State Space Control

SSC

State-space control, also known as state-space representation or state-space modeling, is a mathematical framework used in control systems engineering to describe and analyze the behavior of dynamical systems. It is a modern and powerful approach to control system design that represents a system using a set of first-order differential equations, known as state equations, instead of higher-order transfer functions commonly used in classical control.

The state-space representation is based on the concept of state variables, which are a set of variables that fully describe the internal state of a system at any given time. These state variables collectively form a vector called the state vector, denoted by "x." The state vector evolves over time, and its time derivative represents how the state changes with respect to time.

The general form of a continuous-time linear state-space representation is:

$$x' = Ax + Bu$$

$$y = Cx + Du$$

Where:

• x is the state vector (n x 1) containing the state variables of the system.

• x' is the time derivative of the state vector (dx/dt).

• A is the state matrix (n x n), which governs how the state evolves over time without any external inputs (control signals).

• B is the input matrix (n x m), which relates the control inputs (u) to the rate of change of the state vector.

• u is the control input vector (m x 1) applied to the system to influence its behavior.

• y is the output vector (p x 1) containing the measurable output variables of the system.

• C is the output matrix (p x n), which relates the state vector to the output vector.

• D is the direct transmission matrix (p x m), which relates the control inputs to the output vector without passing through the state dynamics.

The state-space representation provides a more flexible and intuitive way to design control systems, especially when dealing with complex systems or when incorporating multiple inputs and outputs. By controlling the system through its state variables, state-space control allows for better controllability and observability.

Advantages of State-Space Control:

1. Multivariable Control: State-space control naturally handles systems with multiple inputs and outputs, making it suitable for complex control tasks.

2. Observability and State Estimation: The state variables can be estimated from system measurements, even when not all states are directly measurable (observability).

3. Robustness: State-space control facilitates robust control design, which is essential for dealing with uncertainties and disturbances in real-world systems.

4. Stability Analysis: Stability analysis of the system can be performed directly using the state matrix, providing insights into system behavior.

Summary...

1. State Representation: In state-space control, a dynamic system is represented in terms of its "state." The state of the system includes a set of variables that fully describe the current condition of the system. For example, if we have a mobile robot, the state variables could include its position, velocity, orientation, and angular velocity.

2. State Evolution: The state variables change over time based on how the system responds to inputs (control signals) and external disturbances. This evolution is described by a set of equations known as state equations. These equations represent how the state variables change over time, given the control inputs and disturbances acting on the system.

3. Control Input: In state-space control, we can apply control inputs to the system to influence its behavior and achieve desired objectives. These control inputs can be forces, torques, voltages, or any other signals that affect the state variables. The control inputs are represented by a vector termed "u."

4. Output: The system might have measurable output variables that indicate its performance or behavior. The outputs could be position measurements, sensor readings, or any other quantities that provide useful information about the system's state. The outputs are represented by another vector termed "y."

5. Feedback Control: In state-space control, feedback plays a crucial role. The control system continuously measures the output "y" and compares it to the desired or reference output. The difference between the measured output and the reference output is called the "error" signal.

6. Controller Design: The objective of state-space control is to design a controller that takes the error signal and produces the appropriate control inputs "u" to drive the system toward the desired state. The controller's task is to maintain or stabilize the system around the desired state, reject disturbances, and track the reference signal accurately.

7. Full System Representation: The state-space representation of the system comprises two sets of equations: the state equations (how the state variables change) and the output equations (how the outputs are related to the state variables and control inputs).

8. Benefits: State-space control provides a powerful and flexible framework for designing control systems for complex dynamic systems with multiple inputs and outputs. It enables engineers to apply advanced control techniques, such as observers and optimal control, to achieve robust and high-performance control.

closed loop state space control

you can use Kalman filter for the sensors..(recommended u dont have to...)

here is a basic implementation of SSC...

6KB

StrafingController.java

State Space Controller

This is a basic controller that uses the imu and 2 sensors one looking at the right and one looking forward. and REV TICKS PER INCH and with that the robot moves everywhere... keep in mind some sensors cant see far enough (change robots orientation)to reach the fields but you could change the API to make the sensor look to the left in autonomous. its really easy to use but its filled with errors.

In the context of the state space control, A, B, C, and K are matrices used to represent the state space model of a control system.

1. A (State Transition Matrix): The A matrix represents the relationship between the state variables of the system at the current time step and the state variables at the next time step. It describes how the state of the system evolves over time in the absence of control inputs and external disturbances.

2. B (Input Matrix): The B matrix represents the effect of control inputs on the state variables of the system. It shows how the control inputs influence the state evolution.

3. C (Output Matrix): The C matrix relates the state variables of the system to the measured outputs. It defines which state variables are being measured or observed by the sensors.

4. K (Kalman Gain Matrix): The K matrix is used in the Kalman filter to optimize the state estimate based on the sensor measurements and the state space model. It helps in reducing the effects of noise and uncertainties in the sensor measurements.

By using these matrices, the state space model can describe the behavior of a control system, and the Kalman filter can be applied to estimate the state variables more accurately using sensor measurements and control inputs. The ABCK matrices are essential components of the state space control and Kalman filter algorithms. this is with Kalman filter PID and State Space control ;)