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Ind. Eng. Chem. Res. 2009, 48, 3098–3111
Enhanced Control of Unstable Cascade Processes with Time Delays Using a Modified Smith Predictor S. Uma, M. Chidambaram, and A. Seshagiri Rao* Department of Chemical Engineering, National Institute of Technology, Tiruchirappalli 620 015, India
This paper presents a simple cascade controller in the enhanced modified Smith predictor structure for control of open loop unstable cascade processes with/without zero. The proposed structure consists of two control loops, a secondary inner loop and a primary outer loop. The method has totally three controllers of which the secondary loop has one controller and the primary loop has two controllers. The secondary loop controller is designed using the internal model control (IMC) technique. The primary loop set point tracking and disturbance rejection controllers are designed using the direct synthesis method. The primary set point tracking controller is designed as a proportional, integral, and derivative (PID) controller with lag filter and the primary disturbance rejection controller is designed as a PID with lead-lag filter. Simulation studies have been carried out on various cascade unstable processes with/without zero. The present method gives significant disturbance rejection both in the inner and outer loops. Also, the proposed method shows significant improvement when compared to the recently reported methods in the literature. 1. Introduction Open loop unstable processes are much more difficult to control than stable processes. Simple proportional, integral, and derivative (PID) control of open loop unstable systems has been discussed by several authors1-9 in the past decades. To reduce the undesirable overshoots, set point weighting has been proposed for unstable systems.10,11 The stability analysis for controlled unstable systems has been carried out elobaratively by Wang12 and Kwak.13 However, the desired performance can not be obtained with simple PID controllers particularly for unstable systems with large dead times.13 Kwak13 has discussed the limitations of PID controllers for unstable first-order processes with time delay which is also applicable for unstable second-order processes with time delay. IMC design for the control of open loop unstable processes is discussed by Yang14 and Tan.15 Smith delay compensation is proven as an effective tool for time delay processes.16 However, the original Smith predictor is not applicable for unstable systems. To overcome this drawback, many authors have proposed various modified Smith predictor structures for control of unstable processes with dead time.17-21 It is well-known that a cascade control scheme can dramatically improve the closed loop performance by rejecting the disturbances very fastly. A cascade control structure consists of two control loops, a secondary intermediate loop and a primary outer loop. The idea of cascade structure is that the disturbances introduced in the inner loop are reduced to a greater extent in the inner loop itself before they extend into the outer loop. Many people have addressed the design of cascade control for stable processes.22,23 However, design for unstable systems is limited. Recently, Liu24 proposed IMC-based control strategy for open-loop unstable cascade processes. Later, Kaya25 proposed a cascade control structure for control of open loop unstable processes using a modified form of the Smith predictor. But, these methods do not consider systems with zeros. Also, the design involves many controllers and the design methods are difficult. * To whom correspondence should be addressed. E-mail: seshagiri@ nitt.edu. Tel.: +91-431-2503115. Fax: +91-431-2500133.
In this paper, a modified enhanced Smith predictor combined with cascade control is proposed for control of open loop unstable systems. The scheme combines the features of both the cascade control and modified Smith predictor structure for the control of open loop unstable systems. The paper is organized as follows. The proposed structure is explained in section 2. Detailed controller design procedures are given in section 3. Section 4 discusses the selection of tuning parameters, set point weighting, and filter parameters. Section 5 discusses the stability and robustness analysis. Simulation results are explained in section 6, and finally, the conclusions are in section 7. 2. Theoretical Developments In process industries, it is well-known that the secondary loop process dynamics in cascade control ate stable in nature and the dynamics of the primary process may be stable, unstable. Hence, in the present work, theoretical developments are done by considering the transfer function of the secondary process with stable poles and the transfer function of the primary process with unstable poles and zeros. Also, usually, time delays are more in the primary loop than that of secondary loop. The block diagram of the proposed scheme is shown in Figure 1 where a simple IMC-based design is incorporated in the secondary loop and a modified Smith predictor structure is
Figure 1. Proposed cascade control scheme using a modified Smith predictor.
10.1021/ie801590e CCC: $40.75 2009 American Chemical Society Published on Web 02/05/2009
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3099
incorporated in the primary loop in which Gp2 is the transfer function of the secondary process, Gp1 is the transfer function of the primary process, Gm2 is the transfer function of the secondary process model, and Gm is the overall transfer function of the model of the primary process. θm is the overall time delay of the primary model. y2 is the secondary process output, and y1 is the primary process output. r2 is the set point for the secondary loop, and r1 is the set point for the primary loop. Gc2 is the secondary loop controller, Gcs is the primary set point tracking controller, and Gcd is the primary disturbance rejection controller. In the literature of the modified Smith predictor scheme,26 it is recommended to use a first-order filter for the output error to improve the closed loop performance. In this work also, a first-order filter Gf ) 1/(τfs + 1) is considered for the output error to improve the performance. The proposed structure has three controllers. Gc2 is an IMC controller in the secondary loop which simplifies the design of the inner loop and is used for rejecting the disturbances entering the secondary loop; the other controllers are a primary set point tracking controller and a primary disturbance rejection controller. The primary set point controller Gcs is designed as a PID with a lag filter which takes care of the set point tracking. The disturbance rejection controller Gcd is designed as a PID with a lead-lag filter for load disturbance rejection and stabilization. A set point filter/weighting parameter reduces the undesirable overshoot as well as the settling time.10 Hence, set point weighting is considered in this work for smooth set point tracking responses. The closed loop transfer functions of the primary loop for servo and regulatory responses are given by y1 ) r1 (1 + GcdGme-θms)(GcsGp2Gp1Gc2)
(1)
(1 - Gc2Gm2 + Gc2Gp2 + GcdGc2Gp1Gp2)(1 + GmGcs GfGcsGme-θms) + (GfGcsGc2Gp1Gp2)(1 + GcdGme-θms) y1 ) d2 Gp1Gp2(1 - Gc2Gm2)(1 + GmGcs - GfGcsGme-θms)
(2)
(1 + GmGcs - GfGcsGme-θms)(1 - Gc2Gm2 + Gc2Gm2Gp2 + GcdGc2Gp1Gp2) + (GfGcsGp1Gp2Gc2)(1 + GcdGme-θms) y1 ) d0 Gp1(1 - Gc2Gm2)(1 + GmGcs - GfGcsGme-θms)
(3)
(1 + GmGcs - GfGcsGme-θms)(1 - Gc2Gm2 + Gc2Gm2Gp2 + GcdGc2Gp1Gp2) + (GfGcsGp1Gp2Gc2)(1 + GcdGme-θm ) s
3. Design of Controllers The structure requires the design of three controllers (Gc2, Gcs, and Gcd). The design of controllers Gcs and Gcd are based on direct synthesis method, and Gc2 is designed as an IMC controller. Design is addressed by considering the secondary loop process as a stable one and the primary loop process as an unstable one. As many processes can be descried by the firstorder plus time delay (FOPTD) processes, in the present work also, design is addressed for FOPTD processes. 3.1. Design of Secondary Loop Controller. Gc2 is an IMC controller in the secondary loop which stabilizes the process by rejecting the disturbances entering the secondary loop. The closed loop transfer function of the secondary loop is given by
Gc2Gp2 y2 ) r2 1 - Gc2Gm2 + Gc2Gp2
(4)
As stated earlier, the secondary process dynamics are stable in nature as an FOPTD process is usually represented; here, the transfer function of the secondary process is considered with a stable pole and is given as Gp2(s) )
kp2e-θp2s (τp2s + 1)
(5a)
Gm2 is the model of the secondary process and is given as km2e-θm2s Gm2 ) (τm2s + 1)
(5b)
As per the IMC strategy, the controller Gc2(s) transfer function is considered as Gc2(s) )
(τm2s + 1) km2(λ2s + 1)
(6)
Assuming the model exactly represents the process (Gm2 ) Gp2) and substituting the values of Gc2, Gp2, and Gm2 in eq 4, the transfer function of the secondary loop is derived as y2 e-θm2s ) r2 λ2s + 1
(7)
where λ2 is the tuning parameter. Selection of λ2 is done such that it rejects the disturbances entering the inner loop faster and gives a stabilized output. Quantitative expression for selection of λ2 is given in section 4. Remark 1. When the time delay to the time constant ratio is small, the original IMC method does not provide good disturbance rejection performance. In the present work, design of a secondary loop controller is addressed by considering that the secondary loop is much faster than the primary loop which is usual in any cascade loop. One can also consider the design of the secondary loop controller corresponding to two cases based on the dynamics of the secondary loop. When the inner loop dynamics are slow, i.e. the secondary process contains poles sufficiently slower than the desired closed loop response, then the disturbance rejection and set point tracking may be poor. If the inner loop dynamics are fast, i.e. secondary loop process contains poles sufficiently faster than the desired closed loop response, then there will be improvement in the closed loop performance for disturbance rejection. Lee et al.22,23 have addressed this problem of designing controllers for both the cases for cascade control systems. Lee et al.27 have recomn mended to consider the IMC filter in the form of f2 ) ∑i)1 Risi + 1/(λ2s + 1)m for the case of slow inner loop dynamics and in the form of f2 ) 1/(λ2s + 1)m for the case of fast inner loop dynamics in which m and n should be selected to make the IMC controller realizable. In the present work, fast secondary loop dynamics are considered which is usual in any cascade loop, and accordingly, the controller design is addressed. In the literature also IMC method is considered for secondary loop tuning in cascade control systems for stable processes and showed improved performance.22,23 By properly choosing the IMC filter parameter, the performance can be improved for the IMC controller in a cascade loop. Also, for the design of primary loop controllers (Gcs and Gcd), the overall primary loop model (Gm) is required which comes from the relation between y1 and r2. With the present design, the expression for Gm (eq 10e) is obtained in a simple way which is straightforward. Hence, in the present work,
3100 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
the IMC method is considered for secondary loop by considering fast secondary loop dynamics. In fact, with the designed IMC controller, the proposed method is able to provide significant disturbance rejection performances. Remark 2. One can also design the secondary loop IMC controller for slow dynamics, but this design is complex when compared with that of fast dynamics. When the secondary loop dynamics are slower, then the design of IMC controller should be changed by considering the IMC filter as explained above with an additional numerator term. In that case, the overall primary process model (eq 10e) will contain one additional term in the numerator and a third order denominator. To design the primary loop controllers, this higher-order model should be reduced as a second-order model (as eq 10e) by using any model reduction technique, and then, the primary loop controllers are designed as explained in sections 3.2.1 and 3.2.2. 3.2. Design of Primary Loop Controllers. It is well-known that the control of unstable systems with positive zeros is more difficult than unstable systems with negative zeros. Hence in the present work, the primary process is considered with positive zero and the transfer function of the primary process is given as Gp1(s) )
kp1(1 - pps) -θp1s e (τp1s - 1)
(8)
The relation between the primary loop output (y1) and the secondary loop set point (r2) is obtained as y1 y2 ) G r2 r2 p1
(9a)
y1 kp1(1 - pps)e-θm2se-θp1s ) r2 (λ2s + 1)(τp1s - 1)
(9b)
y1 GcsGme-θms ) r1 1 + GcsGm
() y1 r1
(ap1s + ap2s + 1) 2
)
(λss + 1)4
(14)
(15)
s
e-θps
(1 + R1s + R2s2)(1 - ps) Gm[(λss + 1)4 - (1 + R1s + R2s2)(1 - ps)]
and θm ) θm2 + θp1
(16)
Substituting Gm from eq 10e, Gcs is obtained as (10b) Gcs ) (10c)
(10d)
(1 + R1s + R2s2)(1 + a2s + a1s2) km[(λss + 1)4 - (1 + R1s + R2s2)(1 - ps)]
(10e)
Where, Gm is the transfer function of the overall process model and θm is the time delay of the overall primary loop process model. Assuming the model exactly represents the process, the closed loop transfer function for primary loop is derived from eqs 1-3 as
(17)
Simplifying gives Gcs )
(1 + R1s + R2s2)(1 + a2s + a1s2) kmh1s(x1s3 + x2s2 + x3s + 1)
(18)
where x1 )
λs4 , h1
x2 )
4λs3 + R2p , h1
where
(a1s2 + a2s + 1)
m
y1 GcsGme-θms (1 + R1s + R2s2)(1 - ps)e-θms ) ) r1 1 + GcsGm (λ s + 1)4
Gcs )
y1 ) Gm(s)e-θms r2
km(1 - ps)
cd
The actual closed loop relation between y2 and r2 is given by eq 11, and hence,
For the purpose of controller design, the corresponding overall primary loop process model can be written as
Gm(s) )
cs
m
(1 + R1s + R2s2)(1 - ps)e-θms
d
(10a)
kp1(1 - pps) -θp1s e (τp1s - 1)
kp(1 - pps)
(13)
cd
The closed loop transfer function for set point tracking (eq 11) consists of only Gcs. Hence, Gcs can be tuned first for achieving good set point tracking, but Gcs also affects the load disturbance rejection performances (eq 12). For that, Gcd can be tuned to alter the disturbance rejection performances only. Thus, after tuning of Gcs for set point tracking, Gcd is designed for load disturbance rejection. 3.2.1. Design of Gcs. From eq 11, it can be observed that the characteristic equation does not contain time delay and hence the primary set point tracking controller (Gcs) is designed without considering the time delay term. Direct synthesis method is considered here to design Gcs. In direct synthesis method, the desired closed loop transfer function should be assumed such that the resulting controller is realizable. As a PID controller in series with additional lead-lag terms gives improved performance, the desired closed loop transfer function is assumed to achieve this form of the controller and is given as
or Gp(s) )
y1 Gp1(1 + GmGcs - GfGcsGme-θms) ) d0 (1 + G G )(1 + G G e-θms) m
where Gp1 )
(12)
cs
From eq 15, the controller Gcs is derived as
y1 ) GpsGp1 r2
e-θm2s , (λ2s + 1)
y1 Gp1Gp2(1 + GmGcs - GfGcsGme-θms) ) d2 (1 + G G )(1 + G G e-θms) m
In generalized form, the above equation can be written as
Gps )
(11)
x3 )
6λs2 - R2 + R1p , h1 h1 ) 4λs - R1 + p
The denominator term x1s3 + x2s2 + x3s + 1 in eq 18 can be factorized as x1s3 + x2s2 + x3s + 1 ) (βss + 1)(a1s2 + a2s + 1)
(19a)
Upon equating the corresponding coefficients on both sides of eq 19a, we get βsa1 ) x1,,
βsa2 + a1 ) x2,
βs + a2 ) x3
(19b)
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3101
d4 ) (βda1 + 0.5pθmkmkdd)/(kmkid)
[a2λs + 4a1 λs + a1 p - 4a1λs + pλs + 4
2
2
3
4
4a1a2pλs + a1a2p2 - 6a1pλs2]
R1 )
a12 + a1a2p + a1p2
d3 ) (a2βd + a1 - pkmkdd - 0.5θmkmkdd + (19c)
0.5pθmkmkcd)/(kmkid)
R2 ) 6λs + R1p - (βs + a2)h1 2
0.5pθmkmkid)/(kmkid) d1 ) (1 + kmkcd - pkmkid - 0.5θmkmkid)/(kmkid)
βs ) λs4/a1h1
According to direct synthesis method, the characteristic equation (eq 25) is considered to follow the desired trajectory in the form of
Thus, the final controller Gcs(s) form is obtained as
(
Gcs ) kcs +
)(
kis 1 + kdss s βss + 1
)
(20)
where kcs ) R1/kmh1,
kis ) 1/kmh1,
(λds + 1)4 ) 0
kds ) R2/kmh1, and
Here λs is the tuning parameter, and as there is always a tradeoff between the nominal performance and robust performance, the tuning parameter (λs) has to be tuned according to the desired choice. 3.2.2. Design of Gcd. Gcd is the primary load disturbance rejection controller and is designed using the direct synthesis method. The closed loop relation between the output (y1) and the load disturbance (d2) assuming the model represents exactly the process (Gp2 ) Gm2) and (Gc2Gp1Gp2 ) Gme-θms) is given by eqs 12 and 13 and is
cd
(
km(1 - ps) a1s + a2s + 1 2
([
kcd +
)
kid (Rds + 1) + kdds s (βds + 1)
]
kid (Rds + 1) + kdds s (βds + 1)
(
)
(23)
)
1 - 0.5θms ) 0 (24) 1 + 0.5θms
Considering Rd ) 0.5θm, after simplification, eq 24 reduces to d4s4 + d3s3 + d2s2 + d1s + 1 ) 0 where
h2kmkid - 1 km
(28a)
-a1a2q2 - a1pq2 - z1q2 - a13 + a1z1p + Rda12p - Rdz1p2 km(q2λd4 + z1pq2 - 6λd2a1q2 - a1h2pq2 - z1h2q2 + a1q1 - Rdpq1) (28b)
kdd )
a12 - z1p + kmkidq1 q2km
(28c)
Rd ) 0.5θm and βd ) (λd4kmkid - 0.5θmkmkddp)/a1 (28d)
(22)
The characteristic equation containing the Gcd term in eq 22 is1 + GcdGme-θms ) 0. Substituting the expressions for Gcd and Gm from eqs 23 and 10e) and using a first-order Pade approximation for time delay results in 1+
kid )
q1 ) a2λd4 - 4λd3a1 + z1ph2
m
It can be observed that the characteristic equation involving Gcd second term in eq 22 contains the time delay, and hence, it is designed based on the unstable second order plus time delay model. The primary goal of the controller Gcd is to stabilize the unstable model and to improve the load disturbance rejection performances. Thus in the design of Gcd, it is considered that Gcd stabilizes the unstable process model (Gme-θms) in a single conventional feedback loop. The structure of Gcd is considered as a PID controller in series with a lead lag compensator. The main advantage of the lead compensator is that it increases the resonant frequency, which results in increasing the upper bound of frequency in the low frequency region and thus provides improved closed loop performances. The PID lead-lag controller is considered as given below. Gcd ) kcd +
kcd )
where
y1 Gp1Gp2(1 + GcsGm - GcsGfGme-θms) ) d2 (1 + G G )(1 + G G e-θms) m
(27)
Expanding eq 27 and equating the corresponding coefficients of s in eq 25 and eq 27, the controller parameters kcd, kid, kdd, Rd, and βd are obtained as
βs ) λs4/a1h1 (21)
cs
(26)
d2 ) (βd + a2 + kmkdd - pkmkcd - 0.5θmkmkcd +
(25)
q2 ) z1 + z2p + a1p z1 ) Rda1,
z2 ) Rda2,
and h2 ) Rd + p + 4λd
Here λd is the tuning parameter. The tuning parameter has to be selected to obtain good nominal and robust control performances. To obtain the phase lead, “Rd” should be always greater than “βd”. The value of βd is considered to be 0.1βd if the system does not have a zero, and for systems with zero, it is considered to be the same value obtained from eq 28d.11 Remark 3. When a zero is present in the process, one can approximate it as an FOPTD process for the purpose of controller design, but in the present method, design of controllers is addressed directly for the zero present in the unstable process. Hence, the present approach is more straightforward and gives more physical understanding in the controllers design because process zeros are directly incorporated in the controller design. 4. Selection of Tuning Parameters Selection of λ2, λs, and λd. It is well-known that the tuning parameters should be selected in such a way that the resulting controllers give both nominal as well as robust performance. In IMC and synthesis methods, a lesser value for tuning parameter gives good nominal performance. Higher value of these tuning parameter gives robust control performance, but under nominal conditions, the responses are slow. Hence, there is a tradeoff in selecting the tuning parameter values. On the basis of many simulation studies, it is observed that the starting values of the tuning parameters λ2, λs, and λd can be taken to be equal to the corresponding process time delay. If the performance achieved with these values is acceptable, then those values can be taken as the final values; if not, the parameters should be tuned around the time delay. After conducting many
3102 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
Figure 2. Closed loop responses for example 1 for perfect parameters: (solid line) proposed, (dotted line) data of Liu et al.,24 (dash-dotted line) data of Kaya.25
Figure 3. Closed loop responses for example 1 for +10% perturbations in both the time delays and -10% in both the time constants: (solid line) proposed, (dashed line) data of Liu et al.,24 (dash-dotted line) data of Kaya.25
simulation studies on different unstable process, quantitatively, the recommended ranges for the tuning parameters are λ2 ) 0.5θm2-θm2, λs ) 0.3θm-θm, and λd ) 0.8θm-θm. It is observed that λs e λd for processes without zero and for processes with zero λs g λd. Set Point Weighting and Filter Parameter. Sree and Chidambaram10 have proposed a method for finding out the set point weighting parameter for a simple PID controller for unstable second order plus time delay (USOPTD) systems. In the present work, after conducting many simulation studies, the recommended value for the set point weighting parameter ε is 0.3. The disturbance rejection is further enhanced with a firstorder filter, and its parameter is τf. It is observed that smaller values of τf results in a lesser peak and settling times in the load disturbance rejection performance. As τf is increased, the load disturbance rejection performance is smooth with larger settling time. This is due to the fact that filter acts as a lag element and hence there will be a lag in the disturbance rejection response, which causes more settling time. Hence, there is a tradeoff in selecting the filter parameter. On the basis of many simulation studies, the recommended values for τf is 6θm.
For clear understanding, the overall tuning procedure is summarized below in a systematic format. 1. With the known secondary process model (Gm2), find the secondary loop IMC controller (Gc2) from eq 6 and select the IMC tuning parameter in the range of λ2 ) 0.5θm2-θm2. 2. Then from the known primary process (Gp1), obtain the overall primary process model (Gm) and overall primary loop time delay (θm) from eq 10e. 3. Find the primary loop set point tracking controller (Gcs) parameters from eq 21 after selecting the tuning parameter in the range of λs ) 0.3θm-θm. Then calculate the disturbance rejection controller (Gcd) parameters from eq 28a-d after selecting the tuning parameter according to λd ) 0.8θm-θm. 4. Use set point weighting (ε) as 0.3 and the filter (Gf) parameter as τf ) 6θm. 5. Stability and Robustness As the design of controllers is based on the process models, when there are modeling errors (uncertainties), one should carry the sensitivity analysis. It is necessary to analyze the stability
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3103
Figure 4. Magnitude plots for complementary sensitivity function (T) for different values of λs and uncertainty norm bound for example 1: (dashed line) uncertainty norm bound |1/exp(-0.5s) - 1|, (solid line) λs ) 0.7042, (dotted line) λs ) 0.9, (dash-dotted line) λs ) 0.5.
Figure 5. Closed loop responses for example 2 for perfect parameters: (solid line) proposed, (dashed line) data of Liu et al.,24 (dash-dotted line) data of Kaya.25
and robustness of systems in the presence of model uncertainties. The sensitivity analysis of the controller which is present in the complementary sensitivity function is shown by considering modeling errors in all the process parameters. The types of uncertainties considered here are the parametric uncertainties such as uncertainty in process gain, time constant, and time delay. The closed loop system is robustly stable if and only if28,29 |lm(jω)T(jω)| < 1
∀ ω(-∞, ∞)
(29)
where T(s ) jω) is the complementary sensitivity function and lm(s ) jω) is the bound on the process multiplicative uncertainty. The process uncertainty can be represented as - G (jω)e |G (jω)eG (jω)e | -θps
lm(jω) )
p
|T(jω)|∞
