New PI-PD Controller Design Strategy for Industrial Unstable and Integrating Processes with Dead Time and Inverse Response

Abstract

Industrial processes of unstable/integrating nature having a dead time and inverse response characteristics are challenging to control. For controlling such processes, double-loop control structures have proven to be more efficient than conventional PID controllers in a unity feedback configuration. Therefore, a new design method to obtain PI-PD controller settings is proposed for a set of unstable/integrating plant models with dead time and inverse response. The stabilizing proportional–derivative (PD) controller is designed using maximum sensitivity considerations and Routh–Hurwitz stability criteria. The PI controller settings are obtained by comparing the first and second derivatives of expected and actual closed-loop transfer functions about the origin of the s-plane. Adjustable parameters of the inner and outer loops are selected such that the desired value of maximum sensitivity is achieved. Simulation studies are conducted on some benchmark linear and nonlinear plant models used in literature. Robustness of the proposed design is analyzed with perturbed plant models, and quantitative performance measures are computed. It is found that the proposed design yields enhanced and robust closed-loop response than some contemporary works.

Appendices

Appendix A

To start with, $G_{A} \left( s \right)$ given in (16) is considered.

$$ G_{A} \left( s \right) = \frac{{K_{c} P_{t} \left( s \right) \tilde{C}_{1} \left( s \right) }}{{s + K_{c} P_{t} \left( s \right) \tilde{C}_{1} \left( s \right)}} $$

(A1)

For simplifying the computation of derivatives, (A1) is written as

$$ G_{A} \left( s \right) = \frac{{ G_{OL} \left( s \right) }}{{s + G_{OL} \left( s \right)}} $$

(A2)

where $G_{OL} = K_{c} P_{t} \left( s \right) \tilde{C}_{1} \left( s \right)$ is the open-loop transfer function. The first and second derivatives of $G_{OL}$ are obtained as follows:

$$ G_{OL}^{'} \left( s \right) = K_{c} \left[ { \tilde{C}_{1} \left( s \right)P_{t}^{'} \left( s \right) + \tilde{C}_{1}^{'} \left( s \right)P_{t} \left( s \right)} \right] $$

(A3a)

$$ G_{\text{OL}}^{\prime\prime} \left( s \right) = K_{c} \left[ { \tilde{C}_{1} \left( s \right)P_{t}^{\prime\prime} \left( s \right) + \tilde{C}_{1}^{\prime\prime} \left( s \right)P_{t} \left( s \right) + 2\tilde{C}_{1}^{\prime} (s)P_{t}^{\prime} \left( s \right)} \right] $$

(A3b)

From (14) and (15), we get $\tilde{C}_{1} \left( 0 \right) = 1/T_{i}$, $\tilde{C}_{1}^{'} \left( s \right) = 1$ and $\tilde{C}''_{1} \left( s \right) = 0$. Therefore, $\tilde{C}_{1}^{'} \left( 0 \right) = 1$ and $\widetilde{{C^{\prime\prime}}}(0) = 0$. Hence, $G_{OL} \left( 0 \right)$, $G'_{OL} \left( 0 \right)$ and $G''_{OL} \left( 0 \right)$ are given by

$$ G_{OL} \left( 0 \right) = \frac{{K_{c} P_{t} \left( 0 \right)}}{{T_{i} }} $$

(A4a)

$$ G'_{OL} \left( 0 \right) = \frac{{K_{c} }}{{T_{i} }}P'_{t} \left( 0 \right) + K_{c} P_{t} \left( 0 \right) $$

(A4b)

$$ G''_{OL} \left( 0 \right) = \frac{{K_{c} }}{{T_{i} }}P^{\prime\prime}_{t} \left( 0 \right) + 2K_{c} P^{\prime}_{t} \left( 0 \right) $$

(A4c)

At the origin of the s-plane, $G_{d}$ and $G_{A}$ can be expanded using the Maclaurin series as given below:

$$ G_{d} \left( s \right) = G_{d} \left( 0 \right) + s G'_{d} \left( 0 \right) + \frac{{s^{2} }}{2!} G''_{d} \left( 0 \right) + \cdots $$

(A5)

$$ G_{A} \left( s \right) = G_{A} \left( 0 \right) + s G'_{A} \left( 0 \right) + \frac{{s^{2} }}{2!} G''_{A} \left( 0 \right) + \cdots $$

(A6)

Equations (A5) and (A6) are truncated up to the second-order term as it is sufficient to obtain the PI controller settings. The first and second derivatives of (A2) are obtained as follows:

$$ G'_{A} \left( s \right) = \frac{{s G'_{OL} \left( s \right) - G_{OL} \left( s \right)}}{{\left[ {s + G_{OL} \left( s \right)} \right]^{2} }} $$

(A7a)

$$ G''_{A} \left( s \right) = \frac{{\left[ {s + G_{OL} \left( s \right)} \right] sG''_{OL} \left( s \right) - 2\left( {s G^{\prime}_{OL} \left( s \right) - G_{OL} \left( s \right)} \right)(1 + G'_{OL} \left( s \right)}}{{\left[ {s + G_{OL} \left( s \right)} \right]^{3} }} $$

(A7b)

Substituting $s = 0$ in (A7) and using (A4), we get

$$ G'_{A} \left( 0 \right) = \frac{{ - T_{i} }}{{K_{c} P_{t} \left( 0 \right)}} $$

(A8a)

$$ G''_{A} \left( 0 \right) = 2\left[ {\frac{{T_{i} }}{{K_{c} P_{t} \left( 0 \right)}}} \right]^{2} \left( {1 + K_{c} P_{t} \left( 0 \right) + \frac{{K_{c} P^{\prime}_{t} \left( 0 \right)}}{{T_{i} }}} \right) $$

(A8b)

Appendix B

From Fig. 1, we can write

$$ \frac{y\left( s \right) }{r'\left( s \right)} = P_{t} \left( s \right) $$

(B1)

For an impulse input $r'\left( t \right)$, $y\left( s \right) = P_{t} \left( s \right)$. As per the definition of Laplace transform,

$$ y\left( s \right) = \mathop \int \limits_{0}^{\infty } y\left( t \right)e^{ - st} {\text{d}}t $$

(B2)

where $y\left( t \right)$ denotes the impulse response of $P_{t}$. Equation (B2) can be further expanded as

$$ \begin{aligned} Y\left( s \right) & = \mathop \int \limits_{0}^{\infty } y\left( t \right)\left[ {1 - st + \frac{{s^{2} }}{2!}t^{2} - \cdots } \right]{\text{d}}t \\ & = \mathop \int \limits_{0}^{\infty } y\left( t \right){\text{d}}t - s\mathop \int \limits_{0}^{\infty } ty\left( t \right){\text{d}}t + \frac{{s^{2} }}{2!} \mathop \int \limits_{0}^{\infty } t^{2} y\left( t \right){\text{d}}t \\ & - \cdots \\ \end{aligned} $$

(B3)

Maclaurin series expansion of $P_{t} \left( s \right)$ is given by

$$ P_{t} \left( s \right) = P_{t} \left( 0 \right) + sP'_{t} \left( 0 \right) + \frac{{s^{2} }}{2!}P''_{t} \left( 0 \right) + \cdots $$

(B4)

where $P_{t} \left( 0 \right) = P_{t} \left( s \right)_{{{\text{at}}\,s = 0}}$, $P'_{t} \left( 0 \right) = \left[ {\frac{d}{{{\text{d}}s}}P_{t} \left( s \right)} \right]_{at s = 0}$ and $P''_{t} \left( 0 \right) = \left[ {\frac{d}{{{\text{d}}s}}P'_{t} \left( s \right)} \right]_{at s = 0}$. Comparing (B3) and (B4) yields

$$ P_{t} \left( 0 \right) = \mathop \int \limits_{0}^{\infty } y\left( t \right){\text{d}}t $$

(B5)

In general,

$$ P_{t}^{n} \left( 0 \right) = \left( { - 1} \right)^{n} \mathop \int \limits_{0}^{\infty } t^{n} y\left( t \right){\text{d}}t $$

(B6)

The first moment of $y\left( t \right)$ (Marlin 2000) is given by

$$ \mu = \frac{{\mathop \int \nolimits_{0}^{\infty } ty\left( t \right){\text{d}}t}}{{\mathop \int \nolimits_{0}^{\infty } y\left( t \right){\text{d}}t}} = - \frac{{P^{\prime}_{t} \left( 0 \right)}}{{P_{t} \left( 0 \right)}} $$

(B7)

$\mu$ is also called the characteristic time of the process. It can be noted that the equations for $K_{c}$ and $T_{i}$ shown in (19) and (20) are functions of the first moment given by (B7).