# Formal Verification and Co-Simulation in the Design of a Synchronous Motor Control Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background on PVS and Co-Simulation

#### 2.1. The PVS Environment

`THEORY BEGIN`’ and an ‘

`END`name’ statement. A user can define types, constants, variables, and functions. A formula is a named logical statement, and it may be either an axiom (declared as such with the

`AXIOM`keyword) or a theorem (declared with such keywords as

`THEOREM`,

`LEMMA`, etc.). Axioms are taken as proved, while theorems are proved using a large set of prover commands based on the sequent calculus. The PVS language has arithmetic and logic types, and structured types, including record types and predicate subtypes. For example, the specification of a particular motor may include the definition of physical magnitudes, possibly with the acceptable ranges of values:

electric_motor_th: THEORY BEGIN Resistance: TYPE = {x: nonneg_real | 1.5 <= x and x <= 3.5} Inductance: TYPE = nnreal Flux: TYPE = nnreal Current: TYPE = nnreal ... electric_motor: TYPE = [# stator_resistance: Resistance, inductance :Inductance, pole_pairs: nat, flux_intensity: Flux #]

`nnreal`) values, that a resistance takes non-negative values in the range $[1.5,3.5]$, and so on. Type

`Resistance`is an example of a predicate subtype, defined by adding a constraint to its supertype. The concept of electrical motor can then be modeled as a record type containing the parameters of a generic motor. A particular motor is then modeled as an instance of type

`electric_motor`, specifying the values of its parameters:

acmePN123: electric_motor = (# stator_resistance := 3.1, inductance := 0.5, pole_pairs := 4, flux_intensity := 0.85 #)

power(R: Resistance, i1, i2, i3: Current): real = R*((i1*i1) + (i2*i2) + (i3*i3))

power_positive: THEOREM FORALL (R: Resistance, i1, i2, i3: Current): power(R, i1, i2, i3) >= 0 END electric_motor_th

`IF ... ENDIF`clause and the

`COND ... ENDCOND`clause, which is composed of $condition\to expression$ pairs and translates to a set of conditional clauses.

`power_positive`theorem in the theory above is shown below:

power_positive : |------- {1} FORALL (R: Resistance, i1, i2, i3: Current): Power(R, i1, i2, i3) >= 0 Rule? (grind) Power rewrites Power(R, i1, i2, i3) to R*i1*i1 + R*i2*i2 + R*i3*i3 Trying repeated Skolemization, instantiation, and~if-lifting, Q.E.D.

`grind`command, which automatically applies simplifications, using type information. More realistic theorems are proved with longer sequences of commands.

#### 2.2. Co-Simulation

## 3. A Development Process Integrating Verification and Co-Simulation

## 4. Design of a Controller to Reduce Cogging Torque in Brushless Motors

#### 4.1. Mathematical Model

#### 4.2. Simulation Model

#### 4.3. Logic Model

re_lambda_3(K_22, theta: real): real = Phi_1(theta)/cubicrt(K_22/2 + Phi_2(theta) + sqrt((K_22/2 + Phi_2(theta))^2 - Phi_1(theta)^3)) + cubicrt(K_22/2 + Phi_2(theta) + sqrt((K_22/2 + Phi_2(theta))^2 - Phi_1(theta)^3)) - C_6

a(K_22, theta: real): real = K_22/2 + Phi_2(theta) X(K_22, theta: real): real = cubicrt(a(K_22, theta) + sqrt(sq(a(K_22, theta)) - Phi_1(theta)^3)) re_lambda_3_rew(K_22, theta: real): real = Phi_1(theta)/X(K_22, theta) + X(K_22, theta) - C_6

#### 4.4. Verification

`re_lambda_3_rew`above. First, this function is shown to be equivalent to the original definition of $\Re ({\lambda}_{3})$:

lem_1: LEMMA FORALL (K_22, theta: real): re_lambda_3_rew(K_22, theta) = re_lambda_3(K_22, theta)

`quadratic`function defined in the NASALIB library, considering the real part of ${\lambda}_{3}$ as a quadratic function of

`X(k_22, theta)`

quad3: LEMMA FORALL (K_22, theta: real): real_lam3(K_22, theta) <= 0 IFF quadratic(1, -C_6, Phi_1(theta))(X(K_22, theta)) <= 0

`X(k_22, theta)`are found, e.g.,

lem_3: LEMMA FORALL (K_22, theta: real): re_lambda_3_rew(K_22, theta) <= 0 IMPLIES X(K_22, theta) >= b_1_lam3(theta) AND X(K_22, theta) <= b_2_lam3(theta)where

`b_1_lam3(theta)`and

`b_2_lam3`are the roots of the quadratic function. From such relationships, conditions on ${K}_{22}$ are derived.

## 5. Co-Simulation and Design-Space Exploration

`Out1`in the legend represents $\overline{\theta}$ and the line labeled

`Theta_m`represents $\theta $; the label prefixes

`track.trackInstance`and

`plant.plantInstance`are the INTO-CPS names of the signal generator and plant FMUs, respectively.

#### 5.1. Assessment

#### 5.2. Design-Space Exploration

`Theta_des`in the legend), and each dotted line represents $\theta $ for the three values of inductance. The results are qualitatively satisfactory since the differences among the curves are small. Clearly, a quantitative statement of the robustness requirements is needed for a definitive assessment.

- the absorbed power, ${P}_{\mathrm{c}}$:$$\begin{array}{cc}\hfill {P}_{\mathrm{c}}& =\frac{1}{\mathrm{N}}\sum _{k=1}^{\mathrm{N}}({U}_{k}{I}_{k})=\frac{1}{\mathrm{N}}\sum _{k=1}^{\mathrm{N}}({U}_{\mathrm{a},k}{I}_{\mathrm{a},k}+{U}_{\mathrm{b},k}{I}_{\mathrm{b},k}+{U}_{\mathrm{c},k}{I}_{\mathrm{c},k})\hfill \end{array}$$
- the mean square deviation of the tracking error on the angular position, ${E}_{\theta}$$$\begin{array}{cc}\hfill {E}_{\theta}& =\frac{1}{\mathrm{N}}\sum _{k=1}^{\mathrm{N}}{({\theta}_{k}-\overline{\theta})}^{2}\phantom{\rule{1.em}{0ex}}\hfill \end{array}$$
- the mean square deviation of the tracking error on the current’s direct component ${E}_{{i}_{d}}$$$\begin{array}{cc}\hfill {E}_{{i}_{d}}& =\frac{1}{\mathrm{N}}\sum _{k=1}^{\mathrm{N}}{({i}_{\mathrm{d},k}-{\overline{i}}_{\mathrm{d}})}^{2}\hfill \end{array}$$

- ${\mathrm{K}}_{11}$ has been fixed at $-1000$ because this variable has a low impact on the system behavior, as shown in a previous work [3].
- ${\mathrm{K}}_{22}$ takes the values $-\mathrm{250,000}$ and $-\mathrm{200,000}$ in order to study the behavior with two values, one inside and one outside the interval found in Section 4.4.
- the angular position takes the values of 10, 20 and 30 radians.
- The inductance L takes the values $42.5$, 50, and $57.5$ mH, i.e., the nominal value and small variations.

## 6. Conclusions

- using the methodology described in the article, one could think of formally analyzing the global state of a mechatronic systems, with a view to safety, mapping the operating states directly with PVS and analyzing the final effects of the unpredictable variations of the parameters via DSE, as well as verify the robustness of the entire dynamic system through co-simulation
- the development of a generalized and more automatic procedure can be used to analyze and verify conditions for the safety of the dynamic system and its users, which is essential in the field of automation and industrial robotics and in the motor vehicle industry, where often the development of a safe and efficient system requires a large number of field tests, which have significant costs
- a systematic development process based on formal methods and co-simulations in the early phases of system design allows reducing risks in the physical prototyping phase.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Block diagram of the equation for the electromagnetic equilibrium of the k-th stator phase.

Symbol | Value | Parameter |
---|---|---|

$\mathrm{Z}$ | 10 | number of stator teeth |

$\mathrm{p}$ | 3 | number of pole pairs |

${\mathrm{T}}_{1}$ | 4.0 $\mathrm{N}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | amplitude of cogging torque’s first harmonic |

${\alpha}_{1}$ | 0.009 rad | phase of cogging torque’s first harmonic |

$\mathrm{R}$ | 3.3 $\Omega $ | resistance |

$\mathrm{L}$ | 0.05 H | inductance |

$\mathrm{k}$ | 0.5 Wb | magnetic flux |

$\mathrm{J}$ | 0.01 $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ | rotational inertia |

$\beta $ | 0.01 $\mathrm{N}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{s}/\mathrm{m}$ | friction coefficient |

Variable | Quantity |
---|---|

${i}_{\mathrm{a}},{i}_{\mathrm{b}},{i}_{\mathrm{c}}$ | a-, b-, c-components of current |

${u}_{\mathrm{a}},{u}_{\mathrm{b}},{u}_{\mathrm{c}}$ | a-, b-, c-components of voltage |

${e}_{\mathrm{a}},{e}_{\mathrm{b}},{e}_{\mathrm{c}}$ | a-, b-, c-components of electromotive force |

${i}_{\mathrm{d}},{i}_{\mathrm{q}}$ | direct and quadrature components of current |

${u}_{\mathrm{d}},{u}_{\mathrm{q}}$ | direct and quadrature components of voltage |

$\theta ,\omega $ | angular position and velocity |

${\overline{\u0131}}_{\mathrm{d}},\overline{\theta}$ | desired values of ${i}_{\mathrm{d}}$ and $\theta $ |

${T}_{\mathrm{em}},{T}_{\mathrm{cog}}$ | electromagnetic and cogging torques |

${v}_{1},{v}_{2}$ | control voltages |

Coefficient | Value | Coefficient | Value |
---|---|---|---|

${\mathrm{C}}_{1}$ | 4040 | ${\mathrm{C}}_{2}$ | 1,237,529 |

${\mathrm{C}}_{3}$ | 9/1000 | ${\mathrm{C}}_{4}$ | 153,666,659/90,000 |

${\mathrm{C}}_{5}$ | 1,597,813,728,139/27,000,000 | ${\mathrm{C}}_{6}$ | 6371/300 |

Rank | ${\mathit{P}}_{\mathbf{c}}$ | ${\mathit{E}}_{{\mathit{i}}_{\mathbf{d}}}$ | L | $\overline{\mathit{\theta}}$ | ${\mathbf{K}}_{22}$ |
---|---|---|---|---|---|

1 | 133.225808994 | 0.4520951516 | 0.0425 | 10 | −200,000 |

2 | 182.338023217 | 0.6207469189 | 0.05 | 10 | −200,000 |

3 | 203.174600076 | 0.7698643094 | 0.0425 | 10 | −250,000 |

4 | 238.510251169 | 0.8146819387 | 0.0575 | 10 | −200,000 |

5 | 284.764456886 | 1.056688425 | 0.05 | 10 | −250,000 |

6 | 389.645017067 | 1.393178146 | 0.0575 | 10 | −250,000 |

7 | 390.725315887 | 2.4496935013 | 0.0425 | 20 | −200,000 |

8 | 536.217360079 | 3.1270257044 | 0.05 | 20 | −200,000 |

9 | 606.28853597 | 3.9126048439 | 0.0425 | 20 | −250,000 |

10 | 681.333349658 | 3.9590184538 | 0.0575 | 20 | −200,000 |

11 | 764.927467024 | 5.9664482285 | 0.0425 | 30 | −200,000 |

11 | 792.274533863 | 5.0855301421 | 0.05 | 20 | −250,000 |

12 | 991.117882891 | 7.5735576985 | 0.05 | 30 | −200,000 |

12 | 1034.82060285 | 6.5027556699 | 0.0575 | 20 | −250,000 |

13 | 1146.74688913 | 9.5474224247 | 0.0425 | 30 | −250,000 |

13 | 1244.20686295 | 9.3341197085 | 0.0575 | 30 | −200,000 |

14 | 1474.30527198 | 12.1749251909 | 0.05 | 30 | −250,000 |

15 | 1844.30406114 | 15.0481912736 | 0.0575 | 30 | −250,000 |

Rank | ${\mathit{P}}_{\mathbf{c}}$ | ${\mathit{E}}_{\mathit{\theta}}$ | L | $\overline{\mathit{\theta}}$ | ${\mathbf{K}}_{22}$ |
---|---|---|---|---|---|

1 | 133.225808994 | 3.0012689667 | 0.0425 | 10 | −200,000 |

1 | 182.338023217 | 2.8554161212 | 0.05 | 10 | −200,000 |

1 | 203.174600076 | 2.6613697359 | 0.0425 | 10 | −250,000 |

1 | 284.764456886 | 2.5666163063 | 0.05 | 10 | −250,000 |

1 | 389.645017067 | 2.5096108736 | 0.0575 | 10 | −250,000 |

2 | 238.510251169 | 2.6658599693 | 0.0575 | 10 | −200,000 |

3 | 390.725315887 | 13.5604089712 | 0.0425 | 20 | −200,000 |

3 | 536.217360079 | 13.3507337866 | 0.05 | 20 | −200,000 |

3 | 606.28853597 | 12.5238422201 | 0.0425 | 20 | −250,000 |

3 | 792.274533863 | 12.0083838635 | 0.05 | 20 | −250,000 |

4 | 681.333349658 | 13.0097947735 | 0.0575 | 20 | −200,000 |

4 | 1034.82060285 | 12.0973710284 | 0.0575 | 20 | −250,000 |

5 | 764.927467024 | 33.6920496 | 0.0425 | 30 | −200,000 |

5 | 991.117882891 | 32.6884214852 | 0.05 | 30 | −200,000 |

5 | 1146.74688913 | 31.0693366505 | 0.0425 | 30 | −250,000 |

5 | 1474.30527198 | 30.6858006581 | 0.05 | 30 | −250,000 |

6 | 1244.20686295 | 32.611319147 | 0.0575 | 30 | −200,000 |

6 | 1844.30406114 | 31.0435733541 | 0.0575 | 30 | −250,000 |

Rank | ${\mathit{E}}_{\mathit{\theta}}$ | ${\mathit{E}}_{{\mathit{i}}_{\mathbf{d}}}$ | L | $\overline{\mathit{\theta}}$ | ${\mathbf{K}}_{22}$ |
---|---|---|---|---|---|

1 | 2.5096108736 | 1.393178146 | 0.0575 | 10 | −250,000 |

1 | 2.5666163063 | 1.056688425 | 0.05 | 10 | −250,000 |

1 | 2.6613697359 | 0.7698643094 | 0.0425 | 10 | −250,000 |

1 | 2.8554161212 | 0.6207469189 | 0.05 | 10 | −200,000 |

1 | 3.0012689667 | 0.4520951516 | 0.0425 | 10 | −200,000 |

2 | 2.6658599693 | 0.8146819387 | 0.0575 | 10 | −200,000 |

3 | 12.0083838635 | 5.0855301421 | 0.05 | 20 | −250,000 |

3 | 12.5238422201 | 3.9126048439 | 0.0425 | 20 | −250,000 |

3 | 13.3507337866 | 3.1270257044 | 0.05 | 20 | −200,000 |

3 | 13.5604089712 | 2.4496935013 | 0.0425 | 20 | −200,000 |

4 | 12.0973710284 | 6.5027556699 | 0.0575 | 20 | −250,000 |

4 | 13.0097947735 | 3.9590184538 | 0.0575 | 20 | −200,000 |

5 | 30.6858006581 | 12.1749251909 | 0.05 | 30 | −250,000 |

5 | 31.0693366505 | 9.5474224247 | 0.0425 | 30 | −250,000 |

5 | 32.611319147 | 9.3341197085 | 0.0575 | 30 | −200,000 |

5 | 32.6884214852 | 7.5735576985 | 0.05 | 30 | −200,000 |

5 | 33.6920496 | 5.9664482285 | 0.0425 | 30 | −200,000 |

6 | 31.0435733541 | 15.0481912736 | 0.0575 | 30 | −250,000 |

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**MDPI and ACS Style**

Bernardeschi, C.; Dini, P.; Domenici, A.; Palmieri, M.; Saponara, S. Formal Verification and Co-Simulation in the Design of a Synchronous Motor Control Algorithm. *Energies* **2020**, *13*, 4057.
https://doi.org/10.3390/en13164057

**AMA Style**

Bernardeschi C, Dini P, Domenici A, Palmieri M, Saponara S. Formal Verification and Co-Simulation in the Design of a Synchronous Motor Control Algorithm. *Energies*. 2020; 13(16):4057.
https://doi.org/10.3390/en13164057

**Chicago/Turabian Style**

Bernardeschi, Cinzia, Pierpaolo Dini, Andrea Domenici, Maurizio Palmieri, and Sergio Saponara. 2020. "Formal Verification and Co-Simulation in the Design of a Synchronous Motor Control Algorithm" *Energies* 13, no. 16: 4057.
https://doi.org/10.3390/en13164057