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第六章习题E6.1 A system has a characteristic equation :s3+3Ks2+(2+K)s+5=0.Determine the range of K for a stable system.E6.7 A negative feedback system has a loop transfer function (a) Find the value of the gain when the of the closed-loop roots is equal to 0.707.(b) Find the value of the gain when the closed system has two roots on the imaginary axis.E6.11 A system with a transfer function Y(s)/R(s) isDetermine the steady-state error to a unit step input.is the system stable?E6.14 A system has a characteristic equationq(s)=s4+9s3+45s2+87s+50=0.(a) Determine whether the system is stable,using the Routh-Hurwitz criterion.(b)Determine the roots of the characteristic equation.E6.23 A closed-loop feedback system is shown in Figure E6.23.For what range of values of the parameters K and p is the system stable?P6.3 Arc welding is one of the most important areas of application for instustrial robots.In most manufacturing welding situation,uncertainties in dimensions of part,geometry of the joint,and the weiding process itself require the use of sensors for maintaining weld quality.Several systems use a vision system to measure the qeometry of the puddle of melted metal,as shown in Figure P6.3. This system uses a constant rate of feeding the wire to be melted.(a) Calculate the maximum value for K for the system that will result in a stable sytem(b) For half of the maximum value of K found in part(a). Determine the roots of the characteristic equation.(c) Estimate the overshoot of the system of part(b) when it is subjected to step input.P6.4 A feedback control system is shown in Figure P6.4. The process transfer function is And the feedback transfer function is H(s)=1/(s+20).(a) Determine the limiting value of gain K for a stable system(b) For the gain that results in marginal stability,determine the magnitude of the imaginary roots(c) Reduce the gain to half the magnitude of the marginal value and determine the relative stability of the system (1)by shifting the axis and using Routh-Hurwitz criterion and (2)by determining the root locations.Show the roots are between -1 and -2.P6.7 The linear model of a phase detector(phase-lock loop) can be represented by Figure P6.7.The phase-lock systems are designed to maintain zero difference in phase between the input carrier signal and a local voltage-controlled oscillator.Phase-lock loops find application in color television,missile tracking,and space telemetry.The filter for a particular application is chosen as We want to minimize the steady-state error of the system for a ramp change in the phase information signal(a) Determine the limiting value of the gain Kak=kv in order to maintain a stable system(b) A steady-state error equal to 1o,Is acceptable for a ramp signal of 100% rad/s.For that value of Kv,determine the location of the roots of the system.P6.8 A very interesting and useful velocity control system has been designed for a wheelchair control system.We want to enable people paralyzed from the neck down to drive themselves about in motorized wheelchairs.A proposed system utilizing velocity sensors mounted in a headgear is shown in Figure P6.8.The headgear sensor provides an output proportional to magnitude of the head movement.There is a sensor mounted at 90o intervals so that forward,left right,or reverse can be commaded.Typical values for the time constant are 1=0.5s 3=1s,and 4=1/4s.(a) Determine the limiting gain K=K1K2K3 for a stable system(b) When the gain K is set equal to one-third of the limiting value,determine whether the setting time(to within 2% of the final value of the system) is less than 4s.(c) Determine the value of
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