Overview of Predictive Control with Constraints and examples of algorithms and implementations.

Overview of Predictive Control with Constraints

Predictive Control with Constraints (Predictive Control) will be a control method for designing control inputs to predict the future behaviour of a system while satisfying constraints. The method aims to optimise system performance under constraints. An overview of the predictive constraints method is given below.

1. predictive model building: the predictive constraint method requires a predictive model to model the behaviour of the system to be controlled. Usually, state-space models or transient response models are used, which predict the state and output of the system over time.

2. defining constraints: the system under control usually has constraints. For example, there are physical constraints (e.g. limits on the rate of increase of control inputs, stability conditions, etc.) and operating constraints (e.g. the output of the system must fall within a certain range), and predictive constraint methods formulate these constraints mathematically.

3. optimisation problem formulation: an optimisation problem is formulated to optimise the performance of the system while satisfying the constraints. In this optimisation problem, the control inputs are adjusted to maximise or minimise predictions of future states and outputs.

4. predictive control execution: future control inputs are calculated by solving the optimisation problem. These control inputs are selected based on predictive models and constraints and are adjusted to optimise system stability and performance.

5. real-time control: predictive control is a method for real-time control. Calculated control inputs are applied to the system and the system behaviour is controlled to meet the constraint conditions.

Predictive constraint methods are widely used in various fields, such as industrial process control, robot control and transport system control, and are a very useful approach to achieving optimal control under constraints.

Algorithms related to Predictive Control with Constraints.

There are several variations of algorithms associated with Predictive Control with Constraints. They are described below.

1. model predictive control (MPC): MPC is a control technique that uses a predictive model of the controlled object to predict its future behaviour and optimises that prediction under constraints. The control inputs are adjusted in real time by periodically solving the optimisation problem.

2. dual receding horizon control (DRHC): DRHC is a type of MPC that employs a dual problem approach. This allows the control inputs that satisfy the constraints to be computed efficiently.

3. control decomposition predictive control (DPC): DPC is a modification of the MPC approach to reduce the computational load in large systems. It improves computational efficiency by decomposing the system into parts and calculating the control for each part separately.

4. Constrained Model Predictive Control (CMPC): CMPC is a method for performing model predictive control by considering constraints. By directly incorporating constraints into the selection of control inputs, it generates control inputs that satisfy the constraints.

Application of the Predictive Control with Constraints method (Predictive Control with Constraints).

Predictive Control with Constraints (Predictive Control) has been widely applied in various industrial and technical areas. Examples of their application are described below. 1.

1. process control: Predictive Control with Constraints can be particularly useful in process control in the chemical and manufacturing industries. For example, the operation of chemical plants and production lines can be optimised and controlled to meet production targets and quality requirements.

2. energy management: in power systems and energy management systems, predictive constraint methods are used to regulate power supply and demand and to integrate renewable energy. This enables efficient energy utilisation while maintaining the stability of the electricity grid.

3. transport systems: in traffic control and transport system management, predictive constraint methods are used to optimise traffic flows and reduce congestion. The aim is to reduce traffic accidents and environmental impact while improving the efficiency of the traffic system.

4. robot control: in the control of robots and autonomous mobile systems, predictive constraint methods address issues such as obstacle avoidance, task execution and attitude control. They enable safe and efficient operation of robots in complex environments.

5. building and building management: predictive constraint methods are used for energy management and comfort improvement in buildings and architectures. The operating schedules of equipment such as air-conditioning and lighting are optimised to reduce energy consumption while maintaining a comfortable environment.

Example implementation of Predictive Control with Constraints

In this section, a simple example of a predictive constraint method implementation is given, using the Python and SciPy optimisation libraries.

import numpy as np
from scipy.optimize import minimize

# prediction model
def predict_model(u):
    # Predictive model calculations (abbreviated).
    return predicted_output

# Objective function (cost function)
def objective_function(u):
    # Predict future output using predictive models.
    predicted_output = predict_model(u)
    
    # Calculation of the cost function (e.g. the square of the difference between the target value of output and the)
    cost = np.sum((predicted_output - target_output) ** 2)
    
    return cost

# constraint
def constraint_function(u):
    # Calculation of constraints (e.g. upper limits of control inputs)
    return upper_limit - u

# initial control input
initial_input = np.zeros(num_control_inputs)

# Perform optimisation
result = minimize(objective_function, initial_input, constraints={'type': 'ineq', 'fun': constraint_function})

# Optimal control input
optimal_input = result.x

In this example, the optimization problem is solved using the scipy.optimise.minimise function, defining functions for the prediction model calculation, cost function and constraints. Constraints are defined using the constraints argument, which in this example is an inequality constraint representing an upper bound on the control input.

Predictive Control with Constraints (Predictive Control with Constraints) challenges and measures to address them.

Predictive constraint methods are powerful methods for optimising system performance while satisfying constraints, but there are several challenges. These challenges and their countermeasures are described below.

1. increased computational load: predictive constraint methods usually need to solve sophisticated optimisation problems in order to calculate the optimal control inputs while satisfying the constraints. This increases the computational load.

Solution: the computational load can be reduced by optimising the optimisation algorithms and calculation methods. Simplification of the problem and relaxation of constraints can also be effective.

2. constraint relaxation: in some cases, the constraints may be too strict or the choice of control inputs may be restricted. This makes it difficult to find an optimal solution.

Solution: the solution search space can be expanded by adjusting the problem setup, e.g. by relaxing the constraint conditions or introducing additional control inputs. Flexibly adjust the constraints to suit the control objectives and system requirements.

3. model uncertainty: predictive models may contain uncertainty. This makes predictions inaccurate and the effects of control inputs uncertain.

Solution: uncertainty can be reduced by using methods such as uncertainty-aware robust control techniques, model updating and sensor data feedback.

4. real-time requirements: some applications require predictive constraint methods to work in real-time. However, solving sophisticated optimisation problems is computationally time-consuming.

Solution: the use of efficient optimisation algorithms and optimisation of hardware and software are needed to improve real-time performance.

Reference Information and Reference Books

For general machine learning algorithms including search algorithms, see “Algorithms and Data Structures” or “General Machine Learning and Data Analysis.

“Algorithms” and other reference books are also available.

“Model Predictive Control: Theory and Design“

“Constrained Optimization and Lagrange Multiplier Methods“

“Optimal Control Theory: An Introduction“