# Controller Design for Unstable Time-Delay Systems with Unknown Transfer Functions

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## Abstract

**:**

## 1. Introduction

_{2}PID controller from an internal model control design for unstable processes with a single right-half-plane pole and a time delay. Hast and Hägglund [12] proposed a method for finding the set-point weights of a PID controller by using convex optimization techniques. Begum et al. [13] developed analytical tuning rules for PID controllers for unstable first order plus dead time processes. Chakraborty et al. [14] proposed an integral–proportional derivative (I-PD) control strategy for integrator plus time delay processes. The proposed scheme comprised an inner proportional derivative (PD) loop and an outer integral loop for controlling a servo and for regulatory actions. Verma and Padhy [15] studied the effects of exponential weights on square-of-error functions for PID control systems. They demonstrated that the performance of any process can be tailored by applying exponential weights in the objective function. Onat [16] presented a graphical method for tuning the proportional integral (PI)-PD controller parameters for unstable time-delay systems. The proposed method used a new concept, namely the centroid of the convex stability region. The method only requires the coordinates of a few special points related to the curve determining the stable controller parameter area. Zhang et al. [17] proposed an optimized robust control algorithm based on the mirror mapping method for a class of industrial unstable processes with time delays. Raja and Ali [18] proposed tuning rules for the PI-PD control structure for a class of unstable time-delay processes. The proposed method required the tuning of only four controller parameters.

## 2. System Description

- PID controller

- I-PD controller

- PI-PD controller

## 3. Design of Controller Parameters

- Integral of absolute error (IAE)$$J={{\displaystyle \int}}_{0}^{{T}_{f}}\left|e\left(t\right)\right|dt$$
- Integral of squared error (ISE)$$J={{\displaystyle \int}}_{0}^{{T}_{f}}{e}^{2}\left(t\right)dt$$
- Integral of time-weighted absolute error (ITAE)$$J={{\displaystyle \int}}_{0}^{{T}_{f}}t\left|e\left(t\right)\right|dt$$
- Integral of time-weighted squared error (ITSE)$$J={{\displaystyle \int}}_{0}^{{T}_{f}}t{e}^{2}\left(t\right)dt$$

## 4. Simulation Results

#### 4.1. Case 1

#### 4.1.1. PID Controller

#### 4.1.2. I-PD Controller

#### 4.1.3. PI-PD Controller

#### 4.2. Case 2

#### 4.2.1. PID Controller

#### 4.2.2. I-PD Controller

#### 4.2.3. PI-PD Controller

#### 4.3. Case 3

#### 4.3.1. PID Controller

#### 4.3.2. I-PD Controller

#### 4.3.3. PI-PD Controller

#### 4.4. Summary of Cases 1 to 3

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**System block diagram of parallel 2-DOF PID control including the external disturbance $d$, measurement noise $v$, and low-pass filter ${C}_{3}\left(s\right)$.

**Figure 3.**Diagram of various possible candidate points during each of N–M simplex iteration: (

**a**) linear search; (

**b**) shrink.

**Figure 4.**Time responses after these four sets of initial PID controller parameters were used to control the system.

**Figure 5.**Time response of System (26) controlled by the PID controller. Gains were found using the N–M simplex method based on the initial vertices given in Table 4.

**Figure 7.**Time response of System (26) controlled by the I-PD controller with gains obtained using the N–M simplex method based on the initial vertices given in Table 5.

**Figure 8.**Time responses after these five sets of initial PI-PD controller gains were used to control System (26).

**Figure 9.**Time response of System (26) controlled by the PI-PD controller. The controller parameters were obtained using the N–M simplex method with the initial vertices given in Table 6.

**Figure 10.**Output responses of System (30) controlled by the four sets of PID parameters listed in Table 7.

**Figure 11.**Time response of System (30) controlled by the PID controller with gains obtained using the N–M simplex method and based on the initial vertices given in Table 7.

**Figure 12.**Time responses of System (30) controlled by the four sets of initial I-PD controller parameters listed in Table 8.

**Figure 13.**Time response of System (30) controlled by the I-PD controller with gains obtained by the N–M simplex method and based on the initial vertices given in Table 8.

**Figure 14.**Time responses using these five sets of initial PI-PD controller gains to control System (30).

**Figure 15.**Time response of System (30) controlled by the PI-PD controller. Controller parameters were obtained with the N–M simplex method using the initial vertices given in Table 9.

**Figure 16.**Output responses of System (34) controlled by the initial PID gains listed in Table 10.

**Figure 17.**Time response of System (34) controlled by the PID controller with gains obtained using the N–M simplex method and based on the initial vertices given in Table 10.

**Figure 18.**Time responses of System (34) controlled by the four sets of initial I-PD controller parameters listed in Table 11.

**Figure 19.**Time response of System (34) controlled by the I-PD controller with gains obtained using the N–M simplex method based on the initial vertices given in Table 11.

**Figure 20.**Time responses using these five sets of initial PI-PD controller gains to control System (34).

**Figure 21.**Time response of System (34) controlled by the PI-PD controller with controller parameters obtained using the N–M simplex method with initial vertices as in Table 12.

**Table 1.**Definitions and descriptions of the variables and symbols in the proposed performance index.

Name | Definition and Description |
---|---|

$r\left(t\right)$ | System reference input (i.e., $r$ in Figure 2). |

$y\left(t\right)$ | The actual output of the system (i.e., $y$ in Figure 2). The allowable range of change is ${y}_{min}\le y\le {y}_{max}$, where ${y}_{min}$ and ${y}_{max}$ are the lower and upper limits of the output allowable change, respectively. If $y$ exceeds the allowable range, the experiment or simulation is halted immediately. |

$e\left(t\right)$ | The output error (i.e., $e$ in Figure 2) is defined as $e\left(t\right)=r\left(t\right)-y\left(t\right)$. |

$d$ | The disturbance to the system (i.e., $d$ in Figure 2). |

${T}_{f}$ | The start time of each simulation is 0. The simulation end time is ${T}_{f}$. |

${T}_{d}$ | The disturbance $d$ is added to the system at time ${T}_{d}$, and the disturbance continues until ${T}_{f}$. |

${W}_{p}$ | The weight of the penalty function. ${W}_{p}$ must be sufficient to punish (or cause to avoid) situations in which the output y exceeds the allowable range. |

Name | Parameter | Equation | Performance Index |
---|---|---|---|

Reflection | $\sigma =\alpha =1$ | ${p}_{r}=\overline{p}+\Delta p$ | ${J}_{r}=J\left({p}_{r}\right)$ |

Expansion | $\sigma =\gamma =2$ | ${p}_{e}=\overline{p}+2\Delta p$ | ${J}_{e}=J\left({p}_{e}\right)$ |

Outer contraction | $\sigma =\beta =0.5$ | ${p}_{c}=\overline{p}+0.5\Delta p$ | ${J}_{c}=J\left({p}_{c}\right)$ |

Inner contraction | $\sigma =-\beta =-0.5$ | ${p}_{cc}=\overline{p}-0.5\Delta p$ | ${J}_{cc}=J\left({p}_{\mathrm{cc}}\right)$ |

Shrink | ${p}_{i}=0.5\left({p}_{0}+{p}_{i}\right)$ $i=1,\cdots ,N$ | ${J}_{i}=J\left({p}_{i}\right)$ $i=1,\cdots ,N$ |

Main Program | |

Step 1 | Choose an initial ${p}_{0},\dots ,{p}_{N}$, and execute Subroutine to obtain ${J}_{0},\dots ,{J}_{N}$. |

Step 2 | Sort ${p}_{0},\dots ,{p}_{N}$ such that ${J}_{0}\le \dots \le {J}_{N}$. |

Step 3 | If the termination criteria are satisfied, go to Step 20. |

Step 4 | Let $\overline{p}=\frac{1}{N}{{\displaystyle \sum}}_{i=0}^{N-1}{p}_{i}$ and $\Delta p=\overline{p}-{p}_{N}$. |

Step 5 | (Reflection) Let ${p}_{r}=\overline{p}+\Delta p$ and execute Subroutine to obtain ${J}_{r}$. |

Step 6 | If ${J}_{r}<{J}_{0}$, go to Step 7; else go to Step 11. |

Step 7 | (Expansion) Let ${p}_{e}=\overline{p}+2\Delta p$ and execute Subroutine to obtain ${J}_{e}$. |

Step 8 | If ${J}_{e}<{J}_{r}$, go to Step 9; else go to Step 10. |

Step 9 | Let ${p}_{N}={p}_{e}$ and ${J}_{N}={J}_{e}$, then go to Step 2. |

Step 10 | Let ${p}_{N}={p}_{r}$ and ${J}_{N}={J}_{r}$, then go to Step 2. |

Step 11 | If ${J}_{r}<{J}_{N}$, go to Step 12; else go to Step 16. |

Step 12 | (Outer contraction) Let ${p}_{c}=\overline{p}+0.5\Delta p$ and execute Subroutine to obtain ${J}_{c}$. |

Step 13 | If ${J}_{c}<{J}_{r}$, go to Step 14; else, go to Step 15. |

Step 14 | Let ${p}_{N}={p}_{c}$ and ${J}_{N}={J}_{c}$, then go to Step 2. |

Step 15 | Let ${p}_{N}={p}_{r}$ and ${J}_{N}={J}_{r}$, then go to Step 2. |

Step 16 | (Inner contraction) Let ${p}_{cc}=\overline{p}-0.5\Delta p$ and execute Subroutine to obtain ${J}_{cc}$. |

Step 17 | If ${J}_{cc}<{J}_{N}$, go to Step 18; else, go to Step 19. |

Step 18 | Let ${p}_{N}={p}_{cc}$ and ${J}_{N}={J}_{cc}$, then go to Step 2. |

Step 19 | (Shrink) Let ${p}_{i}=0.5\left({p}_{0}+{p}_{i}\right)$ and execute Subroutine to obtain ${J}_{i}$ where $i=1,\dots ,N$, then go to Step 2. |

Step 20 | Print out ${p}_{0}$ and ${J}_{0}$. |

Subroutine | |

Simulate or experiment using the given parameter $p$ to obtain the corresponding index $J$. |

**Table 4.**Initial PID controller parameters of System (26) and their corresponding performance indexes sorted according by the performance indexes.

$i$ | ${p}_{i}=\left[\begin{array}{ccc}{K}_{P}& {K}_{I}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{ccc}6& 12& 0.8\end{array}\right]$ | ${J}_{0}=7.8990\times {10}^{6}$ |

1 | ${p}_{1}=\left[\begin{array}{ccc}2& 8& 1.0\end{array}\right]$ | ${J}_{1}=8.6430\times {10}^{6}$ |

2 | ${p}_{2}=\left[\begin{array}{ccc}4& 10& 1.2\end{array}\right]$ | ${J}_{2}=8.6670\times {10}^{6}$ |

3 | ${p}_{3}=\left[\begin{array}{ccc}8& 6& 0.2\end{array}\right]$ | ${J}_{3}=9.3660\times {10}^{6}$ |

**Table 5.**Four sets of initial I-PD controller parameters and their corresponding performance indexes for System (26).

$i$ | ${p}_{i}=\left[\begin{array}{ccc}{K}_{I}& {K}_{P}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{ccc}7& 6& 0.5\end{array}\right]$ | ${J}_{0}=5.6286\times {10}^{-1}$ |

1 | ${p}_{1}=\left[\begin{array}{ccc}3& 7& 0.5\end{array}\right]$ | ${J}_{1}=1.6343\times {10}^{0}$ |

2 | ${p}_{2}=\left[\begin{array}{ccc}5& 2& 0.2\end{array}\right]$ | ${J}_{2}=5.3141\times {10}^{1}$ |

3 | ${p}_{3}=\left[\begin{array}{ccc}15& 6& 0.2\end{array}\right]$ | ${J}_{3}=6.0510\times {10}^{6}$ |

**Table 6.**Initial PI-PD controller gains and their corresponding performance indexes for the system (26).

$i$ | ${p}_{i}=\left[\begin{array}{cc}{K}_{C}& {K}_{I}\end{array}\begin{array}{cc}{K}_{P}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{cc}3& 2\end{array}\begin{array}{cc}0.5& 0.5\end{array}\right]$ | ${J}_{0}=6.2384\times {10}^{-1}$ |

1 | ${p}_{1}=\left[\begin{array}{cc}3& 10\end{array}\begin{array}{cc}0.8& 0.3\end{array}\right]$ | ${J}_{1}=1.9547\times {10}^{0}$ |

2 | ${p}_{2}=\left[\begin{array}{cc}5& 10\end{array}\begin{array}{cc}0.4& 1.0\end{array}\right]$ | ${J}_{2}=1.2092\times {10}^{1}$ |

3 | ${p}_{3}=\left[\begin{array}{cc}5& 8\end{array}\begin{array}{cc}1.0& 1.0\end{array}\right]$ | ${J}_{3}=5.1920\times {10}^{6}$ |

4 | ${p}_{4}=\left[\begin{array}{cc}3& 5\end{array}\begin{array}{cc}2.5& 2.5\end{array}\right]$ | ${J}_{4}=9.1870\times {10}^{6}$ |

**Table 7.**Four sets of initial PID controller parameters used by System (30) and their corresponding performance indexes.

$i$ | ${p}_{i}=\left[\begin{array}{ccc}{K}_{P}& {K}_{I}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{ccc}2.0& 0.1& 2\end{array}\right]$ | ${J}_{0}=1.0225\times {10}^{2}$ |

1 | ${p}_{1}=\left[\begin{array}{ccc}1.5& 0.2& 4\end{array}\right]$ | ${J}_{1}=1.9751\times {10}^{2}$ |

2 | ${p}_{2}=\left[\begin{array}{ccc}1.0& 0.2& 3\end{array}\right]$ | ${J}_{2}=2.5865\times {10}^{8}$ |

3 | ${p}_{3}=\left[\begin{array}{ccc}2.5& 0.1& 5\end{array}\right]$ | ${J}_{3}=4.2925\times {10}^{8}$ |

**Table 8.**Four sets of initial I-PD controller parameters and their corresponding performance indexes for System (30).

$i$ | ${p}_{i}=\left[\begin{array}{ccc}{K}_{I}& {K}_{P}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{ccc}0.5& 3& 3\end{array}\right]$ | ${J}_{0}=3.2987\times {10}^{1}$ |

1 | ${p}_{1}=\left[\begin{array}{ccc}2.0& 5& 2\end{array}\right]$ | ${J}_{1}=2.8433\times {10}^{3}$ |

2 | ${p}_{2}=\left[\begin{array}{ccc}1.0& 1& 4\end{array}\right]$ | ${J}_{2}=2.9420\times {10}^{8}$ |

3 | ${p}_{3}=\left[\begin{array}{ccc}1.5& 7& 1\end{array}\right]$ | ${J}_{3}=4.2865\times {10}^{8}$ |

**Table 9.**Initial PI-PD controller gains and their corresponding performance indexes for System (30).

$i$ | ${p}_{i}=\left[\begin{array}{cc}{K}_{C}& {K}_{I}\end{array}\begin{array}{cc}{K}_{P}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{cc}0.1& 1.0\end{array}\begin{array}{cc}3& 1.0\end{array}\right]$ | ${J}_{0}=1.1415\times {10}^{1}$ |

1 | ${p}_{1}=\left[\begin{array}{cc}1.0& 2.0\end{array}\begin{array}{cc}4& 2.0\end{array}\right]$ | ${J}_{1}=4.6450\times {10}^{7}$ |

2 | ${p}_{2}=\left[\begin{array}{cc}0.2& 2.0\end{array}\begin{array}{cc}2& 2.0\end{array}\right]$ | ${J}_{2}=2.2615\times {10}^{8}$ |

3 | ${p}_{3}=\left[\begin{array}{cc}0.5& 0.5\end{array}\begin{array}{cc}5& 4.0\end{array}\right]$ | ${J}_{3}=3.8950\times {10}^{8}$ |

4 | ${p}_{4}=\left[\begin{array}{cc}2.0& 3.0\end{array}\begin{array}{cc}1& 0.5\end{array}\right]$ | ${J}_{4}=4.7660\times {10}^{8}$ |

**Table 10.**Four sets of initial PID controller parameters used by System (34) and their corresponding performance indexes.

$i$ | ${p}_{i}=\left[\begin{array}{ccc}{K}_{P}& {K}_{I}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{ccc}2.0& 0.1& 1\end{array}\right]$ | ${J}_{0}=1.0739\times {10}^{2}$ |

1 | ${p}_{1}=\left[\begin{array}{ccc}1.5& 0.2& 2\end{array}\right]$ | ${J}_{1}=2.0745\times {10}^{8}$ |

2 | ${p}_{2}=\left[\begin{array}{ccc}1.0& 0.2& 3\end{array}\right]$ | ${J}_{2}=4.5060\times {10}^{8}$ |

3 | ${p}_{3}=\left[\begin{array}{ccc}0.5& 0.1& 4\end{array}\right]$ | ${J}_{3}=4.5880\times {10}^{8}$ |

**Table 11.**Four sets of initial I-PD controller parameters and their corresponding performance indexes for System (34).

$i$ | ${p}_{i}=\left[\begin{array}{ccc}{K}_{I}& {K}_{P}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{ccc}0.2& 3& 0.3\end{array}\right]$ | ${J}_{0}=3.8643\times {10}^{1}$ |

1 | ${p}_{1}=\left[\begin{array}{ccc}0.5& 2& 0.2\end{array}\right]$ | ${J}_{1}=1.4464\times {10}^{2}$ |

2 | ${p}_{2}=\left[\begin{array}{ccc}0.4& 4& 0.1\end{array}\right]$ | ${J}_{2}=3.3170\times {10}^{8}$ |

3 | ${p}_{3}=\left[\begin{array}{ccc}0.3& 1& 0.4\end{array}\right]$ | ${J}_{3}=3.9800\times {10}^{8}$ |

$i$ | ${p}_{i}=\left[\begin{array}{cc}{K}_{C}& {K}_{I}\end{array}\begin{array}{cc}{K}_{P}& {K}_{D}\end{array}\right]$ | ${J}_{i}=J\left({p}_{i}\right)$ |

0 | ${p}_{0}=\left[\begin{array}{cc}0.1& 1.0\end{array}\begin{array}{cc}3.0& 1\end{array}\right]$ | ${J}_{0}=1.4203\times {10}^{1}$ |

1 | ${p}_{1}=\left[\begin{array}{cc}0.4& 1.0\end{array}\begin{array}{cc}1.5& 2\end{array}\right]$ | ${J}_{1}=1.5060\times {10}^{8}$ |

2 | ${p}_{2}=\left[\begin{array}{cc}0.2& 2.0\end{array}\begin{array}{cc}2.5& 2\end{array}\right]$ | ${J}_{2}=1.5095\times {10}^{8}$ |

3 | ${p}_{3}=\left[\begin{array}{cc}0.5& 0.5\end{array}\begin{array}{cc}1.0& 1\end{array}\right]$ | ${J}_{3}=2.1820\times {10}^{8}$ |

4 | ${p}_{4}=\left[\begin{array}{cc}0.3& 1.5\end{array}\begin{array}{cc}2.0& 3\end{array}\right]$ | ${J}_{4}=4.5125\times {10}^{8}$ |

System | Controller | Performance |
---|---|---|

Case 1: $P\left(s\right)=\frac{1}{s-1}{e}^{-0.2s}$ | PID | $J=0.32556$ |

I-PD | $J=0.52350$ | |

PI-PD | $J=0.11277$^{#} | |

Case 2: $P\left(s\right)=\frac{1}{\left(2s-1\right)\left(0.5s+1\right)}{e}^{-0.5s}$ | PID | $J=7.9370$ |

I-PD | $J=8.1780$ | |

PI-PD | $J=2.1582$^{#} | |

Case 3: $P\left(s\right)=\frac{-0.4s+1}{\left(3s-1\right)\left(0.2s+1\right)}{e}^{-0.5s}$ | PID | $J=38.623$ |

I-PD | $J=15.254$ | |

PI-PD | $J=5.5301$^{#} |

**Relative minimum of the performance indexes obtained for a system with three different controllers.**

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## Share and Cite

**MDPI and ACS Style**

Tsai, H.-H.; Fuh, C.-C.; Ho, J.-R.; Lin, C.-K.; Tung, P.-C.
Controller Design for Unstable Time-Delay Systems with Unknown Transfer Functions. *Mathematics* **2022**, *10*, 431.
https://doi.org/10.3390/math10030431

**AMA Style**

Tsai H-H, Fuh C-C, Ho J-R, Lin C-K, Tung P-C.
Controller Design for Unstable Time-Delay Systems with Unknown Transfer Functions. *Mathematics*. 2022; 10(3):431.
https://doi.org/10.3390/math10030431

**Chicago/Turabian Style**

Tsai, Hsun-Heng, Chyun-Chau Fuh, Jeng-Rong Ho, Chih-Kuang Lin, and Pi-Cheng Tung.
2022. "Controller Design for Unstable Time-Delay Systems with Unknown Transfer Functions" *Mathematics* 10, no. 3: 431.
https://doi.org/10.3390/math10030431