Control Theory: Precision Control: How Degrees of Freedom Shape Modern Control Theory

1. Introduction to Control Theory and Degrees of Freedom

Control theory is a fascinating and intricate field that sits at the intersection of mathematics, engineering, and computer science. It is fundamentally concerned with the behavior of dynamical systems and the ability to direct these systems to behave in a desired manner. At the heart of control theory is the concept of degrees of freedom (DOF), which refers to the number of independent parameters that define the state of a system. In essence, the degrees of freedom are the "knobs" or "levers" that a control engineer can adjust to influence the system's behavior. The more degrees of freedom a system has, the more complex it is to control, but also the more capable it is of performing intricate and varied tasks.

From the perspective of a mechanical engineer, degrees of freedom might refer to the number of axes along which a robotic arm can move. For an aerospace engineer, it could mean the various flight control surfaces on an aircraft. Meanwhile, an electrical engineer might consider the voltage and current levels in a circuit as degrees of freedom. Each viewpoint brings its own set of challenges and insights into how control theory is applied.

Here are some in-depth points about control theory and degrees of freedom:

1. Fundamental Principles: At its core, control theory uses mathematical models to characterize and analyze systems. These models often involve differential equations that describe how the system evolves over time.

2. Feedback Loops: A key concept in control theory is the feedback loop, which allows the system to adjust its behavior based on its performance. For example, a thermostat uses feedback to maintain a room's temperature by switching the heating or cooling system on and off.

3. Stability and Response: Control theory is concerned with the stability of a system, which is its ability to return to a desired state after being disturbed. The response of a system—how quickly and accurately it reaches the desired state—is also crucial.

4. Optimization: Often, control systems aim to optimize a certain performance criterion, such as minimizing energy consumption or maximizing speed, within the constraints set by the degrees of freedom.

5. Adaptive and Robust Control: These are advanced areas of control theory that deal with systems that can adjust their behavior in response to changing conditions (adaptive) or maintain performance despite disturbances or uncertainties (robust).

To illustrate these concepts, consider an unmanned aerial vehicle (UAV) with six degrees of freedom: three translational (movement along the x, y, and z axes) and three rotational (pitch, yaw, and roll). The control system of the UAV must be able to manage these degrees of freedom simultaneously to navigate through the air successfully. If the UAV encounters wind turbulence (a disturbance), a robust control system would adjust the control surfaces to maintain its flight path without deviating from its intended trajectory.

In summary, control theory and degrees of freedom are integral to designing systems that can perform complex tasks reliably and efficiently. By understanding and manipulating the degrees of freedom, control engineers can create systems that respond predictably to inputs and maintain stability even in the face of uncertainty. This interplay between control and freedom is what enables the precision control that is so vital in modern technology.

2. The Role of Degrees of Freedom in System Dynamics

In the realm of system dynamics, the concept of degrees of freedom plays a pivotal role in determining the behavior and controllability of a system. Essentially, degrees of freedom refer to the number of independent parameters that define the state of a system. In control theory, these parameters are crucial as they dictate the possible movements and responses of a system to external inputs. The more degrees of freedom a system has, the more complex its potential behavior can become, offering both challenges and opportunities for precision control.

From an engineering perspective, each degree of freedom corresponds to an independent motion or force that can be controlled or observed. For instance, a robotic arm with three degrees of freedom can move in three-dimensional space—up and down, left and right, forward and backward—but cannot rotate without an additional degree of freedom. Here's an in-depth look at how degrees of freedom shape modern control theory:

1. Defining System Capabilities: The number of degrees of freedom directly influences a system's capabilities. A vehicle with a steering wheel and pedals has fewer degrees of freedom than an aircraft with multiple control surfaces and thrust vectors, which allows for more complex maneuvers.

2. control System design: Engineers must consider the degrees of freedom when designing control systems. For a drone, which typically has six degrees of freedom (three translational and three rotational), a six-axis control system is necessary to manage its flight dynamics.

3. Stability and Performance: Systems with multiple degrees of freedom can be more challenging to stabilize. Control strategies like PID (Proportional-Integral-Derivative) control must be carefully tuned for each degree of freedom to ensure optimal performance.

4. Observability and Controllability: The concepts of observability and controllability are tied to degrees of freedom. A system is fully controllable if it can be driven from any initial state to any final state, given sufficient time. Similarly, it is fully observable if the state of the system can be determined by observing outputs over time.

5. Redundancy and Fault Tolerance: In safety-critical systems, redundancy in degrees of freedom can enhance fault tolerance. For example, an airplane may have multiple control surfaces for the same movement to ensure it can still be controlled if one fails.

6. Optimization and Efficiency: Degrees of freedom allow for optimization of system performance. In a multi-joint robotic arm, the degrees of freedom can be exploited to find the most efficient path to perform a task.

7. human-Machine interaction: In systems designed for human interaction, such as virtual reality controllers, degrees of freedom can enhance the user experience by providing more natural and intuitive control.

Example: Consider a spacecraft docking procedure, which is a delicate operation requiring precise control over multiple degrees of freedom. The spacecraft must align its position and orientation with the docking port, which involves translational and rotational movements. Advanced control algorithms utilize the spacecraft's degrees of freedom to achieve a successful dock without collision.

Degrees of freedom are fundamental to understanding and designing dynamic systems. They influence everything from the basic capabilities of a system to the complexity of the control strategies required to manage it. As technology advances, the interplay between degrees of freedom and control theory will continue to evolve, leading to more sophisticated and capable systems.

3. Fundamentals of Precision Control in Engineering

Precision control in engineering is a critical aspect of modern control theory, particularly when it comes to systems with multiple degrees of freedom. The ability to manage and manipulate these degrees of freedom with high accuracy is what sets precision control apart from more traditional control methods. This approach is essential in fields where even the slightest deviation can lead to significant errors or system failures, such as in aerospace engineering, robotics, and semiconductor manufacturing.

From the perspective of mechanical engineering, precision control involves the fine-tuning of mechanical components and systems to achieve desired motion or force outcomes. For instance, in robotics, each joint or actuator represents a degree of freedom, and precise control is necessary to ensure the robot's arm moves smoothly and accurately to the intended position.

In electrical engineering, precision control often relates to the accurate regulation of current and voltage to achieve stable system performance. For example, in power electronics, controlling the switching of semiconductor devices with precision is vital for the efficient conversion and distribution of electrical power.

From a computer science standpoint, precision control is about the algorithms and software that process sensor data and command actuators. The development of advanced algorithms for real-time control is a key factor in achieving high precision, especially in systems like autonomous vehicles, where decision-making speed and accuracy are paramount.

Here are some in-depth points on the fundamentals of precision control in engineering:

1. Feedback Loops: The cornerstone of precision control is the feedback loop, which allows the system to adjust its actions in response to sensor data. For example, a temperature control system in a chemical reactor uses feedback from temperature sensors to modulate the heating element, maintaining the reaction at the optimal temperature.

2. Sensor Fusion: Combining data from multiple sensors can enhance the precision of control systems. In autonomous drones, sensor fusion might involve integrating data from GPS, accelerometers, and cameras to accurately determine the drone's position and orientation.

3. Control Algorithms: Advanced control algorithms, such as Proportional-Integral-Derivative (PID) controllers or state-space controllers, are employed to manage complex systems with high precision. For instance, PID controllers are widely used in industrial processes to maintain setpoints with minimal overshoot and steady-state error.

4. Actuator Design: The design and selection of actuators play a significant role in precision control. For example, in precision machining, the choice of motors and drive systems determines the accuracy and repeatability of the cutting tool's movements.

5. System Modeling: Accurate mathematical models of the system are essential for precision control. These models help predict system behavior and design control strategies. In satellite attitude control, detailed models of the satellite's dynamics are used to design thruster firing sequences that keep the satellite oriented correctly.

6. Noise Reduction: Minimizing noise and disturbances is crucial for precision control. Techniques such as filtering and shielding are used to reduce the impact of electrical noise on sensor readings and control signals.

7. Calibration: Regular calibration ensures that sensors and actuators perform accurately over time. In medical imaging equipment, calibration is necessary to ensure that the images produced are precise and reliable for diagnostic purposes.

8. Redundancy: Implementing redundancy can enhance the reliability and precision of control systems. In aircraft control systems, redundancy is built into critical components to ensure that the aircraft can be controlled precisely even in the event of a component failure.

To illustrate these principles, consider the example of a high-precision CNC (Computer Numerical Control) machine. The machine's control system uses feedback from position sensors to adjust the speed and position of the cutting tool, ensuring that the final product matches the design specifications with micrometer-level accuracy. The system employs advanced control algorithms to compensate for factors like tool wear and thermal expansion, which could otherwise affect the precision of the machining process.

In summary, the fundamentals of precision control in engineering revolve around the meticulous management of system components and the intelligent application of control strategies to achieve the desired level of accuracy and reliability. As technology advances, the pursuit of even greater precision continues to drive innovation across multiple engineering disciplines.

4. Constraints and Capabilities

In the realm of control theory, the interplay between constraints and capabilities is a dance of precision and possibility. This delicate balance dictates the efficacy of any control system, whether it's guiding a spacecraft to a distant planet or maintaining the temperature in your home. Constraints, in this context, are not merely limitations but rather the defined boundaries within which the system must operate. They are the rules of the game, setting the stage for the capabilities to shine. Capabilities, on the other hand, represent the potential of the system to achieve its goals within those constraints. They are the moves in the dance, executed with precision to navigate the complex space of possibilities.

1. Degrees of Freedom: The concept of degrees of freedom in control systems refers to the number of independent parameters that define the state of the system. For instance, a robotic arm with three joints has three degrees of freedom. The more degrees of freedom, the more complex the control strategy, as each one adds a layer of decision-making and potential movement.

2. Constraints: Constraints can be physical, such as the range of motion of a robotic joint, or they can be performance-related, such as the speed at which a system can respond. In an aircraft control system, physical constraints might include the aerodynamic properties of the wings, while performance constraints could involve the reaction time needed to adjust to turbulence.

3. Capabilities: A system's capabilities are defined by how well it can perform within its constraints. This includes its accuracy, efficiency, and adaptability. For example, a drone's flight control system must be capable of precise movements to navigate through tight spaces, despite the constraints of battery life and payload capacity.

4. Optimization: optimization is the process of finding the best solution within the given constraints. Control theorists use mathematical models to predict system behavior and optimize performance. An example is the tuning of a PID controller to minimize the error in a temperature control system.

5. Adaptability: A key capability in modern control theory is adaptability, which allows a system to adjust to changing constraints in real-time. Adaptive control systems can learn from past performance and adjust their parameters for optimal future performance. This is seen in autonomous vehicles that adjust their driving strategy based on traffic conditions.

6. Robustness: Robustness is the ability of a control system to maintain performance despite disturbances or uncertainties. A robust control system for a satellite might maintain its orbit despite the gravitational pull of other celestial bodies or solar radiation pressure.

7. feedback loops: Feedback loops are essential for maintaining control within constraints. They allow the system to monitor its own performance and make adjustments as needed. A thermostat uses a feedback loop to maintain room temperature, constantly adjusting the heating or cooling output based on the difference between the desired and actual temperatures.

8. Limitations: Understanding the limitations of a control system is crucial for safe operation. For example, the maximum speed of a motor might limit how quickly a robotic arm can move, which in turn affects the overall performance of the manufacturing process it's a part of.

Exploring the limits of control systems through the lens of constraints and capabilities offers a rich tapestry of challenges and innovations. It's a field where theoretical knowledge meets practical application, and where the boundaries of what's possible are continually pushed by the advancements in technology and understanding. Whether it's the graceful arc of a satellite's orbit or the precise cut of a CNC machine, control theory remains at the heart of these marvels, orchestrating the symphony of variables that define our modern world.

5. Precision Control in Robotics and Aerospace

In the realm of control theory, precision control stands as a cornerstone, particularly in the fields of robotics and aerospace. The intricate dance of variables and constraints in these domains is a testament to the sophistication of modern control systems. Robotics, with its myriad applications from manufacturing to medicine, demands a level of precision that can only be achieved through meticulous control of multiple degrees of freedom. Similarly, aerospace applications, where the stakes are astronomically high, rely on precision control to navigate the unforgiving environment of space. Both fields exemplify the critical importance of precision in ensuring the safety, efficiency, and success of complex systems.

From the perspective of robotics, consider the robotic arms used in assembly lines. These marvels of engineering must perform repetitive tasks with unerring accuracy, a feat made possible by advanced control systems that monitor and adjust every movement in real-time. In aerospace, precision control is epitomized by the thrusters of a spacecraft, which must fire with exact timing and intensity to achieve the desired trajectory and orientation.

1. Robotic Surgery:

- Precision: Sub-millimeter accuracy in movements.

- Degrees of Freedom: Multiple arms with 6-7 degrees each.

- Example: The da Vinci Surgical System, which allows surgeons to perform delicate procedures with enhanced vision, precision, and control.

2. Autonomous Vehicles:

- Precision: Centimeter-level positioning.

- Degrees of Freedom: Steering, acceleration, braking, and gear shifting.

- Example: Self-driving cars using LIDAR and complex algorithms to navigate traffic.

3. Spacecraft Attitude Control:

- Precision: Arcsecond-level orientation accuracy.

- Degrees of Freedom: Three-axis control for pitch, yaw, and roll.

- Example: The Hubble Space Telescope, which uses gyroscopes and reaction wheels to maintain precise pointing for astronomical observations.

4. Industrial Robots:

- Precision: Consistent repeatability within fractions of a millimeter.

- Degrees of Freedom: Typically 4-6 for tasks like welding, painting, and assembly.

- Example: KUKA robots, which are integral to automotive manufacturing lines.

5. Drone Flight Control:

- Precision: Stable hovering and maneuvering within tight spaces.

- Degrees of Freedom: Propeller speed, tilt angles, and GPS positioning.

- Example: Consumer drones like the DJI Phantom, which can capture stable aerial footage even in windy conditions.

The case studies from robotics and aerospace sectors highlight the pivotal role of precision control. They showcase how control theory is not just a set of mathematical principles but a practical tool that, when applied with finesse, can lead to groundbreaking advancements in technology and science. The future of these fields hinges on the continuous evolution of control systems that can handle an ever-increasing number of variables with grace and accuracy. The journey of control theory is far from over; it is, in fact, propelling us towards a future where the line between the possible and the impossible becomes increasingly blurred.

6. Advanced Techniques for Managing Multiple Degrees of Freedom

In the realm of control theory, managing multiple degrees of freedom (DOF) is akin to orchestrating a symphony where each instrument must be in perfect harmony with the others. The complexity of this task increases exponentially as the number of DOFs rises, presenting a formidable challenge for engineers and researchers alike. This is particularly true in systems where precision control is paramount, such as in robotics, aerospace, and advanced manufacturing processes. The key to mastering this complexity lies in the deployment of advanced techniques that can handle the intricate interplay between various DOFs.

1. State-Space Representation: One of the most powerful tools in handling multiple DOFs is the state-space representation. This mathematical model captures the dynamics of a system by describing its state variables and their evolution over time. For example, in a robotic arm with six DOFs, the state-space model would include variables for each joint's position, velocity, and acceleration, providing a comprehensive view of the system's behavior.

2. Model Predictive Control (MPC): MPC is a sophisticated control strategy that uses a model of the system to predict future states and optimize control inputs accordingly. It's particularly useful in scenarios where multiple DOFs must be controlled simultaneously, as it can anticipate and mitigate the effects of interactions between them. Consider an autonomous vehicle navigating a complex environment; MPC would enable it to adjust its trajectory, speed, and steering angle in real-time to avoid obstacles while maintaining stability.

3. Feedback Linearization: This technique involves transforming a nonlinear system into an equivalent linear one through feedback, making it easier to apply linear control methods. In the context of a drone with multiple DOFs, feedback linearization could simplify the control of its position and orientation by decoupling these variables, allowing for independent control of each axis.

4. Adaptive Control: Adaptive control methods adjust the controller parameters in real-time to cope with uncertainties and changes in the system's dynamics. This is crucial for systems with multiple DOFs where the interactions can lead to unpredictable behavior. For instance, a satellite with adjustable solar panels and antennas would benefit from adaptive control to maintain optimal orientation despite varying environmental conditions.

5. Decentralized Control: In large systems with numerous DOFs, decentralized control allows for the distribution of the control effort across multiple controllers, each responsible for a subset of DOFs. This approach enhances scalability and robustness. An example is the control of a swarm of drones, where each drone autonomously controls its own DOFs while coordinating with the others to achieve a collective objective.

6. Intelligent Control: Leveraging artificial intelligence and machine learning, intelligent control systems can learn and adapt to complex multi-DOF systems. They can identify patterns and predict outcomes, enabling more efficient and effective control strategies. A humanoid robot, for example, could use intelligent control to learn new tasks by observing and mimicking human movements.

By employing these advanced techniques, engineers can achieve a level of precision and flexibility that was previously unattainable, opening up new possibilities in the design and control of complex systems. As technology continues to advance, the management of multiple degrees of freedom will remain a central theme in the evolution of control theory, driving innovation and enabling the creation of more sophisticated and capable machines.

7. The Impact of Computational Power on Modern Control Systems

The advent of advanced computational power has revolutionized the field of control systems, enabling unprecedented levels of precision and efficiency. In modern control theory, the concept of degrees of freedom plays a pivotal role, allowing for the design of systems that can manage multiple inputs and outputs simultaneously. Computational power is the backbone of this complexity, providing the necessary resources to process and analyze vast amounts of data in real-time. This has led to the development of sophisticated algorithms and models that can predict system behavior, compensate for disturbances, and optimize performance.

From the perspective of an engineer, the increase in computational power means the ability to implement more complex control strategies such as Model Predictive Control (MPC) or adaptive control. These strategies require significant computational resources as they involve solving optimization problems or updating models in real-time based on incoming data.

1. real-time processing: Modern control systems often operate in real-time, with computations that need to be completed within a fraction of a second. For instance, anti-lock braking systems (ABS) in vehicles must rapidly adjust brake pressure to prevent wheel lockup.

2. Complex Simulations: Before implementing a control system, engineers use computational power to run simulations. These simulations can model how a system behaves under various conditions, which is crucial for systems like autonomous drones, which must operate safely in unpredictable environments.

3. Data Handling Capabilities: With the rise of the Internet of Things (IoT), control systems are now expected to handle and process data from a multitude of sensors. This is evident in smart grids, where computational power is used to balance electricity supply and demand across an entire network.

4. Algorithm Development: The development of algorithms for control systems has seen a significant boost. For example, machine learning algorithms are now used to enhance the performance of control systems in robotics, allowing robots to learn and adapt to new tasks.

5. Fault Tolerance and Recovery: Increased computational power allows for the implementation of fault-tolerant control systems that can detect failures and reconfigure themselves to maintain operation. This is particularly important in aerospace applications, where system reliability is critical.

6. Optimization: Computational power enables the use of optimization techniques in control systems to achieve the best performance. This is seen in industrial automation, where systems are optimized for speed, accuracy, and energy efficiency.

7. Integration of Multiple Systems: Modern control systems often need to integrate with other systems and platforms. In automated manufacturing, computational power allows for the seamless integration of control systems with enterprise resource planning (ERP) systems for better coordination and efficiency.

8. user Interface and experience: Finally, computational power has improved the user interface and experience of control systems. In home automation systems, users can interact with their devices more intuitively, thanks to the sophisticated software that processes user inputs and controls the environment accordingly.

The impact of computational power on modern control systems is profound and multifaceted. It has enabled systems to be more responsive, adaptable, and efficient, ultimately leading to innovations that were once thought to be beyond reach. As computational capabilities continue to grow, the potential for further advancements in control theory remains vast and exciting.

8. Challenges and Solutions in High-Degree Freedom Systems

In the realm of control theory, high-degree freedom systems represent a pinnacle of complexity and potential. These systems, characterized by a large number of independent movements or variables, offer unparalleled precision and versatility in applications ranging from robotics to aerospace engineering. However, this complexity also introduces a myriad of challenges that can stymie even the most sophisticated control strategies. The intricacies of managing multiple degrees of freedom (DoF) necessitate a deep understanding of the system's dynamics and the interplay between its various components.

One of the primary challenges in high-degree freedom systems is the curse of dimensionality. As the number of DoFs increases, the space of possible states that the system can occupy expands exponentially. This makes it difficult to model, predict, and control the system's behavior. To address this, engineers and researchers employ various strategies:

1. Model Simplification: By identifying and focusing on the most influential DoFs, control theorists can simplify the model without significantly compromising performance. For example, in a humanoid robot, while it may have dozens of joints, the primary ones affecting balance and locomotion might be prioritized for control.

2. Decomposition: Breaking down a complex system into smaller, more manageable subsystems can make control more feasible. Each subsystem can be controlled independently, and their interactions can be managed through a higher-level controller. This approach is akin to how the human nervous system controls limbs.

3. Advanced Control Algorithms: utilizing cutting-edge algorithms like model predictive control (MPC) or adaptive control can help manage the complexity. These algorithms can anticipate future states and adjust control inputs accordingly, or adapt to changes in the system's dynamics over time.

4. Machine Learning: AI and machine learning techniques, such as reinforcement learning, can be employed to develop control policies that would be too complex to design manually. For instance, Google's DeepMind used reinforcement learning to teach robotic arms to manipulate physical objects with a high degree of dexterity.

5. Sensor Fusion: Integrating data from multiple sensors can provide a more comprehensive understanding of the system's state, which is crucial for precise control. In autonomous vehicles, sensor fusion combines inputs from cameras, LIDAR, and radar to create a detailed picture of the environment.

6. Robust Design: Designing systems that are inherently robust to variations and uncertainties can reduce the burden on the control system. This might involve creating mechanical linkages that are less sensitive to external forces or disturbances.

7. Feedback and Feedforward Control: Implementing both feedback (reactive) and feedforward (predictive) control mechanisms can enhance the system's responsiveness and stability. Feedback control can correct for unexpected disturbances, while feedforward control can preemptively adjust for known disturbances.

By leveraging these strategies, control theorists and engineers can overcome the daunting challenges posed by high-degree freedom systems. The solutions not only enable the precise control of complex systems but also pave the way for innovations that can transform industries and improve our interaction with technology. As we continue to push the boundaries of what's possible, the insights gained from tackling these challenges will undoubtedly lead to even more sophisticated and capable systems in the future.

9. Trends and Innovations

As we delve into the future of control theory, we stand on the precipice of a technological revolution that promises to redefine the boundaries of precision and efficiency in various systems. The field of control theory, which has been instrumental in shaping the modern world through its applications in everything from aerospace to consumer electronics, is poised for transformative changes. Innovations in computational power, algorithmic design, and cross-disciplinary integration are converging to create a new paradigm in how we approach the concept of control in complex systems.

1. Adaptive Control Systems:

The advent of adaptive control systems marks a significant milestone in control theory. These systems are designed to adjust their parameters in real-time, responding to changes in the environment or system dynamics. For example, modern drones utilize adaptive control mechanisms to stabilize flight in varying weather conditions, ensuring smooth operation without manual recalibration.

2. machine Learning integration:

Machine learning algorithms are increasingly being integrated into control systems, enabling them to learn from data and improve performance over time. An example of this is the use of reinforcement learning in robotics, where robots can optimize their movements and decision-making processes through trial and error, leading to more efficient and autonomous operation.

3. Quantum Control:

Quantum control is an emerging field that explores the control of quantum systems, such as qubits in quantum computers. This involves precise manipulation of quantum states to perform computations, which could revolutionize computing power and speed. Researchers are developing algorithms that can control quantum bits with high fidelity, a critical step towards the realization of practical quantum computing.

4. Networked Control Systems:

The rise of the Internet of Things (IoT) has led to the development of networked control systems, where multiple devices are interconnected and work collaboratively. Smart grids are a prime example, where various energy sources and consumers are managed in real-time to optimize energy distribution and consumption.

5. Robust Control in Uncertain Environments:

Control systems are increasingly required to operate in uncertain and unpredictable environments. Robust control theory focuses on ensuring system stability and performance despite these uncertainties. Autonomous vehicles, for instance, must navigate complex and dynamic road conditions while maintaining safety and reliability.

6. Human-in-the-loop Control:

Human-in-the-loop control systems blend human intuition and decision-making with automated control processes. In advanced prosthetics, sensors and algorithms interpret neural signals to control artificial limbs, but the user's feedback is crucial for fine-tuning the system's response.

7. Environmental and Sustainable Control:

Control theory is also playing a pivotal role in environmental sustainability. Smart irrigation systems, for example, use sensors and control algorithms to optimize water usage based on soil moisture levels and weather predictions, reducing waste and conserving resources.

The future of control theory is not just about technological advancements; it's about the integration of these technologies into society, the economy, and the environment. It's a future that holds the promise of enhanced precision, adaptability, and sustainability, shaping the way we interact with and control the world around us. As we continue to push the boundaries of what's possible, control theory will undoubtedly remain at the forefront of innovation, driving progress across a multitude of domains.

As a kid, I grew up middle class, but my father was a great innovator with an entrepreneurial spirit, and it wasn't long before my family became part of the infamous 1%.

Betsy DeVos