

# Continuous-time and discrete-time models



Continuous-time models describe the system's behavior over an infinite time interval. They are commonly used to model physical systems, such as mechanical systems or electrical circuits. On the other hand, discrete-time models describe the system's behavior in discrete time intervals. They are often used in digital systems, such as digital filters or control systems.

Consider a simple continuous-time system described by the following differential equation:

$$ \dot{x} = -2x + u $$

This equation describes the evolution of the system's state variable $x$ over time, where $u$ is the input to the system. The corresponding discrete-time model can be obtained by discretizing the continuous-time system using the Z-transform.

## Exercise
Derive the discrete-time model for the continuous-time system described in the example.

# State-space representation for linear systems



The state-space representation for a linear system consists of the system's state variables, input, and output. The state variables are represented by a vector $x$, the input by a vector $u$, and the output by a vector $y$. The system's dynamics are described by a set of ordinary differential equations (ODEs) or difference equations (DEs) that relate these variables.

Consider the following continuous-time system:

$$ \dot{x} = \begin{bmatrix} -2 & 1 \\ 1 & -2 \end{bmatrix} x + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u $$

This system can be represented in state-space form as:

$$ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u $$

## Exercise
Write the discrete-time model for the continuous-time system described in the example.

# Linear-quadratic optimization



LQ optimization is typically used to design controllers for linear systems. The goal is to find the optimal control input $u$ that minimizes the quadratic cost function $J = \frac{1}{2} x^T Q x + u^T R u + c^T x + d^T u$ subject to the linear constraints $Ax + Bu \leq b$.

Consider a discrete-time linear system with the state-space representation:

$$ x_{k+1} = A x_k + Bu_k $$

The quadratic cost function is:

$$ J = \frac{1}{2} x^T Q x + u^T R u + c^T x + d^T u $$

The linear constraints are:

$$ Ax \leq b $$

## Exercise
Solve the LQ optimization problem for the discrete-time system described in the example.

# Design and implementation of linear-quadratic regulator in MATLAB/Simulink



To design an LQR controller in MATLAB/Simulink, we first need to define the system's state-space representation and the desired cost function. Then, we can use MATLAB's `lqr` function to compute the optimal control input $u$. Finally, we can implement the controller in Simulink to control the system in simulation.

Consider the following discrete-time system with the state-space representation:

$$ x_{k+1} = A x_k + Bu_k $$

The quadratic cost function is:

$$ J = \frac{1}{2} x^T Q x + u^T R u + c^T x + d^T u $$

The linear constraints are:

$$ Ax \leq b $$

## Exercise
Design and implement an LQR controller for the discrete-time system described in the example using MATLAB/Simulink.

# Performance analysis and comparison with other control methods



The performance of the LQR controller can be analyzed using various metrics, such as the tracking error, stability, and robustness. We will also compare the LQR controller with other control methods, such as PID control or model predictive control, based on their advantages and disadvantages.

Consider a discrete-time system with the state-space representation:

$$ x_{k+1} = A x_k + Bu_k $$

The LQR controller is designed using the following cost function:

$$ J = \frac{1}{2} x^T Q x + u^T R u + c^T x + d^T u $$

The linear constraints are:

$$ Ax \leq b $$

## Exercise
Analyze the performance of the LQR controller for the discrete-time system described in the example. Compare it with other control methods and discuss the trade-offs between different control methods.

# Applications of linear-quadratic regulator design



The linear-quadratic regulator design method has found wide applications in control systems, including robotics, automotive systems, and aerospace systems. We will also discuss the challenges and limitations of the LQR method, such as the need for accurate modeling of the system and the computational complexity of the optimization problem.

Consider a discrete-time system with the state-space representation:

$$ x_{k+1} = A x_k + Bu_k $$

The LQR controller is designed using the following cost function:

$$ J = \frac{1}{2} x^T Q x + u^T R u + c^T x + d^T u $$

The linear constraints are:

$$ Ax \leq b $$

## Exercise
Discuss the applications of the linear-quadratic regulator design method in various control systems. Address the challenges and limitations of the LQR method and suggest ways to overcome them.

# Case studies and real-world examples



Case studies and real-world examples will provide insights into the practical application of the linear-quadratic regulator design method in control systems. They will also highlight the importance of considering the system's dynamics, constraints, and objectives when designing a control system using the LQR method.

Consider a discrete-time system with the state-space representation:

$$ x_{k+1} = A x_k + Bu_k $$

The LQR controller is designed using the following cost function:

$$ J = \frac{1}{2} x^T Q x + u^T R u + c^T x + d^T u $$

The linear constraints are:

$$ Ax \leq b $$

## Exercise
Present a case study or real-world example of the linear-quadratic regulator design method in a control system. Discuss the advantages and challenges of using the LQR method in this application and how the method can be adapted to different scenarios.