oxictl 0.1.1

Pure Rust Real-Time Control Systems Framework
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306

//! GP covariance kernel functions.
//!
//! Each kernel implements the [`Kernel`] trait which computes k(x1, x2) for
//! input vectors of compile-time dimension `D`.

#![allow(clippy::needless_range_loop)]

use crate::core::scalar::ControlScalar;

// ──────────────────────────────────────────────────────────────────────────────
// Core trait
// ──────────────────────────────────────────────────────────────────────────────

/// Trait for GP covariance kernel functions k(x1, x2) → S.
///
/// Kernels must be symmetric and positive semi-definite.
pub trait Kernel<S: ControlScalar, const D: usize>: Clone {
    /// Evaluate the kernel at the pair of D-dimensional input vectors.
    fn eval(&self, x1: &[S; D], x2: &[S; D]) -> S;
}

// ──────────────────────────────────────────────────────────────────────────────
// Helper: squared Euclidean distance
// ──────────────────────────────────────────────────────────────────────────────

#[inline]
fn squared_dist<S: ControlScalar, const D: usize>(x1: &[S; D], x2: &[S; D]) -> S {
    let mut acc = S::ZERO;
    for d in 0..D {
        let diff = x1[d] - x2[d];
        acc += diff * diff;
    }
    acc
}

// ──────────────────────────────────────────────────────────────────────────────
// Squared Exponential (RBF) kernel
// ──────────────────────────────────────────────────────────────────────────────

/// Squared Exponential (RBF) kernel:
///
/// k(x, x') = σ² · exp(−‖x − x'‖² / (2 l²))
///
/// where σ² = `variance` and l = `length_scale`.
#[derive(Debug, Clone, Copy)]
pub struct RbfKernel<S: ControlScalar> {
    /// Output variance σ².
    pub variance: S,
    /// Length-scale l.
    pub length_scale: S,
}

impl<S: ControlScalar, const D: usize> Kernel<S, D> for RbfKernel<S> {
    fn eval(&self, x1: &[S; D], x2: &[S; D]) -> S {
        let r2 = squared_dist(x1, x2);
        let two_l2 = self.length_scale * self.length_scale * S::TWO;
        let exponent = -(r2.to_f64() / two_l2.to_f64());
        self.variance * S::from_f64(libm::exp(exponent))
    }
}

// ──────────────────────────────────────────────────────────────────────────────
// Matérn 5/2 kernel
// ──────────────────────────────────────────────────────────────────────────────

/// Matérn 5/2 kernel:
///
/// k(x, x') = σ² (1 + √5 r/l + 5r²/(3l²)) · exp(−√5 r/l)
///
/// where r = ‖x − x'‖.
#[derive(Debug, Clone, Copy)]
pub struct Matern52Kernel<S: ControlScalar> {
    /// Output variance σ².
    pub variance: S,
    /// Length-scale l.
    pub length_scale: S,
}

impl<S: ControlScalar, const D: usize> Kernel<S, D> for Matern52Kernel<S> {
    fn eval(&self, x1: &[S; D], x2: &[S; D]) -> S {
        let r2 = squared_dist(x1, x2).to_f64();
        let r = libm::sqrt(r2);
        let l = self.length_scale.to_f64();
        let sqrt5 = libm::sqrt(5.0_f64);
        let sr_over_l = sqrt5 * r / l;
        let poly = 1.0 + sr_over_l + 5.0 * r2 / (3.0 * l * l);
        let k = poly * libm::exp(-sr_over_l);
        self.variance * S::from_f64(k)
    }
}

// ──────────────────────────────────────────────────────────────────────────────
// Linear (polynomial) kernel
// ──────────────────────────────────────────────────────────────────────────────

/// Linear / polynomial kernel:
///
/// k(x, x') = σ² (x · x' + c)^p
///
/// where σ² = `variance`, c = `bias`, p = `degree`.
#[derive(Debug, Clone, Copy)]
pub struct LinearKernel<S: ControlScalar> {
    /// Output variance σ².
    pub variance: S,
    /// Bias term c.
    pub bias: S,
    /// Polynomial degree p.
    pub degree: u32,
}

impl<S: ControlScalar, const D: usize> Kernel<S, D> for LinearKernel<S> {
    fn eval(&self, x1: &[S; D], x2: &[S; D]) -> S {
        let mut dot = S::ZERO;
        for d in 0..D {
            dot += x1[d] * x2[d];
        }
        let inner = (dot + self.bias).to_f64();
        let k = libm::pow(inner, self.degree as f64);
        self.variance * S::from_f64(k)
    }
}

// ──────────────────────────────────────────────────────────────────────────────
// Additive kernel
// ──────────────────────────────────────────────────────────────────────────────

/// Additive kernel: k(x, x') = k1(x, x') + k2(x, x').
#[derive(Debug, Clone, Copy)]
pub struct AdditiveKernel<K1, K2> {
    /// First sub-kernel.
    pub k1: K1,
    /// Second sub-kernel.
    pub k2: K2,
}

impl<S, K1, K2, const D: usize> Kernel<S, D> for AdditiveKernel<K1, K2>
where
    S: ControlScalar,
    K1: Kernel<S, D>,
    K2: Kernel<S, D>,
{
    fn eval(&self, x1: &[S; D], x2: &[S; D]) -> S {
        self.k1.eval(x1, x2) + self.k2.eval(x1, x2)
    }
}

// ──────────────────────────────────────────────────────────────────────────────
// Tests
// ──────────────────────────────────────────────────────────────────────────────

#[cfg(test)]
mod tests {
    use super::*;

    const EPS: f64 = 1e-10;

    fn approx_eq(a: f64, b: f64, tol: f64) -> bool {
        (a - b).abs() < tol
    }

    // ── RBF kernel ────────────────────────────────────────────────────────────

    #[test]
    fn rbf_symmetry() {
        let k = RbfKernel::<f64> {
            variance: 1.0,
            length_scale: 1.0,
        };
        let x1 = [1.0_f64, 2.0];
        let x2 = [3.0_f64, 4.0];
        assert!(approx_eq(k.eval(&x1, &x2), k.eval(&x2, &x1), EPS));
    }

    #[test]
    fn rbf_diagonal_nonneg() {
        let k = RbfKernel::<f64> {
            variance: 2.5,
            length_scale: 0.5,
        };
        let x = [1.0_f64, -1.0, 3.0];
        assert!(k.eval(&x, &x) >= 0.0);
    }

    #[test]
    fn rbf_at_zero_distance() {
        let k = RbfKernel::<f64> {
            variance: 3.0,
            length_scale: 1.0,
        };
        let x = [0.0_f64];
        // k(x,x) = variance * exp(0) = variance
        assert!(approx_eq(k.eval(&x, &x), 3.0, 1e-12));
    }

    #[test]
    fn rbf_known_value() {
        // l=1, σ²=1 → k([0],[1]) = exp(-0.5)
        let k = RbfKernel::<f64> {
            variance: 1.0,
            length_scale: 1.0,
        };
        let x1 = [0.0_f64];
        let x2 = [1.0_f64];
        let expected = libm::exp(-0.5_f64);
        assert!(approx_eq(k.eval(&x1, &x2), expected, 1e-12));
    }

    // ── Matérn 5/2 kernel ─────────────────────────────────────────────────────

    #[test]
    fn matern52_symmetry() {
        let k = Matern52Kernel::<f64> {
            variance: 1.0,
            length_scale: 1.0,
        };
        let x1 = [1.0_f64, 2.0];
        let x2 = [3.0_f64, 4.0];
        assert!(approx_eq(k.eval(&x1, &x2), k.eval(&x2, &x1), EPS));
    }

    #[test]
    fn matern52_at_zero_distance() {
        let k = Matern52Kernel::<f64> {
            variance: 2.0,
            length_scale: 1.0,
        };
        let x = [0.0_f64];
        // k(x,x) = variance * (1 + 0 + 0) * exp(0) = variance
        assert!(approx_eq(k.eval(&x, &x), 2.0, 1e-12));
    }

    #[test]
    fn matern52_diagonal_nonneg() {
        let k = Matern52Kernel::<f64> {
            variance: 1.5,
            length_scale: 0.7,
        };
        let x = [2.0_f64, -3.0];
        assert!(k.eval(&x, &x) >= 0.0);
    }

    // ── Linear kernel ─────────────────────────────────────────────────────────

    #[test]
    fn linear_symmetry() {
        let k = LinearKernel::<f64> {
            variance: 1.0,
            bias: 1.0,
            degree: 2,
        };
        let x1 = [1.0_f64, 2.0];
        let x2 = [3.0_f64, 4.0];
        assert!(approx_eq(k.eval(&x1, &x2), k.eval(&x2, &x1), EPS));
    }

    #[test]
    fn linear_known_value() {
        // σ²=1, c=0, p=1: k(x,y) = x·y
        let k = LinearKernel::<f64> {
            variance: 1.0,
            bias: 0.0,
            degree: 1,
        };
        let x1 = [2.0_f64, 3.0];
        let x2 = [4.0_f64, 5.0];
        // dot = 2*4 + 3*5 = 8+15 = 23; (23+0)^1 * 1 = 23
        assert!(approx_eq(k.eval(&x1, &x2), 23.0, 1e-12));
    }

    // ── Additive kernel ───────────────────────────────────────────────────────

    #[test]
    fn additive_is_sum() {
        let k1 = RbfKernel::<f64> {
            variance: 1.0,
            length_scale: 1.0,
        };
        let k2 = Matern52Kernel::<f64> {
            variance: 0.5,
            length_scale: 2.0,
        };
        let k_add = AdditiveKernel { k1, k2 };
        let x1 = [1.0_f64];
        let x2 = [2.0_f64];
        let expected = k_add.k1.eval(&x1, &x2) + k_add.k2.eval(&x1, &x2);
        assert!(approx_eq(k_add.eval(&x1, &x2), expected, EPS));
    }

    #[test]
    fn additive_symmetry() {
        let k_add = AdditiveKernel {
            k1: RbfKernel::<f64> {
                variance: 1.0,
                length_scale: 1.0,
            },
            k2: LinearKernel::<f64> {
                variance: 0.5,
                bias: 1.0,
                degree: 2,
            },
        };
        let x1 = [1.0_f64, 0.5];
        let x2 = [0.3_f64, 2.1];
        assert!(approx_eq(k_add.eval(&x1, &x2), k_add.eval(&x2, &x1), EPS));
    }
}