-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathk202111.v
586 lines (507 loc) · 17.6 KB
/
k202111.v
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
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
Require Import Reals.
Require Import List.
Import ListNotations.
Require Import Matrix.Mat_def.
Require Import Matrix.Mat_trans.
Require Import Matrix.Mat_trans_lemma.
Require Import Matrix.Matrix_Module.
Require Import Matrix.FMatrix.
Require Import Matrix.RSupport.
(*矩阵和向量维度*)
Variable n m: nat.
(* xk是k时刻的状态向量
xk1 是k-1时刻的状态向量
uk1是k-1时刻的控制向量
wk1是k-1时刻的系统噪声*)
(* Variable xk: @Mat (R->R) n 1. *)
Variable xk1 uk1 wk1: FMat n 1.
(*zk是k时刻的观测向量
vk是k时刻的观测噪声*)
Variables vk:FMat m 1.
(*phik1是k-1时刻到k时刻的系统转移矩阵 是不变的常数矩阵
gammak1是系统噪声矩阵 表征各噪声分别影响个状态的程度
Hk是k时刻的观测矩阵*)
Variable phik1 gammak1: FMat n n.
Variables Hk : FMat m n.
(**连续时间线性系统**)
Section Chap8_2_3_Discrete_time_linear_system.
(*离散时间线性系统模型 8.38*)
(* xk = Φ *xk1 + uk1 + Γ*wk1 *)
(* zk = H *xk + vk*)
(*
Axiom Discrete_time_linear_system_xk : xk FM= (phik1 FM* xk1) FM+ uk1 FM+ (gammak1 FM* wk1).
Axiom Discrete_time_linear_system_zk : zk FM= (Hk FM* xk) FM+ vk.
*)
(* xk是k时刻的状态向量 *)
Definition xk := (phik1 FM* xk1) FM+ uk1 FM+ (gammak1 FM* wk1).
Definition zk := (Hk FM* xk) FM+ vk.
End Chap8_2_3_Discrete_time_linear_system.
Section Kalman_filtering.
(* Exkk是k时刻状态的最小方差无偏估计 ESxkk1 是在上一时刻状态估计ESxk1k1 的基础上对当前时刻状态xk的一步预测 8.5.7*)
(* Variable ESxkk : @Mat (R->R) n 1. *)
(* Variable ESxkk1 : @Mat (R->R) n 1. *)
Variable ESxk1k1 : FMat n 1.
(*----------------------------8.57-----------------------------*)
(* ESxkk1 = phi*ESxk1k1 + u *)
(* ESzkk1 = H*ESxk1k1 *)
(* ESxkk1 是在上一时刻状态估计ESxk1k1 的基础上对当前时刻状态xk的一步预测*)
(* ESzkk1 是在ESxkk1的基础上对当前时刻zk的一步预测 *)
(*-------------------------------------------------------------*)
Definition ESxkk1 := phik1 FM* ESxk1k1 FM+ uk1.
Definition ESzkk1 := Hk FM* ESxkk1.
Variable Kk' Kk'': FMat n n.
Variable Kk: FMat n m.
Definition In:= FMI n .
(*由于是线性估计器，估计EXxkk(estimate x_(k|k))可以表示未先验估计和观测值的线性组合形式 8.45
ESxkk = Kk' * Exk1k1 + Kkzk + Kk'' * uk1 *)
Definition ESxkk := Kk' FM* ESxk1k1 FM+ Kk FM* zk FM+ Kk'' FM* uk1.
(*------------------------*)
(* 2 参数Kk' 和 Kk''的推导 *)
(*------------------------*)
(**ERxkk为状态误差 8.47 **)
(**ERxkk = xk - ESxkk **)
(* Axiom ERxkk_847 :ERxkk FM= xk FM- ESxkk.
Axiom ERxk1k1_847 :ERxk1k1 FM= xk1 FM- ESxk1k1.
Axiom ERxkk1_847 :ERxkk1 FM= xk FM- ESxkk1. *)
Definition ERxkk := xk FM- ESxkk.
Definition ERxk1k1 := xk1 FM- ESxk1k1.
Definition ERxkk1 := xk FM- ESxkk1.
(*ESxkk 是xkk的无偏估计，则误差ERxkk=0 (Error x_(k|k)) 8.46*)
(* 8.46 *)
Axiom ERxkk_eq0 : ERxkk FM= FMO n 1.
Axiom ERxk1k1_eq0 : ERxk1k1 FM= FMO n 1.
Lemma error_x_kk_848 : ERxkk FM= xk FM- (Kk' FM* ESxk1k1 FM+ Kk FM* zk FM+ Kk'' FM* uk1).
Proof.
reflexivity.
Qed.
(*-----------3.1 8.49---------------*)
Lemma error_xkk_3_1: ERxkk FM=
phik1 FM* xk1 FM+ uk1 FM+ gammak1 FM* wk1 FM-
Kk' FM* ESxk1k1 FM-
Kk FM* (Hk FM* (phik1 FM* xk1 FM+ uk1 FM+ (gammak1 FM* wk1)) FM+ vk) FM-
Kk'' FM* uk1.
Proof.
unfold ERxkk.
unfold ESxkk.
unfold zk.
unfold xk.
rewrite <- ?FMsub_assoc.
reflexivity.
Qed.
(*----------------------------3.2 8.50---------------------------------*)
(* ER = (phi-KHphi)xk1 - K'ES + (gamma-KHgamma)w + (I-KH-K'')u - Kv *)
Lemma error_xkk_3_2: ERxkk FM=
(phik1 FM- Kk FM* Hk FM* phik1) FM* xk1 FM-
Kk' FM* ESxk1k1 FM+
(gammak1 FM- Kk FM* Hk FM* gammak1) FM* wk1 FM+
(In FM- Kk FM* Hk FM- Kk'' ) FM* uk1 FM-
Kk FM* vk.
Proof.
rewrite error_xkk_3_1.
rewrite ?FMmul_sub_distr_l.
rewrite ?FMmul_add_distr_r.
rewrite <- ?FMmul_assoc.
rewrite <- ?FMsub_assoc.
rewrite <- ?FMadd_assoc1.
rewrite FMmul_unit_l.
unfold FMsub.
rewrite FMsub_comm1 .
apply Fmat_sub_compat. 2: { reflexivity. }
apply Fmat_sub_compat. 2: { reflexivity. }
rewrite FMsub_comm1 .
apply Fmat_sub_compat. 2: { reflexivity. }
rewrite <- FMadd_comm1.
apply Fmat_sub_compat. 2: { reflexivity. }
rewrite FMsub_comm1 .
repeat rewrite <- FMadd_comm1.
rewrite FMadd_comm2.
reflexivity.
Qed.
(*------------------------4.22----8.51-------------------------------------------*)
(* ER = (phi-KHphi-K')xk1 + K'(xk1-ES) + (gamma-KHgamma)w + (I-KH-K'')u - Kv *)
Lemma error_xkk_4_22: ERxkk FM=
(phik1 FM- Kk FM* Hk FM* phik1 FM- Kk') FM* xk1 FM+
Kk' FM* (xk1 FM- ESxk1k1) FM+
(gammak1 FM- Kk FM* Hk FM* gammak1) FM* wk1 FM+
(In FM- Kk FM* Hk FM- Kk'' ) FM* uk1 FM-
Kk FM* vk.
Proof.
rewrite error_xkk_3_2.
apply mat_sub_compat'. 2: { reflexivity. }
repeat apply mat_add_compat';try reflexivity.
rewrite ?FMmul_sub_distr_l.
rewrite ?FMmul_sub_distr_r.
rewrite <- FMadd_assoc1.
rewrite <- FMadd_comm1.
rewrite FMadd_assoc1.
rewrite FMsub_self.
rewrite FMadd_zero_r.
reflexivity.
Qed.
Lemma error_x_kk_852: ERxkk FM=
(phik1 FM- Kk FM* Hk FM* phik1 FM- Kk') FM* xk1 FM+
(In FM- Kk FM* Hk FM- Kk'' ) FM* uk1 FM+
Kk' FM* (xk1 FM- ESxk1k1) FM+
(gammak1 FM- Kk FM* Hk FM* gammak1) FM* wk1 FM-
Kk FM* vk.
Proof.
rewrite error_xkk_4_22.
rewrite FMadd_comm2.
apply mat_sub_compat'. 2: { reflexivity. }
apply mat_add_compat'. 2: { reflexivity. }
rewrite FMadd_comm2.
reflexivity.
Qed.
(*卡尔曼滤波器要求wk和vk是互不相关的零均值高斯白噪声序列，即 vk=0 wk=0 8.39*)
Axiom v_eq0: vk FM= FMO m 1.
Axiom w_eq0: wk1 FM= FMO n 1.
(*-----------------------4.23-----8.53---------------------------------------*)
(* (phi-KHphi-K')xk1 + (I-KH-K'')u=0 *)
(*---------------------------------------------------------------------------*)
Lemma eq0_423: (phik1 FM- Kk FM* Hk FM* phik1 FM- Kk') FM* xk1 FM+ (In FM- Kk FM* Hk FM- Kk'' ) FM* uk1 FM= FMO n 1.
Proof.
rewrite <- ERxkk_eq0.
rewrite error_x_kk_852.
assert(ERxk1k1 FM= xk1 FM- ESxk1k1). {reflexivity. }
rewrite <- H.
rewrite v_eq0,w_eq0,ERxk1k1_eq0.
rewrite ?FMmul_zero_l.
rewrite FMsub_zero_r.
rewrite ?FMadd_zero_r.
reflexivity.
Qed.
(*xk1 和 uk1 分别为系统的状态向量和控制向量，不能时刻为0 *)
Axiom noteq0_xk1 : not(xk1 FM= FMO n 1).
Axiom noteq0_uk1 : not(uk1 FM= FMO n 1).
(* A*B + C*D = 0 .B D不等于0 -> A C 等于0*)
Axiom eq0:forall {n m p:nat} (A C:@Mat (R->R) m n) (B D:@Mat (R->R) n p) ,
A FM* B FM+ C FM* D FM= FMO m p ->
not(B FM= FMO n p) -> not(D FM= FMO n p) -> A FM= FMO m n /\ C FM= FMO m n.
(*----------------------------8.54-------------------*)
(* (phi-KHphi-K')xk1 =0 *)
(* (I-KH-K'')u=0 *)
(*---------------------------------------------------*)
Lemma eq0_854_1: phik1 FM- Kk FM* Hk FM* phik1 FM- Kk' FM= FMO n n.
Proof.
apply (eq0 (phik1 FM- Kk FM* Hk FM* phik1 FM- Kk') (In FM- Kk FM* Hk FM- Kk'') xk1 uk1).
apply eq0_423. apply noteq0_xk1. apply noteq0_uk1.
Qed.
Lemma eq0_854_2: In FM- Kk FM* Hk FM- Kk'' FM= FMO n n.
apply (eq0 (phik1 FM- Kk FM* Hk FM* phik1 FM- Kk') (In FM- Kk FM* Hk FM- Kk'') xk1 uk1).
apply eq0_423. apply noteq0_xk1. apply noteq0_uk1.
Qed.
(*----------------------------8.55-------------------*)
(* K'=phi-KHphi *)
(* K''=I-KH *)
(*---------------------------------------------------*)
Lemma Kk'_4_5: Kk' FM= phik1 FM- Kk FM* Hk FM* phik1.
Proof.
assert(phik1 FM- Kk FM* Hk FM* phik1 FM- Kk' FM+ Kk' FM= FMO n n FM+ Kk').
{
apply mat_add_compat'.
apply eq0_854_1. reflexivity.
}
rewrite FMadd_zero_l in H.
rewrite <- FMadd_comm1 in H.
rewrite FMadd_assoc1 in H.
rewrite FMsub_self in H.
rewrite FMadd_zero_r in H.
symmetry.
apply H.
Qed.
Lemma Kk''_4_5: Kk'' FM= In FM- Kk FM* Hk.
Proof.
assert(In FM- Kk FM* Hk FM- Kk'' FM+ Kk'' FM= FMO n n FM+ Kk'').
{
apply mat_add_compat'. apply eq0_854_2. reflexivity.
}
rewrite FMadd_zero_l in H.
rewrite <- FMadd_comm1 in H.
rewrite FMadd_assoc1 in H.
rewrite FMsub_self in H.
rewrite FMadd_zero_r in H.
symmetry.
apply H.
Qed.
(*-----------------4.26-------8.56-------------------*)
(* ESxkk = ESxkk1 + K(z - ESzkk1) *)
(*---------------------------------------------------*)
Lemma estimate_xkk_4_26: ESxkk FM= ESxkk1 FM+ Kk FM*(zk FM- ESzkk1).
Proof.
unfold ESxkk.
rewrite Kk'_4_5,Kk''_4_5.
unfold ESxkk1,ESzkk1,ESxkk1.
rewrite ?FMmul_sub_distr_l.
rewrite ?FMmul_sub_distr_r.
rewrite ?FMmul_add_distr_r.
rewrite <- ?FMmul_assoc.
rewrite FMmul_unit_l.
rewrite <- FMsub_assoc.
rewrite <- ?FMadd_assoc1.
apply mat_sub_compat'. 2: { reflexivity. }
rewrite FMadd_comm2.
rewrite FMadd_comm1.
apply mat_add_compat'. 2: { reflexivity. }
symmetry.
apply FMadd_comm1.
Qed.
(*-------------------*)
(* 3 参数Kk的推导 *)
(*-------------------*)
Definition Pkk := ERxkk FM* T(ERxkk).
Definition Pkk1 := ERxkk1 FM* T(ERxkk1).
Definition Rk := vk FM* T(vk).
(*----------------4.31--------8.59-------------------*)
(* ERxkk = (I-KH)ERxkk1 - Kv *)
(*---------------------------------------------------*)
Lemma error_xkk_4_31: ERxkk FM= (In FM- Kk FM* Hk) FM* ERxkk1 FM- Kk FM* vk.
Proof.
unfold ERxkk.
rewrite estimate_xkk_4_26.
unfold ESzkk1.
rewrite <- FMsub_assoc.
unfold zk.
rewrite FMadd_comm1.
rewrite FMmul_add_distr_r.
rewrite FMmul_sub_distr_r.
rewrite <- ?FMmul_assoc.
rewrite <- FMsub_assoc.
rewrite FMmul_sub_distr_l.
rewrite FMmul_unit_l.
unfold ERxkk1.
rewrite FMmul_sub_distr_r.
reflexivity.
Qed.
(*----------------------------8.60------------------------*)
(* ERxkk * T_ERxkk = (I-KH) * ERxkk1 * T_ERxkk * T_(I-KH) *)
(* - (I-KH)ERxkk1 * T_v * T_K *)
(* - K * v * T_ERxkk*T_(I-KH) *)
(* + K * v * T_v * T_K *)
(*--------------------------------------------------------*)
Lemma ERxkk_mul_T_ERxkk_860: ERxkk FM* T(ERxkk) FM=
(In FM- Kk FM* Hk) FM* ERxkk1 FM* T(ERxkk1) FM* T(In FM- Kk FM* Hk)
FM- (In FM- Kk FM* Hk) FM* ERxkk1 FM* T(vk) FM* T(Kk)
FM- Kk FM* vk FM* T(ERxkk1) FM* T(In FM- Kk FM* Hk)
FM+ Kk FM* vk FM* T(vk) FM* T(Kk).
Proof.
assert(T(ERxkk) FM= T(ERxkk1) FM* T(In FM- Kk FM* Hk) FM- T(vk) FM* T(Kk)).
{
rewrite error_xkk_4_31.
rewrite FMteq_sub.
rewrite ?FMteq_mul.
reflexivity.
}
rewrite H.
rewrite error_xkk_4_31.
rewrite ?FMmul_sub_distr_r.
rewrite ?FMmul_sub_distr_l.
rewrite <- FMsub_assoc1.
rewrite FMsub_comm1.
rewrite <- ?FMmul_assoc.
reflexivity.
Qed.
(*----------------------------8.61------------------*)
(* ERxkk 与噪声vk 无关 *)
(* ERxkk * T_v = 0 *)
(* v * T_ERxkk =0 *)
(*---------------------------------------------------*)
Axiom ERxkk1_Tv_eq0:ERxkk1 FM* T(vk) FM= FMO n m.
Axiom v_TERxkk1_eq0:vk FM* T(ERxkk1) FM= FMO m n.
(*----------------4.33--------8.62------------------------*)
(* Pkk = (I-KH) * Pkk1 * T_(I-KH) *)
(* + K * R * T_K *)
(*--------------------------------------------------------*)
Lemma Pkk_4_33: Pkk FM= (In FM- Kk FM* Hk) FM* Pkk1 FM* T(In FM- Kk FM* Hk) FM+ Kk FM* Rk FM* T(Kk).
Proof.
unfold Pkk,Pkk1,Rk.
rewrite ERxkk_mul_T_ERxkk_860.
rewrite FMmul_assoc with(middle:=ERxkk1)(right:=T( vk)).
rewrite FMmul_assoc with(middle:=vk)(right:=T( ERxkk1)).
rewrite ERxkk1_Tv_eq0,v_TERxkk1_eq0.
rewrite ?FMmul_zero_l.
rewrite ?FMmul_zero_r.
rewrite ?FMsub_zero_r.
rewrite ?FMmul_assoc.
reflexivity.
Qed.
Definition Pk1k1 := ERxk1k1 FM* T(ERxk1k1).
Definition Qk1 := wk1 FM* T(wk1).
(*----------------------------8.67------------------------------------------------------*)
(*Pkk = Pkk1 - Pkk1 * T(H) * T(K) - K * H * Pkk1 + K *(H * Pkk1 * T(H) + R) * T(K) *)
(*--------------------------------------------------------------------------------------*)
Lemma Pkk_867: Pkk FM= Pkk1 FM-
Pkk1 FM* T(Hk) FM* T(Kk) FM-
Kk FM* Hk FM* Pkk1 FM+
Kk FM* ( Hk FM* Pkk1 FM* T(Hk) FM+ Rk) FM* T(Kk).
Proof.
rewrite Pkk_4_33.
assert(T(In FM- Kk FM* Hk) FM= In FM- T(Hk) FM* T(Kk)).
{
rewrite FMteq_sub.
rewrite FMteq_mul.
apply mat_sub_compat'. 2: { reflexivity. }
unfold In.
unfold FMI.
unfold FMatrix.MMI.
apply trans_MI.
}
rewrite H.
rewrite ?FMmul_add_distr_r.
rewrite ?FMmul_add_distr_l.
rewrite ?FMmul_sub_distr_r.
rewrite ?FMmul_sub_distr_l.
rewrite <- ?FMmul_assoc.
rewrite ?FMmul_unit_l.
rewrite ?FMmul_unit_r.
rewrite <- FMsub_assoc1.
rewrite FMsub_comm1.
rewrite FMadd_assoc.
reflexivity.
Qed.
(*tr(Pkk) = tr(Pkk1) - tr(P*T_H*T_K) - tr(KHP) + tr(K(HPT_H)T_K) *)
Lemma tr_Pkk_867: Tr(Pkk) = Tr(Pkk1) -f
Tr(Pkk1 FM* T(Hk) FM* T(Kk)) -f
Tr(Kk FM* Hk FM* Pkk1) +f
Tr(Kk FM* ( Hk FM* Pkk1 FM* T(Hk) FM+ Rk) FM* T(Kk)) .
Proof.
assert(Tr(Pkk) = Tr(Pkk1 FM- Pkk1 FM* T(Hk) FM* T(Kk) FM-
Kk FM* Hk FM* Pkk1 FM+
Kk FM* ( Hk FM* Pkk1 FM* T(Hk) FM+ Rk) FM* T(Kk))).
{ apply FMat_tr_compat. apply Pkk_867. }
rewrite H.
rewrite FMat_tr_add.
rewrite ?FMat_tr_sub.
reflexivity.
Qed.
(* tr P*T_H*T_K = tr KHP*)
Lemma tr_KHP_eq: Tr(Pkk1 FM* T(Hk) FM* T(Kk)) = Tr(Kk FM* Hk FM* Pkk1) .
Proof.
assert(Pkk1 FM* T( Hk) FM* T( Kk) FM= T(Kk FM* Hk FM* Pkk1)).
{
rewrite ?FMteq_mul.
unfold Pkk1.
rewrite FMteq_mul.
rewrite FMtteq.
rewrite <-FMmul_assoc.
reflexivity.
}
unfold FMtr.
rewrite FMat_tr_trans with (Kk FM* Hk FM* Pkk1).
apply FMat_tr_compat.
apply H.
Qed.
(*tr(Pkk) = tr(Pkk1) -2* tr(KHP) + tr(K(HPT_H)T_K) *)
Lemma tr_Pkk_867_1 :Tr(Pkk) = Tr(Pkk1) -f
kmul 2 (Tr(Kk FM* Hk FM* Pkk1)) +f
Tr(Kk FM* ( Hk FM* Pkk1 FM* T(Hk) FM+ Rk) FM* T(Kk)) .
Proof.
rewrite tr_Pkk_867.
rewrite tr_KHP_eq.
unfold kmul .
rewrite fun_minus_assoc.
reflexivity.
Qed.
(*Deri(tr(Pkk))/Deri Kk = 2K(HPT(H)+R) -2*T(P)T(H) *)
Lemma Deri_Pkk_867: Deri ( Tr(Pkk) ) Kk FM=
2 kF* (Kk FM* ( Hk FM* Pkk1 FM* T(Hk) FM+ Rk)) FM-
2 kF* (T(Pkk1) FM* T(Hk)).
Proof.
rewrite FMat_Deri_compat with (mx':= (Pkk1 FM- Pkk1 FM* T(Hk) FM* T(Kk) FM-
Kk FM* Hk FM* Pkk1 FM+ Kk FM* ( Hk FM* Pkk1 FM* T(Hk) FM+ Rk) FM* T(Kk))).
2:{ apply Pkk_867. }
rewrite ?FMat_Deri_add.
rewrite ?FMat_Deri_sub.
rewrite FMat_Deri_not_correlation.
rewrite FMat_Deri_compat' with (x':=Tr(Kk FM* Hk FM* Pkk1)).
2:{ apply tr_KHP_eq. }
rewrite FMat_Deri_compat with (mx':= Kk FM* (Hk FM* Pkk1)).
rewrite FMat_Deri_mul.
2:{ apply FMmul_assoc. }
rewrite FMat_Deri_trans_eq.
2:{
unfold Rk.
unfold Pkk1.
rewrite FMteq_add.
rewrite ?FMteq_mul.
rewrite ?FMtteq.
rewrite <-FMmul_assoc.
reflexivity.
}
rewrite FMat_Deri_compat with (mx':= Kk FM* (Hk FM* Pkk1)).
2:{ apply FMmul_assoc. }
rewrite FMat_Deri_mul.
rewrite ?FMteq_mul.
rewrite FMadd_comm.
rewrite <- ?FMadd_assoc1.
rewrite FMadd_zero_r.
unfold FMtwoMat.
unfold FMatrix.twoMat.
unfold twoMat.
rewrite <- FMsub_assoc.
reflexivity.
Qed.
End Kalman_filtering.
Require Import Matrix.inv.
Section KF_implement.
Print pair.
Definition FMn1 := FMat n 1. (*x u w*)
Definition FMm1 := FMat m 1. (*v *)
Definition FMnn := FMat n n. (*phi gamma *)
Definition FMmn := FMat m n. (*H*)
Definition FMOn1 := FMO n 1.
Definition FMOm1 := FMO m 1.
Definition FMOnn := FMO n n.
Definition FMOmn := FMO m n.
Definition FMIn := FMI n.
Parameter inv : forall {n m :nat}, @Mat (R->R) n m->@Mat (R->R) m n.
Fixpoint KF_k (k:nat) (u_l w_l: list FMn1) (v_l z_l: list FMm1) (phi_l gamma_l: list FMnn)
(H_l: list FMmn) (x0: FMn1) (P0: FMnn):=
match k with
|O => (x0,P0)
|S n => let x' := fst (KF_k n u_l w_l v_l z_l phi_l gamma_l H_l x0 P0) in
let P' := snd (KF_k n u_l w_l v_l z_l phi_l gamma_l H_l x0 P0) in
let phi := nth n phi_l FMOnn in
let u := nth n u_l FMOn1 in
let gamma := nth n gamma_l FMOnn in
let w := nth n w_l FMOn1 in
let H := nth (S n) H_l FMOmn in
let v := nth (S n) v_l FMOm1 in
let z := nth (S n) z_l FMOm1 in
let x_predict := gamma FM* x' FM+ u in
let z_predict := H FM* x_predict in
let p_predict := phi FM* P' FM* T(phi) FM+ gamma FM* w FM* T(w) FM* T(gamma) in
let K := p_predict FM* T(H) FM* inv(H FM* p_predict FM* T(H) FM+ v FM* T(v)) in
let x := x_predict FM+ K FM* (z FM- z_predict) in
let P := (FMIn FM- K FM* H) FM* p_predict in
(x,P)
end.
End KF_implement.
From Coq Require Extraction.
Extraction "KF.ml" KF_k.
Fixpoint KF_k (k:nat) (phi_l u_l gamma_l w_l H_l v_l: list R) (x0 P0: R):=
match k with
|O => (x0,P0)
|S n => let x' := fst (KF_k n phi_l u_l gamma_l w_l H_l v_l x0 P0) in
let P' := snd (KF_k n phi_l u_l gamma_l w_l H_l v_l x0 P0) in
let phi := nth n phi_l 0%R in
let u := nth n u_l 0%R in
let gamma := nth n gamma_l 0%R in
let w := nth n w_l 0%R in
let H := nth (S n) H_l 0%R in
let v := nth (S n) v_l 0%R in
let z := H * x' + v in
let x_predict := gamma * x' + u in
let z_predict := H * x_predict in
let p_predict := phi * P' * phi + gamma * w * w * gamma in
let K := p_predict * H / (H * p_predict * H + v * v) in
let x := x_predict + K * (z - z_predict) in
let P := (1%R - K * H)* p_predict in
(x,P)
end.
Definition phi_l := [0;1;2;3].
Definition u_l := [0;1;2;3].
Definition gamma_l := [0;1;2;3].
Definition w_l := [0;1;2;3].
Definition v_l := [0;1;2;3].
Definition H_l := [0;1;2;3].
Compute KF_k 2 phi_l u_l gamma_l w_l H_l v_l 1 1.