Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Loop Shaping

Outline of Today’s Lecture Review PID Theory Integrator Windup Noise Improvement Static Error Constants (Review) Loop Shaping Loop Shaping with the Bode Plot Lead and Lag Compensators Lead design with Bode plot Lead design with root locus Lag design with Bode plot

PID: A Little Theory Consider a 1 st order function where the 1 st method of Ziegler Nichols applies The general transfer function for this system is The term is the transport lag and delays the action for t 0 seconds. Therefore The term T a is the time constant for the system. T measured on the graph is an estimate of this.

PID: A Little Theory The method 1 PI controller applied to the loop equation is

PID: A Little Theory In Method 2, the gain was increased until the system was nearly a perfect oscillatory system. Since the gain changes the oscillatory patterns, the lowest order system that this could represent would by a 3 rd order system. For this system to oscillate, there must be a solution of the characteristic function for K real and positive where s=±  i

PID: A Little Theory Applying the PI Controller:

Integrator Windup We have tacitly assumed that the controlled devices could meet the demands of the controls that we designed. However real devices have limitations that may prevent the system from responding adequately to the control signal When this occurs with an integrating controller, the error which is used to amplify the control signal may build up and saturate the controller. We refer to this as “integrator windup”: the system can’t respond and the integrator signal is extremely large (often maxed out on a real controller) the result is an uncontrolled system that can not return to normal operating conditions until the controller is reset

Integrator Windup To avoid windup, a possible solution is to provide a correcting error from the actuator by adding another loop: (the actuator has to be extracted from the plant)

Derivative Noise Improvement A major problem with using the derivative part of the PID controller that the derivative has the effect of amplifying the high frequency components which, for most systems, is likely to be noise. Without PID With PID

Derivative Noise Improvement One way to improve the noise rejection at higher frequencies is to apply a second order filter that passes low frequency and rejects high frequency The natural frequency of the filter should be chosen as with N chosen to give the controller the bandwidth necessary, usually in the range of 2 to 20 The controller then has the design

Static Error Constants If the system is of type 0 at low frequencies will be level. A type 0 system, (that is, a system without a pole at the origin,) will have a static position error, K p, equal to If the system is of type 1 (a single pole at the origin) it will have a slope of -20 dB/dec at low frequencies A type 1 system will have a static velocity error, K v, equal to the value of the -20 dB/dec line where it crosses 1 radian per second If the system is of type 2 ( a double pole at the origin) it will have a slope of -40 dB/dec at low frequencies A type 2 system has a static acceleration error,K a, equal to the value of the -40 dB/dec line where it crosses 1 radian per second

Static Error Constants K v (dB)

Loop Shaping We have seen that the open loop transfer function, has profound influences on the closed loop response The key concept in loop shaping designs is that there is some ideal open loop transfer (B(s)) that will provide the design specifications that we require of our closed loop system Loop shaping is a trial and error process: Everything is connected and nothing is independent What we gain in one area may (usually?) causes loss in other areas Often times, our best controller is a compromise between demands To perform loop shaping we can used either the root locus plots or the Bode plots depending on the type of response that we wish to achieve We have already considered an important form of loop shaping as the PID controller Error signal E(s) + + Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor

Loop Shaping with the Bode Plot The open loop Bode plot is the natural design tool when designing in the frequency domain. For the frequency domain, the common specifications are bandwidth, gain cross over frequency, gain margin, resonant frequency, resonant frequency gain, phase margin, static errors and high frequency roll off. Bandwidth rps -3 db Gain cross over frequency rps Roll off Rate dB/dec Resonant peak frequency rps Resonant peak gain, dB

Loop Shaping with the Bode Plot Increase of gain also increases bandwidth and resonant gain Break frequency corresponds to the component pole or zero Poles bend the magnitude and phase down Zeros bend the magnitude and the phase up

Lead and Lag Compensators The compensator with a transfer function is called a lead compensator if a a The lead and the lag compensator can be used together Note: the compensator does add a steady state gain of that needs to be accounted for in the final design There are analytical methods for designing these compensators (See Ogata or Franklin and Powell) xixi x0x0 y b1b1 b2b2 k Mechanical Lead Compensator

Lead Compensator The lead compensator is used to improve stability and to improve transient characteristics. The lead compensator can be designed using either frequency response or root locus methods Usually, the transient characteristics are better addressed using the root locus methods Addressing excessive phase lag is better addressed using the frequency methods The pole of the system is usually limited by physical limitations of the components use to implement the compensator In the lead compensator, the zero and pole are usually separated in frequency from about 0.4 decades to 2 decades depending on the needs

Lead Compensation (Root Locus) Objective: place the dominant poles near a given position to improve transient characteristics and possibly stability Performance criteria for the design: Rise time Settling time Overshoot limits Damping Ratio Natural Frequency Damped Frequency Design strategy: place a negative real valued zero to shape the curve into the desired position and refine the position by moving the a negative real valued pole

Example An aircraft has a pitch rate control as shown. The response of the pitch control is under damped, sluggish has an objectionable transient vibration mode. Design a lead compensator that provides a damping ratio of between 0.45 to 0.50 and 5% settling time of 150 seconds which reduces the vibration mode as much as possible. C(s) R Y

Example The root locus from the complex poles has very little damping and causes the vibration seen in the response. There is a pole at on the real axis that is dominant and causes the sluggish behavior. Strategy: Use a lead compensator to bend the curves to the left and into the 0.6 damping region. The zero of the compensator should counteract the vibrational mode

Example The initial design with the pole at -5 and the zero at -1 had the desired effect of bending the root locus to the left and removing most of the vibration. However the pole is still too close to the origin such that 0.6 damping can not be achieved.

Example We achieved the specifications once the pole of the compensator was moved out to -9 and we adjusted the gain for the 0.6 damping.

Lead Compensator (Frequency Design) Note: 1)the lead compensator opens up the high frequency region which could cause noise problems 2)The Lead compensator adds phase  mm xixi x0x0 y b1b1 b2b2 k Mechanical Lead Compensator

Example An aircraft has a pitch rate control as shown. Design a lead compensator for this system for a static velocity error of 4/sec, and a phase margin of 40 degrees. C(s) + + Aircraft Pitch Rate Dynamics Compensator R Y

Example -33dB Current System: Reading the plot: When design a lead compensator first adjust the gain to meet the static error condition In this case the gain needs the be increase by 180 or 20Log = 45.1 dB added

Example Gain Adjusted System Then noting where the phase currently is, that is the desired location for the peak of the lead phase  spec – P m = a small safety = 55 deg Finally adjust them as necessary

Example Initial design: Phase good but K v not Gain needs to be increased by about 20 dB Final design:

Lag Compensator Lag compensators are used to improve steady state characteristics where the transient characteristics are adequate and to attenuate high frequency noise In order to not change the transient characteristics, the zero and pole are located near the origin on the root locus plot The starting point for the design on a root locus is to start with a pole location at about s = and then locate the pole as needed for the desired effect In order to not give up too much phase, the zero and pole are located away from the phase margin frequency

Lag Compensator a b Note that the lag compensator causes a drop in the magnitude and phase This could be useful in reducing bandwidth, and improving gain margin; however it might reduce phase margin xixi x0x0 b1b1 b2b2 k Mechanical Lag Compensator

Example A linear motor has an open loop rate transfer function of It is desired that the system have a static velocity error constant greater than 20/sec, a phase margin of 45 degrees plus or minus 5 degrees and gain crossover frequency of 1 radian/sec. Design a lag compensator for this system. C(s) + + Linear Motor Rate Dynamics Compensator R Y

Example Current System: Phase margin is low and the static velocity error constant must be improved. Start by correcting the static velocity error constant 4.9dB

Example Gain Adjusted: Gain and Phase Margin problems try placing a pole at rps and adjust the zero Need to move PM to here Need to shape the curve like this

Example Final Design

Summary Static Error Constants (Review) Loop Shaping Loop Shaping with the Bode Plot Lead and Lag Compensators Lead design with Bode plot Lead design with root locus Lag design with Bode plot Next: Sensitivity Analysis