1{
  2  "parameters": {
  3    "a": 1.3,
  4    "b": 0.7,
  5    "d": 0.4,
  6    "p": 0.02,
  7    "w0": 3.0,
  8    "dt": 0.002,
  9    "horizon": 80.0
 10  },
 11  "pito_sweep": [
 12    {
 13      "epsilon": 0.19999999999999998,
 14      "V": 5.384615384615384,
 15      "V_times_epsilon": 1.0769230769230766,
 16      "predicted_T_bound": 0.4858853402374508,
 17      "observed_T": 0.48090000000000005,
 18      "exact_T": 0.48104226142663914,
 19      "rate_predicted": 7.0,
 20      "rate_observed": 7.0024511439339205,
 21      "rate_relative_error": 0.00035016341913150583
 22    },
 23    {
 24      "epsilon": 0.39999999999999997,
 25      "V": 2.692307692307692,
 26      "V_times_epsilon": 1.0769230769230766,
 27      "predicted_T_bound": 0.7737286288863457,
 28      "observed_T": 0.7539,
 29      "exact_T": 0.7540163798900739,
 30      "rate_predicted": 3.5,
 31      "rate_observed": 3.500612642954197,
 32      "rate_relative_error": 0.00017504084405623694
 33    },
 34    {
 35      "epsilon": 0.7999999999999999,
 36      "V": 1.346153846153846,
 37      "V_times_epsilon": 1.0769230769230766,
 38      "predicted_T_bound": 1.1513731545955799,
 39      "observed_T": 1.0696,
 40      "exact_T": 1.0696012439437665,
 41      "rate_predicted": 1.75,
 42      "rate_observed": 1.7501531428669415,
 43      "rate_relative_error": 8.751020968083318e-05
 44    },
 45    {
 46      "epsilon": 1.2000000000000002,
 47      "V": 0.8974358974358974,
 48      "V_times_epsilon": 1.076923076923077,
 49      "predicted_T_bound": 1.3795182106578003,
 50      "observed_T": 1.1882000000000001,
 51      "exact_T": 1.1882523095313346,
 52      "rate_predicted": 1.1666666666666665,
 53      "rate_observed": 1.1667347275159077,
 54      "rate_relative_error": 5.833787077813569e-05
 55    }
 56  ],
 57  "fixed_gain_sweep": {
 58    "0.2": {
 59      "final_x": 0.10000000071451517,
 60      "max_x": 0.10636961687321454,
 61      "final_w": 0.2,
 62      "unbounded_by_80": false,
 63      "predicted_regime": "bounded"
 64    },
 65    "0.39": {
 66      "final_x": 1.1491438284960964,
 67      "max_x": 1.1491438284960964,
 68      "final_w": 0.39,
 69      "unbounded_by_80": false,
 70      "predicted_regime": "bounded"
 71    },
 72    "0.4": {
 73      "final_x": 1.7063696168743616,
 74      "max_x": 1.7063696168743616,
 75      "final_w": 0.4,
 76      "unbounded_by_80": false,
 77      "predicted_regime": "unbounded (marginal at equality)"
 78    },
 79    "0.6": {
 80      "final_x": 1827965.8278582732,
 81      "max_x": 1827965.8278582732,
 82      "final_w": 0.6,
 83      "unbounded_by_80": true,
 84      "predicted_regime": "unbounded (marginal at equality)"
 85    },
 86    "0.8": {
 87      "final_x": 12190451378361.576,
 88      "max_x": 12190451378361.576,
 89      "final_w": 0.8,
 90      "unbounded_by_80": true,
 91      "predicted_regime": "unbounded (marginal at equality)"
 92    }
 93  },
 94  "regulated_at_nominal_gamma_0.8": {
 95    "final_x": 0.5499999999662464,
 96    "max_x": 1.4438561056544417,
 97    "final_w": 0.3636363636048379,
 98    "unbounded_by_80": false,
 99    "predicted_regime": "unbounded (marginal at equality)"
100  },
101  "quantitative_checks": {
102    "PITO_rate_prediction": {
103      "predicted": "a*V",
104      "max_relative_error": 0.00035016341913150583
105    },
106    "PITO_threshold_prediction": {
107      "predicted": "V*epsilon=2*b/a",
108      "observed_values": [
109        1.0769230769230766,
110        1.0769230769230766,
111        1.0769230769230766,
112        1.076923076923077
113      ],
114      "target": 1.0769230769230769
115    },
116    "crossing_bound": {
117      "predicted": "observed_T <= predicted_T_bound",
118      "maximum_observed_minus_predicted": -0.0049853402374507305
119    },
120    "plant_boundary": {
121      "predicted": "gamma=d=0.4; gamma<d bounded, gamma>=d unbounded",
122      "observed": {
123        "0.2": false,
124        "0.39": false,
125        "0.4": false,
126        "0.6": true,
127        "0.8": true
128      }
129    },
130    "regulated_boundedness": {
131      "predicted": "feedback reduces gain and bounds x",
132      "observed_max_x": 1.4438561056544417,
133      "observed_final_w": 0.3636363636048379
134    }
135  }
136}