Unformatted text preview: Chapter 1: Control Systems
Physical process: Input
- Plant Output
- Open-loop control:
Ref. Output Actuator
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.
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 @ [email protected] A + [email protected] 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 , s0 = , L s; u s0 = 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
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
1 Ls I(s) Ls V(s) I(s) Current and voltage and sources:
+ + I(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
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
1 y (i) x a
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 ...
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- Fall '07
- Trigraph, forward path, path gain