Power of State Space Analysis

Hi!

Recently I was reading the book Digital Signal Processing by Proakis
and Manolakis, and came across their state space analysis of single
input, single output LTI systems.
The authors do mention that the real power of state space analysis can
be seen in the analysis of MIMO systems, but there's no example. Can
someone please give me a pointer (via an example) to such an analysis?

thanks,
Kunal

Kunal wrote:
> Hi!
>
> Recently I was reading the book Digital Signal Processing by Proakis
> and Manolakis, and came across their state space analysis of single
> input, single output LTI systems.
> The authors do mention that the real power of state space analysis can
> be seen in the analysis of MIMO systems, but there's no example. Can
> someone please give me a pointer (via an example) to such an analysis?

Not that I have ever used state space analysis, let alone seen an
example of MIMO systems (which basically translates to that I don't
know what I'm talking about), but I wouldn't be surprised if the
power lies in the compact notation of the system.

A state-space model uses matrix-vector notation, where the
matrix is very sparse in the case of a single-channel LTI system.
If you take advantage of all that "free" storage space, you can
incorporate the contributions from several inputs to one output.

But I agree with you, it would be very interesting to actually
see a state-space system in practice.

Rune

Rune Allnor wrote:

> 
> Kunal wrote:
> 
>>Hi!
>>
>>Recently I was reading the book Digital Signal Processing by Proakis
>>and Manolakis, and came across their state space analysis of single
>>input, single output LTI systems.
>>The authors do mention that the real power of state space analysis can
>>be seen in the analysis of MIMO systems, but there's no example. Can
>>someone please give me a pointer (via an example) to such an analysis?
> 
> 
> Not that I have ever used state space analysis, let alone seen an
> example of MIMO systems (which basically translates to that I don't
> know what I'm talking about), but I wouldn't be surprised if the
> power lies in the compact notation of the system.
> 
> A state-space model uses matrix-vector notation, where the
> matrix is very sparse in the case of a single-channel LTI system.
> If you take advantage of all that "free" storage space, you can
> incorporate the contributions from several inputs to one output.
> 
> But I agree with you, it would be very interesting to actually
> see a state-space system in practice.
> 
> Rune
> 
It's not just that it's compact (although that's not a bad thing), it 
also shows explicitly how things are tied together.  If, for instance, I 
told you I had a single-input, two-output system with transfer functions:

Y_2   (1-a)z    Y_1   (1-a)(1-b)z
--- = -------,  --- = ------------.
  U     z - a     U     (z-a)(z-b)

You have no way of knowing if this is a two-state system where the first 
output is coupled to the second, or if this is a three-state system with 
two independent filters.

Telling you that the system's state equations are

       [ a   1-a ]           [  0  ]
x_n = [         ] x_{n-1} + [     ] u_n,
       [ 0    b  ]           [ 1-b ]

y_n = x_n

on the other hand, immediately tells you that it's a coupled system, and 
that you have the system states fully available for you to play with.


-- 
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Kunal wrote:

> Hi!
> 
> Recently I was reading the book Digital Signal Processing by Proakis
> and Manolakis, and came across their state space analysis of single
> input, single output LTI systems.
> The authors do mention that the real power of state space analysis can
> be seen in the analysis of MIMO systems, but there's no example. Can
> someone please give me a pointer (via an example) to such an analysis?
> 
> thanks,
> Kunal
> 
I also find state-space representations to be very useful in nonlinear 
systems.  They're handy when designing IIR filters, to make sure the 
states won't overflow for normal inputs, and to accurately predict how 
processor saturation will affect the states if they do.

-- 
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott wrote:

> I also find state-space representations
> to be very useful in nonlinear systems.

Can you give an example of what that would look like?

-- 
Quidquid latine dictum sit, altum viditur.

Martin Eisenberg wrote:
> Tim Wescott wrote:
> 
> 
>>I also find state-space representations
>>to be very useful in nonlinear systems.
> 
> 
> Can you give an example of what that would look like?
> 
The extended Kalman Filter

Martin Eisenberg wrote:
> Tim Wescott wrote:
> 
> 
>>I also find state-space representations
>>to be very useful in nonlinear systems.
> 
> 
> Can you give an example of what that would look like?
> 

Stan's example of the extended Kalman filter is one.

I tend to use it most in control systems that have one (or at most two) 
nonlinearities that dominate the large-signal response -- and then it's 
almost always actuator saturation.

At any rate, the cannonical representation you'll find in textbooks for 
a nonlinear time-invariant system is:

x_n = f(x_{n-1}) + g(u_n), y_n = h(x_{n-1}, u_n)

where in general f, g and h are all nonlinear vector functions.  This is 
all too wierd to actually handle in practice, so what I usually end up 
with is something like

x_n = a * x_{n-1} + g(u_n), y_n = c * x_{n-1},

where everything is linear except for g, the reaction to the control 
signal, and even that is constrained to:

               { -F     u_n <= -F
g(u_n) = b *  { u_n    -F < u_n < F
               {  F     u_n >= F

You can also represent the action of saturation of state variables by 
doing something similar with f(x) -- usually when I actually write that 
out I do it by representing f(x) as a * x with some additional text 
pointing out that x is limited.

-- 
-------------------------------------------
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

"Kunal" <kunal.yadwadkar@gmail.com> wrote in message 
news:1118583596.182276.87560@z14g2000cwz.googlegroups.com...
> Hi!
>
> Recently I was reading the book Digital Signal Processing by Proakis
> and Manolakis, and came across their state space analysis of single
> input, single output LTI systems.
> The authors do mention that the real power of state space analysis can
> be seen in the analysis of MIMO systems, but there's no example. Can
> someone please give me a pointer (via an example) to such an analysis?
>
> thanks,
> Kunal
>

Well ..... I think it's fair to say that "state space analysis" in practice 
really means more than the notation conventions.  Actually there are many 
forms it can take.  Personally I like the form that explicitly shows each 
state as the state of an energy storage device such as the current in an 
inductor, the voltage on a capacitor, the velocity of a mass, the tension of 
a spring, etc.  However, this form may not be the best for numerical reasons 
or computational efficiency - that's not my area of expertise.

The implication of the notation is that the control system will be 
implemented via a digital computer.  It very quickly gets to a stage where 
mere humans can't control complex systems (e.g. MIMO) but well-designed 
automatic control system can.  One can envision the difference between 
driving a car from A to B and skippering a sailboat with the same objective. 
A sailboat is a good example of a MIMO system.  The inputs are wind, 
current, rudder position, jib and main sail trim and position.  The state is 
velocity.  When the difference becomes even greater then an automated system 
makes a lot of sense.

Ever try to sail a very small, light boat?  How do you bring it about into 
the wind?  How many rudder positions are necessary and why?  (Hint: 4 if you 
start counting before the turn)

Fred

Fred 

Fred Marshall wrote:

> Ever try to sail a very small, light boat?  How do you bring it about into 
> the wind?  How many rudder positions are necessary and why?  (Hint: 4 if you 
> start counting before the turn)

That depends on the boat.

My 10-foot catboat comes into the wind if you release the rudder and 
haul the sail amidships.

With my 12-foot sloop, I can simply put the helm down and return it 
slowly. Of course if I hold it down until I'm smart into the wind, then 
some reverse rudder is needed to stop the rotation. (I can use the jib 
instead of the rudder, but that's a cheat.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:r_WdnRJMc-k1LDDfRVn-sw@centurytel.net...
>>
> Ever try to sail a very small, light boat?  How do you bring it about into
> the wind?  How many rudder positions are necessary and why?  (Hint: 4 if
you
> start counting before the turn)
>
> Fred
>
> Fred
>
>

In control engineering circles it was all the fashion in the 1960s and 1970s
to use state-space methods,Optinal control etc and many missile problems are
well suited to this form of analysis ie MIMO systems.

However, in the 1980s there was a return to polynomial and transfer function
matrix methods at least outside of the US.

Rimmer