Robust Design Method for Discrete-Time Controllers with Simple

Ind. Eng. Chem. Res. 2002, 41, 2705-2715

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Robust Design Method for Discrete-Time Controllers with Simple Structures Min-Lang Lin and Shyh-Hong Hwang* Department of Chemical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.

This paper first presents the design of a discrete-time general-structure controller, the order of which is determined by the process model order. Through the modified internal model control and dominant pole placement algorithms, the direct design is achievable by specifying one to three readily understandable parameters. Simple-structure controllers with user-specified orders can then be derived based on ingenious approximations of the general controller design. For example, a low-order proportional-integral-derivative and a high-order controller are obtainable in the same fashion. Furthermore, with a newly developed index for stability robustness, a methodology is proposed to optimize the controller designs of various structures. It is concluded that the resultant simple-structure controllers are vastly superior to the general-structure controller in almost all respects. The method is demonstrated with a wide range of process dynamics including time delay, right-half-plane and left-half-plane zeros, low- to high-order lags, integrators, and unstable poles. 1. Introduction With advances in computer technology, many studies of discrete-time control techniques have been carried out. The original concept of model predictive heuristic control was expounded by Richalet et al.,1 and the algorithm of dynamic matrix control was developed by Cutler and Ramaker.2 The performance of these methods was found to be superior with respect to robustness.3 Garcia and Morari3 defined the internal model control (IMC) structure to allow a rational controller design procedure for both control quality and robustness. Clarke and Gawthrop4,5 proposed a generalized minimum variance control law based on minimization of a simple one-step criterion to implement a self-tuning controller. Clarke et al.6,7 developed a generalized predictive controller (GPC) to minimize a cost function with a prediction range M and a weighting parameter Q0 specified by users. Pole assignment is a widely used algorithm for the design of discrete-time controllers.8-12 The resulting methods often need to solve the Diophantine equation. An advantage of the pole assignment algorithms is that it can explicitly take into account the stability margin and the performance of the closed-loop system. Isermann12 discussed in detail structurally optimal controller designs through pole assignment. Yet, simple direct synthesis methods for linear controllers are possible through specifications for the closed-loop behavior followed by pole-zero cancellation. Zafiriou and Morari13 examined several available control algorithms (such as Dahlin’s controller, the output and state deadbeat controllers, the Vogel-Edgar controller, etc.) and described them as pole-zero cancellation controllers. The above general-structure controller designs may possess some or all of the following drawbacks. First, because the order of a general-structure controller is directly connected with the process model order, it may be difficult to apply them to high-order plants in practice * Corresponding author. Fax: 886-6-2344496. Tel: 886-62086969 ext. 62661. E-mail: [email protected].

or understand the physical meaning of the associated control parameters. Second, the general control laws could result in sudden changes in the controller output to step set-point changes, namely, derivative kick and proportional kick. Such responses are often detrimental to the process operation. Third, the design of cancellation controllers is not applicable to open-loop unstable processes. In addition, the controller that cancels process zeros inside the unit circle but close to -1 would cause ringing of the manipulated variable, and its effect would be rippling of the system output between the sampling instants.13 Finally, they often lack explicit considerations of stability robustness. Because of the complicated nature of the controller, the frequency response of the open-loop transfer function may not behave well so that the Nyquist curve decreases continuously in magnitude with frequency. Specifically, the Nyquist curve may enlarge for high frequencies if the controller settings are tight. This implies that the commonly used indices, the GM and PM, may not evaluate fairly the stability robustness of the closedloop system. The use of a controller with a simple structure can avoid most of the aforementioned drawbacks. For example, proportional-integral-derivative (PID) control algorithms such as the heuristic Ziegler-Nichols rules have been widely used for real control systems.14,15 Recently, there was a trend to combine the PID controller design with modern control algorithms because the direct introduction of sophisticated control algorithms in the industry might encounter resistance and difficulties from the operators.16-19 In contrast with general-structure controllers, the direct design of simple-structure controllers is restricted to special types of process models.12 To gain a simplestructure controller of arbitrary order for a general process model, the control parameters are often determined through numerical optimization of a performance criterion. Because the computational effort augments considerably with an increasing number of control parameters, low-order control structures, such as a PID

10.1021/ie0106276 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/30/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002

The transfer function of the general-structure controller is then

D(z) ) P(z-1)/∆Q(z-1)

In general, the process model order determines the structure of the controller (6). The use of the operator ∆ is to introduce integral action to eliminate steadystate offset.

Figure 1. Digital control system.

controller, are then preferred. However, for high-order processes, improved performance may demand a highorder simple-structure controller.20 This would result in an increase in the number of control parameters and a time-consuming optimization procedure. In this paper, a discrete-time controller with a general structure is first considered. The direct design of the general-structure controller through pole placement is achieved by specifying one to three parameters that are closely related to the closed-loop response. On the other hand, a study reveals that the closed-loop system resulting from the general controller may have an excellent performance but very poor stability robustness. This poor robustness can only be uncovered via a newly developed index MGH. On the basis of clever approximations of the general controller design, simplestructure controllers with user-specified orders can be derived. For instance, a PID or a higher-order controller can be easily tuned to achieve specific responses. Moreover, a methodology is presented to optimize the simple-structure controller design under various constraints on stability robustness. 2. General-Structure Controller A discrete-time feedback control system shown in Figure 1 is considered. The process model with L(k) ≡ 0 is described by

A(z-1) y(k) ) B(z-1) u(k)

A(z-1) ) 1 + a1z-1 + ... + anz-n

(2a)

B(z-1) ) z-d(b1z-1 + ... + bmz-m)

(2b)

where n and m denote the orders of the process model and the time delay d is an integer multiple of the sample time T. The discrete process transfer function is given by

G(z) ) B(z-1)/A(z-1) The general control law can be

∆Q(z-1) A(z-1) + P(z-1) B(z-1) ) C(z-1) ) 1 + c1z-1 + ... + cncz-nc (7) where nc (em + n + d) denotes the order of the desired characteristic polynomial. Supposing that C(z-1) is provided, eq 7 contains totally n + m + d unknown coefficients pi and qi. We then collect like powers of z-1 and equate the sum of the coefficients of each occurring power of z-1 to zero. This yields n + m + d relations from which the unknown coefficients can be calculated. The pole placement controller design can then be realized by specifying the desired characteristic polynomial C(z-1) and solving the resulting equation (7). Specification of C(z-1) Based on the Modified IMC Algorithm. The polynomial C(z-1) can be easily chosen based on a modification of the IMC algorithm.3 First, factor B as

B(z-1) ) B+(z-1) B-(z-1)

(8)

where B- is monic and B+ is given by nb

B+(z-1) ) b1z-d-1

(1 - νiz-1) ∏ i)1

(9)

Note that B+ contains all of the time delays and all of the (nb) zeros outside the unit circle, i.e., those referred to as nonminimum-phase characteristics in classical frequency response theory. If the process is not openloop stable, also factor A is

A(z-1) ) A+(z-1) A-(z-1)

(10)

where A- is monic and A+ contains all of the (na) integrators and unstable poles as given by

A+(z-1) ) 1 + a1+z-1 + ... + ana+z-na

(11)

employed:12

(4)

where ∆ ()1 - z-1) is a differential operator, w(k) is the command variable, and

P(z ) ) p0 + p1z

Because A(z-1) and Q(z-1) are monic (the leading coefficient is 1), the desired characteristic polynomial of the closed-loop system can be written as

(3)

∆Q(z-1) u(k) ) P(z-1) [w(k) - y(k)]

-1

3. Pole Placement Algorithms

(1)

where u(k) and y(k) are the input and output variables, respectively. The polynomials A(z-1) and B(z-1) are expressed as

-1

(6)

+ ... + pnz

-n

(5a)

Q(z-1) ) 1 + q1z-1 + ... + qm+d-1z-(m+d-1) (5b)

The pole placement design is then achieved by substituting

C(z-1) ) nb

[1 - (1/νi)z-1] ∏ i)1

(1 - Rz-1)1+naA-(z-1) B-(z-1)

(12)

in eq 7 and solving the resultant equation for P and Q:

Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2707

∆Q1(z-1) A+(z-1) + P1(z-1) B+(z-1) ) nb

(1 - Rz-1)1+na

[1 - (1/νi)z-1] ∏ i)1

(13)

where

P(z-1) ) P1(z-1) A-(z-1)

C(z-1) ) (1 - 2e-σ cos($) z-1 + e-2σz-2)(1 - λe-σz-1)n-1 (15)

Q(z-1) ) Q1(z-1) B-(z-1) In the derivation of eq 13, the invertible parts of the process model, A- and B-, are canceled throughout the equation and P1 and Q1 are left out of the original P and Q. Consequently, for open-loop stable processes (na ) 0 and P1 ) p0), the closed-loop transfer function is identical to that obtained by the IMC algorithm:3 nb

y(k) w(k)

)

P(z-1) B(z-1) C(z-1)

p0b1z

-d-1

)

(1 - νiz ∏ i)1

-1

)

nb

(1 - Rz-1)

[1 - (1/νi)z-1] ∏ i)1

(14)

where p0b1 is given by substituting z ) 1 in eq 13 as nb

[1 - 1/νi] ∏ i)1

(1 - R) p 0 b1 )

be of order being at least n + 1. We refer to these n + 1 poles as the dominant poles of a general closed-loop system. It is also postulated that some degree of oscillation is required to deal with a wide range of process dynamics and external disturbances, meaning that C(z-1) should contain at least two complex conjugated roots. Consequently, the polynomial C(z-1) is specified by dominant pole placement as

nb

(1 - νi) ∏ i)1 It is known that the IMC design is not suited for openloop unstable processes because it requires that the controller cancel exactly the unstable poles of the process, which is impossible in practice. For integrating processes, the IMC design would yield a controller without integral action that causes an offset problem for any static input disturbance. Conversely, our approach retains the A+ term in the characteristic equation (13) to allow the controller to shift its position to the stable region, thus avoiding the deficiencies of the conventional IMC design for those process dynamics. The modified IMC algorithm inherits the simple feature of the IMC design. This fact is both an advantage and a disadvantage. It is an advantage in that it allows the engineer to obtain a specified response by adjusting a single parameter R. The disadvantage is that the optimal design is often unavailable for certain process dynamics and for disturbance inputs. Specification of C(z-1) Based on Dominant Pole Placement. For high-order processes, it may be too restrictive to specify all closed-loop poles. Astrom and Wittenmark11 suggested specifying only two dominant poles and requiring that the remaining poles be close to the origin. Such an assignment, however, seems to be too simple to deal with a wide variety of process dynamics. Here, we first consider the appropriate order of the desired characteristic polynomial. It is postulated that the closed-loop system for an nth-order process is composed of n + 1 poles if m ) 1 and d ) 0 in eq 7. The additional pole is induced by integral action. This implies that the desired characteristic polynomial should

where

σ ) γ cos(θ) $ ) γ sin(θ) Here, -σ/T and $/T correspond respectively to the real and imaginary parts of the mapped complex conjugated poles in the s domain. The tuning parameter θ must be selected between 0 and π/2 to ensure stability. A suitable value of γ depends on the speed of the process response. The above specification results from the postulation that two complex conjugated poles together with n - 1 real poles dominate the desired closed-loop response. Moreover, it is assumed, for simplicity, that all of the real poles (if any) are the same and their values are λ times the magnitude of the complex poles. The parameter λ is chosen so that it reasonably reflects the relative importance of the dynamic modes represented by the real poles. The smaller the λ value, the more oscillatory the response can become. The optimal design is often obtainable by adjusting three readily understandable tuning parameters γ, θ, and λ. On the other hand, the performance of the closedloop system is not a strong function of the tuning parameters. Therefore, if the computation burden is a major concern, an approximately optimal design can be achieved by adjusting only two parameters, γ and θ, with the value of λ chosen according to eq 32 given in section 7. 4. Simple-Structure Controllers The general control law (4) is faced with the problem of large sudden changes in the controller output to step changes in w(k), i.e., derivative kick and proportional kick. Such responses are often detrimental to the process operation. A remedy to this is to simplify eq 4 to

∆u(k) ) R(z-1) w(k) -

P(z-1)

y(k)

Q(z-1)

(16)

where R(z-1) contains merely the I or PI mode and is approximated by

R(z-1) ) k0 P(z-1) for I mode (structure 1) ≈ (17) k0 + k1∆ for PI mode (structure 2) Q(z-1)

{

The control law (16) possesses a structure that is simple with respect to the command variable w(k) but still general with respect to the output variable y(k). As a result, the partially simple control law (16) would inherit the closed-loop characteristic polynomial ob-

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tained by the general control law (4). In the simplestructure portion, k0 and k1 denote respectively the integral and proportional gains. The controller of structure 1 removes both the derivative and proportional kicks, whereas the controller of structure 2 eliminates only the derivative kick. A good approximation can be simply achieved by the long division procedure as follows. The polynomials P(z-1) and Q(z-1) are rearranged as

P(z-1) )

n

controller with an arbitrary order r. Specifically, if a PID controller is desired, one can set r ) 2 for K(z-1) and calculate the controller parameters ki as follows:

k0 ) k1 )

p′1 p′0q′1 - 2 q′0 q′

n

piz-i ) ∑p′i∆i ∑ i)0 i)0

(18) k2 )

m+d-1

Q(z ) ) 1 +

∑ i)1

qiz

-i

m+d-1

)

∑ i)0

The relations between pi and p′i are

1 1 1 1 ... 1 0 -1 -2 -3 ... -n n(n - 1) 0 0 1 3 ... 2! n(n - 1)(n - 2) 0 0 0 -1 ... 3! l l l l l l 0

0

... (-1)n

i

q′i∆

(19)

][ ] [ ] p′0 p0 p′1 p1 p′2 p ) 2 p′3 p3 l l p′n pn

(20)

and the parameters q′i are obtainable from qi in a similar fashion. Using long division, k0 and k1 can then be computed as

k0 ) k1 )

p′0 q′0

(21b)

It is sometimes preferable to use a simple-structure controller with respect to both the command and output variables. For example, the controller (16) can be further simplified to the following form:

∆u(k) ) R(z-1) w(k) - K(z-1) y(k)

(22)

where r

n

m+d-1

ki∆i ≈ (∑p′i∆i)/( ∑ ∑ i)0 i)0 i)0

q′i∆i)

(23)

{

Q(z-1) R(z-1) B(z-1) w(k) C(z-1)

k structure 3 R(z ) ) k0 + k ∆ structure 4 0 1

(26)

and that for load disturbances is

∆Q(z-1) A(z-1) GL(z-1) y(k) ) L(k) C(z-1)

(27)

Recall that the closed-loop characteristic polynomial for the partially simple control law (16) (structure 1 or 2) is exactly the same as the desired one obtained by the general-structure controller shown in eq 7. Hence, the partially simple control law can produce the specific responses given above. On the other hand, the simple control law (22) (structure 3 or 4) can only approximate the desired responses. Nevertheless, the approximations are rather satisfactory as verified via the following process:

GP(s) )

(-1.5s + 1)(0.5s + 1)e-s , (s + 1)4 T ) 0.25

The pulse transfer function of the controlled process, G(z), is generated from taking the z transform on the combination of the continuous process GP(s) and the zero-order hold H(s) with T ) 0.25. The specific characteristic polynomial is specified by deliberately letting γ ) 0.175, θ ) 0.45, and λ ) 0.95. Figure 2 compares the set-point and load responses obtained using the controller of structure 1 (resulting in the desired responses) and the controller of structure 3 with r ) 4. It appears that, even though an approximation is involved, the controller of structure 3 gives rise to responses that are in excellent agreement with the specified ones. 5. Robustness Considerations

In eq 22, R(z-1) is also assumed to be -1

y(k) )

Example 1

p′1 p′0q′1 - 2 q′0 q′

(25c)

0

where k2 is the derivative gain. For both the control laws (16) and (22), the desired transfer function for set-point changes is

(21a)

0

K(z-1) )

(25b)

p′2 p′0q′2 p′1q′0q′1 - p′0q′12 - 2 q′0 q′ q′ 3 0

-1

0 0

(25a)

0

and

[

p′0 q′0

(24)

Note that the controllers of structures 3 and 4, like those of structures 1 and 2, differ only for set-point changes. They are the same for load disturbances where w(k) ) 0. In eq 23, the long division procedure elucidated above can still be employed to obtain a simple-structure

Although the aforementioned general-structure controller and simple-structure controller are capable of producing desirable responses, they often encounter different problems of stability robustness as elaborated in this section. Such a problem is also observed for the GPC.6,18 Gain and phase margins (GM and PM) are known as good measures of the system’s stability robustness based on the frequency response of the open-loop transfer


Figure 2. Comparison of the desired responses with those obtained by the controllers of structures 1 and 3.

Figure 3. Characteristics of Nyquist curves for different control laws.

function.15,21,22 For the discrete-time systems under consideration, the open-loop transfer function GOL(z) is expressed as

time control systems, as illustrated also with example 1. Parts a and b of Figure 3 show the Nyquist curves of the closed-loop systems resulting from the control laws (16) and (22) with the previous settings, respectively. Although the two control laws produce almost the same closed-loop responses, they exhibit widely different features of stability robustness. From points A and B in the figures, the GM and PM for the two control laws are calculated respectively as

{

GOL(z) ) P(z-1) B(z-1)

∆Q(z-1) A(z-1) K(z-1) B(z-1) ∆A(z-1)

for control laws (4) and (16) (28) for control law (22)

and the frequency response is obtained by letting z ) ejωT for 0 eω < π/T.23 The GM represents the factor by which the total loop gain must increase to destabilize the control loop. The PM represents the additional amount of phase angle required to destabilize the control loop. Changes in phase angle of the control loop are due mainly to changes in its dynamic terms, such as time delay. The conventional use of two points on the Nyquist curve (GM and PM) to evaluate the system’s robustness is based on the premise that the Nyquist curve does not enlarge significantly in magnitude for higher frequencies. However, this is not true for the present discrete-

GM ) 2.30, PM ) 61°

for control law (16)

GM ) 1.06, PM ) 61°

for control law (22)

The above information indicates that the partially simple control law (16) exhibits good stability robustness with respect to model uncertainties, while the simple control law (22) is quite poor in terms of stability robustness. However, the fact is that both control laws are faced with serious robustness problems. For the control law (16), the Nyquist curve in Figure 3a enlarges significantly at high frequencies and intersects the unit circle several times. This clearly reveals that points A and B alone are not sufficient to evaluate the system’s robustness. In fact, points C and D have more severe implications in robustness than point B. The phase

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law (16) produces a very desirable response, it is extremely sensitive to changes in delay. Such a phenomenon of poor robustness is also encountered in the general control law (4) and cannot be revealed by the conventional GM and PM. For the simple control law (22), the poor GM is also caused by the enlarging effect of the Nyquist curve at high frequencies, as depicted in Figure 3b. A 7% increase in the steady-state gain could easily destabilize the entire loop at high frequencies, as evidenced in Figure 4b. Although the conventional GM is able to predict the lack of stability robustness, it still cannot uncover this high-frequency feature. The above results imply that the sole use of GM and PM is not sufficient for the design of discrete-time controllers. A remedy to this is to propose an index to evaluate the degree of expansion of the Nyquist curve for high frequencies. Here, the index MGH is defined as the maximum gain for high frequencies ω gωLB. The lower bound frequency ωLB is the frequency at which the Nyquist curve, starting from ω ) 0, intersects for the first time the circle with a radius of the reciprocal of the specified GM (GMS). This corresponds to point E in Figure 3 with GMS ) 1.7. Point E′ then gives the value of MGH. Hence, a criterion can be introduced to restrain the Nyquist curve from expanding at high frequencies to a significant extent:

MGH e MGHS

(30)

Note that the design specification for MGH, MGHS, must be greater than or equal to 1/GMS to make the criterion meaningful. The MGH can be regarded as a complement to the conventional GM (GM ) -1/A1) according to a single point in the left-half plane. Note also that if the index MGH is equal to 1/GMS, the system could never have a condition of stability robustness worse than that required by the GM specification. Figure 4. Load responses produced by two controller designs subject to (a) an increase in the delay by one and (b) a 7% increase in the steady-state gain.

angles and corresponding frequencies for the three points are

Point B:

φ ) -119° and ω ) 0.221

Point C:

φ ) -80.2° and ω ) 7.80

Point D:

For a fair comparison, the additional amount of time delay required to destabilize the control loop, d h , can be calculated as

d h)

{

(φ - 180)π for 0 e φ < 180° 180ωT

The conventional definition for GM is suited for openloop stable processes. This definition, however, demands some modification for open-loop unstable processes, as illustrated with a process containing one unstable pole:

GP(s) )

φ ) 90.7° and ω ) 6.74

(180 + φ)π for -180° e φ < 0 180ωT

6. GM for Open-Loop Unstable Processes

(29)

Accordingly, point B indicates that a very large amount of delay (d h ) 19.3) is required to destabilize the control loop. However, points C and D reveal that an increase in delay by 1 (d h ) 0.893) or a decrease in delay by 1 (d h ) -0.925) is sufficient to destabilize the entire loop at high frequencies under the control law (16), as elucidated in Figure 4a. In other words, although the control

e-0.1s , T ) 0.1 (s - 1)(0.5s + 1)

The partially simple control law (16) can be used to stabilize the unstable process. The Nyquist curve (0 e ω < π/T) of the resulting open-loop transfer function is delineated in Figure 5. The two corresponding limit points are (A′1, 0) and (A1, 0). A large number of openloop unstable processes would exhibit such a feature of conditional stability. Unlike open-loop stable processes, the loop gain can increase by a factor of -1/A1 or decrease by a factor of -A′1 to make the closed-loop system unstable again. It is then plausible to redefine the GM for open-loop unstable processes as

GM ) Min{-A′1, -1/A1}

(31)

The definition of MGH can be easily applied to such systems as those elucidated with points E and E′ in Figure 5.


Set-point changes λ)

{

0.95 - 0.025(n - 2) for structures 1 and 3 1.05 - 0.025(n - 2) for structures 2 and 4 (32b)

The resultant algorithm contains only two readily tunable parameters, γ and θ, whose search ranges are often known a priori. (5) Calculate the parameters in the control law (16) or (22) for a range of tuning parameters (one to three). For each set of control parameters, compute GM, PM, MGH, and OS (overshoot, if required) as well as the error sum. The optimal control law is then chosen as that resulting in the minimum error sum subject to the inequality constraints on robustness and/or overshoot. Figure 5. Definition of GM for open-loop unstable processes.

8. Simulation Study

7. Design Procedure for a Robust Controller

A simulation study has been conducted to demonstrate the validity of the proposed design method for a wide range of process dynamics. They include time delay, minimum-phase and nonminimum-phase zeros, low- to high-order lags, integrators, and unstable poles. In the simulation work, both cases of set-point changes and load disturbances are considered. The error sum criterion is chosen to be SAE and the inequality constraints on robustness are

We are now in a position to design a robust controller in the following steps: (1) Choose the type of the input signal (set-point change or load disturbance) and the corresponding error sum criterion, such as SAE (sum of absolute error), STAE (sum of time-weighted absolute error), or SSE (sum of squared error), to be minimized.

GM g GMS ) 1.7

∞

SAE )

∑ |e(k)|

PM g PMS ) 30°

k)0

MGH e MGHS ) 0.67

∞

STAE )

∑ |ke(k)|

In addition, an inequality constraint is imposed on overshoot for a step set-point response:

k)0

OS e 5%

∞

SSE )

∑

[e(k)]2

k)0

(2) Specify the inequality constraints on stability robustness in terms of GM, PM, and MGH. (3) Select the controller structure. If the input signal is a load disturbance, the controller of structure 1 (eq 3) is the same as the controller of structure 2 (eq 4). For the controllers of structures 3 and 4, the value of r is to be set by the designer. It is generally recommended to use r ) n or n + 1 for nth-order processes. However, the setting of r ) 2 can be chosen so as to generate a PID controller despite the process model order. (4) There are three algorithms to arrive at the final design. If the design is based on the modified IMC algorithm, only a single parameter R is to be tuned. This method is the simplest but is not suitable for a range of process dynamics and for disturbance inputs. The best design can be achieved by adjusting three tuning parameters, γ, θ, and λ, based on dominant pole placement. This algorithm, however, demands a rather heavy computation burden. To relieve this, an extensive simulation reveals that the present design is not sensitive to the choice of λ and the best one can be well approximated by presetting λ according to

Load disturbances λ ) 1 - 0.025(n - 2)

for all structures

(32a)

The proposed design method is compared with the IMC controller,3 the GPC,6 and the simplified GPC (SGPC).18 The IMC design can be obtained via the general-structure controller (4) with the modified IMC algorithm for open-loop stable processes. The GPC design also results in a general-structure controller. The SGPC design gives rise to a simple controller that has the same structure as the controller of structure 3 with r ) 2, i.e., a PID controller. For a fair comparison, the prediction range M and the weighting parameter Q0 used by each of the latter two available methods are tuned to give the minimum SAE value under the same constraints on robustness and/or overshoot. In the simulation, the four examples are continuous processes under digital control. Taking the z transform on the combination of the continuous process and the zero-order hold generates all z-transfer functions of the processes. Instead of showing the sampled output, Figures 6-9 plot the continuous closed-loop responses versus time to check if the process output y(t) ripples between the sampling instants. The present design procedure is first applied to the previous example 1 with GL(z) ) G(z). The example is a high-order process with a significant right-half-plane (RHP) zero and a left-half-plane (LHP) zero. Correspondingly, the discrete transfer function G(z) consists of a significant nonminimum-phase zero (z ) 1.1814, outside the unit circle) and two minimum-phase zeros (z ) 0.6065 and -0.801, inside the unit circle). Note that

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Figure 6. Comparison of set-point responses obtained by various controller designs for example 1.

Figure 7. Comparison of set-point and load responses obtained by various controller designs for example 2.

rippling is possible because of the zero of z ) -0.801.13 Tables 1 and 2 enumerate the design results of the controllers of various structures for set-point changes and load disturbances, respectively. Two algorithms are used to find the controller design: the modified IMC algorithm with a single adjustable parameter and the dominant pole placement algorithm with three adjustable parameters. It appears that for set-point changes the smallest SAE value is obtained by the generalstructure controller using the dominant pole placement algorithm, and the second best is produced by the IMC design. However, both designs demand large changes of the controller output u because of derivative kick, as seen in Figure 6. Furthermore, the IMC design causes significant ringing of the controller output. It then follows from Table 1 and Figure 6 that using the dominant pole placement algorithm, the simple controller of structure 4 with r ) 4 results in the best design for set-point changes; it produces a sufficiently small SAE value with mild control action. Also, the best design for load disturbances is obtained by using that algorithm for the same simple-structure controller, as evidenced in Table 2. The GPC design is extremely poor because of the constraint on MGH. It is interesting to note that for set-point changes the performance of the controllers of structure 4 with r ) 2, a PID controller, is almost as

good as the high-order controller. The parameters of the four controllers discussed above are enumerated in Table 3. Generally speaking, the general-structure controller (4) is capable of producing better output responses for set-point changes, but it is often accomplished at the cost of excessive control action. Hence, the partially simple-structure controller (16) and the completely simple-structure controller (22) are preferred. In the sequel, the controllers of structures 2 and 4 (without derivative kick) are applied to the stable process to produce a faster set-point response. As for the integrating and unstable processes, the controllers of structures 1 and 3 (without proportional and derivative kicks) are used to produce an acceptable overshoot for a step setpoint change.

Example 2

GP(s) )

(0.5s + 1)e-0.5s , s2 + s + 1 T ) 0.1, GL(z) ) G(z)

Example 2 is composed of underdamped poles and a LHP zero. The corresponding G(z) contains a minimumphase zero (z ) 0.8183) inside the unit circle. The presence of the minimum-phase zero would accelerate




Table 1. Design Results for Example 1 Subject to Set-Point Changes

Table 2. Design Results for Example 1 Subject to Load Disturbances

tuning param IMC general structure structure 2 structure 2 structure 4 (r ) 2) structure 4 (r ) 2) structure 4 (r ) 4) structure 4 (r ) 4) GPC SGPC

OS SAE (%) GM

tuning param PM MGH

R ) 0.796 γ ) 0.200, θ ) 0.917, λ ) 0.957 R ) 0.881 γ ) 0.276, θ ) 0.910, λ ) 1.04 R ) 0.884

20.9 0.0 2.27 61.6 19.1 1.5 2.01 59.0

0.67 0.67

26.6 5.0 2.60 63.6 25.4 5.0 2.47 65.1

0.38 0.63

26.5 5.0 2.28 64.0

0.44

γ ) 0.276, θ ) 1.26, λ ) 0.971 R ) 0.881

25.8 4.7 1.95 67.1

0.51

26.6 5.0 2.59 63.6

0.39

γ ) 0.309, θ ) 1.43, λ ) 0.931 M ) 29, G0 ) 173.1 M ) 30, G0 ) 126.7

24.0 5.0 1.70 68.0

0.67

51.9 0.0 3.44 75.8 34.2 0.0 1.70 60.5

0.67 0.60

the system’s response. However, the partially simplestructure controller is not best suited for processes containing significant minimum-phase zeros. The reason is that the polynomial Q(z-1) in the control law would tend to cancel the minimum-phase zero, thus wasting the benefit of this accelerating effect. This condition becomes more serious if the modified IMC algorithm is employed, which explicitly cancels the

IMC R ) 0.796 general structure γ ) 0.180, θ ) 1.03, λ ) 0.917 structure 4 R ) 0.776 (r ) 2) structure 4 γ ) 0.176, θ ) 0.836, (r ) 2) λ ) 0.945 structure 4 R ) 0.693 (r ) 4) structure 4 γ ) 0.197, θ ) 1.10, (r ) 4) λ ) 0.922 GPC M ) 29, G0 ) 173.1 SGPC M ) 29, G0 ) 113.6

SAE GM

PM MGH

23.0 2.27 61.6 19.9 1.82 55.1

0.67 0.67

23.1 1.70 61.7

0.59

22.6 1.70 59.2

0.59

21.4 1.70 60.7

0.59

19.2 1.70 55.1

0.59

39.9 3.44 75.8 22.3 1.70 60.5

0.67 0.60

minimum-phase zero by a stable pole in the controller. Conversely, the simple-structure controller in conjunction with the dominant pole placement algorithm is a much better choice for such dynamics. Figure 7 depicts the responses of the controllers of structures 2 and 4 employing the modified IMC algorithm with a single adjustable parameter and the dominant pole placement algorithm with two adjustable parameters. As expected, the controller of structure 4 (r ) 2) using two adjustable parameters gives rise to the best design. The results are comparable to those obtained using three adjustable

2714


Table 3. Four Sets of Controller Parameters for Example 1 controller parameters general-structure controller for set-point changes high-order (r ) 4) simple-structure controller for set-point changes PID controller for set-point changes high-order (r ) 4) simple-structure controller for load disturbances

parameters (SAE ) 23.55 versus SAE ) 23.25 for setpoint changes and SAE ) 9.49 versus SAE ) 9.33 for load disturbances).

Example 3

GP(s) )

e-0.5s , s(s + 1) T ) 0.1, GL(z) ) G(z)

This example is an integrating process. The controller of structure 1 and the controller of structure 3 with r ) 3 are both ideal for such simple dynamics, i.e., those dynamics containing no significant zero in the s domain. The designs obtained by adjusting two parameters, γ and θ, give comparable responses for both set-point changes and load disturbances, as seen in Figure 8. The GPC controller is not able to produce an overshoot of less than 5% no matter how M and G0 are tuned. The proposed PID controller (structure 3 with r ) 2) is far superior to the PID controller obtained by the SGPC law.

Example 4

GP(s) )

e-0.1s , (s - 1)(0.5s + 1) T ) 0.1, GL(z) ) G(z)

Example 4 contains one unstable pole. The design results of various controllers are shown in Figure 9. The dominant pole placement algorithm with two adjustable parameters is employed. It follows that the controller of structure 1 and the controller of structure 3 with r ) 3 are far superior to the other controllers. The proposed PID controller (structure 3 with r ) 2) still outperforms the SGPC controller. 9. Conclusions Two discrete-time control laws have been proposed to cope with a wide range of stable, integrating, and unstable processes. The control law (16), having a partially simple structure, and the control law (22), having a completely simple structure with r ) n + 1, are ideal for continuous processes with no significant zeros. For processes containing significant zeros, the simple control law (22) with r ) n is a much better alternative. Both simple control laws can eliminate the derivative and/or the proportional kick and avoid ringing of the manipulated variable. The controller parameters can be easily determined by prescribed closed-loop poles to produce a specified response. Moreover, two algorithms are presented to facilitate the design of an optimal or approximately optimal controller that satisfies various specifications on stability robustness. The advantage of the modified IMC algorithm is that it contains only a single tuning parameter. This simplicity is due to the use of pole-zero cancellation. Consequently, it has the disadvantages that the optimal

P(z-1)

) 11.846 - 32.689z-1 + 32.684z-2 - 13.774z-3 + 1.9859z-4 Q(z-1) ) 1 - 0.1756z-1 - 0.0934z-2 - 0.0383z-3 - 0.0019z-4 + 0.2424z-5 + 0.1181z-6 - 0.0577z-7 R(z-1) ) 0.0457 + 0.7173∆ K(z-1) ) 0.0457 + 0.7173∆ + 4.5432∆2 + 12.377∆3 + 0.5912∆4 R(z-1) ) 0.0413 + 0.5927∆ K(z-1) ) 0.0413 + 0.5927∆ + 3.2153∆2 K(z-1) ) 0.0598 + 0.8718∆ + 4.589∆2 + 9.24∆3 + 0.2667∆4

design is not necessarily obtainable and ringing of the manipulated variable can occur. On the contrary, the dominant pole placement algorithm does not rely on pole-zero cancellation and hence mitigates possible ringing of the controller output. At the cost of increased computational effort, an approximately optimal design is achievable by adjusting two or three parameters. It is worth noting that the ringing problem can only occur for general-structure controllers. While the two algorithms are applied as tools to design a simple-structure controller, those drawbacks specific to the general structure, such as the ringing problem, are automatically eliminated. The partially simple control law (16) and the GPC law are liable to induce a peculiar feature on the open-loop frequency response; that is, the Nyquist curve may enlarge in magnitude for high frequencies. This could lead to very poor stability robustness that is not detectable merely by the conventional GM and PM. A new index MGH has been proposed to uncover this feature and prevent its happening. The simple-structure controller (22) is less likely to produce such a peculiar feature on the Nyquist curve under the constraints on the GM and PM. Another advantage is that it allows the user to specify the controller order, e.g., a PID controller (r ) 2) or a higherorder controller (r g 3). Nevertheless, the final design can be achieved by adjusting one to three parameters despite a large number of control parameters associated with a high-order controller. As stated in the introductory section, a number of procedures have been introduced to tune the parameters of a simple-structure controller by minimizing performance criteria via numerical optimization techniques. If a high-order controller is required, the computation burden increases with the number of parameters that must be adjusted. The proposed method, on the other hand, designs a simple-structure controller of any order through approximations of the general-structure controller by assigning at most three parameters. This is an advantage in that the computational effort is greatly reduced for a high-order controller. However, if a loworder PID control structure is considered, both the proposed method with the dominant pole algorithm and the parameter optimization method require three parameters to be tuned. Hence, judging merely on the basis of the computation burden in arriving at the minimum performance criterion cannot completely reveal the advantage of the proposed method over the optimization techniques for PID design. From our viewpoint, the merit of the proposed method also lies in the fact that the three parameters γ, θ, and λ can be easily adjusted to give any desired closed-loop response, which is not a simple task for the parameter optimization techniques. Furthermore, if the modified IMC


algorithm is incorporated, only one parameter R needs to be adjusted. Acknowledgment This work was supported by the National Science Council of R.O.C. under Grant NSC-90-2214-E006-005. Literature Cited (1) Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Model Predictive Heuristic Control: Applications to Industrial Processes. Automatica 1978, 14, 413-428. (2) Cutler, C. R.; Ramaker, B. L. DMCsA Computer Control Algorithm. AIChE 1979 Houston Meeting; AIChE: New York, 1979; Paper 516. (3) Garcia, C. E.; Morari, M. Internal Model Control. 1. A Unifying Review and Some New Results. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 308-323. (4) Clarke, D. W.; Gawthrop, P. J. Self-Tuning Controller. Proc. IEE 1975, 122, 929-934. (5) Clarke, D. W.; Gawthrop, P. J. Self-Tuning Control. Proc. IEE 1979, 126, 633-640. (6) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized Predictive Control. Automatica 1987, 23-2, 137-160. (7) Clarke, D. W.; Mohtadi, C. Properties of Generalized Predictive Control. Automatica 1989, 25-6, 859-875. (8) Wellstead, P. E.; Prager, D.; Zanker, P. Pole Assignment Self-Tuning Regulator. Proc. IEE 1979, 126, 781-787. (9) Allidina, A. Y.; Hughes, F. M. Generalized Self-Tuning Controller with Pole Assignment. Proc. IEE 1980, 127, 13-18. (10) Lelic, M. A.; Zarrop, M. B. Generalized Pole-Placement Self-Tuning Controller: Part 1. Basic Algorithm. Int. J. Control 1987, 46, 547-568. (11) Astrom, K. J.; Wittenmark, B. Self-Tuning Controllers Based on Pole-Zero Placement. Proc. IEE 1980, 127, 120-130.

(12) Isermann, R. Digital Control Systems, Volume 1: Fundamentals, Deterministic Control; Springer-Verlag: New York, 1989. (13) Zafiriou, E.; Morari, M. Digital Controllers for SISO Systems: A Review and a New Algorithm. Int. J. Control 1985, 42, 855-876. (14) Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 64, 759-768. (15) Astrom, K. J.; Hagglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 1984, 20, 645-651. (16) Cameron, F.; Seborg, D. E. A Self-Tuning Controller with a PID Structure. Int. J. Control 1983, 38-2, 401-417. (17) Gawthrop, P. J. Self-Tuning PID Control: Algorithms and Implementation. IEEE Trans. Autom. Control 1986, AC31-3, 201-209. (18) Yamamoto, T.; Omatu, S.; Kaneda, M. A Design Method of Self-Tuning PID Controllers. Proceedings of the American Control Conference, Baltimore, MD, June 1994; pp 3263-3267. (19) Yamamoto, T.; Inoue, A.; Shah, S. L. Generalized Minimum Variance Self-Tuning Pole-Assignment Controller with a PID Structure. Proceedings of the 1999 IEEE International Conference on Control Applications, Hawaii, Aug 1999; pp 125-130. (20) Smith, C. A.; Corripio, A. B. Principles and Practice of Automatic Process Control; John Wiley & Sons: New York, 1997. (21) Coughanowr, D. R. Process Systems Analysis and Control; McGraw-Hill: New York, 1991. (22) Ho, W. K.; Lim, K. W.; Xu, W. Optimal Gain and Phase Margin Tuning for PID Controllers. Automatica 1998, 34, 10091014. (23) Kuo, B. C. Digital Control Systems; Holt-Saunders: Tokyo, 1980.

Received for review July 23, 2001 Revised manuscript received December 14, 2001 Accepted March 20, 2002 IE0106276

Robust Design Method for Discrete-Time Controllers with Simple

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