Tracking Differentiator In The Application Of Signal Processing And Theory Research

Tracking differentiator in the application of signal processing andtheory researchMa Zhonghua Directed by Professor Li YongThe data fusion early applies to military, along with the development of computation science, its application has seeped to many domains. The target motion state estimation is the important norm of military command and decision-making for reference, also is the data fusion initial goal and important content.The existing target measure information processing method, including in 1960, the Kalman filter (KF) which Kalman proposed , as well as in 1975, Bar-shalom proposed probability data association filter (PDAF) and so on. All is established on the target motion state model and the measure noise statistics characteristic supposition. But in fact, the majority of the interested targets are uncooperative, so the establishment of the target motion model is very difficult. In other words, the practical application environment of the traditional data fusion is very difficult to be satisfied .In order to make up the above disappointment, the methods which suppose many target motion models appeared, including the multi-model filter (MMF) and the interacting multi-model filter (IMM), as well as the methods which calculate many joint probabilities, including joint probability data association filter (JPDAF) and multi-hypothesis tracker (MHT). And most other algorithms are based on the above algorithms or several algorithms fusion, essentially all do not have any change. In fact, the traditional method makes up the insufficiency which includes the target motion modelling and the measure noise statistics characteristic supposition with large computation quantity. When target mobility is quite high, the target motion model is difficult to be established. This urges people to seek some method, of which the computation quantity is smaller, and the target motion model and measure noise supposition is not requested.In military applications, it is needed to track target in three-dimensional space, which makes it necessary to deal with the multi-dimensional measure signal. Here we make radar-target tracking as an example to illustrate.The target motion state is given in cartesian coordinates, and the normal form target motion formula is given below,X = F(t,X),where X=(x,y,z),F:[0,+∞)×R3→R3. The radar measure signal is given in polar coordinates or spherical coordinates,r=r+ωr,θ=θ+ωθ,α=α+ωα,whereωr,ωθ,ωαare measure noise.If the coordinate origin translation is not considered, thenx=r cosθcosα, y=r cosθsinα, z=r sinθ.The problem is to estimate the target motion x,y,z with the measure r,θ,α.This article proposes the tracking differentiator (TD) which presented byHanJingqing, to use in the military target measure information processing. Tracking differentiator does not rely on the target motion model, also does not need the measure noise statistics characteristic supposition. It has the superiority which the traditional filter does not have .The so-called tracking differentiator is such an organization [38]: inputs signal v(t), it will output two signals z1 and z2, and z1 tracks v(t), z2 = z1, thus z2 is the "approximate differential " of v(t). The relationship of the input signal and the signal which obtained by the tracking differentiator is given by the theorem below.Theorem 1 If the solution ofis approximately stable on zero, where f is continue and satisfies Lipschitzcondition, |f(z11,z21)-f(z12,z22)| 0, the solution ofsatisfies2) x2(t) weakly converges to the" generalized derivative " of v(t).The above theorem showed, when R is large enough, x1(t) tracks v(t), and x2(t) is the generalized derivative of v(t) under weakly convergence significance. Therefore, x2(t) is approximate differential of v(t) . (1) is called the tracking differentiator, generally function f(x1,x2) takes nonlinear function, therefore it is also called the non-linear tracking differentiator. Based on the theoretical analysis of the tracking differentiator , the above theorem will be promoted to multi-dimensional as well as the non-linear measure situation, separately will be produced by the following three theorems.Suppose the system state variableu(t)=(u1(t),…,un(t))T,the svstem measure variabley(t)=(y1(t),…,yn(t))T,satisfies yi corresponding the measure ui(t). Theorem 2 If the solution ofsatisfies: z1(t)=(z11(t), z12(t),…, z1n(t))T→0n, z2(t)= (z21(t),z22(t),…, z2n(t))T→0n(t→∞),where (f1(0n,0n),f2(0n,0n),…,fn(0n,0n))T = 0n,On=(0,0,…,0)T,f(z1,z2)=(f1(z1,z2),f2(z1,z2),…,fn(z1, Z2))T is continue and satisfies Lipschitz condition, that is‖f(z11, z21)-f(z12,z22)‖ then for any y(t) = (y1(t),y2(t),…,yn(t))T, yi(t), i= 1,2,…, n is bounded integrable, and any constant T > 0, the solution ofThe above two theorems have goive the linear measure tracking differentiator (1) (2), we have the non-linear measure tracking differentiator through the two theorems below.Suppose system state variable u(t), measure variable y(t) = g(u(t)), g(·) continuously differential, and g'(·)≠0. When g(·) is nonlinear function, tracking the target motion through y(t) is very significant, because in the military, for example the radar-target tracking, all need to carry on the non-linear coordinates transformation.Theorem 3 If the solution of is approximately stable on zero, where f is continue and satisfies Lipschitz condition, |f(z11, z21)-f(z12, z22)| < L(|z11 - z12 + |z21-z22|), f(0,0)=0, then for any bounded integrable function u(t), the solution ofMulti-dimensional non-linear measure tracking differentiator is given through the theorem below.Suppose the system state variable u(t)=(u1(t),…,un(t))T, the measure variable y(t) = g(u(t))=(g1(u(t)),…,gn(u(t)))T, and det(g'(u(t)))≠0,Theorem 4 If the solution of satisfies: z1(t) = (z11(t), z12(t),…,Z1n(t))T→0n, z2(t) = (z21(t), z22(t),…, z2n(t))T→0n(t→∞,where (f1(0n,0n),f2(0n,0n),…,fn(0n, 0n))T =0n, 0n = (0,0,…,0)T, f(z1,z2) = (f1(z1,z2),f2(z1,z2),…,fn(z1,z2))T is continue and satisfies Lipschitz condition,‖f(z11, z21) - f(z12, z22)‖< L(‖z11 - z12‖+‖z21-z22‖)> then for any bounded integrable function u(t) and for any constant T, the solution ofsatisfieswhereThe system (1), and the systems we reduced in this paper (2) (3) (4) are all called tracking differentiator. The above theoretical results and simulation example all show the superiority of tracking differentiator applied in signal processing.In order to consider the precision problem of the tracking differentiator, we give some theoretic analysis and the time of tracking signal,wheres=sign(x10-v0+(|x20|x20)/2r)this result shows, any solution can track the input signal within finite time, and the time is decided by the tracked signal and the initial value.We give the continue dependence of tracked signal of the tracking differentiator through the following two theorems.Theorem 5 For systemif f(x1-v,x2) is continue, and satisfies Lipschitz condition,|f(x1- v,x2)-f(x'1- v',x'2)| < L(x1-v- (x'1 - v')| + |x2 - x'2|),where L is Lipschitz constant, then the solution of (5) x1(t,v) is continue with respect to v.In fact, the above theorem is the continue dependance on parameter of the solution, so we can have the conclusion through ordinary differential equation theory. If we use the same tracking differentiator tracking two signal v1(t), v2(t), the relation of the two solutions is given through the theorem below.Theorem 6 For any bounded integrable function v1(t) and v2(t), and two tracking differentiators which satisfy theorem 1, initial valueinitial valueif f satisfies Lipschitz condition, with Lipschitz constant L, then |x1(t) - y1(t)|≤N(h(|ξ1-μ1|,|ξ2-μ2)+superv),whereξi,μi, i = 1,2,N are constant, h is linear function of |ξ1-μ1|,|ξ2-μ2|, superv = lim sups∈[τ,t] |v1(s) - v2(s)|.The above theoretical analysis is based on continue function, but commonly, the tracking differentiator takes the formIn practice, the function fhan which is promoted by Han Jingqing is used, But it is not fitted for theoretical analysis for its complex form. In this paper, we consider the approximation function,f(k,x)=kx/1+k|x|andthen we get the tracking differentiator with continue right-hand side,where s = x1 + bx2.The theoretical proof of the relation between (7) and (6) is given by the theorem below.Theorem 7 If the solution of (6) is given according to the definition of Filippov, the limit of the vector field direction of (7) is given by the following formula,then the solution of (7) approximates the solution of (6), when k→∞.Next, We give the continuous dependence of the input signal for the solution of the tracking differentiator with discontinue right-hand side.where s =x1 -v(t) + |x2|x2/2r.Theorem 8 The solution of (8) is continue of input signal v, that is, for some given initial value (x1(0),x2(0)),we have (?)ε> 0,(?)δ> 0, s.t. when |v1 - v2| <δ,|x1(v1,x1(0),x2(0),t)-x1(v2,x1(0),x2(0),t)|<ε.Finally, we promote the self-stable region for uncertain systems control, and construct continual uncertain system control item from this structure. If the systemhas only one stable point at zero, f(0,0) = 0. h1(x1), h2(x1) is continue, and satisfyh1(x1)> 0,x1≤x11;h1(x1)<0,x1≥x11;h2(x1)<0,x1≥x12;h2(x1)>0,x1≤x12;h1(x1)>h2(x1),where x11 and x12 are finite values. The region G which is decided by x2 = h1(x1) and x2 = h2(x1) is given below,G={(x1,x2)|h1(x1)≥x2≥h2(x1)}.Theorem 9 If the function f(x1,x2) in (9) satisfiesorandf(x1, x2)≤0, when x1≥0, x2≥0, (x1, x2)∈G; f(x1,x2)≥0, when x1≤0,x2≤0, (x1, x2)∈G, the solution (x1(t),x2(t)) satisfies(x1(t),x2(t))∈G,t≥T,then we haveThe above theorem showed that, for the second order system which satisfies certain condition , there is region G, and all solutions in the region will converge to the stable point (0,0), and this region is the self-stable region. Prom this we construct system feedback control.The tracking differentiator took from auto-disturbance rejection control theory, has already received very good effect in many projects domain application, but the characteristic that it does not rely on the system state model and the measure noise statistics characteristic has not brought to people enough attention. This article applies it to the military target measure information processing, and has carried on the promotion to the tracking differentiator, has produced the track time, as well as signal fundamental research result, and so on continual dependence, finally has considered the uncertain systems control from self-stable region method. Hopes the work here can advance the further development of the tracking differentiator theory and broader application.