Coondoo I. (ed.) Ferroelectrics

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An Exact Impedance Control of DC Motors Using

Casimir Function

Satoru Sakai

Shinshu University

Japan

1. Introduction

This chapter gives a new and exact impedance control of DC motor. From Hogan’s original

work, the control input for impedance control is torque input since the impedance control is

designed for Lagrangian systems. However, in actual situation, there exist dynamics between

the torque and control input and this dynamics can be dominant in certain scale. In such

situation, if we neglect the dynamics or try to cancel the dynamcis, the standard impedance

control can lose the stability or the control performance at least.

To overcome this problem, we need an new impedance control which takes the dynamics into

account wihtout canceling any dynamics. In this chapter we give a solution for this problem

by focusing on Casimir function which is rarely used in the conventional robotics.

The rest of this chapter is organized as follwows. In Section 2, we give a new model of

DC motor with dynamics between the torque and control input. In Section 3, we propose

a new impedance control which is based on Casimir function. Casimir function is one of the

properties of port-Hamiltonian systems. In Section 4, we conﬁrm the proposed method in

numerical simulation and we conclude this chapter in Section 5.

2. Modeling

Let us start from a well-known model of DC motor:

⎡

⎣

⎤

⎦

⎡

⎣

00K

−

⎤

⎦

⎡

⎣

⎤

⎦

⎡

⎣

⎤

⎦

v (1)

where the displacement θ, the velocity ω and the current i are the states , the voltage v is the

control input with the torque constant K and the inductance L.

Although the system (1) is a third-order system and thus not mechanical system, the system (1)

has a mechanical-like structure, that is, can be modeled as a port-Hamiltonian system van der

Schaft (2000), Maschke & van der Schaft (1996) with a Hamilonian H

=(1/2J)p

+(1/2)Kr

⎡

⎣

⎤

⎦

⎡

⎣

010

−10

−

⎤

⎦



 

∇

H +

⎡

⎣

⎤

⎦

(2)

2 Ferroelectrics

where x =



qpr





θ Jω

√



is the new state,

v = v/

√

L is the new input

with

√

and

= ∇

is taken as the passive output. It is conﬁrmed that this system is now passive Takegaki &

Arimoto (1981) with respect to the Hamiltonian H, that is,

≤ y

holds. This passive property is studied in many systems and used as a structural property

to drive robust and nonliear controllers for stabilization Ortega & Garcia-Canseco (2004)

Stramigioli et al. (1998), trajectory tracking Fujimoto & Sugie (2001) and motion generation

problems Sakai & Stramigioli (2007).

However, in this chapter, we consider a different problem, namely, impedance control

problems and we do not focus on the passivity but focus on another structural property:

Lemma 1 Consider the system (1) with zero-input v

≡ 0 in the case of no dissipation R = 0.

Let the skew-symmetric part of the matrix A be J

(x)=J(x)

. Then the system has a solluton

of the following PDE

∇

C(x)J(x)=0

and the solution is characterized as

(x)=

+ r.

Proof This is confrimed by a direct calculation. (Q.E.D.)

This means that, in the case of no dissipation R

= 0, not only the Hamiltonian function H but

also the Casimir function C are constant

= 0 (u ≡ 0)

for any the value of the Hamiltonian function H. Then we can express the system (1) by using

the Casimir function (with respect to J) explicity.

Lemma 2 (Modeling) Consider the system (1) with zero-input v

≡ 0 in the case of no

dissipation R

= 0. Then the coordinate transformation convert the system (1) into

⎡

⎣

⎤

⎦

⎡

⎣

010

−100

000

⎤

⎦

∇

⎡

⎣

⎤

⎦

(3)

with the state x

=(qpC)

and the Hamiltonian function

(

)

−

KqC.

Proof This is also confrimed by a direct calculation although the old Hamiltonian H is not

equal to the new Hamiltonian

H. (Q.E.D.)

400

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400

Ferroelectrics

An Exact Impedance Control of DC Motors Using Casimir Function 3

3. Exact impedance control

In this section, we give an exact impedance control for DC motor by using Casimir functions.

Proposition 1 (Main result) Consider the system (1) with the velocity input. Then the following

controller

⎧

⎨

⎩

−(1 + k

)(

+ r)

v =

C/J

(4)

converts the close-loop system into the mechanical system with the impedance parameters J

, k

> 0.

Proof First we introduce an artiﬁcial Casimir function

C and via the following dynamic

extension

⎡

⎢

⎣

⎤

⎥

⎦

⎡

⎢

⎣

0 100

−1000

0 000

⎤

⎥

⎦

∇

⎡

⎢

⎣

⎤

⎥

⎦

(5)

where

v is the input corresponding to the (artiﬁcial) Casimir function.

Then the Hamiltonian function

H is replaced by the following new Hamiltonian function

which has a special structure suitable for impedance design with any parameters k

> 0 and

> 0 as follows:

mec

+ K



−



(6)

due to the deﬁnition of the Casimir function. Finally the dynamic controller



=+∇

mec

= −∇

mec

(7)

converts the system (1) with a dissipation R

≥ 0 into the the following Hamiltonian system

⎡

⎢

⎣

⎤

⎥

⎦

⎡

⎢

⎣

0100

−10 0 0

−R 1

−10

⎤

⎥

⎦

∇

mec

(8)

Fig. 1. A port-Hamiltonian system with ﬂow inputs.

401

An Exact Impedance Control of DC Motors Using Casimir Function

401

An Exact Impedance Control of DC Motors Using Casimir Function

4 Ferroelectrics

0 2 4 6 8 10

0.05

0.1

0.15

0.2

Time [sec]

=0.5, a=0.03

=0.5, a=3

=15, a=0.03

Fig. 2. Time response of H

mec

with

=(qpC

. (Q.E.D.)

The proposed impedance control does not input the torque but the velocity, unlike the

conventional impedance control for the mechanical systems. This difference is ullustrated

in Fig.1. The spring coefﬁcient k between the real mass and the virtual mass is not design

parameter unlike the spring coefﬁcient k

between the environment and the virtual mass.

Note that there is no canceling action in the controller.

4. Numerical simulations

Fig.2 shows the time response of the Hamiltonian function H

mec

in the case of no dissipation

(the Adams method) in the case of the parameters J

= 1.5 L = 0.165 K = 0.47, J

= 0.5, a = 0.03

and the initial conditions r

(0)=q(0)=0 p(0)=0.5. It is confrimed that the value is constant

as in the acutal Hamiltonian systems

Figs.3-5 show the time responses of the Casmir function and the state q and p in the case of

dissipation R

= 3.2. The parameters have changed J

→ 15 and a → 3.

In all cases, the nonliner behavior has changed intuitively due to the mechanial strucutre in

the closed-loop system. The validity of our methods are conﬁrmed.

5. Conclusions

there exist dynamics between the torque and control input and this dynamics can be dominant

in certain scale. In such situation, if we neglect the dynamics or try to cancel the dynamcis,

the standard impedance control can lose the stability or the control performance at least.

0 2 4 6 8 10

0.5

1.5

2.5

Time [sec]

=0.5, a=0.03

=0.5, a=3

=15, a=0.03

Fig. 3. Time response of C(t)

402

Ferroelectrics

402

Ferroelectrics

An Exact Impedance Control of DC Motors Using Casimir Function 5

0 2 4 6 8 10

0.5

1.5

Time [sec]

=0.5, a=0.03

=0.5, a=3

=15, a=0.03

Fig. 4. Time response of q(t)

0 2 4 6 8 10

0.5

Time [sec]

=0.5, a=0.03

=0.5, a=3

=15, a=0.03

Fig. 5. Time response of p(t)

To overcome this problem, we need an new impedance control which takes the dynamics into

account wihtout canceling any dynamics. In this chapter we give a solution for this problem

by focusing on Casimir function which is rarely used in the conventional robotics.

First we give a new model of DC motor with dynamics between the torque and control

input. Second, we propose a new impedance control which is based on Casimir function.

Casimir function is one of the properties of port-Hamiltonian systems. Finally, we conﬁrm the

proposed method in numerical simulation.

The generalization of the proposed method and appilications to other systems (such as

hydraulic systems and mascle-skelton systems) are next works in near future.

6. Acknowledgement

The author would like to show the appreciation to Mr. Ida Shinya for his helps in simulation

works.

7. References

Fujimoto, K. & Sugie, T. (2001). Canonical transformation and stabilization of generalized

Hamiltonian systems, Systems & Control Letters 42(3): 217–227.

Maschke, B. M. J. & van der Schaft, A. J. (1996). Interconnection of systems: the network

paradigm, pp. 207–212.

Ortega, R. & Garcia-Canseco, E. (2004). Interconection and damping assignment

passivity-based control: A survey, European Journal of Control pp. 1–27.

403

An Exact Impedance Control of DC Motors Using Casimir Function

403

An Exact Impedance Control of DC Motors Using Casimir Function

6 Ferroelectrics

Sakai, S. & Stramigioli, S. (2007). Port-hamiltonian approaches to motion generations for

mechanical systems, Proc. of IEEE Conference on Robotics and Automation, pp. 69–74.

Stramigioli, S., Maschke, B. M. J. & van der Schaft, A. J. (1998). Passive output feedback and

port interconnection, Proc. 4th IFAC Symp. Nonlinear Control Systems, pp. 613–618.

Takegaki, M. & Arimoto, S. (1981). A new feedback method for dynamic control of

manipulators, Trans. ASME, J. Dyn. Syst., Meas., Control 103: 119–125.

van der Schaft, A. J. (2000). L

-Gain and Passivity Techniques in Nonlinear Control,

Springer-Verlag, London.

404

Ferroelectrics

404

Ferroelectrics

Stabilization of Networked Control Systems with

Input Saturation

Sung Hyun Kim and PooGyeon Park

Pohang University of Science and Technology

Pohang, Kyungbuk, 790-784

Korea

1. Introduction

The technologies on multi-accessible communication networks have been recently received

considerable attention in the area of networked control systems (NCSs) since the

multi-accessibility feature leads to ﬂexibility, cost reduction, and distributiveness. However,

since the multi-accessibility sometimes causes random network-induced delays that

deteriorate the stability and control performance of closed-loop control systems, it is necessary

to always handle the delay problem when implementing a feedback control loop closed

through multi-accessible communication networks. As a result, numerous investigation and

research efforts are underway to deal with the delays (see e.g., Kim et al. (2003), Tipsuwan

and Chow (2003), Kim et al. (2004), Seiler and Sengupta (2005), Yue et al. (2005), Yang

(2006), and references therein), recent results of which have focused on the problem of

attenuating not only the effect of the network-induced delay but also that of unknown

disturbances. According to the trend, Kim et al. (2004) designed a network-delay-dependent

∞

controller for discrete-time systems over communication networks using a deterministic

approach. And Seiler and Sengupta (2005) established necessary and sufﬁcient linear matrix

inequality (LMI) conditions for the synthesis of the

∞

controller for discrete-time systems

with Markovian jumping parameters. And Yue et al. (2005) designed a robust

∞

controller

for uncertain continuous-time NCSs with the effects both of the network-induced delay and

of data dropout.

In addition to the networked-induced delay and the external disturbance problem, this paper

is interested in addressing the input saturation problem since every physical actuator is

subject to saturation, and moreover the input saturation deteriorates the stability of the

feedback control. Already for the traditional point-to-point communication network, various

research results of explicitly handling the input saturation have been published in the system

and control literature. Of them, several important results of directly handling the input

saturation have recently appeared well in Nguyen and Jabbari (2000), Gomes da Silva et al.

(2001), Hu and Lin (2001), Hu et al. (2002), and references therein. However, to the best of our

knowledge, there has been almost no results of considering the input saturation as a constraint

in designing the NCSs. Hence, in this paper, we intend to design an

∞

control capable of

reducing the adverse effects both of the input saturation as well as of the network-induced

delay.

The goal of this paper is to design a networked-delay-dependent switching controller

for systems with input saturation using the deterministic information from the current

2 Ferroelectrics

timestamp. The structural assumption of the networked system follows most of Kim et al.

(2004). Particulary, we are to employ a reliable transport protocol that guarantees data

delivery and supports that transmitted data have their time-stamp information. However,

taking a step forward in Kim et al. (2004), we insert a saturator between the controller

and the communication network, and extract built-in memory from the controller for

clarity of explanation (see Fig. 1). Based on the modiﬁed structure, we propose a

Fig. 1. Networked control system.

systematic method for designing a network-delay-dependent switching controller that

achieves the

∞

disturbance-rejection performance under an L

∞

performance representing

componentwise input saturation. Methodologically, we ﬁrst construct a suitable networked

control system (NCS) with asymmetric path-delay, and build up the conditions for set

invariance, involved in local stabilization, and then incorporate these conditions in the

synthesis of a network-delay-dependent

∞

controller. In the procedure, we use the polytopic

representation method, proposed by Hu and Lin (2001), as one way to deal with the saturation

nonlinearity, and employ a quasi-linear parameter-varying (QLPV) structure for the control

system as in Wu et al. (2005). Then the resultant convex solvability conditions are expressed

as a ﬁnite number of linear matrix inequalities (LMIs), whose solutions are applied to

the construction of the switching controller depending on the previous and current modes

characterized by network trafﬁc.

The chapter is organized as follows: Section 2 states target systems and assumptions. Section

3 proposes the condition for set invariance, and explains the procedure of constructing an

∞

dynamic state-feedback controller. Section 4 illustrates the performance of the proposed

algorithm through an example. Finally, in Section 5, concluding remarks are made.

Notation: Notations in this paper are fairly standard. For x

∈R

, ||x|| is taken to be the

standard Euclidian norm., i.e. ,

||x|| =(x

1/2

. The Lebesgue space L

= L

[0, ∞) consists

of square-integrable functions on

[0, ∞). L

∞

denotes the space of bounded vector sequences

(k) , equipped with the norm ||u||

∞

= sup

{sup

(k)|},andL

∞,e

denotes the space of

bounded vector sequences w

(k) , with the norm ||w||

∞,e

= sup

(k)w(k)}. The notation

≥Y and X > Y means that X −Y is positive semi-deﬁnite and positive deﬁnite, respectively.

Inequalities between vectors mean componentwise inequalities, svd

(·) means a singular value

decomposition function, and the saturation function sat

(u,

u) : R

→R

denotes

sat

(u,

u)=

[

··· s

]

, s

= sign( u

) · min{

,|u

|},(1)

where sign

(·) returns the signs of the corresponding argument, and u

and

stand for the

i-the element of u

∈R

and

u ∈R

, respectively. For a matrix V ∈R

r×s

, L(V) denotes a

406

Ferroelectrics

406

Ferroelectrics