Editor's note: InPart 1 of this series, the basic structural elements and grammar of ablock diagram methodology was described. In this part, methods formanipulating block diagrams as an aid to analyzing system behavior areoutlined.
Manipulating block diagrams
Initially, one usually draws a structural block diagram. This is adiagram that shows how a system is put together. At some point, onewill wish to reduce this structural block diagram into a behavioraldiagram.
While this can be done by the techniques shown in Part1, such techniques immediately sever the connection betweenthe block diagram and the behavioral model, and can be verycounter-intuitive to use. It is often better to reduce a block diagram using themanipulation rules presented here.
There are four tools that you have on hand to manipulate blockdiagrams. Given a block diagram that is described fully in the z domain or theLaplace domain, these tools will allow you to fully analyzethe block diagram to extract the overall system behavior.
If you observe their limitations, you can also use these tools on avariety of other block diagrams. The four tools that you have are:cascading gainblocks, moving summing junctions, combining summingjunctions , and reducing loops.
The block diagram manipulations shown here will always work if theblocks in question contain pure transfer functions. All of the examplesbelow show block manipulations on blocks containing transfer functionsin the z domain ;however,these manipulations can be carried out onblocks in the Laplace domain as well.
Stating when a block operation cannot be carried out is moredifficult. In general, these operations depend on superposition andcannot be performed when the blocks contain nonlinear operation.
In addition, time varying operations like sampling and zero-orderholds can sometimes be moved around and sometimes cannot—generallyifyou can review this section and make the mathematical equations fit,then you can perform the operation with a block diagram.
When a block diagram indicates a feedback loop,you can reduce the loopto a single transfer function block as shown in Figure 4.12. below . If you look atthe equations that govern the behavior of this block diagram, you cansee that the output signal is a function of the forward gain G and the errorsignal e:
The error signal, in turn, is a function of the input and outputsignals and the feedback gain:
By substituting expressions, we get
which reduces to
|Figure4.12 Reducing a loop|
Using Loop Reduction
A feedback control system has a forward gain
and negative feedback with a gain of
Draw its block diagram, and find the overall transfer function forthe system. The block diagram is a simple feedback loop with thespecified gains:
|Figure4.13 Feedback loop|
From the formula for loop reduction, the transfer function for thesystem is
This reduces down to the block diagram:
When two blocks are cascaded directly, the transfer function of thecombination is the product of the two transfer functions, as in Figure 4.14, below . Looking at theequations that govern the behavior of the block diagram you can seethat
From this, it is easy to see that
|Figure4.14 Cascading gain blocks|
Example. A feedback controlsystem has a plant with a gain of
a controller with a gain of
and unity feedback. Draw its block diagram, and find the overalltransfer function for the system.
The block diagram is:
From the gain cascade rule the forward gain is
Using the loop reduction rule with H(z) = 1, the transfer functionfor the system is
This reduces down to the block diagram:
If a loop contains more than one summing junction, it cannot be reducedby simple loop reduction, and cascading gains will not eliminate theextra junction.
In order to reduce such a loop, the summing junctions must be movedaround until there is a loop with just one summing junction. This isdone by propagating a transfer function backward through the junction,or its inverse forward through the junction, as shown in Figure 4.15 below.
|Figure4.15. Moving swimming junctions|
Keep in mind that the inverse transfer function 1/G(z) may not bephysically realizable – this is not a concern unless you aretrying touse your intermediate results in a real system.
Any manipulations you do here are just for the purposes of makingthe math easy, and if the system starts as a physically realizable onethen any contradictions will disappear by the time you get yoursolution.
In the top case of Figure 14.5 ,the input/output relationship on the left is
Using the commutative property of multiplication, this translates to
which corresponds to the input/output relationship on the right.
In the bottom case of Figure 4.15 the input/output relationship on the left is
By using the inverse of the transfer function, the right sidebecomes
which is functionally the same as the left.
When a block diagram contains a pair of signal paths that originatefrom the same signal and terminate on a summing junction, the set canbe reduced to a single block, as shown in Figure 4.16, below. In this case itcan beseen that
From this it is easy to get
|Figure4.16 Removing parallel paths|
Example using Cascading Gains
Feedforward is often used in control systems to increase the systemresponse speed without having to change a “safe” set of tuningparameters for the loop. Figure 4.17 below shows one such system. We can reduce the block diagram in Figure4.17 down to a single block.
|Figure4.17. A system with feedforward|
The loop in Figure 4.17 contains two summing junctions, whichprevents the use of the loop reduction rule. We can either move the G1leg back to the left summing junction, cascading G1 with 1/G2, or wecan move the feedback leg forward to the right summing junction,putting a G2 block in the feedback path.
I'll do the latter, to avoid the inverse:
Using loop reduction for the right half and the parallel summationrule for the left gives us a transfer function of:
Multiple Input Systems
So far I've presented systems with just one input and one output (oftencalled SISO systems for “Single Input, Single Output”). Real systemsaren't restricted to having just one input and one output, and neitherare block diagrams.
Even when you're working with a system that you'd like to treat as aSISO system, you'll find that reality may place additional requirementson your analysis.
It is often very useful to model a disturbance to a system in ablock diagram. Figure 4.18 below shows a block diagram(ignoring the continuous time nature of the actuator and plant) withthe plant split up into an actuator and a mechanism, and a disturbanceforce added to the force of the actuator on the mechanism.
This is an example of a multiple input, single output (MISO) system.When dealing with such a system, you are often interested in itsability to reject disturbances, so you often want to find the transferfunction from the intended input (u in this case) as well as the disturbance input (u d ).
|Figure4.18. A block diagram showing a disturbance input|
Both of the transfer functions in question can be found almost byinspection from Figure 4.18. This is done by appealing to the fact that we're using a linear systemmodel, which means that we can use superposition. To find the transferfunction from the intended input to the output, we assume that thedisturbance is set to zero and solve the system normally:
Similarly, to find the transfer function from the disturbance inputto the output, we set the intended input to zero and solve theresulting system:
Notice that the denominators in (4.37) and (4.38) are thesame. This is a general characteristic of feedback control systems: nomatter how many inputs or outputs the system has, no matter how thesystem behavior may vary when you choose different points to injectsignals or observe them, the underlying behavior of the system in termsof the poles of the transfer function will remain the same.
The only times that you will see two different characteristicpolynomials in one system will either be because you have two disjointsystems that happen to be on the same page, or because the numeratorand denominator of the system share some roots.
(This is called pole-zerocancellation. The cancelled poles haven't ceased to exist—they are justhidden, often to the detriment of real system behavior. Systems withpole-zero cancellation should be treated with all due caution .)
There is one other thing to notice about the system: it is often notnecessary to derive all of the transfer functions directly. In thiscase one could have observed that by moving the summing junction forthe disturbance back to the input summing junction the disturbancetransfer function could be found from Eq.(4.37), earlier:
If you do the math, you'll see that (4.39) and (4.38) are equal.In many cases it is much easier to derive the desired function by thismethod rather than solving the block diagram twice.
Multiple Output Systems
A dual of the case with multiple input, single output (MISO) systemsare systems with single inputs and multiple outputs (SIMO). As with themultiple input case, one often finds the multiple output case when wewould rather be doing a simpler analysis.
Take the case of the command-following system in Figure 4.19, below (which is just Figure 4.18 with the extra summingblock removed and some extra signal labels ).
With such systems, we're often concerned not only with how well theoutput is going to follow the input, but what size the control signalswill be for a given input.
|Figure4.19 A command-following system|
If we have a known input and we wish to know the drive signal, weonly need to find the transfer function from input to drive and applythe input signal. In this case the block diagram can be reduced byinspection:
This transfer function can be used to predict the drive signal tothe actuator for the purposes of checking on actuator heating, actuatordrive requirements, and to see if any system parameters are beingexceeded…
… This chapter has presenteda method ofdescribing a system using blockdiagrams. It showed how these block diagrams may vary within thecontrol systems community and in the larger signal processingcommunity. I presented methods for bringing some formalism to a blockdiagram description so that it could be used to analyze systembehavior, and methods for manipulating block diagrams to discover how aparticular system will behave in a larger sense.
To read Part 1 in this series of articles go to “The language of blocks diagrams and how touse them for embedded control system analysi s .”
Usedwith the permission of the publisher, Newnes/Elsevier, this series oftwo articles is based on Chapter 4 – “Block Diagrams” from TimWescott's new book “AppliedControl Theory For Embedded Systems.”
Tim Wescott of WescottDesign Services isanexpert in the application ofcontrol system theory to embedded designs and is a contributor toEmbedded Systems Design Magazine and to Embedded.com. In addition to “ Sigma-deltatechniques extend DAC resolution,”he is the author of “ PID without aPhD,” published in EmbeddedSystems Design and on-line at Embedded.com, the latter which continuesto be oneof the most frequently accessed articles on the site.