3.7 Hybrid Control System Performance Measures
3.7.1 Discrete Event Measures
While the sequence of DES-plant events triggered by traversal of the state-space drives and is driven by the evolution of DES-controller state transitions, it also describes
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the accuracy of the DES-plant’s response to the DES-controller output. Each DES- controller output symbol, resulting in a CT-plant input, if followed even with a broad degree of precision would produce a well-behaved sequence of subsequent plant events since DES-controller states operate over regions of the state-space.
h1(x) h2(x) h5(x) h6(x) QA QB
Figure 3.15: Diagram that shows two terminal point equivalent trajectories, QA and QB, originating from within the lower left quadrant of the image-space
The diagram in Figure 3.15 shows two possible alternate, parallel trajectories, QA and QB, occurring between times τe[n] and τe[n +k]. Both originate in the region of the state-space that is driven under DES-Controller state ˜s6 and terminate in the region driven by ˜s1. Also included, in green, is the ideal trajectory that should evolve from the same initial point as QA if the plant (user) DES-plant response to the DES-Controller output symbols was highly accurate and precise. While eachQA and QB cross the same four hypersurfaces, they do so in a differing order. Thus the events triggered and the corresponding state transitions will differ by producing the following plant events / controller states / output symbol sets:
QA: X˜A ={·, ε6, ε18,x˜5,x˜13}, ˜SA ={˜s5,s˜5,s˜5,˜s4,s˜1}, andφ(˜s)A ={˜r2,˜r2,˜r2,˜r4,˜r1}
QB X˜B ={·, ε18, ε6,x˜17,x˜1}, ˜SB ={˜s5,s˜5,s˜5,˜s2,s˜1}, andφ(˜s)B ={˜r2,˜r2,r˜2,r˜2,r˜1}
In terms of a region basis the accuracy of the two trajectories is equivalent. Yet examining the individual DES-controller state transitions that differ between the two trajectories shows thatQA has only ideally-behaved transitions, whereasQB has one
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well-behaved trajectory. That difference is at the third state transition in QB where ˜
s5 → s˜2 on x˜17 ˜
r2 as the non-ideal transition. So we can define a measure to evaluate the accuracy behaviour of the trajectory evolution by evaluating each sequential state transition by
δ[n] =m(˜x[n],s˜[n],s˜[n−1]) (3.55) and then generate an accuracy behaviour score for the entire trajectory throughout the reaching task by
D= 1 N −1 N−1 X n=1 m(˜x[n],˜s[n],s˜[n−1]) (3.56)
withm(˜s[n],s˜[n−1]) being a discrete function the produces values from{1,0,−1} for state transitions that are ideally-behaved, well-behaved, or ill-behaved, respec- tively. Equation (3.55) produces the trajectory segment based sequence of state tran- sition scores that is a term in the reaching task performance metric described later. The state transition metric, −1 ≤ D ≤ 1, given in equation (3.56) describes how accurately, on average, the plant reproduced the set of state transitions indicated by the DES-controller through the motion cues issued. A value of D= 1 indicates that the fully correct set of state transitions were followed during the trajectory. A value ofD= 0 indicates that the sequence of state transitions replicates the equivalent of a random set of trajectory segment movements: some portion driven directly towards the target, some driven indirectly towards the target, and some portion directly away from the target. A value ofD=−1 indicates that the trajectory followed was in op- position to the intend motion cues. While theoretically possible, scores of D <0 are impractical as it would require an artificial termination of the reaching task at some arbitrary point, assuming an infinite field of view for the camera or at the point the target leaves the field of view, which would not be considered a successful execution of the reaching task. Returning to the example provided in Figure 3.15, the sample trajectories of QA and QB produce state transition accuracy scores of DA = 1 and DB = 0.75, respectively. With this measure, the state transitions that occur dur- ing the evolution of each trajectory can be quantified in terms of accuracy towards appropriate state transitions for the reaching task solution.
The DES-Controller state transitions can be filtered based on various subsets of silent events that occur during the evolution of a trajectory. The diagram in
Chapter 3: Hybrid System Model for Visual Control 70 Ix 2 Iy 2 h1(x) h3(x) h2(x) h4(x) h5(x) h7(x) h6(x) h8(x) ε5 ε7 ε1 ε3 ε9 ε11 ε17 ε19 ε13 ε15 ε21 ε23
Figure 3.16: Region partitioning of the image space that shows silent events triggering equivalent state transitions.
Figure 3.16 shows three subsets of silent events: {X˜α} = {˜ε1,ε˜3,ε˜13,ε˜15}, {X˜β} = {˜ε5,ε˜7,ε˜9,ε˜11}, and {X˜γ}={˜ε17,ε˜19,ε˜21,ε˜23}. To aid the reader in visualizing the boundaries, the corresponding regions of the state-space in which {X˜α}, {X˜β}, and {X˜γ} occur are colour coded as green, blue, and red, respectively.
Figure 3.16 show the bounded regions that maintain. The subset{˜ε1,ε˜3,ε˜13,ε˜15} under the l2 hypersurface model causes the DES-Controller state to maintain a ˜s1 in comparison to l1 hypersurface model while would trigger non-silent events and a corresponding DES-Controller transition to either one of {˜s2,s˜3,s˜4,˜s5}