CN102540904B - Crystallizer auto regression with extra inputs (ARX) model identification method based on recursive instrumental variable (RIV) - Google Patents

Abstract

The invention provides a crystallizer auto regression with extra inputs (ARX) model identification method based on a recursive instrumental variable (RIV). The method specifically comprises the following steps of: taking the aperture of an oil cylinder valve of a crystallizer as an input (u) and the position of the crystallizer as an output (y); constructing the least square and an index function of a crystallizer ARX model on the basis of sampled data; resolving the least square and the index function by using a quick response (QR) resolving method to acquire unknown parameters of the ARX model without considering colored noise interference; filtering the aperture (u) of the oil cylinder valve of the crystallizer by using the parameters acquired by using the QR resolving method, and acquiring a middle instrumental variable (x); acquiring iterative variables Pk and Lk by using the instrumental variables (x) and (y); and calculating the unknown parameters of the model by gradually iterating Pk and Lk. According to the method, the global optimal solution of the unknown parameters of the crystallizer ARX model can be approximated accurately; updated values of the parameters of the model can be acquired by simply calculating the current sampled input and output data; and when the system of the crystallizer is changed, the parameters of the model can be adjusted timely.

Description

Crystallizer ARX identification Method based on recursion method of instrumental variable RIV

Technical field

The present invention relates to conticaster crystallizer Control System Design field in iron and steel metallurgical industry, relate in particular to a kind of crystallizer ARX (Auto Regression with eXtra inputs) identification Method based on recursion method of instrumental variable RIV (recursive instrumental variable).

Background technology

Mold oscillation has direct, important impact to the strand demoulding and surface quality; in the actual casting cycle of sheet billet continuous casting; pulling rate is normally along with the variation of working condition (as cast temperature) changes; for guaranteeing to obtain good strand stripping result and strand table quality; should guarantee that under the basicly stable prerequisite of vibratory process parameter, suitably adjust frequency, amplitude etc. vibrated basic parameter.But, obtain good frequency, amplitude control effect, crystallizer control system that must be reasonable in design is with quick, accurate tracking frequencies, amplitude set-point, and outstanding control system is carried out system analysis and design take model as basis, in view of the PID controller design method of current crystallizer control system based on experience, be necessary first crystallizer to be carried out to Model Distinguish, on rational model basis, carry out again Control System Design to obtain good control effect.Because the impact of coloured noise is not considered in traditional ARX Model Distinguish, so proposing a kind of method of utilizing iteration instrumental variable, the present invention carries out ARX Model Distinguish, not only can obtain coloured noise and disturb lower rational model, but also can calculate online, there is very important using value for the crystallizer on-line identification in engineering reality and adjustment.

Summary of the invention

Technical matters to be solved by this invention is: a kind of crystallizer ARX identification Method based on recursion method of instrumental variable is provided, the method provides quick, easy method for on-line identification crystallizer coloured noise interference model, and the crystallizer control system good for design performance provides scientific and rational mathematical model.

The present invention solves its technical matters and adopts following technical scheme:

Crystallizer ARX identification Method based on recursion method of instrumental variable RIV provided by the invention, specifically: take crystallizer oil cylinder valve aperture as input u, be output y take crystallizer position, on sampled data basis, set up crystallizer ARX model least square and target function, first utilize QR decomposition method to decompose to obtain to least square and target function the ARX unknown-model parameter of not considering under coloured noise interference, the parameter that recycling QR decomposition method obtains is carried out filtering to obtain intermediate tool variable x to crystallizer oil cylinder valve aperture u, then utilize instrumental variable x and y to obtain iteration variable Pk and Lk, by Pk and progressively iterative computation unknown-model parameter of Lk.

The above-mentioned crystallizer ARX identification Method based on recursion method of instrumental variable RIV provided by the invention, concrete steps comprise:

(1) Gather and input output data, take crystallizer oil cylinder valve aperture as input u (t), take crystallizer position as output y (t) gathers N to data sample Z ⁿ;

(2) the ARX model building under the interference of crystallizer white noise is A (q) y (t)=B (q) u (t)+e (t), wherein A (q)=1+a ₁q ^-1+ a ₂q ^-2+ L+a _naq ^-na, B (q)=b ₁q ^-1+ b ₂q ^-2+ L+b _nbq ^-nb, q ^-1for backward mobile operator, q is forward movement operator, and na, nb are arithmetic number, and e (t) is white Gaussian noise, and accompanying drawing 1 is ARX modular concept figure;

(3) make θ=[a ₁a ₂l a _nab ₁b ₂l b _nb] be ARX model parameter to be identified;

(4) order

for the model prediction of output value based on parameter θ, wherein prediction expression is:

In formula:

(5) order with the objective function of the ARX Model Distinguish process of white Gaussian noise is:

(6) obtain estimates of parameters for the objective function utilization in step (5) based on QR decomposition method

(7) by the parameter obtaining in step (6)

front na element assignment is to a ₁, a ₂, L, a _na, rear nb element assignment is to b ₁, b ₂, L, b _nb, build:

A(q)＝1+a ₁q ^-1+a ₂q ^-2+L+a _naq ^-na，B(q)＝b ₁q ^-1+b ₂q ^-2+L+b _nbq ^-nb；

(8) order

build variable x;

(9) make ζ (t)=[x (t-1)-x (t-2) L-x (t-na) u (t-1) u (t-2) L u (t-nb)] ^t, build intermediate variable ζ;

(10) calculate

require R (t ₀) reversible;

(11) calculate

(12) make P (t ₀)=R (t ₀) ^-1,

(13) order

(14) unknown parameter iterative formula is

Through above-mentioned steps, realize the crystallizer ARX Model Distinguish based on recursion method of instrumental variable RIV.

In above-mentioned steps (10), calculate R (t ₀) time require R (t ₀) reversible.

The present invention compared with prior art has advantages of following main:

One. accurately approach crystallizer ARX unknown-model parameter global optimum solution;

They are two years old. and can utilize current sampling input and output data to carry out simple computation and can obtain model parameter renewal value;

They are three years old. adjustment model parameter in time in the time that crystallizer system changes.

Accompanying drawing explanation

Fig. 1 is ARX model structure schematic diagram.

Fig. 2 is iterative instrumental variable method RIV process flow diagram.

Fig. 3 is that in embodiment 1, crystallizer ARX model obtains the comparison diagram between systematic parameter prediction output valve and actual samples data based on IV and RIV algorithm.

Embodiment

Crystallizer ARX identification Method based on recursion method of instrumental variable RIV provided by the invention, specifically: take crystallizer oil cylinder valve aperture as input u, be output y take crystallizer position, on sampled data basis, set up crystallizer ARX model least square and target function, first utilize QR decomposition method to decompose to obtain to least square and target function the ARX unknown-model parameter of not considering under coloured noise interference, the parameter that recycling QR decomposition method obtains is carried out filtering to obtain intermediate tool variable x to crystallizer oil cylinder valve aperture u, then utilize instrumental variable x and y to obtain iteration variable Pk and Lk, by Pk and progressively iterative computation unknown-model parameter of Lk.

The above-mentioned crystallizer ARX identification Method based on recursion method of instrumental variable RIV provided by the invention, referring to Fig. 1 and Fig. 2, comprises the following steps:

(1) Gather and input output data, take crystallizer oil cylinder valve aperture as input u (t), take crystallizer position as output y (t) gathers N to data sample Z ⁿ;

(2) the ARX model building under the interference of crystallizer white noise is A (q) y (t)=B (q) u (t)+e (t), wherein A (q)=1+a ₁q ^-1+ a ₂q ^-2+ L+a _naq ^-na, B (q)=b ₁q ^-1+ b ₂q ^-2+ L+b _nbq ^-nb, q ^-1for backward mobile operator, q is forward movement operator, and na, nb are arithmetic number, and e (t) is white Gaussian noise, and accompanying drawing 1 is ARX modular concept figure;

(3) make θ=[a ₁a ₂l a _nab ₁b ₂l b _nb] be ARX model parameter to be identified;

(4) order for the model prediction of output value based on parameter θ, wherein prediction expression is:

In formula:

(5) order with the objective function of the ARX Model Distinguish process of white Gaussian noise is:

(6) obtain estimates of parameters for the objective function utilization in step (5) based on QR decomposition method

(7) by the parameter obtaining in step (6)

front na element assignment is to a ₁, a ₂, L, a _na, rear nb element assignment is to b ₁, b ₂, L, b _nb, build:

A(q)＝1+a ₁q ^-1+a ₂q ^-2+L+a _naq ^-na，B(q)＝b ₁q ^-1+b ₂q ^-2+L+b _nbq ^-nb；

(8) order

build variable x;

(9) make ζ (t)=[x (t-1)-x (t-2) L-x (t-na) u (t-1) u (t-2) L u (t-nb)] ^t, build intermediate variable ζ;

(10) calculate

require R (t ₀) reversible;

(11) calculate

(12) make P (t ₀)=R (t ₀) ^-1,

(13) order

(14) unknown parameter iterative formula is:

Through above-mentioned steps, realize the crystallizer ARX Model Distinguish based on recursion method of instrumental variable RIV.

Below in conjunction with concrete application example, the invention described above method is described further, but does not limit the present invention.

Embodiment 1:

Certain steel mill's one slab caster mould sampled data is as shown in table 1, its sampling time interval Ts=0.003 second, number of data points N=250.

Select 2 rank ARX crystallizer models, make A (q)=1+a ₁q ^-1+ a ₂q ^-2, B (q)=b ₁q ^-1+ b ₂q ^-2, the parameter to be identified under white Gaussian noise disturbed condition is: ${\widehat{\mathrm{\θ}}}_0=\left[\begin{array}{cccc}{a}_1& a_2& b_1& b_2\end{array}\right];$

Can obtain according to QR decomposition method: ${\widehat{\mathrm{\θ}}}_0=\left[\begin{array}{cccc}-1.336872& 0.336427& 0.003197& 0.003147\end{array}\right];$

Can obtain according to the invention described above method step (7):

A(q)＝1-1.336872q ^-1+0.336427q ^-2，B(q)＝0.003197q ^-1+0.003147q ^-2。

Can obtain according to the invention described above method step (8)-(12):

$R\left(t_0\right)={10}^3\×\left[\begin{array}{cccc}0.021252006240668& 0.020963325099721& -0.028565022623149& -0.041817680533719\\ 0.015612524222447& 0.015402809149859& -0.020934782237607& -0.025973181784350\\ -0.426076927481673& -0.420217256948533& 0.574291815319505& 0.990848318866577\\ -0.727951029790948& -0.717553334782507& 0.990848318866577& 2.622103985444999\end{array}\right],$

The inverse matrix of R is:

$R{\left(t_0\right)}^{-1}={10}^4\×\left[\begin{array}{cccc}-1.923555180660084& 1.031997592383443& -0.065414677428459& 0.004264387779973\\ 1.067797283356591& -0.508316117355490& 0.039904834233594& -0.003085087094074\\ -0.656826446001522& 0.400543500059693& -0.019657256101700& 0.000920554573693\\ 0.006392922181487& -0.003957916520287& 0.000187835511254& -0.000008191516612\end{array}\right]$

F(t ₀)＝10 ²×[-0.207620397431495 -0.152121982791586 4.176839631176190 7.196773492811511] ^T，

Initial parameter value is:

${\widehat{\mathrm{\θ}}}_{t_0}={10}^2\×\left[\begin{array}{cccc}-1.570131052662982& 1.029793910169938& -0.410872321303568& 0.003913826480786\end{array}\right],$

Can obtain the final estimated value of parameter according to the invention described above method (13)-(14) is:

${\widehat{\mathrm{\θ}}}_t=\left[\begin{array}{cccc}-1.227665234724697& 0.223595516873930& 0.009100274852882& -0.001632011009896\end{array}\right];$

The final estimated value of above-mentioned parameter is the estimates of parameters of the crystallizer ARX model based on recursion method of instrumental variable RIV.

Fig. 3 is model prediction output and the actual correlation curve of exporting between sampled data that adopts IV method and the identification of RIV method to obtain.From Fig. 3, can find that RIV method there will be sustained deviation in part in the time that system exporting change is larger, and IV rule is all vibrated around real output value left and right at whole output interval, prediction effect is better compared with RIV method, but RIV is simple owing to calculating, and can carry out online System Discrimination, therefore RIV is more suitable for the on-line identification application in engineering reality.

Above embodiment is only for calculating thought of the present invention and feature are described, its object is to make those skilled in the art can understand content of the present invention and implement according to this, and protection scope of the present invention is not limited to above-described embodiment.So the disclosed principle of all foundations, equivalent variations or the modification that mentality of designing is done, all within protection scope of the present invention.Subordinate list

Crystallizer sample data in table 1 embodiment 1

Sequence number	1	2	3	4	5	6	7	8	9	10
											Input	46.875	12.02619	11.91768	11.75492	12.22512	11.91406	1190683	1187066	11.54876	11.61386
Output	8.583912	8.715567	8.860388	8.91305	9.071036	9.189525	9308015	9466001	9.571325	9.663484
											Sequence number	11	12	13	14	15	16	17	18	19	20
Input	11.71152	11.2594	11.13643	10.78559	10.61198	10.21412	1004413	1001157	9.939236	9.595631
											Output	9.834635	9.953125	10.11111	10.2296	10.38759	10.50608	1059823	1070356	10.83521	10.96687
Sequence number	21	22	23	24	25	26	27	28	29	30
											Input	9.255642	9.320747	8.915654	8.814381	8.977141	8.289931	8.289931	7.97526	7.722078	7.515914
Output	11.0327	11.17752	11.25651	11.30917	11.46716	11.53299	11.63831	11.73047	11.80946	11.88845
											Sequence number	31	32	33	34	35	36	37	38	39	40
Input	7.309751	7.143374	6.71658	6.5068	6.072772	5.703848	5.645978	5.298756	4.774306	4.481337
											Output	11.95428	12.05961	12.12543	12.21759	12.29659	12.33608	12.40191	12.49407	12.54673	12.62572
Sequence number	41	42	43	44	45	46	47	48	49	50
											Input	4.000289	3.642216	3.096065	2.871817	2.654803	2.177373	1.884404	1.381655	0.907841	0.719763
Output	12.67839	12.75738	12.78371	12.81004	12.8627	12.88903	12.9417	12.98119	12.98119	13.00752
											Sequence number	51	52	53	54	55	56	57	58	59	60
Input	-0.24233	-1.54803	-3.35286	-5.12514	-6.89742	-8.87948	-10.5288	-12.5977	-14.5616	-16.2218
											Output	12.99436	13.03385	13.04702	13.04702	13.06018	13.00752	13.00752	12.98119	12.9022	12.83637
Sequence number	61	62	63	64	65	66	67	68	69	70
											Input	-18.0194	-19.4734	-21.0576	-22.2186	-23.1156	-24.66	-25.6402	-26.3853	-27.2244	-27.4993
Output	12.71788	12.61256	12.44141	12.24392	12.1386	11.95428	11.74363	11.55932	11.29601	11.07219
											Sequence number	71	72	73	74	75	76	77	78	79	80
Input	-27.8827	-27.8501	-28.3095	-28.4252	-28.2661	-27.7416	-27.0906	-26.4685	-25.3328	-24.5913
											Output	10.79572	10.6114	10.38759	10.15061	9.874132	9.610822	9.360677	9.071036	8.860388	8.583912
Sequence number	81	82	83	84	85	86	87	88	89	90
											Input	-23.2458	-21.8967	-20.3631	-18.6921	-16.6667	-14.7931	-13.2198	-10.8579	-8.65885	-6.30064
Output	8.333767	8.083623	7.833478	7.570168	7.346354	7.188368	6.951389	6.753906	6.556423	6.411603
											Sequence number	91	92	93	94	95	96	97	98	99	100

Input	-4.07624	-2.48119	-1.37442	-0.40509	0.596788	1.312934	1.902488	2.470341	2.933304	3.504774
											Output	6.240451	6.121962	6.029803	5.937645	5.884983	5.858652	5.83232	5.83232	5.819155	5.819155
Sequence number	101	102	103	104	105	106	107	108	109	110
											Input	3.978588	4.466869	5.038339	5.349392	5.78342	6.047454	6.394676	6.940828	7.143374	7.273582
Output	5.819155	5.819155	5.858652	5.871817	5.924479	5.963976	5.977141	6.042969	6.121962	6.174624
											Sequence number	111	112	113	114	115	116	117	118	119	120
Input	7.606337	7.671441	8.036748	8.062066	8.445457	8.658854	8.626302	8.969907	9.1182	9.197772
											Output	6.266782	6.319444	6.424768	6.47743	6.556423	6.674913	6.740741	6.832899	6.938223	6.990885
Sequence number	121	122	123	124	125	126	127	128	129	130
											Input	9.693287	9.671586	9.982639	10.09115	10.01157	10.4239	10.44922	11.04601	11.09303	11.2377
Output	7.109375	7.188368	7.293692	7.425347	7.491175	7.609664	7.649161	7.76765	7.872974	7.951967
											Sequence number	131	132	133	134	135	136	137	138	139	140
Input	11.54514	11.48727	11.79832	11.76939	11.95747	12.18533	12.01895	12.01895	12.0298	12.04789
											Output	8.083623	8.162616	8.294271	8.386429	8.478588	8.623408	8.741898	8.860388	8.978877	9.110532
Sequence number	141	142	143	144	145	146	147	148	149	150
											Input	11.93215	11.85619	11.78024	11.83449	11.38238	11.216	10.84708	10.93388	10.88686	10.60113
Output	9.229022	9.360677	9.466001	9.637153	9.768808	9.926794	10.01895	10.12428	10.2691	10.38759
											Sequence number	151	152	153	154	155	156	157	158	159	160
Input	10.43475	10.05498	9.805411	9.733073	9.443721	9.42202	9.190538	8.846933	8.922888	8.470775
											Output	10.53241	10.66406	10.75622	10.88788	10.96687	11.08536	11.20385	11.26968	11.40133	11.46716
Sequence number	161	162	163	164	165	166	167	168	169	170
											Input	8.449074	8.098235	8.083767	7.826968	7.273582	7.24103	6.872106	6.720197	6.579138	6.061921
Output	11.58565	11.63831	11.73047	11.86212	11.91479	12.00694	12.07277	12.12543	12.23076	12.28342
											Sequence number	171	172	173	174	175	176	177	178	179	180
Input	5.877459	5.389178	5.005787	4.680266	4.134115	3.870081	3.43967	3.002025	2.672888	2.267795
											Output	12.37558	12.45457	12.50723	12.59939	12.65205	12.70472	12.77054	12.81004	12.8627	12.88903
Sequence number	181	182	183	184	185	186	187	188	189	190
											Input	1.974826	1.548032	1.063368	0.831887	0.57147	-0.63657	-1.84823	-3.67115	-5.26982	-7.18678
Output	12.92853	12.98119	12.98119	12.99436	13.03385	13.00752	13.04702	13.03385	13.04702	13.04702
											Sequence number	191	192	193	194	195	196	197	198	199	200
Input	-9.08203	-10.7964	-12.8906	-14.5942	-16.4605	-17.9905	-19.7085	-21.3252	-22.3307	-23.6256
											Output	13.00752	13.00752	12.9417	12.88903	12.78371	12.70472	12.59939	12.41507	12.27025	12.08594
Sequence number	201	202	203	204	205	206	207	208	209	210
											Input	-24.66	-25.293	-26.3346	-26.8012	-27.7742	-28.1648	-28.1829	-28.125	-28.0852	-27.8067
Output	11.86212	11.69097	11.45399	11.29601	11.07219	10.80888	10.54557	10.30859	10.05845	9.900463
											Sequence number	211	212	213	214	215	216	217	218	219	220
Input	-28.125	-27.2569	-26.5878	-25.4376	-24.3634	-22.9167	-21.394	-20.5548	-18.8368	-16.6522
											Output	9.610822	9.360677	9.071036	8.807726	8.53125	8.254774	8.109954	7.859809	7.570168	7.346354
Sequence number	221	222	223	224	225	226	227	228	229	230
											Input	-14.7678	-12.5651	-10.4637	-8.0693	-6.29702	-3.94965	-2.27865	-1.24421	-0.18446	0.499132
Output	7.109375	6.911892	6.688079	6.582754	6.411603	6.227286	6.121962	6.016638	5.963976	5.911314
											Sequence number	231	232	233	234	235	236	237	238	239	240
Input	1.240596	1.945891	2.452257	3.088831	3.653067	3.880932	4.58261	4.95515	5.414497	5.996817
											Output	5.871817	5.858652	5.83232	5.819155	5.858652	5.83232	5.858652	5.871817	5.871817	5.924479
Sequence number	241	242	243	244	245	246	247	248	249	250
											Input	6.268084	6.604456	6.77445	7.204861	7.526765	7.642506	8.018663	8.025897	8.098235	8.532263
Output	5.963976	6.029803	6.0693	6.121962	6.200955	6.253617	6.358941	6.451099	6.503761	6.609086

Claims (2)

1. the crystallizer ARX identification Method based on recursion method of instrumental variable RIV, it is characterized in that take crystallizer oil cylinder valve aperture as input u, be output y take crystallizer position, on sampled data basis, set up crystallizer ARX model least square and target function, first utilize QR decomposition method to decompose to obtain to least square and target function the ARX unknown-model parameter of not considering under coloured noise interference, the parameter that recycling QR decomposition method obtains is carried out filtering to obtain intermediate tool variable x to crystallizer oil cylinder valve aperture u, then utilize instrumental variable x and y to obtain iteration variable Pk and Lk, by Pk and progressively iterative computation unknown-model parameter of Lk,

The method comprises the following steps:

(1) Gather and input output data, take crystallizer oil cylinder valve aperture as input u (t), take crystallizer position as output y (t) gathers N to data sample Z ⁿ;

(2) the ARX model building under the interference of crystallizer white noise is A (q) y (t)=B (q) u (t)+e (t), wherein A (q)=1+a ₁q ^-1+ a ₂q ^-2+ ... + a _naq ^-na, B (q)=b ₁q ^-1+ b ₂q ^-2+ ... + b _nbq ^-nb, q ^-1for backward mobile operator, q is forward movement operator, and na, nb are arithmetic number, and e (t) is white Gaussian noise;

(3) make θ=[a ₁a ₂... a _nab ₁b ₂... b _nb] be ARX model parameter to be identified;

(4) order

for the model prediction of output value based on parameter θ, wherein prediction expression is:

In formula:

(5) order with the objective function of the ARX Model Distinguish process of white Gaussian noise is:

(6) obtain estimates of parameters for the objective function utilization in step (5) based on QR decomposition method

(7) by the parameter obtaining in step (6) front na element assignment is to a ₁, a ₂..., a _na, rear nb element assignment is to b ₁, b ₂..., b _nb, build:

A(q)＝1+a ₁q ^-1+a ₂q ^-2+…+a _naq ^-na，B(q)＝b ₁q ^-1+b ₂q ^-2+…+b _nbq ^-nb；

(8) order $x\left(t\right)=\frac{B\left(q\right)}{A\left(q\right)}u\left(t\right),$ Build variable x;

(9) make ζ (t)=[x (t-1)-x (t-2) ...-x (t-na) u (t-1) u (t-2) ... u (t-nb)] ^t, build intermediate variable ζ;

(10) calculate

(11) calculate

(12) make P (t ₀)=R (t ₀) ^-1,

(13) order

(14) unknown parameter iterative formula is:

Through above-mentioned steps, realize the crystallizer ARX Model Distinguish based on recursion method of instrumental variable RIV.

2. crystallizer ARX identification Method according to claim 1, is characterized in that in step (10), calculates R (t ₀) time require R (t ₀) reversible.

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