# Extending the Model-Based Controller Design to Higher-Order Plant Models and Measurement Noise

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- well motivated,
- preferably model-based,
- analytically derived,
- simple and easy to remember,
- work well for a variety of processes,
- provide fast tracking speed and good disturbance rejection,
- provide stability and robustness with lower variance of process inputs, and
- reduce sensitivity to measurement noise.

## 2. Exploring the SIMC Method for Different Plant and Dead-Time Approximations

#### 2.1. Controllers Based on FOTD Models

#### 2.2. Controllers Based on SOTD Models

**Remark**

**1**

**Definition**

**1**

#### 2.3. Controller Based on TOTD Models

#### 2.4. Controller Based on QOTD Models

#### 2.5. Why Just the Multiple Plant Time Constants?

**Definition**

**2**

#### 2.6. Low-Pass Noise Attenuation Filters

#### 2.7. Original Half Rule Method

- the largest neglected (denominator) time constant (lag) has been distributed evenly to the effective delay and the smallest retained time constant,
- the effective delay has summarized (besides of above contribution) the original plant delay and different shorter loop delays.

#### 2.8. Modified Half Rule for Multiple Time Constants

**Definition**

**3**

**Remark**

**2**

## 3. Refined Performance Measures

#### 3.1. Ideal Shapes of Step Responses at the Plant Input and Output

**Theorem**

**1**

**Lemma**

**1.**

**Proof.**

**Definition**

**4**

- continuous for $t\in (0,T),T\to \infty $,
- with possible discontinuity at $t={0}^{+}$ and
- with initial value ${u}_{0}=u\left({0}^{-}\right)$ and final steady-state value ${u}_{T}=u\left(T\right)$.

**Definition**

**5**

**Definition**

**6**

#### 3.2. Shape Related Performance Measures for Useful/Excessive Output Increments

#### 3.3. Shape Related Performance Measures for Useful/Excessive Input Increments

**Remark**

**3**

#### 3.4. First Evaluation Step—Idealized Situation with No Noise

#### 3.5. Optimization Problem

- Traditional optimization based on quadratic cost functions (LQ control design) does not distinguish useful and excessive signal increments which significantly limits effectiveness of its application.
- Similarly, the use of TV to evaluate control efforts does not distinguish between useful and redundant increments of control signal. This can cause a problem especially when controlling higher order systems and requiring several active impulses of control.
- Separation of the excessive and useful increments (both at the input and output) enables to focus fully on an effective minimization of the superfluous changes.
- In application to evaluation of the setpoint step responses of the plant output ${y}_{s}$, the modified performance measure $T{V}_{0}\left({y}_{s}\right)$ (42) has a clear mathematical and physical interpretation as a deviation from monotonicity.
- Optimal controller and filter tuning is expected to depend on the noise parameters. Thus, without considering filtration properties, a “generally” optimal PID tuning becomes questionable.

#### 3.6. Speed-Effort and Speed-Wobbling Characteristics

- the shape related deviations at the input or output, that is, the measures expressing, how far are the measured transients from their ideally required shapes (variable $\xi $) and
- IAE measure characterizing the speed of the control error attenuation (variable $\eta $)

#### 3.7. IAE-Optimization-Based Tuning of Noisy FOTD Plants

## 4. Modified Controllers with Reduced Initial Control Signal Peak

**Lemma**

**2**

**Proof.**

**Theorem**

**2**

**Proof.**

**Definition**

**7**

**Definition**

**8**

#### 4.1. Integrative Controllers for the Simplest Pure Dead-Time Plant Models

#### 4.2. Pre-Filter Design for the SIMC PID Controller

## 5. Illustrative Examples

#### 5.1. Example 1: SIMC and Newly Proposed Control of FOTD System with the 2nd Order Low-Pass Filter

#### 5.2. Example 2:SE/SW Based Analysis of Controller + Filter Tuning—No Noise

**Remark**

**4**

#### 5.3. Example 3: SE and SW Characteristics—External Noise

#### 5.4. Example 4: SIMC and Newly Proposed Control of Fourth-Order System, No Noise

**Remark**

**5**

#### 5.5. Example 5: Comparing SIMC PI and PID Controllers with the Newly Proposed Solutions Applied to Fourth-Order System

**Remark**

**6**

#### 5.6. Example 6: SIMC and Newly Proposed Control of QOTD System with 3rd-Order Noise Attenuation Filters

## 6. Discussion: Everything Should Be Made as Simple as Possible, but Not Simpler

- obtaining a perfect process model is frequently associated with trial-and-error approaches enabling to achieve the highest possible match between theoretical and experimentally obtained results (underpinned by appropriate identification results).
- These expectations are usually interpreted in terms of multicriteria cost functions, instead of a single general-purpose cost function and a single all-encompassing optimization.
- Diverse requirements led to the birth of fuzzy control based on the use of linguistically formulated conditions of optimality [45]. However, similar objectives can be easily achieved with simple analytical and modular approaches offering more direct relation to the tuning parameters, especially when they are designed to optimally cover the specific requirements.
- When looking for the optimal solution for a wider class of problems, the price to be paid is a wide range of existing and newly emerging controllers and methods for evaluating them. A simple list of existing solutions (as offered by [5]), with their ever-growing number, may not lead to a simplification and clarity of the situation. From this point of view, it seems more efficient to classify existing solutions into dynamic classes of control [17,39].
- From this perspective, the clear structure, openness, flexibility of adaptation and compatibility with the concept of dynamic classes can be considered as the main advantages of the newly proposed modifications to the original SIMC method.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1P | One-Pulse, response with 2 monotonic segments (1 extreme point) |

2P | Two-Pulse, response with 3 monotonic segments (2 extreme points) |

ART | Average Residence Time |

BIBO | Bounded-Input-Bounded-Output |

FOTD | First-Order Time Delayed |

HO | Higher Order |

HR | Half-Rule |

I | Integrative |

IMC | Internal Model Control |

jOTD | j-Order Time Delayed |

MHR | Modified Half-Rule |

mP | m-Pulse, response with $m+1$ monotonic segments (m extreme points) |

PI | Proportional-Integrative |

PID | Proportional-Integrative-Derivative |

PIDA | Proportional-Integrative-Derivative-Accelerative |

PIDAJ | Proportional-Integrative-Derivative-Acceleration-Jerk |

${}^{j}$R${}^{m}$ | mth-order controller for jth-order stable plant, $m\ge j$ |

${}^{j}$R${}_{n}^{m}$ | mth-order controller for jth-order stable plant combined with nth-order filter ${Q}_{n}\left(s\right)$, $m\ge j,n\ge 0$ |

SE | Speed - Effort |

SIMC | Simple Control/Skogestad IMC |

SPI | SIMC PI controller |

SPID | SIMC PID controller |

SW | Speed - Wobbling |

SOTD | Second-Order Time Delayed |

QOTD | Fourth-Order Time Delayed (with quadruple time constant) |

TOTD | Third-Order Time Delayed |

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**Figure 1.**The controller R and the process F in the closed-loop configuration with a possible control signal limitation; P is the pre-filter, d-disturbance, $\delta $-measurement noise.

**Figure 2.**Unit setpoint step responses of system (33) for ${T}_{c4}=0.18$ with a monotonic increase of the output $y\left(t\right)$ and four pulses (5 monotonic intervals) of the input $u\left(t\right)$, the first interval of $u\left(t\right)$ increase between $t={0}^{-}$ and $t={0}^{+}$ shorten to zero.

**Figure 3.**Unit input disturbance step responses of system (33) for ${T}_{c4}=0.18$ with 1P output $y\left(t\right)$ and monotonic input $u\left(t\right)$.

**Figure 4.**4P control signal response with extreme values ${u}_{1},{u}_{2},{u}_{3}$ and ${u}_{4}$ outlining 5 monotonic intervals.

**Figure 5.**Static feedforward with input disturbance reconstruction and compensation and a hypothetical process dynamics decomposition into the feedforward and feedback path drawn for a FOTD system; δ-measurement noise.

**Figure 6.**Performance of the loops with FOTD plant, SIMC PI ad PID controllers (SPI and SPID) tuned for (79), ${}^{1}$PI (6) and ${}^{1}$PID (12) and ${}^{1}$PIDA (15) controllers tuned for model (81). All controllers are using the 2nd order filter (35) with parameters (82). The performance is calculated for no external noise ($\delta =0$) and for noise amplitudes $\left|\delta \right|\le 0.2$; $K=1$; ${T}_{1m}=1$; ${T}_{m}=1$; ${T}_{s}=0.001$; ${t}_{sim}=12$.

**Figure 9.**SE and SW characteristics of the loop with FOTD plant, ${Q}_{n}\left(s\right)$ (35) with $n\in [1,4]$ and ${}^{1}$PI (6) controller with parameters ${T}_{c1}=\{0.9,1.0,1.1,1.2,1.3\}$ and ${T}_{f0}=0.01$ (black), ${T}_{f0}=0.1$ (red) and ${T}_{f0}=0.2$ (blue) $K=1$; ${T}_{m}=1$; ${T}_{1m}=1$; $\delta =0$; ${T}_{s}=0.001$; ${t}_{sim}=20$

**Figure 10.**SE and SW characteristics of the loop with FOTD plant, ${Q}_{n}\left(s\right)$ (35) with $n\in [1,4]$ and ${}^{1}$PID (12) controller with parameters ${T}_{c1}=\{0.3,0.4,0.5,0.6,0.7\}$ and ${T}_{f0}=0.01$ (black), ${T}_{f0}=0.1$ (red) and ${T}_{f0}=0.2$ (blue), $K=1$; ${T}_{m}=1$; ${T}_{1m}=1$; $\delta =0$; ${T}_{s}=0.001$; ${t}_{sim}=20$.

**Figure 11.**SE and SW characteristics of the loop with FOTD plant, ${Q}_{n}\left(s\right)$ (35) with $n\in [1,4]$ and ${}^{1}$PIDA (15) controller with parameters ${T}_{c1}=\{0.30.4,0.5,0.6,0.7\}$ and ${T}_{f0}=0.01$ (black), ${T}_{f0}=0.1$ (red) and ${T}_{f0}=0.2$ (blue), $K=1$; ${T}_{m}=1$; ${T}_{1m}=1$; $\delta =0$; ${T}_{s}=0.001$; ${t}_{sim}=20$.

**Figure 12.**SE and SW characteristics of the loop with FOTD plant, ${Q}_{n}\left(s\right)$ (35) with $n\in [1,4]$ and with ${}^{1}$PI (6), ${}^{1}$PID (12) and ${}^{1}$PIDA (15) controller with parameters (84), $K=1$; ${T}_{m}=1$; ${T}_{1m}=1$;$\left|\delta \right|\le 0.2$; ${T}_{s}=0.001$; ${t}_{sim}=20$.

**Figure 13.**Setpoint and input disturbance unit step responses of the system (33) by SPI, SPID, ${}^{1}$PI, ${}^{1}$PID, ${}^{2}$PID, ${}^{2}$PIDA, ${}^{3}$PIDA and ${}^{4}$PIDAJ controllers, no noise.

**Figure 15.**SE and SW characteristics corresponding to transients with with SPI controller (red) in Figure 14 correspond mostly to higher IAE values than the transients with ${}^{1}$PI controller (blue), no noise.

**Figure 18.**SE and SW characteristics corresponding to transients in Figure 17 with SPID (red) and ${}^{2}$PID controllers (blue), no noise.

**Figure 19.**Setpoint and input disturbance unit step responses of the system (33) with SPID+${Q}_{2}\left(s\right)$ (red) and ${}^{3}$PIDA+${Q}_{4}\left(s\right)$ controllers (blue), ${T}_{c}$ (96), ${T}_{f}=0.15$, pre-filters (78) and ${}^{3}P\left(s\right)=1/{(1+{T}_{3}s)}^{2}$ and the plant approximations (85) and (91).

**Figure 20.**SE and SW characteristics of SPID and ${}^{3}$PIDA controllers corresponding to transients in Figure 19.

**Figure 21.**Setpoint and input disturbance unit step responses of the system (33) with SPID+${Q}_{2}\left(s\right)$ (red) and ${}^{4}$PIDAJ+${Q}_{4}\left(s\right)$ controllers (blue), ${T}_{c}$ (98), ${T}_{f}=0.15$, pre-filters (78), ${}^{4}{P}^{3}\left(s\right)=1/{(1+{T}_{4}s)}^{3}$ (97) and the plant approximations (85) and (92).

**Figure 22.**SE and SW characteristics of SPID and ${}^{4}$PIDAJ controllers corresponding to transients in Figure 21.

**Figure 23.**Setpoint and input disturbance unit step responses of the system (33) for SPI, SPID, ${}^{1}$PI, ${}^{2}$PID, ${}^{3}$PIDA and ${}^{4}$PIDAJ controllers combined with the noise attenuation filters and tuning parameters for a measurement noise with an amplitude $\left|\delta \right|\le 0.1$ (i.e., up to 10% of the setpoint step) generated in Matlab/Simulink by a Uniform Random Number block.

**Figure 24.**Performance measures corresponding to transients in Figure 23.

Controller | SPI | SPID | ${}^{1}$PI | ${}^{2}$PID | ${}^{3}$PIDA | ${}^{4}$PIDAJ |
---|---|---|---|---|---|---|

$IA{E}_{s}$ | 9.3005 | 6.3096 | 7.2040 | 6.4126 | 5.2836 | 4.7447 |

$T{V}_{4}\left({u}_{s}\right)$ | 3.9871 | 13.5908 | 12.7076 | 1.8372 | 1.8740 | 3.9900 |

$T{V}_{0}\left({y}_{s}\right){10}^{3}$ | 1.3649 | 5.5289 | 3.4414 | 8.7998 | 13.5847 | 22.1266 |

${J}_{1}\left({u}_{s}\right)$ | 37.0818 | 85.7518 | 91.5451 | 11.7814 | 9.9014 | 18.9313 |

${J}_{1}\left({y}_{s}\right){10}^{2}$ | 1.2695 | 3.4885 | 2.4792 | 5.6429 | 7.1777 | 10.4984 |

${J}_{5}\left({u}_{s}\right){10}^{-5}$ | 2.7745 | 1.3591 | 2.4656 | 0.1992 | 0.0772 | 0.0959 |

${J}_{5}\left({y}_{s}\right)$ | 94.9803 | 55.2887 | 66.7736 | 95.4184 | 55.9382 | 53.2052 |

Controller | SPI | SPID | ${}^{1}$PI | ${}^{2}$PID | ${}^{3}$PIDA | ${}^{4}$PIDAJ |
---|---|---|---|---|---|---|

$IA{E}_{d}$ | 7.8032 | 4.8093 | 4.6555 | 3.3136 | 2.8161 | 2.5935 |

$T{V}_{4}\left({u}_{d}\right)$ | 3.9458 | 13.4804 | 12.6906 | 1.7935 | 1.7950 | 3.8461 |

$T{V}_{1}\left({y}_{d}\right){10}^{2}$ | 0.0428 | 2.4693 | 1.9083 | 18.7217 | 8.5841 | 15.8702 |

${J}_{1}\left({u}_{d}\right)$ | 30.7900 | 64.8315 | 59.0811 | 5.9429 | 5.0548 | 9.9748 |

${J}_{1}\left({y}_{d}\right){10}^{2}$ | 0.0334 | 1.1876 | 0.8884 | 6.2037 | 2.4174 | 4.1159 |

${J}_{5}\left({u}_{d}\right){10}^{-5}$ | 1.1416 | 0.3468 | 0.2775 | 0.0072 | 0.0032 | 0.0045 |

${J}_{5}\left({y}_{d}\right)$ | 1.2392 | 6.3532 | 4.1732 | 7.4794 | 1.5203 | 1.8621 |

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**MDPI and ACS Style**

Huba, M.; Vrancic, D.
Extending the Model-Based Controller Design to Higher-Order Plant Models and Measurement Noise. *Symmetry* **2021**, *13*, 798.
https://doi.org/10.3390/sym13050798

**AMA Style**

Huba M, Vrancic D.
Extending the Model-Based Controller Design to Higher-Order Plant Models and Measurement Noise. *Symmetry*. 2021; 13(5):798.
https://doi.org/10.3390/sym13050798

**Chicago/Turabian Style**

Huba, Mikulas, and Damir Vrancic.
2021. "Extending the Model-Based Controller Design to Higher-Order Plant Models and Measurement Noise" *Symmetry* 13, no. 5: 798.
https://doi.org/10.3390/sym13050798