May 26, 2023

# What is sliding mode control?

In the field of control systems engineering, sliding mode control is a technique that has been gaining increasing attention in recent years. This control methodology is based on the concept of sliding mode, which involves the design of a feedback control law capable of driving the system state onto a constant sliding surface. Once on this sliding surface, the system state is kept in motion on the surface, ensuring tracking of the desired trajectory even in the presence of uncertainties and disturbances. In this article, we will provide an in-depth analysis of what sliding mode control is, its advantages and applications, mathematical foundations, and practical implementation.

## Understanding the basics of sliding mode control

### Definition and principles

Sliding mode control is a feedback control technique that is used to regulate the behavior of a dynamical system by pushing the state of the system onto a predefined sliding surface. The key operating principle is to make the sliding surface attractive for the system trajectory, by using a control law that switches the system dynamics onto the sliding surface. Once on the sliding surface, the system dynamics must remain in motion on it, leading to robustness against system uncertainties and perturbations.

Sliding mode control is particularly useful for systems that are highly nonlinear and have uncertain dynamics. It can also be used for systems with external disturbances that need to be rejected. The technique is widely used in aerospace, automotive, and robotics applications.

### Key components and terminology

The implementation of sliding mode control requires the development of a sliding surface, which is defined as a hypersurface that partitions the state space into two disjoint regions: the sliding region and the non-sliding region. The sliding region is defined as the area where the system dynamic is constrained to be along the sliding surface, while the non-sliding region is where the system dynamics are free to move.

Another important component is the control law, which is the feedback scheme that drives the system dynamics onto the sliding surface. The control law usually switches the system between two different modes of operation: sliding mode and reaching mode. The reaching mode achieves the desired trajectory, and the sliding mode maintains the trajectory on the sliding surface, resulting in a controlled output.

Sliding mode control also involves the use of a switching function, which is a scalar function that changes sign when the system crosses the sliding surface. The switching function is used to design the control law and to determine the stability of the system.

### Historical development

The development of sliding mode control can be traced back to the pioneering work of Russian mathematician Lev Pontryagin in the 1950s. Pontryagin proposed a control scheme based on the discontinuous feedback law to deal with control problems of non-linear systems.

However, the early version of sliding mode control proposed by Pontryagin was unstable due to chattering, a high-frequency oscillation in the control signal, leading to control signal saturation and system instability. In the following decades, several researchers, including Utkin, Young, and Khalil, contributed to the development of sliding mode control techniques that are more robust, and with less chattering.

One of the key breakthroughs in the development of sliding mode control was the introduction of the sliding mode observer, which is a state observer that estimates the state variables of the system and is used to design the control law. The sliding mode observer is particularly useful for systems with unknown or uncertain parameters.

Another important development was the introduction of the super-twisting algorithm, which is a control algorithm that reduces chattering and improves the transient response of the system. The super-twisting algorithm has been widely used in aerospace and robotics applications.

In recent years, sliding mode control has been combined with other control techniques, such as adaptive control and fuzzy control, to further enhance its performance and robustness. The technique continues to be an active area of research and development in the field of control systems engineering.

## Advantages and applications of sliding mode control

### Robustness against uncertainties

One of the main advantages of sliding mode control is its robustness against system uncertainties and perturbations. This control technique is designed to keep the system trajectory on the sliding surface, even in the presence of external disturbances or model uncertainties. The sliding mode behavior ensures that the output of the system is stable, even if the model used for the design of the control law is inaccurate or incomplete.

For example, in the control of unmanned aerial vehicles, sliding mode control can handle the uncertainties caused by wind gusts or turbulence during flight. The controller can adjust the aircraft's attitude and altitude to maintain a stable flight path, even in the presence of these disturbances.

Similarly, in the control of chemical reactors, sliding mode control can handle uncertainties in the reaction kinetics, temperature, and concentration. The controller can adjust the input variables to maintain a desired temperature or concentration, even if the model used for control design is not accurate.

### Chattering reduction

The early version of sliding mode control was characterized by chattering, which made it impractical for the control of many engineering systems. Chattering is a high-frequency oscillation in the control signal, leading to control signal saturation and system instability. However, new techniques have been developed to reduce chattering, such as higher-order sliding mode control, variable structure control, and sliding mode observer. These techniques have made sliding mode control a more practical and attractive option for industrial applications.

For instance, in the control of electric drives, chattering can cause excessive heating and wear in the motor, leading to premature failure. By using higher-order sliding mode control, the control signal can be made smoother, reducing the chattering and improving the motor's reliability.

Similarly, in robotics, chattering can cause erratic motion and instability, making it difficult to achieve precise control. By using sliding mode observer, the controller can estimate the state variables of the robot, reducing the chattering and improving the control accuracy.

### Examples of real-world applications

Sliding mode control has been successfully applied in various real-world applications, such as aerospace engineering, electric drives, robotics, and process control. In aerospace engineering, sliding mode control has been used for the attitude control of spacecraft, flight control of unmanned aerial vehicles, and guidance of missiles. Electric drives such as motor control, power electronics, and renewable energy generation have also benefited from sliding mode control. Other applications include the stabilization of inverted pendulums, the control of quadrotor helicopters, and the regulation of temperature in chemical reactors.

For example, in the control of quadrotor helicopters, sliding mode control can handle the nonlinear and uncertain dynamics of the system, allowing for precise control of the vehicle's position and orientation. This makes it possible to perform complex maneuvers, such as aerial photography, search and rescue, and inspection of infrastructure.

Similarly, in the control of renewable energy generation, sliding mode control can handle the variability and uncertainty of the power source, ensuring a stable and reliable power output. This makes it possible to integrate renewable energy sources into the power grid, reducing reliance on fossil fuels and mitigating climate change.

## Mathematical foundations of sliding mode control

### Sliding surface design

The design of a sliding surface is one of the key components of sliding mode control. The sliding surface is defined as a hypersurface that separates the system state into two disjoint regions: the sliding region and the non-sliding region. The sliding region is the area where the system trajectory is constrained to be along the sliding surface, while the non-sliding region is where the system dynamics are free to move. The design of a sliding surface is based on Lyapunov stability theory and is aimed at ensuring that the system state converges onto the sliding surface.

### Control law formulation

The control law for sliding mode control is based on a discontinuous feedback scheme that is designed to push the system onto the sliding surface and maintain it there. The control law switches the system between two modes: sliding mode and reaching mode. The reaching mode is used to drive the system to the sliding surface, while the sliding mode is used to keep the system trajectory on the sliding surface. A common control law used in sliding mode control is the sign function-based controller, which depends on the sign of the sliding function. Other control laws, such as the sliding mode observer, have also been proposed to reduce chattering and improve the robustness of the sliding mode control.

### Stability analysis

The stability of sliding mode control is analyzed using the Lyapunov stability theory. The stability analysis involves showing that the system dynamics converge onto the sliding surface and remain there. A common criterion used for the stability analysis is the existence of a Lyapunov function, which is a scalar function that satisfies certain properties and is used to prove that the system is stable. The Lyapunov function is used to bound the system state, ensuring that it remains in the sliding region in the presence of uncertainties and disturbances.

## Implementing sliding mode control in practice

### Selecting appropriate parameters

The implementation of sliding mode control in practice requires careful selection of control parameters, such as the sliding surface design, control gain, and switching frequency. The selection of these parameters is crucial to ensure that the sliding mode controller achieves the desired performance, while minimizing chattering and control signal saturation. The tuning of these parameters is usually done through simulation and experimental testing, where the performance of the sliding mode controller is evaluated for different parameter values.

### Dealing with system limitations

The implementation of sliding mode control in practical applications requires dealing with system limitations such as non-linearity, time-varying dynamics, and actuator saturation. The design of the sliding surface should be able to handle the non-linearities and time-varying dynamics of the system. The control law should also be designed to deal with actuator saturation, which could result in chattering and instability in the system.

### Simulation and testing

The implementation of sliding mode control requires extensive simulation and experimental testing. The simulation tests are used to verify the design and tuning of the sliding mode controller, while the experimental testing is used to evaluate the performance of the controller in real-world conditions. The simulation and testing phase also help to identify and correct any errors or limitations in the design of the sliding mode controller.

## Conclusion

Sliding mode control is a robust and effective feedback control technique that is gaining increasing attention in the field of control systems engineering. The design and implementation of sliding mode control require careful consideration of the control parameters, the robustness of the system, and the practical limitations of the system. With its advantages of robustness, chattering reduction, and real-world applications, sliding mode control is a promising avenue for advanced control of non-linear and uncertain systems.