Copyright FL Lewi 999 All right reerved EE - Control Sytem LECTURE 6 Updated: Sunday, February, 999 BLOCK DIAGRAM AND MASON'S FORMULA A linear time-invariant (LTI) ytem can be repreented in many way, including: differential equation tate variable form tranfer function impule repone block diagram or flow graph Each decription can be converted to the other In thi lecture we hall ee how to repreent ytem in term of block diagram, and how to determine the tranfer function of a block diagram ytem uing Maon' Formula SYSTEM INTERCONNECTIONS Sytem can be interconnected in a variety of way, including the following Serie Interconnection () () The overall tranfer function in Y ( ) ( ) U ( ) i given by the product ) ( ) ( ) (
Parallel Interconnection () () The overall tranfer function in Y ( ) ( ) U ( ) i given by the um ) ( ) + ( ) ( Feedback Interconnection () () The overall tranfer function in Y ( ) ( ) U ( ) i given by ( ) ( ) + ( ) ( ) We hall ee later how to derive thi uing Maon' Formula Example - Tranfer Function of Feedback Configuration An untable plant ha tranfer function ( ) which ha one pole in the right-half plane at To tabilize the plant, one may ue a feedback configuration with compenator 0( + ) ( ) +
The cloed-loop tranfer function i ( ) ( ) + + ( ) ( ) 0( + ) ( )( + ) + 0 + 0 + + + + + + ( + )( + ) Thi i table with two left-half plane pole The compenator ha tabilized the plant Superpoition for Linear Sytem For LTI ytem uperpoition hold, o that the effect of different input can imply be added together The formula above can be applied everal time and mixed together to derive the tranfer function in many ytem Example - Superpoition input In thi ytem, might repreent the control input while d(t) i a diturbance d(t) () () () () The ytem can be viewed a two ytem, one driven by input and one driven by input d(t) Setting to zero one can determine that Y ( ) ( ) D( ) Setting to zero d(t), one can conider the tranfer from to a two parallel branche of ytem in erie Thu, the formulae above yield Y ) [ ( ) ( ) + ( ) ( )] U ( ) ( Since the ytem i LTI, the overall output i the um (uperpoition) of the effect of the two input Y ) [ ( ) ( ) + ( ) ( )] U ( ) + ( ) D( ) (
One can define two tranfer function to capture thi If ( ) ( ) ( ) + ( ) ( ) Y D ( ) ( ) then Y ( ) ( ) Y ( ) ( ) D( ) Y + D Thi i a very imple example with no feedback loop In more complex ituation one mut ue Maon' Formula to evaluate the overall tranfer function Example - Superpoition loop It i eay to take into account multiple feedforward path if there are no feedback () u (t) () () () u (t) () One may write U U U U + Y U Y ( + + U U U + U U + + ) U ( U + U ) MASON'S FORMULA Maon' Formula allow one to determine the tranfer function of general block diagram with multiple loop, including feedback loop, and multiple feedforward path It relie on ome idea that we now define A block diagram conit of path and loop A loop i any path where one can go in a circle and return to the beginning point by following arrow in the direction in which they point Note that the figure in Example and do not contain any loop Two loop are aid to be dijoint if they have no element in common, ie if they do not touch
The determinant of a block diagram i defined a () - (um of tranmiion of all loop) + (um of product of tranmiion of all pair of dijoint loop) - (um of product of tranmiion of all triple of dijoint loop) - + The cofactor of a block diagram with repect to the i-th path i defined a ι () - (um of tranmiion of all loop that are dijoint from path i) + (um of product of tranmiion of all pair of dijoint loop that are dijoint from path i ) - (um of product of tranmiion of all triple of dijoint loop that are dijoint from path i) + The i-th cofactor i the ame a the determinant, but doe not include any loop touching path i Alo required i g i tranmiion along path i In general, thee are all function of o that one write ( ), i ( ), gi ( ) In term of the contruction, one may write the tranfer function of any block diagram a Maon' Formula ( ) g i ( ) i ( ) ( ) i It i intriguing to realize that thi i imply the formula for the invere of a matrix in term of the cofactor in a Laplace expanion Example - Feedback Connection Uing Maon' Formula () ()
Let u ue Maon' Formula to determine the tranfer function of the Feedback Interconnection There i one loop, which ha loop tranmiion Therefore, the determinant i ( ) loop gain + There i one path from input to output, which ha tranmiion g The loop touche the path, o there are no loop dijoint from the path Therefore one ha Applying now Maon' formula one obtain ( ) g( ) ( ) ( ) ( ) ( ) + ( ) ( ) Example - Redo Example Uing Maon' Formula () u (t) () () () u (t) () There are no loop o the determinant i ( ) There are three feedforward path from input to output One ha g g g There are no loop dijoint from the path, o all cofactor are equal to Therefore, Maon' Formula reveal that ( ) g + g + g + + 6
Example 6- Multiloop Sytem Even complicated multiloop ytem are uceptible to traightforward olution uing Maon' Formula L p p L 0 L 9 L There are two path from input to output and four loop, all labeled Note that one could manufacture another path from to by going along part of p, then circuiting loop L, then completing path p owever, we conider only independent path and loop, not direct compoition of impler path and loop The pair of dijoint loop are (L,L ), (L,L ), (L,L ), (L,L ) The triple (L,L,L ) i dijoint Therefore the determinant i ( ) ( + + + ) + ( ( + The two path have tranmiion g g ) 9 9 0 0 + + It i eay to determine the cofactor To find the cofactor of path, imply go to the formula for the determinant and cro out all loop touching path Loop L touche path, o we cro out all term containing to obtain ( ) ( + 90 + ) + ( ) To find the cofactor of path, imply go to the formula for the determinant and cro out all loop touching path Loop L touche path, o we cro out all term containing to obtain )
( ) ( + 90 + ) + ( + ) 9 0 The tranfer function i now given by Maon' Formula + For any pecific example given, all the tranfer function will be precribed (cf Example ), and one ubtitute into thi equation and implifie to find () Then one may find the tep repone, pole, zero, output given any input, etc Example 6- Sytem in Obervable Canonical Form a Tranfer Function p p L 6 L L 6 Thi block diagram i in obervable canonical form, and i typical of many we hall ee in analyzing tate pace ytem We hall ee later that the output of each integrator i a tate There are loop and feedforward path All loop touch, and each loop touche both path In fact, all loop and path contain the rightmot integrator The determinant i 6 6 + 6 + + 6 ( ) The path tranmiion are g g
Both cofactor are equal to a there are no loop dijoint from either path The tranfer function i given by + g + g ( ) + + 6 + + 6 + 6 + + 6 Note that the denominator of () i the characteritic polynomial which we by denote () The quantity denoted by () in Maon' Formula i the determinant, which i the characteritic polynomial divided by the highet power of Note that ( + ) ( ) ( + )( + )( + ) ( + )( + ) o there i pole/zero cancellation Thi ytem i actually of econd order, and it can in fact be realized uing two integrator intead of three The block diagram i aid to be nonminimal ince it realize it tranfer function uing too many integrator Noe that if the feedforward gain in path p i changed from to, then one ha + ( ) ( + )( + )( + ) where there i no pole/zero cancellation Thi ytem i of order three and the block diagram realization i minimal b Different Output p p p z(t) L 6 L L 6 p 9
Different input and output can be defined for the ame block diagram Let u find the tranfer function from to the new output z(t) hown in the figure Selecting new input or output doe not change the baic tructure of the block diagram, o the loop tructure i the ame and the determinant i the amea in part a Though it i not obviou at firt glance, there are two path from to z(t) They are labeled in the figure Path p i a circuitou journey coniting of the element, /, /, 6, -,/ The path tranmiion are g g All loop touch path p o that owever, loop L and L are dijoint from path p o that 6 + 6 + ( ) According to Maon' Formula one ha the tranfer function from to z(t) given by + 6 + + g + g ( + 6 + ) ( ) zu + 6 + + 6 + 6 + + 6 Note that ( + )( + ) ( + ) zu ( ), ( + )( + )( + ) ( + )( + ) o the block diagram i nonminimal with repect to the I/O pair (,z(t)) MINIMALITY, POLES, AND ZEROS A block diagram i aid to be minimal if it realize it tranfer function with the minimum number of integrator For ingle-input/ingle-output (SISO) ytem, thi occur if and only if there i no pole/zero cancellation Note that the ame block diagram can be minimal with repect to one input/output (I/O) pair but nonminimal with repect to another 0
The pole are determined by the loop and the zero by the feedforward path Note that the zero change a the input/output pair i changed, but the pole depend on the baic loop tructure and are independent of the election of input and output Once Maon' Formula ha been ued to determine the tranfer function, one may determine for any block diagram the tep repone, pole, natural mode, output given a precribed input, and o on