Performance

In terms of stability and robustness, the aim of the controller is to compensate for disturbances and plant uncertainties. Another motivation for using a feedback con- troller is to get consistent performance over a wide range of conditions, or to improve performance compared to the open-loop plant. The performance is usually defined by a number of performance criteria. This section explains the definitions of such criteria and evaluates them for their application to the control loop of the Collision Prevention System.

In the time domain, the way a system responds can be quantified by examining the output response, y(t), when an input or disturbance variable changes from one

steady level to another steady level. Such a change is typically produced by a step function and the corresponding response of y(t) is then called step response. If the height of the step is one, the step function is called unit step. Based on the step response, it is possible to define the following commonly used performance criteria: Steady-state error: The steady-state error defines the difference between the ex-

pected value and the final value of the output response, i.e. the steady-state output. The aim of the controller’s design is to minimise this difference. The expected value depends on the variable that is stimulated by the step function. For the reference input of a single-input-single-output (SISO) system, we expect the output response to reach the height of the step function, i.e. the output value follows the desired reference. In contrast, if the unit step is applied to the disturbance value (output disturbance), the controller should compensate for this disturbance, so that the disturbance does not influence the output. The aim of the Collision Prevention System is not to keep a specific steady-state output value, but only to prevent a potential collision. Thus we use the peak value of the output variable as an alternative criteria. This value represents the relative distance at which a braking manoeuvre is finished. It is positive if the collision was prevented, cf. Figure 4.4(a), and negative if not, cf. Figure 4.4(b). In addition, an arbitrary offset should compensate for inaccuracies of the sensor or perturbances of the plant, so that the collision is even prevented if ∆s reaches zero. It is internally added to the value of ∆s at the interface of the final software system, i.e. the distance which is received from the radar sensor. ∆s in [m] 0 2 4 6 8 10 t in [s] 10 8 6 4 2 0 no collision

(a) Positive peak value

∆s in [m] 10 8 6 4 2 0 0 2 4 6 8 10 t in [s] collision

(b) Negative peak value

Rise time / Settling time: The rise time is the shortest time required for the step response to achieve some specified percentage of the steady-state value. It is usually measured between 10 percent and 90 percent of the steady-state value, because this range is applicable for overshooting and non-overshooting responses. For the stimulation by a disturbance, the rise time can be interpreted as a fall time, i.e. it is measured while the output value falls from 90 percent to 10 percent of the disturbance’s amplitude. In contrast, the settling time specifies the time taken for the response to reach and remain within some specified range of its final value. This range is called the allowable tolerance and is normally expressed as a percentage of the step’s height. The general difference between both times is that the rise time does not consider the progress of the output signal after it reached the specified percentage, while the settling time is only defined if the signal stays within constant boundaries.

For the CPS, we define the rise time as the time at which the minimal relative distance is reached. It describes how fast the system reacts to a certain initial condition to prevent a collision.

Peak overshoot / Peak time: The peak overshoot specifies the amplitude of the first peak which overshoots the steady-state value, and is normally expressed as a percentage of the steady-state value. The peak time is the time from the initiation of the response to the overshoot.

In terms of the our CPS system, an overshoot of the output value is equal to a collision. Such situations are regarded as unsafe. The value of the overshoot might be used to represent the impact energy, but more common for this purpose is the relative speed at the time of the impact.

Another way to describe the performance in the time domain are so-called per- formance indices. Typically, such indices express the system’s performance by inte- grating the area between the time axis and one or more variables of the closed-loop system. Such a common performance index is the integral of the squared control signal:

J = ! t2

t1

u2(t)dt (4.13)

With t1 specifying the time of application of a step function or a disturbance config-

uration respectively, and t2 specifying the time when the minimal relative distance is

reached, the value of J states how much effort was used by the controller to prevent the collision. Thus, we use this function to represent the control effort. In addition, the minimal value of the control signal, i.e. the maximal deceleration applied by the

controller, can provide further information about the characteristics of the control process, especially because the variable is bounded to certain limits.