Ind. Eng. Chem. Res. 1999, 38, 4309-4316
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A Time Delay Compensation Strategy for Uncertain Single-Input Single-Output Nonlinear Processes Qiuping Hu and Gade Pandu Rangaiah* Department of Chemical and Environmental Engineering, The National University of Singapore, Singapore 119260, Singapore
The dynamic behavior of many processes is characterized by time delays due to transportation lags and measurement delays, which put severe limitations on the performance of control systems. In this paper, a new time delay compensation strategy for single-input single-output nonlinear processes subject to modeling uncertainty is proposed. At first, controller design based on input-output feedback linearization for a class of nonlinear systems with an input time delay is presented. Then, the input-output linearization controller is used in an internal model control (IMC) structure with time delay compensation. Finally, IMC with feedback compensation is proposed to reduce the effect of modeling errors on the performance and robustness. An adjustable parameter in the feedback compensation can be tuned to satisfy a particular specification. The effectiveness of the proposed method is illustrated via simulation on the temperature and concentration control of a nonisothermal chemical reactor with an input delay. 1. Introduction
proposed control strategy is analyzed and implemented on a chemical reactor to demonstrate its effectiveness.
Time delays existent in many processes degrade control performance and can sometimes cause instability of the control system. Although a number of time delay compensation techniques have been proposed to improve the control performance of linear processes with time delays, only a few time delay compensation strategies have been developed for nonlinear systems. The Smith predictor1 was investigated for nonlinear systems with input time delays by Kravaris and Wright2 and Huang and Wang.3 It was also analyzed from an internal model control (IMC) perspective.4 Though the Smith predictor can help to overcome the time delay problem, it requires an accurate model of the process being controlled. Henson and Seborg5 developed a time delay compensation strategy for nonlinear processes using predicted state variables at the future time. However, measured state variables are needed for these predictions. Variable structure control was employed to stabilize uncertain systems with state delay.6,7 Velasco et al.8 studied the approximate disturbance decoupling problem of nonlinear processes with an input time delay. Wu and Chou9 presented a simple design method for output tracking control of nonlinear processes with an input time delay. All of the above methods assume that state variables are available or can be accurately estimated. However, state variables are often not measured and estimation of state variables in nonlinear systems is an open problem that has not yet been solved. This paper proposes a new time delay compensation strategy for uncertain nonlinear processes. Because an IMC control structure is used in this strategy, neither measured state variables nor a separate state variable estimation is needed, which makes the strategy more practical. The * To whom correspondence should be addressed. Phone: (65) 8742187. Fax: (65) 7791936. E-mail:
[email protected]. The manuscript was prepared when G.P.R. was at Curtin University of Technology, Western Australia, as Visiting Professor.
2. Input-Output Linearization In this work, a single-input single-output (SISO) openloop stable nonlinear system with input time delay P of the form
x˘ (t) ) f(x) + g(x) u(t-θ)
(1)
y(t) ) h(x) is considered. In the above equation, x ∈ Rn is the state variable vector, u ∈ R is the manipulated variable, y is a scalar controlled output, and f, g, and h are assumed to be smooth. If the relative degree of eq 1 is welldefined, the system can be transformed into normal form via a diffeomorphism [ξT,ηT]T ) φ(x) defined as
h(x), 1 e k e γ ξk ) φk(x) ) Lk-1 f
(2)
and ηk ) φγ+k(x), 1 e k e n - γ where Lgφγ+k(x) ) 0. The normal form can be written as
ξ˙ i ) ξi+1, i ) 1, 2, ..., γ - 1
(3)
ξ˙ γ ) b(ξ,η) + a(ξ,η) u(t-θ)
(4)
η˘ ) q(ξ,η), ξ ∈ Rγ, η ∈ Rn-γ
(5)
y ) ξ1
(6)
h(x)|x)φ-1(ξ,η) a(ξ,η) ) LgLγ-1 f
(7)
b(ξ,η) ) Lγf h(x)|x)φ-1(ξ,η)
(8)
where
qk(ξ,η) ) Lfφγ+k(x)|x)φ-1(ξ,η), k ) 1, 2, ..., n - γ (9) The input-output linearizing control law is
10.1021/ie990274z CCC: $18.00 © 1999 American Chemical Society Published on Web 10/08/1999
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u(t-θ) )
v(t) - b(ξ,η) a(ξ,η)
(10)
where all the quantities in the right-hand side are at time t. The above equation can be rewritten as
u(t) )
v(t+θ) - b[ξ(t+θ),η(t+θ)] a[ξ(t+θ),η(t+θ)]
(11)
where v(t+θ) is the transformed input to the controller given by
v(t+θ) ) -R1ξ1(t+θ) - R2ξ2(t+θ) ... Rγξγ(t+θ) + R1r(t) (12) where r is the reference input and Ri, i ) 1, ..., γ, are the tuning parameters. 3. Nonlinear Internal Model Control The control law in eq 11 cannot be implemented directly because a perfect model is rarely available in practice. Hence, it was implemented in a nonlinear IMC structure.10-12 Assume the model available for designing the nonlinear controller is
x˜ (t) ) ˜f(x˜ ) + g˜ (x˜ ) u(t-θ)
(13)
y˜ (t) ) h ˜ (x˜ ) where x˜ ∈ Rn is the state variable vector, u ∈ R is the manipulated variable, and y˜ is the model output. To ensure that the model inverse is a well-defined dynamic system, we assume that the model has stable inverse and the vector fields ˜f and g˜ and scalar function h ˜ have continuous derivatives of all order. This implies that all derivatives of ˜f and g˜ and h ˜ are bounded. Following the transformation of eqs 3-6, eq 13 can be changed to
Figure 1. Nonlinear internal model control with time delay compensation.
v˜ (t+θ) )-R1ξ˜ 1(t+θ) - R2ξ˜ 2(t+θ) ... Rγξ˜ γ(t+θ) + R1rj(t) (20) where rj(t) ) r(t) - y(t) + y˜ (t) and Ri, i ) 1, ..., γ, are the tuning parameters which are chosen such that sγ + Rγsγ-1 + ... + R2s + R1 (where s is the Laplace transform parameter) is a Hurwitz polynomial. Equations 14, 15, and 17 can now be written as
where
[
0 0 Ac ) . 0 -R1
ξ˜˙ ) Acξ˜ + Bcrj
(21)
y˜ ) Ccξ˜
(22)
1 0 . 0 -R2
0 1 . 0 -R3
... ... ... ... ...
]
0 0 . ∈ Rγ×γ (23) 1 -Rγ
Bc ) [0 ... 0 R1]T ∈ Rγ×1
(24)
Cc ) [1 0 ... 0] ∈ R1×γ
(25)
ξ˜˙ i ) ξ˜ i+1, i ) 1, 2, ..., γ - 1
(14)
Stability of the nonlinear IMC can be shown by considering the quadratic function, Vm ) 1/2ξ˜ Tξ˜ . The time derivative of this function along the trajectory of eq 21 is V˙ m ) ξ˜ TAcξ˜ + rjξ˜ TBc. Using the definition of norms,
ξ˜˙ γ ) b˜ (ξ˜ ,η˜ ) + a˜ (ξ˜ ,η˜ ) u(t-θ)
(15)
V˙ m e λmax(Ac)|ξ˜ |2 + |rjBc||ξ˜ |
η˜˙ ) q˜ (ξ˜ ,η˜ ), ξ˜ ∈ Rγ, η˜ ∈ Rn-γ
(16)
y˜ ) ξ˜ 1
(17)
where ξ˜ , η˜ , a˜ , b˜ , and q˜ are defined similarly as ξ, η, a, b, and q. The prediction model Mp is constructed by removing the time delay in eq 13:
x˘ *(t) ) ˜f[x*(t)] + g˜ [x*(t)] u(t)
(18)
y*(t) ) h ˜ [x*(t)] By comparison of eqs 13 and 18, it follows that x*(t) ) x˜ (t+θ) if the predictor is initialized as x*(0) ) x˜ (θ). Note that Mp is an open-loop predictor and therefore is restricted to stable processes. In this work, a modified IMC with time delay compensation shown in Figure 1 is studied. The current input u(t) is computed as
u(t) )
v˜ (t+θ) - b˜ [ξ˜ (t+θ),η˜ (t+θ)] a˜ [ξ˜ (t+θ),η˜ (t+θ)]
The future input v˜ (t+θ) is calculated as
(19)
|rjBc|2 3 2 e λmax(Ac)|ξ˜ | + 4 -λmax(Ac)
(26)
For V˙ m e 0, we obtain
|rj| e -
x3 λ (A )|ξ˜ | 2R1 max c
(27)
because λmax(Ac) < 0 and |Bc| ) R1. Without loss of generality, assume r ) 0, then rj ) -y(t) + y˜ (t), and
|e1| e -
x3 λ (A )|ξ˜ | 2R1 max c
(28)
where e1 ) y(t) - y˜ (t). Therefore, if eq 28 is satisfied (i.e., modeling error is absent or small), the nonlinear IMC will be stable. However, if the modeling error is large, the system could become unstable. Hence, reducing the modeling error is beneficial for the stability of the IMC system. In the absence of plant/model mismatch, it can be shown that the overall transfer function of the IMC
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[β1, ..., βγ]T ∈Rγ are selected such that sγ + βγsγ-1 + ... + β2s + β1 is a Hurwitz polynomial. Substituting uc and um into eqs 4 and 15 rspectively, we obtain
[
e˘ ) Ae + ∆A
A) 0 0 . 0 Figure 2. Nonlinear internal model control with feedback compensation and time delay compensation.
control system is2 -θs
R1e y(s) ) γ r(s) s + Rγsγ-1 + ... + R1
(29)
If Ri, i ) 1, ..., γ, are appropriately selected, eq 29 can be written as
y(s) 1 e-θs ) r(s) (s + 1)γ
]
∆A ) [0 ... 0 b(ξ,η) - b˜ (ξ˜ ,η˜ ) + [a(ξ,η) - a˜ (ξ˜ ,η˜ )]um(t-θ)]T ∈ Rγ (35) Theorem. The tracking error between the process and the model outputs, e1, satisfies
Bd
(36)
λmax(A)
(31)
(32)
where e ) ξ - ξ˜ ∈ Rγ and the tuning parameters β )
(37)
|∆A| e Bd
Proof. Consider the Lyapunov equation V(e) ) 1/2eTe. Differentiating V(e) along the trajectory of the system (eq 33) gives
V˙ e λmax(A)|e|2 + Bd|e| e λmax(A) V -
Bd2 2λmax(A)
(38)
which implies that e will converge to the ball
{
B(e) ) e: |e| e -
Bd λmax(A)
}
(39)
Since |e1| e |e|, eq 36 follows from eq 39. The bound Bd exists if the original IMC is stable. If required, simulation of the IMC system can be used to find Bd,9 which is independent of β. Intuitively, to reduce the effects of the uncertainties on the control system, β can be selected such that -λmax(A) is large. In the limiting case, -λmax(A) can be made arbitrarily very large such that perfect tracking can be achieved.
lim
-λmax(A)f∞
This is the same as eq 19 except that v˜ is replaced by v˜ c given by
v˜ c(t+θ) ) v˜ (t+θ) - βTe(t+θ)
0 0 . 1
where Bd is the bound on modeling error
As shown by eq 30, the IMC strategy produces desirable setpoint tracking behavior only if the model is perfect. However, in the presence of modeling errors, the performance of the IMC control system will generally degrade, and the control system could sometimes become unstable. In this section a nonlinear internal model control with feedback compensation (IMC-FC) shown in Figure 2 is proposed and analyzed such that a good control performance is maintained and the effects of uncertainties in the model are attenuated. As noted in section 3, Mp acts as an open-loop observer. The use of an open-loop observer to reconstruct unmeasured variables has the disadvantage that the estimation error cannot be specified. With the proposed structure of IMCFC, the estimation error can be analyzed and a bound on it can be obtained. Further, the additional feedback in the IMC-FC can be regarded as a corrective action for the error in Mp, leading to a better performance. In the IMF-FC, control action to the model (um in Figure 2) is still eq 19, while control action to the process uc is modified to
a˜ [ξ˜ (t+θ),η˜ (t+θ)]
... ... ... ...
a(ξ,η) a(ξ,η) a(ξ,η) a(ξ,η) β1 β2 β3 ... β a˜ (ξ˜ ,η˜ ) a˜ (ξ˜ ,η˜ ) a˜ (ξ˜ ,η˜ ) a˜ (ξ˜ ,η˜ ) γ ∈ Rγ×γ (34)
(30)
4. Nonlinear Internal Model Control with Feedback Compensation
uc(t) )
0 1 . 0
|e1| e -
where is the closed-loop time constant. The effect of on performance is clear from eq 30, and its effect on robustness has been shown by Kravaris and Wright.2
v˜ c(t+θ) - b˜ [ξ˜ (t+θ),η˜ (t+θ)]
-
1 0 . 0
(33)
|e1| ) 0
(40)
Generally, very large -λmax(A) may lead to system poles to be far placed in the left half of the complex plane which can cause a fast transient response. An impulsivelike behavior, known as the peaking phenomenon, may also happen and destroy the closed-loop stability.13 Because a robust IMC-FC with good response is desired, it is possible to choose a reasonable β to achieve both robust stability and performance.
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Table 1. Nominal Operating Conditions of the CSTR variable q CAf Tf V UA k0 E/R
nominal value
variable
nominal value
-∆H F Cp θ Tc T CA
5 × 104 J mol-1 1000 g L-1 0.239 J g-1 K-1 0.2 min 309.9 K 383.7 K 0.1 mol L-1
min-1
100 L 1 mol L-1 350 K 100 L 5 × 104 J min-1 K-1 7.2 × 1010 min-1 8750 K
For IMC-FC control (eq 32), e(t+θ) is needed. In this study, e(t+θ) is approximated by first-order Euler formula
e(t+θ) = e(t) + θ
de(t) dt
(41)
Figure 3. Open-loop responses to step changes in the coolant temperature.
Hence, the derivative of the output is required for implementing IMC-FC. Although eq 41 is valid only if θ is small because the higher order derivative terms are omitted, it can still be used because these derivatives are bounded and can be incorporated into modeling error (eq 35). To avoid aggressive control action, a rate limiter which limits the rate of change of the compensation signal can be put into the feedback loop for the IMCFC (Figure 2). 5. Application In this section, the time delay compensation strategy proposed above is applied to a continuous stirred tank reactor (CSTR) where a first-order reaction, A f B, is occurring. There is a time delay (θ) in the manipulated coolant temperature (Tc).5 The dynamic model of the CSTR is
q E C C˙ A ) (CAf - CA) - k0 exp V RT A
(
)
(42)
(-∆H) E q k exp C + T˙ ) (Tf - T) + V FCp 0 RT A UA [T (t-θ) - T] (43) VFCp c
(
)
where CA is the effluent concentration, T is the reactor temperature, q is the feed flow rate, and CAf is the feed concentration. The remaining variables are defined in the Nomenclature section. Table 1 contains the nominal operating conditions for the reactor. Open-loop temperature responses to (3 K step changes in Tc are shown in Figure 3. Even for this small input change, the reactor exhibits severe nonlinear dynamic behavior. Similar responses (not shown here) were obtained for CA. Two cases were considered for CSTR control: in the first case T is controlled, while in the second case CA is the output variable. Performance results for CSTR control by IMC and IMC-FC in the presence of unmeasured disturbances and modeling errors in the time delay are presented and discussed below. 5.1. Temperature Control. In this case, the state variable x is [CA T]T and the controlled output y is T. It can be seen that the CSTR model (eqs 42 and 43) can be represented as in eq 1 and the relative degree is 1. The nonlinear controller C in eq 19 is
u(t) )
R1(r - y) - f2[x(t+θ)] g2
(44)
Figure 4. Temperature control using IMC without time delay compensation for a -10% unmeasured disturbance in the feed flow rate.
where
(
)
(-∆H) q E UA f2(x) ) (Tf - x2) + k exp x x V FCp 0 Rx2 1 VFCp 2 (45) and
g2 )
UA VFCp
(46)
The controller was tuned with R1 ) 10 (i.e., ) 0.1 min), which is approximately half of the open-loop time constant for the +3 K change in Figure 3. For the IMCFC, β ) 1.5 was selected. Figure 4 shows the regulatory performance of IMC without time delay compensation for a -10% unmeasured disturbance in the feed flow rate (q). As expected, a good performance is obtained when θ ) 0. However, the system becomes unstable when θ ) 0.1 min. Stable but oscillatory and slow responses can be obtained by detuning . Thus, the performance of IMC without time delay compensation can be significantly deteriorated by time delay, and time delay compensation is necessary to obtain satisfactory performance. Hence, subsequent simulations were done for IMC and IMC-FC with time delay compensation only. The abbreviations “IMC” and “IMC-FC” hereafter mean IMC and IMC-FC with time delay compensation. The tracking performance of the IMC is shown in Figure 5 for (10 K step changes in the setpoint. Because there are no disturbances and the model is assumed to be perfect, both IMC and IMC-FC are equivalent in this case and yield identical responses. Further, the response (Figure 5) corresponds to the closed-loop transfer function in eq 30. Note that the
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Figure 5. Setpoint tracking by IMC for a +10 K change (continuous line) and a -10 K change (dashed line).
Figure 7. Temperature control using IMC and IMC-FC with time delay compensation for a +10% unmeasured disturbance in the feed flow rate in the presence of measurement noise.
Figure 6. Temperature control using IMC and IMC-FC with time delay compensation for a +10% unmeasured disturbance in the feed flow rate.
responses for (10 K changes are symmetrical, indicating that the input-output behavior is linear. Except for the delayed response, the time delay does not affect the closed-loop performance. IMC and IMC-FC are compared in Figure 6 for a +10% unmeasured disturbance in the feed flow rate. The IMC-FC provides excellent control compared to IMC. Control moves for the two controllers are satisfactory. In practice, the derivative action in eq 41 may result in difficulties if there is high-frequency measurement noise. To avoid these, the high-frequency gain of the derivative term should be limited, which can be achieved by implementing the derivative term along with a first-order filter. To evaluate the performance of IMC and IMF-FC in the presence of measurement noise, the disturbance in Figure 6 was considered along with measurement noise (a white noise with a correlation time of 0.006 min). The derivative action in eq 41 was filtered by a first-order system with a time constant of 0.04 min. The responses shown in Figure 7 demonstrate that IMC-FC gives a better performance than IMC even
Figure 8. Temperature control using IMC and IMC-FC with time delay compensation for a -10% unmeasured disturbance in the feed flow rate.
in the presence of measurement noise. The performance of the IMC and IMC-FC is compared in Figure 8 for a -10% unmeasured disturbance in the feed flow rate. As expected, IMC-FC is superior to IMC. The performance of the two controllers for modeling error ((20%) in θ is shown in Figure 9. In these tests, the actual time delay θ is 0.2 min and the process is subjected to a -10% unmeasured disturbance in the feed flow rate. In Figure
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Figure 9. Temperature control using IMC and IMC-FC with time delay compensation for a -10% unmeasured disturbance in the feed flow rate in the presence of (a) +20% and (b) -20% modeling error in the time delay.
Figure 11. Concentration control using IMC and IMC-FC with time delay compensation for a +10% unmeasured disturbance in the feed flow rate.
tively, and
(
)
q E f1(x) ) (CAf - x1) - k0 exp x V Rx2 1 a(x) )
(
b(x) ) -k0x1 exp -
Figure 10. Composition control using IMC without time delay compensation for a -10% unmeasured disturbance in the feed flow rate.
9a, the time delay in the model is overestimated as θ˜ ) 0.24 min while it is underestimated as θ˜ ) 0.16 min in Figure 9b. In both cases, the performance of IMC-FC is better than that of IMC. 5.2. Concentration Control. To illustrate the effectiveness of the proposed approach for higher relative order nonlinear processes, concentration control of the CSTR (eqs 42 and 43) is considered. As before, the process time delay (θ) is 0.2 min. The output variable y is now CA, the relative degree is 2, and the nonlinear controller C is
u(t) )
R1(rCA - y) - {R2 - a[x(t+θ)]}f1[x(t+θ)] b[x(t+θ)]
-
f2[x(t+θ)] (47) g2 where f2 and g2 are defined in eqs 45 and 46, respec-
(
E q + k0 exp V Rx2
)
)
E E g Rx2 Rx 2 2 2
(48) (49) (50)
The nonlinear IMC was designed with R ) [100 20]T, i.e., ) 0.1 min, and β ) [10 2]T was used in the IMCFC. To avoid aggressive control action, a rate limiter with rising slew rate and falling slew rate between -2 and +2 is inserted into the feedback loop (Figure 2). In Figure 10, the regulatory performance of the IMC controller without time delay compensation is shown for a -10% unmeasured disturbance in the feed flow rate. The response is stable and acceptable for θ ) 0 and unstable for θ ) 0.1 min. The performance of IMC and IMC-FC (with time delay compensation) is shown in Figure 11 for a +10% disturbance in the feed flow rate. Both IMC and IMC-FC give a satisfactory performance, and IMC-FC is better than IMC. For a -10% disturbance in the feed flow rate (Figure 12), a very oscillatory and sluggish response is obtained with IMC, while IMCFC provides a greatly improved performance. The closed-loop performance of the two controllers for a -10% unmeasured disturbance in the feed flow rate in the presence of (20% modeling error in the time delay is shown in Figure 13 which indicates that the performance of IMC is extremely oscillatory or even unstable, and the IMC-FC provides significantly better control.
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back compensation (FC) based on differences between process and model outputs. It employs state variables from the model block in the IMC structure, and hence measured state variables are not required. The beneficial effect of IMC-FC with time delay compensation in reducing the effect of modeling uncertainties is shown theoretically. Results from temperature and concentration control of a CSTR via simulation demonstrate that IMC-FC with time delay compensation yields good regulatory performance even if only the process output is available. Nomenclature
Figure 12. Concentration control using IMC and IMC-FC with time delay compensation for a -10% unmeasured disturbance in the feed flow rate.
A ) heat-transfer area CA ) concentration of component A CAf ) feed concentration of component A Cp ) heat capacity e ) error signal E ) activation energy f, g ) vector fields h ) output function k0 ) preexponential factor Lifh ) ith-order Lie derivative of h with respect to f q ) feed flow rate r ) reference input T ) reactor temperature Tc ) coolant temperature u ) manipulated variable U ) heat-transfer coefficient v ) transformed input V ) volume of tank x ) state variables y ) controlled output Greek Letters Ri ) controller tuning parameter βi ) tuning parameter of IMC-FC γ ) relative degree ∆H ) heat of reaction ξ ) transformed state variables
Literature Cited
Figure 13. Concentration control using IMC and IMC-FC with time delay compensation for a -10% unmeasured disturbance in the feed flow rate in the presence of (a) +20% and (b) -20% modeling error in the time delay.
6. Conclusions In this paper, a new time delay compensation strategy for uncertain, single-input single-output nonlinear processes with input time delays is proposed and studied. The strategy is developed using input-output linearization, internal model control (IMC) structure, and feed-
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(12) Hu, Q.; Rangaiah, G. P. Strategies for Enhancing Nonlinear Internal Model Control of pH Processes. J. Chem. Eng. Jpn. 1999, 32, 59. (13) Sussmann, H. J.; Kokotovic, P. V. The Peaking Phenomenon and Global Stabilizing of Nonlinear Systems. IEEE Trans. Autom. Control 1991, 36, 424.
Received for review April 14, 1999 Revised manuscript received August 12, 1999 Accepted August 16, 1999 IE990274Z
