【Advanced Edition】MATLAB Model Predictive Toolbox: Model Predictive Control Toolbox User Guide

# **Advanced篇** MATLAB Model Predictive Control Toolbox: A User Guide

MATLAB Model Predictive Control (MPC) Toolbox is a powerful tool for designing, simulating, and deploying MPC controllers. It offers a range of functionalities, including:

- **Model Establishment and Verification:** The toolbox supports various model types, including state-space models, output-error models, and nonlinear models. It provides tools for model identification, verification, and fitting.
- **Controller Design and Parameter Optimization:** The toolbox offers a series of optimization algorithms for designing MPC controllers. It allows users to specify objective functions, constraints, and weights, and optimize controller parameters for optimal performance.
- **Simulation and Deployment:** The toolbox provides a simulation environment for testing and validating MPC controllers. It also supports the deployment of controllers to embedded systems or cloud platforms.

# 2. Theoretical Foundation of Model Predictive Control

### 2.1 Fundamental Principles of Model Predictive Control

Model Predictive Control (MPC) is an advanced control technique that controls dynamic systems by predicting future system behavior and optimizing control inputs. The basic principles of MPC are as follows:

1. **Establishing the System Model:** First, a model that accurately describes the dynamic behavior of the system needs to be established. This model can be a state-space model, an output-error model, or another suitable model.
2. **Predicting Future States:** Using the system model, MPC predicts the future states of the system over a period of time, typically the prediction horizon (H) number of sampling times.
3. **Optimizing Control Inputs:** Based on the predicted future states, MPC optimizes control inputs to minimize an objective function, such as tracking a reference trajectory or minimizing system error.
4. **Implementing Control Inputs:** The optimized control inputs are applied to the system to perform actual control.
5. **Updating the Model and Predictions:** As new data is acquired, the system model and predictions are continuously updated to improve control performance.

### 2.2 Mathematical Models of Model Predictive Control

The mathematical models of MPC typically take one of the following two forms:

#### 2.2.1 State-Space Model

The state-space model describes the relationship between the state variables (x) and the output variables (y) of the system, as shown below:

x[k+1] = A*x[k] + B*u[k]
y[k] = C*x[k] + D*u[k]

Where:

* x[k]: System state vector
* u[k]: Control input vector
* y[k]: System output vector
* A, B, C, D: System matrices

#### 2.2.2 Output Error Model

The output error model describes the relationship between the system's output error (e[k]) and past inputs and outputs, as shown below:

e[k] = y[k] - r[k]
A(z)*e[k] = B(z)*Δu[k] + C(z)*ε[k]

Where:

* e[k]: System output error
* r[k]: Reference trajectory
* Δu[k]: Control input increment
* ε[k]: Process noise
* A(z), B(z), C(z): Polynomial matrices

### 2.3 Optimization Algorithms ***

***mon optimization algorithms include:

***Linear Programming (LP):** Used for Linear Model Predictive Control (LMPC)
***Quadratic Programming (QP):** Used for Quadratic Model Predictive Control (QMPC)
***Nonlinear Programming (NLP):** Used for Nonlinear Model Predictive Control (NMPC)

The objective function of the optimization problem is typically a quadratic function, measuring the trade-off between system output error and control input. Constraints include limits on state variables, control inputs, and outputs.

# 3.1 Design of Model Predictive Controllers

#### 3.1.1 Model Establishment and Verification

**Model Establishment**

Model establishment is a key step in model predictive control and requires selecting an appropriate model structure based on the characteristics of the controlled object and the control objectives. The MATLAB MPC Toolbox provides various model establishment methods, including:

- **State-Space Model:** Describes the dynamic relationship between system states and outputs.
- **Output Error Model:** Describes the error dynamics between system output and reference input.
- **Transfer Function Model:** Describes the frequency response relationship between system input and output.

**Model Verif