Asian Journal of Control, Vol. 6, No. 4, pp. 483-495, December

2004

483

Manuscript received May 27, 2003; revised August 8, 2003;

accepted November 19, 2003.

Yongpeng

Zhang

and

Leang-San

Shieh

are

with

the

Department of Electrical and Computer Engineering, Uni-

versity of Houston, Houston, TX 77204-4005, U.S.A.

Cajetan M. Akujuobi and Warsame Ali are with

the Cen-

ter of Excellence for Communication Systems Technol-

ogy Research (CECSTR), Department of Electrical Engi-

neering, Prairie View A&M University, Prairie View, TX

77446, U.S.A.

This work was supported in part by the US Army Researc

h

Office under Grant DAAD 19-02-1-0321, and Texas Instru-

ments under Grant #410171-03001, and NASA-JSC unde

r

Grant #NNJ04HF32G.

DIGITAL PID CONTROLLER DESIGN FOR DELAYED

MULTIVARIABLE SYSTEMS

Yongpeng Zhang, Leang-San Shieh, Cajetan M. Akujuobi, and Warsame Ali

ABSTRACT

A new methodology is proposed to design digital PID controllers for

multivariable systems with time delays. Except for a few parameters that are

preliminarily selected, most of the PID parameters are systematically tuned

using the developed plant state-feedback and controller state-feedforward

LQR approach, such that satisfactory performance with guaranteed

closed-loop stability is achieved. In order to deal with the modeling error

owing to the delay time rational approximation, an IMC structure is utilized,

such that robust stability is achieved, without need for an observer, and with

improved online tuning convenience. Using the prediction-based digital re-

design method, the digital implementation is obtained based on the

above-proposed analog controller, such that the resulting mixed-signal sys-

tem performance will closely match that of the analog controlled system. An

illustrative example is given for comparison with alternative techniques.

KeyWords: PID, multivariable system, IMC, digital redesign, mixed- signal

system.

I. INTRODUCTION

The Proportional-Integral-Derivative (PID) controller

is the most popular form of controller utilized in the

control industry today, due to its simplicity in controller

structure, robustness to constant disturbances and avail-

ability of many tuning methods [1]. To unify and en-

hance the recent progress in PID control, several special

issues [9,20] and monograph [24] on advances in PID

control have been published over the past few years.

The PID controller structure comes in two forms, as

standard Single-Input-Single-Output (SISO) PID con-

trollers and Multi-Input-Multi-Output (MIMO) PID

controllers. The design, tuning and implementation of

MIMO PID controllers are relatively more complex

compared with those of SISO PID controllers [5],

mostly due to the renowned difficulty arising from loop

interaction (or coupling). One favorable controller

structure in multivariable system design is the decen-

tralized controller, which means all the off-diagonal

elements of the transfer function matrix of the controller

are zero. Although this constraint on the controller

structure may lead to performance deterioration when

compared with the centralized controller, the decentral-

ized controller remains popular in applications [19]. The

underlying reasons can be summarized as follows: (i)

SISO PID controller design methods can be directly

exploited in decentralized controller design [7]; (ii) The

hardware simplicity considerably reduces the complex-

ity and cost in implementing the decentralized controller

[17].

With the rapid progress in microelectronics tech-

nology, the digital controller is now widely applied in

industry for better reliability, lower cost, smaller size,

and better performance. In addition, it can provide sub-

484

Asian Journal of Control, Vol. 6, No. 4, December 2004

stantial convenience in implementing the controller with

sophisticated interconnected structure, which can be

easily realized or modified through programming [2].

To exploit the advantages of digital controllers over

analog controllers, several design methods [10,11,21-23,

28] have been developed in the discrete-time domain via

the direct digital design approach, that is to discretize

the analog plant and then determine a digital controller

for the discretized plant in the discrete-time domain.

Nevertheless, the direct digital design methods (such as

the deadbeat design method), in general, cannot take

care of the inter-sample behavior [2], which may lead to

performance deterioration, especially when the sampling

period is relatively long. In addition, if the continuous-

time plant ‘A’ matrix has a physical parameter of inter-

est in only one or two entries, then these parameters are

transposed to other elements in the corresponding dis-

crete-time system matrix ‘exp (AT)’, where T is a sam-

pling period. As a result, we may lose physical insight

of these parameters and design for these parameter

variations becomes very difficult, especially for a

high-order multivariable control system.

In this paper, we choose the digital redesign ap-

proach [6,16] to design the MIMO PID controller. In

this approach, a centralized analog MIMO PID control-

ler is designed in the continuous-time domain, then re-

designed to its digital counterpart, while keeping the

essential system performance unchanged. The resulting

digitally controlled analog system becomes a hybrid

system (i.e., mixed-signal system), which consists of a

continuous-time plant and a discrete-time controller [16].

The digital redesign approach is adopted here based on

the following considerations: (i) Continuous-time design

theory and tools are well developed, and intuitively

more familiar to most engineers; (ii) Digital redesign

separates the problems of basic controller design from

that of sampling period selection and adjustment [15];

(iii) The digitally redesigned controllers are able to take

care of the inter-sample behavior [8] of the designed

closed-loop mixed-signal (sampled-data) system, and

they can be easily implemented using digital electronics.

However, the analysis of the designed hybrid multivari-

able system is not a simple matter in comparison with

the direct digital design approach.

This paper proposes a feedback/feedforward design

methodology, in which the PID tuning problem is

transformed into a Linear Quadratic Regulator (LQR)

design via proper arrangement of the state space equa-

tion of the cascaded system. Thus, the tuning of most

PID parameters is automatically carried out by solving

LQR design problems, except for a few parameters that

are pre-selected. Compared with existing methods, our

proposed method offers the following advantages: (i)

The stability of the MIMO PID controlled system is

guaranteed during the tuning process [30]; (ii) There are

no specific requirements on system stability, mini-

mum-phase property, low-order model and plant de-

coupling [31]; (iii) The designed centralized analog

MIMO PID controllers can be conveniently imple-

mented using digital electronics after digital redesign.

The presence of time delays in many industrial

processes is a well recognized phenomenon [23]. The

achievable performance of conventional unity-feedback

control processes can be significantly degraded if a

process has a relatively large time delay compared to the

dominant time constant [12]. Predicted delay time com-

pensation is hard to implement since only a causal and

rational controller is realizable. Hence delayed processes

pose a serious challenge to the design of robust feedback

controls for such systems [18]. The most popular

scheme for time delay compensation is possibly the

Smith Predictor Control [22]. Under the Smith predictor

configuration, the control signal is computed on the ba-

sis of the predicted system states and current prediction

error. Thus the controller can be designed with respect

to the delay-free portion of the system. Some other

method with the similar Smith structure like GPC (Gen-

eralized Predictive Control) has been successfully used

in PID controller design [21,22] for delayed SISO sys-

tems. It has been reported that GPC can also be applied

to delayed MIMO system design [10,11]. However, it

has to rely on multivariable system decoupling, that is to

roughly regard the decoupled MIMO system as a series

of SISO subsystems. In addition, in order to reinstate the

performance in the presence of load disturbances, and/or

the effects of modeling error and parameter uncertainty,

some other robustness consideration is needed such as

online self-tuning [10,21], which will considerably in-

crease the real-time computation burden.

In this paper, the pure delay terms are first ap-

proximated by a rational Pade approximation model

[27], then the proposed feedback/feedforward design

method is applied to get the PID controller parameters

based on the approximated system. As some modeling

error is inevitably brought in through the approximation,

an Internal Model Control (IMC) structure is utilized, in

which an approximate model of the plant is constructed

as part of the controller. The control signal is applied to

the approximate plant model, as well as the plant itself.

By comparing the process output with the expected

process response, modeling error and disturbances can

be identified and responded to accordingly [28]. The

advantages of utilizing such an IMC structure in the

proposed design methodology are the following: (i) An

IMC-based control system is more robust, which is

widely recognized in circumstances of imperfect mod-

eling [3]; (ii) The proposed IMC structure can provide

direct access to the internal approximate system states

Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems

485

[22], thus making it unnecessary to design an observer

for implementing the proposed feedback/feedforward

PID controller; (iii) Since it is very convenient for the

digital controller to realize the complicated

intercon-

nected

structure,

the

designed

IMC-based PID controller

is not necessary to endure a reduced-order processing

[25]. As a result, some potential performance deteriora-

tion is avoided.

To utilize the advantages of digital controllers over

analog controllers, the proposed analog multivariable

PID controller is transformed into the equivalent digital

implementation through the prediction-based digital

redesign method [8], such that the mixed-signal system

performance closely matches that of the analog con-

trolled system.

The paper is organized as follows. Section 2 dis-

cusses the problem formulation, while the MIMO PID

controller tuning with state-feedback and state-feed-

forward LQR are discussed in section 3. The IMC-based

controller structure, the robust stability analysis and the

digital redesign of the analog multivariable PID con-

troller are discussed in sections 4, 5, and 6, respectively.

An illustrative example is given in section 7, followed

by conclusions in section 8.

II. PROBLEM FORMULATION

Time delays are common phenomena in many in-

dustrial processes [29]. Suppose a multivariable delayed

system is defined as

ˆ

1

G (s)

, in which every component

is described as g

ij

(s)e

−sT

ij

, for i = 1, 2, …, p1 and j = 1, 2,

…, m1. The structure of the delayed multivariable sys-

tem is shown in Fig. 1.

In order to obtain a rational transfer function, the

time delay term can be approximated by a first order

Pade approximation model [7,26] as,

2

2

sT

s

T

e

s

T

−

−

≈

+

. (1)

Then, the resulting approximate rational model is de-

fined as G

1

(s), in which every component is repre −

a

y

22

)(

22

sT

esg

−

11

)(

11

sT

esg

−

b

y

a

u

b

u

12

)(

12

sT

esg

−

21

)(

21

sT

esg

−

Fig. 1. A delayed multivariable (2

×

2) system structure.

sented as

2

( )

2

ij

ij

ij

s

T

g s

s

T

−

+

, for i = 1, 2, …, p1 and j = 1, 2,

…, m1.

It should be mentioned that the rational approxima-

tion inevitably introduces some modeling error [26],

which could lead to performance deterioration, or even

destabilize the entire system, especially when the delay

time is relatively long. For convenience, we skip over

the modeling error problem in sections 2 and 3, and

propose the corresponding solving method for the mod-

eling error problem in section 4.

Consider a unity-output feedback MIMO analog

plant G

1

(s) ∈ R

p1 × m1

, in cascade with a MIMO analog

PID controller G

2

(s) ∈ R

m1 × p1

. Also, suppose output

disturbance d(t) ∈ R

p1

exists at the output point as

shown in Fig. 2.

)(tr

)(ty

PID controller

c

E

)(td

)(

1

ty

plant model

)(

2

sG

)(

1

sG

)(

2

tu

)(

1

tu

)(

2

ty

Fig. 2. Continuous-time cascaded system.

Let the minimal realization of the analog plant G

1

(s)

be

=

+

1 1 1 1 1

x (t) A x (t) B u (t)

,

=

1 10

x (0) x

,

=

1 1 1

y (t) C x (t)

, (2a)

where x

1

(t) ∈ R

n1

, u

1

(t) ∈ R

m1

, y

1

(t) ∈ R

p1

, and A

1

, B

1

,

C

1

are constant matrices of appropriate dimensions.

Let the entire system output be the sum of the plant

output and load disturbance,

=

+

1

y(t) y (t) d(t)

, (2b)

where y(t) ∈ R

p1

, d(t) ∈ R

p1

.

Each component of the MIMO PID controller G

2

(s)

∈ R

m1 × p1

is described as

2

( )

I ij Dij

ij Pij

ij

K

K s

G s K

s s

= + +

+

α

, (3)

for i = 1, 2, …, m1 and j = 1, 2, …, p1. Parameters of

the PID controller, K

Pij

, K

Iij

, K

Dij

, and filter factor α

ij

are

constants to be determined.

Our proposed MIMO controller design methodology

requires a preliminary design procedure to provide an

initial setting for the PID parameters. A subsequent tune

up process then adjusts most parameters based on the

plant and controller properties. Any available method

can be applied in the preliminary design, or the existing

486

Asian Journal of Control, Vol. 6, No. 4, December 2004

PID controllers can be used as the initial setting. Alter-

natively, the following initial design method can be

adopted.

The filter factor α for the derivative term is used for

physical implementation of the PID controller, and it

can be chosen by the manufacturer or design specifica-

tions [1]. In the presence of load disturbance in the con-

trolled system, it is convenient to achieve steady state

compensation by properly selecting the filter factor.

Based on the Internal Model Principle (IMP) [7], steady

state disturbance compensation requires that the distur-

bance generating polynomial be included as part of the

controller denominator. Thus, we can determine de-

nominators of the controller through setting the filter

factors (i.e., s + α

ij

in (3)) as those of the load distur-

bance denominators. When the load disturbance is dif-

ferent for each subsystem, we can adjust the filter fac-

tors accordingly. Then, the transfer function matrix of

the controller can be rewritten from (3) as follows

1 1 1

1

1

( )

p p

p

j

j

s s s

s

s s

+

+ +

=

+ + + +

=

⋅ +α

∏

1 2 p1 1 p1 2

2

E E E E

G ( )

, (4)

where parameter matrices E

1

, E

2

,

…, E

p1+2

are un-

knowns to be determined.

Although our proposed methodology for MIMO

PID controller design does not require that the plant be

of, or can be decoupled into a form of SISO systems, we

let the plant be diagonally dominant, so that the de-

signed controller would be near to being decentralized.

The static decoupler [4,7] is widely used as a

pre-compensator in MIMO system design for achieving

approximate decoupling. In general, this decoupler is

defined as D

2

= G

1

−1

(0), where G

1

(0) is non-singular.

We can make the plant statically decoupled by setting

E

1

= D

2

= G

1

−1

(0) in (4), which then gives

1 1 1

1

1

( )

p p

p

j

j

s s s

s

s s

−

+ +

=

+ + + +

= +

⋅ +α

∏

2 3 p1 1 p1 2

2 2

F F F F

G ( ) D

, (5a)

where F

2

, F

3

,…, F

p1+2

are constant matrices to be deter-

mined. For simplicity, we can first select the MIMO

PID controller as

1

( )

j

diag

s s

⎧ ⎫

⎪ ⎪

= +

⎨ ⎬

+α

⎪ ⎪

⎩ ⎭

2 2

G (s) D

,

for

1,2,...,1j p=

, (5b)

and leave further parameter adjustment to the next tun-

ing phase.

Then, the minimal realization of the preliminarily

designed cascaded PID controller G

2

(s) can be written

as

=

+

2 2 2 2 2

x (t) A x (t) B u (t)

,

=

2

x (0) 0

, (6a)

=

+ =

2 2 2 2 2 1

y (t) C x (t) D u (t) u (t)

, (6b)

=

− +

2 c

u (t) y(t) E r(t)

, (6c)

where x

2

(t) ∈ R

n2

, u

2

(t) ∈ R

p1

, y

2

(t) ∈ R

m1

, r(t) ∈ R

p1

,

and A

2

, B

2

, C

2

, D

2

, E

c

, are constant matrices of appro-

priate dimensions.

III. THE MIMO PID CONTROLLER TUN-

ING WITH STATE-FEEDBACK AND

STATE-FEEDFORWARD LQR

Having developed a framework of the problem to

be solved in the previous section, along with prelimi-

nary controller parameter selection, we now address the

problem of fine-tuning the PID controller parameters in

the context of an LQR formulation. This further tuning

step is necessary to achieve satisfactory closed-loop

performance.

Based on the pre-designed controller, the basic idea

of our methodology is to transform the PID tuning

problem to that of optimal design. To achieve this aim,

we formulate the closed-loop cascaded systems in Fig. 2

into an augmented system as

=

+ +

e e e e 1 e

x (t) A x (t) B u (t) E r(t)

,

=

=

e 1 e e

y (t) y (t) C x (t)

, (7)

where

⎡

⎤ ⎡ ⎤ ⎡ ⎤

= = =

⎢

⎥ ⎢ ⎥ ⎢ ⎥

⎢

⎥⎢⎥⎢⎥

−

⎣

⎦ ⎣ ⎦ ⎣ ⎦

1 1

e e e

2 1 2 2 c

A 0 B 0

A,B,E,

B C A 0 B E

[ ]

,

⎡ ⎤

= =

⎢ ⎥

⎢ ⎥

⎣ ⎦

1

e e 1

2

x (t)

x,C C 0

x (t)

and

=

− − −

1 1 1 2 2 2 2

u (t) K x (t) K x (t) D u (t)

, (8)

where the state feedback control gains K

1

and K

2

are to

be designed as shown below. For the convenience of

LQR design of u

1

(t), given the existence of the feedfor-

ward input term D

2

u

2

(t) in (8), an alternative representa-

tion of the augmented system (7) can be described as

ˆ

ˆ

ˆ

= + +

e e e e 1 e

x (t) A x (t) B u (t) E r(t)

, (9a)

Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems

487

= =

1 e e e

y (t) y (t) C x (t)

, (9b)

where

ˆ

= +

1 1 2 2

u (t) u (t) D u (t)

, (9c)

= − +

2 1 c

u (t) y (t) E r(t)

, (9d)

and

ˆ

+

⎡ ⎤

=

⎢ ⎥

⎢ ⎥

−

⎣ ⎦

1 1 2 1

e

2 1 2

A B D C 0

A

B C A

,

⎡ ⎤

=

⎢ ⎥

⎢ ⎥

⎣ ⎦

1

e

B

B

0

,

ˆ

−

⎡ ⎤

=

⎢ ⎥

⎢ ⎥

⎣ ⎦

1 2 c

e

2 c

B D E

E

B E

,

⎡ ⎤

=

⎢ ⎥

⎢ ⎥

⎣ ⎦

1

e

2

x (t)

x

x (t)

,

[

]

=

e 1

C C 0

.

The state-feedback LQR for the augmented system (9)

can be chosen as

ˆ

= −

1 e e

u (t) K x (t)

, (10)

where

K

e

= [K

1

, K

2

]

with

K

1

∈ R

m1 × n1

and

K

2

∈ R

m1 × n2

.

Let the quadratic cost function for the system (9) be

ˆ ˆ

∞

= +

∫

T T

e e 1 1

0

J [x (t)Qx (t) u (t)Ru (t)]dt

, (11)

where Q ≥ 0, R > 0, (Â

e

, B

e

) is controllable and (Â

e

, Q)

is observable. The optimal state-feedback control gain

K

e

in (10) that minimizes the performance index (11) is

given by

1−

=

T

e e

K R B P

, (12)

in which the matrix P > 0 is the solution of the Riccati

equation [14],

1

ˆ ˆ

T T

−

+ − + =

e e e e

PA A P PB R B P Q 0

. (13)

The resulting closed-loop system becomes

ˆ

ˆ

= − +

e e e e e e

x (t) (A B K )x (t) E r(t)

, (14)

which is asymptotically stable due to the property of

LQR design.

Defining

K

1

as the plant state feedback gain,

K

2

as

the controller feedforward gain in (10), then the desired

state-feedback and state-feedforward control law

u

1

(t)

in

(8) can be indirectly determined from the state-feedback

control law (10) and the relationships shown in (9c) and

(9d) as

ˆ

= − −

1 1 2 2

u (t) u (t) D u (t)

[ ]

= − − − −

1 1 2 2 2 c 1

K x (t) K x (t) D E r(t) y (t)

= − − − −

1 2 1 1 2 2 2 c

(K D C )x (t) K x (t) D E r(t)

, (15)

where

ˆ

1

K

= K

1

−D

2

C

1

,

ˆ ˆ

[,]=

e 1 2

K K K

and Ê

c

= −D

2

E

c

.

Substituting the control law in (15) into (7), the de-

signed closed-loop system becomes

− − −

⎡

⎤ ⎡ ⎤ ⎡ ⎤

=

⎢

⎥ ⎢ ⎥ ⎢ ⎥

⎢

⎥⎢ ⎥⎢ ⎥

−

⎣

⎦ ⎣ ⎦ ⎣ ⎦

1 1 1 1 2 1 1 2 1

2 2 1 2 2

x (t) A B (K D C ) B K x (t)

x (t) B C A x (t)

−

⎡ ⎤

+

⎢ ⎥

⎢ ⎥

⎣ ⎦

1 2 c

2 c

B D E

r(t)

B E

, (16)

=

1 1 1

y

(t) C x (t)

. (17)

The block diagram of the designed augmented system

(7) with the controller (15) is shown in Fig. 3.

)(

2

tx

)(tr

)(

1

ty

)(

1

tu

)()()(

2222

2

tuBtxAtx +=

•

)()()(

1111

1

tuBtxAtx +=

•

2

K

)(

2

tu

2

D

1

K

1

C

plant model

PID controller

c

E

)(

1

tx

Fig. 3. PID controlled system.

Compared with existing methods, advantages of our

method are: (i) MIMO PID parameters are systemati-

cally tuned with respect to both plant and controller,

while the closed-loop system stability is guaranteed;

(ii) There are no specific requirements on system stabil-

ity, simplification model and plant decoupling.

IV. IMC-BASED CONTROLLER

STRUCTURE

The first order rational Pade approximation of the

time-delay element discussed in section 2 inevitably

introduces some modeling error [26]. To deal with this,

the Internal Model Control (IMC) structure, which is

recognized as very usefully for handling imperfect mod-

eling [3], is introduced below.

The typical IMC structure is shown as Fig. 4, in

which a nominal (approximate) plant model is con-

structed as part of the controller. The control signal from

q is applied to the nominal plant model p, as well as the

plant

p

itself. By comparing the process output with

the expected process response, the difference caused by

modeling error and disturbance is fed back as shown.

If the model is exact (

p p

=

), then the above block

diagram of Fig. 4 will become open loop. It can be ar-

gued that the lack of model uncertainty (error) is an arti-

ficial assumption. However, in any practical situation it

is unacceptable to rely on model uncertainty to construct

the closed-loop structure [13]. Besides, some other

transformation is needed to consider the specific struc-

488

Asian Journal of Control, Vol. 6, No. 4, December 2004

ture of the proposed control system shown in Fig. 3.

Based on these considerations, we present the im-

plementation block diagram for the proposed IMC based

feedback/feedforward PID controlled system in the fol-

lowing Fig. 5. Following the definitions given in section

2, the approximate plant model is G

1

(s), and its rational

transfer function matrix is G

1

(s) = C

1

(sI – A

1

)

−1

B

1

+ D

1

.

On the other hand, the true plant model with time-delay

is described as Ĝ

1

(s). Besides, the PID controller trans-

fer function matrix is G

c

(s) = − [K

2

(sI – A

2

)

−1

B

2

+ D

2

].

The advantages of such an IMC structure are: (i)

We do not need to design a plant state observer to ob-

serve plant states, which are directly accessible in the

internal approximate plant model; (ii) Model uncertain-

ties can be detected through comparing process output

and simulation result, thus providing great convenience

for on-line tuning.

)(tr

)(ty

controller

plant model

)(td

q

p

~

plant

p

Fig. 4. Internal Model Control (IMC) structure.

r

y

PID

1

B

1

C

∫

1

K

1

u

1

x

1

A

d

1

y

)(

ˆ

1

sG

)(sG

c

y

∆

y

plant

Fig. 5. Implementation structure of the proposed

analogously controlled system.

V. ROBUST STABILITY ANALYSIS

Next, we develop the robust stability analysis for

the modeling error. Since the external output distur-

bance has previously been considered for steady state

compensation, and it has no effect on the stability

analysis, we assume there is no external disturbance in

the following.

The additive modeling error is defined as the dif-

ference between the true plant model and the approxi-

mate (nominal) plant model as

ˆ

= −

1 1 1

∆G (s) G (s) G (s)

. (18)

Assume the additive modeling error can be interpreted

as a scalar weight on a normalized perturbation,

( )

L s

=

⋅

1

∆G (s) ∆

, (19)

where ∆ is the normalized perturbation, that is ||∆||

∞

≤

1,

and

( )

L s

is a scalar weight variable. Then, we get the

modeling uncertainty description shown in Fig. 6 below,

which is transformed from Fig. 5, where G

x

(s) = [I + K

1

(sI − A

1

)

−1

B

1

]

−1

.

r

PID

1

B

1

C

∫

1

A

1

K

1

u

)(sG

c

∫

1

A

1

B

)(sG

x

1

y

a

b

∆

y

∆

)(

1

sG

y

)(sL

Fig. 6. Modeling error structure for robust analysis.

If the approximate model G

1

(s) has the same num-

ber of unstable poles as that of the real model Ĝ

1

(s),

then it follows from robust stability analysis [32] that

the above system in Fig. 6 is robustly stable if

|| || || ( )L s

∞

= ⋅ ⋅

ab x c

T (s) G (s) G (s)

||

−

∞

⋅

+ <

1

P c

[I G (s)G (s)] 1

, (20)

where

1 x

−

= − − + = ⋅

1

P 1 1 1 1 1 1

G (s) C [sI (A B K )] B D G (s) G (s)

.

The modeling error due to the delay time rational ap-

proximation will increase with frequency increasing.

The filter factor in the PID controller can be regarded as

a kind of low-pass filter, which if not needed in the

output disturbance steady state compensation, can be

adjusted in the tuning up process until satisfactory per-

formance and robust stability are achieved.

Actual modeling error is not limited to the time de-

lay approximation; there will always be some mismatch

between the mathematical model and the physical sys-

tem. It is still possible for the designed PID controller to

guarantee a zero tracking error for step input. Assume

( )

L

ω

is continuous, the qualitative analysis for the

condition is shown as below. Steady state “perfect con-

trol” [13] means

−

⋅

⋅ + =

1

P c P c

G (0) G (0) [I G (0)G (0)] I

, (21)

which, when substituted into (20) gives the following

condition

|| (0) || || (0) 0 || 1

L

−

∞ ∞

=

⋅ <

1

ab 1

T G ( )

. (22)

This means the designed system will demonstrate a zero

steady state error despite modeling error, provided con-

dition (22) is satisfied.

Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems

489

VI. DIGITAL REDESIGN OF THE ANALOG

MULTIVARIABLE PID CONTROLLER

Finally, we now transform the obtained multivari-

able analog PID controller to its digital implementation,

on the condition that the resulting mixed-signal (sam-

pled-data) system response closely matches that of the

analogously controlled system.

The prediction-based digital redesign method [8]

can be briefly introduced as follows. Consider a linear

controllable continuous-time system, described by

= +

c c c

x (t) Ax (t) Bu (t)

,

=

c 0

x (0) x

, (23)

where x

c

(t) ∈ R

n

, u

c

(t) ∈ R

m

, A and B are constant ma-

trices of appropriate dimensions. Let the continuous-

time state-feedback control law be

= − +

c c c c

u (t) K x (t) E r(t)

, (24)

where K

c

∈ R

m × n

and E

c

∈ R

m × n

have been designed

to satisfy some specified goals, and r(t)

∈ R

m

is a

piecewise-constant reference input vector.

The closed-loop state of the digitally controlled

system will closely match that of the analogously con-

trolled system at all the sampling instants, if the equiva-

lent discrete-time state-feedback control law is defined

as

= − +

*

d d d d

u (kT) K x (kT) E r (kT)

, (25)

where

−

= +

1

d c c

K (I K H) K G

,

−

= +

1

d c c

E (I K H) E

,

= +

*

r (kT) r(kT T)

,

=

AT

G e

,

−

= −

1

H (G I)A B

.

For details, readers are referred to the Appendix. As the

above redesign method cannot be applied directly to the

unity output feedback cascaded system, some transfor-

mation as shown below is necessary.

From (16), the analogously controlled system has

the form as:

− − −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

−

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 1 1 1 2 1 1 2 1

2 2 1 2 2

x (t) A B (K D C ) B K x (t)

x (t) B C A x (t)

−

⎡ ⎤

+

⎢ ⎥

⎢ ⎥

⎣ ⎦

1 2 c

2 c

B D E

r(t)

B E

,

(t)xC(t)y

111

=

. (26)

This can be transformed to the following structure,

⎡ ⎤ ⎡ ⎤

⎛ ⎞

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= −

⎢ ⎥ ⎢ ⎥

⎜ ⎟

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎜ ⎟

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎝ ⎠

⎣ ⎦ ⎣ ⎦

1 1

1 1 1 2

2 2

2 2

x (t) x (t)

A 0 B 0 K K

0 A 0 B 0 0

x (t) x (t)

( )

⎡ ⎤

−

⎡ ⎤

+ ⋅ −

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

⎣ ⎦

1

2

2

B 0

D

r(t)

y

(t)

I

0 B

. (27)

Define A

f

= block diag. {A

1

, A

2

}, B

f

= block diag. {B

1

,

B

2

},

⎡

⎤

=

⎢

⎥

⎣

⎦

1 2

f

K K

K

0 0

,

−

⎡ ⎤

=

⎢ ⎥

⎣ ⎦

2

f

D

E

I

. The discrete-time

conversion gives the following terms, G

f

= block diag.

{G

d1

, G

d2

}, H

f

= block diag. {H

d1

, H

d2

}, in which G

d1

=

e

A

1

T

, H

d1

= [G

d1

− I]

−

1

1 1

A B

, and G

d2

= e

A

2

T

, H

d2

= [G

d2

− I]

−

1

2 2

A B

. Applying the afore-mentioned prediction-

based digital redesign method, gives the corresponding

digital gains from (27) as

−

−

⎧

⎡ ⎤

= + ⋅ ⋅ =

⎪

⎢ ⎥

⎣ ⎦

⎪

⎪

⎨

⎪

⎡ ⎤

= + ⋅ =

⎪

⎢ ⎥

⎪

⎣ ⎦

⎩

d1 d2

1

d f f f f

d1

1

d f f f

K K

K (I K H ) K G

0 0

E

E (I K H ) E

I

. (28)

Then, we get the discrete-time controlled system model

as:

+

⎡ ⎤

⎛

⎡

⎤

=

⎢ ⎥

⎜

⎢

⎥

⎜

⎢ ⎥

+

⎣

⎦

⎝

⎣ ⎦

d1

d1

d2

d2

x (kT T)

G 0

0 G

x (kT T)

⎡ ⎤

⎞

⎡ ⎤ ⎡ ⎤

−

⎢ ⎥

⎟

⎢ ⎥ ⎢ ⎥

⎟

⎢ ⎥

⎣ ⎦ ⎣ ⎦

⎠

⎣ ⎦

d1

d1 d1 d2

d2

d2

x (kT)

H 0 K K

0 H 0 0

x (kT)

( )

⎡ ⎤

⎡ ⎤

+ ⋅ −

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

⎣ ⎦

d1

d1

d2

H 0

E

r(kT)

y

(kT)

I

0 H

, (29)

from which, we can get the corresponding digital con-

trol law as

= − −

d1 d1 d1 d2 d2

u (kT) K x (kT) K x (kT)

( )

+

−

d1

E r(kT)

y

(kT)

. (30)

Thus, we get the resulting mixed-signal system as

shown below in Fig. 7, where G

c

(z) = −K

d2

(zI –

G

d2

)

−1

H

d2

, z.o.h. represents the zero-order-hold, and the

part in the dotted line is the digital implementation

structure.

)(kTr

)(ty

1

C

)(

1

kTu

d

)(

1

kTx

d

)(td

)(

1

kTy

)(zG

c

1−

z

1d

K

...hoz

T

)(kTy

∆

)(kTy

1d

E

plant

1d

G

1d

H

Fig. 7. Implementation structure of the proposed mixed-signal

system.

490

Asian Journal of Control, Vol. 6, No. 4, December 2004

VII. AN ILLUSTRATIVE EXAMPLE

Consider the well known Wood and Berry empiri-

cal model Ĝ

1

(s) of a pilot-scale distillation column that

is used to separate a methanol-water mixture [4],

3

1

7 3

1

12.8 18.9

( )

( )

16.7 1 21 1

( ) ( )

6.6 19.4

10.9 1 14.4 1

s s

a

a

s s

b b

e e

u s

y s

s s

y s u s

e e

s s

− −

− −

⎡ ⎤

−

⎢ ⎥

⎡ ⎤

⎡ ⎤

+ +

⎢ ⎥

⎢ ⎥

=

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

−

⎣ ⎦

⎣ ⎦

⎢ ⎥

⎢ ⎥

+ +

⎣ ⎦

8.1

3.4

3.8

14.9 1

( )

4.9

13.2 1

s

s

e

s

d s

e

s

−

−

⎡ ⎤

⎢ ⎥

+

⎢ ⎥

+

⎢ ⎥

⎢ ⎥

⎢ ⎥

+

⎣ ⎦

, (31)

where y

a

(s) and y

b

(s) are the overhead and bottom com-

positions of methanol, respectively; u

1a

(s) is the reflux

flow rate and u

1b

(s) is the steam flow rate to the reboiler;

d(s) is the feed flow rate, a disturbance variable.

Due to the existence of the dead time, a first order

Pade approximation model (1) is introduced. Substitut-

ing it into (31), we get the parameter matrices of the

minimal realization of the rational approximation plant

model G

1

(s) as

1 0

1 0

0 1

0 1

1 0

1 0

0 1

⎡ ⎤

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

=

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

1

B

,

The initial design for the PID controller G

2

(s) can be

carried out as (5b) of section 2 to get

1

( )

j

diag

s s

⎧

⎫

⎪

⎪

= +

⎨

⎬

+α

⎪

⎪

⎩ ⎭

2 2

G (s) D

, for

j

= 1, 2,

…

,

p

1,

(33a)

where D

2

is selected as the static decoupler, i.e. D

2

=

−

1

1

G (0)

. The filter factor α

j

can be chosen as the load

disturbance denominator for steady state disturbance

compensation according to IMP. Particularly note that

the time-delay term in the load disturbance is not neces-

sarily considered in the PID controller design, because it

makes no difference for the closed-loop performance if

a load disturbance enters directly at the plant output or

after passing through a time delay [13]. The preliminar-

ily designed PID controller G

2

(s) is given as

1

0

( 0.0671)

0.1572 0.1531

0.0535 0.1036

1

0

( 0.0758)

s s

s s

⎡

⎤

⎢

⎥

+

−

⎡ ⎤

⎢

⎥

= +

⎢ ⎥

⎢

⎥

−

⎣ ⎦

⎢

⎥

⎢

⎥

+

⎣

⎦

2

G (s)

, (33b)

with its corresponding minimal realization as

0 0 0 0

0 0 0 0

0 0 0.0671 0

0 0 0 0.0758

⎡

⎤

⎢

⎥

⎢

⎥

=

⎢

⎥

−

⎢

⎥

−

⎢

⎥

⎣

⎦

2

A

,

1 0

0 1

1 0

0 1

⎡

⎤

⎢

⎥

⎢

⎥

=

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

2

B

,

14.9031 0 14.9031 0

0 13.1926 0 13.1926

−

⎡

⎤

=

⎢

⎥

−

⎣

⎦

2

C

,

0.1572 0.1531

0.0535 0.1036

−

⎡

⎤

=

⎢

⎥

−

⎣

⎦

2

D

. (34)

Based on the preliminarily designed PID

controller, a further tuning process with re-

spect to the plant (31) is necessary to guar-

antee satisfactory performance. Following

the proposed methodology, the PID tuning

problem has been converted to that of LQR

design. Formulating

the

plant

model

(32)

and

controller

(34) into a cascaded system

as (7), the optimal controller u

1

(t) in (8) can

be obtained through the afore-mentioned

method shown in section 3, by choosing Q =

diag {3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1}, R = I, as

=

− − −

1 1 1 2 2 2 2

u (t) K x (t) K x (t) D u (t)

, (35)

where

2 0 0 0 0 0 0

0 0.0599 0 0 0 0 0

0 0 0.6667 0 0 0 0

,

0 0 0 0.0476 0 0 0

0 0 0 0 0.2857 0 0

0 0 0 0 0 0.0917 0

0 0 0 0 0 0 0.0694

−

⎡ ⎤

⎢ ⎥

−

⎢ ⎥

⎢ ⎥

−

⎢ ⎥

=

−

⎢ ⎥

⎢ ⎥

−

⎢ ⎥

⎢ ⎥

−

⎢ ⎥

−

⎢ ⎥

⎣ ⎦

1

A

1.5802 0.8138 1.9385 1.0385 0 0 0

.

0 0 3.0078 0 1.7838 1.1783 1.6605

− −

⎡ ⎤

=

⎢ ⎥

− −

⎣ ⎦

1

C

(32)

Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems

491

0.9640 0.2657 0.3056 0.0757

0.2657 0.9640 0.0860 0.2794

− −

⎡ ⎤

=

⎢ ⎥

⎣ ⎦

2

K

,

0.1572 0.1531

0.0535 0.1036

−

⎡ ⎤

=

⎢ ⎥

−

⎣ ⎦

2

D

.

This indicates the tuned PID controller (A

2

, B

2

, K

2

, D

2

)

has the form of

1

( ) ( )s s

−

= + −

2 2 2 2 2

G D K I A B

Eigenvalues of the optimally designed closed-loop sys-

tem (16a) are

indicating that the system is asymptotically stable. Note

that this is an advantage of the proposed methodology as

the closed-loop system stability is guaranteed during

tuning, so that the designer is free from stability post

check.

Simulation results are shown in Fig. 8,

comparing different PID controller settings. Unit

setpoint changes were introduced in y

a

(t) (at t =

0) and y

b

(t) (at t = 100min), with a unit feed-flow

step disturbance occurring at t = 200min.

The solid line represents the output per-

formance of the real system (31), with the pro-

posed feedback/feedforward PID controller (35),

and the corresponding approximate system re-

sponse shown with the dash-dot line. The best

performance as cited in reference [4], with PID

setting (β = 1, f = 1), is shown with the dotted

line in Fig. 8. It is observed that the proposed

method successfully achieves a better result for

every set-point response. The limitation of the

proposed methodology is that its primary focus

is on the set-point response, with disturbance

compensation only considered at steady state.

Some additional control action has to be added if

the disturbance rejection is needed during the

dynamic process. We also find that performance

mismatch

between

the

real

system and the appro-

Fig. 8. Analogously controlled closed-loop

system responses.

ximate system, which is ∆y in Fig. 5, is

reflected whenever the setpoint changs,

but that the real system output quickly

returns to the trajectory of the approxi-

mate system nevertheless.

Selecting the sampling period as T

= 1min, and digitally redesigning the

analog controller (35) by using the de-

veloped cascaded prediction-based

digital redesign method in (30), we get

the digital control law as

=

− −

d1 d1 d1 d2 d2

u (kT) K x (kT) K x (kT)

( )

+

−

d1

E r(kT)

y

(kT)

, (37)

where

0.9037 4.1269 2.1009 4.9944 1.6426 1.3113 2.6740

,

0.3072 0.8485 6.7723 1.3565 6.0612 5.7394 8.8842

− − −

⎡ ⎤

=

⎢ ⎥

− − −

⎣ ⎦

1

K

0.9640 4.5544 0.2657 0.9987

4.3972 0.8456

0.0671 0.0758

.

0.2657 1.2817 0.9640 3.6860

1.3352 3.5824

0.0671 0.0758

s

s

s s s s

s

s

s s s s

− −

⎡ ⎤

− + + + +

⎢ ⎥

+ +

=

⎢ ⎥

− −

⎢ ⎥

+ + + +

⎢ ⎥

+ +

⎣ ⎦

3.7847 2.9582 1.5687 0.6927 0.1945i

,

0.4847 0.0923 0.0396i 0.0380 0.0475 0.0541

− − − − ±

⎧ ⎫

⎪ ⎪

⎨ ⎬

− − ± − − −

⎪ ⎪

⎩ ⎭

(36)

0.0250 0.8095 0.1359 0.9832 0.3882 0.3987 0.7596

,

0.0032 0.0115 0.5623 0.0681 0.8224 0.9382 1.5045

− − −

⎡

⎤

=

⎢

⎥

− − −

⎣

⎦

d1

K

0.1989 0.0831 0.0589 0.0221

,

0.0144 0.1742 0.0046 0.0468

− −

⎡

⎤

=

⎢

⎥

⎣

⎦

d2

K

0.2950 0.1416

.

0.0049 0.2455

−

⎡

⎤

=

⎢

⎥

− −

⎣

⎦

d1

E

The output responses of the proposed system with the analog control

law (35), and those of the proposed system with digital control law (37)

are compared in Fig. 9.

It is observed that the digitally controlled system responses

closely match those of the analogously controlled ones. Obviously,

our proposed digital redesign method gives a smooth transition from

the analog control law to the digital control law. The analog control

law u

c1

(t) in (35), and the corresponding digital control law u

d1

(kT)

in (37) are shown in Fig. 10.

492

Asian Journal of Control, Vol. 6, No. 4, December 2004

VIII. CONCLUSION

In this paper, a new methodology has been pre-

sented to design digital MIMO PID controllers for mul-

tivariable analog systems with time delay. Compared

with existing methods, the proposed method offers the

following

advantages:

(i)

MIMO

PID

parameters

are

systematically tuned, while the closed-loop system sta

bility is guaranteed; (ii) There are no specific require-

ments on system stability, simplification model and

plant decoupling; (iii) The proposed system is very ro-

bust due to the IMC structure utilized, and it provides

some convenience for online tuning; (iv) Designing a

plant state observer is unnecessary, as the plant states

are directly accessible in the internal approximate plant

model;

(v)

Digital

implementation

can

substantially

Fig. 9. Output responses of analogously controlled system

and digitally controlled system.

Fig. 10. Comparison between analog control law and digital

control law.

reduce the hardware complexity in realization and ad-

justment. The simulation results indicate that the pro-

posed methodology provides good performance in case

of considerable time-delay. Further research extending

the presented methodology for providing additional

control action for dynamic disturbance rejection, as well

as the physical realization of the proposed methodology

for mixed-signal system design is currently ongoing.

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APPENDIX

The development of the prediction-based digital re-

design method [8].

Consider a linear controllable continuous-time sys-

tem, described by

( ) ( ) ( )

c c c

x

t Ax t Bu t

=

+

,

0

(0)

c

x

x=

, (A1)

where x

c

(t) ∈ R

n

, u

c

(t) ∈ R

m

, and A and B are constant

matrices of appropriate dimensions. Let the continuous-

time state-feedback controller be

( ) ( ) ( )

c c c c

u t K x t E r t

=

− +

, (A2)

where K

c

∈ R

m

×

n

and E

c

∈ R

m

×

m

have been designed to

satisfy some specified goals, and r(t) ∈ R

m

is a piece-

wise-constant reference input vector. The controlled

system is

( ) ( ) ( )

c c c c

x

t A x t BE r t

=

+

,

0

(0)

c

x

x=

, (A3)

where A

c

= A − BK

c

. Let the state equation of a corre-

sponding hybrid model be

( ) ( ) ( )

d d d

x

t Ax t Bu t

=

+

,

0

(0)

d

x

x=

, (A4)

where u

d

(t) ∈ R

m

is a piecewise-constant input vector,

satisfying

( ) ( )

d d

u t u kT

=

for

( 1)kT t k T≤ < +

,

and T > 0 is the sampling period. Let the discrete-time

state-feedback controller be

*

( ) ( ) ( )

d d d d

u kT K x kT E r kT

= − +

, (A5)

where K

d

∈ R

m

×

n

is a digital state-feedback gain, E

d

∈

R

m

×

m

is a digital feedforward gain, and r

*

(kT) ∈ R

m

is a

piecewise-constant reference input vector to be deter-

mined in terms of r(t) for tracking purpose. The digitally

controlled closed-loop system thus becomes

*

( ) ( ) [ ( ) ( )]

d d d d d

x

t Ax t B K x kT E r kT

= + − +

,

0

(0)

d

x

x

=

for

( 1)kT t k T

≤

< +

. (A6)

A zero-order-hold device is used for (A5). The

digital redesign problem is to find digital controller

gains (K

d

, E

d

) in (A5) from the analog gains (K

c

, E

c

) in

494

Asian Journal of Control, Vol. 6, No. 4, December 2004

(A2), so that the closed-loop state x

d

(t) in (A6) can

closely match the closed-loop state in (A3) at all the

sampling instants, for a given r(t) ≡ r(kT), k = 0, 1, 2,….

The state x

c

(t) in (A1), at t = t

v

= KT + vT for 0 ≤ v

< 1, is found to be

( ) exp( ( )) ( )

c v v c

x

t A t kT x kT

= −

exp( ( )) ( )

v

t

v c

kT

A

t Bu d

+ −τ τ τ

∫

. (A7)

Let u

c

(t

v

) be a piecewise-constant input. Then equation

(A7) reduces to

( ) exp( ) ( )

c v c

x

t AvT x kT

≈

exp( ( )) ( )

kT vT

c v

kT

A

kT vT Bd u t

+

+ + −τ τ

∫

( ) ( )

( ) ( )

v v

c c v

G x kT H u t

= +

, (A8)

where

( )

exp( ( )) exp( )

v

v

G A t KT AvT

= − =

(exp( )) ( )

v v

A

T G

= =

( )

exp( ( ))

v

t

v

v

kT

H

A t Bd

= −τ τ

∫

( ) 1

0

exp( ) [ ]

vT

v

n

A

Bd G I A B

−

= τ τ = −

∫

.

Here, it must be noted that [G

(v)

− I

n

]

A

−

1

is a shorthand

notation, which is well defined as can be verified by a

cancellation of A

−

1

in the series expansion of the term

[G

(v)

− I

n

]. This convenient notation for an otherwise

long series will be used throughout the paper.

Also, the state x

d

(t) of (A4), at t = t

v

= kT + vT for 0

≤ v ≤ 1, is obtained as

( ) exp( ( )) ( )

d v v d

x

t A t kT x kT

= −

exp( ( )) ( )

v

t

v d

kT

A

t Bd u kT

+ −τ τ

∫

( ) ( )

( ) ( )

v v

d d

G x kT H u kT

= +

. (A9)

Thus, from (A8) and (A9) it follows that to obtain the

state x

c

(t

v

) = x

d

(t

v

), under the assumption of x

c

(kT) =

x

d

(kT), it is necessary to have u

d

(kT) = u

c

(t

v

). This leads

to the following prediction-based digital controller:

( ) ( ) ( ) ( )

d c v c c v c v

u kT u t K x t E r t

= = − +

( ) ( )

c d v c v

K

x t E r t

= − +

, (A10)

where the future state x

d

(t

v

) (denoted as the predicted

state) needs to be predicted based on the available causal

signals x

d

(kT) and u

d

(kT).

Substituting the predicted state x

d

(t

v

) in (A9) into

(A10) and then solving it for u

d

(kT) results in

( ) 1 ( )

( ) ( ) [ ( ) ( )]

v v

d m c c d c v

u kT I K H K G x kT E r t

−

= + − +

.

(A11)

Consequently, the desired predicted digital controller

(A5) is found, from (A11), to be

( ) ( ) *

( ) ( ) ( )

v v

d d

d d

u kT K x kT E r kT

= − +

, (A12)

where, for tracking purpose, r

*

(kT) = r(kT + vT), and

( ) ( ) 1 ( )

( )

v v v

m c c

d

K

I K H K G

−

= +

,

( ) ( ) 1

( )

v v

m c c

d

E I K H E

−

= +

.

In particular, if v = 1, then the pre-requisite x

c

(kT) =

x

d

(kT) is ensured. Thus, for any k = 0, 1, 2,…, the con-

troller is given by

*

( ) ( ) ( )

d d d d

u kT K x kT E r kT

= − +

, (A13)

where

1

( )

d m c c

K

I K H K G

−

= +

,

1

( )

d m c c

E I K H E

−

= +

,

*

( ) ( )r kT r kT T

=

+

,

in which G = exp(AT) and

H = (G − I

n

)A

−

1

B.

In selecting a suitable sampling period for the digi-

tal redesign method, a bisection searching method is

suggested to find an appropriate long sampling period,

so that the reasonable tradeoff between the closed-loop

response (i.e., the matching of the states x

c

(kT) in (A8)

and x

d

(kT) in (A9)) and the stability of the closed-loop

system can be achieved.

Yongpeng Zhang received his B.S.

degree from Xi’an University of

Technology, Xi’an, China in 1994, and

M.S. and Ph.D. degree from Tianjin

University, Tianjin, China, and Uni-

versity of Houston, Houston, Texas,

USA, in 1999 and 2003 respectively,

all in electrical engineering. He was a post-doctoral re-

searcher at CECSTR, Prairie View A&M University.

Currently he is an Assistant Professor in Engineering

Technology Department, Prairie View A&M University

Texas, U.S.A. His main research interests are digital

control, mixed-signal systems, and DSP solutions for

motor direct drive.

Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems

495

Leang-San Shieh

received his B.S.

degree from the National Taiwan Uni-

versity, Taiwan in 1958, and his M.S.

and Ph.D. degrees from the University of

Houston, Houston, Texas, in 1968 and

1970, respectively, all in electrical engi-

neering. He is a Professor in the De-

partment of Electrical and Computer

Engineering and the Director of the Computer and Systems

Engineering. He was the recipient of more than ten College

Outstanding Teacher Awards, the 1973 and 1997 College

Teaching Excellence Awards, and the 1988 College Senior

Faculty Research Excellence Award from the Cullen Col-

lege of Engineering, University of Houston, and the 1976

University Teaching Excellence Award and the 2002 El

Paso Faculty Achievement Award from the University of

Houston. He has published more than two hundred articles

in various referred scientific journals. His fields of interest

are digital control, optimal control, self-tuning control and

hybrid control of uncertain systems.

Cajetan M. Akujuobi

received the B.S.

degree from Southern University, Baton

Rouge, Louisiana in 1980, and his M.S.

and Ph.D. from Tuskegee University,

Tuskegee, Alabama and George Mason

University, Fairfax, Virginia in 1983 and

1995 respectively, all in Electrical Engi-

neering. He has also an MBA Degree from Hampton Uni-

versity, Hampton, Virginia, in 1987 in Business Admini-

stration. He is an Associate Professor in the Department of

Electrical Engineering and is the founding Director of the

Analog & Mixed Signal (AMS), DSP Solutions and High

Speed (Broadband) Communication Programs at Prairie

View A&M University. He is also the founding Director

of the Center of Excellence for Communication Systems

Technology Research (CECSTR). He has been a member

of the IEEE for over 24 years and is one of the founding

corporate members of the IEEE Standards Association

(IEEE-SA), Industry Advisory Committee (IAC). He is a

Senior Member of IEEE, Senior Member of ISA, ASEE,

SPIE, and Sigma XI, the Scientific Research Society.

His current research interests include all areas of signal

and image processing and communication systems

(Broadband Telecommunications) using such tools as

wavelet and fractal transforms. Other research interests are

in the areas of DSP Solutions, analog mixed signal system,

control system-based communications. He was a partici-

pant and collaborative member of ANSI T1E1.4 Working

Group that had the technical responsibility of developing

T1.413, Issue 2 ADSL Standard. He has been published

extensively and has also written many technical reports. He

was selected as one of U.S. representatives for engineering

educational and consultation mission to Asia in 1989.

Warsame H. Ali

was born in Moga-

disho, Somalia, in 1955. He received

the Bsc degree in electrical engineering

from King Saud University of Riyadh,

Saudi Arabia, in 1986. He received MS

from Prairie View A&M University,

Prairie View, Texas, USA in 1988.

Mr. Ali is currently pursuing the PhD

degree in electrical engineering at University of Houston,

Houston, Texas, USA. Presently, he is an instructor of

electrical engineering at Prairie View A&M University

since 1988. His main research interests are concerned with

the application of digital PID controllers, digital methods to

electrical measurements, mixed signals testing techniques,

and hybrid vehicles.

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