# U-Model-Based Two-Degree-of-Freedom Internal Model Control of Nonlinear Dynamic Systems

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## Abstract

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## 1. Introduction

- Design control systems in a universal procedure, separate two dynamic inversions, invariant controller implementation by inversing specified system performance in a feedback configuration and plant utilization by plant inversion. These two designs are parallel and separable;
- The difference in U-control between linear models and nonlinear models is the solution with the first-order or higher-order polynomial root-solving. The difference in U-control between polynomial models and state-space models is the one-layer or multi-layer polynomial root-solving;

- Propose a general U-model-based Two-Degree-of-Freedom IMC (UTDF-IMC) structure for controlling a class of open-loop stable polynomial/state-space modeled linear and nonlinear dynamic plants. The new control system structure accommodates both linear and nonlinear plants consistently and separate the tracking control filter design from robust control filter design.
- Tailor the UM-dynamic inversion platform [31] in conjunction with IMC, which removes the necessity of either linearizing the nonlinear model, or converting it to a quasi-linear parameter-varying (quasi-LPV) model in advance. This UM-dynamic inversion platform directly provides algorithms dealing with all types of inversions in IMC structured systems.
- Analyze the designed UTDF-IMC control system properties to provide a valid reference for future study expansions and applications.
- Verify the control system performance through benchmark tests of simulated case studies and illustrate application procedure from an industrial case demonstration.

## 2. Preliminaries

#### 2.1. Internal Model Control (IMC)

**Remark 1.**

- Dual stability: For $P={P}_{0}$ and $d=0$, and $y={y}_{m}$, the feedback error signal $e$ is obviously zero. IMC system becomes an open-loop structure and both controller $Q$ and plant $P$ stable.
- Perfect control: This requests plant $P={P}_{0}$ minimum-phase and invertible and controller as the model inverse $Q={P}_{0}{}^{-1}$. Accordingly (2) becomes:$$y=\frac{{P}_{0}{}^{-1}P}{1+{P}_{0}{}^{-1}P-{P}_{0}{}^{-1}{P}_{0}}r+\frac{1-{P}_{0}{}^{-1}{P}_{0}}{1+{P}_{0}{}^{-1}\left(P-{P}_{0}\right)}d=r,\text{}\mathrm{and}\text{}\alpha =1,\beta =0$$
- Augmented robust IMC is shown in Figure 2. It decomposes model and dynamic inversion by factorizing ${P}_{0}$ into ${P}_{0+}$ and ${P}_{0-}$, namely: ${P}_{0}={P}_{0+}{P}_{0-}$, where ${P}_{0+}$ is the part containing pure delay and uncertain zero, and ${P}_{0-}$ is the minimum-phase part. There are certain factorization techniques, such as simple factorization, all-pass factorization [32]. Hence, the controller is kept as the inverse of the plant/process model with invertible portion, i.e.,$${Q}_{1}=\frac{1}{{P}_{0-}}$$

**Remark 2.**

#### 2.2. U-Model-Based Control (U-Control)

#### 2.2.1. U-Models

**Remark 3.**

**Remark 4.**

#### 2.2.2. UM-Dynamic Inversion

#### 2.2.3. U-Control

- Model of ${G}_{p}$ is invertible, i.e., ${G}_{P}^{-1}$ exists
- Meet the continuity of Lipschitz, ${G}_{p}$ and its inverse ${G}_{P}^{-1}$ are globally unified and diffeomorphic in ${\mathbb{R}}^{n}$, i.e.,$$\Vert {G}_{p}\left({x}_{1}\right)-P\left({x}_{2}\right)\Vert \le {\gamma}_{1}{G}_{p}\Vert {x}_{1}-{x}_{2}\Vert ,\text{}\forall {x}_{1},{x}_{2}\in {\mathbb{R}}^{n}\Vert {G}_{P}^{-1}\left({x}_{1}\right)-{G}_{P}^{-1}\left({x}_{2}\right)\Vert \le {\gamma}_{2}\text{}{G}_{P}^{-1}\Vert {x}_{1}-{x}_{2}\Vert ,\text{}\forall {x}_{1},{x}_{2}\in {\mathbb{R}}^{n}$$

- Designed dynamic inverter ${G}_{P}^{-1}$ to achieve ${G}_{P}^{-1}{G}_{P}=1$. This gives $\sum =\left(F,{G}_{c1}\right)$
- Design invariant controller ${G}_{c1}$, which is a typically linear controller. Let the specified closed-loop performance in transfer function $G$, in form of $G=\frac{{G}_{c1}}{1+{G}_{c1}}$, which can be comfortably assigned using damping ratio $\zeta $ and undamped natural frequency ${\omega}_{n}$ for linear system dynamic/steady-state response.

## 3. U-Model-Based Two-Degree-of-Freedom IMC (UTDF-IMC)

#### 3.1. Classical Two-Degree-of-Freedom IMC (TDF-IMC) Structure

#### 3.2. U-Model-Based Two-Degree-of-Freedom IMC (UTDF-IMC) Structure

#### 3.3. UTDF-IMC Design Procedures

- Assume the controlled plant or process $P$ is stable and bounded, and its inverse ${P}^{-1}$ exists. Use U-model to describe $P$ or convert the plant model ${P}_{0}$ into its U-realization ${P}_{u}$. The specific U-modeling process can refer to Section 2.1. In contrast to the classical IMC or classical TDF-IMC, U-realization of the original model ${P}_{0}$ can comfortably cover nonlinear dynamics, therefore, remove linearization restrictions.
- Design filters $f$ and $F$ according to system control performance requirements, then re-optimize the parameters of the filters according to the controller output limit. The feedforward filter determines the system’s set-point tracking ability (response time) while the feedback filter determines the system’s robustness. Because the control system performance is completely designed according to the two filters independently, designers can select the appropriate filters according to performance requirements, hardware limitations, controller output limitations, etc.

#### 3.4. Property Analysis

- Property 1 (Dual stability): Assume the plant model is perfectly matched (${P}_{u}=P$) and system disturbance is absent $d=0$, then from Table 1, the closed-loop stability is characterized by the stability of the plant $P$(${P}^{-1}$) and the feedforward filter $f$. In this case, the system output signal will be: $y=rf$.
- Property 2 (Perfect control): Assume that the dynamic inverter ${P}_{u}{}^{-1}$ is satisfied with ${P}_{u}=P$ and $P$ stable, then the closed-loop system is stable and perfectly controlled. In this case, the system output is $y=rf+\left(1-F\right)d$. The faster the response speed of feedback filter $F$, the better the system robustness.
- Property 3 (Zero offset): Assume that the steady-state gain of the controller equals to steady-state gain of the inverse model, and this closed-loop system is input-output stable with this controller, then offset free control is obtained asymptotically to step or ramp type inputs and disturbances.
- Property 4: Separability of designing the tracking filter and the robust filter: This is shown in the tables, which UTDF-IMC has no product of the two filters $Ff$.

- Classical TDF-IMC structure can make tracking ability and robustness be designed separately but not wholly independent due to the product of Ff in robustness specification. The UTDF-IMC overcomes this shortcoming without resorting to a more complex structure. Therefore, when the robustness performance of the system is determined, UTDF-IMC structure will have a faster response speed than the classical TDF-IMC structure.
- U-model is used to facilitate control system design, which can be easily to form an inversion of the plants to cancel both dynamic and nonlinearities. Accordingly, it converts the nonlinear control system into a linear model-based control with a nonlinear dynamic inverter.
- UM-dynamic inversion algorithm is used to design the inversion part in UTDF-IMC structure, which has a faster convergence speed and allows the inversion part exists alone properly without the feedforward filter.
- This structure where feedforward filter $f$ from outside the control loop allows the tracking ability and robustness performance to be completely independently designed.
- The improved control performance is not complicating the system structure and/or increasing the additional computation burden throughout the design process.

## 4. Simulation Demonstrations

#### 4.1. Linear Internal Model (Also Called Nominal Model in the Study)

- Convert plant model (25) into its corresponding U-model:$$\left\{\begin{array}{c}{P}_{u}\left(s\right):\text{}\ddot{y}=u-3\dot{y}-y={\alpha}_{0}{f}_{0}\left(u\right)+{\alpha}_{1}{f}_{1}\left(u\right)\\ {\alpha}_{0}=-3\ddot{y}\frac{1}{s}-\ddot{y}\frac{1}{{s}^{2}},\text{}{f}_{0}\left({u}^{0}\right)=1\\ {\alpha}_{1}=1,\text{}{f}_{1}\left(u\right)=u\end{array}\right.$$
- Design the inverter of the plant model ${\mathrm{P}}_{u}\left(\mathrm{s}\right)$:$$u=\ddot{y}+3\ddot{y}\frac{1}{s}+\ddot{y}\frac{1}{{s}^{2}}$$
- Design feedforward filter $f\left(s\right)$ and feedback filter $F\left(s\right)$

#### 4.2. Nonlinear Internal Model

- Convert plant model (29) into its corresponding U-model:$$\left\{\begin{array}{c}{\mathrm{P}}_{u}\left(\mathrm{s}\right):\dot{y}={\alpha}_{0}{f}_{0}\left(\dot{u}\right)+{\alpha}_{1}{f}_{1}\left(\dot{u}\right)+{\alpha}_{2}{f}_{2}\left(\dot{u}\right)+{\alpha}_{3}{f}_{3}\left(\dot{u}\right)\\ {\alpha}_{0}=-0.5\dot{y}\frac{1}{s}+{e}^{u},\text{}{f}_{0}\left({\dot{u}}^{0}\right)=1\\ {\alpha}_{1}=1,\text{}{f}_{1}\left(\dot{u}\right)=-\dot{u}\\ {\alpha}_{2}=1,\text{}{f}_{1}\left({\dot{u}}^{2}\right)={\dot{u}}^{2}\\ {\alpha}_{3}=1,\text{}{f}_{1}\left({\dot{u}}^{3}\right)={\dot{u}}^{3}\end{array}\right.$$
- Design the inverter of the plant model ${\mathrm{P}}_{u}\left(\mathrm{s}\right)$:$$u=root\left({\alpha}_{0}{f}_{0}\left(\dot{u}\right)+{\alpha}_{1}{f}_{1}\left(\dot{u}\right)+{\alpha}_{2}{f}_{2}\left(\dot{u}\right)+{\alpha}_{3}{f}_{3}\left(\dot{u}\right)-\dot{y}=0\right)$$

- 3.
- Design feedforward filter $f\left(s\right)$ and feedback filter $F\left(s\right)$

#### 4.3. Control of PMSM System

#### 4.3.1. Modeling of PMSM System

#### 4.3.2. Simulation Test

- IMC: The filter time parameter shown in equation (9) is chosen as $\lambda =0.01$, use linearization to approximate the inverse of PMSM.
- TDF-IMC: Based on the structure in Figure 4, the feedforward filter and feedback filter are chosen as $f=\frac{1}{{\left(1+0.1s\right)}^{\gamma}},\text{}F=\frac{1}{{\left(1+0.01s\right)}^{\gamma}}$, use UM-dynamic inversion to design the inverse of PMSM.
- UTPF-IMC: To test the performance of UTDF-IMC fairly, based on the structure in Figure 5, the feed forward filter and feedback filter are chosen as $f=\frac{1}{{\left(1+0.1s\right)}^{\gamma}},\text{}F=\frac{1}{{\left(1+0.01s\right)}^{\gamma}}$, use UM-dynamic inversion to design the inverse of PMSM.

#### 4.3.3. Matched Model with System Disturbance

#### 4.3.4. Mismatched Model with System Disturbance

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**Simulation results of plant1 (

**a**) System outputs; (

**b**) Tracking errors; (

**c**) Controller outputs.

**Figure 9.**Simulation results of plant2 (

**a**) System outputs; (

**b**) Tracking errors; (

**c**) Controller outputs; (

**d**) Tracking reference.

**Figure 11.**Simulation results with only system disturbance (

**a**) Output angular position ${\theta}_{r}$; (

**b**) Output rotor speed ${\omega}_{r}$; (

**c**) Output current ${i}_{d}$; (

**d**) Controller output voltage ${V}_{d}$; (

**e**) Controller output voltage ${V}_{q}$.

**Figure 13.**Simulation results with modeling error and disturbance (

**a**) Output angular position ${\theta}_{r}$; (

**b**) Output rotor speed ${\omega}_{r}$; (

**c**) Output current ${i}_{d}$; (

**d**) Controller output voltage ${V}_{d}$; (

**e**) Controller output voltage ${V}_{q}$.

$\mathbf{Controller}\text{}\mathbf{Output}\text{}\mathit{u}$ | $\mathbf{System}\text{}\mathbf{Output}\text{}\mathit{y}$ | |
---|---|---|

IMC | $u=\frac{1}{{P}_{0}+\left(P-{P}_{0}\right)f}\left(rf-df\right)$ | $y=\frac{fP}{{P}_{0}+\left(P-{P}_{0}\right)f}r+\frac{{P}_{0}\left(1-f\right)}{{P}_{0}+f\left(P-{P}_{0}\right)}d$ |

TDF-IMC | $u=\frac{1}{{P}_{0}+\left(P-{P}_{0}\right)Ff}\left(rf-dFf\right)$ | $y=\frac{fP}{{P}_{0}+\left(P-{P}_{0}\right)Ff}r+\frac{{P}_{0}\left(1-fF\right)}{{P}_{0}+\left(P-{P}_{0}\right)Ff}d$ |

UTDF-IMC | $u=\frac{1}{{P}_{u}+\left(P-{P}_{u}\right)F}\left(rf-dF\right)$ | $y=\frac{fP}{{P}_{u}+\left(P-{P}_{u}\right)F}r+\frac{{P}_{u}\left(1-F\right)}{{P}_{u}+\left(P-{P}_{u}\right)F}d$ |

where ${P}_{u}$ is the U-realization of ${P}_{0}$ |

$\mathbf{System}\text{}\mathbf{Output}\text{}\mathit{y}$ | |
---|---|

IMC | $y=rf-\left(1-f\right){y}_{e}$ |

TDF-IMC | $y=rf-\left(1-Ff\right){y}_{e}$ |

UTDF-IMC | $y=rf+\left(1-F\right){y}_{e}$ |

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**MDPI and ACS Style**

Li, R.; Zhu, Q.; Narayan, P.; Yue, A.; Yao, Y.; Deng, M.
U-Model-Based Two-Degree-of-Freedom Internal Model Control of Nonlinear Dynamic Systems. *Entropy* **2021**, *23*, 169.
https://doi.org/10.3390/e23020169

**AMA Style**

Li R, Zhu Q, Narayan P, Yue A, Yao Y, Deng M.
U-Model-Based Two-Degree-of-Freedom Internal Model Control of Nonlinear Dynamic Systems. *Entropy*. 2021; 23(2):169.
https://doi.org/10.3390/e23020169

**Chicago/Turabian Style**

Li, Ruobing, Quanmin Zhu, Pritesh Narayan, Alex Yue, Yufeng Yao, and Mingcong Deng.
2021. "U-Model-Based Two-Degree-of-Freedom Internal Model Control of Nonlinear Dynamic Systems" *Entropy* 23, no. 2: 169.
https://doi.org/10.3390/e23020169