note2

note2 - Chapter 1: Control Systems Physical process: Input...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 1: Control Systems Physical process: Input - Plant Output - Open-loop control: Ref. Output Actuator Closed-loop control: Ref. Comparison Controller Process Output Process Measurement Closed-loop control system uses a measurement of the output and feedback of this signal to compare it with desired signal, so the output will be regulated to desired reference. 1 Why Use Feedback? Under modeling: The mathematical model can not describe complete dynamics of the process, or it is too complex to be useful. Uncertain parameters: Some physical parameters may change with respect to operating conditions, or di cult to measure accurately. External disturbance: Physical process can't be insolated completely from outside world, and disturbance may a ect the output performance. The use of feedback suppresses uncertainties due to modeling, parameter variations, and external disturbances. The price: feedback devices. 2 Example 1: Operational ampli er. I2 I1 Vi R1 Ve - R2 A + Vo Usually A 1 and is very uncertain: A = 1; 000 10; 000; and Vo = ,AVe: With connection of R1 and R2 we have a feedback system. Since A 1, Solving Ve gives 0 V Ve = @ Ri 1 Vi , Ve Ve , Vo : R1 R2 Vo 1 0 1 + 1 1,1 = 0 R1R2 1 0 Vi + Vo 1 : A @ A@ A + A@ R2 R1 R2 R1 + R2 R1 R2 - n - R1R2 R1+R2 6 Vi - 1 R1 Ve - ,A Vo - 1 R2 3 Hence the gain from Vi to Vo is Vo = 1 ,RAR1RR22 = , R2 : 1+ , Vi R1 1 + R1AR22 R1 + R1+R2 +R A If R2 = 10R1, and A = 1; 000 10; 000, then Vo = ,9:881 9:989: Vi The relative error is only 1:09 after using feedback. Without feedback, the error is 900. If gain = 1; 000 is required, then 3 ampli ers can be connected together in series that will give gain of 1; 000, and an error 2:9 which is again considerably smaller. The feedback quality is determined by R2 , Vo : R1 Vi The use of feedback eliminates signi cantly the uncertainty. Other advantages include enlargement of bandwidth, and linear operating range of the ampli er. 4 Mathematical Modeling Model is a mathematical description of physical process, often in terms of input output signals. Models for Electric Circuits x2.3: Consider the following circuit: v(t)+ i(t) R L C - At any node, all currents add to zero: Assume iL0 = v0 = 0. Then the model is vt + 1 Z t v d + C dvt it = 0 which is an integro-di erential equation. Its transfer function is s V s = 1 C T s = I s 1 + 1 + sC = s2 + 1 s + 1 : R sL RC LC 5 it = iRt + iLt + iC t; t iRt = vR ; iC t = C dvt ; dt Zt 1 iLt = L 0 v d + iL0: R L dt Continuation of Modeling: Two methods dominate modeling: 1. Using principles in physics to derive models where reasonable approximation is employed. 2. Experimental methods. Consider pendulum oscillation model on the right hand side: The length is L, and the mass is m point mass. With angle , the force is: f = mg sin . The torque is: T = fL = mgL sin . Newton's law gives L The minus sign is caused by opposite direction of torque and angle . The above gives di erential equation: d2 ; J = mL2: T = ,J = ,J dt2 f if is small. d2 = ,mL2 d2 ; mgL sin = ,J dt2 dt2 d2 = , g sin , g ; dt2 L L 6 Applying Laplace transform with 0 = 0 6= 0 and _0 = 0, which is an oscillation with amplitude 0. 0 g s2s , s 0 = , L s; u s 0 = s ; ! = v g ; u t s = s2 + g 0 s2 + !2 L L v ug t = t = 0 cos!t = 0 cosu t L Linearization: yt = gxt. At each xed t, xt , x0t yt = gx t + dg + high order terms dx x=x0 1! dg gx0t + mxt , x0t; m = dx ; x=x0 = y0 + mx , x0 = y , y0 mx , x0: yt , y0 = M xt , x0; where M is the Jacobian matrix: 2 6 6 6 6 6 6 =6 6 6 6 6 4 @g1 @x1 @g2 @x1 @g1 @x2 @g2 @x2 The above can be generalized to a vector of yt = gxt: M @gn @gn @x1 @x2 .. . . . . . . @g1 @xn @gn @xn @gn @xn : .. 3 7 7 7 7 7 7 7 7 7 7 7 5 7 Circuit Elements: Resistor: vt = itR. It can be represented by SFG signal ow graph as: i(t) + v - 1 R i R v(t) v(t) i(t) 1 due to it = R vt. Capacitor: it = C dt Zt 1 vt = C 0 i d + v0. For zero initial condition, Laplace transform gives dv + v - i 1 I s = sCV s; or V s = sC I s V(s) sC I(s) 1 sC V(s) I(s) 8 Inductor: vt = L dt Zt 1 it = L 0 v d + i0. For zero initial condition, Laplace transform gives di + v - i 1 V s = sLI s; or I s = sL V s V(s) 1 Ls I(s) Ls V(s) I(s) Current and voltage and sources: + + I(s) _ V(s) _ 9 Signal Flow Graph Cause-e ect relation: Input is cause and output is e ect. y = ax: y = akxk: x a1 x1 a x2 . 2 . . an xn a y y SFG is a collection of nodes, representing summed signals, and branches, representing gains. For dynamic systems, gain can be transfer functions. De nitions of SFG: 1. Input node: a node has only outgoing branches source. 2. Output node: a node has only incoming branches sink. 3. Path: a collection of a continuous succession of branches, traversed in the same direction. Note that an SFG can have many paths. 10 4. Forward path: a path starts at input node, and ends at the output node, along which no node is traversed more than once. d x1 a1 a2 x2 b x3 1 x3 Example 1: c a1 b 1 Path: x1 ,! x2 ,! x3 ,! x3 forward path a2 b 1 x1 ,! x2 ,! x3 ,! x3 forward path a1 b c x1 ,! x2 ,! x3 ,! x3 not a forward path a2 b c x1 ,! x2 ,! x3 ,! x3 not a forward path 5. Loop: a path which begins and ends at the same node, for which no other node can be traversed more than once see loop bd and self-loop c of Example 1. 6. Path gain: the product of the branches gain in the path. 7. Forward path gain: the path gain of the forward path. 8. Loop gain: the path gain of the loop. 11 Example 2: x a c z b d y 1 y0 Input node: Output node: Forward path: Path: Loop: none. y0 y is not an output node. none. a ! b ! 1. ac, bd. Question: Is a ! b ! d ! c a loop? Answer: No. Because z is traversed twice, and it is not the beginning, nor the end node. SFG Algebra: Cascade connection: By x4 = cx3 = bcx2 = abcx1. x1 a x2 x1 b a bc x3 c x4 x4 Parallel connection: y = ax + bx + cx = a + b + cx. a x b c y x a+b+c y 12 x1 a b c z Self-Loop: It is noted that d e y 1 x2 x3 y2 g z = ax1 + bx2 + cx3 + gz 1 , gz = ax1 + bx2 + cx3 b c a z = 1 , g x1 + 1 , g x2 + 1 , g x3: Conclusion: All incoming branches to a node having a self1 loop will have their gains scaled by a factor of 1,g , but all outgoing branches do not change their gains. Summary: 1. SFG is not unique, and applies to only linear systems. 2. Equations must be in the form of e ects as functions of cause. 3. Nodes represent variables that are arranged from left to right, following a succession of causes and e ects. 4. Signals travel in the direction of arrows. 13 Example 3: With input u, output y, draw SFG for x1 = cx2; x2 = bx3; x3 = ax4; x4 = u + dx1 + ex2 + fx3; y = x1 Since y depends only on x1, which depends only on x2, which in turns depends only on x3, which depends only x4, we have u 1 x4 a f x3 b e x2 c x1 1 y d Forward path: abc, Loops: af , abe, abcd. N k Maison's Formula: M = yyout = X Mk . in k=1 yout: output node, and yin: input node; N = total number of forward path; Mk = path gain of the kth forward path; = 1 , Pm1 pm1 + Pm2 pm2 , Pm3 pm3 + where pmr = gain product of r nontouching loops, or 14 =1 + sum of all gain products of all possible combinations of two nontouching loops , sum of all gain products of all possible combinations of three nontouching loops + ; , sum of all gain product of all individual loops Nontouching: if two parts of SFG do not share a common node, then they are called nontouching. k = the value for the part of the SFG after removing the kth forward path. If there is no loop, then k = 1. Example 4: For i two loops are touching, x a c b d 1 y (i) x a c b d y (ii) = 1 , ac , bd: For ii, two loops are nontouching, = 1 , ac , bd + abcd: 15 Example 5: a32 a43 a45 a23 a24 a25 a34 a44 1 u a12 y Forward path: i M1 = a12a23a34a45; 1 = 1; ii M2 = a12a24a45; 2 = 1; iii M3 = a12a25; 3 = 1 , a34a43 , a44: Individual loops: a23a32; a34a43; a44; a24a43a32: Two nontouching loops: a32a23a44. Overall gain: 2 M = M11 + M 2 + M33 a a a a + a a a + a a 1 , a34a43 , a44 : = 12 23 34 45 12 24 45 12 25 1 , a23a32 , a34a43 , a44 , a24a43a32 + a32a23a44 16 Example 6: g u a b c -h -j d -i e f y Forward path: i M1 = abcdef; 1 = 1; ii M2 = abcgf; 2 = 1: Individual loops: i , cdh; ii , ide; iii v , bcdej; iv , bcgj Overall gain: , gi; + M = M11 M22 abcdef + abcgf = : 1 + cdh + ide + gi + bcdej + bcgj 17 Example 7: Find gain of y for y 5 2 b a y1 y2 c -g y3 -i d y4 e -h y5 1 y6 Answer: y y =y It is noted that 5 = 5 1 . y2 y2=y1 For gain of y5=y1, we have that Forward path: M1 = acde, 1 = 1, M2 = abe, 2 = 1. Single loops: ,cg, ,eh, ,bei, ,cdei. Two-nontouching loops: cgeh. = acde + abe M15 = 1 + cg + eh + bei + cdei + cgeh : For gain of y2=y1, we have that Forward path: M1 = a with 1 = 1 + eh. Single and two-nontouching loops are the same. = 1 + M12 = 1 + cg + eha+ beiehcdei + cgeh : + Hence the overall gain for y5=y2 is + abe M = M15 = acde+ eh = cde + be : M12 a1 1 + eh 18 Equivalence Between SFG and Block Diagram: Block diagrams are similar to SFG. They can converted to each other. The following summarizes the equivalence: A block in a a block diagram is equivalent to a branch in an SFG with the same gain. The summed signal at the summation symbol is equivalent to a node. u e y u 1 e y H H 19 ...
View Full Document

Ask a homework question - tutors are online