The MEMS Handbook Introduction & Fundamentals (2nd Ed) - M. Gad el Hak Part 13 pptx

For the block diagram in Figure 14.2 if Cs , As 1, Ps , Ss 1 and Rs a unit step input, then: Control system analysis and design consider primarily stability and performance.. Becaus

For more examples, see Ogata (1997), Raven (1995), Kuo (1995), and Franklin et al (1994).

Due to the convolution property of Laplace transforms, a convenient representation of a linear controlsystem is the block diagram illustrated in Figure 14.1 In such a block diagram, each block contains theLaplace transform of the differential equation representing that component of the control system thatrelates the block’s input to its output Arrows between blocks indicate that the output from the precedingblock is transferred to the input of the subsequent block The output of the preceding block multipliesthe contents of the block to which it is an input Simple algebra will yield the overall transfer function of

a block diagram representation for a system

unstable because it will correspond to an exponentially increasing solution Given a reference input R(s),

determine the response of the system by multiplying the transfer function by the reference input, and perform a partial fraction expansion (i.e., expand):

where each term in the sum on the right-hand side of the equation is similar to one of the entries in Table14.1 The contribution to the response of each individual term can be determined by referring to aLaplace transform table and can be superimposed to determine the overall solution:

FIGURE 14.1 Typical block diagram representation of a control system

FIGURE 14.2 Generic block diagram including transfer functions

For the block diagram in Figure 14.2 if C(s)
, A(s)
1, P(s)
, S(s)
1 and R(s)

(a unit step input), then:

Control system analysis and design consider primarily stability and performance The stability of a system with the closed-loop transfer function (note that in such a case a controller has already been specified):

is determined by the roots of the denominator, or characteristic equation It is possible to determinewhether the system is stable without actually computing the roots of the characteristic equation A nec-

essary condition for stability is that each of the coefficients a iappearing in the characteristic equation be

positive Because this is a necessary condition, if any of the a iare negative, then the system is unstable, but

the converse is not necessarily true Even if all the a iare positive, the system may still be unstable Routh(1975) devised a method to check necessary and sufficient conditions for stability

The method is to construct the Routh array, defined as follows:

in which the a i are from the denominator of Equation (14.14) b i and c iare defined as:

The basic result is that the number of poles in the right-half plane (i.e., unstable solutions) is equal

to the number of sign changes among the elements in the first column of the Routh array If they are allpositive, the system is stable When a zero is encountered, it should be replaced with a small positive con-stantεwhich will then be propagated to lower rows in the array The result can be obtained by taking thelimit as ε→0

the time-domain solution of the previous example, as t →∞, y(t) → 1 However, the final value theorem

can be used to determine this without actually solving for the time-domain solution

Example

Determine the steady-state value for the time-domain function y(t) if its Laplace transform is given by

Y(s)
ω2/s(s2 2ζω

n s  ω2) Because all the solutions of s2 2ζω

n s  ω2
0 have a negative real part,

all the poles of sY(s) lie in the left half of the complex plane Therefore, the final value theorem can be

14.2.2.1 Proportional–Integral–Derivative (PID) Control

Perhaps the most common control implementation is so-called proportional–integral–derivative (PID)control, where the commanded control input (the output of the “controller” box in Figures 14.1 and 14.2)

is equal to the sum of three terms: one term proportional to the error signal (the input to the “controller”box in Figures 14.1 and 14.2), the next term proportional to the derivative of the error signal, and the

third term proportional to the time integral of the error signal From Figure 14.2, C(s)
K P  (K I /s) 

K d s, where K P is the proportional gain, K I is the integral gain, and K dis the derivative gain A simple

analy-sis of a second-order system shows that increasing K P and K Igenerally increases the speed of the response

at the cost of reducing stability Increasing K dgenerally increases damping and stability of the response

With K I
0, there may be a nonzero steady-state error, but when K Iis nonzero, the effect of the integralcontrol effort is to typically eliminate steady-state error

Example — PID Control of a Robot Arm

Consider a robot arm illustrated in Figure 14.3 Linearizing the equations of motion about θ
0 (theconfiguration in Figure 14.3) gives:

Iθ

where I is the moment of inertia of the arm, m is the mass of the arm,θis the angle of the arm, and u is

a torque applied to the arm For PID control,

Figure 14.5 illustrates the step response of the system for proportional control (K P
1, K I
0,

K d
0), PD control (K P
1, K I
0, K d
1), and PID control (K P
1, K I
1, K d
1) Note that forproportional and PD controls, there is a final steady-state error that is eliminated with PI control (Alsonote that both of these facts could be verified analytically using the final value theorem.) Finally, note that

the system response for pure proportional control is oscillatory, whereas with derivative control theresponse is much more damped.

The subjects contained in the subsequent sections consider controller synthesis issues For PID trollers, tuning methods exist Refer to the undergraduate texts cited previously or to the papers by Zieglerand Nichols (1942, 1943)

con-14.2.2.2 The Root Locus Design Method

As mentioned previously in the discussion of PID control, various rules of thumb can be determined torelate system performance to changes in gains, however, a systematic approach is more desirable Becausepole locations determine the characteristics of the response of the system (recall the partial fractionexpansion), one natural design technique is to plot how pole locations change as a system parameter orcontrol gain is varied [Evans, 1948, 1950] Because the real part of the pole corresponds to exponentialsolutions, if all the poles are in the left-half plane, the poles closest to the jω-axis will dominate the system response If we focus a second-order system of the form:

1.4

Proportional control

PD control PID control

Time (sec)

FIGURE 14.5 PID control response

the poles of the system are as illustrated in Figure 14.6 The terms ω

n,ω

d, and ζare the natural frequency,

the damped natural frequency, and the damping ratio, respectively Multiplying H(s) by 1/s (unit step),

and performing a partial fraction expansion give:

Time (sec)

FIGURE 14.7 Step response for various damping factors

Because the natural frequency and damping are directly related to the location of the poles, one tive approach to designing controllers is picking control gains based upon desired pole locations A rootlocus plot is a plot of pole locations as a system parameter or controller gain is varied Once the root locushas been plotted, pick the location on the root locus with the desired pole locations to give the desiredsystem response There is a systematic procedure to plot the root locus by hand (refer to the cited under-graduate texts), and computer packages such as Matlab (using the rlocus() and rlocfind() func-tions) make it even easier Figure 14.9 illustrates a root locus plot for the previously noted robot arm with

effec-the block diagram as effec-the single gain K is varied from 0 to∞as illustrated in Figure 14.8 Note that for the usual root locus plot, only one gain can be varied at a time In the previous example, the ratio of theproportional, integral, and derivative gains was fixed, and a multiplicative scaling factor was varied in theroot locus plot

Because the roots of the characteristic equation start at each pole when K
0 and approach each 0 of the characteristic equation as K→ ∞, the desired K can be determined from the root locus plot by find-

ing the part of the locus that most closely matches the desired natural frequencyω

nand damping ratio ζ

(recall Figure 14.7)

Typically, control system performance is specified in terms of time-domain conditions, such as risetime, maximum overshoot, peak time, and settling time, all of which are illustrated in Figure 14.10.Rough estimates of the relationship between the time-domain specifications and the natural frequencyand damping ratio are given in Table 14.2 [Franklin et al., 1994]

1+ s + 1 s

FIGURE 14.8 Robot arm block diagram

Returning to the robot arm example, assume the desired system performance has a system rise timeless than 1.4 sec, a maximum overshoot less than 30%, and a 1% settling time less than 10 sec From thefirst row in Table 14.2, the natural frequency must be greater than 1.29 From the third and fourth rows,the damping ratio should be greater than approximately 0.4.Figure 14.11 illustrates the root locus plot,the pole locations and corresponding gain, and K (rlocfind() is the Matlab command for retrievingthe gain value for a particular location on the root locus) These results provide a damping ratio ofapproximately 45 and a natural frequency of approximately 1.38 Figure 14.12 illustrates the stepresponse of the system to a unit step input verifying these system parameters

14.2.2.3 Frequency Response Design Methods

An alternative approach to controller design and analysis is the so-called frequency response method.Frequency response controller design techniques have two main advantages They provide good controller

TABLE 14.2 Time-Domain Specifications as aFunction of Natural Frequency, Damped NaturalFrequency, and Damping Ratio

1.4

Maximum overshoot

Rise time Peak time

Settling time Steady-state error

FIGURE 14.10 Time domain control specifications

design even with uncertainty with respect to high-frequency plant characteristics, and using experimentaldata for controller design purposes is straightforward The two main tools are Bode and Nyquist plots (see[Bode, 1945] and [Nyquist, 1932] for first-source references), and stability analyses are considered first.

A Bode plot is a plot of two curves The first curve is the logarithm of the magnitude of the response

of the open-loop transfer function with respect to unit sinusoidal inputs of frequency ω The second

− 1.5

− 1

− 0.5 0 0.5 1 1.5

FIGURE 14.12 Robot arm step response

curve is the phase of the open-loop transfer function response as a function of input frequency ω Figure14.13 illustrates the Bode plot for the transfer function:

As the frequency of the sinusoidal input is increased, the magnitude of the system response decreases Thephase difference between the sinusoidal input and system response starts near 90° andapproaches 270° as the input frequency becomes large

An advantage of Bode plots is that they are easy to sketch by hand Because the magnitude of the tem response is plotted on a logarithmic scale, the contributions to the magnitude of the response due toindividual factors in the transfer function add together Due to basic facts related to the polar represen-tation of complex numbers, the phase contributions of each factor add as well Recipes for sketching Bodeplots by hand can be found in any undergraduate controls text, such as Franklin et al (1994), Raven(1995), Ogata (1997), and Kuo (1995)

sys-For systems where the magnitude of the response passes through the value of 1 only one time and forsystems where increasing the transfer function gain leads to instability (the most common, but not exclu-sive, scenario), the gain margin and phase margin can be determined directly from the Bode plot to pro-vide a measure of system stability under unity feedback Figure 14.13 also illustrates the definition of gainand phase margin Positive gain and phase margins indicate stability under unity feedback Conversely,negative gain and phase margins indicate instability under unity feedback The class of systems for whichBode plots can be used to determine stability are called minimum phase systems A system is minimumphase if all of its open-loop poles and zeros are in the left-half plane

Bode plots also determine the steady-state error under unity feedback for various types of referenceinputs (steps, ramps, etc.) In particular, if the low-frequency asymptote of the magnitude plot has a slope

Positive gain margin

Positive phase margin

FIGURE 14.13 Bode plot

of zero and if the value of this asymptote is denoted by K, then the steady-state error of the system under

unity feedback to a step input is

t→∞p(t).Figure 14.15 illustrates the unity feedback closed-loop step response of the system, verifying that the

steady-state value for y(t) is the same as computed from the Bode plot.

A Nyquist plot is a more sophisticated means to determine stability and is not limited to cases whereonly increasing gain leads to system instability A Nyquist plot is based on the well-known result fromcomplex variable theory called the principle of the argument Consider the (factored) transfer function:

i (s  z i)

j are the angles between s and the poles p j Thus, a plot of G(s) as s follows a closed contour (in the

clockwise direction) in the complex plane will encircle the origin in the clockwise direction the same

number of times that there are zeros of G(s) within the contour minus the number of times that there are

FIGURE 14.14 Bode plot for example problem

poles of G(s) within the contour Therefore, an easy check for stability is to plot the open loop G(s) on a contour that encircles the entire left-half complex plane Assuming that G(s) has no right-half plane poles (poles of G(s) itself, in contrast to poles of the closed-loop transfer function), an encirclement of 1 by the plot will indicate a right-half plane zero of 1  G(s), which is an unstable right-half plane pole of the

unity feedback closed-loop transfer function:

Lead–lag controller design is another popular compensation technique In this case, the compensator (the

C(s) block in Figure 14.2) is of the form:

where α 1 and β 1 The first fraction is the lead portion of the compensator and can provide

increased stability with an appropriate choice for A The second term is the lag compensator and provides

decreased steady-state error.Figure 14.18 plots the Bode plot for a lead compensator for various values of

the parameter A Because the lead compensator shifts the phase plot up, by an appropriate choice of the parameter A, the crossover point where the magnitude plot crosses through the value of 0 dB can be

shifted to the right, increasing the gain margin

FIGURE 14.15 Step response for example problem

Lag compensation works in a similar manner to increase the magnitude plot for low frequencies, whichdecreases the steady-state error for the system Lead and lag controllers can be used in series to increasestability and decrease steady-state error Systematic approaches for determining the parameters α,β, A, and B can be found in the references, particularly Franklin et al (1994).

Various other topics are typically considered in classical control but will not be outlined here Such ics include, but are not limited to, systematic methods for tuning PID regulators, lead–lag compensation,and techniques for considering and modeling time delay Interested readers should consult the references,particularly Franklin et al (1994), Ogata (1997), Kuo (1995), and Raven (1995)