoided (Feliu

et al, 2001). The modelling of flexible manipulators has been developed almost 35 years ago

(Mirro, 1972; Whitney et al, 1974), where, almost in all cases, they used electric or hydraulic

actuators, and pneumatic cylinders are discouraged due to their non linear behavior.

Pneumatic control started in 1968 with Burrows (1968), and recent works are focused mainly

with adaptive control methods (Suarez & Luis, 2005; Quiles et al, 2004), some of them use a

computer to implement the control (Burbano et al, 2003). Other researches are focused on

mechanical system modelling using pneumatic actuators (Perez, 2003), from these kind of

works, a Flexible Manipulator Model with pneumatic cylinder -called Thermo-Mechanical

model- was developed, then the mechanical system is involved to give the movement for

the flexible arm (Kiyama & Vargas, 2004). By other hand, electric actuators are used in the

development of flexible manipulators (Feliu & Garcia, 2001), where the motor speed is

considered for the control implementation along with the motor effects and the mechanical

structure.

In our prototype we are using a flexible manipulator robot with a pneumatic actuator,

where the damping systems in both sides and the mechanical dynamics for control are

considered. The full Thermo-Mechanical model is used as a starting point, later it is

simplified and the results are used for the control development. One contribution of this

work is the position control of a flexible manipulator using a pneumatic actuator and a

simplified Thermo-Mechanical model (Ramos et al, 2006).

The simplified Thermo-Mechanical model of pneumatic actuators allows us to predict its

behavior, considering the air compressibility effects, internal friction forces, damping effects

in both extremes of the cylinder, massic flow and energy conservation; also, gives us the

instant pressure that depends on the rod position. From the engineering control point of

view, this model let us predict the variable behavior, involved in the physical process, and

can be used for control purposes.

This chapter is important, because the innovation of this work is the application of three

research lines to obtain a flexible manipulator light, ecological and fast with great power-

weight rate. The contribution of this chapter is the reported behaviour of the one-link

flexible manipulator with pneumatic actuator with practical control results. Simulations of

several controllers have been reported to learn about the pneumatic manipulator behaviour

with one-link. The simulation and practical results are discussed. Later, the graphical

simulation is important, because let us to obtain the control parameters in few time, and

learn about the pneumatic manipulator behaviour before implementation. Finally, the

control algorithm is implemented in Matlab language, and digital interfaces are

implemented into field programmable gate array (FPGA), obtaining an embedded digital

system with serial communication (RS232) protocol. The control algorithm implemented is a

PD controller and results are compared with Fuzzy-Controller simulation results.

2. The pneumatic system

The complete system is showed in figure 1, and we call the PLANT. The output plant is θ

corresponding to the arm elevation. The arm movement depends of the rod displacement