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Mathematical modeling is critical for engineers to understand and accurately predict the behavior of all types of complex systems. MATLAB is one of the most popular object-oriented programming languages used by engineers and researchers for performing mathematical operations and building mathematical models by applying the concepts of linear algebra and matrices. In fact, all the data entered into MATLAB as input is perceived by the program only as a matrix. For example, when a number is entered, MATLAB will perceive it as a matrix of rows and columns. Therefore, it contains many functions that are inbuilt and embedded for matrix operations such as matrix addition and multiplication. Mathematical models created with MATLAB allow users to perform critical tasks such as optimizing and forecasting system behavior, designing advanced control systems, and characterizing the response of those systems.

Other than linear algebra and matrix functions, MATLAB provides tools such as calculus, optimization, statistics, curve fitting, and ODE and PDE solving to support the development of mathematical models using both symbolic and numeric modeling approaches. Using Simulink as a complementary program to MATLAB allows users to add a programming environment to build embedded systems and model and simulate the behavior of various types of multi-domain systems.

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MATLAB and **Simulink** programming provide a range of advanced tools to develop models for virtually all types of systems including linear and nonlinear, static and dynamic, deterministic and stochastic, and continuous and discrete systems. Depending on specific requirements, users can choose from a variety of modeling environments which enable them to describe their system symbolically, programmatically, or with state machines and block diagrams. MATLAB provides many benefits to users when they create models using data-driven and scientific-based (such as physics) principles. Key benefits of using scientific and data principles include:

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- Allowing for virtual commissioning of a system.
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- Computational finance including economic forecasting, risk estimation and portfolio optimization.
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- Power-train calibration and modeling.

When physical insights are available to users, they can create mathematical models by applying first principles using symbolic or analytic approaches. However, in cases where sufficient amount and types of information are not available, techniques of data-based modeling become especially useful. Accuracy of the model in such cases can be ensured by selecting a modeling technique which is suitable for the historical or experimental data being used. MATLAB tools for statistical curve fitting allow users to explore and understand the relationships that exist or may develop in the future among the data. Relationships can be explored using surface fitting, linear and nonlinear models, clustering and classification tools. System identification and neural network techniques provided by MATLAB can be used to build dynamic models that help in expressing the effects of past experiences of a system on both its current and expected future behavior. Data-based techniques also allow users to adjust the coefficients of their models built using first principles. This helps them in fitting their experimental data using response optimization and grey-box modeling techniques.

Users can select from several approaches that MATLAB supports for the creation of first principles-based mathematical models. Using MATLAB allows them to:

- Derive analytical models and equations which describe a system by using the symbolic computing technique.
- Create block diagrams that can be used to demonstrate complex multi-domain systems.
- Apply finite element methods to systems that have been described using partial differential equations.

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Once the models have been developed in the MATLAB environment, users can exercise them under multiple conditions, visualize and manage the results of simulation, and optimize their fidelity. MATLAB also provides tools that can be used for documenting the work and sharing the models with other users.

After building the models, users can optimize their chosen parameters and validate the models vis-à-vis the behavior of the actual system. Optimization tools in MATLAB allow users to optimize the design of a new system or refine the model of any existing system by making adjustments to the design variables in order to fulfill the given performance criteria.

Simulations in MATLAB are used to predict how a system will behave under different conditions and to validate a model by comparing the test data with simulation results. Using MATLAB tools makes it quick and easy for users to manage each aspect of simulating a model. Users can:

- Define conditions of the simulation using tools such as probability distribution, design of experiments (DOE) and test vectors.
- Run their simulations using parallel computing and advanced numeric solvers.
- Post-process the simulation results using MATLAB’s visualization, data management and data analysis capabilities.

Reporting tools provided in MATLAB and Simulink programming can be used to automatically document simulation results and model derivation steps and update them with the design. Deployment tools designed for desktop and the web can be used to share the optimized models and related applications with several users simultaneously.

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