Control PID

Actividad. Control PID

1. Representation of systems in State-Space

The state space refers to the space of dimensions whose coordinate axes are formed by state variables. This representation is a mathematical model of a physical system described by a set of inputs, outputs, and state variables related by differential equations of any order in the time domain that are combined into a matrix differential equation of first order. The variables are conveniently expressed as vectors according to their requirement. Besides the fact that if the dynamic system is linear and invariant in time, the algebraic equations are written in matrix form.

• State: The state of a dynamic system is the set of state variables, such that knowledge of these variables and the input value completely determine the behavior of the system at any time . The state of the system can be represented as a vector within the state space.[pic 1]
• State variables: These are the smallest set of variables that determine the dynamic behavior of a system. Representing the linear dynamic system (LTI), as a block (Black Box) that has a number of inputs and outputs.

[pic 2]

In order to know the evolution that the system will have over time when disturbing it with a signal at the input, there are different ways of modeling its behavior, as is the case of the representation of state variables. The inputs can be a set of signals that command the behavior of the system, while the outputs can be made up of a set of responses to the inputs.

[pic 3]

• States vector: It’s the one that uniquely determines the state of the system x(t) for any time , once the state is obtained in  and the input u(t) is specified for .[pic 4][pic 5][pic 6]
• State spaces: The representation of state spaces allows to model and analyze linear and nonlinear systems with multiple inputs and outputs in a compact and convenient way to model.

Equations for the state space: There are three types of variables that appear in the modeling of dynamical systems for state space analysis: input variables, output variables, and state variables. The dynamic system must contain elements that remember the values of the input to , since the integrators in a continuous-time control system serve as a memory device and the outputs of such integrators can be considered as the variables that describe the internal state of the dynamic system, so the outputs of the integrators serve as control variables. state. The number of state variables to completely define the dynamics of the system is equal to the number of integrators that appear in it.[pic 7]

Example 1:

For the diagram, the equation of the system is: [pic 8][pic 9]

Since it is a second order system and contains two integrators, state variables are defined as:[pic 10]

Then we get:                                                        Or well:

[pic 11]                        [pic 12]

The output equation is: [pic 13]

The output equation is written as:                 Se estandarizan las ecuaciones:

[pic 14]                                                [pic 15]

Where:

[pic 16]

Example 2:

Considering the mechanical system above, and the state space equations obtained, the transfer function for this system will be obtained from the state space equations.

Substituting A, B, C and D:

[pic 17]

As: [pic 18]

The inverse of a 2x2 matrix is calculated:

[pic 19]

1. Ziegler-Nichols rules for tuning PID controllers

The Ziegler-Nichols method allows to adjust a PID controller empirically, without the need to know the equations of the plant or the controlled system. These tuning rules proposed by Ziegler and Nichols were published in 1942, constituting one of the most widely spread and used tuning methods.

This method makes it possible to define the constants or proportional, integral and derivative gains (Kp, Ki and Kd) based on the response of the system in open loop or closed loop, depending on the one that best fits the type of system.

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