Analysis via State Space

● We were introduced to the concepts of state-space analysis and system modelling before. Where we showed that the state-space mothods, like transform methods, are simply tools for analysing feedback control systems.

● However, the state-space techniques can be applied to a wider class of systems. Such as nonlinear systems, multiple input and multiple output systems.

● In this lecture we aill concetrate on developing control solutions with state-space approaches in time domain (for linear systems only)

A little reminder

A typical DE is in the form

This type of system (linear, causal time invariant) can also be represented in the form

}is

is

is

where y(t) denotes the system output, u(t) is the system inputx(t) is the state vector (used to account for the „derivatives“) in the system

An Example

Rewrite the following differential equation in space-space form :

Notice that the transfer function of the system described by the given differential equation is

Using a similar representation as we have done before define

We can then define and write the eqaution in matrix form as

Note that depending on the definition of the states there are infinite number of state space forms for a given differential equation

Reachable Canonical Form

● While there are infinite number of state space forms for a given differential equation, the Reachable Cannonical Form (RCF) also called the Phase Variable Form is one of the easiest to setup

● Each subsequent state variable is defined to be the derivative of the previous state variable.

● Finally it is assembled in a matrix form.

Another Example

Put into state space form

which can be written as

Input side of DE

Output side of DE

}}

The corresponding matrix entries can be obtained as :

??? How

It can be done following these steps

● Step 1 : Insert zeros to all of the matrices except

– Super diagonal and last row of A

– Last entry of B

– The entire C matrix● Step 2: Put 1 to the super diagonal of A and the last entry of B

● Step 3: Take the last n coefficients of the output equation and put the negative values of them in the last row of A in reverse order

● Step 4 : Take the coefficients of the input side of the equation and put them in C in reverse order

Example

● Find the state-space representation of the circuit below

for

● Start with the transfer function approach

where

Example cont.

● Calculate Z(s)

● Then the transfer function

● Plug in for the values of capasitor inductor and resistors

Example cont.

● Then

Solution of Linear State Space equations

● The solution to the state equation

is

And the output y(t)

Note that is a matrix!!!

For details look at the slides from week 2

State Transition Matrix

By definition

There are also some other ways for calculating the state transition matrix but we will stick with the above one

Lets try to calculate the state transition matrix for

So we need to calculate

In steps

Taking inverse Laplace transsform leads

Example

Find the output when the input is a unit step for

with

Solution : From the solution of the output we found before

that is

From our previous calculations we know that

which yields

Summary

Definitions

Time Domain Solution

} State-space Eq

} State solution

} Output solution

} State Transition Matrix

Definitions

Frequency Domain SolutionState-space Eq}

State solution

Output solution

Open loop transfer Function

Defintions

Open loop transfer function in pole-zero form

Poles of the open loop system

Zeros of the open loop system

More Definitions

● A state variable realization (A, B, C) can be obtained from the transfer function representation or directly from the differential equations representing the system

● The transfer function representation satisfies

– has real coefficients

– is Rational in s

– Strictly proper● A realization is minimal if it has the smallest possible

number of states

● A realization is minimal if has no pole-zero cancelation

Cont.

● If the modes or the poles of the system can be arbitrary changed then it is controllable

● A realization is controllable if the n x n matrix

has full rank

● If the modes or the poles of the system can be observed at the output then the system is observable

● A realization is observable if the nxn matrix

has full rank

● If a realization is both controllable and observable, then it is minimal

● If a realization is minimal it is both controllable and observable

Back to our example

● The system

Full rank : controllable

Full rank : observable

● The system is minimal (also from )

no pole-zero cancellation

● Note that if we have the transfer function the minimality test cab be used to determine controllability and observability as opposed to the matrix checks given in the previous slide