Direct Design Method of Ragazzini - Evaluation of the Design

Evaluation of the Design

7.5 Direct Design Method of Ragazzini

+

10 1

•

_ _ _

() Ij

ltJlTIrad)

interesting to look at the it)' for designs. plotted in log of linear frequency toillustrate the balance between and for such plots for stable systems. On this onc can sec

results in the highest and the highest maximum of the or gain margin. Controller robustness (lower maximum of the ^I bm band\\ idth. Finally. the design given splits the difference and

Much of the style of the transform design techniques we have been discussing in this chapter grew out of the limitations of technology that was available for realization of continuous-time compensators with pneumatic components01

electric networks and amplifiers. In particular. many constraints were imposed in order to assure the realization of electric compensator networksD(s)consisting only of resistors and capacitors.' With controllers realized by digital computer.

such limitations on realization of course. not and one can ignore these particular An alternative design method that ignores constraints of

In book i much of theory collected about of first

of de,elopment. a chapler toRe

7.5 Direct Design Method of Ragazzini

Figure 7.36 Sensitivity plots of designs with controllers 0₃.0_{4 ,}and 05 for Example7.15

17.661

Solution. discrete characteristic equation according to the specifications is

- +IU6788

=

^O.

Let us therefore ask for a design that is stable. = 1. and has poles at the roots of Eq, (7.66) plus, if additional poles at = where the transient is as as possible. The form of thus

+ + + +

= .'

The causality design constraint. using Eq. Jrequires that

Direct Design Method of 267

=(O.5820)[h,+ + .J - +

(0.5820)(0.5820)

b_l + = h, + =0.5318.

Equations0.59)and 17.60) add no constraints has all poles and zeros inside the unit circle except for the singleLeroatx.which is taken care of by Eq. (7.68.1. The

error requirement leads to

Therefore

and

_TdHI K,.

in this case hothTand areI.we use Eq. 17.691 and the respect to to obtain

Because we only equations to need and can

truncate at resulting equations arc' .

which have the solution

-In other words-perhaps it was obvious from the starr-a common factorremains a factor of the characteristic polynomial. If this factor is outside the unit circle.

the system is unstable! How do we avoid such cancellation'! Considering again Eq.(7.54).we that if is not to cancel a pole of then that factor of must also be a factor of1 - Likewise. if not to cancel a zero of such zeros must be factors of Thus we write the constraints⁶

1 - must contain as zeros all the of that are outside the unit circle.

H must contain as zeros all the of that are outside the unit circle.

Consider finally the constraint of steady-state accuracy. Because H the overall transfer function. the error transform is given by

HO)= 1.

which implies

Thus if the system is to be Type1 with velocity constant K

r•we must have zero steady-state error to a step and1/ error to a unit ramp. The first requirement is

e(oo)

=

^{1)--[1 -}1

=

^O. ^(7.62)

+s+I= O.

From Eq.0.63)we know that 1 - is zero = I. so that to evaluate the limit in Eq. (7.64), it is necessary to use L'Hopital's with the result

dHI

T -The constant requirement is that

r-

e(oo)= - = -K .

1)-with a sampling periodT= I sec.

Roots on the unit circle are also by definitions. good practice indicates that should not singularities outside the of desired time.

• Example7.16 bythe

Consider again the plant described by the transfer function of Eq. (7.53) and suppose we ask for a digital design that has the characteristic equation that is the discrete equivalent of continuous characteristic equation

266 Chapter 7 Design Transform Techmques

D I+DG

• Discrete controllers can be designed by emulation. root locus, or frequency response methods.

• Successful design by emulation typically requires a sampling frequency at least30 times the expected closed-loop bandwidth.

• Expressions for steady-state error constants for discrete systems have been given in Eq.0.12)and Eq. 0.14)in terms of open-loop transfer functions and in Eg.0.18)in terms of closed-loop poles and zeros.

• Root locus rules for discrete system characteristic equations are shown to be the same as the rules for continuous system characteristic equations.

• Step response characteristics such as rise time and overshoot can be corre-lated with regions of acceptable pole locations in the as sketched in Fig.7.10.

that cause a large ripple in the system response between samples. How can this be for a system response transfer function.

- =

I+DG

that is designed to have only two well-damped roots" The answer lies in the fact that the control response is determined from

Summary 269

Summary

which for this example is

- 0.0793

- - = 13.06

+

0.3679

+

0.9672

The pole at = -0.9672. very near to the unit circle. is the source of the oscil-lation in the control response. The poor transient due to the pole did not show up in the output response because it was exactly canceled by a zero in the plant transfer function. The large control oscillation in causes the ripple in the output response. This pole was brought about because we allowed the controller to have a pole to cancel a zero at this position. The poor response that resulted could have been avoided if this nearly unstable zero had been included in the stabilitv constraint list. In that case we would introduce another term in

require that be zero at = -0.9672. so this zero of is not canceled by The result will be a simpler with a slightly more complicated

In chapter we have reviewed the philosophy and specifications of the design of control systems by transform techniques and discussed three such methods.

7.6

18 20 16 t2 t4

J

10 Time

= - 0.0501

- +0.3679

I _ = - I - OA180)

- +0.3679

0.5 - ,

'I.J __

(: - 1)(: - 0.90-18)(06321). - 0.(7932)

+0.9672) - - 0.4180)

= - 0.07932).

I:+ - OA180)

•

Thus the final design gives an overall transfer function

We shall also need

We know thatH ( I)= I that I - H I must a zcro at = 1. Now. turning10the design formula. Eg. (7.541. we compute

A plot of the step response of the resulting design for this example is provided in Fig.7.37 and verifies that the response samples behave as specified by However. as can he seen also from the figure. large oscillations occur in the 268 Chapter 7 Design Using Transfonn Techlllques

Figure7.37 Step response of antenna system from direct design

270 Chapler 7 DesIgn lsing Transform

7.5 Repeat the design for the satellite attitude control of Problem including method of choosing sampling periods but using the matched pole-zero method to obtain the discrete compensations.

7.6 Repeat the design of the satellite attitude control of Problem including method of choice of sampling period but using the triangle hold equivalent Inoncausal first-order hold) to design the discrete compensations.

1+K =0

- PI

-(d) For (b) The locus for

illustrates the use of complex zeros to compensate for the presence of complex poles duetostructural flexibility. Be sure to estimate the angles of departure and arrival. Sketch the loci for = Iand = 3 Which case unconditionally stable (stable for all positiveK less than the design valuer'

i, Plot the step response of the design and note the rise time and the percent overshoot.

7.7 Problems 271

show that the locus is a circle of radius centered at the origin (location of the zero). Can this result be translated to the case of poles and zero on the

1+K =0

The basic transfer function ofa sateHite attitude control Gis)= I

s-(a) Design continuous lead network compensation so as to give closed-loop poles corresponding to =0.5 and natural frequency =1.0, The ratio of pole to zero of the lead tobe no more than 10,

is a typical thaI includes poles and the \'alue of departure angles. Plot the locus for =0, I. and 2. Be sure to note the departurc from the complex in each case.

I 1+ K

(c) Select sampling period that will give 5 samples per time. compute the discrete equivalent using method, and compare rise time and of this design the continuous case.

ii. What is the system type and corresponding error

(b) Select a sampling periodto 10 samples in a rise time and compute the discrete equivalenttothe lead using the Tustin bilinear transformation. Plot the step response of the discrete system and compare rise time and overshoot to those of the continuousdesign.

7.4

l+K =0

s-(s

+

P,)

typical or the behavior near = ()of a double integrator with lead compensation or a single integration with lag network and one additional real pole. Sketch the locus Kfor values ofP, of 5.9. and 20. Pay close attention to the real break-in and break-away points.

• Asymptotes as used in continuous system frequency response plotsdoIIOf

for discrete frequency response.

• Nyquist's stability criterion and gain and phase margins were developed for discrete systems,

• The sensitivity function was showntobe useful to develop specifications on performance robustness as expressed in Eq, (7.34).

• Stability robustness in terms of the overall transfer function. the complemen-tary sensitivity function. is expressed in Eq. (7.47).

• Limitations on the frequency response of closed-loop discrete designs made more severe by poles and zeros outside the unit circle as expressed Eq. (7.50) and Eq. (7,51).

Lead and lag compensation can be used to improve the steady-state and transient response of discrete systems.

• The direct design method of Ragazzini can be used to realize a closed-loop transfer function limited only by causality and stability constraints.

Problems

7.1 Use the = mapping function and prove that the curve of constant in5 is a logarithmic spiral in

7.2 A servomechanism system is e"peeted to have a rise-time of no mOre than I0 milliseconds and an overshoot of no more than 5 ,

(a) Plot in the s-plane the corresponding region of acceptable closed-loop pole locations.

tb) What is the estimated Bode gain frequenc)' (e) What is the estimated phase margin in degrees"

(d) What is the sample period.T.if the estimated phase doe to the sample and hold is to be no more than 10' at the gain crossover"

(e) What is the sample periodTif there are to be samples per rise time"

7.3 ROOllocl/s The following root loci illustrate important features of the root locus technique, All are capable or being sketched hand. and it is recommended that they be done that way in order to develop skills in a computer's output. Once roughly hand. it useful to till in the details with computer.

(a) The locus for

7.7

istobe comrolled with a controller using a sample period of T=IJ.I

(a) Design using the :-plane ro01locus that will respond to a step with a rise time 1sec and an Plot the step response and "erify that

the meets the

(bl What is the system type and corresponding error constan!"! 'Vhat can be done to reduce the steady-state error to a input?

Plot the step response and compare complete to the transient and steady-state error specifications.

7 273

III X= +f(X.II.

where the force on the ball duetothe electromagnet given X.I).It is found that the magnet force balances the when the magnet currem is and the hall at If write I= +ⁱ^and^X= and expandf aboutX= andJ= and then neglect higher-order terms. we ohtain a linear approximation

= +

Values measured for a particular in the Stanford areIII=

kg.k, = = N/A.

(a) Compute the transfer function fromito and draw the (continuous 1root locus for proportional feedbacki=

(b) Let the sample period be T= and the plant discrete transfer function when used with a and zero-order hold.

7.12 The plant transfer function

7.13 It is tosuspend a steel ball bearing means of an electromagnct whose current is controlled by the position of the [Woodson and Melcher11968)1. Aschematic of a possible setup shown in Fig. The equations of motion are

Figure 7.38

A steel ball balanced by

an electromagnet

/10.

(e)

S+ 2/T

then redesign and find the discrete equiv'alem with the matched pole-zero emulation method. Plot the step and compare with the continuous design done on the unaugmented plant.

7.10 For the satellite with transfer function design a lead compensation10give c1osed-loop poles with damping = 0.5 and natural frequency = 1.0. The pole-to-zero ratio of the compensation should not be more than 10. Plot the step response of the closed loop and note the rise time and overshoot.

(a) Let sampling period be =0.5 and compute the discrete model of the piant with a sample and zero-order hold. Using this model. design a discrete lead compensation with pole at = -0.5 and a zero so as to closed loop at the mapped place from the continuous = 0.5 and = 1.0. What is the ratio of for this problem'} How many samples do you to find per rise time? Plot the step response and compare result with expectation. Compare the discrete design ith the continuous design.

tb) Repeat the discrete design with sampling period Ts= sec. plot the step response. and rise time and overshoot the continuous case.

7.11 the region in the of discrete pole locations correspcnding to 0.5 and

1-before doing the continuous design. Once the design of the lead compensation is com-pleted. continue with the discrete as in Problem 4. including the method of choosing sampling periuds. the matched pole-zero method to obtain the discrete compensallons. Compare the design with the continuous

7.8 a discrete for the antenna control system as specified in with a sample period ofT= 0.1 using a matched pole-zero equivalent for the discrete compensation. Plot the response and compare rise time and overshoot with those of the continuous design.

7.9 Design the antenna system specified in with a sample period of T= 0.5 sec.

tal Use the zero-pole mapping emulation method.

Augment the plant model with an approximation of the sample-hold delay consisting of

7.7 Repeat the design for the satellite attitude control of Problem but augment the plant with a Pade approximation to the delay ofT which is to say. multiply the plant transfer function by

272 Chapter Transform Techniques

Prohlems 275

Electronic Head lag

Spacecraft

---Tether

Ie) Do a design for the specifications and its step response. Compare these three designs with to meetlOg the transient

7.17The tethered satellite system shown in Fig, Ihas a moveable tether attachment point so that .torgues can be produced for allitude control. The block diagram of the system shown¹⁰ 7.42. Note that integrator in the actuator block indicates that a constant-voltage command to the servomotor will produce a constant v'elocit\' of the attachment

tal Is it to stabilize this system with feedback to a PlD Support

your with root argument.

(b) Suppose is possble to and as well Select the variable(s) on which you would like to a augment the feedback.

(e) Design compensation for the system using the that you selected in part (bl so that it has a 2-sec rise time and equivalent closed loop damping of = 0.5.

Figure 7,40 A disk drive head assembly

Figure 7,41 A tethered satellite system

- I

= .

where the pole at = [)adds some destabilizing phase lag, It therefore seems that it would be advantageous it and to use derivative control of the fonn

Sensor

= 1)

Can this be done" Support your answer with the difference equation that would required and the requirements to it.

7.15 For the automotive cruise-control system shown in Fig, 7.39. the sample period isT= 0.5sec.

(a) Design a PO controllertoachieve a of 5 sec with no overshoot.

(b) Detennine the speed error on a grade (i.e.. = 3in Fig, 7.39),

(e) Design a PID controllertomeet tbe same specifications as in part (a) and that has zero steady-state error on constant grades, What is the velocity constant of your

7.16 For the disk read/write head assembly described in Fig. 7.40, you are to design a compensation that will result in a closed-loop settling time t,=20msec and with overshoot a step input

(a) Assume no sampling and use a continuous compensation. Plot the step response and verify that your design meets tbe specifications,

(b) Assume a sampling periodT= 1msec and use matched pole-zero emulation. If you wish. you can include a Pade approximation to the delay and do a redesign of the continuous compensation before computing tbe discrete equivalent. Plot the step response and compare it with the continuous design's response.

(d) Plot a root locus of your design versus and the possibility of balancing balls of masses.

(e) Plot a step response of your design to an initial disturbance displacement on the ball and show both and the control currenti.If the sensor can measure a range of only em. and if the amplifier can provide a maximum current of IA. what the initial displacement. that will keep the variables within limits. using =

7.14 A discrete transfer function for approximate derivative control is

Figure 7.39 An automotive cruise-control system

274 Chapter Deslgn Using Transform Techniques

276 Chapter 7 Design Lsing Transform Techniques

(a) Show that the steady· error. (= - is

For further on damping the oil-mass

sis+a)

Problems 277

T =0.02 sec

when is ramp.

(bl Determine the highestK (i.e.. at the stability boundary) for proportional control(K, = =0).

Ie) Determine the K I .. at the stahility houndaryIfor PD =0) control.

Id) Determine the K (i.e.. at the stability boundary) for PD plus acceleration 0'

7.19For Ihe described in Problem7.18with transfer function IO()()

= sis+ + +

plot the Bode plot and measore the gain and phase margins with just unity feedback.

(a) a compensation that will a margin of 50 . a gain margin measured at the peakof at least and a of at least 1.0.

Plor the step response of the design and note the rise time and the

(bJ sample frequency = a discrete controller to meet the specifications for the continuous Plot the Bode plor of the design and verify that the specifications are met.

7.20 Asimple model of a satellite attitude control has the transfer function1 (a) Pial the Bode pial for this syslem and design a lead to a

margin of 50 and a crosso\'er ...of 1.0 rad/'ec. Plot the step response and note the rise time and overshoot.

(bl With sample period of T= sec. togil'e 50

phase margin and w" = 1.0.Plot the step and compare rise time and overshoot the continuous design.

Ie) With T =0.5 sec. design a discrete compen,ation to 50 phase margin and w" = 1.0.Plot the response and compare rise time and overshoot with the continuous design.

7.21 For a system by figure7.44

Schematic diagram for the control system of the excavator

Servo

?

=displacement of tether attachment point To=tether tension

M=moment on satellite 6 =satellite attitude

7.18 The shown in Fig. has a measuring the angle of the stick part01 a control system to control automatically the motion of bucket through the earth.

sensed angle is to be used to the control signal to the hydraulic actuator mo\'ing the stick. The schematic diagram for this control system is shown in Fig where GIs) is the system transfer function gi\'cn by

](jOO

+lOl(s'+ + .

The is implemented in a control computer sampling at = 50 Hz

of the form = Kil+Kill - + + The oscillatory

in arise from the compressibility of the hydraulic fluid (with some entrained air' and often referred to the

figure 7.43 An excavator with an automatic control system figure7.42

Block diagram for the tethered satellite system

278 Design Techniques

determine conditions under which theK, of the continuous

to K orthe preceded a ZOH and represented by its discretc function.

7.22 a controller for

Gis)= +

by a ZOH so that has a rise time of approximately 0.5 scc.

< and zero steady-state error to a step command,[Him:Cancclthe plant pole

= with a zero; a second-order closed-loop result.

response comparison and

(a) a using emulation with mapping

Do two forT=100 msee and one forT=250 (b) part (a) using root locus method for the two sample

In document Digital Control of Dynamic Systems (1998) (Page 144-151)