Error budgeting for control system design

 

 

Johan Gouws

Associate Professor, Department of Electrical and Electronic Engineering, Rand Afrikaans University, P.O. Box 524, Auckland Park, 2006 South Africa

 

 

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ABSTRACT

System accuracy is an important issue in control system design. The design goals for the different subsystems of a control system are often derived from 'rolling down' the total allowable error in the response of the control system to the different subsystems. Once the attainable subsystem errors are known (from analysis and/or tests), these values need to be 'rolled up' in order to determine the overall attainable system accuracy. The iterative process of rolling down the requirements, rolling up the attainable values, rolling down reviewed requirements, etc., is termed error budgeting. This paper defines different error budgeting approaches and illustrates some of the most important aspects by means of error budgeting for a satellite inertial measurement system.

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Nomenclature

A/D analogue to digital [converter]

hr hour

IMS inertial measurement system

mrad milliradian

mV millivolt

μrad microradian

ppm parts per million

RSS root-sum-square

SF scale factor

temp.comp, temperature compensation

V/F voltage-to-frequency [converter]

 

Introduction

Accuracy is a critical issue in the development of control systems. In many complex systems, zero error is not attainable and some error in the controlled variable has to be tolerated. System design can be done in a top-down manner (where the maximum allowable system error is defined, and then rolled down to the different subsystems); or it can be done in a bottom-up manner (where the attainable accuracies of subsystems are determined and then rolled up to determine the attainable overall system accuracy); or it can be done in an iterative manner (where requirements are first rolled down, and attainable values are then rolled up, until an optimum balance between technical performance, cost, and schedule is reached).This iterative process is termed error budgeting. The first part of this paper is aimed at providing a concise, but useful, description of the principles and procedures involved in error budgeting - aspects which are often neglected in existing literature on the subject (e.g. [2] and [4]). Thereafter, some of the most important aspects of error budgeting are illustrated for a satellite inertial measurement system - which is an example of a subsystem in a high-accuracy control system.

 

System design approaches

Table 1 defines the top-down, bottom-up, and iterative approach to system design in more detail. From the table it is clear that the iterative approach is the most flexible of the three, and that it is the most suitable when an optimum balance between technical performance, cost, and development schedule is pursued.

 

Error budgeting

Once a system's desired accuracy has been specified (typically based on the user requirement and the system's mission), the next step in the iterative error budgeting process is to determine the attainable accuracy, for comparison with the requirement. Then the iteration can start towards the point where an optimum balance between the goal and the attainable value is reached - within technical, financial, and schedule constraints. The major steps for determining a system's attainable accuracy by means of analysis during the design phase, are:

1. Define the system's major functional blocks.

2. Identify potential contributors to the system error in each of the system's functional blocks.

3. Perform a qualitative analysis of the contributors' potential effects on the system accuracy.

4. Identify the dominant contributors to the error, from the qualitative analysis.

5. Quantify the dominant error sources, and determine their combined contribution to the overall system error. This implies a translation from the values of errors in the functional blocks to values representing their effects on the overall error. For this translation process, three analysis routes can be utilised: static analysis, dynamic analysis, and system simulation.

These three techniques are defined in more detail in Table 2.

 

Error budgeting example: Satellite inertial measurement system

Attitude control of earth observation satellites (e.g. for earth resource management) is an example of a control system which requires a high accuracy. Such a satellite's maximum allowable pointing error (or desired pointing accuracy) is therefore one of its major design parameters.

For a satellite making use of an inertial measurement system (IMS) as part of its attitude control loops, part of the satellite's desired pointing accuracy translates (rolls down) to a desired IMS measurement accuracy; which in turn translates to accuracy requirements at consecutive lower levels in the IMS hierarchy. Conversely, the attainable measurement accuracies of the different IMS subsystems combine to form the attainable IMS measurement accuracy; which in turn rolls up to form part of the attainable satellite pointing accuracy. Finding the optimum balance between the desired accuracy and the attainable accuracy clearly calls for iterative error budgeting. In this section, an overview is provided of the configuration and dominant errors of a typical satelliteTMS, after which the principles of static and simulation-based error budgeting are illustrated for such an IMS.

 

Configuration of a typical satellite IMS

A typical satellite IMS is configured to measure satellite angular rates and incremental angles around three orthogonal measurement axes (X, Y, and Z).[3] For this purpose the IMS consists of a sensor subsystem and an electronics subsystem. The major components of a typical sensor subsystem are three dynamically tuned gyroscopes, a sensor block in which the gyros are mounted orthogonally, and shock mounts by means of which the sensor block and gyros are isolated from shocks and vibrations occurring on the satellite. For each gyro, the electronics subsystem contains components to perform electrical power conversion (power supply), to drive and control the gyro, to measure and process data, and to communicate with other satellite subsystems. Figure 1 shows a high level block diagram of one of the three measurement axes (i.e. around X, or Y, or Z) of a typical satellite IMS. Each gyro measurement axes uses a control loop to keep the gyro rotor's motion the same as the motion imposed around (and thus sensed by) that specific gyro axis. (With reference to Figure 1, the purpose of the gyro control loop is to maintain a zero rotor angle relative to the gyro casing.) The rotor motion, in reaction to the satellite's motion, is caused by torquer coils in the gyro, and the current in these coils is proportional to the angular rate experienced by the specific gyro measurement axis. The rotor motion control current is measured as a voltage across a sense resistor, and is used to derive the incremental angle about the specific measurement axis, by counting pulses from a voltage-to-frequency (V/F) converter; and to derive the angular rate by means of an analogue-to-digital (A/D) converter and digital processor.

 

Importance of iterative system design

To illustrate the importance of the iterative approach to system design, consider the following example: For an IMS in a land-based vehicle, temperature control within narrow bounds, is often used to reduce the IMS temperature sensitive errors (thus leaving residual errors only). In satellites - where the available power is limited - it is more attractive, however, to use temperature compensation instead, whereby angles and angular rates measured by the IMS are adjusted according to the temperature of the IMS during the specific measurement. Since gyros are normally the most temperature-sensitive components in an IMS, restrictions on system complexity typically dictate that the temperature measurements for use in the temperature compensation, be restricted to the gyros only. It is clear that for an IMS with limited temperature compensation, decreased system accuracy is accepted in lieu of lower power consumption and lower system complexity.

The only way to determine how far this trade-off can be taken, is to calculate the effects of the temperature sensitive errors on the overall system accuracy; and this is done as part of the error budgeting process. It is obvious that if the desired measurement accuracy were allocated in a purely top-down fashion, the system complexity, power consumption, and cost could escalate beyond reasonable limits. The error budget can therefore enable the IMS designer to compare temperature control, with different levels of temperature compensation and this information can then be fed back to the higher level system engineer for decision making.

 

Static error budget

Static error budgeting is useful to determine the effects of different error sources on the overall measurement accuracy of a satellite IMS. Typically, such an error budget includes fixed, random, temperature-dependent, and angular acceleration dependent error sources. (Although the effects of fixed errors can largely be reduced by prelaunch calibration, some residual error values remain due to factors such as non-ideal calibration, fixed errors introduced during satellite launching, ageing of components, and mechanical creeping.) Not only can the contribution to the total IMS measurement error be determined for each error source, the effects of combinations of different error sources can also be predicted. The static error budget can furthermore be used to evaluate specific circumstances, such as the effect on the measurement error, when neither temperature control nor temperature compensation is used, or when only gyro temperature compensation is used.

Two important steps in static error budgeting are identification of the dominant error sources and translation of its individual contributions to the overall system error. These steps are illustrated in Table 3, which contains:

• Typical dominant static error sources for an IMS measuring angular increments of a satellite. (These error sources are mainly due to one or more of the following: residual fixed errors after system calibration, launch-induced fixed errors, angular acceleration dependent effects, and temperature-dependent effects.)

• Typical values of the error sources. (These figures are typical for a high-accuracy IMS, but since they were rounded-off for use here, they do not represent any specific commercially available IMS.)

• Descriptions of the broad principles according to which the errors are translated (rolled up) to total system error.

• Contribution of each error source to the error in the measured satellite angular increment (i.e. the values of the error sources, translated to system level error). Two sets of results are shown: one set for the case where no temperature compensation or control is used on the IMS (i.e. temperature-sensitive errors have a large influence) and one set for the case where gyro temperature compensation is used (i.e. the influence of all gyro parameters which are temperature sensitive is reduced to only a residual effect - typically as if a gyro temperature variation of only 1^oC occurs.)

The numbers obtained in the translation from error source values to system level measurement error were often rounded-off and are therefore not extremely accurate (some answers are in milliradians whilst others are in microradians). The intention was however not to reproduce the most accurate IMS error budget here, but rather to illustrate the principles and the trends that can be observed from such an exercise.

The individual error values, as used in Table 3, are statistical figures (typically σ values), with the total error being calculated as the root-sum-square (RSS) of the individual values. (RSS is used because the individual errors are considered to be independent.[1] The results in Table 3 indicate that, for the IMS considered here, gyro temperature compensation causes a twenty-fold increase in angular measurement accuracy. The same procedure can be used to verify what the effect on system accuracy would be if temperature compensation is also used, for example, for the most temperature sensitive electronic components. The error budgeting process can thus be used to investigate trade-offs between system complexity and system accuracy and, when used iteratively, it can largely contribute to optimization of system performance, cost, and development schedule.

 

Simulation model

Instead of the procedure used to compile Table 3, a complete simulation model can be used for determining the contribution of the IMS dynamics and the dominant static error sources to the total IMS measurement error. The simulation model creates the ability to evaluate the effects of individual and different combinations of error sources, whilst taking the IMS nominal and dynamic characteristics into account. The effects of different levels of temperature compensation can easily be verified by means of such a simulation model. Normal system modelling principles are used in this process, i.e.:

1. Identify the system's different functional blocks and compile a model which includes the nominal characteristics of each functional block.

2. Identify and quantify any error sources present in each of the functional blocks, and incorporate them into the model. Different approaches are possible for including the different error sources in the model:

(a) In order to determine the worst-case overall measurement error, the positive or the negative maximum values of all error sources can be used in the simulation model. However, this approach causes unrealistic simulation results, because in reality some errors cancel the effects of others.

(b) Random signs can be allocated to all error sources in the simulation model. This renders more practical simulation results than approach (a) above, since errors negating each other are thus introduced.

(c) In order to simplify the simulation program, errors occurring in the same functional block of the system can be added in root-sum-square fashion - provided they are independent.[1] Random signs are then allocated to the RSS errors of the different functional blocks. The effect of errors negating each other is still introduced, but the simulation model becomes less complex. In this case, the repeatability of the simulation results is better than for approach (b) above, since less randomness is incorporated into the model.

(Since the signs of errors are allocated randomly in both approaches (b) and (c), the simulation ought to be repeated a number of times - in Monte-Carlo fashion - in order to derive average and standard deviation values for the overall measurement error.)

3. Independently verify the model by means of careful reviews with the system's design experts.

4. Implement the simulation model by means of a digital computer program and verify the implementation by means of standard inputs (e.g. step functions) to, and expected outputs from, the simulation (first for individual functional blocks, then for different combinations of functional blocks, and then for the complete system).

Table 4 describes the major characteristics (nominal and error sources) used for compiling a simulation model of a typical satellite IMS. With the simulation model, there is no need for a separate translation of errors to the overall IMS measurement error - the simulation takes care of that automatically. Because the individual error values are (as for the static error budget) statistical figures, the predicted overall error value is also a statistical value.

By making use of an IMS simulation model - including similar error sources to those used in Table 3, including gyro temperature compensation, and with the dominant dynamic characteristics (transfer functions) of the shock mounts, the gyro electronics, and the gyro included - the predicted measurement error was less than 1.1 milliradian. The question can arise: Why should the more complex simulation model be used if the results are similar to those obtained with the simpler static error budget? The answer is:

1. With the static error budget, in Table 3, only the maximum IMS measurement error (statistical value) can be calculated, whilst with the simulation model the designer can evaluate the measurement error as a function of time, or in the presence of noise signals, or for different frequencies of satellite motion, etc.

2. The simulation model can be used in different roles - such as for static error budgeting when the system's dynamic characteristics are removed from the model, or for dynamic error budgeting when the system's static errors are all made zero in the model, or for a combined static and dynamic analysis (such as in a Monte-Carlo type of statistical analysis).

3. With the simulation model, the translation of error sources to its system level effects are performed automatically when the simulation program is executed, whilst it requires complex calculations in the static error budget.

Although the simulation model is the most versatile of the three techniques described in Table 2, it is also the most complicated, because of the requirement for an accurate system model.

 

Conclusions

System accuracy is an important issue in control system design. In this paper, the iterative (error budgeting) approach to system design was defined alongside the pure top-down and pure bottom-up approaches. Whereas the pure top-down approach can lead to unrealistic design goals being imposed on lower level subsystems and the pure bottom-up approach can lead to the predicted overall system performance being poorer than the true attainable performance, the iterative approach renders an opportunity to make trade-offs between technical performance, cost, and schedule on the different system levels.

Three different analysis routes, static, dynamic, and simulation-based analysis, can be used for translating error sources in a system to their effect on overall system accuracy. The static and simulation-based routes were addressed in terms of the principles involved in error budgeting for the inertial measurement system of a high-accuracy satellite. Since a major portion of a satellite's required pointing accuracy is typically allocated to the IMS, measurement accuracy of the IMS is of paramount importance. Both analysis routes used here are very valuable for evaluating accuracy requirements, for establishing attainable accuracies, and for adapting requirements and/or designs to the extent that neither an over-design nor neglect of the system's accuracy requirements results. Static error budgeting provides less quantitative answers than simulation-based error budgeting, but the former is less complex to perform. Which of the two approaches should be followed is therefore dictated by a trade-off between error budget complexity on the one hand and accuracy of results on the other.

Although the iterative error budgeting approach was illustrated in this paper mainly by making use of a specific configuration of an inertial measurement system for satellite applications, the procedures and principles are widely applicable. Not only can accuracy requirements of other satellite subsystems be handled in similar fashion, but a wide variety of control systems can be designed as such.

 

References

[1] Doebelin EO. Measurement Systems - Application and Design, 3rd edn. McGraw-Hill, Tokyo, 1983.

[2] Marley R &Dungate DG. Managing satellite pointing accuracy - a systems engineering approach. Journal of the British Interplanetary Society, 45:63-68.         [ Links ]

[3] Britting KR. Inertial Navigation Systems Analysis. Wiley-Interscience, New York, 1971.

[4] Cannon WW & Sweeney JJ. Error Budget Analysis of Automatic Cannons on Armored Combat Vehicles. MS thesis, Naval Postgraduate School, Monterey, California, 1977.         [ Links ]

 

 

First received November 1994
Final version May 1995