# Linear relationship between the input and output of a system

### functions - How To Create a Non-Linear Output from a Linear Input? - Mathematics Stack Exchange

A non-linear system is one where such a proportional relationship between input and output does not hold, as shown in the corresponding graph. The equation is linear, because the relevant variable, y(t), and its derivative(s) all . mathematical relationship between a system's input and output. In abstract. Transfer Function of Linear Time-Invariant (LTI) Systems. 3. . The transfer function between a pair of input and output variables is the ratio of the. Laplace.

### State-space representation - Wikipedia

For a system to be modeled using the state-space method, the system must meet this requirement: The system must be "lumped" "Lumped" in this context, means that we can find a finite-dimensional state-space vector which fully characterises all such internal states of the system.

This text mostly considers linear state space systems, where the state and output equations satisfy the superposition principle and the state space is linear. However, the state-space approach is equally valid for nonlinear systems although some specific methods are not applicable to nonlinear systems.

State[ edit ] Central to the state-space notation is the idea of a state.

A state of a system is the current value of internal elements of the system, that change separately but not completely unrelated to the output of the system. In essence, the state of a system is an explicit account of the values of the internal system components.

## Control Systems/State-Space Equations

Here are some examples: Consider an electric circuit with both an input and an output terminal. This circuit may contain any number of inductors and capacitors. The state variables may represent the magnetic and electric fields of the inductors and capacitors, respectively.

Consider a spring-mass-dashpot system. The state variables may represent the compression of the spring, or the acceleration at the dashpot.

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- Representing functions as rules and graphs
- State Space Representations of Linear Physical Systems

Consider a chemical reaction where certain reagents are poured into a mixing container, and the output is the amount of the chemical product produced over time. The state variables may represent the amounts of un-reacted chemicals in the container, or other properties such as the quantity of thermal energy in the container that can serve to facilitate the reaction. When modeling a system using a state-space equation, we first need to define three vectors: We need to define all the inputs to the system, and we need to arrange them into a vector.

## State-space representation

We can define 4 new variables, q1 through q4. In many cases e.

Such cases are explained in the discussion of transformations between system representations. The state space representation is not unique Case 1: Alternate State Space Representation Another important point is that the state space representation is not unique. As a simple example we could simply reorder the variables from the example above the new state variables are labeled qnew.

This results in a new state space representation Case 2: Alternate State Space Representations In the previous case careful examination of the original and modified state space system reveals that they represent the same system.

**How to match function input to output given the graph (example) - Algebra I - Khan Academy**

However we can make entirely new state variables by forming linear combination of the original state variables in which this equality is not obvious. Consider the state variable qnew defined as follows: In this case the new state space variables are given by the details of how these matrices are determined are not important for this discussion.

### functions - How To Create a Non-Linear Output from a Linear Input? - Mathematics Stack Exchange

They are given here if you are interested: This new state space system is quite different from the original one, and it is not at all obvious that they represent the same system. It can be shown that the systems are identical by transforming the state space representation to a transfer function. Techniques for doing so are discussed elsewhere. Defining a State Space Representation A nth order linear physical system can be represented using a state space approach as a single first order matrix differential equation: The first equation is called the state equation and it has a first order derivative of the state variable s on the left, and the state variable s and input smultiplied by matrices, on the right.

The second equation is called the output equation and it has teh output on the left and the state variable s and input smultiplied by matrices, on the right. No other terms are allowed in the equation. Developing a state space model from a system diagram Mechanical Translating Another, powerful, way to develop a state space model is directly from the free body diagrams.

If you choose as your state variables those quantities that determine the energy in the system, a state space system is often easy to derive. This is best illustrated by several examples, two rotating and one electrical.

The input is fa and the output is z. We can write free body equations for the system at x and at y.