matlab.rar_MATLAB 实例_N4SID_matlab实例

% This demo describes the group of RECURSIVE (ON-LINE) % algorithms. The recursive M-files include: RPEM, RPLR, % RARMAX, RARX, ROE, and RBJ. These algorithms implement % all the recursive algorithms described in Chapter 11 of % Ljung(1987). % % RPEM is the general Recursive Prediction Error Algorithm % for arbitrary multiple-input-single-output models % (the same models as PEM works for). % % PRLR is the general Recursive PseudoLinear Regression method % for the same family of models. % % RARX is a more efficient version of RPEM (and RPLR) for the % ARX-case. % % ROE, RARMAX and RBJ are more efficient versions of RPEM for % the OE, ARMAX, and BJ cases (compare these functions to the % off-line methods). % ADAPTATION MECHANISMS: Each of the algorithms implement the % four most common adaptation principles: % % KALMAN FILTER approach: The true parameters are supposed to % vary like a random walk with incremental covariance matrix % R1. % % FORGETTING FACTOR approach: Old measurements are discounted % exponentially. The base of the decay is the forgetting factor % lambda. % % GRADIENT method: The update step is taken as a gradient step % of length gamma (th_new=th_old + gamma*psi*epsilon). % % NORMALIZED GRADIENT method: As above, but gamma is replaced by % gamma/(psi'*psi). The Gradient methods are also % known as LMS (least mean squares) for the ARX case. % Let's pick a model and generate some input-output data: u = sign(randn(50,1)); e = 0.2*randn(50,1); th0 = idpoly([1 -1.5 0.7],[0 1 0.5],[1 -1 0.2]); y = sim(th0,[u e]); z = iddata(y,u); figure, plot(z) % Press a key to see data. % First we build an Output-Error model of the data we just % plotted. Use a second order model with one delay, and apply % the forgetting factor algorithm with lambda = 0.98: thm1 = roe(z,[2 2 1],'ff',0.98); % It may take a while .... % The four parameters can now be plotted as functions of time. figure, plot(thm1), title('Estimated parameters')% Press a key to see plot. % The true values are as follows: (Press a key for plot) hold on, plot(ones(50,1)*[1 0.5 -1.5 0.7]), title('Estimated parameters and true values') hold off % Now let's try a second order ARMAX model, using the RPLR % approach (i.e ELS) with Kalman filter adaptation, assuming % a parameter variance of 0.001: thm2 = rplr(z,[2 2 2 0 0 1],'kf',0.001*eye(6)); figure,plot(thm2), title('Estimated parameters') axis([0 50 -2 2]) % The true values are as follows: (Press any key for plot) hold on, plot(ones(50,1)*[-1.5 0.7 1 0.5 -1 0.2]), title('Estimated parameters and true values') hold off % So far we have assumed that all data are available at once. % We are thus studying the variability of the system rather % than doing real on-line calculations. The algorithms are % also prepared for such applications, but they must then % store more update information. The conceptual update then % becomes: % 1. Wait for measurements y and u. % 2. Update: [th,yh,p,phi] = rarx([y u],[na nb nk],'ff',0.98,th',p,phi) % 3. Use th for whatever on-line application required. % 4. Go to 1. % Thus the previous estimate th is fed back into the algorithm % along with the previous value of the "P-matrix" and the data % vector phi. % We now do an example of this where we plot just the current % value of th. The code is as follows: figure, [th,yh,p,phi] = rarx(z(1,:),[2 2 1],'ff',0.98); for k = 2:50 [th,yh,p,phi] = rarx(z(k,:),[2 2 1],'ff',0.98,th',p,phi); plot(k,th(1),'*',k,th(2),'+',k,th(3),'o',k,th(4),'*'),hold on end hold off % SEGMENTATION OF DATA % % The command SEGMENT segments data that are generated from % systems that may undergo abrupt changes. Typical applications % for data segmentation are segmentation of speech signals (each % segment corresponds to a phonem), failure detection (the segments % correspond to operation with and without failures) and estimating % different working modes of a system. We shall study a system % whose time delay changes from two to one. load iddemo6m.mat z = iddata(z(:,1),z(:,2)); % First, take a look at the data: figure, idplot(z) % Press any key for plot. % The change takes place at sample number 20, but this is not % so easy to see. % % We would like to estimate the system as an ARX-structure model % with one a-parameter, two b-parameters and one delay: % y(t) + a*y(t-1) = b1*u(t-1) + b2*u(t-2) % The information to be given is the data, the model orders % and a guess of the variance (r2) of the noise that affects % the system. If this is entirely unknown, it can be estimated % automatically. Here we set it to 0.1: nn = [1 2 1]; [seg,v,tvmod] = segment(z,nn,0.1); % Let's take a look at the segmented model. Blue(yellow) line % is for the a-parameter. Green(magneta) is for b1 and Red(cyan) % for the parameter b2. figure,plot(seg) % Press any key for plot. hold on % We see clearly the jump around sample number 19. b1 goes from % 0 to 1 and b2 vice versa, which shows the change of the % delay. The true values can also be shown: plot(pars) % Press any key for plot. hold off % The method for segmentation is based on AFMM (adaptive % forgetting through multiple models), Andersson, Int. J. % Control Nov 1985. A multi-model approach is used in a first % step to track the time varying system. The resulting tracking % model could be of interest in its own right, and are given by % the third output argument of SEGMENT (tvmod in our case). They % look as follows: figure, plot(tvmod) % Press any key for plot. % The SEGMENT M-file is thus an alternative to the recursive % algorithms RPEM, RARX etc for tracking time varying systems. % It is particularly suited for systems that may change rapidly. % From the tracking model, SEGMENT estimates the time points when % jumps have occurred, and constructs the segmented model by a % smoothing procedure over the tracking model. % The two most important "knobs" for the algorithm are r2, as % mentioned before, and the guessed probability of jumps, q, the % fourth input argument to SEGMENT. The smaller r2 and the larger % q, the more willing SEGMENT will be to indicate segmentation % points. In an off line situation, the user will have to try a % couple of choices (r2 is usually more sensitive than q). The % second output argument to SEGMENT, v, is the loss function for % the segmented model (i.e. the estimated prediction error variance % for the segmented model). A goal will be to minimize this value. % OBJECT DETECTION IN LASER RANGE DATA % % The reflected signal from a laser (or radar) beam contains % information about the distance to the reflecting object. The % signals can be quite noisy. The presence of objects affects % both the distance information and the correlation between % neighbouring points. (A smooth object increases the % correlation between nearby points.) % % In the following we study some quite noisy laser range data. % They are obtained by one horizontal sweep, like one line on % a TV-screen. The value is the distance to the reflecting object. % We happen to know that an object of interest hides between % sample numbers 17 and 48. figure, plot(hline) % Press any key for plot. % The eye is