Control Engineering Fundamentals Control Engineering Fundamentals Dr. aadi AlObaidi DUC
Nonlinear? linearize Analytical model development Time domain t models Input-output tate-space Linear ordinary differential equation Transfer function Laplace domain: d ds Frequency transfer function frequency esponse analysis Performance specification speed of response, stability, error, sensitivity Performance analysis Controller design PID control; position velocity feedback; lead and lag compensation domain: jw oad Map TEMINOLOY Plant or process: system to be controlled Inputs: Excitations known, unknown to the system Outputs: esponses of the system ensors: They measure system variables excitations, responses, etc. Actuators: They drive various parts of the system. Controller: Device that generates control signal. Control Law: elation according to which the control signal is generated Control ystem: Plant controller, at least can include sensors, signal conditioning, etc. Feedback Control: Control signal is determined according to plant "response" Open-Loop Control: No feedback of plant response to controller. Feed-forward Control: Control signal is determined according to plant "inputs" not "outputs". Control Engineering Fundamentals Dr. aadi AlObaidi DUC
Control Engineering Fundamentals Dr. aadi AlObaidi DUC
HITOY OF CONTOL ENINEEIN 00B.C reece Float valves and regulators for liquid level control 770 James Watt team engine: overnor for speed control 868 James Maxwell Cambridge University, Theory of governors 877 E.J. outh tability criterion 89 A.M. Laypunov oviet Union, tability theory, basis of state space formulation 97 H. Black and H.W. Bode AT&T Bell Labs, Electronic feedback amplifier 90 Norert Wiener MIT, Theory of stochastic processes 9 H. NyquistAT&T Bell Labs, tability criterion from Nyquist gain/phase plot 96 A.Callender, D.. Hartee, and A. Potter England, PID Control 948 Claude hannon MIT, Mathematical Theory of Communication 948 W. evans oot locus method 940s Theory and applications of servomechanisms, cybernetics, and control MIT, Bell Labs, etc. 959s H.M. Paynter MIT, Bond graph techniques for system modeling 960s apid developments in tate-space techniques, Optimal control, pace applications.bellman and.e. Kalman in UA, L.. Pontraygin in U, NAA 965 Theory of fuzzy sets and fussy logicl.a. Zadeh 970s Intelligent control; Developments of neural networks; widespread developments of robotics and industrial automation North America, Japan, Europe 980s obust control; widespread applications of robotics, and flexible automation 990s Widespread application of smart products; Developments in mechatronics, MEM Challenges: Nanotechnology, embedded, distributed, and integrated sensors, actuators, and controllers; Intelligent multiagent systems; mart and adaptive structures; Intelligent multiagent systems; mart and adaptive structures; Intelligent vehiclehighway systems, etc. Control Engineering Fundamentals Dr. aadi AlObaidi DUC 4
. Open-loop and Closed loop ystems Definition.: control system: Is an arrangement of physical components connected or related in such a manner as to command, direct, or regulate itself or another system. Definition.: Open-loop control systems: Those systems are which the output has no effect on the control action; it is neither measured nor fed back for comparison with the input. input Plant output Example thermostat etting A/C or heater indoor Temperature Control Engineering Fundamentals Dr. aadi AlObaidi DUC 5
Definition.: Closed-loop control systems: Feedback control systems are often reformed as closed-loop control systems. In such systems, the actuating error signal which is the difference between the input signals and the feedback signals is fed to the controllers so as to reduce the error and bring the output of the system to the desired value. input plant output measurement Example desired temp Amplifier K change to thermostat A/C or heater indoor temp Desired course of travel A MANUAL DIVIN CONTOL YTEM Error Driver teering mechanism Automobile Actual course of travel Measuremental visual Actual direction of travel Desired direction of travel Control Engineering Fundamentals Dr. aadi AlObaidi DUC 6
Control Engineering Fundamentals Dr. aadi AlObaidi DUC 7.Transfer Functions TFs:. Open-loop transfer function and feed forward T.F: H E B looptf open E C FFTF If H =, then the OLTF and the FFTF are the same.. Closed-loop transfer function: eferring to the figure above: B E E C C H Eliminating Es from these equations gives: ] [ H C C H C C.L.T.F Thus the output of the closed-loop system clearly depends on both the closed-loop transfer function and the nature of the input C H B E
Black's formula: is a way of achieving this same result directly, without any equation writing. It can be directly applied to general feedback configuration as follows: Over all transfer function forward pathain loopgain forward path X A B Y loop C forward path A B loop A B C The ve sign in the "loop" is due to the negative feedback. A more complex system: X A B Y C Y X A B A A B C A Control Engineering Fundamentals Dr. aadi AlObaidi DUC 8
. Closed-loop system subjected to disturbance: disturbance N C H Fig 6 eference input N Distrubance In examining the effect of the disturbance N, we may assume initially that the system is at rest with zero error. We may calculate the response C N to the disturbance only. C N N H Now, consider the response to the reference input assuming that the disturbance is zero, then the response C to the reference input can be obtained from C H The response to the simultaneous application of the reference input and the disturbance can be obtained by adding the two individual responses. C C CN [ N H Consider now the case where H and H ] Control Engineering Fundamentals Dr. aadi AlObaidi DUC 9
C N In this case the CLTF N becomes almost zero, and the effect of the disturbance is suppressed. This is an advantage of the closed-loop system. On the other hand, the CLTF C approaches H as the gain of H increases. C This means that if H then the CLTF becomes independent of and and becomes inversely proportional to H so that the variation of and do not affect the CLTF C. This is another advantage of the close loop system. If H, unity feedback. The input and the output will be equalized. Control Engineering Fundamentals Dr. aadi AlObaidi DUC
.4 Procedures for drawing a block diagram: To draw a block diagram for a system, first write the equation that describe the dynamic behavior of each component. Then take the laplace-transform of equation individually in block form. Finally, assemble the elements into a complete block diagram. Example: consider the C circuit shown ei i C eo The equations for t his circuit are: ei eo i e o i t dt c The Laplace transform of the above two equations become: Ei s Eo s I E i I E o I E o C I E O C Control Engineering Fundamentals Dr. aadi AlObaidi DUC
Assembling these two elements, we obtain the overall diagram for the system E i C E O E o.5 Block diagram reduction: A complicated block diagram involving many feedback loops can be simplified by a step by step rearrangements using rules of block diagram algebra, some of them are given in table. In simplifying a block diagram, remember the following:. The product of the transfer functions in the feed forward must remain the same.. The product of the transfer functions around the loop must remain the same. Example: simplify the diagram by using the rules given in table. H C H Control Engineering Fundamentals Dr. aadi AlObaidi DUC
By moving the summing point of the negative feedback loop containing outside the positive feedback loop containing H, we obtain the follow figure H H C H H H C H H C H H C Control Engineering Fundamentals Dr. aadi AlObaidi DUC
The denominator of C is equal to - product of the transfer functions around each loop H H H H the positive feedback loop yields a negative term in the denominator. C Note that numerator of the closed-loop function transfer functions of the feed forward path. is the product of the Control Engineering Fundamentals Dr. aadi AlObaidi DUC 4
Control Engineering Fundamentals Dr. aadi AlObaidi DUC 5
. outh's stability criterion: The outh's stability criterion is a way of determining whether a given feedback system, as modelled by a transfer function in the frequency domain, is "asymptotically stable". That is to say, according to the Lyapunov concept of stability, that the output and all internal variables never become unbounded and they go to zero as time goes to infinity for sufficiently small initial conditions. This occurs when all the poles roots p i of the system's characteristic equation strictly lie in the left half of the complex plane LHP: e {p i } < 0. The beauty of this technique is that you don't actually have to solve for the roots of the nthorder characteristic polynomial, which can be quite difficult for higher-order systems to say the least. This test is sometimes referred to as the outh-hurwitz tability Criterion, since equivalent tests were proposed independently by Edward outh in 874 and Adolf Hurwitz in 895.. Method: iven a closed-loop transfer function see Black's formula in the s-domain for a linear time-invariant LTI feedback system, the characteristic equation shows up as its denominator: Bs b 0 s m b s m-... b m s = ---- = -------------------------- ;m< or = n As s n a s n-... a n Characteristic equation: ps = As = s n a s n-... a n = 0 The first step is to begin to construct the outh array by arranging the coefficients of the characteristic polynomial in two rows, beginning with the first and second coefficients followed by the even-numbered and odd-numbered coefficients respectively, like so: s n : a a 4... s n- : a a a 5... The table is then completed with a bit of calculation as follows the following 'b' is unrelated to the 'b' in the transfer function shown above: sn : a a4... sn- : a a a5... sn- : b b b... Control Engineering Fundamentals Dr. aadi AlObaidi DUC 6
sn- : c c c...... :... Where det[ a; a a] a*a - a b = - ---------------- = ---------- a a det[ a4; a a5] a*a4 - a5 b = - ---------------- = ---------- a a det[ a6; a a7] a*a6 - a7 b = - ---------------- = ---------- a a det[a a; b b] b*a - a*b c = - ---------------- = -------------- b b det[a a5; b b] b*a5 - a*b c = - ---------------- = -------------- b b det[a a7; b b4] b*a7 - a*b4 c = - ---------------- = -------------- b b Once you've completed the table, you then look at the first column and count the number of sign changes e.g., -, = changes. Each sign change counts as one pole in the right half of the complex plane HP and, as mentioned earlier, poles in the HP make the system unstable! Therefore, you want all the elements in the first column to be positive and if you get that, your poles are safely in the LHP and your system will be stable. Example: ps = s 4 s s - 4s 5 -------- s4 5 s 4 0 s b b s c Control Engineering Fundamentals Dr. aadi AlObaidi DUC 7
s0 d b = -det[ ; 4]/ = b = -det[ 5; 0]/ = 5 c = -det[ 4; 5]/ = -6 d = -det[ 5; -6 0]/-6 = 5 b = c = c = d = d = 0 by inspection; trailing zeros are normally not written in the outh array since we just care about the first column -------- s4 5 s 4 0 s 5 s -6 s0 5 sign changes => system has poles in the HP => system is unstable. pecial Cases: If a first-column term in any row is zero, but the remaining terms are not zero or there is no remaining term, then the zero term is replaced by a very small positive number and the rest of the array is evaluated. For example, consider the flowing equation: The array of coefficients is 0 0 0 Control Engineering Fundamentals Dr. aadi AlObaidi DUC 8
If the sign of the coefficient above the zero is the same as that below it, it indicates that there are a pair of imaginary roots. Actually, the above equation has two roots at J. If however, the sign of the coefficient above the zero is opposite that below it, it indicates that there is one sign change. For example, for the flowing equation, 0 the array of coefficients is one sign change one sign change - 0 0 There are two sign changes of the coefficients in the first column. This agrees with the correct result indicated by the factored form of the polynomial equation. If all the coefficients in any derived row are zero, it indicates that there are roots of equal magnitude lying radially opposite in the s plane, that is, two real roots with equal magnitudes and opposite signs and/or two conjugate imaginary roots. In such a case, the evaluation of the rest of the array can be continued by forming an auxiliary polynomial with the coefficients of the last row and by using the coefficients of the derivative of this polynomial in the next row. uch roots with equal magnitudes and lying radially opposite in the plane can be found by solving the auxiliary polynomial, which is always even. For a n -degree auxiliary polynomial, there are n pairs and opposite roots. For example, consider the following equation: 5 4 4 48 5 50 0 The array coefficients is 5 4-5 4 48-50 Auxiliary polynomial P 0 0 The terms in the row are all zero. The auxiliary polynomial is then forme from the 4 coefficients of the row. The auxiliary polynomial P is P 4 48 50 Control Engineering Fundamentals Dr. aadi AlObaidi DUC 9
Which indicates that there are two pairs of roots of equal magnitude and opposite sign. These pairs are obtained by solving the auxiliary polynomial equation P 0. The derivative of P with respect to is dp 8 96 ds The terms in the row are replaced by the coefficients of the last equation, that is 8 and 96. The array of coefficients then becomes 5 4-5 4 48-50 8 96 Coefficients of dp ds 4-50.7 0 0-50 We see that there is one change in sign in the first column of the new array. Thus, the original equation has one root with a positive real part. By solving for roots of the auxiliary polynomial equation, 4 48 50 0 we obtain 5 or j5 The two pairs of roots are a part of the roots of the original equation. As a matter of fact, the original equation can be written in factored form as follows: j5 j5 0 Clearly, the original equation has one root with a positive real part.. elative stability analysis: outh's stability criterion provides the answer to the question of absolute stability. This, in many practical cases, is not sufficient. We usually require information about the relative stability of the system. A useful approach for examining relative stability is to shift the s-plane axis and apply outh's stability criterion. That is, we substitute ˆ constan t Into the characteristic eqation of the system, write the polynomial in the terms of Ŝ, and apply outh's stability criterion to the new polynomial in Ŝ. The number of changes of sign in the first column of the array developed for the polynomial in Ŝ is equal to the number of roots that are located to the right of the vertical line. Thus, this test reveals the number or roots that lie to the right of the vertical line..4 Application of outh's stability criterion to control system analysis: Control Engineering Fundamentals Dr. aadi AlObaidi DUC
outh's stability criterion is of limited usefulness in linear control system analysis mainly because it does not suggest how to improve relative stability or how to stabilize an unstable system. It is possible, however, to determine the effect of changing one or two parameters of a system by examining the values that cause instability. In the following, we shall consider the problem of determining the stability range of a parameter value. Consider the system shown in figure below. Let us determine the range of K for stability. The closed-loop transfer function is The characteristic equation is C K K 4 The array of the coefficients becomes K 0 4 K 0 0 7 9 K 7 K K K C For stability, K must be positive, and all coefficients in the first column must be positive. Therefore, When 4 K 9 4 9 K 0, the system becomes oscillatory and, mathematically, the oscillation is sustained at constant amplitude. Control Engineering Fundamentals Dr. aadi AlObaidi DUC
4. oot Locus Method: A pole-zero plot is simply a plot of the open-loop poles in the complex plane. A plot of the closed-loop poles can be similarly helpful. ince the closed-loop poles depend on the controller parameters, we don't get single points; instead, we get curves showing the pole position as a function of controller gain. uch plots are called oot Locus Plots. Quickly sketched root locus plots can be made using a little bit of algebra and following a few basic rules. It usually isn't necessary to have exact numerical values. The plots that result provide some very useful qualitative understanding of the closed loop response. An Initial Example Consider a feedback control loop with a forward patch process transfer function P actuator, valve, process, etc. and a return path transfer function measurement given by: When a proportional only controller is added, the open loop transfer function for this system becomes which corresponds to the general pole/zero form equation Control Engineering Fundamentals Dr. aadi AlObaidi DUC
When K=0, this is the open loop transfer function and the poles are easily plotted on the complex plane to obtain a pole-zero plot. To make a root-locus plot, we just pick several values of K and replot the poles for each. The result will be a set of curves, each beginning at an open loop pole. Look at the plots that result from the example. The system has poles, and the root locus plot has branches. Each branch begins at an open loop pole the "X"s. The plot is symmetrical. Complex roots of equations always appear as conjugate pairs. The branches roots of the CLCE stay in the LHP as long as a certain critical value of the gain is not exceeded. That gain is thus a stability limit. The branches follow the real axis roots are real for gains less than another trigger value. For gains larger than that, the response will be oscillatory. Control Engineering Fundamentals Dr. aadi AlObaidi DUC
oot locus plots are calculated by solving a complex valued polynomial equation -- but it isn't really necessary to do the math. The beauty of the root locus method is that L plots can be sketched by following a set of simple rules that require only a little algebra. oot Locus Plotting ules ee handout ummary of the steps for constructing root loci. Locate the poles and zeros of H on the s plane. The root-locus branches start from open-loop poles and terminate at zeros finite zeros or zeros at infinity.. Determine the root-locus on the real axis.. Determine the asymptotes of the root-locus branches. 4. Find the breakaway and break-in pints. 5. Determine the angles of departure angle of arrival of the root locus from a complex pole at a complex zero. 6. Find the points where the root loci may cross the imaginary axis. 7. Taking a series of test points in the bread neighborhood of the origin of the s plane, sketch the root loci. Other Examples Next, we'll look at some example systems to see what some typical root locus plots look like. First Order Lag The open loop transfer function for a first order lag is: It has one real pole Control Engineering Fundamentals Dr. aadi AlObaidi DUC 4
A first order system is never underdamped the pole is always on the real axis and is always stable. econd Order Lag A second order lag has two poles. Control Engineering Fundamentals Dr. aadi AlObaidi DUC 5
The nd order system becomes underdamped as the gain is increased, but is always stable since the poles never cross the imaginary axis. We can calculate the center of gravity and the breakaway point but don't need to. ince the system is always stable, these numbers don't tell us much. Third Order Lag Our initial example showed a third order system. The center of gravity calculation is needed to draw the asymptotes. We can easily calculate the breakaway point but probably don't need it. econd Order Lag with Zero Consider the open loop transfer function Control Engineering Fundamentals Dr. aadi AlObaidi DUC 6
Note how one of the branches ends at the zero. This is the rare case where the center of gravity provides no value -- because the asymptote is 80 degrees. Control Engineering Fundamentals Dr. aadi AlObaidi DUC 7