🔄Nonlinear Control Systems Unit 6 – Sliding Mode Control

Key Concepts and Fundamentals

• Sliding mode control (SMC) is a robust nonlinear control technique that can handle uncertainties and disturbances in the system
• SMC aims to drive the system states onto a predefined sliding surface and maintain the states on the surface thereafter
• The sliding surface is a hyperplane in the state space, designed based on the desired system dynamics and performance specifications
• Once the system reaches the sliding surface, it exhibits reduced-order dynamics and becomes insensitive to matched uncertainties and disturbances
• The control law in SMC consists of two components: the equivalent control and the switching control
  • The equivalent control maintains the system on the sliding surface once reached
  • The switching control drives the system towards the sliding surface
• SMC can handle systems with nonlinearities, uncertainties, and external disturbances, making it suitable for a wide range of applications (robotics, automotive systems, power electronics)
• The main advantages of SMC include robustness, fast response, and the ability to handle constraints and nonlinearities in the system

Mathematical Foundations

• SMC relies on the concept of variable structure systems (VSS), where the system structure changes based on the state feedback
• The sliding surface is defined as a linear combination of the system states, represented by $s(x) = 0$
• The sliding surface is designed to ensure the desired system dynamics and convergence properties
• The equivalent control is derived using the invariance condition $\dot{s}(x) = 0$, which ensures the system remains on the sliding surface once reached
• The switching control is designed to satisfy the reaching condition $s(x)\dot{s}(x) < 0$, which guarantees the system converges to the sliding surface
• The signum function $\text{sgn}(s(x))$ is commonly used in the switching control to provide a discontinuous control action
• Lyapunov stability theory is employed to analyze the stability of the closed-loop system and ensure convergence to the sliding surface
• The concept of equivalent dynamics describes the system behavior on the sliding surface, which is of reduced order and independent of matched uncertainties

System Modeling for Sliding Mode Control

• Accurate system modeling is crucial for the design and implementation of SMC
• The system model should capture the essential dynamics, nonlinearities, and uncertainties of the plant
• Consider a general nonlinear system represented by $\dot{x} = f(x) + g(x)u + d(x, t)$, where $x$ is the state vector, $u$ is the control input, and $d(x, t)$ represents uncertainties and disturbances
• The system model can be transformed into the regular form, separating the states into two components: the sliding variable and the remaining states
• The regular form facilitates the design of the sliding surface and the control law
• Uncertainties and disturbances can be modeled as bounded functions or as perturbations to the system parameters
• The concept of matched and unmatched uncertainties is important in SMC
  • Matched uncertainties are those that enter the system through the same channel as the control input and can be directly compensated by the control action
  • Unmatched uncertainties cannot be directly compensated and require additional techniques (higher-order sliding modes, adaptive control)

Sliding Surface Design

• The sliding surface is a crucial element in SMC, as it determines the desired system dynamics and performance
• The sliding surface is designed as a linear combination of the system states, $s(x) = c^T x$, where $c$ is a vector of design parameters
• The design parameters are chosen to ensure the desired convergence properties and system behavior on the sliding surface
• The sliding surface can be designed based on various criteria, such as pole placement, linear quadratic regulator (LQR), or model reference adaptive control (MRAC)
• The relative degree of the sliding variable with respect to the control input should be one to ensure the existence of a sliding mode
• The sliding surface should be designed to minimize the reaching phase, during which the system converges to the sliding surface
• The sliding surface can incorporate additional objectives, such as tracking error minimization or constraints on the system states or control input
• In some cases, multiple sliding surfaces can be designed to handle different objectives or to achieve a hierarchical control structure

Control Law Formulation

• The control law in SMC consists of two components: the equivalent control and the switching control
• The equivalent control is derived using the invariance condition $\dot{s}(x) = 0$, assuming the system is already on the sliding surface
• The equivalent control maintains the system on the sliding surface and compensates for the known dynamics and uncertainties
• The switching control is designed to drive the system towards the sliding surface and satisfy the reaching condition $s(x)\dot{s}(x) < 0$
• The switching control typically includes a discontinuous term, such as the signum function $\text{sgn}(s(x))$, to provide a high-frequency switching action
• The discontinuous term in the switching control can be replaced by continuous approximations (saturation, hyperbolic tangent) to reduce chattering
• The control law can incorporate additional terms, such as a proportional-integral (PI) term, to improve the system performance and robustness
• The control gains in the switching control should be chosen to ensure a fast reaching phase and robustness against uncertainties and disturbances
• The control law can be modified to handle input constraints, such as saturation or rate limits, using techniques like control allocation or anti-windup

Stability Analysis

• Stability analysis is essential to ensure the convergence of the system to the sliding surface and the desired performance
• Lyapunov stability theory is commonly used to analyze the stability of SMC systems
• A Lyapunov function candidate, such as $V(s) = \frac{1}{2}s^2$, is chosen to represent the energy of the system
• The time derivative of the Lyapunov function, $\dot{V}(s)$, is evaluated along the system trajectories
• The reaching condition $s(x)\dot{s}(x) < 0$ ensures that the Lyapunov function decreases along the system trajectories, guaranteeing convergence to the sliding surface
• The equivalent control is assumed to maintain the system on the sliding surface, resulting in reduced-order dynamics
• The stability of the reduced-order dynamics on the sliding surface is analyzed separately, often using linear system techniques
• The robustness of the SMC system against uncertainties and disturbances can be assessed by considering the bounds on the uncertainties and the control gains
• Input-to-state stability (ISS) and integral sliding mode control (ISMC) can be used to analyze the stability and robustness of SMC systems in the presence of unmatched uncertainties

Chattering and Its Mitigation

• Chattering is a common problem in SMC, characterized by high-frequency oscillations of the control input and system states around the sliding surface
• Chattering is caused by the discontinuous nature of the switching control and the presence of unmodeled dynamics, delays, or discretization effects
• Chattering can lead to excessive control effort, actuator wear, and excitation of unmodeled high-frequency dynamics
• Several techniques can be employed to mitigate chattering in SMC systems:
  • Boundary layer approach: Replace the discontinuous signum function with a continuous approximation (saturation, hyperbolic tangent) within a boundary layer around the sliding surface
  • Higher-order sliding modes: Use higher-order derivatives of the sliding variable to design a continuous control law that maintains the robustness properties of SMC
  • Adaptive gain tuning: Adjust the control gains online based on the system states or the magnitude of the sliding variable to reduce chattering while maintaining robustness
  • Disturbance observer-based control: Estimate and compensate for the uncertainties and disturbances using a disturbance observer, reducing the required switching control effort
• The choice of the chattering mitigation technique depends on the specific application, the available computational resources, and the acceptable level of chattering
• It is important to strike a balance between chattering reduction and the preservation of the robustness properties of SMC

Applications and Case Studies

• SMC has been successfully applied to a wide range of nonlinear systems and control problems
• Robotics: SMC is used for robust trajectory tracking, force control, and compliance control of robotic manipulators and mobile robots
  • Example: SMC for precise position control of a robotic arm in the presence of joint friction and payload variations
• Automotive systems: SMC is employed for vehicle stability control, traction control, and engine management systems
  • Example: SMC-based electronic stability control (ESC) for maintaining vehicle stability during extreme maneuvers
• Power electronics: SMC is applied to control power converters, inverters, and electrical drives
  • Example: SMC for robust current control of a three-phase inverter under load variations and parameter uncertainties
• Aerospace systems: SMC is used for attitude control, guidance, and fault-tolerant control of aircraft and spacecraft
  • Example: SMC-based attitude control of a satellite in the presence of external disturbances and actuator faults
• Process control: SMC is employed for robust control of chemical processes, HVAC systems, and industrial plants
  • Example: SMC for temperature regulation in a chemical reactor under varying operating conditions and disturbances
• Biomedical systems: SMC is applied to control medical devices, prosthetics, and rehabilitation systems
  • Example: SMC-based control of a powered prosthetic leg for robust and adaptive gait assistance
• These case studies demonstrate the effectiveness and versatility of SMC in handling nonlinearities, uncertainties, and disturbances in real-world applications