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Ind. Eng. Chem. Res. 2007, 46, 540-545
Novel Disturbance Controller Design for a Two-Degrees-of-Freedom Smith Scheme Qing-Guo Wang,*,† Xiang Lu,† Han-Qin Zhou,‡ and Tong-Heng Lee† Department of Electrical and Computer Engineering, National UniVersity of Singapore, Singapore, and Department of Electrical and Computer Engineering, UniVersity of Houston, Houston, Texas 77204-4005
A two-degrees-of-freedom Smith control scheme is investigated for improved disturbance rejection. The resulting set-point and disturbance responses can be tuned by two controllers separately. A novel disturbance controller design is presented with easy tuning and greatly improved performance. The internal and robust stability issues are discussed. Examples are provided for illustration. 1. Introduction
2. The Proposed Method
In process control, the Smith predictor1 is a well-known and very effective dead-time compensator. One major concern with the normal Smith control is that its disturbance rejection performance is usually limited, because of its one-degree-offreedom nature. To cater to disturbance rejection and robustness as well, a double-controller scheme is presented in ref 2 for stable first-order processes with dominant delay. However, the improvement of disturbance rejection is not significant, and its performance deteriorates when the process time delay is relatively small. Recently, several “modified Smith predictor” control schemes have been proposed3-5 to extend applicability of the Smith predictor to unstable processes. They handle integral or first-order unstable plants through the use of more controllers, and they can be applied to stable processes as well, through scheme simplification. However, it is noted that their characteristic equations are all delay-dependent, which is in contrast to the delay-free relationship enjoyed by the normal Smith control; this keeps the stabilization problem a complicated task. Also, they do not pay enough attention to disturbance rejection. Undoubtedly, disturbance rejection is most important in process control and good solutions have been sought for a long time.
In this paper, our goal is to seek a new control design that can keep the nominal delay-free stabilization of the delay system, such as that in the normal Smith control, yet provide some additional means to improve disturbance rejection and, hopefully, tune the set-point and disturbance responses separately and easily. After many trials, we decided to use the twodegrees-of-freedom Smith control scheme, as depicted in Figure 1. In this figure, G(s) and G ˆ (s) represent a stable and minimal phase process and its model, respectively (G(s) ) G0(s)e-Ls and G ˆ (s) ) G ˆ 0(s)e-Lˆ s. Suppose that the model matches the plant dynamics perfectly, i.e., G ˆ 0 ) G0 and Lˆ ) L. It follows that the closed-loop transfer function from the set point to the output is given by
In this paper, the two-degrees-of-freedom Smith predictor control scheme6-8 is investigated for improved disturbance rejection. This scheme is featured by delay-free nominal stabilization. The resulting set-point response remains the same as that in the normal Smith scheme. However, the disturbance response can be tuned by one additional controller separately, with no effects on the set-point response. Furthermore, a novel method is presented to design this disturbance controller easily and yield substantial control performance improvement. The remainder of the paper is organized as follows. In section 2, the proposed disturbance controller design is presented. Stability analysis is given in section 3. Typical designs are detailed for first-order plus dead time (FOPDT) and secondorder plus dead time (SOPDT) processes in section 4. In section 5, two examples are provided to demonstrate our methods. Finally, the paper is concluded in section 6. * To whom all correspondence should be addressed. Tel: (+65) 6516 2282, Fax: (+65) 6779 1103. E-mail address:
[email protected]. † ECE Department, National University of Singapore. ‡ ECE Department, University of Houston.
Hr )
G0C1 -Ls e ( Hr0e-Ls 1 + G0C1
(1)
where Hr0 denotes the delay-free part of Hr. For the disturbance path from D(s) to Y(s), it can be shown that the transfer function is
1 + G0C1 - G0C1C2e-Ls Hd ) 1 + G0C1
(2)
which shares the same delay-free denominator as that in Hr. To compare this scheme with the Smith system, letting C2 ) 1 reduces the scheme to the normal Smith system, which has the same set-point transfer function as in eq 1 but a different disturbance transfer function:
Hd1 )
1 + G0C1 - G0C1e-Ls 1 + G0C1
Obviously, with C1 designed for closed-loop stability and the set-point response, the normal Smith scheme simply does not have any freedom to manipulate the disturbance response. Because of the great importance of disturbance rejection in the process control industry, it is definitely desirable to have a means to improve it. In the scheme of Figure 1, C2 appears in the numerator of Hd and, thus, can be utilized to reduce or minimize Hd. It is also noted that C2 is not in the set-point transfer function (eq 1). Hence, C1 and C2 can be tuned separately as follows. C1 is designed to have the desired stability and set-point response. This is a standard task and there are many solutions
10.1021/ie0605130 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/16/2006
Ind. Eng. Chem. Res., Vol. 46, No. 2, 2007 541
which requires the compensating response ydb(t) to be
ydb(t) )
{
}
0 (for 0 < t < L) ) yda(t)1(t - L) (for t g L) yda(t)
(6)
as displayed in Figure 2. We now derive an analytical solution for C2(s) to meet eq 6. In view of eq 4, Yda can be expressed using the partial fraction expansion as, for example,
already. Our focus here is on C2; that is, design it to achieve best disturbance rejection. In ref 7, C2 is composed of a firstorder lag and a delay term to approximate the inverse of time delay in low-frequency range; however, its disturbance performance improvement is not significant, and a novel design for C2 is proposed in this paper. Given eq 2, intuitively, one might attempt to determine C2 via frequency response fitting, i.e., by minimizing
R0
Yda(s) )
Figure 1. Schematic illustration of two-degrees-of-freedom Smith control.
Ri
∑i s + β
+
s
i
and its time domain form is yda(t) ) R0 + ∑iRie-βit. It follows that
∑i Rie-β t]1(t - L)
yda(t)1(t - L) ) [R0 +
i
∑i Rie-β Le-β (t-L)]1(t - L)
) [R0 +
i
i
( yˆ da(t - L)1(t - L)
(7)
-jωL
|Hd| ) |1 -
G0C1e C | ) |1 - HrC2| 1 + G0C1 2
where yˆ da(t) ) R0 + ∑iRie-βiLe-βit, with
for some working frequency range ω ∈ [ω, ω j ], so that the disturbance response is attenuated. Such optimization falls into the model matching category and sounds reasonable. However, it is actually difficult to produce an expected performance. This is because the optimization has a tendency to get C2 as C2 ) 1/Hr over [ω, ω j ]. The resulting C2 would mimic the behavior of 1/Hr, which contains a pure time leading ejωL term with a counter-clockwise Nyquist curve, and would exhibit large magnitude for ω > ω j . This increases the corresponding |Hd| and may even make the scheme more susceptible to unmodeled high-frequency dynamics or uncertainties. To attain better disturbance rejection in face of the delay term in the numerator of Hd, our novel method proceeds as follows. For a given type of disturbance (for example, D(s)), it follows from eq 2 that the disturbance response is
Yˆ da(s) )
Laplace transform of eq 6, with the help of eqs 5 and 7, gives Hr0C2De-Ls ) Yˆ da(s)e-Ls, and its solution is
C*2 )
Yˆ da(s) -1 H D(s) r0
(4)
(5)
Suppose that the disturbance occurs at t ) 0. The nonzero responses in yda(t) and ydb(t) then enter at t ) 0 and t ) L, respectively. Obviously, the disturbance response during t ) 0 to t ) L is solely from yda(t) and is fixed. Any effort to change it during this time period is useless but causes problems in regard to controller design. The best achievable disturbance rejection is to zero the disturbance response from t ) L onward:
{
yda(t) (for 0 < t < L) 0 (for t g L)
(9)
Because C*2 generally is improper, a low-pass filter should be added for practical implementation, so that the actual C2 is given by
C2 )
(3)
and Ydb can be manipulated using C2:
yd(t) ) yda - ydb )
(8)
i
Yˆ da(s)Hr0-1
(10)
(τs + 1)nD(s)
[
Yˆ da(s)
]
(τs + 1)n
and
Ydb ) Hr0C2De-Ls
∑i s + β
+
s
Ydb(s) )
where Yda is fixed,
Yda ) D
Rie-βiL
The actual Ydb(s) and Yd(s) then are given as
1 + G0C1 - G0C1C2e-Ls D Yd ) 1 + G0C1 ) Yda - Ydb
R0
Yd(s) ) Yda(s) -
[
e-Ls
(11)
]
Yˆ da(s)
(τs + 1)n
e-Ls
(12)
respectively. Detailed controller design will be provided for several typical industrial processes in section 4. Before concluding this section, we would highlight the advantage of our design over the standard two-degrees-offreedom control scheme (either single-loop-based or Smithpredictor-based), where a pre-filter is added between the reference input and the negative feedback. In the standard twodegrees-of-freedom control scheme, obviously, the pre-filter does not affect the disturbance response and could only be utilized to tune the set-point response. Then, this leaves its primary controller responsible for both closed-loop stabilization and disturbance response, and, thus, it limits disturbance rejection performance. On the other hand, in our design, it is easier to design and achieve superior disturbance rejection
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Ind. Eng. Chem. Res., Vol. 46, No. 2, 2007
according to the small gain theorem.10 By invoking eqs 1 and 10, eq 14 reduces to
|
|
D(jω) 2 2 (τ ω + 1)n/2 > |∆G| ∀ω > 0 Yˆ da(jω)
(15)
To be specific, when the disturbance is of step type, Yˆ da(s) ) D(s) and, thus, the robust stability requirement turns out to be
(τ2ω2 + 1)n/2 > |∆G| ∀ω > 0 or Figure 2. Illustration of the desired disturbance rejection.
|∆G| (τ ω2 + 1)n/2 2
performance by tuning C2. In the extreme case, where the process is bi-proper, C2 may eliminate the disturbance response completely from t ) L, which is impossible for the standard two-degrees-of-freedom control scheme and any other scheme where the controller that addresses disturbance rejection also must cope with closed-loop stability and/or pole placement.
< 1 ∀ω > 0
(16)
It can been seen from eqs 10 and 15 that a tradeoff is to be made by C2, or tuning of the parameter τ must occur: a decrease in τ will improve the disturbance rejection performance but reduce the robust stability, and vice versa. 4. Typical Design Cases
3. Stability Analysis Stability is a prerequisite for any control systems. In this section, both the internal and robust stability of the two-degreesof-freedom scheme are investigated. The two-degrees-of-freedom structure in Figure 1 is an interconnected system that consists of five subsystems, and each of them is of the single input and single output (SISO) type. Such a system is internally stable9 if and only if the relation pc(s) ( ∆∏ipi(s) has all its roots in the open left half of the complex plane, where pi(s) are the denominators of the respective subsystem transfer functions and the symbol “∆” represents the system determinant as defined in the Mason’s ˆ 0, formula. The five subsystems in Figure 1 are C1(s), C2(s), G G0e-Ls, and e-Lˆ s. Let C1(s) ) f1(s)/g1(s), C2(s) ) f2(s)/g2(s), and G ˆ 0(s) ) G0(s) ) f(s)/g(s). The respective pi values of the five subsystems are given as p1 ) g1(s), p2 ) g2(s), p3 ) p4 ) g(s), and p5 ) 1. The system determinant ∆ is given by the relation ∆)1+G ˆ 0C1 + (G0e-Ls - G ˆ 0e-Lˆ s)C1C2. It follows that
pc(s) ) g2(s)g(s)[g1(s)g(s) + f1(s)f(s)] The polynomial g1(s)g(s) + f1(s)f(s) reflects stabilization of the delay-free G0 term by controller C1, which is always possible, for example, via pole placement. Controller C2 must be stable for the stability of g2(s), and it is used to achieve the best disturbance response. With the aforementioned two constraints, the overall system is internally stable. For robust stability analysis, let the total uncertainty be given in the form of a multiplicative one, as
G(s) - G ˆ (s) ∆G(s) ) G ˆ (s)
C1C2 G ˆ∆ 1 + C1G ˆ0 G
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