June 8, 2023

# What is a model predictive controller?

Model predictive control (MPC) is an advanced control strategy that is commonly used in engineering and industrial applications. It is a mathematical optimization technique that predicts the future behavior of a system and takes control actions based on that prediction. MPC algorithms use a model of the system being controlled and optimize a control sequence over a finite time horizon, subject to constraints on the system inputs and outputs. This approach allows MPC to handle complex and nonlinear systems more effectively than traditional control strategies.

## Understanding the concept of model predictive control

To understand how MPC works, it is helpful to start with the basics. At its core, MPC is a feedback control strategy that uses a model of the system to generate predictions of the future behavior. These predictions are then used to calculate the optimal control action to achieve a desired outcome. The optimization process takes into account the constraints of the system, including physical limitations and performance metrics, to determine the optimal control sequence.

But how does this translate to real-world applications? Let's take an example of an automated car. The car's MPC system uses a mathematical model of the car's dynamics to predict its future behavior, such as its speed, position, and acceleration. Based on these predictions, the system calculates the optimal control action, such as accelerating or braking, to achieve a desired outcome, such as maintaining a safe distance from other vehicles on the road.

Over time, the optimization algorithm can learn more about the system and improve its predictions, leading to better control performance. This means that the car's MPC system can adapt to changing road conditions, such as traffic congestion or slippery roads, to ensure safe and efficient driving.

### The history of model predictive control

The roots of MPC can be traced back to the 1960s, where it was first developed as a control strategy for chemical process plants. It was initially used to control variables such as temperature, pressure, and flow rate, and improve the efficiency of industrial processes.

Since then, MPC has been widely adopted in various fields, including automotive, aerospace, energy management, and robotics. Many industries have recognized the benefits of MPC, and research is continuously being conducted to improve its capabilities and applications.

For example, in the aerospace industry, MPC is used to control the flight of aircraft, including altitude, speed, and direction. In energy management, MPC is used to optimize the operation of power plants and renewable energy systems, such as wind turbines and solar panels.

### Basic principles of model predictive control

MPC is based on a fundamental principle of feedback control - adjusting the control input based on the current state of the system to achieve a desired output. In traditional feedback control, a controller reacts to the current state of the system and makes adjustments to the input. However, in MPC, a model of the system is used to predict future behavior, and the control input is adjusted accordingly.

Let's take the example of a heating system in a building. The MPC system uses a mathematical model of the building's thermal dynamics to predict its future temperature. Based on these predictions, the system calculates the optimal control action, such as adjusting the heating or cooling, to maintain a comfortable temperature for the occupants.

MPC algorithms use a mathematical model of the system to predict future states, and a cost function is used to measure the performance of the control sequence. The optimization process balances the cost function with the constraints of the system, such as input and output limits. The output of the optimization process is the optimal control sequence for the given prediction horizon.

### Key components of model predictive control

There are several key components that make up an MPC system. These include:

• Mathematical model of the system being controlled
• Cost function to evaluate the control sequence
• Optimization algorithm to determine the optimal control sequence
• Prediction horizon to define the length of time over which predictions are made
• Control horizon to define the length of time over which control actions are taken

Each of these components plays a crucial role in the performance and effectiveness of an MPC system. For example, the choice of the prediction horizon and control horizon can affect the trade-off between control performance and computational complexity. A longer prediction horizon can lead to better control performance but requires more computational resources.

Overall, MPC is a powerful control strategy that has been successfully applied in various industries to improve control performance and efficiency. As research continues to advance, we can expect to see even more applications of MPC in the future.

## Types of model predictive controllers

Model predictive control (MPC) is a powerful control technique that uses a model of the system being controlled to predict future behavior and optimize a control sequence. There are several types of MPC controllers, each with its strengths and weaknesses. Some of the most common types of MPC include:

### Linear model predictive controllers

Linear MPC is the most basic type of model predictive control and is suitable for systems with linear dynamics. It uses a linear model of the system and a quadratic cost function to optimize the control sequence. Linear MPC is often used in industrial processes for control of variables such as temperature and pressure.

For example, in a chemical plant, linear MPC can be used to control the temperature of a reactor. The controller uses a model of the reactor's dynamics and a cost function that penalizes deviations from the desired temperature. The controller then calculates a control sequence that minimizes the cost function while keeping the temperature within acceptable limits.

### Nonlinear model predictive controllers

Nonlinear MPC is used when the system being controlled has nonlinear dynamics. It uses a nonlinear model of the system to predict future behavior and optimize the control sequence. Nonlinear MPC is commonly used in automotive control systems and robotics, where the system dynamics are highly nonlinear.

For example, in a self-driving car, nonlinear MPC can be used to control the vehicle's speed and steering. The controller uses a model of the vehicle's dynamics and a cost function that penalizes deviations from the desired speed and heading. The controller then calculates a control sequence that minimizes the cost function while keeping the vehicle on the desired path.

### Robust model predictive controllers

Robust MPC is used when the system being controlled is subject to uncertainties, such as disturbances or measurement errors. It uses a robust optimization algorithm to account for these uncertainties and produce a control sequence that is resilient to disturbances.

For example, in a wind turbine, robust MPC can be used to control the pitch angle of the blades. The controller uses a model of the turbine's dynamics and a cost function that penalizes deviations from the desired power output. The controller then calculates a control sequence that minimizes the cost function while accounting for wind gusts and other disturbances.

Adaptive MPC is used when the system being controlled is subject to changes in behavior over time. It uses an adaptive model of the system that can learn and adapt to changing conditions. Adaptive MPC is often used in energy management, where the system dynamics can change over time due to weather and other factors.

For example, in a smart grid, adaptive MPC can be used to control the power output of a wind farm. The controller uses a model of the wind farm's dynamics and a cost function that penalizes deviations from the desired power output. The controller then calculates a control sequence that minimizes the cost function while adapting to changes in wind speed and direction.

## Applications of model predictive control

Model predictive control (MPC) is a control strategy that uses a mathematical model of the system to predict its future behavior and optimize its performance. MPC has numerous applications in various fields, ranging from industrial process control to robotics and autonomous systems. Some of the most common applications of MPC include:

### Industrial process control

MPC is widely used in industrial processes to control variables such as temperature, pressure, and flow rate. By predicting the behavior of the system and optimizing its performance, MPC can help improve the efficiency of the process and reduce costs. For example, in a chemical plant, MPC can be used to control the temperature and pressure of a reactor, ensuring that the reaction proceeds as efficiently as possible while minimizing the risk of safety hazards.

### Automotive control systems

MPC is used in automotive control systems to regulate various vehicle dynamics such as speed, acceleration, and braking. By predicting the future behavior of the vehicle and optimizing its performance, MPC can aid in improving the overall performance and safety of the vehicle. For example, in an autonomous vehicle, MPC can be used to control the vehicle's speed and trajectory, ensuring that it stays on the road and avoids obstacles.

### Energy management and optimization

MPC is used in energy management to optimize the energy consumption of a system, such as a building or a power plant. By predicting the future energy demand and optimizing the energy supply, MPC can help reduce energy costs and improve the sustainability of the system. For example, in a smart grid system, MPC can be used to control the energy consumption of households and businesses, ensuring that the energy supply is balanced and stable.

### Robotics and autonomous systems

MPC is used in robotics and autonomous systems to control the movement and behavior of the robot or system. By predicting the future behavior of the robot and optimizing its performance, MPC can help improve the accuracy and speed of the robot's movements, as well as its ability to adapt to changing conditions. For example, in a warehouse automation system, MPC can be used to control the movement of robots that transport goods from one location to another, ensuring that the goods are delivered efficiently and on time.

In conclusion, MPC is a powerful control strategy that has numerous applications in various fields. By predicting the future behavior of the system and optimizing its performance, MPC can help improve efficiency, reduce costs, and enhance safety and sustainability. As technology continues to advance, it is likely that the applications of MPC will continue to expand, making it an increasingly important tool for control and optimization.

### Benefits of using model predictive controllers

• Improved control performance in complex and nonlinear systems
• Ability to handle constraints and uncertainties in the system
• Optimization of the control sequence over a finite time horizon
• Flexibility to incorporate additional performance metrics or constraints

### Limitations and challenges in implementing model predictive control

While MPC has many benefits, there are also several challenges and limitations to its implementation, including:

• Complexity and computational requirements of the optimization algorithm
• Difficulty in selecting and tuning the appropriate model for the system
• Sensitivity to modeling errors and uncertainty in the system
• Cost and feasibility of deploying the necessary hardware and software

## Conclusion

Model predictive control is an advanced control strategy that has been widely adopted in various fields due to its ability to handle complex and nonlinear systems. It uses a mathematical optimization algorithm to predict future behavior and optimize the control sequence over a finite time horizon. MPC has numerous applications in industrial process control, automotive control systems, energy management, and robotics. While MPC has many benefits over traditional control strategies, there are also challenges and limitations to its implementation that need to be considered. As technology continues to improve and evolve, the capabilities and applications of MPC are likely to expand and grow in the future.