Multi-input Multi-output Control Research Articles

Optimal Design of I-PD and PI-D Industrial Controllers Based on Artificial Intelligence Algorithm

This research aims to apply Artificial Intelligence (AI) methods, specifically Artificial Immune Systems (AIS), to design an optimal control strategy for a multivariable control plant. Two specific industrial control approaches are investigated: I-PD (Integral-Proportional Derivative) and PI-D (Proportional-Integral Derivative) control. The motivation for using these variations of PID controllers is that they are functionally implemented in modern industrial controllers, where they provide precise process control. The research results in a novel solution to the control synthesis problem for the industrial system. In particular, the research deals with the synthesis of I-P control for a two-loop system in the technological process of a distillation column. This synthesis is carried out using the AIS algorithm, which is the first application of this technique in this specific context. Methodological approaches are proposed to improve the performance of industrial multivariable control systems by effectively using optimization algorithms and establishing modified quality criteria. The numerical performance index ISE justifies the effectiveness of the AIS-based controllers in comparison with conventional PID controllers (ISE1 = 1.865, ISE2 = 1.56). The problem of synthesis of the multi-input multi-output (MIMO) control system is solved, considering the interconnections due to the decoupling procedure.

A new multi-input multi-output control design method for a large-scale web transport system

The paper introduces an innovative multi-input multi-output (MIMO) control design approach tailored for the complex dynamics of large-scale web transport systems in printed electronics manufacturing technology. First, a novel hybrid control design is proposed to a MIMO strict-feedback nonlinear dynamic system, in which a modified tracking error is introduced in term of a sliding surface for design purpose. Then, a combined use of Lyapunov-based control design and sliding mode control is employed to formulate a novel control law, called Backstepping Sliding Mode Control (BSMC) and a BSMC based control structure is provided. Second, application of this control strategy is implemented to control web tension and velocity of a three-span web transport system. In this study, a series of assumptions are provided to transform an original system dynamics of web transport system into a standard strict-feedback form. Then, the proposed theory is applied for the achieved system, a BSMC for the system is synthesized for simulation study. Finally, numerical simulations are conducted under different operating conditions to validate the effectiveness, reliability, and robustness of the proposed control method.

ControlPULP: A RISC-V On-Chip Parallel Power Controller for Many-Core HPC Processors with FPGA-Based Hardware-In-The-Loop Power and Thermal Emulation

High-performance computing (HPC) processors are nowadays integrated cyber-physical systems demanding complex and high-bandwidth closed-loop power and thermal control strategies. To efficiently satisfy real-time multi-input multi-output (MIMO) optimal power requirements, high-end processors integrate an on-die power controller system (PCS). While traditional PCSs are based on a simple microcontroller (MCU)-class core, more scalable and flexible PCS architectures are required to support advanced MIMO control algorithms for managing the ever-increasing number of cores, power states, and process, voltage, and temperature variability. This paper presents ControlPULP, an open-source, HW/SW RISC-V parallel PCS platform consisting of a single-core MCU with fast interrupt handling coupled with a scalable multi-core programmable cluster accelerator and a specialized DMA engine for the parallel acceleration of real-time power management policies. ControlPULP relies on FreeRTOS to schedule a reactive power control firmware (PCF) application layer. We demonstrate ControlPULP in a power management use-case targeting a next-generation 72-core HPC processor. We first show that the multi-core cluster accelerates the PCF, achieving 4.9x speedup compared to single-core execution, enabling more advanced power management algorithms within the control hyper-period at a shallow area overhead, about 0.1% the area of a modern HPC CPU die. We then assess the PCS and PCF by designing an FPGA-based, closed-loop emulation framework that leverages the heterogeneous SoCs paradigm, achieving DVFS tracking with a mean deviation within 3% the plant’s thermal design power (TDP) against a software-equivalent model-in-the-loop approach. Finally, we show that the proposed PCF compares favorably with an industry-grade control algorithm under computational-intensive workloads.

Kinematics, dynamics and control of stiffness-tunable soft robots

Modeling and control methods for stiffness-tunable soft robots (STSRs) have received less attention compared to standard soft robots. A major challenge in controlling STSRs is their infinite degrees of freedom, similar to standard soft robots. In this paper, demonstrate a novel STSR by combing a soft-rigid hybrid spine-mimicking actuator with a stiffness-tunable module. Additionally, we introduce a new kinematic and dynamic modeling methodology for the proposed STSR. Based on the STSR characteristics, we model it as a series of PRP segments, each composed of two prismatic joints(P) and one revolute joint(R). This method is simpler, more generalizable, and more computationally efficient than existing approaches. We also design a multi-input multi-output (MIMO) controller that directly adjusts the pressure of the STSR’s three pneumatic chambers to precisely control its posture. Both the novel modeling methodology and MIMO control system are implemented and validated on the proposed STSR prototype.

Design of 3×3 Multi Input Multi Output (MIMO) Decoupling on Coupled Tank System with PID Controller

Multivariable processes are the general classification for process features in the chemical industry. The variables in a single process unit interact with one another through more than one input and output. Since parameters interact with one another, changing one parameter's variable requires changing another parameter's variable as well. Further research is needed to determine how to create controllers that can handle interactions in the process. A MIMO control system is exemplified by the coupled tank system. Coupled tanks are present, and they communicate with one another by horizontal pipes. An interacting system can be represented using a coupled tank setup. Decoupling can be added to reduce the interactions that take place between variables. According to the results of the simulation, adding decoupling reduces the IAE value compared to not adding it. When the setpoint for tank level 1 (H1) was changed, the system's IAE value was 17.35 without the addition of decoupling, whereas it was 11.71 with it.

A novel MIMO SMC for comprehensive unified control of the cascaded DC‐DC and DC‐AC converters in grid connected photovoltaic systems

SummaryIt is well known that DC‐DC boost converters are cascaded with DC‐AC inverters for grid connection of the photovoltaic (PV) systems. In the traditional control approaches, the mentioned DC‐DC and DC‐AC converters are controlled separately to facilitate the controller design problem. However, from the controller design viewpoint, the overall structure of the grid connected PV generator is a multi‐input multi‐output (MIMO) system. The duty cycle of the DC‐DC converter and inverter modulation index are the control inputs and, on the other hand, generated photovoltaic DC power and exported power to the grid are control outputs. Moreover, the inverter DC link voltage should be stabilized by the closed‐loop controller as well as an internal control output. If controllers are designed separately, it means that the interaction between DC‐DC and DC‐AC controllers isn't considered accurately, and since the isolated models of DC‐DC converter and DC‐AC inverter are extracted based on some approximated assumptions, separate controller design cannot guarantee stability and robustness of the whole system in a wide range of operation. To cope with these problems, in this paper, a novel MIMO sliding mode controller (SMC) is developed for comprehensive closed‐loop control of the DC‐DC boost converter cascaded with a single‐phase DC‐AC grid connected photovoltaic inverter. In the proposed approach, the dynamic model of whole system is developed comprehensively at first, and then, a unique MIMO controller is designed to control both DC‐to‐AC and DC‐to‐DC converters together. To cope with the nonlinear characteristic of the system and uncertainty of model parameters in a wide range, a fixed‐frequency SMC is developed using the comprehensive state space model of the closed loop system. In the proposed MIMO‐SMC controller, the AC power (which is exported to the grid) and operating point of the PV source are controlled via inverter modulation index and duty cycle of the DC‐DC boost converter, respectively. Another major advantage of the proposed system is mitigating the non‐minimum phase characteristic of the boost converter through the indirect control of inverter DC link capacitor. To evaluate the performance of the designed control system, simulation results are compared with a standard linear PI controller. It is shown that the developed system has zero steady‐state error and enjoys faster dynamic response during the start‐up and step changes of AC and DC current references. Moreover, it can maintain the stability of closed‐loop systems in a wide range of operations.

On design of power sharing for VSC‐based islanded AC microgrids: An adaptive MIMO controller equipped with a predictor

AbstractIn an AC microgrid, the active/reactive power is usually shared among its distributed generators (DGs) based on the frequency‐active power () droop and the voltage‐reactive power () droop. By increasing the resistant/inductance ratio () of feeder lines; however, adverse effects of interactions between these two control loops are intensified. In this paper, an adaptive multi‐input multi‐output (MIMO) current control structure is proposed to tackle this problem in AC microgrids with arbitrary numbers of DGs in the primary control level. A deep analysis based on the relative gain array (RGA) matrix and the diagonal dominance concept is provided to systematically design MIMO controllers. The proposed technique is based on the Lyapunov's stability theory, and the asymptotic stability of the whole microgrid is guaranteed. For each DG, the suggested design procedure is started by defining a model reference in which the desired control objectives, including the settling time and the steady‐state error, are considered. Then, a feedback‐feedforward controller is established where its gains are adaptively tuned by some rules derived from a Lyapunov function. Moreover, a predictor is used to estimate the adverse effects of other DGs which are taken into account as external disturbances during the design process of the adaptive controller. By considering some realistic scenarios through time‐domain simulations in MATLAB/SIMULINK, it is shown that the proposed strategy can be successfully used to solve the power sharing problem in AC microgrids.

Systematic design of a multi-input multi-output controller by model-based decoupling: a demonstration on TCV using multi-species gas injection

In this paper, we present the first results of a systematically designed multi-input multi-output gas-injection controller on Tokamak á Configuration Variable (TCV). We demonstrate the simultaneous real-time control of the NII emission front position and line-integrated electron density using nitrogen and deuterium gas injection. Injection of nitrogen and/or deuterium affects both the NII emission front position and line-integrated electron density. This interplay between control loops is termed interaction and, when strongly present, makes designing a controller a significantly more complex problem. Interaction between the control loops can be reduced to an acceptable level by redefining inputs, decoupling the multi-input multi-output control problem to separated single-input single-output problems. We demonstrate how to achieve this by defining virtual control inputs from linear combinations of the actuators available. For the demonstration on TCV, linear combinations of deuterium and nitrogen gas injection are computed from transfer-function models to obtain these virtual inputs. The virtual inputs reduce the interaction in the control-relevant frequency range to a point where control of the NII emission front position and line-integrated electron density can be considered decoupled, allowing for the much simpler design of single-input single-output controllers for each loop. Implementing the controllers with the virtual inputs gives the multi-input multi-output gas-injection controller. This approach is well established in the control community, and is presented here as a demonstration to drive developments of multi-input multi-output control strategies. In particular, the envisioned control of particle- and heat fluxes impacting the divertor targets by injection of multiple gas species.

Cross-Coupled Dynamics and MPA-Optimized Robust MIMO Control for a Compact Unmanned Underwater Vehicle

A compact, 3-degrees-of-freedom (DoF), low-cost, remotely operated unmanned underwater vehicle (UUV), or MicroROV, is custom-designed, developed, instrumented, and interfaced with a PC for real-time data acquisition and control. The nonlinear equations of motion (EoM) are developed for the under-actuated, open-frame, cross-coupled MicroROV utilizing the Newton-Euler approach. The cross-coupling between heave and yaw motion, an important dynamic of a class of compact ROVs that is barely reported, is investigated here. This work is thus motivated towards developing an understanding of the physics of the highly coupled compact ROV and towards developing model-based stabilizing controllers. The linearized EoM aids in developing high-fidelity experimental data-driven transfer function models. The coupled heave-yaw transfer function model is improved to an auto-regressive moving average with exogenous input (ARMAX) model structure. The acquired models facilitate the use of the multi-parameter root-locus (MPRL) technique to design baseline controllers for a cross-coupled multi-input, multi-output (MIMO) MicroROV. The controller gains are further optimized by employing an innovative Marine Predator Algorithm (MPA). The robustness of the designed controllers is gauged using gain and phase margins. In addition, the real-time controllers were deployed on an onboard embedded system utilizing Simulink′s automatic C++ code generation capabilities. Finally, pool tests of the MicroROV demonstrate the efficacy of the proposed control strategy.

On-line automatic controller tuning of a multivariable grinding mill circuit using Bayesian optimisation

Process controllers are abundant in the industry and require attentive tuning to achieve optimal performance. While tuning controllers by the most primitive method of trial and error is possible, it often leads to sub-optimal performance if not conducted by a skilled expert. It is much more appealing to develop an on-line, sample efficient, automated tuner which can optimise the performance of a given controller to the task at hand. The automatic tuning procedure can be conducted during commissioning, when poor controller performance is observed or when process conditions have changed. The problem statement is formulated as the minimisation of an objective function constructed to achieve the desired controller performance. In this context the automatic tuning problem of multi-input multi-output (MIMO) controllers is considered within the framework of Bayesian optimisation and applied in simulation to an ore milling circuit with three manipulated and three controlled variables. Regulatory and set point tracking controllers are tuned automatically and are shown to achieve better performance than a reference controller.

MIMO Multi-Frequency Active Vibration Control for Aircraft Panel Structure Using Piezoelectric Actuators

Undesirable vibration of aircraft panel structures has a significant impact on the safety of their occupants and the operational reliability of various systems. The traditional time-domain controller based on the parallel multi-frequency filtered-x least mean square (FxLMS) control algorithm is simple and effective. However, the mutual influence of convergence steps and the induced harmonic components will limit its performance. An improved parallel multi-input multi-output multi-frequency FxLMS (MIMOMF-FxLMS) algorithm is proposed, considering the multi-input multi-output (MIMO) control requirement of the aircraft panel structure active vibration control (AVC) system and its inevitable channel coupling phenomenon. The feedback error signal and the control channel model are processed by frequency division in the MIMOMF-FxLMS algorithm, so that the FxLMS algorithm of each control channel runs independently, effectively avoiding mutual interference between convergence steps and the generation of induced harmonic components. Simulation and experimental research on multi-frequency AVC of the panel structure model is performed using a new type of piezoelectric stack actuator. The test results show that the proposed controller reduces the vibration level of each observation point by more than 98%, and the vibration responses of the first two-order frequency component are significantly suppressed, which also has strong adaptability and anti-noise ability.

Machine learning-based ethylene and carbon monoxide estimation, real-time optimization, and multivariable feedback control of an experimental electrochemical reactor

Electrochemical reduction of CO2 gas is a novel CO2 utilization technique that has the potential to mitigate the global climate crisis caused by anthropogenic CO2 emissions, and enable the large-scale storage of energy generated from renewable sources in the form of carbon-based chemicals and fuels. However, due to the complexity of the electrochemical reactions, the explicit first-principles models for CO2 reduction are not available yet, and there has been a limited effort to develop process modeling, optimization and control of CO2 electrochemical reactors. To this end, a rotating cylinder electrode (RCE) reactor has been constructed at UCLA to understand the mass transfer and reaction kinetics effects separately on the productivity. In the RCE reactor, the applied potential strongly influences the reaction energetics and the electrode rotation speed affects the hydrodynamic boundary layer and modifies the film mass transfer coefficient, which involves convective and diffusive transport. The present work aims to develop a multi-input multi-output (MIMO) control scheme for the RCE reactor that integrates techniques from artificial and recurrent neural network modeling, nonlinear optimization, and process controller design. Specifically, production rates of two products from the experimental reactor, ethylene and carbon monoxide, are controlled by manipulating two inputs, applied potential and catalyst rotation speed. Process dynamics and controllability are analyzed, a feedback control strategy is designed and the controllers are tuned accordingly. The experimental electrochemical cell is employed to gather data for process modeling and implement the multivariable control system. Finally, the experimental results are presented which demonstrate excellent closed-loop performance by the control system and regulation of the outputs at three different set-points including an economically-optimal set-point.

Multi-Input Multi-Output Sliding-Mode Control of LCL-based Grid-Connected Modified Y-Source Inverter for Power Conditioning of Photovoltaic Generation

Multi-Input Multi-Output Sliding-Mode Control of LCL-based Grid-Connected Modified Y-Source Inverter for Power Conditioning of Photovoltaic Generation

State Feedback Reshaping Control of Voltage Source Converter

Admittance reshaping is a widely used strategy to address the converters low-frequency stability issues in weak grid, caused by the PLL and its interaction with the dc and ac voltage controls. However, the asymmetric control of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> - and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -axis current references and the coupling between the converter ac and dc sides restricts the damping capability of single-input single-output feedbacks. This phenomenon gets even worse in the presence of nearby converters. This article extends the concept of admittance reshaping to multi-input multi-output control. A full state feedback is added to the current reference of the converter to increase the damping of the conventional multiloop control. A systematic offline algorithm is delegated to design the feedback, and a scalar coefficient is employed to activate/deactivate online the reshaping feedback, making the proposed solution user-friendly. The proposed control is analyzed both in time and frequency domains and tested in parallel-operation with other converters, and shows higher damping capability than conventional solutions and good robustness with respect to grid impedance and operating point variations. Experimental tests under ac and dc disturbances are conducted both in lab setup and in hardware-in-the-loop.

Control and Stability Analysis of Current-Controlled Grid-Connected Inverters in Asymmetrical Grids

In symmetrical grids, single-input single-output (SISO) analysis/design tools are applicable for the analysis and design of grid-tied inverter systems as their average <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">αβ</i> - or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> -frame models are to a high extent decoupled (in the high-frequency range). In this way, designing the control system could be done in a straightforward manner. However, this is not the case under asymmetrical grids. In this condition, the interaction between different phases makes the average model of the grid-tied inverter a multi-input multioutput (MIMO) system, and therefore, complicated to analyze and design. To cope with this challenge, this article deals with deriving the MIMO control system considering the impact of the asymmetrical grid. Then, the MIMO system turns into two decoupled SISO systems. Thus, a simple yet accurate model is developed which intuitively demonstrates the effect of the grid asymmetry on creating coupling between current control loops. In this way, the stability analysis could be carried out irrespective of the MIMO system difficulties. The experimental results validate the accuracy of the proposed approach in asymmetrical grid conditions.

Modeling and Flight Control of Small UAV with Active Morphing Wings

In recent research works, morphing wings were studied as an interesting field for a small unmanned aerial vehicle (UAV). The previous studies either focused on selecting suitable material for the morphing wings or performing experimental tests on UAVs with morphing wings. Though, the dynamic modeling of active flexible morphing wings and their involved interactions with the aerodynamics of the UAV body are challenging subjects. Using such a model to control a small UAV to perform specific maneuvering is not investigated yet. The dynamic model of UAV with active morphing wings generates a multi-input multi-output (MIMO) system which rises the difficulty of the control system design. In this paper, the aeroelastic dynamic model of morphing wing activated by piezocomposite actuators is established using the finite element method and modal decomposition technique. Then, the dynamic model of the UAV is developed taking into consideration the coupling between the wing and piezocomposite actuators, as well as the dynamic properties of the morphing actuators with the aerodynamic wind disturbances. A model predictive control (MPC) is designed for the MIMO control system to perform specific flight maneuvering by tracking desired trajectories of UAV altitude and yaw angle. Additionally, the MPC achieves constrained behavior of pitch and roll angles to get satisfactory UAV motion. Also, the behaviors of the UAV control system using MPC are evaluated after adding Dryden wind turbulence to the UAV outputs. Finally, a UAV flight simulation is conducted which shows that the control system successfully rejects the applied disturbances and tracks the reference trajectories with acceptable behavior of pitch and roll angles.

Virtual pre-test analysis for optimization of multi-channel control strategies in direct field acoustic testing

Direct field acoustic excitation testing, or DFAX, is gaining the attention of the space industry as a reliable and flexible method to perform acoustic qualification tests of spacecrafts. The test aims at reproducing the sound conditions of the space launch using an array of high-powered loudspeakers placed around the test specimen. To achieve this end, advanced multi-input multi-output (MIMO) control strategies are employed to design the independent signals which drive the different speakers based on the responses acquired at the control locations over the test volume. Consequently, the performance and overall efficiency of the test campaign is then determined by a number of factors, among them stack configuration, test specimen orientation, number of speakers, control matrix and sensor location. To assist engineers in the design of such tests, this work proposes the use of numerical models to simulate the entire test set-up before the on-site campaign takes place. State-of-the-art vibro-acoustic solvers are employed to predict the pressure responses at all points in the test volume for various design choices, allowing test engineers to perform a virtual pre-test analysis based on numerical solutions to optimize their control strategy. The results included in this work illustrate the methodology followed to account for all the transfer functions of the electro-acoustic system as well as the validation of the levels of accuracy which can be achieved in a full-scale DFAX test via direct comparison with experimental measurements.

DBFact: A better approach to calculate the minimum variance control law for nonminimum phase MIMO systems

AbstractThis paper presents the diagonal with Blaschke products factorization (DBFact) approach to factor out multivariable time delay and nonminimum phase zeros from multi‐input multi‐output (MIMO) systems. Based on that, a new output‐order independent minimum variance (MV) control law for MIMO systems is proposed. The DBFact method is a two‐step factorization procedure, relying on the diagonal and Blaschke factorization methods. This method has the advantage of being a direct and non‐iterative procedure. This new factorization approach allows the calculation of an MV control law considering the multivariable time delay as a limiting‐performance factor and the nonminimum phase zeros and their corresponding directions. Based on the proposed MV control law, a performance benchmark is introduced, which can be calculated by the DBFact filters and routine operating data. The DBFact methodology was applied to two control structures of the linear plant model of Linde's heat integrated air separation, in which the MV control law output‐order dependency property and the suitability of the performance benchmark were evaluated. Some results were compared with those obtained by admitting the generalized interactor matrix instead of the DBFact filters. The results show the capability of the DBFact methodology to factor the nonminimum phase terms to provide a reliable MIMO controller performance benchmark and illustrate the importance of considering the nonminimum phase zeros and their actual directionality in the MV control law.

Synthetic Lyapunov stabilization technique for designing actuation-constrained multi-input multi-output control systems

The main objective of this paper is to present a non-predictive method in the design of nonlinear multi-input multi-output (MIMO) control systems with the presence of constraints that are determinant in practical conditions, namely, the frequency bandwidth limitation of the actuation system and saturation boundaries in control commands. If these constraints are applied in the non-predictive control design problem, it is not possible to simultaneously satisfy Lyapunov stability and actuation constraints, analytically. Instead of model-predictive-based algorithms, which in most cases are computationally expensive, this paper proposes an algorithm based on synthetic Lyapunov stability. In this technique, by defining an intelligent filter applied to the system desired trajectories, defining intelligent proximity coefficients in decoupled inequalities resulting from Lyapunov stability, and determining the admissible boundaries of control commands, a space of regulatory parameters is generated. By appropriately adjusting these parameters based on statistical analysis conducted on the overall dynamics of the system, the Lyapunov stability is guaranteed, and the mentioned control constraints are not violated. In summary, the proposed control algorithm includes the formulation of discrete-time dynamics of sliding functions, the presentation of the procedure of defining and adjusting the control algorithm parameters with the proposed synthetic stability criterion, and the calculation of control inputs based on constraints imposed on the problem. Finally, the algorithm is applied to a cart moving in the X-Y plane, including two rigid cooperative arms that are carrying a load. The most important features of synthetic Lyapunov stability compared to the model predictive-based method are its small computational load and its acceptable performance in satisfying both the Lyapunov stability conditions and determinant control constraints in more realistic situations.

Layer-by-layer model-based adaptive control for wire arc additive manufacturing of thin-wall structures

Improving the geometric accuracy of the deposited component is essential for the wider adoption of wire arc additive manufacturing (WAAM) in industries. This paper introduces an online layer-by-layer controller that operates robustly under various welding conditions to improve the deposition accuracy of the WAAM process. Two control strategies are proposed and evaluated in this work: A PID algorithm and a multi-input multi-output model-predictive control (MPC) algorithm. After each layer of deposition, the deposited geometry is measured using a laser scanner. These measurements are compared against the CAD model, and geometric errors are then compensated by the controller, which generates a new set of welding parameters for the next layer. The MPC algorithm, combined with a linear autoregressive (ARX) modelling process, updates welding parameters between successive layers by minimizing a cost function based on sequences of input variables and predicted responses. Weighting coefficients of the ARX model are trained iteratively throughout the manufacturing process. The performance of the designed control architecture is investigated through both simulation and experiments. Results show that the real-time control performance is improved by increasing the complexity of implemented control algorithm: controlled geometric fluctuations in the test component were reduced by 200% whilst maintaining fluctuations within a 3 mm limit under various welding conditions. In addition, the adaptiveness of designed control strategy is verified by accurately controlling the fabrication of a part with complex geometry.