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Ind. Eng. Chem. Res. 2006, 45, 1408-1419
Optimal Design of the Refined Ziegler-Nichols Proportional-Integral-Derivative Controller for Stable and Unstable Processes with Time Delays† Weidong Zhang* Institute for Systems Theory in Engineering, UniVersity of Stuttgart, Stuttgart 70550, Germany
In this paper, the well-known refined Ziegler-Nichols (RZN) method is analyzed and extended to the control of unstable processes with time delays. It is shown that both the RZN control structure and the inner stabilizing control structure can be equivalent to the unity feedback loop with a set-point filter or, equivalently, a twodegree-of-freedom control structure. This implies that those analysis and design techniques developed for the classical control structure can be applied to the two special control structures. The original formulas for the RZN proportional-integral-derivative (PID) controller are empirical and can only be used in restricted scope. A new procedure is developed for designing the RZN PID controller. Suboptimal formulas without the restriction are analytically derived. In the procedure, the load response can be independently optimized. Nominal performance of the closed-loop system is discussed, and sufficient and necessary conditions for internal stability and robust stability are given. Numerical examples are provided to illustrate the proposed method and compare it with the original RZN method. 1. Introduction The proportional-integral-derivative (PID) controller remains the most popular controller for industrial control applications despite continual advances in control theory. This is not only due to the simple structure, which is conceptually easy to understand and makes manual tuning possible, but also due to the fact that the algorithm provides adequate performance in the vast majority of applications. The PID controller has been implemented in many different forms in industries: as standalone regulators, as parts of soft packages and hierarchical distributed control systems, as built-in controllers in robots, and so on. The controller can also be combined with other technologies to build complex automation systems (ref 1). In the PID controller, the integral, proportional, and derivative parameters are based on past, present, and future control errors, respectively. To implement such a controller, the three parameters must be determined for the given process. Usually, process engineers must tune PID controllers manually. The most wellknown method is the Ziegler-Nichols (ZN) method developed by Ziegler and Nichols.2 The method determines the parameters by observing the gain at which the system begins to oscillate and the frequency of this oscillation. Many other similar methods have been developed since then. Since the manual tuning is laborious and time-consuming and requires the close attention of the process engineer, autotuning methods that can automatically tune PID controllers were later developed. These methods automatically generate a special input to the process, and by the measurement of the process response, the PID parameters are computed with the ZN formulas.3-5 The simplicity of these methods has made it possible to develop a wide range of such autotuning instruments. † The original version of this paper was presented at the 42nd IEEE Conference on Decision and Control. This is a modified and extended version of that paper. * To whom correspondence should be addressed. This work was finished when Weidong Zhang was a Humbold Research Fellow at the IST, University of Stuttgart, with Prof. F. Allgower. His permanent address is Department of Automation, Shanghai Jiaotong University, Shanghai 200240, P.R. China. Tel.: +86.21.34202019. Fax: +86.21.54260762. E-mail:
[email protected].
Other works concerning the tuning and autotuning methods include the Cohen-Coon method,6 the gain-phase margin method,7,8 the Ziegler-Nichols complementary method,9,10 the no-overshoot method,11 the refined Ziegler-Nichols (RZN) method,18-20 and the integral of squared time weighted error method.12 References to these methods and some newly developed methods can be found in refs 1, 13, and 14. Among these tuning methods, the ZN method and the RZN method are very important. This is because they have been widely accepted in practice and have been used in many papers as benchmarks. In this paper, we explore the RZN method in several aspects: (1) The nominal performance and robust stability are analyzed. Sufficient and necessary conditions for internal stability and robust stability are provided. A simple tuning rule is developed. (2) The original formulas for the RZN PID controller are empirical and can only be used in restricted scope. In this paper, an optimal design procedure is developed and analytical formulas without the restriction are derived. The tuning of four parameters is reduced to that of only two parameters. (3) Many design methods are developed for the control of stable processes. These methods can be used for the control of unstable process in the following way: Stabilize the unstable process by an inner stabilizing loop, and then, control the augmented stable process. The control structure with the inner stabilizing loop is analyzed, and its equivalence to the unity feedback loop is discussed. (4) On the basis of the discussion of aspect 3, the original RZN method is extended to the control of unstable processes with time delays. The nominal performance and robust stability are analyzed, an optimal design procedure is developed, and analytical design formulas are obtained. This paper is organized as follows. Several related methods are reviewed and compared in section 2. In section 3, an optimal RZN design procedure is developed and the nominal performance and robust stability are analyzed. In section 4, the original RZN method is extended to the control of unstable processes with time delays. Numerical examples are given to illustrate the proposed method in section 5. Finally, conclusions are given in section 6.
10.1021/ie0507981 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/19/2006
Ind. Eng. Chem. Res., Vol. 45, No. 4, 2006 1409
is designed to give a phase specification of 25 degrees, by reducing the gain to 66.2% of it, one can obtain the same results as that of the ZN method. The procedure is used by Mantz and Tacconi9 to estimate the location of the dominant poles, and the two weights are obtained as follows:
Figure 1. Modified PID control structure.
2. Review and Comparison of Several Related Methods The PID controller is usually implemented as follows:
(
u ) kc e +
∫
)
1 dy e dt + Td Ti dt
(4)
(1) R)
The controller output, process output, and set-point are u, y, and r, respectively. The variables kc, Ti, and Td are positive real constants. To implement it physically, a lag of 1/(1 + sTd/10) is usually introduced to the derivative term or all the terms of u. The ZN tuning method has been widely known as a fairly accurate heuristic method to determine good settings for PID and PI controllers for a wide range of common industrial processes. The method is based on the empirical knowledge of the ultimate gain ku and ultimate period tu. One shortcoming of the ZN method is that it usually gives a large overshoot for the set-point response. The excessive overshoot can be overcome by tuning the gain. This will, however, decrease the speed of both the set-point and load responses. Eitelberg15 and Hippe et al.16 showed that the following form of the PID controller could reduce the set-point overshoot (Figure 1):
)
ded 1 ei dt + Td Ti dt
∫
R ) 0.654
2|σ| (σ + ω2)Ti
(5)
1 (σ + ω2)TiTd
(6)
β)
e)r-y
(
(3)
Let the dominant pole be denoted by σ + jω. The empirical formulas for systems with time delays are10
where
u ) kc ep +
β ) 0.17
(2)
where
ep ) βr - y
Here, R, β, and γ are weights. If R ) β ) γ ) 1, the modified structure is reduced to the classical one. Usually, γ ) 1 has to be chosen so that the steady error is zero. Then, the key to overcoming the large overshoot is choosing the two other weights. Eitelberg15 and Hippe et al.16 developed a trial and error procedure to determine the two weights. Mantz and Tacconi9,10 also proposed a simple direct method. Suppose that the closed-loop transfer function of the system controlled by a classical PID controller can be modeled by a pair of complex dominant poles. The two weights are chosen so that the zeros of the closed-loop transfer function of the system controlled by a modified PID controller are assigned to reduce the effect of the complex dominant poles. The residues corresponding to the complex poles are substantially reduced, while those corresponding to the poles placed near to the original zeros are increased. Thus, the overshoot of the set-point response will be substantially reduced or eliminated. If a PID controller
2
In ref 17, only the weight β is used and R is assumed to be 1. Astrom and Hagglund17 point out that the classical PID controller introduced a zero in the closed-loop transfer function. This zero will causes excessive overshoot if it is too close to the real part of the dominant pole. The modified PID controller gives a modified zero at -1/(βTi). By selecting a suitable value for β, the zero can be properly placed. Astrom and Hagglund17 recommended the following choice
β)
1 3σTi
(7)
Astrom et al.,18 Hang et al.,19 and Hang and Sin20 point out that the single weight modification is sufficient to reduce the overshoot to a specified value for PID tuning, and the original ZN formulas can be retained. The application of the weight and the modification of the tuning formulas are based on the knowledge of the normalized gain and the normalized time delay. Assume that the process is
kpe-θps G(s) ) τps + 1
(8)
Let Go(s) be the minimum phase part of G(s), that is, G(s) ) Go(s) exp(-θps). The normalized gain is defined as the product of the ultimate gain and the process gain. Formally,
ei ) γr - y ed ) Rr - y
2
k ) kpku
(9)
and the normalized time delay is defined as the ratio of the time delay to the time constant of the process. Formally,
Θ)
θp τp
(10)
When 2.25 < k < 15 and 0.16 < Θ < 0.57, the ZN formulas are retained and only the set-point weight is applied. For 10% overshoot,
15 - k 15 + k
(11)
36 27 + 5k
(12)
β) and for 20% overshoot,
β)
If 1.5 < k < 2.25 and 0.57 < Θ < 0.96, both the set-point
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Figure 2. Unity feedback control loop.
Figure 3. Equivalent of the modified PID control structure.
weight and the modified integral time constant should be applied. The new integral time constant is given as
Hence, the closed-loop system can be considered as two blocks in a serial connection (Figure 3): one is the unity feedback loop and the other can be regarded as a set-point filter. This is, in fact, a two-degree-of-freedom control structure. The structure in Figure 3 provides a generalized extension of that in Figure 1, because F(s) can be in a more general form rather than that of (19) and it is possible to optimize the load response independently. The control system design problem can usually be stated as follows: given a linear, time invariant plant, design a controller such that the closed-loop system is internally stable and satisfy some other performance specifications. It is evident that the closed-loop system shown in Figure 3 is internally stable if and only if (1) the set-point filter is stable and (2) the unity feedback loop is internally stable. Since there is no constraint on the set-point filter, one can choose it as any stable set of transfer functions. With respect nominal performance, however, we hope to choose a set-point filter such that the nominal performance is optimal or suboptimal. Morari and Zafriou21 show that if the set-point filter is chosen as the inverse of the minimum phase part of the total unity feedback loop, that is,
Ti ) 0.5µtu
(13)
where µ is defined as the ratio of the new integral time constant to the ZN integral time constant. For 20% overshoot, the formulas are
8 4 k+1 17 9
(
β)
)
(14)
4 µ) k 9
(15)
Equations 11-15 are the well-known RZN formulas for PID controller tuning. 3. Analysis and Design for Stable Processes The control structure shown in Figure 1 is not a unity feedback loop. Thus, it is difficult to directly analyze the nominal performance and robust stability. As a result, some important questions cannot be answered, for example, (1) how to design the optimal control system, (2) how to tune the controller for the required set-point response, (3) how to optimize the load response, and (4) how to estimate the robust stability. In this section, the relationship between the structure and the classical unity feedback loop is studied. It is shown that the structure is closely related to the unity feedback loop. Let C(s) be the standard PID controller:
(
)
1 + Tis + TiTds2 1 + Tds ) kc C(s) ) kc 1 + Tis Tis
(16)
The closed-loop transfer function of the unity feedback loop (Figure 2) is
Hu(s) )
G(s) C(s) 1 + G(s) C(s)
(17)
The closed-loop transfer function of the modified control structure (Figure 1) is
Hm(s) )
G(s) F(s) 1 + G(s) C(s)
(18)
with
(
F(s) ) kc β +
)
1 + βTis + RTiTds 1 + RTds ) kc Ti s Ti s
2
F(s) G(s) C(s) C(s) 1 + G(s) C(s)
)
-1
)
1 + G(s) C(s) Go(s) C(s)
(21)
the optimal nominal performance can be obtained. However, this may not be physically implemented, because the delay free part of the block is usually strictly proper. In this case, a lowpass element has to be introduced and then a suboptimal controller will be obtained:
1 + G(s) C(s) F(s) 1 ) C(s) (λfs + 1)m Go(s) C(s)
(22)
Here, m is an integer and should be chosen such that the setpoint filter is biproper. λf is a positive constant and is defined as the performance degree, because it can be used to tune the shape of the closed-loop system response (overshoot, settling time, etc.). When λf f 0, the optimality is revived. We will introduce how to tune the performance later on. A question of interest is that when the process contains a time delay, the above set-point filter is, in fact, of infinity dimension and, thus, cannot be directly implemented. In this case, we have two choices. When there is no restriction on the form of F(s), we can implement the set-point filter by two parallel blocks:
(19) F(s) 1 1 e-θps + ) C(s) (λfs + 1)mGo(s) C(s) (λfs + 1)m
Hm(s) can then be rewritten as
Hm(s) )
(
Go(s) C(s) F(s) ) C(s) 1 + G(s) C(s)
(20)
The new control structure is shown in Figure 4.
(23)
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Figure 4. New control structure for the modified PID controller.
Sometimes, a RZN PID controller may have been installed. We require the control structure in Figure 1 with F(s) in the form of (19). Then, rational approximations or model reduction techniques have to be used. Without loss of generality, assume that the process is described by a first-order plus time delay model. We have
τps + 1
kc(1 + Tis + TiTds2) -θps + e F(s) ) kp(λfs + 1) (λfs + 1)Tis )
(τps + 1)Tis + kpkc(1 + Tis + TiTds2)e-θps kp(λfs + 1)Tis
(24)
The Taylor series might as well be used to expand the above F(s). For the convenience of calculation, let
not independent. They are internally related to the performance degree. By tuning the performance degree, these parameters are simultaneously adjusted and a suboptimal set-point response can be obtained. A limitation of this method is that some PID term may be negative.22 A simple way to overcome the limitation is to choose the first two or four terms as a controller. In the former case, we obtain a PI controller, while in the latter case we obtain a four-term controller or a PID controller with a lag. To obtain a PID controller with a lag, we can let F(s) ) {(f(s)[1 - f ′′′(0)/ 3f ′′(0)])/(s[1 - f ′′′(0)/3f ′′(0)])}{1/(λfs + 1)} and expand the numerator in a Taylor series. This makes the third-order term become zero, as in ref 22. To make it more clear, the design procedure is summarized as follows: (1) Design a PID controller by (27) and (28). (2) If one chooses the control structure in Figure 4, design the setpoint filter by (23), or if one chooses the control structure in Figure 3, design the set-point filter by (19), (29), and (30). It is the dual purpose of control to change the dynamics of the given system for a better response and to be able to do this even if the system is not as well modeled as one would wish. How the closed-loop system is affected by the model-plant mismatch can be exploited according to more recent robustness criteria (see, for example, refs 21 and 23). Suppose the normbounded uncertainty is defined by
|
(τps + 1)Tis + kpkc(1 + Tis + TiTds2)e-θps (25) f(s) ) kpTi
|
G ˜ (s) - G(s) e ∆(ω) G(s)
(31)
where G ˜ (s) is the real process and ∆(ω) is the uncertainty profile. Suppose that the real plant is in the form of
Then,
F(s) )
f(s) 1 s λfs + 1
(26) G ˜ (s) )
Expanding this in a Taylor series gives
(
)
f′′(0) 2 1 1 s + ... F(s) ) f(0) + f′(0)s + s 2! λfs + 1
k˜ pe-θ˜ ps τ˜ ps + 1
(32)
with
(27)
where
+ + + ˜ p ∈ [θ˜ p ∈ [τk˜p ∈ [kp , kp ], θ p , θp ], τ p , τp ]
The midrange plant is chosen to be the model:
f(0) ) f′(0) )
kc Ti
Ti + kpkcTi - kpkcθp Tikp
kpe-θps G(s) ) τps + 1 (28)
where
2τpTi + 2kpkcTiTd - 2kpkcTiθp + kpkcθp2 f′′(0) ) Tikp
kp )
The first three terms consist of a PID controller. With this controller, F(s) is in the form of (19) and the weights are
f′(0) kc
(29)
f′′(0) R) 2kcTd
(30)
β)
(33)
+ + + kθτp + kp p + θp p + τp , θp ) , τp ) 2 2 2
Let
∆k ) |k+ p - kp| ∆τ ) |τ+ p - τp| ∆θ ) |θ+ p - θp|
Compared with that of the original RZN method, the new formulas are suboptimal for the set-point response, are analytical, and have no restriction on the scope of the normalized gain and the normalized time delay. It is also noticed that in the original RZN method we have to tune four to five parameters. However, in the new PID controller, all these parameters are
{
The uncertainty profile can be directly obtained as follows:24
∆(ω) )
| |
|
|kp| + ∆k jτω + 1 e-j∆θω - 1 ω < ω* |kp| j(τ - ∆τ)ω - 1
|
|kp| + ∆k jτω + 1 +1 |kp| j(τ - ∆τ)ω - 1
ω gω* (34)
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Where ω* is determined by
∆θω* + arctan
∆τω* π e ∆θω* e π ) π, 2 2 1 + τ(τ - ∆τ)ω* (35)
Especially, when only the gain is uncertain, that is, ∆τ ) ∆θ ) 0, the expression simplifies to
∆(ω) )
∆k |kp|
(36)
When only the time constant is uncertain, that is, ∆K ) ∆θ ) 0, the expression simplifies to
∆(ω) )
|
jτpω + 1
j(τp - ∆τp)ω + 1
-1
|
(37)
When only the time-delay is uncertain, ∆τ ) ∆K ) 0. We get ω* ) π/∆θ. In this case,
{
-j∆θpω - 1| ω < ω* ∆(ω) ) |e 2 ω g ω*
(38)
The robust stability is met if and only if21
||∆(ω) Hu(jω)||∞ < 1, ∀ ω
(39)
The PID controller in the equivalent control structure can also be designed by any number of methods. In practice, besides the performance and stability, one important factor is the simplicity of the design and tuning of a control system. One extreme case can be found in process control, where one engineer may be responsible for several hundred loops. In such a case, it is not possible to devote much effort to each loop. Simplicity of handling is then one of the primary requirements. Due to this reason, it is recommended to use the H infinity design method proposed by Zhang et al.25. Suppose that the PID controller is in the form of
(
C(s) ) kc 1 +
)
1 1 + Tds Tis Tfs + 1
(40)
For the first-order plus time delay model, the tuning rules are
Tf )
θp θpτp λ2 , Ti ) + τp, Td ) , 2λ + θp/2 2 2Ti kc )
Ti kp(2λ + θp/2)
(41)
where λ is the performance degree. In this case, the load response is suboptimal. The derivation of the tuning rules is briefly stated in Appendix A. At times, only the ultimate gain and ultimate period are obtained in the autotuning procedure. We have to convert the model described by the ultimate gain and ultimate period into the first-order plus time delay model. The relationship is
tux(kukp)2 - 1 τp ) 2π θp )
tu tutg-1x(kukp)2 - 1 2 2π
(42)
(43)
Assume that the load loop is deigned by the ZN method; we only need to tune the set-point response. In case the load loop is designed by other methods, for example, the above method, we should tune the load response first and then tune the setpoint response. As a matter of fact, in the above methods, it is normally not necessary to use the set-point filter. The overshoot can be calculated as follows overshoot ) -0.86(λ/θp)3 + 14.21(λ/θp)2 - 8.72λ/θp + 1.86 0.1 e λ/θp e 0.59 0.59 e λ/θp e 1.2 0
{
(44)
or one can use a rule such as the following: increase the performance degree monotonically until the required response is obtained. Other indices, such as rise time, stability margin, integral squared error (ISE), etc., are also determined only by the ratio of the performance degree to θp and can be similarly calculated. There always exists model uncertainty in practice. It can be verified that there is a monotonic relationship between the performance degree and the robustness, and good robustness can be obtained by increasing the performance degree. For an uncertain system, we usually wish to estimate the “worst case” response (i.e., the gain and time delay are at their upper limits, while the time constant is at its lower limit) or the range of the closed-loop response. Assume that we hope to achieve the worst case overshoot of 5%. A simple quantitative tuning procedure proposed by Zhang et al.25 is as follows: (1) Design the controller for the nominal process. For 5% overshoot, λf ) 0.5θp. (2) Substitute the nominal process with the worst case process. (3) Increase the performance degree λf monotonically, until the overshoot is equal to 5%. The typical step is 0.01θp (or smaller). If the process has no time delay, the step can be chosen as 0.01τp. If the set-point filter is used, the tuning procedure is similar. Even for the robustness, we can use the following rule: increase the performance degree monotonically until the required response is obtained. Leva and Colombo26 proposed an optimization procedure for stable plants. Compared with the approach of Leva and Colombo26, the beneficial features of the proposed method are that the result of the optimal design is analytic, the tuning of four parameters is reduced to that of only two parameters, and the tuning rule is very simple. These features make it possible, as indicated above, to simultaneously tune the four parameters of the PID controller in a simple way to obtain a quantitative closed-loop response (both in performance and robustness) for a system with a time delay, especially when the uncertainty profile is unknown. 4. Unstable Processes with Time Delays Most industrial processes are stable and can be described adequately by a first-order plus time delay model. For the control of such systems, many methods have been developed. One of the difficult problems is the control of unstable processes with time delays (including the special case, integrating processes with time delays). This has been the subject of many studies. As early as 1966, Koppel27 studied sampled-data control of a first-order, open-loop, unstable process. Luyben and Melcic28 explored the effect of a time delay on the stability of an unstable process. Since the ZN method cannot be directly used for the control of an unstable process, De Paor and O’Malley29 proposed
Ind. Eng. Chem. Res., Vol. 45, No. 4, 2006 1413
Figure 5. Control structure with an inner stabilizing loop.
Figure 7. Equivalent 2 of the control structure with an inner stabilizing loop.
There exists another arrangement for the closed-loop system. Rewrite the closed-loop transfer function as follows:
Hm(s) )
G(s) C(s) 1 + G(s) C(s) 1 + G(s) K(s) + G(s) C(s) 1 + G(s) C(s)
(47)
Let
F(s) ) Figure 6. Equivalent 1 of the control structure with an inner stabilizing loop.
a modified ZN method. In the past 10 years, this problem has appealed to more and more researchers.30-32 So far, two major methods have been developed for the control of unstable processes. One is to directly control the unstable processes by a controller, and the other is to stabilize the unstable process with an inner loop and then to control the augmented stable process. The latter has been adopted by many other control strategies such as model predictive control. In this section, the relationship between the two control structures, as well as the relationship between the inner stabilizing control structure and the RZN control structure, will be discussed. As a result, the RZN method is extended to the control of unstable processes. A typical control structure with an inner stabilizing loop is shown in Figure 5, where K(s) is the stabilizer. The augmented stable process can be written as
Ga(s) )
G(s) 1 + G(s) K(s)
(45)
Ga(s) C(s) 1 + Ga(s) C(s)
)
G(s) C(s) 1 + G(s) K(s) + G(s) C(s)
Hm(s) )
It is claimed that the function of the inner stabilizing loop is in fact equivalent to that of a set-point filter. To illustrate this, rearrange the block diagram, as shown in Figure 6. Then, a twodegree-of-freedom control structure is obtained. An alternative arrangement is shown in Figure 7. It is seen that the closedloop system can be considered to be two blocks in a serial connection: one is the unity feedback loop and the other can be regarded as a set-point filter. This is just as the case shown in Figure 2. This implies that the inner stabilizing loop structure can be equivalent to the RZN structure.
F(s) G(s) C(s) C(s) 1 + G(s) C(s)
(49)
When implemented as Figure 2, the arrangement usually cannot stabilize the unstable process. Therefore, such an arrangement cannot be equivalent to the original system. The reason for this is that only the set-point response is considered in the scheme and the characteristic equation may be changed. Assume that the process is described by the following transfer function
G(s) )
kpe-θps τps - 1
(50)
Usually, for simplicity, K(s) is taken to be a proportional gain ks. It can be designed by the rule given by De Paor and O’Malley29:
ks ) (46)
(48)
We get
The closed-loop transfer function is
Hm(s) )
(1 + G(s) C(s))C(s) 1 + G(s) K(s) + G(s) C(s)
1
x|G(jωu)||G(0)|
(51)
Then, the PID controller C(s) can be designed for the stabilized process. Let us see how the inner stabilizing control structure is related to the RZN structure. Let
Ca(s) ) K(s) + C(s) If C(s) is a PID controller, Ca(s) is also a PID controller:
Ca(s) ) kc Obviously,
1 + (ks/kc + 1)Tis + TiTds2 Tis
(52)
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(ks + kc)Ti kcTd , Tad ) kc k s + kc
kac ) kc + ks, Tai )
C(s) )
(
1 + Tis + TiTds2 C(s) ) Ca(s) 1 + (ks/kc + 1)Tis + TiTds2
-Tai + kpkcTai - kpkcθp Taikp
(64)
f(0) )
(53) f′(0) )
In this case, for the equivalent RZN structure, we have
R)1 1 β) 1 + ks/kc
(54)
1 + G(s) Ca(s) C(s) 1 ) m Ca(s) (λfs + 1) Go(s) Ca(s)
Taking the first three terms as the required PID controller, we get the following formulas
kp(λ + θp)2
f′(0) kac
(65)
f′′(0) 2kacTad
(66)
β) R)
(56)
Consider the norm-bounded uncertainty. If the real plant is
Similar to the case of stable processes, C(s) can be in a general form. For the first-order integrating process, the design result for Ca(s) has been given by Zhang et al.33:
2λ + θp
kc Tai
2τpTai + 2kpkcTaiTad - 2kpkcTaiθp + kpkacθp2 Taikp
f′′(0) )
(55)
To obtain better performance, one can also design the control system by optimal methods. The design procedure involves two steps: (1) Ca(s) is designed for the desired load response. (2) The set-point filter C(s)/Ca(s) is designed so that the suboptimal set-point response is obtained, that is,
G ˜ (s) )
k˜pe-θ˜ ps τ˜ ps - 1
(67)
with
, Tai ) 2λ + θp
(57)
For first-order unstable processes, the result is given as follows
kac )
(63)
The coefficients of the Taylor series are
The set-point filter is
kac )
)
f′′(0) 2 1 1 f(0) + f′(0)s + s + ... s 2! λf s + 1
a + θp aθp λ3 , Tai ) a + θp, Tad ) , Taf ) kpb a + θp τpb (58)
+ + + ˜ p ∈ [θ˜ p ∈ [τk˜p ∈ [kp , kp ], θ p , θp ], τ p , τp ]
the midrange plant is chosen to be the model
G(s) )
(68)
where
The detailed design procedure is provided in Appendix B. In analogy to the case of stable processes,
C(s) 1 1 + e-θps (59) ) Ca(s) (λfs + 1)mGo(s) Ca(s) (λfs + 1)m
kpe-θps τps - 1
+ + + kθτp + kp p + θp p + τp , θp ) , τp ) 2 2 2
kp ) Let
∆k ) |k+ p - kp|
Then,
∆τ ) |τ+ p - τp|
τ ps - 1
kac(1 + Tais + TaiTads2) -θps ) + e C(s) ) kp(λfs + 1) (λfs + 1)Tais (τps - 1)Tais + kpkac(1 + Tais + TaiTads2)e-θps kp(λfs + 1)Tais
∆θ ) |θ+ p - θp| (60)
Let
f(s) )
(τps - 1)Tais + kpkac(1 + Tais + TaiTads2)e-θps kpTai (61)
{
The uncertainty profile can be directly obtained as follows:24
∆(ω) )
| |
|
|kp| + ∆k jτω - 1 e-j∆θω - 1 ω < ω* |kp| j(τ - ∆τ)ω - 1
|
|kp| + ∆k jτω - 1 +1 |kp| j(τ - ∆τ)ω - 1
ω gω* (69)
where ω* is determined by
We have
C(s) )
f(s) 1 s λfs + 1
Expanding it in a Taylor series gives
(62)
-∆θω* + arctan
-∆τω* ) -π, 1 - τ(-τ + ∆τ)ω*2 π e ∆θω* e π (70) 2
Ind. Eng. Chem. Res., Vol. 45, No. 4, 2006 1415
Especially, when only the gain is uncertain, that is, ∆τ ) ∆θ ) 0, the expression simplifies to
∆(ω) )
∆k |kp|
(71)
When only the time constant is uncertain, that is, ∆K ) ∆θ ) 0, the expression simplifies to
∆(ω) )
|
jτpω - 1
j(τp - ∆τp)ω - 1
-1
|
(72)
When only the time delay is uncertain, that is, ∆τ ) ∆K ) 0, we get ω* ) π/∆θ. In this case,
{
-j∆θpω - 1| ω < ω* ∆(ω) ) |e 2 ω gω*
(73)
The robust stability is met if and only if
||
G(jω) Ca(jω) ∆(ω) 1 + G(jω) Ca(jω)
||
∞
< 1, ∀ ω
Figure 8. Nominal responses of the closed-loop system in Example 1: (solid line) RZN; (dashed line) New1; (dotted line) New2; (dash-dotted line) New3.
(74)
With the same tuning rule as that for stable process, the PID controller for unstable processes can also be tuned for quantitative performance and robustness. 5. Numerical Examples Example 1. Consider the following process:
G(s) )
1 -2s e s+1
The ultimate gain is 1.52, and the ultimate period is 5.49. Then, the PID parameters given by the ZN method are kc ) 0.91, Ti ) 2.75, and Td ) 0.68. The parameters of the RZN method are β ) 0.79 and µ ) 0.68. The RZN control structure is in fact equivalent to that of a unity feedback loop with the set-point filter
1.26s2 + 1.47s + 1 1.26s2 + 1.86s + 1 The filter can also be chosen to be a simpler one, such as 1/(λs + 1), for most of the process. We take λf ) 0.7 for the proposed controller. The parameter can be determined by the tuning rule given in section 3. If the PID controller is designed by the ZN method, f(0) ) 0.33, f ′(0) ) 1.25, and f ′′(0) ) 0.92. The parameters of the generalized RZN method with an approximated set-point filter (New1) are β ) 1.37 and R ) 0.74. If the PID controller is tuned by the RZN method, f(0) ) 0.49, f ′(0) ) 0.93, and f ′′(0) ) 1.55. The parameters of the proposed modified RZN method with an approximated set-point filter (New2) are β ) 1.02 and R ) 1.26. In the generalized RZN method with the exact set-point filter (New3), the filter is in a more general form and, thus, there are no such parameters. Assume that a step set-point is added at t ) 0 and a step load is added at t ) 30. The closedloop responses are shown in Figure 8. For the set-point response, New1 has almost the same overshoot as that of the RZN method. New2 has a smaller overshoot than the RZN method. New3 provides significant improvement on the overshoot, and the response is very steady. The load response of the RZN method is better than that of New1. This implies that the modification on the integral time
Figure 9. Robust responses for θp ) 2.4: (solid line) RZN; (dashed line) New1; (dotted line) New2; (dash-dotted line) New3.
by the RZN method is efficient, and thus, it is recommended to use it. Since the set-point filter has no effect on the load response, the load responses of New2 and New3 are optimized by the RZN rule. The resulted load responses are identical to that of the RZN method. The ratio of the time delay to the time constant in this example is very large. If the ratio is small, that is, less than 0.5, the proposed three methods can give very good set-point responses. The difference of set-point responses between New2 and New3 is very small. Now, suppose that there is 20% error in the estimation of the time delay. The worst case is θp ) 2.4. In this case, the closed-loop responses are shown in Figure 9. The uncertainty profile is
{
-j0.4ω - 1| ω < 2.5π ∆(ω) ) |e 2 ω g 2.5π
The closed-loop system shows robust stability if and only if
G(jω) C(jω) ||∆(ω) || < 1, ∀ ω 1 + G(jω) C(jω) ∞ Example 2. Consider an unstable process described by
G(s) )
1 -0.5s e s-1
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Figure 10. Nominal responses of an unstable system: (solid line) DO; (dotted line) IS; (dash-dotted line) New1.
If a PID controller is directly used, the parameters given by De Paor and O’Malley29 (the DO parameters) are kc ) 1.357, Ti ) 4.6083, and Td ) 0.217. For the inner stabilizing method (IS), the stabilizing gain is ks ) 1.5927. The augmented stable process is
Ga(s) ≈
1.6872e-0.34s 0.3913s2 + 0.4939s + 1
The PID parameters for the augmented stable process are kc ) 1.0073, Ti ) 0.6906, and Td ) 0.6384. This is equivalent to the unity feedback loop with the following PID controller
Ca(s) ) 2.6000
1 + 1.7826s + 0.4409s2 1.7826s
The set-point filter is
C(s) 1 + 0.6906s + 0.4409s2 ) Ca(s) 1 + 1.7826s + 0.4409s2 The structure is equivalent to the RZN structure with the parameters β ) 0.3874 and R ) 1. Take λf ) 0.2. Suppose that the DO PID controller is adopted for Ca(s). The parameters of the extended RZN method are β ) 0.1546 and R ) 2.2168. In this case, the set-point filter is
C(s) 1.357 + 0.9667s + 3.0028s2 ) Ca(s) 1.357(1 + 4.608s + 0.9982s2) Assume that a step set-point is added at t ) 0 and a step load is added at t ) 30. The responses of the closed-loop system are shown in Figure 10. The proposed method provides significant improvement for the set-point response. Since the DO PID controller is used in the RZN method, they have the same load responses. Suppose that there is 20% error in estimating the time delay. The worst case is θp ) 0.6. In this case, the IS has been unstable and the closed-loop responses of the DO and the proposed method are shown in Figure 11. The uncertainty profile is
{
-j0.1ω - 1| ω < 10π ∆(ω) ) |e 2 ω g 10π
Figure 11. Robust responses of an unstable system: (solid line) DO; (dash-dotted line) New1.
6. Conclusions Although the PID controller is always very important, practically, it has only received moderate interest from theoreticians. Many important issues have not been well-documented in the literature. In this paper, the well-known RZN method is discussed and extended to the control of unstable processes with time delays. In the past, both the RZN control structure and the inner stabilizing control structure have been regarded as two special structures that have no relationship with the unity feedback loop. It is demonstrated by this paper that the two structures can be equivalent to the unity feedback loop with a set-point filter or, equivalently, a two-degrees-of-freedom control structure. This result makes it possible to analyze and design the corresponding control systems by techniques developed for the classical control structure. The original formulas for the RZN PID controller are empirical and can only be used in restricted cases. In this paper, a new design procedure is developed. Suboptimal formulas without restriction are analytically derived, and the tuning of four parameters is reduced to that of only two parameters. In the new design procedure, the load response can be independently optimized. With those techniques developed for the classical control structure, we analyze the nominal performance and robustness and sufficient and necessary conditions are provided. These results provide insight into tuning installed RZN PID controllers and designing new PID control systems. Acknowledgment This paper is supported by the National Science Foundation of China (60474031), NCET (04-0383), SRSP (04QMH1405), and the Alexander von Humboldt Research Fellowship. The author would like to thank Frank Allgower for the good working environment and interesting discussions. Appendix A: H Infinity PID Controllers for Stable Processes31 Consider the unity feedback control system. Define the transfer function
The closed-loop system shows robust stability if and only if
||
G(jω) Ca(jω)
∆(ω) 1 + G(jω) Ca(jω)
||
∞
Q(s) )
< 1, ∀ ω We have
C(s) 1 + G(s) C(s)
(A.1)
Ind. Eng. Chem. Res., Vol. 45, No. 4, 2006 1417
C(s) )
Q(s) 1 - G(s) Q(s)
(A.2)
The sensitivity transfer function of a closed-loop system can be written as
S(s) )
1 ) 1 - G(s) Q(s) 1 + G(s) C(s)
(A.3)
A central objective in automatic control is that a physical quantity is made to behave in a prescribed way by using the error between the system output and the set-point input. This gives rise to optimal control. Assume that the optimal performance index is H infinity optimal, i.e., min||W(s) S(s)||∞, where W(s) is a weighting function. W(s) ) 1/s for a unit step setpoint. With a Pade approximation, we have
1 - θps/2 G(s) ) K (τps + 1)(1 + θps/2)
(A.4) Figure A1. Quantitative overshoot for a stable process.
It is seen that W(s) S(s) is analytical in the open right half plane. According to the well-known maximum modulus theorem, ||W(s) S(s)|| does not attain its maximum value at an interior point of the open right half plane. On the other hand, G(s) has a zero at s ) 2/θ in the open right half plane. Thus, for all Q(s)
||W(s)(1 - G(s) Q(s))||∞ g |W(2/θp)|
(A.5)
Then we have
If the practical PID controller is in the form of
(
C(s) ) kc 1 +
lim(1 - G(s) Q(s)) ) 0 sf0
(A.7)
In other words, we must guarantee that S(s) ) (1 - G(s) Q(s)) has a zero at s ) 0 to cancel the pole of W(s). With the constraint, the unique optimal Qim(s) is obtained as follows
Tf )
θp θpτp λ2 , , Ti ) + τp, Td ) 2λ + θp/2 2 2Ti kc )
J(s) )
Q(s) ) Qim(s) J(s) )
(τps + 1)(1 + θps/2) kp(λs + 1)2
Q(s) )
1 (τps + 1)(1 + θps/2) kps λ2s + 2λ + θ /2
C(s) 1 + G(s) C(s)
(B.1)
The controller can be computed by the inverse relation
Q(s) 1 - G(s) Q(s)
(B.2)
Under the nominal condition, the closed-loop transfer function is
G(s) C(s) ) G(s) Q(s) 1 + G(s) C(s)
(B.3)
The closed-loop system is said to be internally stable if all the transfer functions in the following matrix are stable21
(A.8)
As λ tends to be zero, the controller tends to be optimal. The corresponding controller of a unity feedback loop is
C(s) )
(A.11)
Consider a unity feedback loop. Define the transfer function
Hm(s) )
then
kp(2λ + θp/2)
Appendix B: H Infinity PID Controllers for Unstable Processes34
C(s) )
1 , λ>0 (λs + 1)2
Ti
The quantitative overshoot is shown in Figure A1.
(τps + 1)(1 + θps/2) Qim(s) ) kp Obviously, Qim(s) is improper. Now, use the following low pass filter to roll Qim(s) off at high frequency:
(A.10)
The parameters of the PID controller are
min||W(s) S(s)||∞ ) min||W(s)(1 - G(s) Q(s))||∞ ) θp/2 (A.6) However, W(s) has a pole on the imaginary axis. To obtain a finite infinity norm, a constraint will be imposed on the design procedure:
)
1 1 + T ds Tis Tfs + 1
(A.9)
[
G(s) Q(s) G(s)(1 - G(s) Q(s)) Q(s) -G(s) Q(s)
]
This is equivalent to the condition that Q(s) is stable and satisfies the following constraints
lim S(s) ) lim 1 - G(s) Q(s) ) 0
(B.4)
lim S(s) ) lim 1 - G(s) Q(s) ) 0
(B.5)
sf1/τp
sf1/τp
p
The suboptimal H infinity PID controller is derived analytically.
sf0
sf0
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Define the optimal performance index as H infinity optimal, that is, min||W(s) S(s)||∞. Here, W(s) is the weighting function. In process control, controllers are normally designed for step inputs. Thus, W(s) can be selected as 1/s.21,25 Theorem 2.1 (Maximum Modulus Theorem).23 Assume that Ω is a nonempty region in the complex plane. If a function that is analytic in Ω is not a constant, then the modulus of the function does not attain its maximum value at an interior point of Ω. Assume that Ω equals the right half plane. W(s) S(s) is analytic in Ω and is obviously not a constant. Let s0 be an interior point of Ω. As |W(s) S(s)| does not attain its maximum value at an interior point of Ω, we have
||W(s) S(s)||∞ ) ||W(s)(1 - G(s) Q(s))||∞ g |W(s0) S(s0)| (B.6) To get a controller such that the closed-loop system has the desired properties, our method is first to drop the properness requirement, find a suitable parameter, for example, Qopt(s), which is stable but usually improper, and then obtain a suitable Q(s) by rolling Qopt(s) off at high frequency. The reason this works is that any improper function can be approximated over any desired frequency range by a proper function. Minimizing the left-hand side of the above inequality with respect to Q(s), the optimal controller is obtained as follows:
Qopt(s) )
W(s) - W(s0) S(s0) W(s) G(s)
K (τs - 1)(1 + θs)
The rational approximation adopted here is very important. With it, a PID controller can be derived. If G(s) has zeros in the right half plane, there is a positive lower bound for the maximum value of |W(s) S(s)|. To see this, let s0 be the zero of G(s) in the right half plane; then
1 s0
|W(s)(1 - G(s) Q(s))|∞ g 0
(B.11)
In this case, the maximum value of |W(s) S(s)| with respect to Q(s) can be identically reduced. Minimizing the left-hand side of the above inequality, the optimal Q(s) is
Qopt(s) )
(τs - 1)(1 + θs) K
(B.12)
Qopt(s) is improper. A filter with the pole-zero excess of 2 should be introduced to roll it off at high frequency. The lowest order
(B.13)
λ3 3λ2 + 3λ + τ τ2
λ is a user-specified positive constant and is defined as the performance degree. It follows that
(τs - 1)(1 + θs)(as + 1)
Q(s) ) Qopt(s) J(s) )
K(λs + 1)3
(B.14)
The corresponding controller of the unity feedback loop can be written as
C(s) )
1 (1 + θs)(as + 1) K λ3 s s+b τ
(
)
(B.15)
where
b)
λ3 3λ2 + τ τ2
Suppose that the PID controller is in the form of
(
C(s) ) KC 1 +
(B.10)
When G(s) has no zeros in the right half plane, there is no such constraint and, thus, we can let
as + 1 (λs + 1)3
where
(B.9)
It is evident that the maximum value of |W(s) S(s)| should be greater than the boundry:
|W(s)(1 - G(s) Q(s))|∞ g
J(s) )
(B.7)
(B.8)
1 |W(s0) S(s0)| ) |W(s0)| ) s0
filter that satisfies eqs B.4 and B.5 is in the form of
a)
It is known that PID controllers cannot be analytically designed if the time delay involved in the process is rigorously treated. Introducing a 0/1 order Pade approximation, the plant becomes
G(s) )
Figure B1. Quantitative overshoot for an unstable process.
)
1 1 + TDs TIs TFs + 1
(B.16)
Elementary computations give
KC )
a+θ aθ λ3 , TI ) a + θ, TD ) , TF ) Kb a+θ τb
(B.17)
Many papers have derived the stability condition as the ratio of the time delay to the time constant being less than 1 for PID controllers (see, for example, ref 29¢). An important advantage of this method is that it can be used for the control of openloop unstable processes with the ratio of the time delay to the time constant being larger than 1. Of course for stabilization, there is an upper bound for the ratio, which will be the content of another paper. The quantitative overshoot is shown in Figure B1.
Ind. Eng. Chem. Res., Vol. 45, No. 4, 2006 1419
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ReceiVed for reView July 6, 2005 ReVised manuscript receiVed October 25, 2005 Accepted November 2, 2005 IE0507981
