Sliding mode control is a powerful nonlinear control technique that handles uncertainties and disturbances. It forces system states onto a predefined sliding surface, offering robustness and finite-time convergence. This method stands out for its ability to simplify control design and handle complex nonlinear systems.

The notes cover key aspects of sliding mode control, including surface design, reaching phase analysis, and chattering reduction techniques. They also explore advanced topics like higher-order sliding modes, observers, and applications in robotics and automotive systems. Understanding these concepts is crucial for designing effective nonlinear control systems.

Sliding mode control fundamentals

• Sliding mode control (SMC) is a robust nonlinear control technique that can handle system uncertainties and disturbances
• SMC is based on the concept of forcing the system state to reach and maintain on a predefined sliding surface in the state space
• Offers advantages over classical control methods, such as insensitivity to matched uncertainties and disturbances, finite-time convergence, and reduced-order dynamics on the sliding surface

Definition of sliding mode control

• SMC is a variable structure control strategy that employs a discontinuous control law to drive the system state towards a sliding surface
• The sliding surface is a hyperplane in the state space, defined by a linear combination of the system states
• Once the system state reaches the sliding surface, it is constrained to slide along the surface towards the desired equilibrium point

Advantages vs classical control methods

• SMC can handle system uncertainties and disturbances that satisfy the matching condition, ensuring robustness
• Provides finite-time convergence to the sliding surface, unlike asymptotic convergence in classical control methods
• Reduces the order of the system dynamics on the sliding surface, simplifying the control design process
• Can be applied to a wide range of nonlinear systems, including those with non-differentiable or discontinuous dynamics

Sliding surface design

• The design of the sliding surface is crucial in SMC, as it determines the system behavior during the sliding mode
• The sliding surface is typically selected as a linear combination of the system states, with coefficients chosen to ensure desired performance
• The existence and stability of the sliding mode depend on the proper design of the sliding surface

Selection of switching function

• The switching function, denoted as $s(x)$, defines the sliding surface in the state space
• Common choices for the switching function include linear, terminal, and integral sliding surfaces
• Linear sliding surface: $s(x) = cx$, where $c$ is a constant vector
• Terminal sliding surface: $s(x) = x_1^{p_1/q_1} + \lambda x_2^{p_2/q_2}$, where $p_i$, $q_i$ are positive odd integers and $\lambda > 0$
• Integral sliding surface: $s(x) = cx + k\int_0^t (cx) d\tau$, where $k > 0$

Existence of sliding mode

• The existence of the sliding mode requires the system state to reach the sliding surface in finite time
• This is ensured by satisfying the reaching condition, which states that the time derivative of the switching function should have the opposite sign of the switching function itself
• Mathematically, the reaching condition is expressed as $s(x)\dot{s}(x) < 0$

Stability of sliding mode

• The stability of the sliding mode depends on the dynamics of the reduced-order system on the sliding surface
• The sliding surface coefficients should be selected such that the reduced-order dynamics are stable
• Lyapunov stability theory can be used to analyze the stability of the sliding mode
• Common Lyapunov function candidates include quadratic forms, such as $V(s) = \frac{1}{2}s^2$

Reaching phase analysis

• The reaching phase refers to the system behavior before it reaches the sliding surface
• Analyzing the reaching phase is essential to ensure that the system state reaches the sliding surface in finite time and to minimize the reaching time
• Techniques for reducing the reaching phase include the use of high-gain control, disturbance observers, and reaching law approaches

Reaching condition for sliding mode

• The reaching condition ensures that the system state moves towards the sliding surface during the reaching phase
• It is mathematically expressed as $s(x)\dot{s}(x) < -\eta |s(x)|$, where $\eta > 0$ is the reaching rate
• The reaching condition guarantees that the distance between the system state and the sliding surface decreases at a rate of at least $\eta$

Techniques for reducing reaching phase

• High-gain control: Increasing the control gain can reduce the reaching time but may lead to excessive control effort and chattering
• Disturbance observers: Estimating and compensating for the system uncertainties and disturbances can improve the reaching phase performance
• Reaching law approaches: Modifying the control law during the reaching phase to ensure faster convergence to the sliding surface
• Examples of reaching law approaches include the constant rate reaching law, the power rate reaching law, and the exponential reaching law

Equivalent control method

• The equivalent control method is a technique used to analyze the system behavior during the sliding mode
• It involves solving for the continuous control input that maintains the system state on the sliding surface
• The equivalent control input is then used to design the sliding mode control law

Derivation of equivalent control

• Consider a system $\dot{x} = f(x) + b(x)u$, where $x$ is the state vector, $u$ is the control input, and $f(x)$ and $b(x)$ are smooth functions
• The sliding surface is defined as $s(x) = 0$
• During the sliding mode, $\dot{s}(x) = 0$
• Solving for the control input that satisfies $\dot{s}(x) = 0$ yields the equivalent control $u_{eq} = -((\partial s / \partial x)b(x))^{-1}(\partial s / \partial x)f(x)$

Role in sliding mode control design

• The equivalent control input represents the average control effort required to maintain the system state on the sliding surface
• It can be used to analyze the reduced-order dynamics of the system during the sliding mode
• The sliding mode control law is designed as a combination of the equivalent control and a discontinuous term to ensure robustness and reaching condition satisfaction
• Example: $u = u_{eq} - k \text{sign}(s(x))$, where $k > 0$ is a constant gain and $\text{sign}(\cdot)$ is the signum function

Chattering phenomenon

• Chattering is a high-frequency oscillation of the system state around the sliding surface, caused by the discontinuous nature of the sliding mode control law
• It can lead to excessive control effort, actuator wear, and excitation of unmodeled high-frequency dynamics
• Techniques for chattering reduction aim to mitigate these negative effects while preserving the robustness properties of SMC

Causes of chattering

• The discontinuous control law in SMC, which switches between positive and negative values based on the sign of the switching function
• Unmodeled dynamics, such as actuator and sensor delays, which can cause the system state to overshoot the sliding surface
• Digital implementation of SMC, which introduces sampling and quantization effects that can lead to chattering

Techniques for chattering reduction

• Boundary layer approach: Replacing the discontinuous control law with a continuous approximation within a thin boundary layer around the sliding surface
• Higher-order sliding modes: Using higher-order derivatives of the sliding variable to design a continuous control law that preserves robustness
• Observers and disturbance estimation: Estimating and compensating for the unmodeled dynamics and disturbances that contribute to chattering
• Adaptive gain tuning: Adjusting the control gain based on the system state or sliding variable to reduce chattering while maintaining robustness

Boundary layer approach

• The boundary layer approach is a widely used technique for chattering reduction in SMC
• It involves replacing the discontinuous control law with a continuous approximation within a thin boundary layer around the sliding surface
• The boundary layer thickness is a design parameter that trades off between chattering reduction and robustness

Continuous approximation of discontinuous control

• The discontinuous term in the sliding mode control law, typically a signum function, is replaced with a continuous approximation within the boundary layer
• Common approximations include the saturation function and the hyperbolic tangent function
• Example: $u = u_{eq} - k \text{sat}(s(x)/\phi)$, where $\phi > 0$ is the boundary layer thickness and $\text{sat}(\cdot)$ is the saturation function

Tradeoff between robustness and performance

• Increasing the boundary layer thickness reduces chattering but also reduces the robustness of the system to uncertainties and disturbances
• Decreasing the boundary layer thickness improves robustness but increases chattering
• The choice of the boundary layer thickness should balance the competing requirements of chattering reduction and robustness
• Adaptive boundary layer approaches can be used to adjust the thickness based on the system state or performance metrics

Higher-order sliding modes

• Higher-order sliding modes (HOSM) are an extension of SMC that aim to reduce chattering while preserving robustness
• HOSM uses higher-order derivatives of the sliding variable to design a continuous control law
• The most common types of HOSM are second-order sliding mode (SOSM) and arbitrary-order sliding mode (AOSM) control

Second-order sliding mode control

• SOSM uses the second-order derivative of the sliding variable to design a continuous control law
• The sliding variable and its first derivative are driven to zero in finite time
• Examples of SOSM algorithms include the twisting algorithm, the super-twisting algorithm, and the sub-optimal algorithm
• SOSM provides robustness to uncertainties and disturbances while reducing chattering compared to conventional SMC

Arbitrary-order sliding mode control

• AOSM generalizes the concept of HOSM to arbitrary orders
• The sliding variable and its derivatives up to a specified order are driven to zero in finite time
• Higher-order derivatives of the sliding variable are used to design a continuous control law
• AOSM can provide even smoother control signals and further reduce chattering compared to SOSM
• The design of AOSM controllers becomes more complex as the order increases, requiring careful selection of the sliding variable and control law parameters

Sliding mode observers

• Sliding mode observers (SMOs) are a class of state estimators that use the principles of SMC to estimate unmeasured states or detect faults in a system
• SMOs are designed to drive the estimation error to zero in finite time, providing robust and accurate state estimates in the presence of uncertainties and disturbances
• SMOs have applications in fault detection and isolation, as well as in the design of output feedback sliding mode controllers

Design of sliding mode observers

• Consider a system $\dot{x} = Ax + Bu + f(x, u, t)$, $y = Cx$, where $x$ is the state vector, $u$ is the control input, $y$ is the output, and $f(x, u, t)$ represents uncertainties and disturbances
• The SMO is designed as $\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x}) + K \text{sign}(y - C\hat{x})$, where $\hat{x}$ is the estimated state, $L$ is the observer gain matrix, and $K$ is the sliding mode gain matrix
• The sliding surface for the SMO is defined as $s_o = y - C\hat{x}$
• The observer gains $L$ and $K$ are selected to ensure the stability of the estimation error dynamics and the reaching condition for the sliding surface

Applications in fault detection and estimation

• SMOs can be used for fault detection and isolation by designing the sliding surface to be sensitive to specific faults
• The occurrence of a fault will drive the system state away from the sliding surface, which can be detected by monitoring the sliding variable
• SMOs can also be used to estimate the magnitude of faults or disturbances by analyzing the equivalent output error injection term
• Example: In a sensor fault detection scheme, the SMO can be designed to estimate the sensor output, and the difference between the estimated and measured outputs can be used to detect and isolate sensor faults

Sliding mode control applications

• SMC has been successfully applied to a wide range of engineering systems, thanks to its robustness, finite-time convergence, and ability to handle nonlinear dynamics
• Some notable application areas include robotics, electric drives, power systems, automotive control, and aerospace systems
• The following sections provide an overview of SMC applications in these domains

Robotic manipulators

• SMC is well-suited for controlling robotic manipulators due to its ability to handle model uncertainties, external disturbances, and varying payloads
• SMC can be used to design joint-level controllers for trajectory tracking, force control, and compliance control
• Example: A sliding mode controller can be designed to track a desired joint trajectory while rejecting disturbances and compensating for model uncertainties in a robot arm

Electric motors and power systems

• SMC has been applied to various types of electric motors, including DC motors, induction motors, and permanent magnet synchronous motors
• SMC can be used for speed control, torque control, and position control of electric motors, ensuring robust performance in the presence of parameter variations and load disturbances
• In power systems, SMC has been used for voltage regulation, frequency control, and power flow management, improving the stability and reliability of the grid

Automotive and aerospace systems

• In automotive systems, SMC has been applied to traction control, anti-lock braking systems (ABS), and electronic stability control (ESC) to improve vehicle safety and handling
• SMC can also be used for engine control, transmission control, and active suspension systems in vehicles
• In aerospace systems, SMC has been used for attitude control, guidance, and fault-tolerant control of aircraft and spacecraft
• Example: An SMC-based attitude controller can be designed for a satellite to maintain a desired orientation while rejecting external disturbances and compensating for actuator faults

Adaptive sliding mode control

• Adaptive sliding mode control (ASMC) combines the robustness of SMC with the adaptability of adaptive control techniques
• ASMC is designed to handle systems with parametric uncertainties, where the exact values of certain system parameters are unknown or time-varying
• The control law and the sliding surface are adapted based on the estimated values of the uncertain parameters

Combining sliding mode with adaptive control

• In ASMC, the sliding mode control law is augmented with an adaptive term that estimates the uncertain parameters
• The adaptive law is designed to ensure the convergence of the parameter estimates and the stability of the closed-loop system
• Example: Consider a system $\dot{x} = (A + \Delta A)x + (B + \Delta B)u$, where $\Delta A$ and $\Delta B$ are unknown parameter matrices
• The ASMC law can be designed as $u = u_{eq} - K \text{sign}(s) - \hat{\Delta A}x - \hat{\Delta B}u$, where $\hat{\Delta A}$ and $\hat{\Delta B}$ are the estimates of the uncertain parameters, updated using an adaptive law

Handling parametric uncertainties

• ASMC can handle parametric uncertainties that satisfy certain conditions, such as the matching condition or the extended matching condition
• The adaptive law is designed to ensure that the parameter estimation errors remain bounded and converge to zero asymptotically
• Lyapunov stability theory is used to analyze the stability of the closed-loop system and derive the adaptation laws
• Example: The adaptation laws for the uncertain parameters can be designed as $\dot{\hat{\Delta A}} = \Gamma_1 x s^T$ and $\dot{\hat{\Delta B}} = \Gamma_2 u s^T$, where $\Gamma_1$ and $\Gamma_2$ are positive definite adaptation gain matrices

Discrete-time sliding mode control

• Discrete-time sliding mode control (DSMC) is the application of SMC principles to sampled-data systems
• DSMC is important for the practical implementation of SMC on digital control systems, where the control inputs are updated at discrete time instants
• The design of DSMC involves the discretization of the sliding surface, the reaching law, and the control law, while considering the effects of sampling and quantization

Sliding mode control for sampled-data systems

• In DSMC, the sliding surface is defined in the discrete-time domain as $s(k) = cx(k)$, where $k$ is the discrete-time index
• The reaching law is modified to ensure that the system state reaches the sliding surface in a finite number of sampling periods
• Example: A discrete-time reaching law can be designed as $s(k+1) - s(k) = -qT_s \text{sign}(s(k)) - \varepsilon T_s s(k)$, where $T_s$ is the sampling period, $q > 0$, and $0 < \varepsilon < 1$

Digital implementation considerations

• The digital implementation of DSMC requires the consideration of several factors, such as the choice of the sampling period, the quantization effects, and the computation delay
• The sampling period should be chosen sufficiently small to ensure the stability and performance of the closed-loop system while considering the available computational resources
• Quantization effects, such as the finite resolution of sensors and actuators, can lead to the deterioration of the DSMC performance and should be accounted for in the design process
• Computation delays, arising from the time required to calculate the control input, can affect the stability and performance of DSMC and may require the use of delay compensation techniques