شهادت امام جعفر صادق(ع) تسلیت باد سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

Recap. State Space Equation: Physical System, Phase Variable, Canonical Forms, Control Systems2

3-نمايش معادلات فضاي حالت توسط فرمهاي كانوليكال هدف : با فرض مشخص بودن تابع تبديل سيستم، تحقق های فضای حالت که از اهميت ويژه ای بر خوردار هستند را بدست می آوريم. اين تحقق ها عبارتند از: الف) فرم كانونيكي كنترل‌پذير ب) فرم كانونيكي مشاهده‌پذير ج) فرم كانونيكي قطري (جردن) تابع تبديل زير را در نظر می گيريم: تبديل لاپلاس ورودي : تبديل لاپلاس خروجي :

فرم کانونيکی کنترل پذير ويژگی ها : 1) اين تحقق همواره کنترل پذير است. 2) در صورتيکه تابع تبديل سيستم، قطب و صفر مشترکی نداشته باشند، اين تحقق رويت پذير خواهد بود. مدل فضای حالت :

فرم کانونيکی رويت پذير مدل فضای حالت : ويژگی ها : 1) اين تحقق همواره رويت پذير است. 2) در صورتيکه تابع تبديل سيستم، قطب و صفر مشترکی نداشته باشند، اين تحقق کنترل پذير خواهد بود.

فرم کانونيکی قطری (جردن) حالت اول : اگر مقادير ويژه سيستم، حقيقی و غير تکراری باشند. مدل فضای حالت : ويژگی ها : 1) اين تحقق همواره کنترل پذير است. 2) در صورتيکه باشند، اين تحقق رويت پذير خواهد بود.

حالت دوم : اگر تعدادی از مقادير ويژه سيستم، حقيقی و تکراری باشند. ز مدل فضای حالت : ويژگی ها : 1) اين تحقق همواره کنترل پذير است اگر و فقط اگر آخرين سطر بلوکهای جردن مربوط به هر مقدار ويزه تکراری، در ماتريس B مستقل خطی (اگر تنها يک بردار باشد، مخالف صفر) باشند. 2) اين تحقق همواره رويت پذير است اگر و فقط اگر اولين ستون بلوکهای جردن مربوط به هر مقدار ويزه تکراری، در ماتريس C مستقل خطی (اگر تنها يک بردار باشد، مخالف صفر) باشند.

بدست آوردن تابع تبديل از معادلات فضاي حالت حالت اول : سيستمهای تک ورودی – تک خروجی (SISO) تبديل لاپلاس تابع تبديل يعني مقادير ويژه ماتريس A في‌الواقع همان قطبهاي سيستم مي‌باشند.

مثال : تابع تبديل سيستم زير را بدست آوريد :

حالت دوم : سيستمهای چند ورودی – چند خروجی (MIMO) اگر بردار ورودي u ، m بعدي و بردار خروجي y ، l بعدي باشد، آنگاه ماتريس G عبارت است از : در واقع عنصر (i, j) ام از تابع G ، ، تبديلي است كه خروجي i ام را به ورودي j ام مربوط مي‌سازد. بنابراين :

Outline Block Diagram Terms and concepts Canonical form of a feedback control system Block diagram transformations 11

Block diagrams Block diagrams consist of unidirectional, operational blocks that represent the transfer function of the variables of interest. The block diagram representation of a given system often can be reduced to a simplified block diagram with fewer blocks than original diagram. 12

Introduction A graphical tool can help us to visualize the model of a system and evaluate the mathematical relationships between their elements, using their transfer functions. In many control systems, the system of equations can be written so that their components do not interact except by having the input of one part be the output of another part. 13

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15 A block diagram is a shorthand, pictorial representation of the cause-and-effect relationship between the input and output of a physical system. It provides a convenient and useful method for characterizing the ‘functional relationships among the various components of a control system. The arrows represent the direction of information or signal flow.

Component Block Diagram 16

Block Diagram It represents the mathematical relationships between the elements of the system. The transfer function of each component is placed in box, and the input-output relationships between components are indicated by lines and arrows. 17

Block Diagram Algebra We can solve the equations by graphical simplification, which is often easier and more informative than algebraic manipulation, even though the methods are in every way equivalent. The interconnections of blocks include summing points, where any number of signals may be added together. 18

1 st & 2 nd Elementary Block Diagrams Blocks in series:Blocks in parallel with their outputs added: 19

Combining blocks in cascade 20

3 rd Elementary Block Diagram Single-loop negative feedback Transfer function Two blocks are connected in a feedback arrangement so that each feeds into the other. 21

Proof: G1G1 x y G2G2 - + b e xy 

G1G1 G2G

Quarter car suspension R(s y + - Series R(s ) + - y Feedback R(s ) y

1 st Elementary Principle of Block Diagram Algebra 25

2 nd Elementary Principle of Block Diagram Algebra 26

3 rd Elementary Principle of Block Diagram Algebra 27

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Example 1 29

U Y Example 2: Find TF from U to Y: No pure series/parallel/feedback Needs to move a block, but which one? Key: move one block to create pure series or parallel or feedback! So move either left or right.

U Y U Y U Y

Feedback Rule The gain of a single-loop negative feedback system is given by the forward gain divided by the sum of 1 plus the loop gain 32

Eliminating a feedback loop 33

Closed-loop transfer function E a (s) = R(s) - B(s) = R(s) - H(s) Y(s) Y(s) = G(s) E a (s) Y(s) = G(s) [ R(s) - H(s) Y(s) ] Y(s) [ 1 + G(s) H(s) ] = G(s) R(s) Y(s)/R(s) = G(s) /(1 + G(s) H(s)) 34

Closed-loop transfer function E a (s) = R(s) - B(s) = R(s) - H(s) Y(s) Y(s) = G(s) E a (s) E a (s) = R(s) - H(s) G(s) E a (s) E a (s) [ 1 + G(s) H(s) ] = R(s) E a (s)= R(s) /(1 + G(s) H(s)) 35

Closed-loop transfer function Y(s) = R(s) G(s)/(1 + G(s) H(s)) E a (s)= R(s) /(1 + G(s) H(s)) 36

All the transformations can be derived by simple algebraic manipulation of the equations representing the blocks. 37

Ex. 3 Block diagram reduction 38

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40

Example 4 41

Can use superposition: First set D=0, find Y due to R Then set R=0, find Y due to D Finally, add the two component to get the overall Y Example 5

First set D=0, find Y due to R

G2 Then set R=0, find Y due to D

Finally, add the two component to get the overall Y

Summary Using transfer function notations, block relationships were obtained. 46

Signal-Flow Graph Models

Control Systems48 Outline Terms and concepts Mason’s signal-flow gain formula Numerical examples

Control Systems49 A signal-flow graph A diagram consisting of nodes that are connected by several directed branches. A graphical representation of a set of linear relations.

Control Systems50 The basic elements of a signal- flow graph branch - a unidirectional path segment, which relates the dependency of an input and an output variable. nodes - the input and output points or junctions. path - a branch or continuous sequence of branches that can be traversed from one node to another node.

Control Systems51 All branches leaving a node will pass the nodal signal to the output node of each branch ( uniderectionally ). The summation of all signals entering a node is equal to the node variable. The relation between each variable is written next to the directional arrow.

Control Systems52 A loop - a closed path that originates and terminates on the same note, and along the path no node is met twice. Two loops are said to be nontouching if they do not have a common node. Two touching loops share one or more common nodes.

Control Systems53 Block and branch of DC motor

Control Systems54 Two-input, two-output system

Control Systems55 Y 1 (s) = G 11 (s) R 1 (s) + G 12 (s) R 2 (s) Y 2 (s) = G 21 (s) R 1 (s) + G 22 (s) R 2 (s) G ik - transfer function relating the i-th output to the k-th input

Control Systems56 Interconnected system

Control Systems57

Control Systems58 The flow graph is a pictorial method of writing a system of algebraic equations so as to indicate the interdependencies of the variables.

Control Systems59 A set of simultaneous equations 1. Write the system equations in the form X 1 = A 11 X 1 + A 12 X 2 + …+ A 1n X n X 2 = A 21 X 1 + A 22 X 2 + …+ A 2n X n ….…………………………………… X m = A m1 X 1 + A m2 X 2 + …+ A mn X n Note: An equation for X 1 is not required if X 1 is an input node.

Control Systems60 2. Arrange the m or n (whichever is larger) nodes from left to right. 3. Connect the nodes by the appropriate branches A 11, A 12, etc. 4. If the desired output node has outgoing branches, add a dummy note and unity branch. 5. Rearrange the nodes and/or loops in the graph to achieve pictorial clarity.

Control Systems61 Set of simultaneous algebraic equations a 11 x 1 + a 12 x 2 + r 1 = x 1 a 21 x 1 + a 22 x 2 + r 2 = x 2 r 1, r 2 - input variables x 1, x 2 - output variables x 1 (1 - a 11 ) + x 2 (- a 12 ) = r 1 x 1 ( - a 21 ) + x 2 (1 - a 22 ) = r 2

Control Systems62

Control Systems63 Mason’s signal-flow gain formula T ij (s) = ∑ k P ijk ∆ ijk /∆ P ijk = k-th path from variable x i to variable x j ∆ = determinant of the graph ∆ ijk = cofactor of the path P ijk ∑ k = all possible k path from x i to x j

Control Systems64 Step by step construction y 2 = a 12 y 1 + a 32 y 3 y 3 = a 23 y 2 + a 43 y 4 y 4 = a 24 y 2 + a 34 y 3 + a 44 y 4 y 5 = a 25 y 2 + a 45 y 4

Control Systems65 The nodes representing the variables y 1,y 2,y 3,y 4 and y 5 are located in order from left to right. The first equation states that y 2 depends upon two signals a 11 y 1 and a 32 y 3. The second equation states that y 3 depends upon the sum of a 23 y 2 and a 43 y 4,therefore a branch of gain a 23 is drawn from node y 2 to y 3 and a branch of gain a 43 is drawn from y 4 to y 3, with directions of the branches indicated by arrows. Similarly, with consideration of the third and fourth equation.

Control Systems66 Mason’s signal-flow gain formula ∆ = 1 - (sum of all different loop gains) + ( sum of the gain products of all combinations of two nontouching loops) - ( sum of the gain products of all combinations of three nontouching loops) + …,

Control Systems67 ∆ ijk = cofactor of the path P ijk is the the determinant with the loops touching the k-th path removed.

Control Systems68 T(s) = Y(s)/R(s) T(s) = Y(s)/R(s) = ∑ k P k ∆ k /∆

Control Systems69 The path gain or transmittance P k is defined as continuous succession of branches that are traversed in the direction of the arrows and with no node encountered more than once. A loop is defined as a closed path in which no node is encountered more than ones per traversal.

Control Systems70 Ex. 2.8 Interacting system

Control Systems71 The paths connecting the input R(s) and output Y(s) path 1 P 1 = G 1 G 2 G 3 G 4 path 2 P 2 = G 5 G 6 G 7 G 8

Control Systems72 Four self-loops L 1 = G 2 H 2 L 2 = H 3 G 3 L 3 = G 6 H 6 L 4 = G 7 H 8 Loops L 1 and L 2 do not touch L 3 and L 4

Control Systems73 The determinant : ∆ = 1 - (L 1 + L 2 + L 3 + L 4 ) + ( L 1 L 3 +L 1 L 4 + L 2 L 3 + L 2 L 4 )

Control Systems74 The cofactor along path 1 is evaluated by removing the loops that touch path 1 from ∆. L 1 = L 2 = 0 ∆ 1 = 1 - (L 3 +L 4 )

Control Systems75 The cofactor along path 2 is evaluated by removing the loops that touch path 2 from ∆. L 3 = L 4 = 0 ∆ 2 = 1 - (L 1 +L 2 )

Control Systems76 The transfer function of the system T(s) = Y(s)/R(s) = (P 1 ∆ 1 + P 2 ∆ 2 )/∆

Control Systems77 Ex. 2.7 Block diagram reduction

Control Systems78

Control Systems79

Control Systems80 Ex. 2.7 Mason’s signal-flow gain P 1 = G 1 G 2 G 3 G 4 L 1 = - G 2 G 3 H 2 L 2 = G 3 G 4 H 1 L 3 = - G 1 G 2 G 3 G 4 H 3

Control Systems81 All the loops have common nodes and therefore are all touching. The path P 1 touches all the loops, so ∆ 1 = 1 T(s) = Y(s)/R(s) = P 1 ∆ 1 /(1 - L 1 - L 2 - L 3 )

Control Systems82 P 1 = G 1 G 2 G 3 G 4 L 1 = - G 2 G 3 H 2 L 2 = G3 G4 H1 L3 = - G1 G2 G3 G4 H3 T(s) = Y(s)/R(s) = P 1 ∆ 1 /(1 - L 1 - L 2 - L 3 ) G 1 G 2 G 3 G 4 /(1+ G 2 G 3 H 2 - G3 G4 H1- G1 G2 G3 G4 H3)

Control Systems83 T(s) = G 1 G 2 G 3 G 4 /(1 - G3 G4 H1 + G 2 G 3 H 2 + G1 G2 G3 G4 H3)

Control Systems84 Ex A complex system

Control Systems85 The forward paths: P 1 = G 1 G 2 G 3 G 4 G 5 G 6 P 2 = G 1 G 2 G 7 G 6 P 3 = G 1 G 2 G 3 G 4 G 8

Control Systems86 Ex A complex system

Control Systems87 The feedback loops: L 1 = - G 2 G 3 G 4 G 5 H 2 L 2 = - G 5 G 6 H 1 L 3 = - G 8 H 1 L 4 = - G 7 H 2 G 2 L 5 = - G 4 H 4 L 6 = - G 1 G 2 G 3 G 4 G 5 G 6 H 3 L 7 = - G 1 G 2 G 7 G 6 H 3 L 8 = - G 1 G 2 G 3 G 4 G 8 H 3

Control Systems88 The determinant and cofactors: ∆ = 1 - (L 1 + L 2 + L 3 + L 4 + L 5 + L 6 + L 7 + L 8 ) + ( L 5 L 7 + L 5 L 4 + L 3 L 4 ) ∆ 1 = ∆ 3 = 1 ∆ 2 = 1 - L 5 = 1 + G 4 H 4 Loop L 5 does not touch loop L 7 or loop L 4, and loop L 3 does not touch loop L 4 ; but all other loops touch.

Control Systems89 The transfer function: T(s) = Y(s)/R(s) = (P 1 ∆ 1 + P 2 ∆ 2 + P 3 ∆ 3 )/∆

Control Systems90 Important properties of SF-G A SF-G applies only to linear systems. The equations for which a SF-G is drawn must be algebraic equations in the form of effects as function of causes. Nodes are used to represent variables. Normally, the nodes are arranged from left to right, following a succession of causes and effects through the system.

Control Systems91 Signal travel along branches only in the direction described by the arrows of the branches. The branch directing from node y k to y j represents the dependence of variable y j upon y k, but not the reverse. A signal y k traveling along a branch between nodes y k and y j is multiplied by the gain of the branch, a kj, so that a signal a kj y k is delivered at note y j.

Control Systems92 In the case when the system is represented by a set of integrodifferencial equations, we must first transform these into Laplace transform equations and then rearrange the latter into the form of Y j (s) = ∑G kj (s) Y k (s) for k, j =1,2,…,N

Control Systems93 Summary An alternative use of T(s) models in S-FG was investigated. Mason’s SF-G formula was found to be useful for obtaining the relationship between system variables in complex feedback system. Mason’s SF-G formula provides the relationship between system variables without any reduction or manipulation of the flow graph.