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Ind. Eng. Chem. Res. 2005, 44, 900-909
IMC-Based Control Strategy for Open-Loop Unstable Cascade Processes Tao Liu,* Weidong Zhang, and Danying Gu Department of Automation, Shanghai Jiaotong University, Shanghai 200030, P.R. China
In this paper, a new three degree-of-freedom control structure is proposed for open-loop unstable cascade processes. Enhanced regulatory capacities for both set point tracking and load disturbance rejection are obtained. By virtue of an open-loop control manner for the set point tracking, the nominal set point response is decoupled from the load disturbance response and thus can be quantitatively regulated by tuning the single adjustable parameter of the set point tracking controller. There are two closed loops for rejecting load disturbances, of which the inner loop is primarily responsible for the load disturbances that seep into the intermediate process and the outer loop for the load disturbances injected into the primary unstable process. Both of the closed-loop controllers are analytically derived by proposing the desired closed-loop complementary sensitivity functions, and meanwhile, the robust stability constraints for tuning the two closed loops are analyzed in the presence of the process multiplicative uncertainties. Simulation examples are included to demonstrate the superiority of the proposed method. 1. Introduction In many industrial and chemical processes concerned with temperature, flow, and pressure control issues, cascade control structures have been preferably adopted for improving load disturbance rejection performance on condition that the measurement for the intermediate process of such a cascade structure can be conveniently obtained in practice. Generally a cascade control structure is composed of two closed loops, i.e., a secondary inner loop embedded within a primary outer loop. The load disturbances that seep into the inner loop are supposed to be largely reduced before they extend to the primary outer loop. Therefore, it is crucial that the inner loop takes on a faster dynamic response in comparison with the outer loop for these load disturbances. Some bibliographies1-3 had provided fundamental tuning methods for conventional cascade control systems. Enhanced tuning methods of proportional-integralderivative (PID) controller within the framework of a conventional cascade control structure had been developed by refs 4-8. In contrast to the conventional cascade control structure, Lestage et al.9 and Semino and Brambilla,10 respectively, proposed two parallel cascade control structures for some chemical processes. Kaya11 suggested a cascade control scheme in combination with a Smith predictor for stable cascade processes with dominant time delay, which led to enhanced system performance compared with some existing PID controller methods, and recently was further improved by Liu et al.12 In addition, Hua and Jutan13 put forward an inferential cascade control method for a class of exothermic fixed-bed reactors. However, for open-loop unstable processes, the water bed effect between the set point tracking and load disturbance rejection using a closed-loop control structure becomes severe no matter how the primary controller is tuned.14 Hence, some two degree-of-freedom (DOF) control schemes15-17 had been proposed to solve the * To whom correspondence should be addressed. Tel./Fax: +86.21.62826946. E-mail:
[email protected].
problem, in each of which one controller was responsible for the set point tracking, and the other controller for the load disturbance rejection. As a result, significant improvement of the system performance had been achieved. In fact, open-loop unstable cascade processes can be regarded as a special class of unstable processes that are often constructed in practice to obtain advanced chemical products such as the chemical continuous stirred tank reactors (CSTRs)2,18,19 and the exothermic reactors,20 of which the intermediate stable processes can be measured and utilized to obtain better control performance. Lee et al.18 modified the conventional cascade control structure by adding two prefilters to the set points of the inner and outer loops, respectively, and tuned the closed-loop controllers in terms of the internal model control (IMC) theory,21 so that the load disturbance rejection performance was enhanced in comparison with some existing methods based on the conventional cascade control structure. Saraf et al.20 proposed a simultaneous relay autotuning approach for some chemical CSTRs so that a considerable drawback of the conventional cascade control structure, i.e., both the primary and secondary controllers had to be tuned in a sequential manner, was effectively overcome. However, each of the processes was deliberately identified in the form of an integrator plus time delay, and therefore, IMC-based proportional-integral (PI) controllers were suggested for obtaining improved system performance in comparison with the Tyreus and Luyben method22 developed earlier. Besides, Nagrath et al.23 proposed a model predictive control approach within the discrete time domain by using the state modeling equations for some chemical CSTRs. In this paper, a new cascade control scheme is proposed for open-loop unstable cascade processes based on the referential process transfer function model in the frequency domain, of which the control structure is shown in Figure 1. In Figure 1, P1m is the process transfer function model of the intermediate stable process P1, and P2m of the primary unstable process P2. Correspondingly P1mo and
10.1021/ie049203c CCC: $30.25 © 2005 American Chemical Society Published on Web 01/14/2005
Ind. Eng. Chem. Res., Vol. 44, No. 4, 2005 901
P2 the primary unstable process. Hence, the following analytical controller design procedure is developed in terms of the above nominal process models in the frequency domain. 2.1. Stabilizing Controller PC. From Figure 1 it can be determined that the set point response transfer function is in the form of
Hr ) Figure 1. Three degree-of-freedom cascade control structure.
P2mo are respectively the delay-free part of the process models P1m and P2m; i.e., P1m ) P1moe-θ1ms and P2m ) P2moe-θ2ms. Note that there are three main controllers in the proposed cascade control structure: C is the set point tracking controller, F1 is the inner loop controller used for rejecting the load disturbances injected into the intermediate process, and F2 is the outer-loop controller used for rejecting the load disturbances injected into the primary unstable process. Therefore, both F1 and F2 are called load disturbance estimator in this paper. Note that Pc is an auxiliary controller set for stabilizing the set point response. It is easily seen from Figure 1, that owing to the open-loop control manner for the set point tracking, the set point response is decoupled from the load disturbance response of either the inner or the outer closed loop and therefore can be independently regulated by means of the set point tracking controller C to meet the system operation requirement in practice. Moreover, when P1m is a perfect model of the intermediate process, the nominal load disturbance response of the primary outer loop is also decoupled from that of the inner loop, so that the nominal performance of the two closed loops can be respectively adjusted by F1 and F2. It should be noted that the stabilizing controller Pc, in fact, does not affect the control system performance and has no relationship with any one of the above three main controllers C, F1, and F2 for tuning the desired set point response and load disturbance response, which may be clearly identified in the controller design procedure provided in the following section. Therefore, the proposed cascade control structure is indeed of a 3-DOF control scheme. For clear interpretation of the proposed control scheme, this paper is organized as follows. Section 2 provides the analytical controller design procedure, and meanwhile, the achievable system performance is discussed. Section 3 provides some analysis on the robust stability constraints of the proposed control structure in the presence of the process multiplicative uncertainties. Accordingly, the on-line general rule for tuning the adjustable parameters of the two closed-loop controllers to hold the control system robust stability is provided. In section 4, illustrative simulation examples are included to show the superiority of the proposed method. Some conclusions are addressed in the last section 5. 2. Controller Design Procedure Usually an open-loop unstable cascade process in industrial and chemical practice is identified in transfer function form as
P1(s) ) k1e-θ1s/(τ1s + 1)
(1)
P2(s) ) k2e-θ2s/(τ2s - 1)
(2)
where P1 denotes the intermediate stable process and
CP1P2 (1 + P1moP2moF2e-(θ1m+θ2m)s)(1 + P1mF1) ‚ 1 + PcP1moP2mo 1 + P1F1 + P1P2F2 + P1P2P1mF1F2 (3) In the nominal case, i.e., P1m and P2m are respectively perfect models of the intermediate process and the primary unstable process, the set point response transfer function can be simplified as
Hr )
)
CP1P2 1 + PcP1moP2mo
k1k2Ce-(θ1+θ2)s τ1τ2s2 + (τ2 - τ1)s + k1k2Pc - 1
(4)
Obviously if the stabilizing controller Pc is not installed in the proposed cascade control structure, i.e., Pc ) 0, the nominal set point response transfer function shown in eq 4 cannot hold stable at all. Hence, by using the Routh-Hurwitz stability criterion for its characteristic equation, a practical form of Pc can be easily ascertained. There are two cases of the intermediate process P1 and the primary process P2 for choosing Pc. For one case, i.e., τ1 < τ2, take Pc ) kc, (kc > 1/k1k2) for simplicity. Correspondingly the characteristic equation of the set point transfer function is
τ1τ2s2 + (τ2 - τ1)s + k1k2kc - 1 ) 0 which is certainly stable in terms of the Routh-Hurwitz stability criterion. For the other case, i.e., τ1 g τ2, take Pc ) kc + kds, (kc > 1/k1k2 and kd > (τ1 - τ2)/k1k2) for simplicity. Correspondingly the characteristic equation of the set point transfer function becomes
τ1τ2s2 + (k1k2kd + τ2 - τ1)s + k1k2kc - 1 ) 0 which also holds stable in terms of the Routh-Hurwitz stability criterion. In fact, the above pure differential term can be physically implemented by cascading it with a low-pass filter in which the time constant can be chosen as (0.01-0.1)kd. It should be noted that Pc may be chosen as a conventional PID controller, which however tends to cause the controller tuning procedure to be much more complicated in order to stabilize the set point response and therefore is not recommended. 2.2. Set Point Tracking Controller C. From the nominal set point response transfer function shown in eq 4, it can be seen that there will definitely exist no dead-time element in its denominator if the set point tracking controller C is designed to be rational and stable, which will certainly contribute to obtaining a good set point response for the nominal system. Here the H2 optimal performance objectivesmin||e||22, i.e., the integral-square-error (ISE) specification, is utilized to design the optimal set point tracking controller C. That
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is, it should be implemented to satisfy the performance specification min||W(1 - Hr)||22, where W is the set point input weight function. In view of the step change of the set point input that occurs in industrial and chemical practice, W should be chosen as 1/s. Using the n/n order all-pass Pade´ approximation for the pure time delay term in eq 4 yields
where λc is an adjustable parameter. When λc is tuned to zero, C recovers the optimality. For the other case, i.e., τ1 g τ2, following a similar deriving procedure yields
C(s) )
τ1τ2s2 + (k1k2kd + τ2 - τ1)s + k1k2kc - 1 k1k2(λcs + 1)2
e-(θ1+θ2)s ) Qnn[-(θ1 + θ2)s]/Qnn[(θ1 + θ2)s]
Remark 1. Substituting eq 5 (or eq 6, if τ1 g τ2) into eq 4, the practical set point transfer function is obtained as
where n
Qnn[(θ1 + θ2)s] )
(2n - j)!n!
[(θ1 + θ2)s]j ∑ j)0(2n)!j!(n - j)!
Hr(s) )
and n is chosen to be an integer large enough so that the introduced approximation error can be neglected in comparison with the process model mismatch in practice. Hence, the set point tracking controller C can be derived in an analytical way. There are two cases of the intermediate process and the primary process for performing the controller design procedure. For one case, i.e., τ1 < τ2, there is Pc ) kc as given in the previous subsection. Hence, it follows from eq 4 that |W(1 - Hr)|22 )
|| (
)||
2
||
2
k1k2C(s)Qnn[-(θ1 + θ2)s] 1 12 s [τ1τ2s + (τ2 - τ1)s + k1k2kc - 1]Qnn[(θ1 + θ2)s] )
||
Qnn[(θ1 + θ2)s]
sQnn[-(θ1 + θ2)s]
2
k1k2C(s)
2
2
s[τ1τ2s + (τ2 - τ1)s + k1k2kc - 1]
Note that Qnn(0) ) 1 and all zeros of Qnn[-(θ1 + θ2)s] are located in the complex right half plane (RHP). Therefore, utilizing the orthogonality property of the H2 norm gives
|W(1 - Hr)|22 ) Qnn[(θ1 + θ2)s] - Qnn[-(θ1 + θ2)s]
|| ||
sQnn[-(θ1 + θ2)s]
||
2
2
+
||
τ1τ2s2 + (τ2 - τ1)s + k1k2kc - 1 - k1k2C(s) s[τ1τ2s2 + (τ2 - τ1)s + k1k2kc - 1]
2
2
Minimizing the right side, i.e., letting its second term be zero, yields the ideally optimal controller
Cim(s) )
τ1τ2s2 + (τ2 - τ1)s + k1k2kc - 1 k1 k2
However, Cim(s) is obviously not proper and cannot be physically realized. A low-pass filter
Jc(s) ) 1/(λcs + 1)2 is introduced to be in series with it, so the practically optimal controller is obtained as
C(s) )
τ1τ2s2 + (τ2 - τ1)s + k1k2kc - 1 k1k2(λcs + 1)2
(6)
(5)
1 e-(θ1+θ2)s (λcs + 1)2
(7)
By using the inverse Laplace transform, we obtain
{
t e θ1 + θ2 0, yr(t) ) 1 - 1 + t e-(t-θ1-θ2)/λc, t > θ + θ 1 2 λc
(
)
(8)
It shows that there is no overshoot in the nominal set point response, and its quantitative time domain specification can be conveniently achieved by tuning the single adjustable parameter λc. For instance, defining the system rising time tr to be the period from the moment that a unit step change is added to the set point input to the moment that the system output reaches 90% of its final value, the tuning formula can be figured out from eq 8 as tr ) 3.8897λc + θ1 + θ2. In addition, eqs 7 and 8 demonstrate that the stabilizing controller Pc really does not affect the performance specification of the set point response in the proposed control structure. It should be noted that when the adjustable parameter λc is tuned to zero, the set point response goes to the ideal case Hr(s) ) e-(θ1+θ2)s; i.e., the system output can reach the set point value just after the overall process time delay. In fact, the set point tracking controller can only provide bounded output energy as can its corresponding actuator in practice. As a result, asymptotic set point tracking can be actually implemented. Hence, tuning the single adjustable parameter λc actually aims at the tradeoff between the nominal performance of the set point response and the output capacities of the set point tracking controller C and its corresponding actuator. That is, when λc is tuned to be small, the nominal set point tracking becomes faster but the output energy of C and its corresponding actuator is larger, which however, will result in more aggressive dynamic behavior of the set point response in the presence of the process uncertainty in practice. On the contrary, when λc is tuned to be large, the nominal set point tracking turns out to be slower but the output energy of C and its corresponding actuator is smaller, and consequently less aggressive dynamic behavior of the set point response will occur in the presence of the process uncertainty. Generally it is recommended to initially tune λc to be approximately equal to the process time delay to achieve a compromise between the nominal performance of the set point response and the output capacities of C and its corresponding actuator. However, if the tuning is not satisfactory, by monotonically increasing (or decreasing) λc on-line, the desirable set point response specification can be conveniently achieved.
Ind. Eng. Chem. Res., Vol. 44, No. 4, 2005 903
2.3. Load Disturbance Estimator F1. In the proposed cascade control structure shown in Figure 1, to evaluate the achievable load disturbance rejection performance of the inner loop separately, it is suggested to leave out the primary outer loop for the present so that the load disturbance response of the inner loop can be taken into account independently, which will allow the analytical design of the load disturbance estimator F1 in a simple way. However, this will inevitably result in actual system performance degradation for the load disturbance rejection due to the interaction arising from the primary outer loop, such as the load disturbance response oscillation caused by the load disturbances that seep into the inner loop, which also inherently occurs in a conventional cascade control structure. Therefore, it can be easily seen from Figure 1 that the nominal load disturbance response transfer functions of the inner loop are
Hd1(s) )
P1(s) y1 ) d1 1 + F1(s)P1(s)
(9)
Hd2(s) )
y1 1 ) d2 1 + F1(s)P1(s)
(10)
Correspondingly the nominal complementary sensitivity function of the inner loop for load disturbance rejection can be obtained as
Td-inner(s) )
F1(s)P1(s) f1 ) d1 1 + F1(s)P1(s)
(11)
In the ideal case, the desired complementary sensitivity function should be of the form Td-inner(s) ) e-θ1s; that is, when a continuous load disturbance d1 shown in Figure 1 is injected into the intermediate process P1, the load disturbance estimator F1 should detect the resultant intermediate process output error just after the intermediate process time delay θ1 and then output an inversely equivalent signal f1 to counteract the load disturbance. However, there actually exists an asymptotic tracking constraint as follows:
limHd2(s) ) 0 sf0
(13)
where λf1 is an adjustable parameter for obtaining the desirable load disturbance response. By using eqs 11 and 13, the load disturbance estimator F1 can be obtained as
Td-inner(s)
1 F1(s) ) 1 - Td-inner(s) P1(s)
which may be reorganized as
F1(s) )
F10(s) 1 - F10(s)P1(s)
(15)
where
F10(s) )
Td-inner(s) P1(s)
)
τ1s + 1 k1(λf1s + 1)
(16)
This indicates that F1 can be implemented by using the control unit shown in Figure 2. It should be noted that the denominator of F1 shown in eq 15 is actually in the form of D(s) ) λf1s + 1 - e-θ1s. It is obvious to see D(0) ) 0 and the first derivation of D(s) is D′(s) ) λf1 + θ1e-θ1s > 0, which denotes that the characteristic equation of F1 includes no RHP zero except the origin. Therefore, the control unit shown in Figure 2 can really act as a special integrator so as to eliminate the intermediate process output error arising from the load disturbances. In consequence, the achievable load disturbance rejection performance can be quantitatively regulated by the single adjustable parameter λf1 of the load disturbance estimator F1. It can be seen from eq 13 that tuning λf1 to be small will result in faster inner loop response for load disturbance rejection, but the output energy of F1 and its corresponding actuator is larger, and vice versa. Hence, tuning λf1 aims at the tradeoff between the nominal load disturbance rejection performance of the inner loop and the output capacities of F1 and its corresponding actuator. 2.4. Load Disturbance Estimator F2. From Figure 1 it can be determined that the nominal load disturbance transfer function of the primary outer loop is in the form of
Hd3(s) )
(12)
It implies that F1 can only provide bounded control energy, and therefore, asymptotic tracking for the load disturbance can actually be realized. Hence, combined with the H2 optimal performance objective of IMC theory,21 the practically desired closed-loop complementary sensitivity function is proposed as
1 e-θ1s Td-inner(s) ) λf1s + 1
Figure 2. Executable control unit for F1.
y2 1 ) d3 1 + F2(s)P1(s)P2(s)
(17)
Accordingly, the nominal complementary sensitivity function of the primary outer loop for load disturbance rejection can be derived as
Td-outer(s) )
f2 F2(s)P1(s)P2(s) ) d1 1 + F2(s)P1(s)P2(s)
(18)
Following an analysis similar to that in the previous subsection, it is known that there exists an asymptotic constraint for rejecting the load disturbances injected into the primary process, i.e.
limHd3(s) ) 0 sf0
(19)
Besides, the following asymptotic constraint
lim Hd3(s) ) lim (1 - Td-outer(s)) ) 0
sf1/τ2
sf1/τ2
(20)
(14) is required to be met in order to reject step load
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disturbances injected into the primary unstable process input, so that the closed-loop internal stability can be guaranteed. Hence, combined with the H2 optimal performance objective of IMC theory, the practically desired closedloop complementary sensitivity function is proposed as
Td-outer(s) )
as + 1 -(θ1+θ2)s e (λf2s + 1)3
(21)
where λf2 is an adjustable parameter for obtaining the desirable load disturbance rejection performance of the primary outer loop, and a is determined by the asymptotic constraint shown in eq 20. Substituting eq 21 into eq 20 yields
[
F2-N/N(s) )
where N is the user-specified order to achieve the desirable closed-loop performance specification, and ci and dj are determined by the following two matrix equations.
]
[
Following a simple calculation gives
a ) τ2
[(
)
]
3 λf2 + 1 e(θ1+θ2)/τ2 - 1 τ2
(22)
[][
d0 b0 d1 b ) 1 l l dN bN
bN bN+1 l b2N-2
(τ1s + 1)(τ2s - 1)(as + 1) 1 ‚ (23) k1k2 (λ s + 1)3 - (as + 1)e-(θ1+θ2)s f2
Note that there exists a RHP zero-pole cancellation at s ) 1/τ2 in eq 23, which, however, cannot be removed directly and may cause the disturbance estimator F2 to work unreliably. Therefore the mathematical Maclaurin expansion formula is utilized to reproduce the ideally desired load disturbance estimator. Let F2(s) ) M(s)/s, then
[
M′′(0) 2 1 s + ‚‚‚ + F2(s) ) M(0) + M′(0)s + s 2! M(i)(0) i s + ‚‚‚ (24) i!
]
Obviously the first three terms of the above expansion constitute exactly a standard PID controller in the form of
F2-PID(s) ) kF +
1 + TDs TI s
‚‚‚ ‚‚‚ l ‚‚‚
bN-1 bN l b2N-3
c0 )
][ ] ][ ] [ ]
0 0 l bN-2
b2 b3 l bN
‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚
0 0 l b1
c1 c2 )l cN-1
c0 c1 l cN-1
(27)
bN+1 bN+2 l b2N-1
(28)
{
if if
1, -1,
ci g 0 ci < 0
Note that there remains an integrator in eq 26 that ensures the resultant Pade´ approximation to be inherently coincident with the ideally desired disturbance estimator shown in eq 23. For instance, taking N ) 2 and using eqs 26-28 yields the PID controller
(
F2-2/2(s) ) kF +
)
1 1 + TDs TI s TF s + 1
(29)
where
kF )
d1 , c0
TI )
d0 ) b0c0,
1 , b0
TD )
d2 , c0
d1 ) b1c0 + b0c1,
TF )
c1 b3 , c1 ) c0 b2
d2 ) b2c0 + b1c1
Taking N ) 3 and using eqs 26-28 yields the thirdorder approximation controller
(25)
where kF ) M′(0), TI ) 1/M(0), and TD ) M′′(0)/2. This is the practically proposed disturbance estimator F2 in the form of PID controller. It should be noted that the pure derivative term in eq 25 can be physically implemented by cascading it with a first-order low-pass filter in which the time constant can be chosen as (0.01 ∼ 0.1)TD. To obtain a better approximation for the ideally desired load disturbance estimator shown in eq 23, a linear fractional approximation formula based on the mathematical Pade´ series expansion is therefore proposed, that is,
0 b0 l bN-1
where bi ) M(i)(0)/i! (i ) 0, 1, ‚‚‚, 2N - 1) is the Maclaurin coefficient of each term in eq 24 and c0 should be taken as
Hence by using eqs 18, 21, and 22, the desired load disturbance estimator is obtained as
F2(s) )
(26)
N-1
cisi+1 ∑ i)0
as + 1 -(θ1+θ2)s e )0 (λf2s + 1)3
lim 1 -
sf1/τ2
N
djsj ∑ j)0
F2-3/3(s) )
d3s3 + d2s2 + d1s + d0 s(c2s2 + c1s + c0)
(30)
where
c1 )
b2b5 - b3b4 , b32 - b2b4
c2 )
b42 - b3b5 b32 - b2b4
d0 ) b0c0, d1 ) b1c0 + b0c1, d2 ) b2c0 + b1c1 + b0c2, d3 ) b3c0 + b2c1 + b1c2 In fact, the above third-order approximation controller can be implemented by using three conventional con-
Ind. Eng. Chem. Res., Vol. 44, No. 4, 2005 905
trollers in practice, that is,
F2-3/3(s) )
d3s2 + d2s + d1 c2s2 + c1s + 1
+
d0 s(c2s2 + c1s + 1)
(31)
where the first part is a conventional second-order leadlag controller, and the second part is an integrator cascaded with a conventional second-order low-pass filter. Remark 2. Note that both the Maclaurin and Pade´ approximation controllers for the ideally desired disturbance estimator are actually tuned by the single adjustable parameter λf2, which is explicitly presented in eq 23. Besides, the choice of c0 is to keep all of ci (i ) 0, 1, ‚‚‚, N - 1) the same sign to exclude the possibility of any RHP zeros from being enclosed in the denominator of the approximation formula shown in eq 26. Accordingly, by using the Routh-Hurwitz stability criterion, it can be shown that the proposed low-order approximation controller forms in eqs 29-31 can be reliably implemented to hold the closed-loop internal stability. However, a high-order approximation controller (e.g., N g 4) may be involved with RHP poles, which can be identified by using the Routh-Hurwitz stability criterion. Therefore, it is recommended to use the RouthHurwitz stability criterion to identify the stability of a high-order approximation controller obtained from eq 26 and then to choose an available approximation in combination with tuning λf2 before using the controller in practice. Obviously, owing to a better approximation for the ideally desired disturbance estimator, a highorder approximation controller is capable of achieving better closed-loop performance for load disturbance rejection compared with a Maclaurin approximation PID controller or a low-order approximation controller. Hence, it depends on the user choice of the compromise between the achievable load disturbance rejection performance and the complexity of the controller approximation form. It should be noted that the proposed high-order approximation controller can be practically and conveniently implemented by means of modern electrical instruments such as a direct digital controller or an industrial processing computer. 3. Control System Robust Stability Analysis The nominal system stability can be easily identified by using the controller design formulas developed in the previous section. By virtue of the open-loop control manner for the set point tracking in the proposed cascade control structure, the control system robust stability analysis can be restricted to the inner and outer closed loops. In industrial and chemical practice, frequently encountered process uncertainties are the perturbation of the process parameters, the actuator output uncertainties, and the measurement uncertainties of the process sensors. A practical way to identify the closed-loop system robust stability in the presence of the abovementioned process uncertainties is to lump multiple sources of uncertainty into a multiplicative form.24 According to the well-known small-gain theorem, a closed-loop control structure holds robust stability if and only if
||∆m(s)Td(s)||∞ < 1
(32)
where ∆m defines the process multiplicative uncertainty bound and Td is the closed-loop complementary sensitivity function. For the inner closed loop of the proposed cascade control structure, substituting eq 13 into eq 32 yields the robust stability constraint
∆m1(s) || || < 1 λf1s + 1 ∞
(33)
where ∆m1(s) defines the multiplicative uncertainty bound of the intermediate process P1 and therefore describes the actual intermediate process family Π1 ) ˆ 1(s) ) (1 + ∆m1)P1(s)}. For instance, for the {P ˆ 1(s): P intermediate process gain uncertainty that can be denoted as ∆m1(s) ) ∆k1/k1, the robust stability constraint for tuning the single adjustable parameter λf1 is
xλf12ω2 + 1 >
|∆k1| , k1
∀ω>0
(34)
For the intermediate process time delay uncertainty ∆θ1, which may be converted to the multiplicative uncertainty ∆m1(s) ) e-∆θ1s - 1, the robust stability constraint for tuning λf1 is
xλf12ω2 + 1 > |e-j∆θ ω - 1|, 1
∀ω>0
(35)
As for the intermediate process uncertainty of both gain and time delay, which may be converted to the multiplicative uncertainty ∆m1(s) ) (1 + ∆k1/k1)e-∆θ1s - 1, the robust stability constraint for tuning λf1 can be similarly derived. For the intermediate process actuator output uncertainty, e.g., ∆m1(s) ) (s + 0.2)/(s + 1), which can be loosely interpreted as the intermediate process input supplied by the corresponding actuator increases by up to 100% uncertainty at high frequencies and by almost 20% uncertainty in the low-frequency range, substituting it into eq 33 yields the constraint for tuning λf1 to hold the inner-loop robust stability, i.e.
xλf12ω2 + 1 >
x
ω2 + 0.04 , ω2 + 1
∀ω>0
(36)
Similarly, for the intermediate process output measurement uncertainty, e.g., ∆m1(s) ) -(s + 0.3)/(2s + 1), which can be physically viewed as the intermediate process output measurement provided by the corresponding sensor decreases by up to 50% uncertainty at high frequencies and by almost 30% uncertainty in the low-frequency range, the robust stability constraint for tuning λf1 can be derived as
xλf12ω2 + 1 >
x
ω2 + 0.09 , 4ω2 + 1
∀ω>0
(37)
Hence, for a specified bound of the process multiplicative uncertainty ∆m1(s) in practice, it is practicable to employ eq 33 to evaluate the inner-loop robust stability, i.e., to determine whether the magnitude plot of the left side of eq 33 with ω ∈ [0, + ∞) falls below unity, which in fact can be conveniently performed by using modern control software packages such as the MATLAB robust control toolbox.25
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Besides, according to the closed-loop performance analysis of IMC theory, the following constraint between the robust stability and nominal performance of the inner loop requires to be coupled with in tuning λf1, i.e.
|∆m1(s)Td-inner(s)| + |W1(1 - Td-inner(s))| < 1
(38)
where W1 is the sensitivity weight function and usually can be chosen as 1/s for the step change of the load disturbances. Substituting eq 13 into eq 38 yields
(
)
∆m1(s) 1 e-θ1s |+| 1|
