Sample-Based MPC (Sample-Based MPC) Overview
Sample-Based Model Predictive Control (Sample-Based MPC) is a type of model predictive control (MPC) that predicts the future behaviour of a system and calculates the optimum control inputs. It is a method characterised by its ease of application to non-linear and high-dimensional systems and its ease of ensuring real-time performance, compared to conventional MPC. An overview of the method is given below.
1. model predictive control (MPC):
– Abstract: MPC is a control method that predicts the future behaviour of a system using a model and optimises the control inputs within a specified prediction horizon. The optimisation is usually based on some objective function (e.g. minimisation of tracking errors or energy consumption).
– Scope of application: applies to a wide range of areas such as industrial processes, robotics, vehicle control, energy management, etc.
2. sample-based MPC:
– Abstract: Sample-based MPC is an approach that samples the behaviour of a system and predicts future behaviour from samples, whereas conventional MPC is model-based and predicts future behaviour.
– Features: because it is based on sampling, it is easily applicable to non-linear and high-dimensional systems. It reduces computational complexity and improves real-time performance.
3. components of sample-based MPC:
Sampling: random or planned sampling of the system’s state space to collect data for predicting future behaviour.
Predictive horizons: predict the behaviour of the system using sampled data over a specific period of time in the future (predictive horizons).
Optimisation: optimise the control inputs based on the system behaviour within the prediction horizon. Objective functions include minimising tracking errors and maximising energy efficiency.
Applying control inputs: applying the optimised control inputs to the system and controlling the system until the next sampling time.
4. benefits of sample-based MPC:
Application to non-linear systems: sample-based MPC can directly address system non-linearity and does not rely on conventional linear models, making it applicable to complex systems.
Computational efficiency: sample-based MPC is computationally efficient and suitable for real-time control, as it uses sampled data and is independent of model complexity.
Adaptability: because it is based on sampling, it can flexibly adapt to environmental fluctuations and uncertainties.
Algorithms associated with Sample-Based MPC (Sample-Based MPC)
The algorithms associated with Sample-Based Model Predictive Control (Sample-Based MPC) use samples to predict the future behaviour of the system and to determine the optimal control inputs. The algorithms commonly used in sample-based MPC and their characteristics are described below.
1. random sampling:
Abstract: Random sampling is a technique that randomly searches the control input space to generate samples. This allows a wide range of input spaces to be covered and the diverse behaviour of the system to be captured.
Features:
– Simple implementation: the random sample generation makes the implementation simple.
– Extensive search: the ability to search a wide input space allows for systems with strong non-linearity.
2. particle filter:
Description: Particle filters are probabilistic algorithms that represent the state of the system as a series of particles (samples) and update these sequentially.
Features:
– Tolerant of non-linear systems: it enables highly accurate estimation even in non-linear systems and noisy environments.
– Dynamic adaptation: adapts to system variations by dynamically adjusting particle weights.
3. Monte Carlo Tree Search (MCTS):
Abstract: MCTS is a method that uses a tree structure to search for possible future states and generate samples. It is mainly used in game AI, but is also applied to MPC. See Detail in “Overview of Monte Carlo Tree Search and Examples of Algorithms and Implementations“
Features:
– Consideration of future scenarios: the use of tree structures allows multiple future scenarios to be considered in detail.
– Balanced exploration: using algorithms (e.g. UCB1) to balance exploration and utilisation and to generate samples efficiently.
4. receding horizon control (Lanein-Bargain):
Abstract: Lenin Bargain is a method that updates the predicted horizon at regular intervals to sequentially find the optimal control input.
Features:
– Sequential optimisation: real-time performance is easily ensured due to the sequential optimisation.
– Adaptability: by dynamically adjusting the horizon, it can respond flexibly to system fluctuations.
5. Gaussian Processes (GP):
Abstract: GP is a probabilistic method for learning dynamic models of systems and using them to generate samples.
Features:
– Uncertainty handling: uncertainty in the model can be explicitly taken into account, improving the accuracy of sample generation.
– Learning-based: can be applied to complex systems as it learns system behaviour from data.
6. Sample-Based Dynamic Programming:
Abstract: a sampling version of dynamic programming, where the state space is explored based on samples to find the optimal policy.
Features:
– Applicable to high-dimensional systems: by using samples of the state space, it can be applied to high-dimensional systems.
– Sequential updating: the state space is updated sequentially, making it suitable for real-time control.
7. hybrid methods:
Abstract: combines multiple sampling methods to provide a more effective approach to sample generation and prediction.
Features:
– Flexibility: the advantages of multiple methods can be exploited to build the best sampling strategy for the characteristics of the system.
– Balancing accuracy and efficiency: balances sampling accuracy and computational efficiency to achieve practical control performance.
Sample-Based MPC (Sample-Based MPC) application examples
Sample-Based Model Predictive Control (Sample-Based MPC) is an effective control method for complex systems with non-linearity and high dimensionality and has been applied in various fields. Some specific applications are described below.
1. route planning and driving control of automated vehicles:
Case study overview: automated vehicles need to navigate and control in a dynamic environment in real time. Sample-based MPC uses predictive models to simulate future vehicle behaviour and calculate optimal driving manoeuvres.
Details:
– Sampling: random sampling of the vehicle’s future position and speed to generate multiple scenarios.
– Predictive horizon: predicts the route several seconds ahead, taking into account obstacle avoidance and compliance with traffic rules.
– Optimisation: determines optimal driving manoeuvres from the predicted path with the aim of avoiding collisions, driving safely and maximising energy efficiency.
Example implementations: self-driving technology developers such as Waymo and Tesla use sample-based MPC for real-time route planning and control of vehicles.
2. drone flight control and obstacle avoidance:
Case study overview: drones need to avoid obstacles in real-time while flying in complex 3D environments to reach their destination. Using sample-based MPC, safe and efficient flight is achieved.
Details:
– Sampling: multiple samples of the drone’s flight path are generated and predicted.
– Predictive horizon: predicts the flight path a few seconds ahead and takes into account the position of obstacles.
– Optimisation: calculates the optimal flight path towards the target point while avoiding obstacles.
Example implementations: sample-based MPCs are used in control systems for commercial drones and delivery drones to ensure safe flight and efficient path planning.
3. precision manipulation of robotic arms:
Case study overview: industrial robot arms are required to operate with high precision in complex work environments. Using sample-based MPC, the optimum movement is planned while taking non-linearity and interference into account.
Details:
– Sampling: the joint angles of the robot arm and the positions of the end-effectors are generated as samples and predicted.
– Predictive horizon: predicts the movements to complete the task, taking into account interferences and constraints.
– Optimisation: calculates the optimum joint angles and forces to achieve highly accurate movements.
Implementation examples: sample-based MPC is applied for precise motion control of robot arms in areas such as automobile manufacturing and electronics assembly.
4. energy management systems:
Case study overview: optimisation of energy consumption is required in building energy management and smart grids. Sample-based MPC enables real-time energy consumption forecasting and optimisation.
Details:
– Sampling: sampling energy consumption patterns to predict future consumption.
– Predictive horizon: predicts daily energy consumption and takes into account peak shifts and demand response.
– Optimisation: calculate optimal energy management strategies with the aim of minimising energy costs and maximising the use of renewable energy.
Implementation examples: sample-based MPCs contribute to energy efficiency in energy management systems in large buildings and in regional smart grids.
5. autonomous marine exploration robots:
CASE SUMMARY: An ocean exploration robot navigates in a dynamic and uncertain ocean environment and uses sample-based MPC to ensure efficient and safe exploration.
Details:
– Sampling: sample-based generation and prediction of navigation paths taking into account currents and obstacles.
– Prediction horizon: predicts navigation paths for the duration of the exploration mission, taking into account oceanographic conditions.
– Optimisation: optimise battery consumption and obstacle avoidance while maximising coverage of the exploration area.
Implementation examples: marine research institutes and energy companies have deployed sample-based MPCs in the navigation systems of autonomous ocean exploration robots.
Sample-Based MPC (Sample-Based MPC) implementation examples
Examples of Sample-Based Model Predictive Control (Sample-Based MPC) implementations can be found in a variety of areas. Specific examples of implementations are described below.
1. route planning for automated vehicles:
Techniques used: Monte Carlo tree search (MCTS) and random sampling
Steps:
1. Environmental modelling: the vehicle’s surrounding environment is acquired using sensors (LIDAR, cameras) and the location of vehicles and obstacles is modelled.
2. Sampling: the possible paths of the vehicle are randomly sampled to generate multiple future scenarios.
3. Prediction horizon: simulation of each sample route within a fixed time horizon.
4. Cost assessment: cost functions such as journey time, safety and energy efficiency are assessed for each route scenario.
5. Optimisation: the optimum route is selected and instructions are sent to the vehicle in real-time.
Example implementation code (Python/Pseudo-code):
import numpy as np
def sample_paths(current_state, num_samples, horizon):
# Functions to generate sample routes.
samples = []
for _ in range(num_samples):
path = [current_state]
for _ in range(horizon):
next_state = simulate_next_state(path[-1])
path.append(next_state)
samples.append(path)
return samples
def evaluate_cost(path):
# Function to evaluate the cost of each route.
cost = 0
for state in path:
cost += compute_cost(state)
return cost
def simulate_next_state(current_state):
# Function to simulate the next state (e.g. random walk)
return current_state + np.random.randn()
def main():
current_state = get_current_state()
num_samples = 100
horizon = 10
paths = sample_paths(current_state, num_samples, horizon)
costs = [evaluate_cost(path) for path in paths]
best_path = paths[np.argmin(costs)]
apply_control(best_path[0])
if __name__ == "__main__":
main()
2. drone flight control:
Technique used: particle filter
Steps:
1. State initialisation: set the initial state of the drone.
2. Particle generation: generate particles (sample states) around the initial state.
3. Predictive update: simulate future state of each particle.
4. Weighting: weights each particle based on its agreement with the measured data.
5. Resampling: generate new samples from the weighted particles.
6. Optimisation: determines the optimal flight path and calculates control inputs.
Example implementation code (Python/Pseudo-code):
import numpy as np
def initialize_particles(num_particles, state_dim):
return np.random.randn(num_particles, state_dim)
def predict_particles(particles):
return particles + np.random.randn(*particles.shape)
def update_weights(particles, observation):
weights = np.exp(-np.linalg.norm(particles - observation, axis=1))
return weights / np.sum(weights)
def resample_particles(particles, weights):
indices = np.random.choice(range(len(particles)), size=len(particles), p=weights)
return particles[indices]
def main():
num_particles = 1000
state_dim = 4 # Number of dimensions, e.g. position and velocity
observation = get_observation()
particles = initialize_particles(num_particles, state_dim)
particles = predict_particles(particles)
weights = update_weights(particles, observation)
particles = resample_particles(particles, weights)
best_particle = np.mean(particles, axis=0)
apply_control(best_particle)
if __name__ == "__main__":
main()3. precision manipulation of the robot arm:
Technology used: Gaussian Processes (GP) and Lenin Bargain
Steps:
1. GP modelling: dynamic model of the robot arm is trained with Gaussian Processes.
2. Sampling: joint angles and end-effector positions are sampled.
3. Predictive horizon: predict the robot arm’s behaviour within a certain horizon.
4. Optimisation: calculates the optimum joint angles and movement paths based on GP.
5. Control input application: applies the calculated control inputs to the robot arm.
Example implementation code (Python/Pseudo-code):
import numpy as np
from sklearn.gaussian_process import GaussianProcessRegressor
def train_gp_model(data, targets):
gp = GaussianProcessRegressor()
gp.fit(data, targets)
return gp
def sample_joint_angles(gp_model, num_samples):
samples = gp_model.sample_y(np.linspace(0, 1, num_samples).reshape(-1, 1))
return samples
def predict_arm_movement(gp_model, joint_angles):
return gp_model.predict(joint_angles)
def main():
# Training data (e.g. historical movement data)
data = np.random.randn(100, 1)
targets = np.sin(data).ravel()
gp_model = train_gp_model(data, targets)
joint_angles = sample_joint_angles(gp_model, 100)
predicted_movements = predict_arm_movement(gp_model, joint_angles)
best_movement = select_best_movement(predicted_movements)
apply_control(best_movement)
if __name__ == "__main__":
main()4. energy management system:
Technology used: lanin bar gain and sample-based dynamic programming
Steps:
1. Initialisation: energy consumption data is collected and an initial model is built.
2. Sampling: sample future energy consumption patterns.
3. Prediction horizon: set up a prediction horizon for consumption patterns.
4. Optimisation: calculate optimal management strategies with the aim of minimising energy costs.
5. Implementation: implement the calculated management strategies and optimise consumption.
Example implementation code (Python/Pseudo-code):
import numpy as np
def sample_energy_consumption(current_consumption, num_samples, horizon):
samples = []
for _ in range(num_samples):
consumption = [current_consumption]
for _ in range(horizon):
next_consumption = simulate_next_consumption(consumption[-1])
consumption.append(next_consumption)
samples.append(consumption)
return samples
def evaluate_cost(consumption_path):
cost = 0
for consumption in consumption_path:
cost += compute_energy_cost(consumption)
return cost
def simulate_next_consumption(current_consumption):
return current_consumption + np.random.randn() * 0.1
def main():
current_consumption = get_current_energy_consumption()
num_samples = 100
horizon = 24 # Predicting 24 hours ahead
consumption_paths = sample_energy_consumption(current_consumption, num_samples, horizon)
costs = [evaluate_cost(path) for path in consumption_paths]
best_path = consumption_paths[np.argmin(costs)]
implement_energy_strategy(best_path[0])
if __name__ == "__main__":
main()Sample-Based MPC (Sample-Based MPC) challenges and measures to address them
Sample-Based Model Predictive Control (Sample-Based MPC) has been used effectively in a variety of areas, but there are some challenges. The main challenges and their respective countermeasures are described below.
1. high computational costs:
Challenge: Sample-based MPC is computationally expensive because it generates a large number of samples and simulates future scenarios for each sample. Rapid computation is required, especially in applications where real-time performance is required.
Solution:
1. parallel computing: reduce computation time by generating and evaluating samples in parallel using GPUs and multi-core CPUs. Distributed computing can also be used to share the computation among multiple machines.
2. improved sampling methods: use efficient sampling methods (e.g. low-difference sequences and Lanein-Bargain) to obtain accurate results with fewer samples. Also, use critical sampling to reduce the computational load by concentrating on important samples.
3. model simplification: simplify the system model to reduce computational costs. For example, linearisation or approximate models could be used.
2. sample quality and diversity:
Challenges: if the generated samples do not cover realistic scenarios, the optimal control inputs may not be obtained. Sample quality and diversity are particularly important in systems with high non-linearity and uncertainty.
Solution:
1. adaptive sampling: dynamically adjust the sampling strategy to generate more samples for critical areas. Also adjust the distribution of sample generation based on historical performance data.
2. integration of reinforcement learning: integrates reinforcement learning algorithms to generate effective samples based on policies learnt from the environment.
3. particle filtering: use particle filters to highlight important areas through resampling of weighted samples.
3. model accuracy and reliability:
Challenge: as MPC is strongly dependent on the system model, model inaccuracy and uncertainty can have a negative impact on control performance. If the model is inaccurate, the optimisation results may also be inaccurate.
Solution:
1. model updating and learning: collect data in real time and update the model through online learning. Also, use Gaussian Processes and neural networks to incorporate uncertainty in the model.
2. robust MPC: use robust MPCs that explicitly account for uncertainty to improve the model’s tolerance to uncertainty.
3. multi-model approach: multiple models are used in parallel and the predictions of each model are integrated to determine the control inputs.
4. ensuring real-time performance:
Challenge: in systems where real-time performance is required, long computation times make proper control difficult. Delays are particularly problematic in systems that require high-frequency control updates.
Solution:
1. adjusting the prediction horizon: shorten the prediction horizon to reduce the calculation volume and ensure real-time performance. If necessary, the length of the horizon can be adjusted dynamically.
2. Sequential optimisation: optimise sequentially and advance the computation step by step to maintain real-time performance.
3. approximate optimisation methods: reduce computation time by using heuristics and meta-heuristics to find an approximate optimal solution, rather than rigorous optimisation.
5. scalability of the system:
Challenge: sample-based MPCs have a huge computational cost due to the exponential increase in the number of samples as the dimension of the system increases. Efficient approaches are needed for high-dimensional systems.
Solution:
1. dimensionality reduction: use principal component analysis (PCA) and autoencoders to reduce the dimensionality of the system and reduce the computational load.
2. hierarchical MPC: decompose the system hierarchically and apply MPC at each level to distribute the computational load.
3. collaborative control: in multi-agent systems, each agent cooperates to perform MPC, thereby distributing the overall computational load.
Reference Information and Reference Books
For details on distributed learning, see “Parallel and Distributed Processing in Machine Learning. For details on deep learning systems, see “About Deep Learning.
Reference books also include”Machine Learning Engineering on AWS: Build, scale, and secure machine learning systems and MLOps pipelines in production“
“Parallel and Distributed Computing, Applications and Technologies“
“Parallel Distributed Processing: Explorations in the Microstructure of Cognition“
“Model Predictive Control: Theory, Computation, and Design” by James B. Rawlings and David Q. Mayne
“Nonlinear Model Predictive Control: Theory and Algorithms” by Lars Grüne and Jürgen Pannek
“Handbook of Model Predictive Control” edited by Saša V. Raković and William S. Levine
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