Volume 13 Number 6
December 2016
Article Contents
Zhuo-Yun Nie, Rui-Juan Liu, Fu-Jiang Jin and Lai-Cheng Yan. Analysis and Design of Scaling Optimal GPM-PID Control with Application to Liquid Level Control. International Journal of Automation and Computing, vol. 13, no. 6, pp. 624-633, 2016. doi: 10.1007/s11633-016-0998-y
Cite as: Zhuo-Yun Nie, Rui-Juan Liu, Fu-Jiang Jin and Lai-Cheng Yan. Analysis and Design of Scaling Optimal GPM-PID Control with Application to Liquid Level Control. International Journal of Automation and Computing, vol. 13, no. 6, pp. 624-633, 2016. doi: 10.1007/s11633-016-0998-y

Analysis and Design of Scaling Optimal GPM-PID Control with Application to Liquid Level Control

Author Biography:
  • Rui-Juan Liu,received the B. Sc. and M. Sc degrees in mathematics from Changsha University of Science and Technology, China in 2004 and 2007, respectively, and Ph. D. degree in control theory and control engineering from Central South University, China in 2014. She is currently a lecturer in the School of Applied Mathematics, Xiamen University of Technology, China.
    Her research interests include robust control, nonlinear control, and fractional-order system.
    E-mail:liuruijuan0313@163.com

    Fu-Jiang Jin, received the M. Sc. and Ph. D. degree in control theory and control engineering from Zhejiang University, China in 1998 and 2002, respectively. He is currently a professor in the School of Information Engineering, Huaqiao University, China.
    His research interests include process modeling and control.
    E-mail:jinfujiang@163.com

    Lai-Cheng Yan,received the B. Sc. and M. Sc degrees in electrical engineering from Chongqing Communication Institute and Chongqing University, China in 2004 and 2007, respectively. He is currently a lecturer in the School of Information Engineering, Huaqiao University, China.
    His research interests include robot modeling and control.
    E-mail:ylaicheng@126.com

  • Corresponding author: Zhuo-Yun Nie received the B. Sc. degree in automation from Central South University, China in 2006, and Ph.D. degree in control theory and control engineering from Central South University, China in 2012. He is currently a lecturer in the School of Information Engineering, Huaqiao University, China.
    research interests include robust control, nonlinear control, and financial forecasting.
    E-mail: yezhuyun2004@sina.com ;
    ORCID iD: 0000-0002-5980-3268
  • Received: 2015-04-27
  • Accepted: 2015-07-17
  • Published Online: 2016-06-20
Fund Project:

This work was supported by National Natural Science Foundation of China 61403149 and 61273069

Natural Science Foundation of Fujian Province 2015J01261

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Analysis and Design of Scaling Optimal GPM-PID Control with Application to Liquid Level Control

  • Corresponding author: Zhuo-Yun Nie received the B. Sc. degree in automation from Central South University, China in 2006, and Ph.D. degree in control theory and control engineering from Central South University, China in 2012. He is currently a lecturer in the School of Information Engineering, Huaqiao University, China.
    research interests include robust control, nonlinear control, and financial forecasting.
    E-mail: yezhuyun2004@sina.com ;
    ORCID iD: 0000-0002-5980-3268
Fund Project:

This work was supported by National Natural Science Foundation of China 61403149 and 61273069

Natural Science Foundation of Fujian Province 2015J01261

Abstract: In this paper,a new analysis and design method for proportional-integrative-derivative (PID) tuning is proposed based on controller scaling analysis.Integral of time absolute error (ITAE) index is minimized for specified gain and phase margins (GPM) constraints,so that the transient performance and robustness are both satisfied.The requirements on gain and phase margins are ingeniously formulated by real part constraints (RPC) and imaginary part constraints (IPC).This set of new constraints is simply related with three parameters and decoupling of the remaining four unknowns,including three controller parameters and the gain margin,in the nonlinear and coupled characteristic equation simultaneously.The formulas of the optimal GPM-PID are derived based on controller scaling analysis.Finally,this method is applied to liquid level control of coke fractionation tower,which demonstrate that the proposed method provides better disturbance rejection and robust tracking performance than some commonly used PID tuning methods.

Zhuo-Yun Nie, Rui-Juan Liu, Fu-Jiang Jin and Lai-Cheng Yan. Analysis and Design of Scaling Optimal GPM-PID Control with Application to Liquid Level Control. International Journal of Automation and Computing, vol. 13, no. 6, pp. 624-633, 2016. doi: 10.1007/s11633-016-0998-y
Citation: Zhuo-Yun Nie, Rui-Juan Liu, Fu-Jiang Jin and Lai-Cheng Yan. Analysis and Design of Scaling Optimal GPM-PID Control with Application to Liquid Level Control. International Journal of Automation and Computing, vol. 13, no. 6, pp. 624-633, 2016. doi: 10.1007/s11633-016-0998-y
  • In the industry and engineering,proportional-integral-derivative (PID) controller still serves as the most widely used controller form,owing to its simple structure and design. An initial work developed by Ziegler and Nichols gave a well-known systematic design of a PID controller and this work was followed and modified to form Ziegler et al.[1, 2] approach. More importantly,this famous tuning method was accepted quickly and used by engineers in the industry. Meanwhile,it raised a lot of attention in PID controller tuning technologies in the automatic control filed. Nowadays,one can still find some latest developments of PID control theory and technology in the recent literatures[3, 4] and books[5, 6].

    Stability robustness plays an important role in the robust control and is often measured by gain and phase margins (GPM). The controller design based on gain and phase margin can be considered as a classical robust control approach that was first introduced in [7] for single-input and single-output (SISO) system,where specification on the exact gain and phase margins are used as a hard target in the design. A well-known GPM-PID design procedure for SISO system is Astrom's[8] method. The fundamental step is estimating the critical gain and the critical frequency by relay test. The GPM-PID controller can be found straightforward by using such critical point. Multi-input and multi-output (MIMO) system design on the basis of gain and phase margins are also considered in [9, 10]. With the decoupling controller designed for the MIMO system to reduce the integration between the loops,single loop GPM-PID controller is utilized for each loop to obtain the desired gain and phase margins. Unlike synthesis method,graphical methods are developed in [11, 12],where exact gain and phase margins can be obtained regardless of the process order,time delay or damping nature.

    Although intensive research has been done on the design of GPM-PID,good closed-loop performance is also required in the system design to achieve the tradeoff between the performance and the robustness[13]. An improvement of GPM-PID is to formulate the gain and phase margins specifications to be inequality constraints in optimal controller design with an objective function introduced. In this regard,the closed-loop bandwidth is maximized for the lower bounds of GPM with constraint on overshoot ratio in [14].

    In this paper,optimal GPM-PID controller tuning methods are proposed for first-order plus dead time processes. The requirements on the stability margins and negative feedback are formulated as real part constraints and imaginary part constraints of characteristic equations which determine all the stabilizing PID controllers and guarantee the robustness against uncertainties. Combined with the real part and imaginary part constraints,one-dimensional search technology is employed to find the optimal controllers for minimum integral of time absolute error (ITAE) index. With a simple process transformation to normal model,controller scaling analysis is employed to obtain optimal GPM-PID controller formulas. Simulation results on the liquid level control of coke fractionation tower are provided for the comparisons.

    The remaining parts of this paper are organized as follows. In Section 2,frequency analysis and the problem statement are given for first order plus time delay (FOPTD). In Section 3,optimal GPM-PID design method is proposed with an illustration example. In Section 4,simulation results on the liquid level control of coke fractionation tower are provided for the performance analysis and comparison. Finally,conclusion is given in Section 5.

  • The proposed GMP-PID controller is derived starting from the structure presented in Fig. 1 where $G(s)$ and $G(s)$ are the transfer functions of the process and PID controller,respectively. In the practice,some real-world physical systems can be well characterized by FOPTD

    Figure 1.  Unity output feedback system

    $G(s)=\frac{K}{Ts+1}{{\text{e}}.{-Ls}}$

    (1)

    where L is time delay,K is the model gain and T is the time constant. The proposed PID controller is

    $C(s)={{k}_{p}}+\frac{{{k}_{i}}}{s}+{{k}_{d}}s$

    (2)

    where ${{k}_{p}}$,${{k}_{i}}$ and ${{k}_{d}}$ are controller parameters. In this paper we always adopt negative feedback by requiring positive controller parameters,${{k}_{p}}>0$,${{k}_{i}}\ge 0$,and ${{k}_{d}}\ge 0$. Then,the closed-loop transfer function of the feedback system is given by

    $H(s)=\frac{G(s)C(s)}{1+G(s)C(s)}.$

    (3)

    GPM-PID control scheme is primarily concerned with the robustness of the designed feedback system. Denote the gain margin and phase margin by A and $\phi $,respectively. We use the inequality constraint on the stability margin,i.e.,

    1) Gain margin: \[A\ge A*\],where we set $A.*=2$ as the default value.

    2) Phase margin: $\pi \ge \phi >{{\phi }.{*}}$,where we set ${{\phi }.{*}}={{45}.\circ}$ as the default value.

    The transient performance is measured in terms of ITAE index,i.e.,

    ${{J}_{ITAE}}=\int_{0}.{\infty }{t\left| e(t) \right|\text{dt}}$

    (4)

    where $e(t)=r(t)-y(t)$ is the error signal in Fig. 1. ITAE in (4) can be obtained approximated by a finite B-spline series[15]. In this paper,we adopt ITAE index as transient performance for the optimal GPM-PID controller tuning. The optimal GPM-PID design problem can be formulated as

    $\begin{array}{l} \min {J_{ITAE}} = \int_0.\infty {t\left| {e(t)} \right|{\rm{d}}t} \\ {\rm{st}}.\left\{ \begin{array}{l} {\rm{A}} \ge {\rm{A}}.{\rm{*}}\\ \pi \ge \phi > \phi {\rm{*}}\\ {{\rm{k}}_{\rm{p}}} > {\rm{0}}\\ {{\rm{k}}_i} \ge {\rm{0}}\\ {{\rm{k}}_d} \ge {\rm{0}}. \end{array} \right. \end{array}$

    (5)

    This problem is to compute the stabilizing and optimal GPM-PID controller,such that the resulting feedback system achieves the optimal control performance in time domain with the guaranteed robustness in terms of gain and phase margins.

  • According to the Nyquist stability criterion,the GMP-PID is determined by following two equations

    $G(\text{j}{{\omega }_{p}})C(\text{j}{{\omega }_{p}})=-\frac{1}{A}$

    (6)

    $G(\text{j}{{\omega }_{g}})C(\text{j}{{\omega }_{g}})=-{{\text{e}}.{\text{j}\phi }}$

    (7)

    where ${{\omega }_{p}}$ and ${{\omega }_{g}}$ are the phase and gain crossover frequencies of the loop,respectively. The relationship of these two crossover frequencies can be approximately formulated as ${{\omega }_{p}}\approx A{{\omega }_{g}}$ [16]. For the convenient derivation later,we formulate them by

    ${{\omega }_{p}}=\gamma {{\omega }_{g}}$

    (8)

    where $\gamma >1$.

  • Substituting (2) and (8) into (6) and (7),the real parts can be rewritten as

    ${{k}_{p}}=\operatorname{Re}\left[ -\frac{1}{AG(\text{j}\gamma {{\omega }_{g}})} \right]=\frac{a}{A(a.2+b.2)}$

    (9)

    ${{k}_{p}}=\operatorname{Re}\left[ -\frac{\text{e}.\text{j}\phi }{G(\text{j}{{\omega }_{g}})} \right]=\frac{c\cos \phi +d\sin \phi }{(c.2+d.2)}$

    (10)

    where $a=\operatorname{Re}(-G(\text{j}\gamma {{\omega }_{g}}))$,$b=\operatorname{Im}(-G(\text{j}\gamma {{\omega }_{g}}))$,$c=\operatorname{Re}(-G(\text{j}{{\omega }_{g}}))$,$d=\operatorname{Im}(-G(\text{j}{{\omega }_{g}}))$.

    From (9) and (10),${{k}_{p}}$ is obtained as

    ${{k}_{p}}=\frac{a}{A(a.2+b.2)}=\frac{1}{\sqrt{c.2+d.2}}\sin (\phi +\alpha )$

    (11)

    where

    $\alpha =\left\{ \begin{align} & \arctan \frac{c}{d}\left( c>0,d>0 \right) \\ & \pi -\arctan \frac{c}{\left| d \right|}\left( c>0,d<0 \right) \\ & -\pi +\arctan \frac{\left| c \right|}{\left| d \right|}\left( c<0,d<0 \right) \\ & -\arctan \frac{\left| c \right|}{d}\left( c<0,d>0 \right). \\ \end{align} \right.$

    In (11),there is the relationship between gain margin A and phase margin $\phi $

    $A=\frac{a\sqrt{c.2+d.2}}{\sin (\phi +\alpha )(a.2+b.2)}.$

    (12)

    Since ${{k}_{p}}>0$,$A\ge {{A}.{*}}$ and $\phi \in \left[{{\phi }.{*}},\pi \right)$,(11) and (12) lead to the following inequalities

    $\left\{ \begin{align} & 0<\sin (\phi +\alpha )\le \frac{a\sqrt{c.2+d.2}}{A*(a.2+b.2)},(a>0) \\ & \phi *\le \phi <\pi \\ \end{align} \right.$

    (13)

    which is named as real part constraints (RPC). Note that,the constraints of ${{k}_{p}}>0$ and $A\ge {{A}.{*}}$ have been fully converted to the constraints on $\phi $ in RPC.

    It follows from the imaginary parts of (6) and (7) that

    ${{k}_{d}}\gamma {{\omega }_{g}}-\frac{{{k}_{i}}}{\gamma {{\omega }_{g}}}=\operatorname{Im}\left[ -\frac{1}{AG(\text{j}\gamma {{\omega }_{g}})} \right]=\frac{-b}{A({{a}.{2}}+{{b}.{2}})}$

    (14)

    ${{k}_{d}}{{\omega }_{g}}-\frac{{{k}_{i}}}{{{\omega }_{g}}}=\operatorname{Im}\left[ -\frac{{{\text{e}}.{\text{j}}}\phi }{G(\text{j}{{\omega }_{g}})} \right]=\frac{c\sin \phi -d\cos \phi }{({{c}.{2}}+{{d}.{2}})}$

    (15)

    which are solved to get

    $\left\{ \begin{align} & {{k}_{i}}=-\frac{b\gamma {{\omega }_{g}}\sin (\phi +\alpha )-a{{\omega }_{g}}{{\gamma }.{2}}\cos (\phi +\alpha )}{a\sqrt{{{c}.{2}}+{{d}.{2}}}\left( {{\gamma }.{2}}-1 \right)} \\ & {{k}_{d}}=-\frac{b\gamma \sin (\phi +\alpha )-a\cos (\phi +\alpha )}{a{{\omega }_{g}}\sqrt{{{c}.{2}}+{{d}.{2}}}\left( {{\gamma }.{2}}-1 \right)}. \\ \end{align} \right.$

    (16)

    For the PID controller,since ${{k}_{d}}\ge 0$ and ${{k}_{i}}\ge 0$,the equation (16) can be simplified and converted to

    $\max (\frac{b\gamma }{a},\frac{b}{a\gamma })\le \frac{1}{\tan (\phi +\alpha )}$

    (17)

    which is named as imaginary parts constraints (IPC). Similarly,the constraint on ${{k}_{d}}\ge 0$ and ${{k}_{i}}\ge 0$,have been converted to the constraint on $\phi $in IPC.

    The design problem in (5) can be formulated again to be

    $\begin{array}{l} \min {J_{ITAE}} = \int_0 . \infty t\left| {e(t)} \right|{\rm{d}}t\\ {\rm{st}}.\left\{ \begin{array}{l} {\rm{0}} < \sin (\phi {\rm{ + }}\alpha ) \le \frac{{{\rm{a}}\sqrt {{{\rm{c}}.{\rm{2}}}{\rm{ + }}{{\rm{d}}.{\rm{2}}}} }}{{{\rm{A*}}({{\rm{a}}.{\rm{2}}}{\rm{ + }}{{\rm{b}}.{\rm{2}}})}},({\rm{a}} > {\rm{0}})\\ {\rm{max}}(\frac{{{\rm{b}}\gamma }}{{\rm{a}}},\frac{{\rm{b}}}{{{\rm{a}}\gamma }}) \le \frac{{\rm{1}}}{{\tan (\phi {\rm{ + }}\alpha )}}\\ \phi {\rm{*}} \le \phi < \pi . \end{array} \right. \end{array}$

    (18)

    In this way,the requirements on the gain and phase margins and on the controller parameters in (5) are fully summarized by RPC and IPC. We can find two advantages of PRC and IPC:

    1) The gain margin A and three controller parameters ${{k}_{p}}$,${{k}_{i}}$ and ${{k}_{d}}$ are decoupled from each other. We collect them by

    $\left\{ \begin{align} & A=\frac{a\sqrt{{{c}.{2}}+{{d}.{2}}}}{\sin (\phi +\alpha )({{a}.{2}}+{{b}.{2}})} \\ & {{k}_{p}}=\frac{a}{A({{a}.{2}}+{{b}.{2}})}=\frac{1}{\sqrt{{{c}.{2}}+{{d}.{2}}}}\sin (\phi +\alpha ) \\ & {{k}_{i}}=-\frac{b\gamma {{\omega }_{g}}\sin (\phi +\alpha )-a{{\omega }_{g}}{{\gamma }.{2}}\cos (\phi +\alpha )}{{{a}.{2}}\sqrt{{{c}.{2}}+{{d}.{2}}}\left( {{\gamma }.{2}}-1 \right)} \\ & {{k}_{d}}=-\frac{b\gamma \sin (\phi +\alpha )-a\cos (\phi +\alpha )}{a{{\omega }_{g}}\sqrt{{{c}.{2}}+{{d}.{2}}}\left( {{\gamma }.{2}}-1 \right)} \\ \end{align} \right..$

    (19)

    2) It is obvious that,RPC and IPC are formulated as trigonometric inequalities on $\phi $. With simple trigonometric calculations,the solutions can be presented by $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\phi }({{\omega }_{g}}.,\gamma )<\phi <\bar{\phi }({{\omega }_{g}}.,\gamma )$,which provides great convenience for the optimization in (18).

  • In (18) and (19),four parameters A,${{k}_{p}}$,${{k}_{i}}$ and ${{k}_{d}}$ are determined uniquely by three unknowns $\phi $,${{\omega }_{g}}$ and $\gamma $. When the gain crossover frequency ${{\omega }_{g}}$ and $\gamma $ are fixed,the controller parameters are dependent on phase margin $\phi $ in the trigonometric inequalities. Some discussions are given for such three parameters:

    1) Gain crossover frequency ${{\omega }_{g}}$. Generally,${{\omega }_{g}}$ can be viewed approximately as the closed-loop bandwidth ${{\omega }_{B}}$. A large bandwidth implies less attenuation of the reference signal and a faster response,but indicates more sensitivity to noise,time delay and unmodelled high-frequency dynamics[17].

    2) Parameter $\gamma $. As Skogestad and Postlethwaite[18] pointed out that,“to get a high bandwidth,we want ${{\omega }_{g}}$ therefore ${{\omega }_{p}}$ large,i.e.,$\left| G({{\omega }_{p}})C({{\omega }_{p}}) \right|<1$. So,with ${{\omega }_{g}}$ specified,$\gamma $ can be viewed as a fine tuning on the bandwidth because the phase crossover ${{\omega }_{p}}$ is determined by $\gamma $.

    3) Phase margin $\phi $. With ${{\omega }_{g}}$ and $\gamma $ specified,A is decreasing with $\phi $ increasing in (12) in most cases. On the other hand,a small gain margin allows big amplitude $\left| G(\omega )C(\omega ) \right|$ near ${{\omega }_{p}}$ on the frequency response,such that the signals around ${{\omega }_{p}}$ can easily pass on to the outputs.

  • Suppose the solutions of RPC and IPC are $\Pi \left( \phi ,{{\omega }_{g}},\gamma \right)$,denoted by $\Pi $ for short,which is concerned in the controller design. In this way,we can represent the solutions $\Pi $ in three levels:

    1) The range of phase $\phi $ with ${{\omega }_{g}}=\omega _{g}.{*}$ and $\gamma ={{\gamma }.{*}}$ fixed is denoted by

    $\begin{align} & \Phi (\omega _{g}.{*},\gamma *)= \\ & q\left\{ \left. \phi \right|(\omega _{g}.{*},\gamma *)<\phi <\bar{\phi }(\omega _{g}.{*},\gamma *),\left( \phi ,\omega _{g}.{*},\gamma * \right)\in \Pi \right\}. \\ \end{align}$

    (20)

    2) The range of $\gamma $ with ${{\omega }_{g}}=\omega _{g}.{*}$ fixed is denoted by

    $\Gamma (\omega _{g}.{*})=\left\{ \left. \gamma \right|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma }(\omega _{g}.{*})<\gamma <\bar{\gamma }(\omega _{g}.{*}),\Phi (\omega _{g}.{*},\gamma )\ne \varnothing \right\}.$

    (21)

    3) The range of frequency ${{\omega }_{g}}$ is denoted by

    $\Lambda =\left\{ \left. {{\omega }_{g}} \right|\underline{{{\omega }_{g}}}<{{\omega }_{g}}<\overline{{{\omega }_{g}}},\Gamma ({{\omega }_{g}}.)\ne \varnothing \right\}.$

    (22)

    Then,the optimization problem (19) can be solved progressively in (20) - (22) by one-dimensional search technology. With the optimal value $\left( {{\omega }_{g}},\gamma ,\phi \right)$ found in $\Pi $,optimal GPM-PID controllers are decided by (19) accordingly as well as the achieved gain and phase margins.

  • To obtain the formulas of the proposed PID controller,the FOPDT process (1) is normalized to be

    ${{G}_{n}}(\tilde{s})=\frac{1}{\tilde{s}+1}\text{e}.-\tau \tilde{s}$

    (23)

    and we have $\tilde{s}=Ts,\tau =\frac{L}{T}$ and $G(s)=K{{G}_{n}}(s)$.

    For the normalized process (23),time delay is an important factor that serves limitations on control performance and robustness of the closed-loop system. To cover the various processes,most of the tuning rules for (23) consider three different cases: lag-dominant process with $\tau =0.1$,balanced process with $\tau =1$ and delay-dominant process $\tau =2$ [19-21]. For the simplicity of analysis,the phase ratio of delay-free item $\frac{1}{(\tilde{s}+1)}$ and ${{G}_{n}}(\tilde{s})$ is defined

    $p=\frac{\angle \frac{1}{(\tilde{s}+1)}}{\angle {{G}_{n}}(\tilde{s})}$

    (24)

    to have an insight into the “dominance” of delay. As shown in Fig. 2,when $\tau >1$,the phase-lag of the process is mainly coming from the delay item.

    Figure 2.  Phase ratio

    To explore the effect of the normalized delay on our method,$\tau $ was varied from 0.1 to 2 in our simulation study. Some of the solutions for optimal PID controllers are plotted in Figs. 3-5. To avoid tedious calculation,explicit PID controller

    Figure 3.  Some solutions of optimal GPM-PID

    Figure 4.  The achieved gain, phase margins and ITAE index

    Figure 5.  Crossover frequencies

    ${{C}_{n}}(\tilde{s}):\left\{ \begin{align} & {{k}_{p}}=21.45{{\text{e}}.{-13.06\tau }}+2.399{{\text{e}}.{-0.7769\tau }} \\ & {{k}_{i}}=15.33{{\text{e}}.{-11.97\tau }}+1.892{{\text{e}}.{-\tau }} \\ & {{k}_{d}}=0.3317{{\text{e}}.{0.02842\tau }}-0.1377{{\text{e}}.{-1.46\tau }} \\ \end{align} \right.$

    (25)

    is provided by curves fitting in the least-squares sense for $0<\tau \le 2$,which stands for the most cases of delay processes.

    According to the properties of scaling system[22],we have

    ${{G}_{n}}(\tilde{s}){{C}_{n}}(\tilde{s})=G(s)C(s).$

    (26)

    Then,

    $C(s)=\frac{{{C}_{n}}(Ts)}{K}=\frac{{{k}_{p}}}{K}+\frac{{{k}_{i}}}{TKs}+\frac{{{k}_{d}}T}{K}s.$

    (27)

    These PID formulas are simple and can be easily adopted in the industry.

    lternately,we give other PID controller formulas based on the calculation results above. From Figs. 4 and 5,it is obvious that when the system achieves good transient performance,the phase margin and the parameter $\gamma $ are always kept constant. Approximately,we set

    $60{}.\circ \le \phi \le 70{}.\circ ,3\le \gamma \le 4.$

    (28)

    We also calculate the time delay margin by

    $TM=\frac{\text{0}\text{.0175}\phi ({}.\circ )}{{{\omega }_{g}}}.$

    (29)

    Fig. 6 shows that TM is almost proportional to $\tau $,i.e.,

    Figure 6.  Time delay margin

    $TM\approx 1.732\tau .$

    (30)

    Combining (29) and (30),we have

    ${{\omega }_{g}}=\frac{\text{0}\text{.6871}}{\tau }.$

    (31)

    Based on the simulation study,we give the recommended value $\left( \phi ,{{\omega }_{g}},\gamma \right)$ for (23) in Table 1.

    Table 1.  Recommended value (φ, ωg, γ) and GPM-PID

    Remark 1. Conventional PID controllers could have poor performance when the process exhibits long dead-time since a significant amount of detuning is required to maintain closed-loop stability[23]. For this case,it is convenient to introduce a dead-time compensating structure,such as Smith predictor,by removing the dead-time term from the characteristic equation of the process.

    Remark 2. The proposed optimal GPM-PID is applicable to $0<\tau \le 2$,but not limited. Usually,the proposed method admits any positive value of $\tau $. An example will be given in the next sub-section to show,with a suitable value $\left( \phi ,{{\omega }_{g}},\gamma \right)$ in $\Pi $,a GPM-PID controller can also be determined uniquely by (19) for the long dead-time process.

  • Example 1. To show the validity of optimal GPM-PID controllers,an example is given in Table 2. One can see that the frequency scaling $\tilde{s}=2$ s in (26) leads that both systems in real case and normalized case achieve the same gain and phase margins with ${{\tilde{\omega }}_{p}}=2{{\omega }_{p}}$ and ${{\tilde{\omega }}_{g}}=2{{\omega }_{g}}$. In this way,the requirements on the gain and phase margins for the real process are still suitable for the normalized process.

    Table 2.  Optimal GPM-PID controller for each process

    In Table 2,the scaling controllers are calculated by the formulas in (25) and Table 1. The responses of the real controllers ${{C}_{1}}(s)$,${{C}_{2}}(s)$ and the scaling controllers ${{C}_{1n}}(\tilde{s})$,${{C}_{2}}(\tilde{s})$ are illustrated in Fig. 7. Note that,frequency scaling in (26) leads to time domain scale in the step response,i.e.,${{y}_{r}}(t)={{y}_{n}}(\frac{t}{T})$. We can also find that the resultant systems achieve the same gain and phase margins and similar closed-loop performances.

    Figure 7.  Step responses and time domain scale

    Example 2. Consider the normalized processes with long dead-time

    ${{G}_{i}}(s)=\frac{1}{s+1}{{\text{e}}.{-{{\tau }_{i}}s}}$

    (32)

    where i=1,2,3 and ${{\tau }_{i}}=5,10,15$. Since $\tau >2$,time delay dominates the main behavior of the process,and the optimization problem in (18) is solved to find GPM-PID controller for each process.

    The resultant controllers and the optimal value of $\left( \phi ,{{\omega }_{g}},\gamma \right)$ are given in Table 3,and the step responses are given in Fig. 8. It is interesting to note that,when $\tau >1$,the achieved phase margin almost keeps constant in the proposed method,which is also agreed with by (28). Simulation study shows that,the achieved phase margin can be set as constant value $\phi =65.5$ for long dead-time process in ITAE tuning rule.

    Table 3.  Optimal GPM-PID controllers for long dead-time processes

    Figure 8.  Step responses of long dead-time processes

    Example 3. In this example,load disturbance rejection performance of the closed-loop system is analyzed for the tuning rule (25). The relevant transfer function between the disturbance and output is

    ${{H}_{d}}(s)=\frac{G(s)}{1+C(s)G(s)}.$

    (33)

    Thus,the requirement on the rejection of disturbance is naturally expressed by

    $\max \left| {{H}_{d}}(j\omega ) \right|.$

    (34)

    Based on the loops haping technique,we know that

    $\left| {{H}_{d}}(\text{j}\omega ) \right|\approx \left\{ \begin{matrix} {{\left| C(\text{j}\omega ) \right|}.{-1}} & \text{if}\left| C(\text{j}\omega )G(\text{j}\omega ) \right|\gg 1, & \omega <{{\omega }_{0}} \\ \left| G(\text{j}\omega ) \right|, & \{\text{if}\left| C(\text{j}\omega )G(\text{j}\omega ) \right|\ll 1, & \omega >{{\omega }_{0}} \\ \end{matrix} \right.$

    (35)

    where ${{\omega }_{0}}$ is the cutoff frequency and ${{\omega }_{0}}\approx {{\omega }_{g}}$. It follows that,the disturbance rejection performance is determined by the magnitude of the frequency response $C({\rm j}\omega )$.

    Consider the normalized processes with different time delay ${{\tau }_{i}}=0.1,0.3,0.5,1,1.5$,where $i=1,\cdots ,6$. GPM-PID controller in (25) is used for the disturbance rejection control. The magnitude of the controller $C({\rm j}\omega )$ and the disturbance responses of the resultant systems are given in Figs. 9 and Figure 10,respectively.

    Figure 9.  Magnitude of the frequency response of controller in (25)

    Figure 10.  Disturbance responses

    It is obvious that,the simulation results prove the loops haping analysis above. The controller $C({\rm j}\omega )$ with large magnitude to a small delay process gives small overshoot and settling time. So,one can see that,in the proposed design method,the closed-loop systems provide better disturbance rejection performance for small time delay.

  • To demonstrate the effectiveness of the proposed optimal GPM-PID design method,this section chooses the liquid level of the coke fractionation tower as test. The fractionation tower system and the process flow are given in Fig. 11. It consists of a fractionating tower,coke furnaces and coke towers. This process is widely used to coke the residual oil in the petrochemical engineering.

    Figure 11.  Coke fractionation tower

    Liquid level is an important variable of fractionation tower and associated with the quality of production. One of the targets of the control system is to maintain the process operation at suitable liquid level around the reference value and meet the requirement of the subsequent equipment. The liquid level is controlled by the flow of the residual oil and has integrating behavior. With a proportional controller in the inner loop,the liquid level process is stabilized to be a generalized stable process. First order plus dead time model can be used to describe the dynamics of this process at its steady operating points and identified by step response[24]

    $G(s)=\frac{6.5}{1000s+1}{{\text{e}}.{-250s}}$

    (36)

    which is normalized to be

    $6.5{{G}_{n}}(\tilde{s})=\frac{6.5}{\tilde{s}+1}{{\text{e}}.{-0.4\tilde{s}}}$

    (37)

    with $\tilde{s}=1000s$. The reference of liquid level is set by the operator to be 70% and the disturbance is assumed to occur in the residual oil flow.

    The proposed optimal GPM-PID is obtained directly by (25) and (27). For this process,ZN tuning method in [1],IMC-PID in [25] and tuning method in [26] are used for the comparisons under step response,load disturbance response and random disturbance response.

    Closed-loop responses to the step change from 70 % to 80 % from different methods are shown in Figs. 12 and 13. Note that,since the gain and phase margins are formulated in inequality,it requires the achieved stability margins larger than the lower bounds,and we consider the achieved ITAE index of the step response in the comparison.

    Figure 12.  Closed-loop responses from 70 % to 80 %

    Figure 13.  Overall residual oil flow (liquid level change from 70% to 80%)

    The PID parameters and simulation results are given in Table 4. Observing Table 4,it is obvious that the resultant system with the proposed optimal GPM-PID gives the minimal ITAE index. When $\lambda$ is set to 0.4,the resultant system with $\lambda$-PID achieves the similar step response performance with our method,but its response is still slower than the proposed optimal GPM-PID in the rising up stage.

    Table 4.  Optimal GPM-PID controllers for long dead-time processes

    Load disturbance would occur in the overall residual oil flow if the operation condition is changed. We assume that the actual residual oil flow is increased by 0.8 t/h. In Figs. 14 and 15,one can see that tuning method is sensitive to the load disturbance and gives a big peak value in the responses. However,the liquid level can be well regulated by the proposed optimal GPM-PID with the smallest liquid level variation.

    Figure 14.  Disturbance response with d = 0.8 (t)

    Figure 15.  Overall residual oil flow with disturbance d = 0.1 (t)

    We also consider the random disturbance occurring in the residual oil flow as shown in Figs. 16 and 17. Similar to IMC-PID and $\lambda $-PID methods,the proposed optimal GPM-PID method can regulate the liquid level around 70 %. The liquid level is almost varying between 69 % and 71 % and the overall residual oil flow is between 194 t/h and 196 t/h.

    Figure 16.  Random disturbance response

    Figure 17.  Overall residual oil flow with random disturbance

    Furthermore,we consider the robustness performance of the control system. The stability margin requirements on $A>2$ and $\phi >{{45}.{\circ}}$ are automatically satisfied for the optimal GPM-PID. The robustness performances using the proposed optimal GPM-PID are also illustrated with the three parameters variations (with loop gain $K=8$,time delay $L=300$ or time constant $T=800$). The step responses and disturbance responses for the parameter variations are presented in Figs. 18-20. It should be emphasized that,even with large uncertainty occurred in the process parameters,the proposed optimal GPM-PID still provides the best response performance than other three methods.

    Figure 18.  Step responses with K = 8

    Figure 19.  Step responses with T = 800

    Figure 20.  Step responses with L = 300

  • This paper has for the first time formulated the specified gain and phase margins with a set of constraint conditions,namely,RPC and IPC. This set of new conditions is simply related with three parameters $\left( {{\omega }_{g}},\gamma ,\phi \right)$,so that the controller parameters and the gain margin are decoupled from each other in the nonlinear and coupled characteristic equation,which is one of the advantage of RPC and IPC. Another advantage of RPC and IPC is that they appear as trigonometric inequalities on $\phi $,such that one-dimensional search technology can be used progressively in (20) - (22).

    We have further introduced ITAE index to optimize the transient performance over all the stabilizing GPM-PID controllers. We have also obtained the formulas of optimal GPM-PID based on optimization results and controller scaling analysis. The proposed method is intensively demonstrated with the simulations of liquid level control for coke fractionation tower. Comparisons are made among some common design methods to illustrate our design results.

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