Nelson R. C.; Flight stability and automatic control

Control engineering is based on the linear systems analysis associated with the development of feedback theory. A control system is constituted as an interconnection between the components which make up the system.

Components may be electrical, mechanical, hydraulic, pneumatic, thermal, or chemical in nature

The well-designed control system will provide the best response of the complete system to external, time-dependent disturbances operating on the system.

System to be controlled

Classification of Control Systems

Engineering control systems are classified according to their application, and these include the following:

1. Servomechanisms: Servomechanisms are control systems in which the controlled variable (or output) is a position or a speed. D.C. motors, stepper motor position control systems, and some linear actuators are the most commonly encountered examples of servomechanisms. These are especially prevalent in robotic arms and manipulators.

2. Sequential control: A system operating with sequential control is one where a set of prescribed operations is performed in sequence. The control may be implemented as event based, where the next action cannot be performed until the previous action is completed. An alternative mode of sequential control is termed time based, where the series of operations are sequenced with respect to time. Event-based sequential control is intrinsically a more reliable fail-safe mode than time based. Consider, for example, a process in which a tank is to be filled with a liquid and the liquid subsequently heated.

Sequential control systems

3. Numerical control: In a system using numerical control the numerical information, in the form of digital codes, is stored on a control medium, which may be a magnetic sensitive tape or a magnetic sensitive disk. This information is used to, operate the system in order to control such variables as position, direction, velocity, and speed. A large variety of manufacturing operations involving machine tools utilize this versatile method of control.

4. Process control: In this type of control the variables associated with any process are monitored and subsequent control actions are implemented to maintain the variables within the predetermined process constrains. The word process is all-encompassing and might include, for example, flight process or electrical power generation. The generation of electricity can be considered as a manufacturing process where the product is kilowatt hours. In the control of power generation the variables which are measured include temperature, pressure, liquid-level, speed flow-rate, voltage, current and a range of various gas concentrations. This is further complicated by the need to satisfy the power demand, and it is apparent that the control of such a system is necessarily complex. Similarly complex examples exist in the oil and paper-making industries, in aerospace assembly plants and in any entity which aspires to the designation of a flexible manufacturing system.

Open- and Closed-Loop Control

Open-loop control system

The input element supplies information regarding the desired value, X of the controlling variable. This information is then acted on by the controller to alter the output, Y External disturbances are fed in as shown and will cause the output to vary from the desired value. The open-loop system may be likened to the driving of a vehicle where the driver constitutes the input element. Essentially, two variables are controlled by the driver-the speed and the direction of motion of the vehicle. The controller, in the case of speed, is the engine throttle valve and in the case of direction, is the steering system.

The system become closed-loop, two further elements must be added: 1. A monitoring element, to measure the output, Y 2. A comparing element, to measure the difference between the actual output and the desired value, X

Closed loop feedback control system

For the purpose of definition, however, any system which incorporates some form of feed-back is termed closed-loop. With no feedback mechanism, the system is categorized as open-loop.

Feedback control

Aircraft control system

control system

control system elements relation

Linear and Nonlinear Control Systems

input X(t) output Y(t) input k*X(t) output k*Y(t)

input X1(t) output Y1(t) input X2(t) output Y2(t) input X1(t) + X2(t) output Y1(t) + Y2(t)

Characteristics of Control Systems

The characteristics of a control system are related to the output behavior of the system in response to any given input. The parameters used to define the control system's characteristics are stability, accuracy, speed of response, and sensitivity.

The system is said to be stable if the output attains a certain value in a finite interval after the input has undergone a change.

When the output reaches a constant value the System is said to be in steady state.

The system is unstable if the output increases with time.

In any practical control system, stability is absolutely essential. Systems involving a time delay or a dead time may tend to be unstable and extra care must be taken in their design to ensure stability. The stability of control system can be analyzed using various analytical and graphical techniques. These include the Routh Hurwitz criteria and the Bode, Nichols, Root Locus and Nyquist graphical methods.

The accuracy of a system is a measure of the deviation of the actual controlled value in relation to its desired value. Accuracy and stability are interactive, and one can in fact be counterproductive to the other. The accuracy of a system might be improved, but in refining the limits of the desired output the stability of the system might be adversely affected.

The speed of response is a measure of how quickly the output attains a steady-state value after the input has been altered.

Sensitivity is an important factor and is a measure of how the system output responds to external environmental conditions. Ideally, the output should be a function only of the input and should not be influenced by undesirable extraneous signals.

Dynamic Performance of Systems

The dynamic performance of a control system is assessed by mathematically modeling (or experimentally measuring) the output of the system in response to a particular set of test input conditions:

1. Step input: This is perhaps the most important test input, since a system which is stable to a step input will also be stable under any of the other forms of input. The step input is applied to gauge the transient response of the system and gives a measure of how the system can cope with a sudden change in the input.

2. Ramp input: A ramp input (is used to indicate the steady-state error in a system attempting to follow a linearly increasing input.

3. Sinusoidal input: The sinusoidal input over a varying range of input frequencies is the standard test input used to determine the frequency response characteristics of the system.

Static and Astatic systems

Static system - G(s) = L0(s) / M0(s)

Astatic system - G(s) = L0(s) / [sl M0(s)]

a static system, b - astatic system (first order)

Time Domain and Frequency Domain

The time domain model of a system results is an output Y(t) with respect to time for an input X(t). The System model is expressed as a differential equation, the solution of which is displayed as a graph of output against time.

In contrast, a frequency domain model describes the system in terms of the effect that the system has on the amplitude and phase of sinusoidal inputs. Typically, the system performance is displayed in plots of amplitude ratio, (Y(t)/X(t)) or 20 log10(Y(t)/X(t)), and phase angle, against input signal frequency. Neither system model has an overriding advantage over the other, and both are used to good effect in describing system performance and behavior.

MATHEMATICAL MODELS OF SYSTEMS TIME DOMAIN ANALYSIS

Differential equations are used to model the relationship between the input and output of a system. The most widely used models in control engineering are based on first- or second-order linear differential equations.

First-Order Systems some simple control systems (which includes the control of temperature, level, and speed) can be modeled as a first-order linear differential equation:

1)(

+=

s

ksG

-first order lag k -gain

Step input response

Response of a first-order system to a step input.

Ramp input response

First-order System response to a ramp input.

Sinusoidal input response

First-order system response to a sinusoidal input.

Second-Order Systems

The second-order differential equation has the general form:

where - is termed the damping ratio and is defined as the ratio of the actual damping in the system to that which would produce critical damping.

n - is the undamped natural frequency of the system k - is the system gain.

The time-domain solution depends on the magnitude of ( and three solutions for a step input are possible:

Response of a second-order system to a step input.

Response of a second-order system to a ramp input.

Analysis of second-order control system

Second-order open-loop control system

Second-order closed-loop control system

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Response curve for an underdamped system to a step input.

The speed of the response is reflected in the, tr - rise time

tp - peak time ts - settling time For uderdamped systems, the rise time is the time taken for the output to reach 100% of the step input. The peak time is that taken to the first maximum in the output response. For critically damped and overdamped systems, the time taken for the output to change between I0% and 90% of the input is used alternatively as a measure of the speed of the response. Settling time is the time taken for the oscillatory response to decay below a percentage of the input amplitude, , often taken as +-2%.

maximum overshoot (peak value of the output)

peak time

delay time

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Astatic system case

Second-order closed-loop control system with astatic element

maximum overshoot (peak value of the output)

peak time

delay time

rise time

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= n *

Frequency domain analysis

Second-order open-loop control system frequency response.

resonance module

resonance frequency

bandwidth

CONTROL STRATEGIES

Basic closed-loop control system.

SP(s) is the set point (required value, r(r), is sometimes used). PV(s) is the process value (corrected value, c(r), is sometimes used).

E(s) is the error signal, which is the difference between SP and PV. U(s) is the control effort output from the controller to the process. C(s) is the controller transfer function. G(s) is the process transfer function. The transfer function for the closed-loop system is obtained as before:

Hence:

ON / OFF Control

In many applications a simple ON/OFF strategy is perfectly adequate to control the output variable within preset limits. The ON/OFF control action results in either full or zero power being applied to the process under control.

ON/OFF controllers usually incorporate a dead band over which no control action is applied, which is necessary to limit the frequency of switching between the ON and OFF states. For example, in a temperature-control system: the ON/OFF control strategy would be: If temperature Tmax then heater is to be switched OFF.

The dead band in the above case is (Tmin -Tmax) and while the temperature remains within the dead band no switching will occur. A large dead band will result in a correspondingly large fluctuation of the process value about the set point. Reducing the dead band will decrease the level of fluctuation but will increase the frequency of switching. The simple ON/OFF control strategy is mostly applicable to processes and systems which have long time constants and in consequence have relatively slow response times (e.g., temperature and level control).

Output variation with ON/OFF control

While simple in concept, ON/OFF control systems are in fact, highly nonlinear and require some complex nonlinear techniques to investigate their stability characteristics.

Three-Term or PID Control

Since complicated transfer functions can be very difficult to model, the most common strategy used to define the controller transfer function is the so-called three-term or PID controller.

PID is the popular short form for Proportional, Integral, and Derivative. The three elements of the controller action, U, based on the evaluated error, E, are as follows:

Proportional Action

Controller output : K* E

where K is the controller gain. Manufacturers of three-term controllers tend to favor the parameter proportional band (PB) in preference to gain. K. The proportional band represents the range of the input over which the output is proportional to the input. The PB is usually expressed as a percentage of the input normalized between 0 and 100%

Illustration of the proportional band.

It must also be noted that for proportional control only, there must always be an error in order to produce a control action. From equation proportional control only gives a transfer function of the form

For steady-state conditions, s tends to 0 and G(s) tends to a constant value. Equation shows therefore that the gain must theoretically tend to infinity if PV=SP and the steady-state error is to approach zero.

This is simply another manifestation of the classical control problem, i.e., stability at the expense of accuracy and vice versa. With a very high gain (i.e., low proportional band) the steady-state error can be very much reduced. A low proportional band, however, tends to ON/OFF control action and a violent oscillation may result in sensitive systems.

Integral Action

The limitations of proportional control can be partly alleviated by adding a controller action which gives an output contribution that is related to the integral of the error value with respect to time, i.e.,

Where Ki is the controller integral gain (= K/Ti) and Ti is the controller integral time or reset. The nature of integral action suggests that the controller output will increase monotonically as long as an error exists. As the error tends to zero the controller output tends towards a steady value. The general behavior of the controller output with integral action is shown in figure

Controller output with integral action

If Ti is very large, the integral action contribution will be low and the error may persist for a considerable time. If, on the other hand, Ti is too small the magnitude of the integral term may cause excessive overshoot in the output response. Unstable operation is also possible when Ti is too small and the controller output value then increases continuously with time.

Derivative Action

The stability of a system can be improved and any tendency to overshoot reduced by adding derivative action. Derivative action is based on the rate of change of the error.

where Kd is the controller derivative gain (=K*Td) and Td is the controller derivative time or rate. Equation indicates that the derivative action is dependent on how quickly or otherwise the error is changing. Derivative action tends therefore only to come into operation during the early transient part of a system's response. The full three-term control strategy may be written as

To summarize, the proportional action governs the speed of the response, the integral action improves the accuracy of the final steady state, and the derivative action improves the stability. Note that derivative action may result in poor performance of the system if the error signal is particularly noisy. In Laplace notation, the three-term controller transfer function is as shown in figure

Three-term or PID control

Empirical Rules for PID Controller Settings

A simple and still popular technique for obtaining the controller settings to produce a stable control condition is due to Ziegler and Nichols (1942). The method is purely empirical and is based on existing or measurable operating records of the system to be controlled.

Open-Loop Reaction Curve Method The process to be controlled is subjected to a step-input excitation and the system open-loop response is measured. A typical open-loop response curve is shown in figure .

Open-loop system response to a step input.

Any system which has a response similar to that given in the figure has a transfer function which approximates to a first-order system with a time delay, i.e.

In general industrial applications, oscillatory open-loop responses are extremely rare and figure is in fact representative of quite a large number of real practical processes. In the figure, N is the process steady-state value for a controller step output of P. The system steady-state gain is

k= N/P From the process response curve the apparent dead time, T1, and the apparent time constant, T2, can be measured directly. The three parameters, k, T1, and T2, are then used in a set of empirical rules to estimate the optimum controller settings. The recommended controller settings are given in Table 6.5.

In fast-acting servomechanisms, where T1 may be very small, the method is none too successful. For moderate response systems, however, the method will yield very reasonable first-approximation controller settings.

Closed-Loop Continuous Cycling Method The process to be controlled is connected to the PID controller and the integral and derivative terms are eliminated by setting Td= 0 and Ti=. In some industrial controllers the integral term is eliminated with Ti=0. A step change is introduced and the system run with a small controller gain value, K. The gain is gradually increased for a step input until constant-amplitude oscillations are obtained as illustrated in figure.

Continuous cycling method.

The gain Ku, which produces the constant-amplitude condition is noted and the period of the oscillation Tu is measured. These two values are then used to estimate the optimum controller settings according to the empirical rules listed in Table 6.6.

The PID settings obtained according to the methods of Ziegler and Nichols are approximate only, and some fine tuning would almost certainly be required in practice.

PID Controller with a First-Order System If a P + I controller is to be used (i.e. no derivative action) the controller transfer function is

The process is modeled as a first-order system and its open-loop transfer function

is given by equation

after some manipulation

Comparing the denominator with that for the generalized second order system

equation it can be shown

For the system being controlled, both k and are known either via a mathematical model or an open-loop test. The controller settings, K and Ti can then be calculated for a chosen damping ratio and natural frequency n. Alternatively a controller gain can be imposed and the corresponding natural frequency evaluated.

For full PID control, an initial value of Td=Ti/ 4 can be used. Other systems can be similarly handled to obtain the approximate PID controller settings. In all cases some fine adjustment would probably be necessary to obtain the optimum output response.

Case T0=3s, T=15s find PID

T=0.3s k=16

STABILITY

The practicing control engineer will use many techniques to assess system stability. These might include the numerical Routh-Hurwitz criterion, which determines only whether a system is stable or not. Alternative graphical methods include the use of Hall charts, Nichols charts, inverse Nyquist plots, and root locus plots. The graphical methods additionally indicate the relative stability of a system.

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Rootlocus method

Three systems with the same characteristic equation

Rootloci travel from poles to zeros

Rootloci with real axis poles and zeros

Rootloci asymptotes for three pole system

Root locus example showing closed loop roots

Example

Time delays are very difficult to handle mathematically when they occur in differential equations, and the inclusion of multiple feedback loops can greatly increase the order of the governing equation. For these two reasons solutions in the time domain become extremely difficult, and frequency domain methods are almost exclusively used to assess the behaviour of the more complex control systems. The main consideration in frequency-domain analyses is the stability of the system and how it can be adjusted if it happens to be unstable. Various graphical methods are used and these include the Bode and Nyquist plots. The Bode plot is a graph of amplitude ratio and phase angle variation with input signal frequency. The resulting normalized plot for an open-loop first-order system is shown in Figure. Note that when the input frequency is equal to the inverse of the system time constant, the output amplitude has been decreased (or attenuated) by 3dB. The phase lag at this point is -45o. This is characteristic of first-order systems.

The Nyquist plot represents the same information in an alternative form. The plot is in polar coordinates and combines the amplitude ratio and phase lag in a single diagram. Figure shows the Nyquist plot for the open loop, first-order system.

Bode and Nyquist Stability Criteria

The Bode criterion for stability is that the system is stable if the amplitude ratio is less than 0 dB when the phase angle is -180". This is illustrated graphically in Figure. This figure represents a stable system since Bode's criterion is satisfied. The gain margin (GM) and phase margin (PM) are used as measures of how close the frequency response curves are to the 0 dB and 180o points and are indicative of the relative stability of the system.

Bode's stability criterion

The Nyquist criterion for stability is that the system is stable if the amplitude ratio is greater than -1 at a phase angle of -180o. In effect, this means that the locus of the plot of amplitude ratio and phase angle must not enclose the point - 1 on the real axis. A stable response curve is shown plotted in Figure. Also indicated in this figure are the gain margin and phase margin in the context of the Nyquist plot.

Nyquist's stability criterion

System Stability with Feedback

In a closed--loop system the transfer function becomes modified by the feedback loop. The first task therefore is to determine the overall transfer function for the complete system.

For simple open-loop systems the transfer functions are combined according to the following rules: 1. For elements in series, the overall transfer function is given by the product of the

individual transfer functions. 2. For elements in parallel, the overall transfer function is given by the sum of the

individual transfer functions. For a system with feedback, the overall transfer function can be evaluated using a consistent step-by-step procedure. series and parallel control elements

Disturbance Sensitivity

The main problem with the classical single-loop control system is that it is not truly representative of the natural environment in which the system operates. In an ideal single-loop control system the controlled output is a function only of the input. In most practical systems, however, the control loop is but a part of a larger system and is therefore subject to the constraints and vagaries of that system. This larger system, which includes the local ambient, can be a major source of disturbing influences on the controlled variable. The disturbance may be regarded as an additional input signal to the control system. Any technique, therefore, which is designed to counter the effect of the disturbance must be based on a knowledge of the time-dependent nature of the disturbance and also its point of entry into the control system. Two methods commonly used to reduce the effect of external disturbances are feedforward and cascade control.

Feedforward control system

Feedforward Control. The principle of a feedback loop is that the output is compared with the desired input and a resultant error signal acted upon by the controller to alter the output as required. This is a control action which is implemented after the fact. In other words, the corrective measures are taken after the external disturbance has influenced the output. An alternative control strategy is to use a feedforward system where the disturbance is measured. If the effect of the disturbance on the output is known, then theoretically the corrective action can be taken before the disturbance can significantly influence the output. Feedforward can be a practical solution if the external disturbances are few and can be quantified and measured.

Feedforward control can be difficult to implement if there are too many or perhaps unexpected external disturbances. In Figure the path which provides the corrective signal appears to go back. The strategy is still feedforward, however, since it is the disturbance which is measured and the corrective action which is taken is based on the disturbance, and not the output signal. Some control systems can be optimized by using a combination of feedforward and feedback control.

Cascade control system

Cascade Control. Cascade control is implemented with the inclusion of a second feedback loop and a second controller embodied within a main feedback loop in a control system see Figure. The second feedback loop is only possible in practice if there is an intermediate variable which is capable of being measured within the overall process. Cascade control generally gives an improvement over single-loop control in coping with disturbance inputs. The time constant for the inner loop is less than that for the component it encloses, and the undamped natural frequency of the system is increased. The overall effects of cascade control are an increase in the system bandwidth and a reduction in the sensitivity to disturbances entering the inner loop. Disturbances entering the outer loop are unaffected. Cascade control works best when the inner loop has a smaller time constant than the outer one.

Hydraulic Servo-actuator

Aircraft flight control Power Control Unit schematic

Actuator block diagram