Compensation of Delay Time Uncertainties on Industrial Control Ethernet Networks using LMI

Based Robust H∞ PID Controller Endra Joelianto, Herman Y. Sutarto and Adi Wicaksono

Instrumentation and Control Research Group, Engineering Physics Study Program Institut Teknologi Bandung, Bandung 40132, Indonesia

e-mail: ejoel@tf.itb.ac.id Abstract-Industrial ethernet and wireless network become a common platform to implement industrial control in industries. Network control systems are subjected to delay time uncertainties due to the load traffic in the networks. As the PID controller is a major control algorithm in industries, this paper proposes a LMI based robust ∞H PID controller using robust integral backstepping method to compensate the delay time uncertainties. To design using robust integral backstepping method, the plant and the PID controller are represented as a full state feedback control system in state space representation. The obtained parameters can then be used in the installed PID controller in the network based industrial control systems.

I. INTRODUCTION With the development of the computer network technology and intelligent sensors and actuators, industrial data communication via ethernet and wireless as part of the process automation is being used more and more widely nowadays. This data communication is a network for connecting field devices: sensors, actuators and field controllers such as distributed control systems (DCS) or programmable logic controller (PLC), regulators, driver controllers and so on. It is a kind of real time communication systems and is based on a layered structure deduced from the seven layers OSI model. For safety-critical systems such as fire alarm system, to avoid data loss is of high importance. While for time-critical control systems such as real-time control system, to reduce message delay is one of the major considerations [1][2]. However, it is known that network based control systems always suffer to delay time uncertainties, known as latency [2]. PID controllers have been successfully used in industries since 1940s and remain the most often implemented controller today. The PID controller can be found in many application areas: petroleum processing, steam generation, polymer processing, chemical industries and many more. This success is a result of the many good features of the PID controller. Survey in Japanese industry shows that, in the future trend, enhanced PID algorithm will be widely used

This work was partially supported by the Incentive Research Program, State Ministry of Research and Technology under Grant 97M/Kp/XI/2007.

with large expectation [3]. The algorithm of PID controller is simple, single equation but it can provide good control performance for many different processes. This flexibility is achieved through several adjustable parameters whose values can be selected to modify the behaviour of the closed loop system. Because of its wide use, the PID controller is available in nearly all commercial digital control systems and becomes the standard controller in DCS and PLC, so that efficiently programmed and well tested implementation are available. Those commercial systems now implement industrial ethernet network as the main communication platform.

Fig. 1. Network Control System with PID Controller

A convenient feature of the PID controller is its compatibility with enhancement that provides capabilities not in the basic algorithm by means of its three adjustable tuning constants. Thus the basic PID controller can be improved without discarding it. Through judicious selection of their values, good control performance can be achieved by a proper choice of tuning constant values, but poor performance and even instability can result from a poor determination of values. Many tuning methods are available in the literature, among with the most popular method the Ziegler-Nichols (Z-N) method [4]. A drawback of many of those tuning rules is that such rules do not consider load disturbance, model uncertainty, measurement noise, and set-point response simultaneously. In general, a tuning for high performance is always accompanied by low robustness [5].

Difficulties arise when the plant dynamics are complex and poorly modelled or, specifications are particularly stringent.

PID Actuator

Sensor

Plant N E T WO R K

-

+

978-1-4244-1980-7/08/$25.00 ©2008 IEEE.

Even if a solution is eventually found, the process is likely to be expensive in terms of design time. Delay time is a difficult problem in the control system designs, hence the presence of uncertainties in the delay time makes the problem harder. This paper considers parameter selection problem of PID controller within the framework of robust control theory. That is, an alternative method to optimize the setting of a PID controller in order to achieve performance limits and to answer existence of satisfactory controllers. By using optimization method, an absolute scale of merits against which any design can be measured. In this case, the selection is made via control system optimization in robust control design by using linear matrix inequality (LMI). In order to design with robust control theory, the PID controller is expressed as a state feedback control law. The robust PID control problem can then be solved by using state feedback robust control methods, such as Guaranteed Cost Design using Quadratic Bound, and Quadratic Dissipative Linear Systems [6]. The robust PID controller has been tested by using Yokogawa's DCS CS1000 to control a mini plant simulator and have been reported on two undergraduate final project reports under the first author supervision [7][8]. Results showed that the robust PID controllers tend to produce good tracking response without overshoot. The robust PID controllers also resulted in good load disturbance response and can handle plant uncertainties caused by non linearity of the control valve. Although any robust control design can be implemented, in this paper, the investigation is focused on the theory for parameter selection using ∞H synthesis [9]. The paper also considers the

extension of ∞H method with an integrator backstepping [10].

II. PID CONTROLLER Consider a single input single output linear time invariant plant described by the linear differential equation

)()()( 1 tuBtAxtx +=& )()( tCxty = (1) with some uncertainties in the plant where x is the solution of (1), the control signal u is assumed to be the output of a PID controller with input y and then of the form. The PID controller for regulator problem is of the form

∫ ++=t

tydtdKtyKtdtyKtu

0321 )()()()()( (2)

which is an output feedback control system and ip TKK /1 = ,

pKK =2 , dpTKK =3 and pK , iT and dT denote proportional gain, time integral and time derivative of the well known PID controller respectively. The structure in equation (2) is known as the standard PID controller or ISA form. For tracking control problem, the output )(ty is replaced by the output error rye −≡ where r is the set-point.

In order to design with robust control theory, the PID controller (2) is expressed as a state feedback control law using (1) and by including the design of integral backstepping [10]. The robust integral backstepping PID control problem can then be solved by using state feedback robust control methods [11]. In this paper, the investigation is focused on computing robust integral backstepping PID controller using LMI approach. By using LMI, the robust integral backstepping PID controller stabilizes the uncertain plant (1), more specifically delay time uncertainties due to the load traffic in the networks. Following [12], the control law (2) is expressed as a state feedback law using (1). From (1), we have that

uCBuCABxCAyuCBCAxy

Cxy

&&&

&

112

1

++=+=

=

This means that the control signal (2) when differentiated may be written in more compact form as

xCKCAKCAKuCBK )()1( 122

313 ++−− & -

uCBKCABK )( 1213 + = 0 (3)

Using the notation K̂ as a normalization of K as follows

TT KKKK ]ˆˆˆ[ˆ321= =

TKKKCBK ][)1( 3211

13−− (4)

or TT cKK =ˆ where c is a scalar. This control law may be written as

xCACACKu TTTTTTT ])([ˆ 2=& +

uCABCBK TTTTTTT ]0[ˆ11 (5)

Equation (5) represents an output feedback law with constrained state feedback. That is, the control signal (2) may be written as

aaa xKu = (6) where

uua &= ,

=

ux

xa

KCABCBCACAC

K TTTTT

TTTTT

aˆ

0)(

11

2

= = K̂Γ (7)

The augmented system equation are given from (1) and (7) as

aaaaa uBxAx +=& (8)

where

=

101BA

Aa ;

=

10

B

Equation (7) and (8) show that the PID controller can be viewed as a state variable feedback law for the original system augmented with an integrator at its input [13].

III. ∞H BACKSTEPPING SYNTHESIS

Recently, a new control strategy called backstepping method has been developed in [10]. The method has received considerable attention and become a well known method for control system design. An effort to combine the advantage of the backstepping method and ∞H optimal control is studied for the case of PID controller with delay time uncertainties. This study is to develop a new robust PID controller by augmenting a system with an integrator backstepping. Consider the augmented plant of the form

)()()()( 21

tCxtyBwBtAxtx

=++= ξ&

(9)

and the objective state

)()()( 121 tuDtxCtz a+= (10) Also, it is given an integrator backstepping dynamic of the form

)()()( 2 tutt a+= ξρξ& ; 02 <ρ The augmented system is then given by the following

auttxBA

ttx

+

=

10

)()(

0)()(

2

2

ξρξ&&

and define the objective state as

auttx

DC

tz

+

=

1

12

1

00

)()(

000

0)(

ρξ

auttx

DC

tz

+

=

1

12

1

00

)()(

000

0)(

ρξ

If there exists a solution 0≥X and 0>γ of the following algebraic Riccati equation (ARE)

XBABA

X T

T

+

=

222

2 00

0ρρ

-

XBB

IX

T

−

−−

000

000 2222

1 γρ +

1212

11

00DD

CCT

T

For a 01 >ρ then the control law is given by

[ ]( )

−= −

ξρ

xXua 102

1

is stabilizing and leads to the infinity norm from w to z , γ≤∞|||| zwR . The PID ∞H -backstepping can then be

computed by using (7) and (4) to compute 21 , KK and

3K .

IV. FIRST ORDER SYSTEM WITH DELAY TIME UNCERTAINTIES

Consider a first order system with delay time which is common assumption in industrial process control

)(1

1)( sUes

sY Ls−

+=τ

In order to write in the state space representation, the delay time is approximated by using first order Pade approximation yields

)(11

11)( sU

dsds

ssY

++−

+=τ

, 2/Ld =

In this example, we choose nominal values of the system are s1=τ and sd 3= . The delay time uncertainties are

assumed in the range ]4,2[∈d . Defining

−

=11

011xx

AA ,

=

10211 xx

x

BBB ,

=

10011x

x

CC ,

=

001211 xx

x

DDD

−=

=

000011

0

00

11112

xx

n

BABA

A

=

=

00

0

111

1

x

n

BB

B ,

=

=

100

02 I

B n

zzn = ,

=

00000

00

12

11

1 x

x

n DC

C ,

=

1

12 00

ρnD

If there exists a solution 0≥X , 01 >ρ and 0>γ of the following algebraic Riccati inequality (ARI)

0)( 1111222 ≤+−−+ −

nTn

Tnn

Tnn

Tn CCXBBBBXXAXA γρ

For a 01 >ρ , then the control law is given by

[ ]( )

−= −

ξρ

xXua 102

1

is stabilizing and leads to the infinity norm from w to z , γ≤∞|||| zwR . The solvability of the ARI implies the solvability

of the ARE [14]. The algebraic Riccati inequality can then be considered as an LMI problem. That is to find 0≥X that satisfies Schur complement

00

00

0

121

121

1

≤

−−

+

I

N

IDBDIXCBXCXAXA

IN R

nTn

nn

nTn

TnnT

x

γγ

The state space representation for the nominal system is given by

−−=

016667.06667.1

nomA ;

=

01

nomB ;

[ ]6667.01−=nomC

The delay time uncertainties can be represented as

ddd nom βα += , 11 <<− δ

After simplification, the delay time uncertainties of any known ranges have Linear Fractional Transformation (LFT) representation as shown in the following figure.

In this representation, the performance of the closed loop system will be guaranteed for the specified delay time range with fast transient response ( zzn = ). With τ = 1 s and

nomd = 3 s, the full state feedback gain of PID is given by the following equation

In this example, we set 02 =ρ and choose different value

of 1ρ and γ . For 1ρ =1 and differentγ , we obtain the following PID parameters and transient performance, such as: settling time ( st ) and rise time ( rt ).

TABLE 1

PARAMETERS OF PID FOR DIFFERENT γ

γ 1ρ Kp Ki Kd ts tr

0,1 1 0,2111 0,1768 0,0695 10,8 12,7

0,248 1 0,3023 0,2226 0,1102 13,2 8,63

0,997 1 0,7744 0,3136 0,2944 4,44 5,79

1,27 1 10,471 0,5434 0,4090 9,27 2,59

1,7 1 13,132 0,746 0,5191 13,1 1,93

TABLE 2

PARAMETERS PID FOR DIFFERENT 1ρ

γ 1ρ Kp Ki Kd ts tr

0,997 0,66 11,019 0,1064 0,3127 39,8 122

0,997 0,77 0,9469 0,2407 0,3113 13,5 39,7

0,997 1 0,7744 0,3136 0,2944 4,44 5.79

0,997 1,24 0,4855 0,1369 0,1886 21,6 56,8

0,997 1,5 0,2923 0,0350 0,1151 94,4 250

The simulation results are shown in Figure 3 and 4, with γ

and 1ρ are denoted by g and r respectively. In order to test the robustness to the specified delay time uncertainties, the obtained robust PID controller with parameter γ =0.1 and

1ρ =1 is tested by perturbing the delay time from the nominal value in the range value of ]4,1[∈d . The results are shown in Figure 5.

Fig. 2. Interconnection delay time with the plant G

yΘ

u

uΘ

y

δ

d G

= δ

αβ,

10uFd

[ ]

−−=

^

^

3

^

2

1

3

3

2

1

01

6667.011

K

K

K

KKKK

Fig. 3. Transient response for different γ

Fig. 4. Transient response for different 1ρ

Fig. 5. Transient response for different d

Note that the LFT representation in Fig. 2 can also be extended to include plant uncertainties, multiplicative perturbation, pole clustering, etc. In which case, the problem will be considered as multi objective LMI based robust ∞H PID controller problem.

V. CONCLUSIONS This paper has considered the optimum setting of PID controller calculated by using LMI based ∞H method and

the extension using backstepping method. The robust ∞H PID controller has been applied as a controller to compensate delay time uncertainties caused by the presence of industrial ethernet network or wireless network in industries. If the range of the delay time uncertainties is predictable, then the stability and the performance of the closed loop system with the proposed method will be guaranteed. Simulation results show the superiority of this controller in dealing with delay time uncertainties. The developed controller has good properties from robust ∞H controller and the parameter flexibility of the backstepping method. Future research will be directed to the multi objective LMI based robust ∞H PID controller problem.

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