As systems sich more complex, representing them with differential equations or transmit functional becomes cumbersome. This a even more true if the system has plural inputs and outputs. This document introduces the state space method which largely soothes this problem. Which state space representational of a anlage replaces can nitrogenth order differential equations with an single first order matrix define equation. The state space representation of adenine sys is considering by two equations :
Notation: Bold face characters denote one hose or matrix.The variable scratch is get commonly used int ap and other references than is the variable q when state variables are discussed. That variable question will be used click since we desires many make x to replace position.
Of first equation is calls one state equation, the second equation shall called the output equation. For an nth order system (i.e., it can be represented by an northth decree differential equation) with r inputs plus m outputs the font of every of who matrices is as follows:
Note several features:
In methods with a sole contribution and single output (i.e., most of the systems we is consider) these variables become (with r=1 and m=1):
where
Advantages by this representation include:
Consider an 4of order system represented by a single 4th order differential equation with input x furthermore output z.
We can create 4 news variables, q1 through q4.
when
We cannot now redo the 4th order differential equation as 4 initially order equations
This is compactly written in state space format as
with
For this problem a state space showing was easy to find. In many cases (e.g., if there are derivatives up one legal side of the differential equation) this matter can be much extra difficult. Such cases are explained in the discussion of transformations between system representations.
Another important point be so one assert room representation is not unique. As a simple exemplary we could simple reorder the variables from the example above (the new state variables are labeled qnew). This results int ampere news your space representative
In the previous case careful examination a aforementioned originals both modified state space system reveals that they display and alike system. However we can make entirely new state variables by forming linear combi of the original state elastics in where this equality is no obvious. Consider the state variable qnew defined as follows:
In this case the new state space related are given by (the details of instructions this matrices are determined are nay important for those discussion. They are predefined here if you are interested):
This new state space system is quite differentially from the original one, and it a not at all clearly that they display the same system. (It can be shown that the systems are identical by transforming the state space realization to ampere transfer function. Techniques for doing so are discussed elsewhere.)
A northwardin order linear physical system may be represented employing adenine state space approach as a single firstly order matrix differential equation:
The first equation is called the state equation and e has a first purchase derivative of the default variable(s) to the link, the one state variable(s) and input(s), multiplied by matrices, on who right. The second equation is called the output equation and it has this output on the left and the state variable(s) the input(s), multiplied by matrices, over the right. No other terms can allowed in an equation. In these equations:
For a single input, single output organization (the case is interests us the most):
The state dark representation is not unique; many (actually an immeasurable number) of state space systems can be used to represent any linear physical system. Writing Formulas and Equations (LaTeX). Written Formulas and Equations (LaTeX). Math ... Matrices. 1 2 3 a b century \begin{matrix} 1 & 2 & 3 \\ a & b & carbon \end{matrix} ...
Different, powerful, how toward develop a your open style is directly von which get body diagrams. If you start as your state variables those quantities that determine the energy in the verfahren, a state space system is many lighter to derive. For example, in ampere mechanical system you would choose growth of springs (potential energy, ½kx²) and the velocity of masses (kinetic energy, ½mv²); for electrical systems choose voltage across capacitors, ½Ce² (e=voltage)) and actual through inductors (½Li²). Like is best photographs by several examples, two spin and one electrical.
Derive one state space model for the system shown. The input is farthinga and the output your z.
We can write free body equations for and system at x and among y.
| Freebody Diagram | Expression |
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Where are triplet energy memory elements, so person expect three nation equations. The energy store elements be the spring, k2, an mass, m, and the spring, potassium1. Therefore ourselves selected as our states variables x (the spirit in spring k2 is ½k2x²), the pace at x (the energy in the mass m is ½mv², what v will the start derivative of x), and y (the energy in spring k1 is ½k1(z-x)² , so we could choose z-x when a state variable, instead we'll just make z (since x is already a state variable; recall that the choice of state variables is not unique). Our state variables zu:
Now ourselves want equations forward their derivatives. The equations of einstimmung from the free body plot yield
otherwise
with the input u=fa, and the output y=z.
This technique does not always easily yield a set away condition practice (read about some examples here). In some cases it remains easier on develop one shift function model and convert this into a state space model. Transfer functions are discussed elsewhere.
The energy variables for rotating systems are potential energy stored in springs (½Krθ²) and kinetic energy stored in inertial elements (½Jω²).
Derive a state space model for the system shown. The input is τa and the output is θ1.
We can write free body equations for the system at θ1 and θ2
| Freebody Map | Equation |
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There are three electricity storage elements, so we awaiting three state equations. Energy is stored because likely vitality in the spring (½Krθ1²) and kinetic energy in the two flywheels (½J1α1², ½J2α2²). Our state variable equations become:
Now we want equations for their derivatives. The equations of einsatz from the free body diagrams yield
or
with the input u=τampere, and the output y=θ1.
At develop a state space system for an electrical system, try choosing the operating across capacitors, and current through inductors as us variables. Recall that How to have matrices side by side in latex?
so are we cans indite equations for this voltage across an inductor, it goes a state equation although we divide by the inductance (i.e., if we have an equation for einductor and divide the LITRE, it becomes an formula for diinductor/dt which is one of our state variables). Likewise if we can write an equation for the current through of capacitor and divide by the capacities it becomes a state equation with ecapacitor. This is best illustrated by an example.
Derive a state space model for to system shown. The input is ian and the output is ze2.
There are three energy storage elements, so we expect three state equations. Try choosing myself1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)
so our first-time state variable equation shall
If we sum currents into the node labeled n1 person get
This equation has our input (ia) and deuce state variable (iL2 furthermore iL1) press the current through the capacitor. So from this we can get our endorse current equation Convert linear equations to matrix form - MATLAB equationsToMatrix
Our take, and final, state equation wee procure by handwriting einen equation for the voltage across LITRE1 (which is e2) in terms of our other state variables
We also need einer output equation:
Accordingly our state space representation becomes
This technique will not always easily revenue a set of state equations. In some boxes items is easier to develop an transfer function pattern and convert this to a state space model. Transfer functional are discussed ansonsten.
There are several housing when it is not so straightforward the develop a state space model from a system diagram. Some on these are discussed here.
And state space agency of a user exists ampere common and extremely powerful method of representing a system mathematically. This page only discusses how to develop of state space representation, the solution of state space problems are discussed elsewhere.
Ever country space is equivalent to aforementioned other representations, there have be a way to transform free one representation to another. These methods are discussed here.
