DIGITAL PID CONTROLLER DESIGN FOR DELAYED MULTIVARIABLE SYSTEMS

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 theDepartment 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, TX77446, U.S.A.

This work was supported in part by the US Army ResearchOffice under Grant DAAD 19-02-1-0321, and Texas Instru-ments under Grant #410171-03001, and NASA-JSC underGrant #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

1G (s) , in which every component is described as gij(s)e−sTij, 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,

22

sT sTesT

− −≈

+. (1)

Then, the resulting approximate rational model is de-fined as G1(s), in which every component is repre −

ay

22)(22sTesg −

11)(11sTesg −

by

au

bu

12)(12sTesg −

21)(21sTesg −

Fig. 1. A delayed multivariable (2 × 2) system structure.

sented as 2

( )2

ijij

ij

sTg s

sT−

+, 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 G1(s) ∈ R p1 × m1, in cascade with a MIMO analog PID controller G2(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

cE

)(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 G1(s) be

= +1 1 1 1 1x (t) A x (t) B u (t) , =1 10x (0) x ,

=1 1 1y (t) C x (t) , (2a)

where x1(t) ∈ Rn1, u1(t) ∈ Rm1, y1(t) ∈ Rp1, and A1, B1, C1 are constant matrices of appropriate dimensions.

Let the entire system output be the sum of the plant output and load disturbance,

= +1y(t) y (t) d(t) , (2b)

where y(t) ∈ Rp1, d(t) ∈ Rp1.

Each component of the MIMO PID controller G2(s) ∈ Rm1 × p1 is described as

2 ( ) I ij D ijij P ij

ij

K K sG s K

s s= + +

+ α, (3)

for i = 1, 2, …, m1 and j = 1, 2, …, p1. Parameters of the PID controller, KPij, KIij, KDij, and filter factor αij 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


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 + αij 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

jj

s s ss

s s

++ +

=

+ + + +=

⋅ + α∏

1 2 p1 1 p1 22

E E E EG ( ) , (4)

where parameter matrices E1, E2, …, Ep1+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 D2 = G1

−1(0), where G1(0) is non-singular. We can make the plant statically decoupled by setting E1 = D2 = G1

−1(0) in (4), which then gives

1 1 1

1

1( )

p p

p

jj

s s ss

s s

−+ +

=

+ + + += +

⋅ + α∏

2 3 p1 1 p1 22 2

F F F FG ( ) D , (5a)

where F2, F3,…, Fp1+2 are constant matrices to be deter-mined. For simplicity, we can first select the MIMO PID controller as

1( )j

diags s

⎧ ⎫⎪ ⎪= + ⎨ ⎬+ α⎪ ⎪⎩ ⎭2 2G (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 G2(s) can be written as

= +2 2 2 2 2x (t) A x (t) B u (t) , =2x (0) 0 , (6a)

= + =2 2 2 2 2 1y (t) C x (t) D u (t) u (t) , (6b)

= − +2 cu (t) y(t) E r(t) , (6c)

where x2(t) ∈ Rn2, u2(t) ∈ Rp1, y2(t) ∈ Rm1, r(t) ∈ Rp1, and A2, B2, C2, D2, Ec, 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 ex (t) A x (t) B u (t) E r(t) ,

= =e 1 e ey (t) y (t) C x (t) , (7)

where

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 1e e e

2 1 2 2 c

A 0 B 0A , B , E ,

B C A 0 B E

[ ] ,⎡ ⎤

= =⎢ ⎥⎢ ⎥⎣ ⎦

1e e 1

2

x (t)x , C C 0

x (t)

and

= − − −1 1 1 2 2 2 2u (t) K x (t) K x (t) D u (t) , (8)

where the state feedback control gains K1 and K2 are to be designed as shown below. For the convenience of LQR design of u1(t), given the existence of the feedfor-ward input term D2u2(t) in (8), an alternative representa-tion of the augmented system (7) can be described as

ˆ ˆˆ= + +e e e e 1 ex (t) A x (t) B u (t) E r(t) , (9a)


= =1 e e ey (t) y (t) C x (t) , (9b)

where

ˆ = +1 1 2 2u (t) u (t) D u (t) , (9c)

= − +2 1 cu (t) y (t) E r(t) , (9d)

and

ˆ+⎡ ⎤

= ⎢ ⎥⎢ ⎥−⎣ ⎦

1 1 2 1e

2 1 2

A B D C 0A

B C A,

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

1e

BB

0,

ˆ−⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

1 2 ce

2 c

B D EE

B E,

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

1e

2

x (t)x

x (t), [ ]=e 1C C 0 .

The state-feedback LQR for the augmented system (9) can be chosen as

ˆ = −1 e eu (t) K x (t) , (10)

where Ke = [K1, K2] with K1 ∈ Rm1 × n1 and K2 ∈ Rm1 × n2. Let the quadratic cost function for the system (9) be

ˆ ˆ∞= +∫ T Te e 1 10J [x (t)Q x (t) u (t)Ru (t)]dt , (11)

where Q ≥ 0, R > 0, (Âe, Be) is controllable and (Âe, Q) is observable. The optimal state-feedback control gain Ke in (10) that minimizes the performance index (11) is given by

1−= Te eK R B P , (12)

in which the matrix P > 0 is the solution of the Riccati equation [14],

1ˆ ˆ T T−+ − + =e e e ePA A P PB R B P Q 0 . (13)

The resulting closed-loop system becomes

ˆ ˆ= − +e e e e e ex (t) (A B K )x (t) E r(t) , (14)

which is asymptotically stable due to the property of LQR design.

Defining K1 as the plant state feedback gain, K2 as the controller feedforward gain in (10), then the desired state-feedback and state-feedforward control law u1(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 2u (t) u (t) D u (t)

[ ]= − − − −1 1 2 2 2 c 1K 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 ˆ1K = K1−D2C1, ˆ ˆ[ , ]=e 1 2K K K and Êc = −D2Ec.

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 Er(t)

B E, (16)

=1 1 1y (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)()()( 22222 tuBtxAtx +=

•

)()()( 11111 tuBtxAtx +=•

2K

)(2 tu

2D

1K

1Cplant model

PID controller

cE)(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-


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 G1(s), and its rational transfer function matrix is G1(s) = C1(sI – A1)−1 B1 + D1. On the other hand, the true plant model with time-delay is described as Ĝ1(s). Besides, the PID controller trans-fer function matrix is Gc(s) = − [K2(sI – A2)−1 B2 + D2].

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 )(tycontroller

plant model

)( td

q p~plant

p

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

r

y

PID1B

1C∫

1K

1u 1x

1A

d

1y

)(ˆ1 sG

)(sGc

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 Gx(s) = [I + K1 (sI − A1)−1 B1] −1.

r

PID

1B 1C∫

1A1K

1u)(sGc

∫

1A

1B

)(sGx

1y

a b∆ y∆

)(1 sG

y

)(sL

Fig. 6. Modeling error structure for robust analysis.

If the approximate model G1(s) has the same num-ber of unstable poles as that of the real model Ĝ1(s), then it follows from robust stability analysis [32] that the above system in Fig. 6 is robustly stable if

|| || || ( )L s∞ = ⋅ ⋅ab x cT (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 1G (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

−⋅ ⋅ + =1P c P cG (0) G (0) [I G (0)G (0)] I , (21)

which, when substituted into (20) gives the following condition

|| (0) || || (0) 0 || 1L −∞ ∞= ⋅ <1

ab 1T G ( ) . (22)

This means the designed system will demonstrate a zero steady state error despite modeling error, provided con-dition (22) is satisfied.


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 cx (t) Ax (t) Bu (t) , =c 0x (0) x , (23)

where xc(t) ∈ Rn, uc(t) ∈ Rm, A and B are constant ma-trices of appropriate dimensions. Let the continuous- time state-feedback control law be

= − +c c c cu (t) K x (t) E r(t) , (24)

where Kc ∈ Rm × n and Ec ∈ Rm × n have been designed to satisfy some specified goals, and r(t) ∈ Rm 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 du (kT) K x (kT) E r (kT) , (25)

where

−= + 1d c cK (I K H) K G , −= + 1

d c cE (I K H) E ,

= +*r (kT) r(kT T) , = ATG e , −= − 1H (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 Er(t)

B E, (t)xC(t)y 111 = . (26)

This can be transformed to the following structure,

⎡ ⎤ ⎡ ⎤⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤= −⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠⎣ ⎦ ⎣ ⎦

1 11 1 1 2

2 22 2

x (t) x (t)A 0 B 0 K K0 A 0 B 0 0x (t) x (t)

( )⎡ ⎤ −⎡ ⎤

+ ⋅ −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

1 2

2

B 0 Dr(t) y(t)

I0 B. (27)

Define Af = block diag. {A1, A2}, Bf = block diag. {B1,

B2}, ⎡ ⎤= ⎢ ⎥

⎣ ⎦

1 2f

K KK

0 0,

−⎡ ⎤= ⎢ ⎥

⎣ ⎦

2f

DE

I. The discrete-time

conversion gives the following terms, Gf = block diag. {Gd1, Gd2}, Hf = block diag. {Hd1, Hd2}, in which Gd1 = eA1T, Hd1 = [Gd1 − I] −1

1 1A B , and Gd2 = eA2T, Hd2 = [Gd2

− I] −12 2A B . Applying the afore-mentioned prediction-

based digital redesign method, gives the corresponding digital gains from (27) as

−

−

⎧ ⎡ ⎤= + ⋅ ⋅ =⎪ ⎢ ⎥

⎣ ⎦⎪⎪⎨⎪ ⎡ ⎤

= + ⋅ =⎪ ⎢ ⎥⎪ ⎣ ⎦⎩

d1 d21d f f f f

d11d f f f

K KK (I K H ) K G

0 0

EE (I K H ) E

I

. (28)

Then, we get the discrete-time controlled system model as:

+⎡ ⎤ ⎛⎡ ⎤=⎢ ⎥ ⎜⎢ ⎥⎜⎢ ⎥+ ⎣ ⎦⎝⎣ ⎦

d1 d1

d2d2

x (kT T) G 00 Gx (kT T)

⎡ ⎤⎞⎡ ⎤ ⎡ ⎤− ⎢ ⎥⎟⎢ ⎥ ⎢ ⎥ ⎟ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎠ ⎣ ⎦

d1d1 d1 d2

d2 d2

x (kT)H 0 K K0 H 0 0 x (kT)

( )⎡ ⎤ ⎡ ⎤

+ ⋅ −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

d1 d1

d2

H 0 Er(kT) y(kT)

I0 H, (29)

from which, we can get the corresponding digital con-trol law as

= − −d1 d1 d1 d2 d2u (kT) K x (kT) K x (kT)

( )+ −d1E r (kT) y(kT) . (30)

Thus, we get the resulting mixed-signal system as shown below in Fig. 7, where Gc(z) = −Kd2(zI – Gd2)−1Hd2, z.o.h. represents the zero-order-hold, and the part in the dotted line is the digital implementation structure.

)(kTr

)(ty

1C

)(1 kTud

)(1 kTxd

)(td

)(1 kTy

)(zGc1−z

1dK

... hoz

T

)(kTy∆

)(kTy

1dE

plant

1dG

1dH

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

system.


VII. AN ILLUSTRATIVE EXAMPLE

Consider the well known Wood and Berry empiri-cal model Ĝ1(s) of a pilot-scale distillation column that is used to separate a methanol-water mixture [4],

3

1

7 31

12.8 18.9( )( ) 16.7 1 21 1

( ) ( )6.6 19.410.9 1 14.4 1

s s

aa

s sb b

e eu sy s s s

y s u se es s

− −

− −

⎡ ⎤−⎢ ⎥ ⎡ ⎤⎡ ⎤ + +⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥+ +⎣ ⎦

8.1

3.4

3.814.9 1

( )4.913.2 1

s

s

es

d ses

−

−

⎡ ⎤⎢ ⎥+⎢ ⎥+ ⎢ ⎥⎢ ⎥⎢ ⎥+⎣ ⎦

, (31)

where ya(s) and yb(s) are the overhead and bottom com-positions of methanol, respectively; u1a(s) is the reflux flow rate and u1b(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 G1(s) as

1 01 00 10 11 01 00 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1B ,

The initial design for the PID controller G2(s) can be carried out as (5b) of section 2 to get

1( )j

diags s

⎧ ⎫⎪ ⎪= + ⎨ ⎬+ α⎪ ⎪⎩ ⎭2 2G (s) D , for j = 1, 2, …, p1,

(33a)

where D2 is selected as the static decoupler, i.e. D2 = −1

1G (0) . The filter factor αj 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 G2(s) is given as

1 0( 0.0671)0.1572 0.1531

0.0535 0.1036 10( 0.0758)

s s

s s

⎡ ⎤⎢ ⎥+−⎡ ⎤ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎢ ⎥⎢ ⎥+⎣ ⎦

2G (s) , (33b)

with its corresponding minimal realization as

0 0 0 00 0 0 00 0 0.0671 00 0 0 0.0758

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥−⎢ ⎥

−⎢ ⎥⎣ ⎦

2A ,

1 00 11 00 1

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

2B ,

14.9031 0 14.9031 00 13.1926 0 13.1926

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

2C ,

0.1572 0.15310.0535 0.1036

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

2D . (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 u1(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 2u (t) K x (t) K x (t) D u (t) , (35)

where

2 0 0 0 0 0 00 0.0599 0 0 0 0 00 0 0.6667 0 0 0 0

,0 0 0 0.0476 0 0 00 0 0 0 0.2857 0 00 0 0 0 0 0.0917 00 0 0 0 0 0 0.0694

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥

= −⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥

−⎢ ⎥⎣ ⎦

1A

1.5802 0.8138 1.9385 1.0385 0 0 0.

0 0 3.0078 0 1.7838 1.1783 1.6605− −⎡ ⎤

= ⎢ ⎥− −⎣ ⎦1C

(32)


0.9640 0.2657 0.3056 0.07570.2657 0.9640 0.0860 0.2794

− −⎡ ⎤= ⎢ ⎥

⎣ ⎦2K ,

0.1572 0.15310.0535 0.1036

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

2D .

This indicates the tuned PID controller (A2, B2, K2, D2) has the form of

1( ) ( )s s −= + −2 2 2 2 2G 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 ya(t) (at t = 0) and yb(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 d2u (kT) K x (kT) K x (kT)

( )+ −d1E 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− − −⎡ ⎤

= ⎢ ⎥− − −⎣ ⎦1K

0.9640 4.5544 0.2657 0.99874.3972 0.84560.0671 0.0758 .

0.2657 1.2817 0.9640 3.68601.3352 3.58240.0671 0.0758

s ss s s s

s ss 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− − −⎡ ⎤

= ⎢ ⎥− − −⎣ ⎦d1K

0.1989 0.0831 0.0589 0.0221,

0.0144 0.1742 0.0046 0.0468− −⎡ ⎤

= ⎢ ⎥⎣ ⎦

d2K

0.2950 0.1416.

0.0049 0.2455−⎡ ⎤

= ⎢ ⎥− −⎣ ⎦d1E

The output responses of the proposed system with the analog controllaw (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 responsesclosely match those of the analogously controlled ones. Obviously,our proposed digital redesign method gives a smooth transition fromthe analog control law to the digital control law. The analog controllaw uc1(t) in (35), and the corresponding digital control law ud1(kT)in (37) are shown in Fig. 10.


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.

REFERENCES

1. Astrom, K.J. and T. Hagglund, PID Controllers: The-ory, Design, and Tuning, Instrument Society of Amer-ica, Research Triangle Park, NC, U.S.A., pp. 1-30 (1995).

2. Astrom, K.J. and B. Wittenmark, Computer Controlled Systems: Theory and Design, Prentice Hall, NJ, U.S.A., pp. 1-29 (1997).

3. Boimond, J.L. and J.L. Ferrier, “Internal Model Con-trol and Max-Algebra: Controller Design,” IEEE Trans. Automat. Contr., Vol. 41, No. 3, pp. 457-461 (1996).

4. Chen, D. and D.E. Seborg, “Multiloop PI/PID Con-troller Design Based on Gershgorin Bands,” IEE Proc. Contr. Theory Appl., Vol. 149, No. 1, pp. 68-73 (2002).

5. Deshpande, P.B., Multivariable Process Control, In-strument Society of America, Research Triangle Park, NC, U.S.A., pp. 1-25 (1991).

6. Fujimoto, H., A. Kawamura, and M. Tomizuka, “Gen-eralized Digital Redesign Method for Linear Feedback System Based on N-Delay Control,” IEEE/ASME Trans. Mechatron., Vol. 4, No. 2, pp. 101-109 (1999).

7. Goodwin, G.C., S.F. Graebe, and M.E. Salgado, Con-trol System Design, Prentice Hall, NJ, U.S.A., pp. 267-268 (2001).

8. Guo, S.M., L.S. Shieh, G. Chen, and C. F. Lin, “Effec-tive Chaotic Orbit Tracker: A Prediction-Based Digital Redesign Approach,” IEEE Trans. Circuits Syst. – I, Fundamen. Theory Appl., Vol. 47, No. 11, pp. 1557-1570 (2000).

9. Isaksson, A. and T. Hagglund, Eds., “PID Control,” IEE Proc. Contr. Theory Appl., Special Issue, Vol. 149, No. 1, pp. 1-81 (2002).

10. Katayama, M., T. Yamamoto, and Y. Mada, “A De-sign of Multiloop Predictive Self-Tuning PID Control-lers,” Asian J. Contr., Vol. 4, No. 4, pp. 472-481 (2002).

11. Moradi, M.H., M.R. Katebi, and M.A. Johnson, “The MIMO Predictive PID Controller Design,” Asian J. Contr., Vol. 4, No. 4, pp. 452-463 (2002).

12. Marshall, J.E., Control of Time-Delay Systems, Institu-tion of Electrical Engineers, London, UK, pp. 1-35 (1979).


13. Morari, M. and E. Zafiriou, Robust Process Control, Prentice Hall, NJ, U.S.A., pp. 1-20 (1989).

14. Ogata, K., Modern Control Engineering, Prentice Hall, NJ, U.S.A., pp. 897-902 (1996).

15. Pierre, D.A. and J.W. Pierre, “Digital Controller De-sign-Alternative Emulation Approaches,” ISA Trans., Vol. 34, No. 3, pp. 219-228 (1995).

16. Shieh, L.S., W.M. Wang, and M.K. Appu Panicker, “Design of PAM and PWM Digital Controllers for Cascaded Analog Systems,” ISA Trans., Vol. 37, No. 3, pp. 201-213 (1998).

17. Sourlas, D.D. and V. Manousiouthakis, “Best Achiev-able Decentralized Performance,” IEEE Trans. Auto-mat. Contr., Vol. 40, No. 11, pp. 1858-1871 (1995).

18. Szita, G. and C.K. Sanathanan, “Robust Design for Disturbance Rejection in Time Delay Systems,” J. Franklin Inst., Vol. 334, No. 4, pp. 611-629 (1997).

19. Szita, G. and C.K. Sanathanan, “A Model Matching Approach for Designing Decentralized MIMO Con-trollers,” J. Franklin Inst., Vol. 337, No. 6, pp. 641-660 (2000).

20. Tan, K.K., Ed., “Advances in PID Control,” Asian J. Contr., Special Issue, Vol. 4, No. 4, pp. 364-508 (2002).

21. Tan, K.K., S.N. Huang, and T.H. Lee, “Development of a GPC-Based PID Controller Design for Unstable Systems with Deadtime,” ISA Trans., Vol. 9, No. 1, pp. 57-70 (2000).

22. Tan, K.K., T.H. Lee, and F.M. Leu, “Optimal Smith- Predictor Design Based on a GPC Approach”, Ind. Eng. Chem. Res., Vol. 41, pp. 1242-1248 (2002).

23. Tan, K.K., T.H. Lee, and F.M. Leu: “Predictive PI Versus Smith Control for Dead-Time Compensation,” ISA Trans., Vol. 40, No. 1, pp. 17-29 (2001).

24. Tan, K.K., Q.G. Wang, C.C. Hang, and T.J. Hagglund, Advances in PID Control, Springer, London, UK, pp. 1-25 (1999).

25. Tan, W., T. Chen, and H.J. Marquez, “Robust Con-troller Design and PID Tuning for Multivariable Proc-esses,” Asian J. Contr., Vol. 4, No. 4, pp. 439-451 (2002).

26. Wang, Q.G., C.C. Hang, and X.P. Yang, “Single-Loop Controller Design via IMC Principles,” Automatica, Vol. 37, No. 12, pp. 2041-2048 (2001).

27. Wang, Q.G., Q. Bi, and Y. Zhang, “Partial Internal Model Control,” IEEE Trans. Ind. Electron., Vol. 48, No. 5, pp. 976-982 (2001).

28. Wassick, J.M. and R.L. Tummala, “Multivariable In-ternal Model Control for a Full-Scale Industrial Distil-lation Column,” IEEE Contr. Syst. Mag., pp. 91-96 (1989).

29. Zhang, W.D., and X.M. Xu, “Optimal Solution, Quan-titative Performance Estimation, and Robust Tuning of the Simplifying Controller,” ISA Trans., Vol. 41, No. 1, pp. 31-36 (2002).

30. Zhang, Y., Q.G. Wang, and K.J. Astrom, “Dominant Pole Placement for Multi-Loop Control Systems,” Automatica, Vol. 38, No. 7, pp. 1213-1220 (2002).

31. Zheng, F., Q.G. Wang, and T.H. Lee, “On the Design of Multivariable PID Controllers via LMI Approach,” Automatica, Vol. 38, No. 3, pp. 517-526 (2002).

32. Zhou, K. and J.C. Doyle, Essentials of Robust Con-trol, Prentice Hall, NJ, U.S.A., pp. 145-146 (1998).

APPENDIX

The development of the prediction-based digital re-design method [8].

Consider a linear controllable continuous-time sys-tem, described by

( ) ( ) ( )c c cx t Ax t Bu t= + , 0(0)cx x= , (A1)

where xc(t) ∈ Rn, uc(t) ∈ Rm, and A and B are constant matrices of appropriate dimensions. Let the continuous- time state-feedback controller be

( ) ( ) ( )c c c cu t K x t E r t= − + , (A2)

where Kc ∈ Rm×n and Ec ∈ Rm×m have been designed to satisfy some specified goals, and r(t) ∈ Rm is a piece-wise-constant reference input vector. The controlled system is

( ) ( ) ( )c c c cx t A x t BE r t= + , 0(0)cx x= , (A3)

where Ac = A − BKc. Let the state equation of a corre-sponding hybrid model be

( ) ( ) ( )d d dx t Ax t Bu t= + , 0(0)dx x= , (A4)

where ud(t) ∈ Rm is a piecewise-constant input vector, satisfying

( ) ( )d du 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 du kT K x kT E r kT= − + , (A5)

where Kd ∈ Rm×n is a digital state-feedback gain, Ed ∈ Rm×m is a digital feedforward gain, and r*(kT) ∈ Rm 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 dx t Ax t B K x kT E r kT= + − + ,

0(0)dx 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 (Kd, Ed) in (A5) from the analog gains (Kc, Ec) in


(A2), so that the closed-loop state xd(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 xc(t) in (A1), at t = tv = KT + vT for 0 ≤ v < 1, is found to be

( ) exp( ( )) ( )c v v cx t A t kT x kT= −

exp( ( )) ( )vtv ckT A t Bu d+ − τ τ τ∫ . (A7)

Let uc(tv) be a piecewise-constant input. Then equation (A7) reduces to

( ) exp( ) ( )c v cx t AvT x kT≈

exp( ( )) ( )kT vTc vkT A kT vT Bd u t++ + − τ τ∫

( ) ( )( ) ( )v vc c vG x kT H u t= + , (A8)

where ( ) exp( ( )) exp( )v

vG A t KT AvT= − =

(exp( )) ( )v vAT G= =

( ) exp( ( ))vtvvkTH A t Bd= − τ τ∫

( ) 10 exp( ) [ ]vT v

nA Bd G I A B−= τ τ = −∫ .

Here, it must be noted that [G(v) − In] 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) − In]. This convenient notation for an otherwise long series will be used throughout the paper.

Also, the state xd(t) of (A4), at t = tv = kT + vT for 0 ≤ v ≤ 1, is obtained as

( ) exp( ( )) ( )d v v dx t A t kT x kT= −

exp( ( )) ( )vtv dkT A t B d u kT+ − τ τ∫

( ) ( )( ) ( )v vd dG x kT H u kT= + . (A9)

Thus, from (A8) and (A9) it follows that to obtain the state xc(tv) = xd(tv), under the assumption of xc(kT) = xd(kT), it is necessary to have ud(kT) = uc(tv). This leads to the following prediction-based digital controller:

( ) ( ) ( ) ( )d c v c c v c vu kT u t K x t E r t= = − +

( ) ( )c d v c vK x t E r t= − + , (A10)

where the future state xd(tv) (denoted as the predicted state) needs to be predicted based on the available causal signals xd(kT) and ud(kT).

Substituting the predicted state xd(tv) in (A9) into (A10) and then solving it for ud(kT) results in

( ) 1 ( )( ) ( ) [ ( ) ( )]v vd m c c d c vu 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 vd dd du kT K x kT E r kT= − + , (A12)

where, for tracking purpose, r*(kT) = r(kT + vT), and

( ) ( ) 1 ( )( )v v vm c cdK I K H K G−= + ,

( ) ( ) 1( )v vm c cdE I K H E−= + .

In particular, if v = 1, then the pre-requisite xc(kT) = xd(kT) is ensured. Thus, for any k = 0, 1, 2,…, the con-troller is given by

*( ) ( ) ( )d d d du kT K x kT E r kT= − + , (A13)

where

1( )d m c cK I K H K G−= + , 1( )d m c cE I K H E−= + , * ( ) ( )r kT r kT T= + ,

in which G = exp(AT) and H = (G − In)A−1B. 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 xc(kT) in (A8) and xd(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.


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.

.

Documents

DIGITAL PID CONTROLLER DESIGN FOR DELAYED MULTIVARIABLE SYSTEMS