Jun 6, 2005 - force-feedback systems in vehicular control. (Van Paassen, 1994) ... (1994):. (A system is) a part of reality which we think to be a separated unit within this reality. ...... Models for Steering Control of Motor Vehicles. Proceedings.
M. Mulder, M. M. Van Paassen, J. M. Flach, and R. J. Jagacinski, “Fundamentals of Manual Control Theory,” in The Occupational Ergonomics Handbook (Second Edition) – Fundamentals and Assessment Tools for Occupational Ergonomics, W. S. Marras and W. Karwowski, Eds. CRC Press, Taylor & Francis, London, 2005, pp. 12.1–12.26, ISBN 0849319374.
Fundamentals of manual control Max Mulder∗, Ren´e (M. M.) van Paassen†, John M. Flach‡ and Richard J. Jagacinski§ June 6, 2005
1 Introduction For their work, for transportation or simply for entertainment, human beings are often involved in the manual control of devices. Vehicles, such as cars, bicycles, ships and airplanes are some examples, but also video games and many work situations involve manual control. Normally, after learning the task, the human operator in such a control situation behaves like a well-designed controller. In fact, in their paper on “Quasi-linear pilot models”, McRuer and Jex make the remark that data of measured pilot behavior so well matches the Primary Rule of Thumb for Frequency Domain Synthesis, a design rule for automatic controllers. It is not surprising, therefore, that many of the models used in modeling manual control situations are based on various control system design techniques. Two of the most commonly applied are the frequency domain design methods, which serve as the basis for the crossover model and variants thereof, the precision model and the simplified precision model (McRuer & Jex, 1967), and optimal control theory, which lies at the basis of the Optimal Control Model (Kleinman, Baron, & Levison, 1970b). The theories on human control behavior have matured by now, and its applications, particularly human vehicle control, have been extensively studied. However, for many application areas these theories and their application are still very relevant today. Some examples are: • Investigation of the roles of multi-modal (visual, vestibular, tactile) feedback on human manual control behavior in virtual environments such as flight and driving simulators. • The design of haptic manipulators in applications like tele-operation or the development of force-feedback systems in vehicular control. (Van Paassen, 1994) • Investigation of vehicular control (aircraft, automobile) in general, including handling qualities research. ∗
Faculty of Aerospace Engineering, Delft University of Technology Faculty of Aerospace Engineering, Delft University of Technology ‡ Psychology Department, Wright State University § Psychology Department, Ohio State University
†
1
• Investigation and evaluation of augmented systems, i.e., the study of the interim between fully manual and automated control. • Studying human perception & action cycles in active psychophysics, for example in the determination and identification of a human’s use of visual cues in multi-cue displays. (Flach, 1991; Mulder, 1999). In this chapter we will provide a short introduction into some fundamentals of control theory. This introduction is very limited, however, and for further study the reader is referred to the many good textbooks that are available. A textbook geared towards human control is “Control Theory for Humans”, by Jagacinski and Flach. Others that are recommended for their “human engineering” perspective are “Man-Machine Systems” by Sheridan and Ferrell, and “Engineering Psychology and Human Performance” by Wickens. More engineering oriented textbooks are “Control Systems Engineering” by Nise, “Control System Design”, by Goodwin, Graebe and Salgado, “Modern Control Systems” by Dorf and Bishop and “Feedback Control of Dynamics Systems”, by Franklin, Powell and Emami-Naeini. Note that this selection is not exhaustive, and that many more excellent textbooks are available. Furthermore, a historic overview of the modeling of human control behavior is given, followed by a more detailed description of two of the most common and widely used approaches to describe human behavior in control-theoretical terms, the crossover model (COM), based on classical control theory, and the optimal control model (OCM), based on modern control theory.
2 Fundamentals of systems and control theory Systems and control theory is a branch of mathematics that studies dynamic processes, i.e., things that evolve in time. Examples of dynamic processes that are the subject of systems and control theory are artifacts like airplanes, power plants, and cars, biological systems like the heart, chemical processes, large scale processes like economics, etc. These can be modeled, described and simulated with dynamical systems theory, and their automated control systems are designed with control theory. The building blocks of systems and control theory are differential equations, linear matrix algebra, complex number theory, and probability theory. In this section we will focus on studying linear, time-invariant (LTI), continuous-time, single-input single-output (SISO) systems. We will briefly study the response of these systems to deterministic signals, in later sections we will also briefly address stochastic signals. Most of the physical processes in the real-world are non-linear, time-varying, multi-input multioutput (MIMO), stochastic systems. A deep understanding of SISO systems, however, is an important first step for understanding more complex MIMO systems. Many of the intuitions gained from working with simple control systems will generalize to more complex systems – even though it might not ever be possible to model the more complex systems with the same confidence that we can model the simpler systems.
2
environment
u1
inputs
u2
outputs system
u3
y1 y2
Figure 1: Representation of a system, with inputs and outputs, and a system boundary that separates the system from the environment.
2.1 Linear, time-invariant systems A useful definition of a system in systems and control theory is given by Olsder and van der Woude (1994): (A system is) a part of reality which we think to be a separated unit within this reality. The reality outside the system is called the surroundings. The interaction between system and surroundings is realized via quantities, quite often functions of time, which are called input and output. This definition stresses the fact that the choice of what is considered to be the system and what belongs to the environment (surroundings) is often a subjective one. However, once the system boundaries are defined, the interaction between the system and its environment, by means of input and output signals, also becomes clear, see Figure 1. A system is linear with respect to the inputs and outputs, if the output to a sum of two or more input signals, is the sum of the outputs each of these inputs would give individually. Most real-world systems are not linear, but in many control situations, where there are only small excursions around an “operating point”, they can be well approximated with linear models. If also the properties of the system do not change over time, thus if we neglect processes such as wear and tear, the change of mass due to use of fuel, etcetera, one obtains a linear, time-invariant (LTI) system. The behavior of a LTI system can be described with an ordinary differential equation (ODE), with constant coefficients. For instance, consider the following ODE, describing the relationship between the input signal of a system, u(t) and the output signal of that system, y(t), see Figure 2: a0 y(t) + a1
dy(t) d2 y(t) du(t) + a2 = b0 u(t) + b1 . 2 dt dt dt
(1)
As in this example, the ODE generally consists of time-derivatives of the input and output signals, characterizing the dynamic response of that system to the input signals.
2.2 Transfer function The calculation of solutions for ODE’s, and the combination of systems described in ODE form, is usually quite laborious. Therefore, in classical control theory, extensive use is made of the 3
u(t)
h(t)
y(t)
Y (jω)
U(jω) H(jω)
or
Figure 2: LTI systems in the time domain (left) and the frequency domain (right). Laplace Transform, to transform the ODE’s to the Laplace domain, where manipulation, such as the combination of systems with other systems and signals, and the solution of the ODE’s, is simpler, since all the equations become algebraic. Textbooks on control theory contain tables of Laplace transforms that show the transformation of signals to and from the Laplace domain, and Laplace transform theorems that can be used to transform system descriptions to and from the Laplace domain. One can use the Laplace real differentiation theorem, which states that; df (t) = sF (s) − f (0) (2) L dt to transform a differential equation to the Laplace domain. Assuming that the system is at rest at t=0, so f (0) = 0 and df (0)/dt = 0, one obtains: a0 Y (s) + a1 sY (s) + a2 s2 Y (s) = b0 U(s) + b1 sU(s),
(3)
This yields a constant relation between the input U(s) and the output Y (S): H(s) =
Y (s) b0 + b1 s = , U(s) a0 + a1 s + a2 s2
(4)
which is known as the system’s transfer function. For an LTI system with one input and one output, the transfer function completely defines the system. The transfer function is a convenient format for manipulation of system models. For example, the transfer function for two systems placed in series, i.e., the output of the first system is the input of the second, is simply the product of the two transfer functions, and, likewise, the transfer function for the total of two systems placed in parallel is the sum of the transfer functions. The stability of an LTI system can be determined from its transfer function, by determining the poles of the transfer function, which are the (complex) numbers for which the denominator equals zero (these solutions are referred to as the “roots” of the denominator). Any poles with a positive real part are associated with a response of the system to an input signal. This response may be oscillating for a pair of complex poles with positive real part, or it may be a-periodic when the corresponding poles are on the positive real axis. A system with such poles is unstable. Poles with negative real part produce responses with exponentially decreasing amplitude. Pole pairs on the imaginary axis produce an undamped oscillatory response of the system, a pole in the origin produces an integration. An experienced control engineer can interpret the location of the poles and zeros of a transfer function in terms of a system’s dynamical behavior.
4
1
amplitude response
0.5 0
phase response
−0.5 −1 0
10
20
30
40
50
60
70
80
90
100
Figure 3: Input sinusoid (blue) and system output (green) as a function of time.
2.3 Frequency response Another method of characterizing a LTI system is by means of its frequency response. If a sine wave input signal is applied to a LTI system, and the system is stable, then after the initial response to the start of the sine wave has faded, the output of the system will be a sine wave with the same frequency as the input signal, but with a different amplitude, and a different phase from the input signal. Figure 3 shows the sine input and the output signal for a system. The ratio of the output sine to the input sine signal, as a function of frequency, is called the gain. The difference in phase between the two sine signals, is called the phase shift. Together they form the systems’ frequency response. For physically implemented systems, the frequency response can be determined with a Frequency Analyzer, a device that can generate sinusoid test signals and measure the response of the system. The frequency response can be determined from the Laplace transfer function, by substituting jω for the Laplace variable s; proof for this is given in most control engineering textbooks. Here j is the imaginary number, and ω is the frequency of the input sine signal in radians per second. The frequency response H(jω) is a complex function of ω. The magnitude of that function, |H(jω)|, is the gain of the frequency response and the angle of the complex number H(jω) is the phase shift. In most cases the phase shift is negative, and called a lag. For LTI systems, the frequency response completely defines the system. Consider an LTI system with the following transfer function: H(s) =
K . (1 + τ s)
(5)
This is known as a ‘first order system’ with time constant τ . Now consider two values of τ (0.1 and 10 s) and look at the system response y(t) for sinusoidal input signals that have various frequencies ω. The result is illustrated in Figure 4. The output of both LTI systems are sinusoids as well, but the amplitudes and phases of these sinusoids are different from the input sinusoids. Whereas the system with τ equal to 0.1 s hardly changes the amplitude and phase and the input and output signals are almost the same, the system with τ =10 s, does not (completely) “pass” the sinusoidal input signals that have a higher frequency ω. It “filters out” these higher-frequency signals, and is therefore known as a low-pass filter system. The value of τ determines which frequencies are “passed” and which frequencies are not. 5
0.10 rad/s
output (τ =0.1 s)
input sinusoid
1
0
0.32 rad/s
−1
1.00 rad/s
50
100
1
0
0
−1
0
50
100
−1
1
1
1
0
0
0
−1
0
50
100
−1
0
50
100
−1
1
1
1
0
0
0
−1
ω0 =3.16 rad/s
0
output (τ =10.0 s)
1
0
50
100
−1
0
50
100
−1
1
1
1
0
0
0
−1
0
50
time, s
100
−1
0
50
time, s
100
−1
0
50
100
0
50
100
0
50
100
50
100
0
time, s
Figure 4: LTI response to sinusoidal input signal with four frequencies ω0 (left column), for two values of the variable τ : τ =0.1 s (middle column) and τ =10 s (right column). Such frequency responses are often shown as Bode plots. The magnitude and phase for different input signal frequencies are plotted separately. The magnitude is plotted on a logarithmic scale, against the frequency, which is also plotted on a logarithmic scale. At the whim of the maker of the plot, the frequency may be given in radians/s, in Hz, or sometimes in octaves. For the magnitude, often a decibel (dB) scale is used. The relation between a magnification M and its equivalent in dB m is: m = 20 · 10 log M
[dB]
(6)
Figure 5 shows the Bode plot for this system. One can see that for low frequencies the gain of the system is 1 (0dB). For high frequencies the gain decreases with 20 dB per decade. The asymptotes for the low and high frequency behavior cross at the corner frequency 1/τ , in this case, with τ = 10 [s], at 0.1 [rad/s].
6
amplitude response
1
10
|H(jω)|
0
10
−1
10
corner frequency −2
10
−2
−1
10
0
10
10
1
10
phase response
0
H(jω)
−20
6
−40 −60
corner frequency
−80 −2
10
−1
10
0
frequency, rad/s
10
1
10
Figure 5: Bode plot for LTI system H(jω) = 1/(1 + 10jω). The circles show the system response for the four frequencies shown in Figure 4.
2.4 Control 2.4.1 Feed-Forward and Feedback In a control system, a controlling element (controller) provides input to a system, often called a plant, normally with the aim of producing outputs of the plant that are equal to given reference values. Unknown disturbances may be acting on the plant, as for example the turbulence acting on an aircraft. The controller may be an automatic device, such as an autopilot, or it may be a human. In feed-forward control, or open-loop control, the controller measures the disturbances on the plant, and based on the knowledge about the plant’s dynamics, creates inputs that produce plant outputs as close to the reference values as possible, see Figure 6(a). In feedback, or closed-loop control, the controller measures the output of the plant, and compares that to the reference values. Control input to the plant is calculated on the basis of this comparison, see Figure 6(b). The advantage of closed-loop control over open-loop control is that in most cases “modeling errors”, i.e., mismatches in the plant model used to tune the controller and the real plant, have little effect on the control performance. Closed-loop control is thus said to be more robust, meaning that it is insensitive to variations in the controlled system and influences from the environment. Robustness is an important property of control systems, and tools exist for the design and tuning of robust controllers. Feedback is a fundamental property of many control systems. The closed loop system can itself be 7
disturbance
disturbance reference
Sensor
Controlled System
Controller reference
Controlled System
Controller
output
output Sensor
(a) Open loop control system
(b) Closed loop control system
Figure 6: Open and closed loop control systems system
controller r(t)
e(t)
+
K(s)
c(t)
G(s)
z(t)
−
Figure 7: The feedback control of a SISO LTI system. considered as a new system, with an input (in most cases the desired output) and an output. If they are part of a larger whole, such systems are often called a servo. For example, the motor, “plant” and feedback system for moving an aircraft’s elevator forms a servo system, and the servo system itself is part of the automatic pilot. The choice for a type of controller applied in a servo system is normally made on the basis of knowledge on the plant dynamics, and on the desired properties of the closed loop system. Common controller types are proportional (P) controllers, that generate a control signal proportional to the error, proportional and differentiating controllers (PD), generating a control signal on the basis of a sum of error and the derivative of the error, proportional and integrating controllers (PI), generating a control signal on the basis of a sum of error and the integrated error, and the combination of the above, PID controllers. After a choice for a particular type of controller has been made, its parameters need to be tuned. Several tuning methods are in use in control system design, as an example, and since it forms the basis for the commonly used Cross-Over model, tuning in the frequency domain with Bode diagrams will be treated here. Consider a linear time-invariant (LTI) system, G(s), Figure 7. The main design requirement is to design the controller K(s) in such a way that the system output z(t) follows the reference signal r(t) as closely as possible. Solving this problem in the Laplace domain is not too difficult, since all combinations of signals and systems can be obtained by algebraic manipulation. From Figure 7 one can derive the following basic equations: E(s) = R(s) − Z(s), (7) 8
Z(s) = G(s)C(s) = G(s)K(s)E(s),
(8)
E(s) = R(s) − G(s)K(s)E(s).
(9)
(1 + G(s)K(s))E(s) = R(s),
(10)
E(s) =
1 R(s), 1 + G(s)K(s)
(11)
Z(s) =
G(s)K(s) R(s). 1 + G(s)K(s)
(12)
Solving for E(s) yields: so:
and:
Thus:
Z(s) K(s)G(s) = . R(s) 1 + K(s)G(s)
(13)
One can consider this solution in the frequency domain, by substituting jω for s, which yields: K(jω)G(jω) Z(jω) = . R(jω) 1 + K(jω)G(jω)
(14)
The design requirement is that the system output Z(jω) equals the reference signal R(jω), so Z(jω) ≈ 1. The solution to this problem would be to achieve a high “open loop gain” K(jω)G(jω). R(jω) When K(jω)G(jω) is very large one can see that Z(jω) ≈ R(jω), Eq. 14 and that E(jω) ≈ 0. Eq. 11. However, one should bear in mind that K(jω)G(jω) is still a function of ω, with a complex-valued outcome. In general, it is not possible, and for many practical reasons not desirable, to obtain a large value for K(jω)G(jω) for all ω. Essentially K(jω)G(jω) determines the “speed” of reaction to an error signal. When time delays are small then a faster response will yield a lower tracking error. When time delays are large, however, it is possible to respond too quickly and cause the system to become unstable. Hence, as will be discussed in more detail below, generally there is a trade-off between response speed and accuracy. Just as transfer functions can be considered in terms of their frequency response, i.e. what (sine signal) frequencies they pass and what frequencies they block, signals can be considered in terms of their frequency content. A reference signal such as a block or sawtooth signal can be seen as constructed from an infinitely large sum of sine signals, (see Figure 8), a much smoother signal has less high-frequency components. In most cases, a smooth response is acceptable, and even desirable from a mechanics standpoint, and thus responses to the high-frequency components in an input signal are not needed. Summarizing, at low frequencies, the gain of K(jω)G(jω), called the open-loop gain, should be high, while at high frequencies it may be low. Note that K(jω)G(jω) is a complex-valued function, and that it can have a value equal to or close to -1. To asses the behavior of the closed loop, the frequency response of the open loop, K(jω)G(jω), is studied near this point where the magnitude 9
3
3
3
0
0
0
−3
−3 0
5
(a) One sine function
10
−3 0
5
(b) Three sine functions
10
0
5
10
(c) Ten sine functions
Figure 8: Triangular function approximated by sums of of sine functions, base frequency 0.1 [Hz], following sine components at 0.3 [Hz], 0.5 [Hz], 0.7 [Hz], etc. of the open loop response, |K(jω)G(jω)|, is one, or 0 [dB]. This point is the cross-over point, and the corresponding frequency is called the cross-over frequency. The phase shift of the open loop at the cross-over frequency determines what the gain of the closed loop system will be at the cross-over frequency. A phase shift near 180 degrees will cause the magnitude of the denominator of the closed-loop system in Eq. 14 to become very small (while the numerator’s magnitude is 1), and the closed-loop frequency response will have what is called an oscillatory peak. The response of the closed-loop system to, for example, a step change in the reference signal will show an oscillation with approximately the cross-over frequency. Investigation of the phase at cross-over is important for the assessment of stability of the closed loop. The difference between the phase shift at cross-over and a phase of -180 degrees (i.e., the point -1) is called the phase margin. For stability of the closed-loop system, the phase margin must be positive, i.e. the phase shift of the system is less than -180 degrees. Usually, a phase margin larger than 40 degrees is chosen. For any feedback system where the open-loop transfer function is not unstable, a positive phase margin is a guarantee for close-loop stability. Stability for a system that is open-loop unstable, i.e. has open-loop poles in the right-half complex plane, can be studied by means of the Nyquist stability theorem, which also constitutes more formal proof of stability by means of the frequency response. Proof of and explanation on the Nyquist stability theorem can be found in the engineering textbooks already mentioned. An exemplary open-loop system, which, when used in a closed loop feedback, is the “single integrator”. For a single integrator, K(jω)G(jω) = 1/(jω). The single integrator has an infinitely high gain at ω = 0, thus the output of the closed loop system will perfectly follow the input for low frequencies. The phase margin is always 90 degrees, whatever gain is chosen for the controller. A simple proportional control results in excellent closed-loop properties. Of course, most systems are not single integrator systems. However, the same principles can be applied, that is that a high gain at low frequencies is needed, and a sufficiently wide frequency range, with a frequency response locally resembling a single integrator and having an acceptable phase margin, can then be chosen for the cross-over frequency. This is summarized in the primary rule of thumb for frequency domain design, which says that near the cross-over frequency, the 10
|H|20 [dB]
|H|20 [dB]
10
10
H(s) =
0
1 + τ2 s 1 + τ1 s
−10
−10
−20
−20
1/τ1
1/τ2
H(s) =
0
ω[rad/s]
H 90 [deg]
H 90 [deg]
45
45
0
0
−45
−45
6
1 + τ1 s 1 + τ2 s
1/τ1
1/τ2
ω[rad/s]
1/τ1
1/τ2
ω[rad/s]
6
−90
−90
1/τ1
1/τ2
ω[rad/s]
Figure 9: Bode diagram of a lag-lead compensating network (left) and of a lead-lag compensating network (right). following must hold: Y OL (jω) = K(jω)G(jω) =
ωc , jω
(15)
For the controlling element, when design in the frequency domain is used, a lead/lag or a lag/lead network is usually chosen. Since a logarithmic scale is used in a bode diagram, the frequency response of the controller K(jω) can simply be added to the frequency response of the controlled system G(jω). Diagrams of the most common “compensating networks” are given in Figure 9.
3 Motivation and overview of human manual control models 3.1 Motivation The motivation for obtaining mathematical models of human control behavior has evolved from the need to explain the behavior of human-vehicle control systems to understanding human behavior 11
in general. The analytical descriptions desired are in control-theoretical terms. The main purposes of the engineering models are: 1. to summarize behavioral data, 2. to provide a basis for rationalization and understanding of human control behavior, 3. to be used in conjunction with vehicle dynamics in forming predictions or in explaining the behavior of combined closed loop human-machine systems. Modeling humans using systems theory has proved to be a tremendous challenge. Humans are complex control and information processing-systems: they are time-varying, adaptive, non-linear, and their behavior is essentially stochastic in nature (McRuer & Jex, 1967). Such systems are difficult to be characterized in mathematical terms because most of the mathematical tools we have apply strictly to stationary, linear and non-adaptive systems.
3.2 Quasi-linear function theory Research in the two decades after WW-II resulted in the successful application of quasi-linear describing function theory to the problem of modeling human control behavior in the single-axis compensatory tracking task (Krendel & McRuer, 1960). In the compensatory tracking task the human operator is controlling a dynamic system in such a way that the output of that system approximates the value of a reference signal. This is essentially the same task as introduced above, Figure 7. In the early days the model structure and model parameters obtained experimentally using the quasi-linear describing function models had predictive significance only in applications that were similar to the experimental conditions. No attempts were conducted to relate the model structure or model parameters to the context in which the task was conducted, the so-called task variables. This changed with the publication of McRuer, Graham, Krendel, and Reisener Jr. (1965). This landmark report provided an overview of earlier experiments, putting them in a general framework, and reported the results of new experiments that were especially conducted to show the relation between human control behavior and the task variables. The report provided convincing empirical evidence that humans systematically adapt their control behavior to the task variables.
3.3 The crossover model This systematic adaptation of human control behavior was generalized in the crossover model (COM). The crossover model allows human control behavior to be predicted, based on knowledge about the main task variables as system dynamics and reference signal bandwidth. Attempts to parameterize the model of human control behavior resulted in the development of the so-called structural-isomorphic models such as the precision model and the simplified precision model (McRuer & Jex, 1967). The parameters of these models can be determined from the verbal adjustment rules, a set of rules that relate the parameters to the task variables.
12
The descriptive models that have become available with the crossover model are still widely applied in human-machine system studies. They are simple to use, require only a few parameters, and have proved to provide a good insight for many situations.
3.4 The optimal control model The advance of “modern” control theory in the mid-1960s resulted in the concepts of optimal filtering (Linear-Quadratic Gaussian, LQG) and optimal control (Linear-Quadratic Regulator, LQR) (Kwakernaak & SiVan, 1972). One of the first attempts to describe human control behavior in the paradigm of optimal control and estimation theory yielded the optimal control model (OCM) (Kleinman et al., 1970b; Kleinman, Baron, & Levison, 1970a). Here, the input-output relation of the human operator is compared with an optimal controller, instead of only a stabilizing controller. The algorithmic model consists of an optimal observer, generating an optimal state estimate of the system to be controlled, and a deterministic regulator which transfers the estimated state into the optimal output. In contrast with the structural models which describe what the human is doing, the optimal control model is a normative model, i.e. it describes what the human should be doing given the inherent human limitations and the task variables at hand. The OCM was believed to be suitable for a more general application area, having a wider objective than the describing function models (Kok & Van Wijk, 1978; Wewerinke, 1989). In spite of this, however, the algorithmic model has not become as widely used as the structural models, which can be attributed primarily to the fact that the algorithmic model is over-parameterized (Van Wijk & Kok, 1977), hampering the model validation significantly.
3.5 More recent models The crossover model can be considered the result of applying “classical” control theory, i.e., theories that emerged from the 1950s and 1960s, control theory to model human behavior, and the optimal control model the result of applying modern control theory, i.e. theories that emerged in the mid-to-late 1960s, to model human behavior. It is no surprise that since then, with other control-theoretical approaches emerging such as fuzzy logic, neural networks, adaptive neurofuzzy models, andsoforth, that many of these new approaches have also been applied to describe human behavior. In this chapter we will concentrate on the two classic approaches, however.
4 The crossover model 4.1 The problem with modeling humans The human operator is an adaptive controller, and the observed control behavior is likely to change, either consciously or unconsciously, to the environment. The ability to change and adapt is un13
doubtedly of the most prominent and valuable human capabilities. Yet, when one attempts to describe the behavior and grasp it into a mathematical model, one encounters a non-trivial problem, as most of the available techniques only work out well in the time-invariant case. The earliest attempts in describing human behavior with control-theoretical models failed to pay much attention to the adaptivity of human behavior. Research showed that when experiments are not done under (almost) exactly the same circumstances, the human will adapt and the observed control behavior will be different. The lack of a systematic approach to this problem resulted in scattered data and many different and unexpected findings, and theory progressed only slowly. This changed when in the late 1950s and early 1960s McRuer and his co-workers started working on this problem. Learning from past experience, they first determined and classified a list of variables that could possibly have an effect on human behavior. These include environmental variables (e.g., conducting the task in real flight or in a fixed-base simulator), procedural variables (e.g., subject instruction, practice) and operator-centered variables (e.g., subject motivation, workload). Most important to understand the adaptation of the human operator are the task variables, however, which include: • the dynamics of the system to be controlled, • the properties (bandwidth) of the forcing function, i.e., the signal to be followed (in a following task) or the disturbance signal acting on the system (in a disturbance task), • the type of display (e.g., a compensatory display or a pursuit display), • the type of manipulator. McRuer et al. conducted a massive number of tracking task experiments. In their approach they tried to very closely control all the variables that could have an effect on human behavior, and systematically varied two of the task variables, i.e., the dynamics of the system and the bandwidth of the forcing function signal. As a result of this approach, human variability decreased significantly and for the first time insight was gained into how and why humans adapt to changing circumstances. In the following paragraphs the main results of this research will be elaborated on.
4.2 Quasi-linear pilot models The earliest work in this field already showed that a human operator establishes, during a learning and skill-development phase, a particular control system structure (Krendel & McRuer, 1960). The feedback connections in this system are similar to those which would be selected for the development of an automatic controller. The loop closures selected will have the following properties (McRuer & Jex, 1967): Skill acquisition
(i) To the extent possible, the feedback loops selected and adjustments made will be such as to allow wide latitude and variation in pilot characteristics. (ii) The loop and equalization structure selected will exhibit the highest pilot rating of all practical loop closure possibilities.
14
system R(jω) + E(jω) -e 6 -
U (jω)
- Hc (jω)
PILOT
Z(jω)
-
(a) single-axis compensatory tracking task
Quasi-linear pilot model remnant N (jω) R(jω) + E(jω) -e 6
-? e+
- Hp(jω)
system U (jω)
+
- Hc (jω)
Z(jω)
-
describing function
(b) quasi-linear pilot model
Figure 10: The quasi-linear pilot model in the elementary single-axis compensatory tracking task. In this figure Hc depicts the dynamics of the system to be controlled, and R, E, U, and Y the reference signal, the displayed error signal, the pilot control signal and the system output signal respectively. The quasi-linear pilot model consists of a describing function Hp and a remnant N. (iii) Delays due to scanning and sampling are minimized. In short, the human will establish, in a learning process, a control system that aims at establishing a trade-off between the requirements of performance and stability, in the same fashion as a control engineer would design an automatic control system. Extensive research has been conducted on the problem of modeling human control behavior in elementary single-axis compensatory tracking tasks, see Figure 10(a). In this task the operator must minimize the difference (error E) between the output (Z) of the system to be controlled and a reference signal (R). This is the same control situation as described earlier, Figure 7. The double-lined block in Figure 10(a) illustrates that pilot behavior in this closed loop is essentially non-linear. Compensatory tracking tasks
Experimental studies concerning the compensatory tracking task showed that, as long as the task variables remain constant, the operator control behavior remains fairly constant too. In this case, it can be described by a deterministic model – a linear differential equation with constant coefficients and a time delay (the describing function) – and a remnant model – a stationary noise process.
15
The result is a quasi-linear pilot model (Figure 10(b)): the describing function accounts for the portion of the human controller’s output that is linearly related to his input and the remnant represents the difference between the causal model output and the experimentally measured output of the human controller. The application of quasi-linear theory allowed human manual control behavior in single-axis compensatory control tasks to be identified. The insights gained led to the postulation of the crossover model theorem.
4.3 The crossover model theorem In a series of experiments the characteristics of two task variables, i.e., the dynamics of the system to be controlled (Hc (jω) in Figure 10(b)) and the bandwidth of the reference signal (R(jω) in Figure 10(b)) were systematically varied. The system dynamics were chosen to be the basic proportional (k), integrator (k/s) and double integrator (k/s2 ) systems. The bandwidth of the forcing function (ωr ) was set at 1.5, 2.5 and 4.0 rad/s. The forcing function consisted of a sum of 10 sinusoids in a block spectrum. These sinusoids have different frequencies and random phases, and their sum results in a quasi-random signal, being quasi-random in the sense that the human operator can not anticipate future values of this signal. The compensatory display shows only the error between the forcing function signal R(jω) and the system output signal Z(jω). Figure 11 summarizes the main findings of McRuer’s experiments. The left column shows the ˆ magnitude of the pilot frequency response function (FRF) Hp (jω) that could be estimated from the experimental data. The identification procedure allows this FRF to be determined at only the frequencies of the sinusoids in the forcing function. The center column shows the magnitude of the system FRF, and because we know the system exactly we can compute this function for all frequencies. The right column shows the magnitude of the estimated open loop, i.e., the product ˆ p (jω)Hc(jω). of the estimated pilot FRF and the system FRF: Yˆ OL (jω) = H The dynamics of the center column clearly show the proportional (top), integrator (center) and double integrator (bottom) properties of the system to be controlled. The left column clearly shows that the pilot FRF is different for all three systems, a clear indication that the pilot is adapting to the dynamics of the system to be controlled. The right column, however, shows that the shape of the open loop is the same. Independent of the system dynamics, the open loop FRF resembles “integrator-like” dynamics near the frequency where the open loop equals one, i.e., near the crossover frequency. As will be discussed later, the value of the crossover frequency depends, among others, on the dynamics of the system and the bandwidth of the forcing function. Apparently, human operators adapt to the system to be controlled in such a way that the open loop, i.e., the human dynamics times the system dynamics, becomes an integrator. McRuer and his colleagues generalized this systematic adaptation of human control behavior with the postulation of their crossover model theorem (McRuer et al., 1965). The crossover model theorem states that human controllers adjust their control behavior to the 16
pilot Hp
1
magnitude
10
0
0
10
0
10
−1
10
−1
−1
10
0
10
1
10
10
1
ωc
−1
−1
10
0
1
10
10
10
1
10
magnitude
open loop Y OL
1
10
k
10
−1
10
0
1
10
10
1
10
10
k/s 0
0
10
0
10
−1
10
10
−1
−1
10
0
10
1
10
10
1
ωc
−1
−1
10
0
1
10
10
10
1
10
magnitude
system Hc
1
10
−1
10
0
10
1
10
1
10
10
k/s2 0
0
10
−1
10
0
10
10
−1
−1
10
0
10 frequency, rad/s
1
10
10
ωc
−1
−1
10
0
10 frequency, rad/s
1
10
10
−1
10
0
10 frequency, rad/s
1
10
Figure 11: Identification of pilot control behavior when controlling the three basis systems k, k/s and k/s2 . The crosses indicate the frequencies of the sinusoids in the forcing function R(jω). The red dashed lines in the right column indicate integrator-like dynamics.
17
dynamics of the controlled element in such a way that the dynamic characteristics of the open loop transfer function in the crossover region can be described by: ωc e−jωτe Y OL (jω) = Hp (jω)Hc (jω) ≈ jω
(16)
with ωc the crossover frequency and τe a time delay lumping the information-processing delays of the human operator. The crossover model is a mathematical statement of the empirical observation that human controllers adjust their control characteristics so that the closed loop system dynamics mimic those of a well-designed feedback system (McRuer & Jex, 1967). Then, when the dynamics of the system to be controlled are known, the crossover model allows a prediction of the human controller characteristics via: ωc e−jωτe (17) Hp (jω) ≈ jωHc(jω) The parameters ωc and τe are task-dependent and can be selected on the basis of the so-called verbal adjustment rules that are on their turn based on an immense amount of experimental data (McRuer & Krendel, 1974).
4.4 Model parametrization: structural-isomorphic models The linear describing function of the quasi-linear pilot model takes on various forms depending on the precision with which one attempts to reproduce the characteristics of the observed control behavior. In its most extensive form, the so-called precision model, the linear describing function can be described by (McRuer & Jex, 1967):1 neuromuscular system
lead−lag
time delay }| z }| {z { z }| { z}|{ 1 + τL jω ωn 2 Hp (jω) = Kp e−jωτ 2 2 1 + τI jω (1 + τN1 jω)(ωn + 2ζn ωn jω + (jω) ) {z }| | {z } gain
pilot equalization
pilot limitations
(18)
The parameters in this model reflect the pilot adaptation characteristics as well as the limitations such as the time delay and the neuromuscular system. The equalization parameters are adjusted by the pilot in such a way that the open loop dynamics satisfy the crossover model (Eq.(16)). The most commonly used approximation of the precision model is the simplified precision model (McRuer & Jex, 1967). 1 + τL jω e Yp (jω) = Kp · · e|−jωτ (19) {z } |{z} 1 + τI jω | {z } time delay gain pilot equalization
with: 1
Skipping the so-called ‘α’-term (McRuer & Jex, 1967).
18
• Kp the pilot gain • τL the lead time constant (in [s]) • τI the lag time constant (in [s]) • τe the effective time delay (in [s]) The simplified precision model (SPM) is widely used and will be discussed in the following.
4.5 Verbal adjustment rules 4.5.1 Empirical rules The adjustment rules of McRuer et al. are based on the experience obtained with a large number of tracking experiments, all SISO following tasks with a compensatory display, conducted over a wide range of dynamic characteristics of controlled elements and input signal bandwidths (McRuer et al., 1965). The adjustment rules enable us to predict the parameters used in the precision or simplified precision models, including their numerical values, for a specific combination of system dynamics and forcing function signal bandwidth. There are 6 rules, only the first and the last will be discussed below. The reader should keep in mind that the adjustment rules have an empirical basis, and should be used with care. 4.5.2 Rule # 1: equalization selection and adjustment The first rule is the most important one, as it allows us to predict how humans will adapt to a particular system, i.e., what equalization parameters of the SPM are needed in order to follow the crossover model. The first rule states that the particular equalization is selected from the SPM such that the following properties occur: 1. The system can be stabilized by proper selection of gain Kp preferably over a very broad range of Kp . 2. Over a wide frequency range, near the crossover region, the magnitude ratio |Hp Hc | has an “integrator-like” shape, i.e. the ratio has approximately a -20 dB/decade slope. 3. |Hp Hc | 1 is obtained at low frequencies to provide good low frequency closed loop response to system commands and suppression of disturbances. Figure 12 illustrates the choice of the pilot equalization parameters τL (lead) and τI (lag) such that, for all three basis systems considered, the open loop magnitude becomes “integrator-like” near the crossover frequency.
19
For the velocity control system (k/s) the pilot equalization parameters can be zero, and the pilot can act like a proportional gain with a time delay: Hc = Kp e−jωτe
(20)
There is no need to “fix” the system dynamics through adaptation because the system dynamics itself already mimics the integrator-like dynamics that are predicted by the crossover model. For the proportional control system (k) the pilot must use the lag equalization (τI ) to make the open loop an integrator: 1 H c = Kp e−jωτe (21) (1 + jωτI ) The lag parameter needs to be chosen such that it creates the integrator-like characteristic well before the crossover frequency, i.e., 1/τI
