MODULE 5

STATE VARIABLE ANALYSIS

A state variable is one of the set of variables that are used to describe the mathematical

"state" of a dynamical system. Intuitively, the state of a system describes enough about the

system to determine its future behavior. Models that consist of coupled first-order differential

equations are said to be in state-variable form.

In mechanical systems, the position coordinates and velocities of mechanical parts are typical state variables; knowing these, it is possible to determine the future state of the objects in the system.

In a thermodynamic system, properties such as temperature,pressure, volume, internal energy, enthalpy, and entropy are state variables.

In electronic circuits, the voltages of the nodes and the currents through components in the circuit are usually the state variables.

In ecosystem models, population sizes (or concentrations) of plants, animals and resources (nutrients, organic material) are typical state variable.

STATE SPACE REPRESENTATION 

A state space representation is a mathematical model of a physical system as a set of

input, output and state variables related by first-order differential equations..

The most general state-space representation of a linear system with   inputs,   outputs

and   state variables is written in the following form.

where:

 is called the "state vector",   ;

 is called the "output vector",   ;

 is called the "input (or control) vector",   ;

 is the "state (or system) matrix",   ,

 is the "input matrix",   ,

 is the "output matrix",   ,

 is the "feedthrough (or feedforward) matrix"

.

In this general formulation, all matrices are allowed to be time-variant (i.e. their elements

can depend on time); however, in the common LTI case, matrices will be time invariant. The

time variable   can be continuous (e.g.  ) or discrete (e.g.  ). In the latter case, the

time variable   is usually used instead of  . Hybrid systems allow for time domains that have both 

continuous   and   discrete   parts.   Depending   on   the   assumptions   taken,   the   state-space   model 

representation can assume the following forms:

System type State-space model

Continuous time-invariant

Continuous time-variant

Laplace domain of

continuous time-invariant

In a state space system, the internal state of the system is explicitly accounted for by an

equation known as the state equation. The system output is given in terms of a combination of

the current system state, and the current system input, through the output equation. These two

equations form a system of equations known collectively as state-space equations. The state-

space is the vector space that consists of all the possible internal states of the system. Because the

state-space must be finite, a system can only be described by state-space equations if the system

is lumped.

Input variables

A SISO (Single Input Single Output) system will only have a single input value, but a

MIMO system may have multiple inputs. We need to define all the inputs to the system,

and we need to arrange them into a vector.

Output variables

This is the system output value, and in the case of MIMO systems, we may have several.

Output variables should be independent of one another, and only dependent on a linear

combination of the input vector and the state vector.

State Variables

The state variables represent values from inside the system, that can change over time. In

an electric circuit, for instance, the node voltages or the mesh currents can be state

variables. In a mechanical system, the forces applied by springs, gravity, and dashpots

can be state variables.

We denote the input variables with u, the output variables with y, and the state

variables with x.

The State Equation shows the relationship between the system's current state and its

input, and the future state of the system.

The Output Equation shows the relationship between the system state and its input, and

the output. These equations show that in a given system, the current output is dependent on the

current input and the current state. The future state is also dependent on the current state and the

current input.