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Ind. Eng. Chem. Res. 2006, 45, 3604-3614
Enhanced Two-Degrees-of-Freedom Control Strategy for Second-Order Unstable Processes with Time Delay A. Seshagiri Rao and M. Chidambaram* Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, 600 036, India
Using the direct synthesis method, a proportional-integral-derivative (PID) controller in series with a leadlag compensator is designed for control of the open-loop unstable second order plus time delay processes with/without a zero. Set-point weighting is considered for reducing the overshoot. The method has two tuning parameters, and guidelines are provided for selecting the two tuning parameters. The method gives significant improvement in load disturbance rejection performances. The designed controller is also robust for uncertainties in the process parameters. Illustrative examples are considered to show the performances of the proposed method. Significant improvement is obtained when compared to recently reported methods. 1. Introduction Open-loop unstable processes arise frequently in the chemical and biological systems and are fundamentally difficult to control. The difficulty increases when the process contains a time delay. Extensive information on the physical significance of the unstable systems in the context of aeroplanes has been given by Stein.1 The open-loop unstable first-order plus time delay (UFOPTD) model is generally used to represent such processes. The performance specifications that are usually achieved for stable systems are difficult to achieve for unstable systems. The performance of the closed loop for such processes exhibits large overshoots and settling times. The controller design methods for the UFOPTD processes have been addressed by many researchers.2-7 Many chemical and biological systems exist whose dynamics also show second-order behavior. These types of systems can be modeled as open-loop unstable second-order plus time delay (USOPTD) models. Controlling these types of processes is difficult, and this difficulty increases when the USOPTD model contains a positive or negative zero. Some of the literature on unstable second-order processes includes work by Jacobsen,8 Huang and Chen,9 Marlin,10 Bequette,11 and Seki et al.12 Jacobsen8 has studied the dynamics of reactor-separator networks and has shown that the transfer function model between the composition of the distillate and the recycle ratio of the distillation column results in an unstable second-order model with one unstable pole and one unstable zero. Huang and Chen9 have shown that the dynamics of an unstable continuous stirred tank reactor (CSTR) that is being used to perform an exothermic reaction can be modeled by a secondorder unstable model with one right half plane (RHP) pole and time delay. Marlin10 showed that a jacketed CSTR that is being used to perform a simple reaction gives USOPTD model that contains two unstable complex conjugate poles and a negative zero, when the model equations are linearized around an unstable operating point. Bequette11 considered a CSTR that was performing a simple reaction and showed that the linearized model around the unstable operating point gives an USOPTD model that contains a zero and two unstable poles. Seki et al.12 have studied an unstable transfer function model of the gasphase polyolefin reactor, where the number of open-loop RHP poles varies from one to two, depending on the operating conditions. * To whom correspondence should be addressed. Fax: 91-044-2257 0509. E-mail:
[email protected].
It is well-known that conventional P/PI controllers alone cannot control USOPTD models with two unstable poles, because the phase angle of these processes is always smaller than π. A derivative action is required in the controller structure for compensation. However, if the time delay in the process is large, such that the process phase is much smaller than π, a PID controller alone cannot stabilize the process. If the process has a RHP zero, then the total phase comes below π/2. Hence, to compensate for that effect, in addition to the PID controller, a lead compensator is required to shift the phase in the phase angle versus frequency curve upward. Also, the lead compensator increases the resonant frequency, which results in increasing the upper bound of frequency in the low-frequency region (D’Azzo and Houpis13). In the present work, efforts have been made to obtain such types of controllers, using the direct synthesis method. The direct synthesis method is chosen here because the desired output behavior of the closed loop can be specified as a trajectory model based on the process to design the required form of the controller (Ogunnaike and Ray14) and it also is a simple design method. In the open literature, recent controller design methods for USOPTD model with either one RHP pole or two RHP poles include reports from Lee et al.,15 Yang et al.,16 Wang and Cai,17 Huang and Chen,9 Kwak et al.,18 Tan et al.,19 Lu et al.,20 and Liu et al.21 The controller design for USOPTD systems with two unstable poles and a negative zero has been addressed by Lee et al.,15 as well as Wang and Hwang;22 however, controller design for USOPTD processes with two RHP poles and a RHP zero has not yet been addressed. Some of the recently reported methods use a two-degrees-of-freedom control structure with a greater number of controllers; also, the design of controllers is complicated. Recently, Rao and Chidambaram23 have proposed a controller design method for USOPTD processes with two RHP poles and a zero. To reduce the overshoot and settling time for servo responses, a set point weighting is suggested for UFOPTD systems by Chidambaram,24 and for USOPTD systems by Sree and Chidambaram.25 With the set-point weighting, the overshoot and settling time are drastically reduced for the servo problems. It is well-known that the set-point weighting does not affect the load disturbance rejection performances. Hence, the set-point weighting is useful when the designed controller gives a good load disturbance rejection. In the present work, the method proposed by Rao and Chidambaram23 is generalized for all types of USOPTD processes and
10.1021/ie051046+ CCC: $33.50 © 2006 American Chemical Society Published on Web 04/08/2006
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3605
should be assumed such that the resulting controller is realizable. As stated previously, to compensate for the total phase lag introduced by the two unstable poles (unstable zero and time delay), a PID controller with additional lead and lag terms is selected. To achieve this form of the controller, the desired closed-loop transfer function is assumed as
Figure 1. Feedback control structure.
also set-point weighting is incorporated for improving the performance for the servo responses. The present paper is organized as follows. Section 2 addresses the proposed controller design method, and section 3 gives the robustness studies and stability. Section 4 deals with the design of the set point weighting parameter. In section 5, simulation results are explained and finally conclusions are given in section 6. 2. Controller Design The closed-loop control structure is shown in Figure 1, where Gp(s) is the process transfer function and Gc(s) is the transfer function of the controller. The typical USOPTD processes exist in most of the chemical and biological systems can be represented by any of the following transfer function models:
Gp(s) ) Gp(s) )
kp e-θs (τ1s + 1)(τ2s - 1) kp e-θs (τ1s - 1)(τ2s - 1)
Gp(s) ) Gp(s) )
kp e-θs s(τs - 1)
kp(1 ( ps) e-θs (τ1s ( 1)(τ2s - 1)
(1)
Gp(s) )
a1s2 + a2s + 1
From eq 6, using the direct synthesis is given by
Gc(s) )
method,14
(y/yr)d
1 Gp [1 - (y/yr)d]
Gc(s) )
[
]
n2s2 + n1s + 1 a1s2 + a2s + 1 kp (λs + 1)3 - (n2s2 + n1s + 1)(1 - ps) e-θs (9) The controller can be approximated in such a way that it should contain PID with lead and lag terms. Using the first-order Pade’s approximation for the time delay term in eq 9, the controller can be approximated in the form of
Gc(s) )
(a1s2 + a2s + 1)(n2s2 + n1s + 1)(1 + 0.5θs) kphs(x1s3 + x2s2 + x3s + 1)
(10)
where
h ) 3λ + θ - n1 + p x1 )
(3) (4)
0.5θλ3 - 0.5θn2p h
(11b) (11c)
3λ2 + 1.5θλ - n2 + n1p + 0.5θn1 - 0.5pθ h
(11d)
The denominator term in eq 10, x1s3 + x2s2 + x3s + 1, can be factorized as
x1s3 + x2s2 + x3s + 1 ) (βs + 1)(a1s2 + a2s + 1)
(12)
Upon equating the corresponding coefficients on both sides of eq 12, we get
βa1 ) x1
(13a)
βa2 + a1 ) x2
(13b)
β + a 2 ) x3
(13c)
From the aforementioned relations (eqs 11 and 13), the coefficients n2, n1, and β are obtained as
(6) n2 )
y3z1 - y1z2 y2y3 - y1y4
(14a)
n1 )
y2z2 - z1y4 y2y3 - y1y4
(14b)
x1 a1
(14c)
the controller
(7)
Here, (y/yr)d is the desired closed-loop transfer function for a set-point change. The desired closed-loop transfer function
(11a)
λ3 + 1.5θλ2 + n2p + 0.5θn2 - 0.5θn1p x2 ) h x3 )
(5)
where a1 > 0, a2 < 0, and the open-loop RHP poles of Gp(s) may be real or complex. The closed-loop transfer function for the set-point changes is given by
Gc(s)Gp(s) y(s) ) yr(s) 1 + Gc(s)Gp(s)
(8)
According to eq 7, the controller is obtained as
(2)
Of all the processes, the one that is difficult to control is the USOPTD process with two RHP poles and a RHP zero (eq 4). Hence, in the present work, this process is considered for the controller design. If the USOPTD process is of the type of eqs 1-3, then the designed controller can be easily extended for those type of processes by neglecting the corresponding terms. For generalization, the process is considered for the design of the controller as
kp(1 - ps) e-θs
2 -θs y(s) (n2s + n1s + 1)(1 - ps) e ) yr(s) (λs + 1)3
β) where
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Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006
y1 ) -a12 + 0.5θpa1
(15a)
y2 ) -0.5θpa2 - a1p - 0.5θa1
(15b)
y3 ) -a1a2 - a1p - 0.5θa1
(15c)
y4 ) -0.5θp + a1
(15d)
z1 ) a1λ3 + 1.5θa1λ2 - 0.5θa2λ3 - 3λa12 - θa12 - pa12 (15e) z2 ) 3a1λ + 1.5θa1λ - 0.5pθa1 - 0.5θλ 3a1a2λ - a1a2θ - a1a2p (15f) 2
3
Thus, the final controller Gc(s) form is obtained as
(
Gc(s) ) Kc 1 +
)(
)
Rs + 1 1 + τ Ds τIs βs + 1
(16)
where Kc ) n1/(kph), τI ) n1, τD ) n2/n1, β ) x1/a1, and R ) 0.5θ, and λ is the tuning parameter. Because there exists always a tradeoff between the nominal performance and robust performance, the tuning parameter (λ) must be tuned according to the desired choice. To obtain the phase lead, R should be always greater than β. Remarks: For the USOPTD process without any zero, it is observed that the designed value of β is too large to obtain robust performances of the closed-loop system when the parametric uncertainties are large. Also, because of the high value of β, the phase lag imposed by the term (βs + 1) in the controller is determined to be greater and, thus, the designed controller with this value of β is not able to give robust control performances when there are large perturbations in the process parameters. To show the effect of β on the robustness of the closed-loop system, gain margin (GM) and phase margin (PM) can be used which are known as good robustness measures. The lower bounds for GM and PM can be calculated from the maximum peak values of the sensitivity and complementary functions. For any closed-loop control system to be robust, the requirements are as follows:26
( ) ( )
MS 1 , PM g 2 sin-1 GM g MS - 1 2MS GM g1 +
1 1 , PM g 2 sin-1 MT 2MT
where MS and MT are the maximum peaks of the sensitivity and complementary sensitivity functions, respectively. In the present work, when sensitivity and complementary sensitivity functions are plotted against the frequency, the peak values of both the sensitivity and complementary sensitivity functions are observed to be significantly high when the actual value for β is used, which results in much lower bounds for the gain and phase margins of the open-loop system (the gain margin should be >1.7 and the phase margin should be >30° for robust control of a process).27 Hence, to get good nominal and robust closedloop performances, the value of β is considered much lower than that obtained by the design procedure so that the resulting lower bounds for GM and PM are increased, thus increasing the robustness. Based on many simulation studies that have been conducted on different USOPTD processes, it is observed that using a value of “0.1β” instead of β gives robust control performances. Justification for considering 0.1β will be explained further in section 5. Thus, in the present work, for the USOPTD processes without any zero, the value of β is
considered directly as 0.1β and for the USOPTD processes with a zero, the value of β obtained directly from the controller design procedure is retained. Selecting the Tuning Parameter λ. It is well-known that there is always a tradeoff in selecting the desired closed-loop tuning parameter (λ). Fast speed of response and good disturbance rejection are favored by choosing a small value of λ; however, stability and robustness are favored by a large value of λ. Hence, the choice of λ is entirely based on the experience of the operator with the control system. In fact, this is the case for any controller design method, based on the closed-loop tuning parameter such as the IMC method/direct synthesis method, etc. Based on many simulation studies, it is observed that the starting value of λ can be considered to be slightly more than the process time delay, i.e., λ can be considered to be 1.2 times greater than the time delay. If both the nominal and robust control performances are achieved with this value, then this value for λ can be taken as the final value. If not, the value should be increased carefully until both the nominal and robust control performances are achieved. 3. Stability and Robustness For any closed-loop control system, it is necessary to analyze the stability and robustness for uncertainties in the process and for load disturbances. The closed-loop system is robustly stable if and only if28
∀ ω ∈ (-∞,∞)
||lm(jω)T(jω)|| < 1
(17)
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
lm(jω) )
|
|
Gp(jω) - Gm(jω) Gm(jω)
(18)
where Gm(jω) is the model of the USOPTD process. The complementary sensitivity function of the closed loop for the USOPTD process with the designed controller (eq 16) is
T(s) ) [kpKc(1 - ps)(1 + Rs)(1 + τIs + τDs2) e-θs]/[τIs(βs + 1)(a1s2 + a2s + 1) + kpKc(1 - ps)(Rs + 1)(1 + τis + τDs2) e-θs] (19) where the controller parameters Kc, τI, τD, and β are the functions of the tuning parameter λ. If uncertainty exists in the time delay, then the tuning parameter should be selected such that
||T(jω)|∞
