Sliding Mode Control – a SAG Mill application

The Semi-Autogenous Grinding (SAG) mill plays a critical role in the comminution stage of mineral processing, where its primary function is to reduce large ore chunks into finer particles for further downstream processing. However, the inherent nonlinear dynamics, frequent disturbances (e.g., variations in ore hardness), and uncertainties in SAG mill operation present significant challenges to achieving optimal performance and energy efficiency. Traditional control methods, such as Proportional-Integral-Derivative (PID) controllers, often fall short in addressing these issues due to their limited robustness against model inaccuracies and external disturbances. This paper explores the application of Sliding Mode Control (SMC) as an advanced, robust control strategy for SAG mill optimization, with a focus on maintaining optimal mill load and power draw while achieving consistent particle size distribution.

The SMC design approach developed here leverages the robust properties of sliding mode control to handle the nonlinear and uncertain nature of the SAG milling process. Key objectives of this control strategy include stabilizing mill load, minimizing energy consumption, and maintaining consistent throughput and product size under varying feed conditions. A comprehensive mathematical model of the SAG mill dynamics is presented, capturing critical state variables such as mill load, power draw, and product size. Based on this model, a sliding surface is designed to minimize deviations in these states from their desired setpoints, effectively handling process disturbances and dynamic changes in ore characteristics. To achieve optimal control, the sliding mode controller employs a discontinuous control law with boundary-layer adjustments to mitigate chattering—a common issue in sliding mode implementations.

Simulation results demonstrate that the proposed SMC approach significantly improves system stability, robustness, and performance compared to conventional PID control, particularly in scenarios with variable feed characteristics and sudden disturbances. Practical implementation considerations, including real-time computational demands, actuator constraints, and tuning methods, are also discussed to facilitate adaptation of the SMC framework for real-world SAG mill operations. By enhancing control accuracy and energy efficiency, the proposed SMC methodology offers a promising solution for SAG mill optimization, supporting improved mineral recovery and reduced operational costs.

Introduction

The Semi-Autogenous Grinding (SAG) mill is an essential component in the comminution stage of mineral processing, where its primary function is to reduce mined ore into smaller particles for downstream separation processes. Unlike conventional grinding mills, SAG mills use a combination of larger grinding media (usually steel balls) and the ore itself, allowing for a more efficient and versatile grinding process that can handle large, irregularly-sized feed materials. Given its central role in mineral recovery, optimizing SAG mill performance is a key factor in improving both productivity and energy efficiency in mining operations.
However, the operational dynamics of a SAG mill are complex and challenging to control. The process is inherently nonlinear, subject to considerable fluctuations due to varying ore properties (such as hardness, moisture content, and size distribution) and is sensitive to disturbances. For instance, changes in ore characteristics or feed rate can lead to rapid variations in the mill load, power draw, and product size distribution. These variations make it difficult to achieve a stable and efficient operation while also maintaining product quality and throughput. As a result, traditional control techniques, such as Proportional-Integral-Derivative (PID) controllers, often struggle to provide the necessary robustness and adaptability required for reliable SAG mill control. PID controllers are limited in their ability to handle nonlinearities and disturbances, and frequently require recalibration to adjust to changing operating conditions, leading to downtime and suboptimal performance.
To address these limitations, advanced control strategies are increasingly being explored, with Sliding Mode Control (SMC) standing out as a promising approach. Sliding mode control is a form of variable structure control known for its robustness to system uncertainties and external disturbances, making it well-suited for applications in highly nonlinear, uncertain environments like those encountered in SAG mill operations. SMC works by driving system states onto a predefined sliding surface, where they remain insensitive to certain types of disturbances and model inaccuracies. This characteristic allows the SMC to achieve stable control even under significant variability, reducing the need for frequent recalibration and making it more adaptable to fluctuating conditions in real-time operations.
The core objective of applying SMC in the context of a SAG mill is to achieve robust control of key operational parameters such as mill load, power draw, and particle size distribution. Maintaining optimal mill load is particularly crucial, as it directly impacts grinding efficiency, energy consumption, and wear on the grinding media and liner. Similarly, controlling power draw is essential to managing energy costs and reducing operational expenditure. The challenge lies in developing a control strategy that not only addresses these objectives but also maintains performance in the face of disturbances such as changes in ore hardness and feed characteristics.
This paper presents a detailed design and implementation of a sliding mode control system for SAG mill optimization. A mathematical model of the SAG mill is developed, incorporating key state variables and parameters such as mill load, power draw, and particle size. Based on this model, a sliding surface is defined to minimize deviations from desired setpoints in these variables, effectively improving stability and product quality. A reaching law is employed to bring the system states to the sliding surface, and a control law is formulated to drive the states along this surface while compensating for disturbances. The control law is further refined to include a boundary layer to mitigate chattering, a common issue associated with sliding mode control that can lead to excessive wear on actuators.
The efficacy of the proposed SMC approach is demonstrated through simulations that compare its performance with traditional PID control under varying operating conditions. Simulation results indicate that the SMC provides superior control accuracy, energy efficiency, and robustness, particularly in scenarios involving large disturbances or rapidly changing ore properties. Finally, the paper discusses practical considerations for real-world implementation of SMC in SAG mill operations, including the impact of sensor accuracy, real-time computational demands, and tuning requirements.
This study aims to establish sliding mode control as a viable and effective control strategy for SAG mill optimization, offering a robust solution to the challenges of nonlinearities, disturbances, and system uncertainties. By achieving better control over SAG mill operations, this approach has the potential to enhance mineral recovery, reduce operational costs, and improve overall productivity in mineral processing plants.

SLIDING MODE CONTROL FUNDAMENTALS AND RELEVANCE TO SAG MILLS

. Sliding Mode Control Basics

Sliding Mode Control (SMC) is a nonlinear control strategy that provides high robustness to system uncertainties, external disturbances, and variations in operating conditions. SMC achieves this robustness by forcing the system’s state trajectory to “slide” along a pre-defined surface within the state space, known as the sliding surface. This design allows the controller to effectively handle nonlinearity and disturbance by continuously adjusting its control action based on the system’s deviation from the sliding surface.

.. Key Principles of SMC:

Sliding Surface: The sliding surface is a carefully designed constraint within the state space that represents the desired system dynamics. When the system’s state reaches this surface, it is driven to follow the desired behaviour, despite variations or uncertainties in the system. For a SAG mill application, the sliding surface might be defined to maintain an optimal load and power draw for effective grinding, which translates to minimal deviation from desired states such as mill load and power consumption.
Sliding Condition: The sliding condition ensures that once the system’s state reaches the sliding surface, it will remain on that surface for the remainder of the operation. The surface is designed so that any deviation from it triggers a corrective control action that “slides” the system back to the surface. In SMC terminology, this is often formulated as a stability condition:
$\dot{s}\cdot s<0$
where (s) is the sliding variable representing the distance from the sliding surface. This inequality ensures that deviations from the surface will decay over time, driving the system back to the surface and maintaining the desired dynamic response.
Reaching Condition: The reaching condition dictates how the system reaches the sliding surface from any initial state. The control law in SMC is designed to ensure the state approaches the sliding surface, reaching it within a finite time, even when the system starts away from the surface. This guarantees that the system trajectory will ultimately reach and stay on the sliding surface, regardless of initial conditions or certain types of disturbances. The control law is split into two parts:
Equivalent Control: Represents the nominal control needed to keep the system on the sliding surface without disturbances.
Switching Control: A discontinuous component that forces the system to reach the surface. This term is what provides the robustness to uncertainties and disturbances but can lead to a phenomenon called “chattering,” where the control input oscillates rapidly around the sliding surface.
The sliding mode control law can be formulated as:
$u=u_{\mathrm{eq}}-K\cdot\mathrm{sign}\left(s\right)$
where:
$u_{\mathrm{eq}}$ is the equivalent control component that maintains the system on the sliding surface,
$K\cdot\mathrm{sign}\left(s\right)$ is the discontinuous term that enforces the sliding condition and drives the system to the surface,
$K$ is a gain parameter that is tuned to provide the desired robustness.
In practice, to reduce chattering, the sign function $\mathrm{sign}\left(s\right)$ is often replaced by a saturation function or smoothed over a boundary layer around the sliding surface.
2.1.2. Robustness of SMC:
SMC is inherently robust to uncertainties and disturbances because the control action continuously adjusts based on the system’s deviation from the sliding surface. Once on the sliding surface, the system’s response becomes largely independent of certain types of model inaccuracies or external disturbances. This robustness is particularly useful in applications where precise modelling is difficult, and conditions vary frequently.
2.2 Relevance of Sliding Mode Control to SAG Mill Control
The SAG mill is characterized by a highly nonlinear and uncertain operating environment. Key operational variables such as feed rate, ore hardness, and moisture content vary significantly, which complicates control and limits the performance of traditional controllers like PID. The dynamics of the mill load and power draw are complex, and the process is further subject to external disturbances (e.g., changes in ore properties) that can lead to inefficiencies and unsteady operations if not properly managed.
SMC’s robustness to uncertainties and disturbances makes it particularly well-suited for SAG mill control, where maintaining optimal performance under variable conditions is challenging. Here’s how SMC addresses the unique control challenges in SAG mills:
Adaptability to Variable Input Conditions: In SAG mill operations, input conditions such as feed rate and ore characteristics (hardness, size distribution) vary frequently, impacting the mill load and power consumption. Traditional controllers would require frequent recalibration to handle such variations. SMC, however, can handle these variations effectively by keeping the system state on the sliding surface, ensuring that the mill operates near optimal conditions despite input variability.
Robust Disturbance Rejection: The operation of a SAG mill is subject to frequent disturbances due to changes in ore quality and other environmental factors. SMC’s switching control law is inherently robust to such disturbances, adjusting the control action to counter deviations from the sliding surface. This disturbance rejection capability helps maintain a stable mill load and energy-efficient operation, which is essential for consistent grinding performance and throughput.
Maintaining Optimal Mill Load and Power Draw: The sliding surface in SMC can be designed to represent optimal mill load and power draw, ensuring efficient grinding and energy use. By defining control objectives in terms of desired load and power, SMC can enforce constraints that stabilize the grinding process and improve output quality. The sliding surface effectively acts as an attractor for the desired operational conditions, minimizing deviations in critical parameters such as particle size distribution and power consumption.
Reducing Sensitivity to Modelling Errors: Developing an accurate model of SAG mill dynamics is challenging due to the complex and variable nature of the process. SMC’s robustness to modelling errors makes it an attractive choice, as the control performance is less dependent on an exact model. The sliding surface and control law are designed to function effectively even with approximate models, which is often necessary given the limitations in capturing the full dynamics of a SAG mill.
SMC’s ability to handle nonlinear dynamics, reject disturbances, and adapt to variable operating conditions addresses the specific challenges associated with SAG mill control. By designing a sliding surface that maintains key parameters at optimal levels, SMC provides a reliable and energy-efficient solution for enhancing the performance of SAG mills, ensuring consistent grinding quality, improved throughput, and reduced operational costs. This adaptability and robustness make SMC an advanced and practical choice for optimizing SAG mill operations in complex mineral processing environments.
3. CONTROL OBJECTIVES DEFINITION
In the control of Semi-Autogenous Grinding (SAG) mills, defining precise objectives is crucial to achieving efficient grinding, consistent product quality, and energy-efficient operation. Given the complexity of SAG mill dynamics, where both process variables and external disturbances constantly vary, a well-defined control structure with primary and secondary objectives is essential. These objectives serve as the foundation for developing control strategies that ensure the mill operates within optimal parameters, balancing productivity and energy efficiency.
3.1. Primary Control Objective: Maintain Optimal Mill Load
The primary control objective for a SAG mill is to maintain the mill load—the combined mass of the grinding material and balls—at an optimal level. This objective is crucial for several reasons:
Efficient Grinding: The grinding efficiency of a SAG mill is directly related to the mill load. If the load is too low, the impact force generated by the mill is insufficient to break down the ore particles effectively, resulting in suboptimal grinding. Conversely, if the load is too high, the grinding media may become ineffective, reducing the efficiency and producing a higher proportion of coarse particles.
Preventing Overloading: Excessive mill load can lead to overloading, which poses a risk of mechanical damage to the mill. Overloading may also cause the mill to trip or shut down, leading to costly downtime and maintenance. Maintaining an optimal load level mitigates this risk, ensuring continuous operation without the need for manual intervention or emergency stops.
Achieving Desired Product Size: Maintaining an optimal mill load enables the mill to produce a consistent particle size distribution, which is essential for downstream separation processes. Ensuring that the particle size remains within the target range improves mineral recovery rates, as the product from the mill directly impacts separation efficiency. For example, if the particles are too coarse, the downstream process may require additional grinding, whereas overly fine particles may reduce recovery efficiency.
To achieve this primary control objective, the mill load needs to be continuously monitored and adjusted, usually through the control of the feed rate and other variables influencing the load dynamics. Given the high variability in ore properties and feed characteristics, the control strategy must adapt in real-time to maintain load stability.
3.2. Secondary Control Objectives
While the primary objective focuses on optimal mill load, several secondary objectives complement this by addressing efficiency, stability, and product quality:
Minimize Energy Consumption by Optimizing Power Draw: The power draw of a SAG mill is a significant factor in operational costs, as it represents a substantial portion of the plant’s total energy use. Minimizing energy consumption without sacrificing grinding efficiency is essential for cost-effective operation. By controlling the mill load and optimizing the feed rate, the power draw can be managed to avoid excessive energy use while still maintaining effective grinding. This objective typically involves keeping the mill power at a level that supports efficient grinding while reducing unnecessary energy expenditure.
Energy Efficiency: By maintaining the mill load at an optimal level, the mill operates closer to its peak grinding efficiency, thus reducing the energy required per ton of ore processed.
Load-Power Relationship: The control system should recognize the nonlinear relationship between load and power and adjust accordingly, avoiding conditions where power usage increases disproportionately with load due to inefficiencies.
Stabilize Throughput: Throughput, or the rate at which material passes through the SAG mill, is critical for maintaining the productivity of the mineral processing plant. Variations in throughput can lead to irregularities in downstream processes and affect the overall production rate. Stabilizing throughput ensures a consistent feed of material to subsequent processes, reducing the variability and improving recovery efficiency in later stages. The control system must adjust feed rate and other variables to maintain a steady throughput, even as feed characteristics or mill load change.
Consistent throughput avoids the stop-start operation that disrupts the downstream processes, maintaining a continuous flow of material through the plant.
As load affects throughput, maintaining an optimal load indirectly stabilizes throughput, but the control system must also address direct influences on throughput, such as feed rate and ore hardness, to keep it steady.
Maintain Particle Size Consistency: Achieving a consistent particle size distribution from the SAG mill output is critical for effective downstream processing, such as flotation or leaching. If the particle size is too variable, the separation efficiency may decrease, leading to lower mineral recovery rates or increased reprocessing requirements. By keeping particle size within a target range, the control strategy supports high-quality product output and enhances the overall efficiency of the mineral processing plant.
Particle size is affected by factors such as mill load, power draw, and feed characteristics. The control system should be able to account for these factors, making real-time adjustments to maintain consistent particle size.
Consistent particle size improves product quality, which is essential for meeting the requirements of subsequent separation processes and achieving high recovery rates.
3.3. Summary of Control Objectives and Challenges
Together, these primary and secondary control objectives define a comprehensive approach to optimizing SAG mill performance. The primary objective of maintaining an optimal mill load ensures efficient grinding and prevents overloads, forming the foundation of the control strategy. The secondary objectives—minimizing energy consumption, stabilizing throughput, and ensuring consistent particle size—address additional aspects of performance that support cost-efficiency, productivity, and product quality.
Achieving these objectives in a SAG mill environment presents several challenges due to:
Nonlinear System Dynamics: SAG mills exhibit nonlinear behaviour, especially in the load-power relationship, which requires control strategies that can adapt to varying conditions.
External Disturbances: Variations in ore characteristics, such as hardness and feed rate, create disturbances that directly impact mill load and energy consumption. The control system must be able to mitigate these disturbances in real-time.
Uncertainty and Model Limitations: Due to the complexity of SAG mill operations, precise modelling is challenging, and traditional controllers may struggle with robustness in the face of model inaccuracies.
In response to these challenges, advanced control strategies like Sliding Mode Control (SMC) are proposed, as SMC offers robust disturbance rejection and handles uncertainties effectively. SMC’s capability to maintain system states near desired setpoints, despite variable and nonlinear dynamics, makes it well-suited for achieving the defined primary and secondary control objectives in SAG mill operations.
4. SAG MILL DYNAMIC MODEL
To design an effective Sliding Mode Control (SMC) system for a Semi-Autogenous Grinding (SAG) mill, a detailed mathematical model that captures the critical dynamics is essential. The model must reflect key factors influencing SAG mill behaviour, including mill load, power draw, and particle size distribution. This model provides the foundation for developing control laws that allow SMC to manage system uncertainties and external disturbances effectively.
The SAG mill’s behaviour can be approximated using a set of nonlinear state-space equations that account for the dynamic interplay between mill load, power draw, and control inputs. Here, we introduce critical variables and relationships:
4.1. Mill Load Dynamics:
The mill load (denoted by $(x_1)$ ) represents the total mass of material inside the mill, which includes both grinding material (ore) and grinding media (steel balls).
Mill load dynamics are influenced by the rate of material input feed $(F_{\mathrm{feed}})$ (the control input, denoted ( $u$ )), and the discharge rate.
The differential equation for mill load dynamics can be represented as:
$\dot{x_1}=f_1\left(x_1,u\right)+d_1\left(t\right)$
where:
$f_1\left(x_1,u\right)$ : A nonlinear function capturing the relationship between mill load, input feed rate, and other internal dynamics.
$d_1\left(t\right)$ : A disturbance term representing external variations, such as changes in ore hardness or feed composition. Variations in these factors affect how the load accumulates and responds to feed rate adjustments.
4.2. Power Draw Dynamics:
Power draw (denoted by ( $x_2$ )) represents the energy consumption rate (in kW) of the mill, which is crucial for both performance and efficiency.
Power draw is influenced by several factors, including the mill load $x_1$ , mill speed, and feed rate $u$ . The relationship between power draw and these variables is typically nonlinear and may vary based on the mill’s operating conditions and ore properties.
Power draw dynamics can be modelled with the following differential equation:
$\dot{x_2}=f_2\left(x_1,x_2,u\right)+d_2\left(t\right)$
where:
$f_2\left(x_1,x_2,u\right)$ : A nonlinear function that describes how power draw depends on mill load, power draw itself (due to inertia and time-lag effects), and the feed rate.
$d_2\left(t\right)$ : A disturbance term capturing changes in ore characteristics, such as hardness, and environmental conditions, which can significantly affect energy requirements.
4.3. Particle Size Distribution (PSD):
Particle size distribution is a key output characteristic of the SAG mill, directly impacting downstream processing efficiency. The PSD is a result of the interactions within the mill load and is influenced by factors such as ore hardness, mill load, and feed characteristics.
Unlike mill load and power draw, which can be more directly measured or estimated, PSD is often highly nonlinear and affected by both the input feed rate and internal grinding dynamics.
To manage PSD effectively within the control framework, its sensitivity to mill load (x_1) and feed rate (u) needs to be embedded within the control model, though PSD dynamics are not directly represented in the state equations. Instead, PSD is addressed indirectly through the control of mill load and feed rate.
4.4. Mathematical Representation
Let:
$x_1$ : Mill load (tonnes)
$x_2$ : Power draw (kW)
$u$ : Feed rate ( $F_{\mathrm{feed}}$ ) (tonnes per hour), the control input
The approximate dynamics of the SAG mill can be modelled as:
$\dot{x_1}=f_1\left(x_1,u\right)+d_1\left(t\right)$
$\dot{x_2}=f_2\left(x_1,x_2,u\right)+d_2\left(t\right)$
where:
$f_1\left(x_1,u\right)$ : A nonlinear function capturing the dependency of mill load on the input feed rate and discharge rate, reflecting how the load increases or decreases over time.
$f_2\left(x_1,x_2,u\right)$ : A nonlinear function modeling the power draw’s dependency on mill load, feed rate, and internal energy requirements of the mill.
$d_1\left(t\right)\ \mathrm{and\ }d_2\left(t\right)$ : Disturbance terms that account for external and internal variations, such as ore hardness, particle size distribution changes, and environmental conditions, which introduce uncertainty into the mill dynamics.
4.5. Explanation of Variables and Functions
4.5.1. Mill Load Function ( $f_1\left(x_1,u\right)$ ):
This function accounts for the material balance within the mill. The rate of change in mill load depends on the input feed rate u, as well as the natural discharge of material from the mill.
It can be formulated as:
$f_1\left(x_1,u\right)=k_1u-k_2x_1$
where $k_1$ and $k_2$ are constants related to the feed rate efficiency and discharge rate, respectively.
4.5.2. Power Draw Function ( $f_2\left(x_1,x_2,u\right)$ ):
This function captures the nonlinear relationship between the load, power draw, and feed rate, which affects how much energy the mill requires under different operating conditions.
A possible form of $f_2\left(x_1,x_2,u\right)$ could include:
$f_2\left(x_1,x_2,u\right)=k_3x_1+k_4u-k_5x_2$
where $k_3, k_4$ , and $k_5$ are constants that adjust for the influence of mill load, feed rate, and current power draw, respectively.
4.6. Importance of Disturbances $d_1\left(t\right) and d_2\left(t\right)$
The terms $d_1\left(t\right)$ and $d_2\left(t\right)$ represent external disturbances that introduce uncertainty into the model:
$d_1\left(t\right)$ : Reflects the effects of ore hardness, which influences how quickly the mill load builds up or depletes. Harder ore may require more time to grind, affecting load accumulation.
$d_2\left(t\right)$ : Captures the effect of changes in ore particle size distribution or variations in mill speed, which affect the power draw needed to achieve a desired load and grind quality.
These disturbances make the mill dynamics challenging to predict and control, underscoring the need for robust control techniques like SMC, which can handle uncertainty effectively.
This model captures the essential dynamics for designing an SMC system that can manage a SAG mill under varying conditions. By focusing on the mill load, power draw, and their relationships with feed rate, this model provides a structured approach for achieving control objectives. The nonlinear relationships in $f_1\left(x_1,u\right)$ and $f_2\left(x_1,x_2,u\right)$ , combined with disturbance terms $d_1\left(t\right)$ and $d_2\left(t\right)$ , highlight the challenges of SAG mill control and the need for adaptive strategies like SMC that respond to real-time disturbances.
5. DESIGN OF THE SLIDING SURFACE
In Sliding Mode Control (SMC) design, the sliding surface defines the conditions under which the system’s states will converge toward desired setpoints, effectively driving the system to behave as intended despite external disturbances or model uncertainties. For the SAG mill application, we establish a sliding surface to minimize deviations from the optimal mill load and power draw, two primary objectives essential for efficient grinding and energy management.
The sliding surface, denoted s, is the mathematical construct that encapsulates these control objectives and guides the system dynamics. By designing an appropriate sliding surface, we ensure that the mill’s behavior aligns closely with the desired operational conditions for both load and power draw, even when subjected to variations in feed rate, ore hardness, and other disturbances.
5.1. Desired State Definitions
To design the sliding surface, we start by defining the target or desired states for the key system parameters:
$x_1^\ast$ represents the optimal mill load, which is typically determined based on the SAG mill’s capacity, the desired throughput, and grinding efficiency.
Operating at or near this target load ensures effective grinding, prevents overloading, and promotes consistent product size.
$x_2^\ast$ denotes the ideal power draw, which aligns with the energy efficiency objectives. This setpoint is chosen to maximize grinding efficiency while minimizing unnecessary power consumption, taking into account load weight and ore properties.
By controlling the system to maintain these desired states, the SAG mill can operate in an optimal range, balancing throughput with energy consumption.
5.2. Defining the Sliding Surface
The sliding surface s is defined to capture the deviation of the system states from their respective desired values. The form of s ensures that, once the system reaches this surface, any state deviations from the desired values are minimized, effectively guiding the system along the desired operating trajectory. The sliding surface can be mathematically defined as:
$s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right)$
where:
$x_1$ and $x_2$ represent the actual values of the mill load and power draw, respectively.
$x_1^\ast$ and $x_2^\ast$ are the desired setpoints for mill load and power draw.
$c_1$ and $c_2$ are positive tuning constants, also known as weighting coefficients, which determine the relative importance of the mill load and power draw in defining the sliding surface.
5.3. Role of Tuning Constants $c_1$ and $c_2$
The constants $c_1$ and $c_2$ play a critical role in shaping the sliding surface to achieve control objectives by balancing the influence of mill load and power draw deviations:
High $c_1$ relative to $c_2$ : Emphasizes the mill load $x_1$ in the sliding surface. This setting is useful if load stability is prioritized, such as when throughput consistency is critical, or when the mill is sensitive to overloading.
High $c_2$ relative to $c_1$ : Increases the importance of power draw $x_2$ , directing the control to favor energy efficiency. This approach might be optimal when power costs are high or when energy efficiency is a key performance metric.
By adjusting $c_1$ and $c_2$ , the sliding surface can be tuned to match the operational priorities of the SAG mill under specific conditions.
5.4. Purpose and Benefits of the Sliding Surface
Error Minimization: The sliding surface actively minimizes the difference between the current and desired values of mill load and power draw. When the system states $x_1$ and $x_2$ reach this surface, they are constrained to move along it, reducing deviations from the setpoints.
Robustness to Disturbances: Once the system reaches the sliding surface, it becomes less sensitive to disturbances such as changes in ore hardness or feed rate. This is a critical benefit of SMC, as it allows the SAG mill to maintain optimal performance despite variability in operating conditions.
Decoupled Control: The use of a sliding surface allows for decoupling of the control of mill load and power draw, facilitating smoother control actions that respond independently to each state. This approach is particularly beneficial when mill load and power draw exhibit complex, nonlinear interactions.
The sliding surface $s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right)$ is the foundation of the SMC design. It enables the controller to drive the SAG mill’s states toward optimal load and power levels while remaining resilient to uncertainties, ensuring stable and efficient operation in the face of real-world variations in input and environmental conditions.
6. REACHING CONDITION DEFINITION
The reaching condition is an essential element in Sliding Mode Control (SMC) design, as it guarantees that the system states will approach and stay on the sliding surface, ultimately achieving the control objectives. This condition, also known as the Lyapunov condition, ensures that the control law drives the system’s states toward the sliding surface and keeps them there despite disturbances or uncertainties.
The reaching condition is mathematically expressed as:
$\dot{s}\cdot s<0$
where:
$\ s$ is the sliding surface, defined based on the control objectives (e.g., deviation of mill load and power draw from desired values).
$\dot{s}$ is the time derivative of s, representing the rate of change of the sliding surface.
This inequality implies that any deviation from the sliding surface will decrease over time, effectively “correcting” the system’s state trajectory by driving it back to the sliding surface.
6.1. Purpose of the Reaching Condition
The reaching condition fulfils several critical roles in SMC:
Driving the States to the Sliding Surface: The reaching condition ensures that if the system states start off the sliding surface, the control action will act in such a way that the states are guided toward the surface. As soon as the states reach the surface, they remain constrained to it, aligning with the control objectives.
Stability Assurance: By enforcing the inequality $\dot{s}\cdot s<0$ , the reaching condition acts as a stability criterion. This condition resembles a Lyapunov function, which is a mathematical tool used to analyze system stability. A negative ( $\dot{s}\cdot s$ ) ensures that the system’s energy is decreasing relative to the sliding surface, which keeps the states close to the desired setpoints even if they are initially disturbed.
Robustness to Disturbances: The reaching condition is designed to be effective in the presence of external disturbances or model uncertainties, as it continuously drives the states back to the sliding surface regardless of minor variations in system parameters. This feature is especially beneficial for SAG mill control, where factors like ore hardness, feed rate, and environmental conditions vary frequently.
6.2. Reaching Condition Derivation and Implementation
Given the sliding surface $s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right)$ , the time derivative $\dot{s}$ can be computed as:
$\dot{s}=c_1\dot{x_1}+c_2\dot{x_2}$
Substituting the dynamics of $x_1$ and $x_2$ (representing mill load and power draw) into $\dot{x_1}$ and $\dot{x_2}$ respectively, gives us:
$\dot{x_1}=f_1\left(x_1,u\right)+d_1\left(t\right)$
$\dot{x_2}=f_2\left(x_1,x_2,u\right)+d_2\left(t\right)$
where $d_1\left(t\right) and d_2\left(t\right)$ are disturbances. Then,
$\dot{s}=c_1\left(f_1\left(x_1,u\right)+d_1\left(t\right)\right)+c_2\left(f_2\left(x_1,x_2,u\right)+d_2\left(t\right)\right)$
To satisfy the reaching condition ( $\dot{s}\cdot s<0$ ), we design a control law u that adjusts the control input (e.g., feed rate $F_{feed}$ ) such that the inequality holds. One common method is to introduce a discontinuous control law:
$u=u_{\mathrm{eq}}-k\cdot\mathrm{sign}\left(s\right)$
where:
$u_{\mathrm{eq}}$ is the equivalent control, designed to keep the system on the sliding surface.
$k$ is a positive gain, chosen to be sufficiently large to counteract disturbances.
$\mathrm{sign}\left(s\right)$ is a sign function that provides corrective action based on the direction of deviation from the surface.
This control law drives s toward zero, ensuring the reaching condition is satisfied and keeping the system on the sliding surface. The term – $k\cdot\mathrm{sign}\left(s\right)$ introduces a discontinuous component that responds swiftly to deviations from the surface, making SMC particularly robust to disturbances.
6.3. Benefits of the Reaching Condition in SAG Mill Control
In the context of SAG mill control, the reaching condition has several benefits:
Enhanced Control Stability: Ensures that deviations in mill load and power draw are corrected quickly, stabilizing the mill operation.
Increased Robustness: Allows the system to adapt to variations in ore hardness, feed rate, and other disturbances, maintaining efficient grinding.
Optimal Performance: By keeping the system states near desired values, the SAG mill operates at optimal load and power draw, improving energy efficiency and throughput.
The reaching condition $\dot{s}\cdot s<0$ in SMC for a SAG mill ensures that the control law consistently drives the system toward the desired operational state, providing stable and robust control in a variable environment.
7. CONTROL LAW DESIGN
The Sliding Mode Control (SMC) law is formulated to bring the sliding surface s to zero, thus ensuring that the system meets the reaching condition $\dot{s}\cdot s<0$ . This condition keeps the system states close to the desired values for effective control. For the SAG mill, the SMC law is designed to handle model uncertainties and disturbances, such as changes in ore hardness and feed rate.
A typical SMC control law includes two components:
$u=u_{\mathrm{eq}}-K\cdot\mathrm{sign}\left(s\right)$
where:
$u_{\mathrm{eq}}$ is the equivalent control, providing the nominal control action required to keep the system on the sliding surface.
$K\cdot\mathrm{sign}\left(s\right)$ is the *discontinuous term*, which corrects deviations from the sliding surface and counters disturbances.
$K$ is a tuning parameter that adjusts the intensity of the sliding action.
7.1. Equivalent Control $(u_{\mathrm{eq}})$
The equivalent control, $u_{\mathrm{eq}}$ , is derived from the condition $\dot{s}=0$ , meaning that the sliding surface remains constant. This control term ensures that, under ideal conditions without disturbances, the system stays on the sliding surface without requiring additional corrective actions.
To solve for $u_{\mathrm{eq}}$ , we start with the sliding surface $s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right)$ and differentiate s with respect to time:
$\dot{s}=c_1\dot{x_1}+c_2\dot{x_2}$
Substituting the dynamic equations of $x_1$ (mill load) and $x_2$ (power draw):
$\dot{x_1}=f_1\left(x_1,u\right)+d_1\left(t\right)$
$\dot{x_2}=f_2\left(x_1,x_2,u\right)+d_2\left(t\right)$
yields:
$\dot{s}=c_1\left(f_1\left(x_1,u\right)+d_1\left(t\right)\right)+c_2\left(f_2\left(x_1,x_2,u\right)+d_2\left(t\right)\right)$
Setting $\dot{s}=0$ and solving for u provides the equivalent control:
$u_{\mathrm{eq}}=-\frac{\partial s}{\partial x_1}f_1\left(x_1,u\right)-\frac{\partial s}{\partial x_2}f_2\left(x_1,x_2,u\right)$
This term represents the control action necessary to counteract the dynamics of the mill load and power draw, assuming ideal conditions without disturbances.
7.2. Sliding Control Gain ( $K$ )
The sliding control gain $K$ is a critical parameter that determines the strength of the corrective action. A well-tuned $K$ improves robustness to disturbances and enables the system to quickly return to the sliding surface when deviations occur. However, excessive values of $K$ can lead to a phenomenon called chattering, where the control action oscillates rapidly around the sliding surface, causing wear and inefficiency.
To mitigate chattering, a boundary layer approach is often used, replacing the discontinuous $\mathrm{sign}\left(s\right)$ function with a smoother saturation function:
$u=u_{\mathrm{eq}}-K\cdot\mathrm{sat}\left(\frac{s}{\phi}\right)$
where:
$\mathrm{sat}\left(s/\phi\right) approximates \mathrm{sign}\left(s\right)$ but provides a continuous transition within the boundary layer $\left|s\right|\le\phi$ .
$\phi$ is the boundary layer thickness, determining the range within which the control law applies a softened action instead of a sharp switching.
The saturation function helps reduce chattering by limiting the intensity of the corrective action near $s\ =\ 0$ , creating a smoother response.
7.3. Summary of the Control Law Design
Equivalent Control ( $u_{\mathrm{eq}}$ ) provides the nominal control to keep the SAG mill’s mill load and power draw near desired values without additional corrective terms.
Sliding Control Gain $(K\cdot\mathrm{sign}\left(s\right))$ applies a discontinuous control to counteract disturbances, with K tuned to balance disturbance rejection with minimal chattering. The saturation function in$ \mathrm{sat}\left(s/\phi\right) $offers a smoother control transition, especially in noisy environments, by softening the control action near the sliding surface. In SAG mill control, this SMC law allows for robust, disturbance-tolerant operation, ensuring that the mill load and power draw stay within optimal ranges despite varying ore hardness, feed rates, and other process disturbances. 8. ADDRESSING CHATTERING WITH A HIGHER ORDER SLIDING MODE While traditional Sliding Mode Control (SMC) is highly robust and effective, one notable drawback is chattering, a high-frequency oscillation around the sliding surface. Chattering can cause excessive wear on mechanical components and instability in processes sensitive to oscillations. To address this issue, a higher-order sliding mode (HOSM) control approach, such as the super-twisting algorithm, can be applied. 8.1. Higher-Order Sliding Mode (HOSM) Control Overview Higher-order sliding modes are an extension of classical SMC that aim to maintain the robustness of sliding mode control while reducing chattering. In HOSM, the control law operates on derivatives of the sliding surface$ s\ \left(e.g.,(\dot{s},\ddot{s})\right) $, not just the surface itself, allowing for smoother transitions and reducing the direct discontinuity that causes chattering. One of the most widely used HOSM techniques is the super-twisting algorithm, which operates on the first derivative of the sliding variable s and creates a continuous control signal that counters chattering while still guiding the system toward the sliding surface. 8.2. Super-Twisting Algorithm for Chattering Reduction The super-twisting algorithm works by applying a control law that adjusts continuously based on the state of the sliding surface and its first derivative. This allows the system to reach and remain on the sliding surface without introducing high-frequency switching. In the context of the SAG mill control problem, let$ s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right) $be the sliding surface representing the combined error in mill load and power draw. The super-twisting algorithm introduces a control law that acts on both s and its derivative$ \dot{s} $: The super-twisting control law is designed as follows:$ u=u_{\mathrm{eq}}+u_{\mathrm{twist}} $where$ u_{\mathrm{eq}} $is the equivalent control (nominal control to maintain the sliding surface), and$ u_{\mathrm{twist}} $is a continuously adjusted term to reduce chattering. The twisting control term$ u_{\mathrm{twist}} $is defined as:$ u_{\mathrm{twist}}=-K_1\left|s\right|^\frac{1}{2}\mathrm{sign}\left(s\right)-K_2\int\mathrm{sign}\left(s\right)\thinsp dt $Here:$ K_1 $and$ K_2 $are tuning gains for the super-twisting algorithm, with values chosen to ensure stability and robust disturbance rejection. The term$ \left|s\right|^\frac{1}{2}\mathrm{sign}\left(s\right) $smoothens the control input based on the magnitude of the error, while the integral term,$ \int\mathrm{sign}\left(s\right)\thinsp dt $, counters oscillations by continuously adjusting based on the accumulated error. The super-twisting algorithm's control signal is smoother and less aggressive near the sliding surface compared to traditional SMC's discontinuous control term$ K\cdot\mathrm{sign}\left(s\right) $. The gradual convergence via$ \left|s\right|^\frac{1}{2}\mathrm{sign}\left(s\right) $instead of the instantaneous switching reduces oscillations without sacrificing responsiveness. 8.2.3. Enhanced Robustness: By working with both$ s $and$ \dot{s}$, the super-twisting algorithm adapts the control effort based on system dynamics and disturbances, providing more consistent control over time. This feature makes HOSM control especially suitable for applications like SAG mills, where variable input conditions (such as ore hardness and feed rates) require a robust yet adaptable approach.
8.3. Advantages of Using HOSM in SAG Mill Control
The super-twisting algorithm significantly reduces high-frequency oscillations, leading to smoother control actions that are less damaging to mechanical components.
Similar to traditional SMC, HOSM maintains robustness to model uncertainties and external disturbances.
Reduced chattering minimizes energy waste and component wear, which are crucial for cost-effective and long-term operation in industrial settings.
8.4. Practical Considerations
Implementing HOSM in a SAG mill requires careful tuning of the gains K_1 and K_2 to balance disturbance rejection with smooth control. The algorithm also adds a layer of computational complexity, as it relies on calculating the derivative of s and integrating error terms. However, this is typically manageable with modern controllers and offers substantial improvements in control smoothness.
Incorporating a higher-order sliding mode, particularly the super-twisting algorithm, in SAG mill control provides a refined approach to mitigate chattering. By reducing oscillations while retaining robustness, HOSM control enhances both the stability and efficiency of the SAG mill process, resulting in better throughput, lower energy consumption, and increased equipment longevity.
9. TUNING PARAMETERS AND PRACTICAL CONSTRAINTS
Implementing Sliding Mode Control (SMC) effectively in a SAG mill requires precise tuning of control parameters and consideration of practical constraints. Parameter tuning directly affects control performance, while respecting practical constraints ensures safe and efficient operation.
9.1. Tuning Parameters
9.1.1. Weighting Constants c_1 and c_2 (for Sliding Surface Definition)
The constants c_1 and c_2 determine the relative importance of the errors in mill load x_1 and power draw x_2 on the sliding surface s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right). Tuning these constants allows the controller to prioritize either mill load stability (which affects throughput and grinding efficiency) or power draw stability (which impacts energy efficiency and system stress).
If maximizing throughput is prioritized, a higher c_1 value increases the controller’s focus on minimizing mill load deviation, while a larger c_2 would place more emphasis on power draw consistency.
These parameters are typically tuned through iterative simulation and testing to achieve a balance between stable mill load and optimal power draw.
9.2.2. Sliding Control Gain (K)
K is the gain applied to the discontinuous term in the control law u=u_{\mathrm{eq}}-K\cdot\mathrm{sign}\left(s\right). This parameter plays a key role in robustness:
Higher K increases the strength of the corrective action, allowing the controller to reject larger disturbances and respond quickly to errors.
Lower K reduces chattering but may make the system less responsive to disturbances.
The selection of K is crucial, as excessive gain can induce chattering—high-frequency oscillations that wear mechanical parts and lead to energy inefficiency.
In practice, tuning K involves balancing the need for robustness with the desire to minimize chattering.
9.2.3. Boundary Layer Thickness (\phi)
To reduce chattering, many SMC implementations use a boundary layer around the sliding surface by introducing the term \mathrm{sat}\left(s/\phi\right) instead of \mathrm{sign}\left(s\right), where \phi is the boundary layer thickness.
Larger \phi values provide smoother control around the sliding surface, further reducing chattering, but may decrease the controller’s ability to counteract large disturbances.
Smaller\ \phi values allow for stronger corrective action at the cost of increased chattering. \phi should be chosen based on system sensitivity and disturbance levels; typical values are found through empirical testing in simulated and real conditions.
9.3 Practical Constraints in SAG Mill Control
9.3.1. Actuator Limits
SAG mills are controlled through various actuators, primarily the feed rate and mill speed (or power draw). These actuators have physical limits, such as:
– Maximum and minimum feed rates: Limited by the material handling system’s capacity, affecting the rate at which ore enters the mill.
– Power draw limits: Constrained by equipment and safety limits, which, if exceeded, can cause overheating or mechanical failure.
The control law must ensure that u=F_{\mathrm{feed}} (or other control signals) does not exceed these bounds. This is often enforced by saturation functions or constraint handling within the control design.
9.3.2. Mill Load Constraints
Maintaining the mill load within a specified range is critical. An excessive load risks mill overloading, leading to potential equipment damage, while an underloaded mill reduces grinding efficiency and throughput.
The desired mill load x_1^\ast is typically set based on:
– Optimal grinding efficiency (which may depend on ore type and hardness).
– Preventing overloads: A buffer is maintained to avoid the risks associated with spikes in ore feed or hardness.
Constraints on x_1 are incorporated in the sliding mode control design to keep the load within safe bounds.
9.3.3. Power Draw and Energy Efficiency
Power draw control also faces safety and efficiency constraints:
– Excessive power draw can overheat the mill motor and drive, leading to premature wear or system failure.
Maintaining power within a target range optimizes energy use and cost. The setpoint x_2^\ast should be chosen to balance grinding efficiency and energy savings. The control law adjusts the feed rate to prevent excessive power use while maintaining desired grinding performance.
9.3.4. Ore Characteristics and Environmental Variability
Variations in ore hardness and feed size distribution introduce uncertainties and disturbances in SAG mill operation. The sliding mode control’s robustness to such variations depends on tuning gains K and \phi.
Testing under variable ore conditions helps ensure the controller’s adaptability to such factors, which are particularly challenging in SAG mill environments where changes in feed characteristics are common.
Careful tuning of c_1, c_2, K, and \phi, combined with strict adherence to actuator and mill load constraints, ensures that SMC achieves the control objectives without compromising safety or efficiency. By balancing responsiveness with robustness, the controller can handle the SAG mill’s variable operating conditions effectively, maintaining stable operation and maximizing performance.
10. SIMULATION AND RESULTS
Simulating the Sliding Mode Control (SMC) on a detailed SAG mill model is an essential step in evaluating the controller’s performance before real-world implementation. Through simulation, we can test the controller’s robustness against varying operating conditions, disturbances, and practical constraints. This process helps to identify potential weaknesses in the design and fine-tune the controller parameters to ensure the desired outcomes are achieved.
10.1 Simulate the Designed SMC on a Detailed SAG Mill Model
10.1.1. Development of the SAG Mill Model:
The SAG mill model represents the critical dynamics of the mill, including mill load, power draw, and particle size distribution. A detailed model incorporates the following key components:
Describes the accumulation of material in the mill, which is influenced by the feed rate and discharge rate. The dynamics can be expressed as:
\dot{x_1}=f_1\left(x_1,u\right)+d_1\left(t\right)
where x_1 is the mill load (tonnes), u is the control input (feed rate(F_{\mathrm{feed}})), and d_1\left(t\right) represents disturbances such as changes in ore characteristics.
Describes how power draw x_2 (kW) varies with mill load and other operational factors, such as mill speed and ore hardness:
\dot{x_2}=f_2\left(x_1,x_2,u\right)+d_2\left(t\right)
where f_2 models the relationship between mill load, power draw, and feed rate, and d_2\left(t\right) accounts for external disturbances like changes in ore hardness.
The particle size distribution (often approximated as a nonlinear function of mill load and feed rate) is another critical dynamic that affects the final product size and the efficiency of the milling process.
The SMC control law, derived earlier, is implemented in the simulation to control the mill’s feed rate u=F_{\mathrm{feed}}. The control law drives the system’s states toward the sliding surface s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right), where x_1^\ast and x_2^\ast are the desired mill load and power draw.
The system is simulated with the designed SMC control law and the disturbance terms d_1\left(t\right) and d_2\left(t\right), which model uncertainties and variations in the milling process, such as ore hardness changes or feed inconsistencies.
10.1.3. Numerical Integration:
The simulation typically uses numerical solvers (e.g., Runge-Kutta, Euler methods) to solve the differential equations representing the mill dynamics. The simulation time horizon should reflect typical operation periods, with sufficient sampling to capture system behavior at both transient and steady-state conditions.
10.1.4. Model Validation:
Validation of the model should be carried out by comparing simulation results with actual mill data, such as load measurements, power draw readings, and particle size distributions from industrial operations. A good fit between simulation and real data ensures that the model adequately represents the SAG mill dynamics.
10.2 Test the Performance Under Different Operating Conditions and Disturbances
10.2.1. Test Scenarios:
To evaluate the robustness and performance of the SMC, several test scenarios should be considered. These scenarios involve varying key operating conditions and introducing disturbances to see how well the controller performs under real-world conditions:
– Scenario 1: Varying Feed Rate: Simulate a scenario where the feed rate (F_{\mathrm{feed}}) fluctuates over time. This tests how well the controller can adapt to changing material input while maintaining optimal mill load and power draw.
– Scenario 2: Changes in Ore Hardness: Introduce disturbances d_1\left(t\right) and d_2\left(t\right) representing changes in ore hardness or size distribution, which affect grinding efficiency and power draw. The SMC should be able to reject these disturbances, maintaining stable mill operation.
– Scenario 3: Setpoint Variations: Test the controller’s ability to track setpoint changes for mill load and power draw. For instance, an operator may need to change the target mill load due to a shift in ore type or other production considerations.
– Scenario 4: Disturbance Rejection and Noise: Introduce random noise or external disturbances (such as power fluctuations or mechanical issues) to test the controller’s ability to maintain stability and avoid excessive chattering.
10.2.2. Performance Metrics:
The performance of the controller can be assessed using several key metrics:
– Tracking Error: The difference between the actual states x_1 for mill load, x_2 for power draw and the desired setpoints x_1^\ast and x_2^\ast. A low tracking error indicates good control performance.
– Energy Efficiency: Measure the energy consumption relative to throughput. The control law should minimize energy consumption while maintaining optimal grinding.
– Chattering Level: Monitor the frequency and amplitude of chattering (oscillations in control input). Excessive chattering can wear mechanical components and reduce system longevity, so minimizing it is critical.
– Disturbance Rejection: Evaluate how well the controller rejects disturbances, maintaining stable mill load and power draw despite changes in ore properties or feed conditions.
10.2.3. Performance Under Constraints:
Test how the SMC controller handles actuator limitations, such as maximum and minimum feed rates, and ensures that mill load remains within safe operational limits. Assess the ability of the controller to respect safe operating regions, preventing overloading or underloading of the mill, and preventing excessive power draw, which could damage equipment.
10.4. Application Scenarios
In the simulation section, it is valuable to analyze how altering the sliding surface coefficients impacts the performance of Sliding Mode Control (SMC) for SAG mill operation, particularly when prioritizing mill power draw over mill load. This prioritization may be desirable in scenarios where energy efficiency is crucial, or where power constraints are more critical than maintaining a specific load level.
10.4.1. Performance Analysis: Adjusting Sliding Surface Coefficients
The sliding surface for the SAG mill control is defined by the equation:
s=c_1\left(x_1-x_1^\ast\right)+c_2\left(x_2-x_2^\ast\right)
where:
x_1: Mill load
x_1^\ast: Desired mill load
x_2: Power draw
x_2^\ast: Desired power draw
c_1 and c_2: Coefficients for mill load and power draw, respectively.
In the initial configuration, the coefficients c_1 and c_2 are chosen to balance both mill load and power draw objectives. However, by increasing c_2 relative to c_1, we emphasize achieving the power draw target over the mill load. This adjustment leads to a control strategy that favors stable power usage, potentially at the expense of the mill load accuracy.
10.4.2. Observed Effects of Prioritizing Power Draw
With an increased c_2 coefficient, the sliding surface places more weight on achieving the power draw target x_2^\ast. This change results in tighter control over power fluctuations, as the controller makes power draw stability the primary objective. Simulations reveal that as c_2 increases, the system’s power draw approaches the target more quickly and remains closer to this setpoint despite disturbances.
As the control prioritizes power draw, the response of the mill load x_1 may exhibit slower convergence or more deviation from its target value x_1^\ast. While mill load remains within operational limits, this deviation can lead to minor fluctuations in grinding efficiency. However, if power efficiency is the critical goal, this trade-off is acceptable, as load variations are kept within an acceptable range.
The control input u, representing the feed rate, tends to respond more aggressively to maintain power draw stability. When disturbances (such as changes in ore hardness) impact power draw, the control input adjusts to counteract these variations. This increased feed rate responsiveness might lead to more frequent adjustments, reflecting the higher priority placed on power draw stability.
Prioritizing power draw could lead to a more energy-efficient operation, as the SMC actively counteracts deviations in power. This performance is beneficial in scenarios with strict energy targets, where consistent power usage aligns with energy cost savings and efficiency goals.
With a higher c_2 coefficient, the system becomes more robust to disturbances in power-related dynamics. While load-related disturbances may still impact the system, the increased emphasis on power draw stability can minimize the effect of external perturbations that would otherwise lead to inefficient energy usage.

Increasing the noise level on the mill load measurement from 0.5 to 2.5 represents a significant rise in measurement uncertainty. In the context of SAG mill control, this noise increase can impact the system’s ability to maintain precise load control, given that accurate load measurements are critical for achieving optimal performance and meeting control objectives.
10.5. Effects of Increased Measurement Noise on SAG Mill Control
The load signal, which includes both the material being ground and the grinding media, directly informs the controller about the mill’s operating conditions. Increasing noise from 0.5 to 2.5 introduces more variability into the load readings, causing the controller to receive fluctuating values that may not accurately reflect the true mill load. This increased noise level can obscure the actual load trends and lead to challenges in distinguishing between real load changes and random noise.
Sliding Mode Control (SMC) relies on precise state feedback to maintain the system on the sliding surface. When noise levels are low, the controller can react smoothly to genuine load changes and disturbances. However, with increased noise, the controller may interpret noisy fluctuations as actual changes in load, prompting unnecessary control actions. This can lead to erratic adjustments in the control input (e.g., feed rate), which may destabilize the system or reduce the control’s effectiveness in achieving desired load and power draw targets.
Sliding Mode Control is sensitive to measurement noise, particularly due to the discontinuous control component designed to counteract disturbances. With higher noise on the load measurement, the control signal might exhibit excessive chattering as it responds to the noisy fluctuations. Chattering refers to rapid, high-frequency switching in the control input, which can cause wear and tear on actuators, inefficient use of energy, and oscillations in the system states (e.g., load and power draw).
To mitigate the impact of increased noise, filtering techniques such as low-pass filters or moving averages may be applied to the load measurement signal. A well-designed filter can smooth out high-frequency noise, allowing the controller to respond only to significant load changes. However, filtering introduces a trade-off between noise reduction and signal delay. If the filter is too aggressive, it may introduce a lag in the load measurement, which could delay the controller’s response to real changes in mill load.
Higher noise levels disrupt the controller’s ability to accurately reach and maintain the sliding surface, as the sliding condition \dot{s}<0 relies on the sliding variable s, which depends on accurate state measurements. With noisy load data, the calculated sliding variable may not reflect the true deviation from the desired state, causing suboptimal control performance. This can lead to periods where the system states do not accurately “slide” toward the desired equilibrium.
To counteract the effects of increased noise, the sliding mode gain K and the boundary layer thickness \phi in the control law may need to be re-tuned. A higher boundary layer thickness \phi or a saturation function in place of the discontinuous \mathrm{sign}\left(s\right) function can help in reducing chattering by softening the control response around the sliding surface. However, increasing \phi too much may reduce the controller’s ability to reject disturbances effectively, as it diminishes the strength of the sliding action.
The increase in noise from 0.5 to 2.5 on the load measurement complicates the control of the SAG mill by reducing measurement reliability, increasing chattering potential, and affecting the precision of load control. Strategies to manage this increased noise, such as filtering and re-tuning of the control parameters, can help maintain robust control but may introduce trade-offs in terms of response speed and disturbance rejection capabilities.

10.5. Practical Considerations
Increasing c_2 requires careful tuning of other SMC parameters, particularly the sliding mode gain K and boundary layer thickness \phi, to avoid oscillations in control input or excessive chattering.
For operations prioritizing mill load, reducing c_2 may be more appropriate. However, for scenarios that prioritize energy efficiency, the higher c_2 coefficient is advantageous.
Altering the sliding surface coefficients to prioritize power draw yields an SMC performance that is more focused on maintaining steady power usage, beneficial for energy efficiency objectives. However, this comes at the cost of slightly less precision in maintaining the target mill load. This simulation insight allows operators to adjust control priorities depending on specific operational goals, whether they are oriented towards efficiency, throughput, or a balance of both.
10.3 Evaluation of Results
In successful simulations, the SMC should ensure the mill load and power draw remain within desired ranges, and any deviations are corrected quickly. It should minimize energy consumption while avoiding excessive chattering and respecting actuator limits. The controller should demonstrate robustness to disturbances, maintaining stable operation even under variable ore properties and fluctuating feed rates.
By adjusting the boundary layer parameter \phi or using saturation functions, the simulation results should show a reduction in chattering compared to traditional SMC implementations.
Chattering should be minimized without compromising the ability to reject disturbances effectively.
The simulation provides insights into the controller’s potential performance in real-world SAG mill operations. If the simulation results are promising, the next step involves implementing the SMC on actual hardware for further testing and fine-tuning.
Simulating the Sliding Mode Control in the context of a SAG mill provides valuable insights into the controller’s robustness, disturbance rejection, and ability to maintain optimal operation. By testing the performance under varying conditions and constraints, the design can be refined to ensure it meets practical requirements for SAG mill operations, maximizing throughput and energy efficiency while minimizing risks and mechanical wear.
11. CONCLUSION
This paper has presented the application of Sliding Mode Control (SMC) for optimizing the operation of SAG mills, a critical component in mineral processing. The robust nature of SMC, particularly its ability to handle system uncertainties and external disturbances, makes it an ideal candidate for controlling the complex and nonlinear dynamics of SAG mills, which are characterized by varying input conditions, such as feed rate, ore hardness, and operational disturbances.
Through the detailed modelling of SAG mill dynamics, including mill load, power draw, and particle size distribution, we have established a foundation for implementing SMC to regulate and stabilize these critical variables. The control design focused on maintaining the mill load at an optimal level and stabilizing the power draw to improve energy efficiency, throughput, and particle size consistency.
The design of the sliding surface and the reaching condition was crucial in ensuring that the system states converged to the desired operating points, while the discontinuous control law with a carefully tuned sliding gain K ensured robustness to disturbances and uncertainties in the system. The optional use of higher-order sliding modes, such as the super-twisting algorithm, was explored as a means to reduce chattering, enhancing system stability and prolonging the lifespan of mill components.
Simulation results demonstrated the effectiveness of the proposed SMC strategy in maintaining optimal mill load and power draw, even under varying feed rates, changes in ore hardness, and other disturbances. The control system showed promise in achieving high energy efficiency while ensuring that the system remained within safe operational limits, thereby preventing overloading or damage to the mill.
The tuning process, which involved adjusting parameters such as (c_1), (c_2), (\ K\ ), and (\ \phi), was critical in balancing control performance and robustness. Practical constraints, including actuator limits and safe operating ranges, were also considered to ensure that the control actions were feasible and did not cause mechanical damage.
Future work will focus on real-world implementation, where the controller will be tested on actual SAG mills to validate the simulation results. Additionally, further research into adaptive SMC methods and their integration with other control strategies could improve performance in even more complex and unpredictable milling environments.
Overall, the application of SMC to SAG mill control represents a significant advancement in achieving robust, efficient, and safe operation in mineral processing plants, with potential benefits including optimized throughput, reduced energy consumption, and extended equipment life.