8 GENERAL FORM FOR LINEAR TIME-INVARIANT SYSTEM 8.1 TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME 8.1.1 Introduction In Section 1.1 we defined the target dynamics model for target having a constant velocity; see (1.1-1). A constant-velocity target is one whose trajectory can be expressed by a polynomial of degree 1 in time, that is, d ¼ 1, in (5.9-1). (In turn, the tracking filter need only be of degree 1, i.e., m ¼ 1.) Alternately, it is a target for which the first derivative of its position versus time is a constant. In Section 2.4 we rewrote the target dynamics model in matrix form using the transition matrix È; see (2.4-1), (2.4-1a), and (2.4-1b). In Section 1.3 we gave the target dynamics model for a constant accelerating target, that is, a target whose trajectory follows a polynomial of degree 2 so that d ¼ 2; see (1.3-1). We saw that this target also can be alternatively expressed in terms of the transition equation as given by (2.4-1) with the state vector by (5.4-1) for m ¼ 2 and the transition matrix by (5.4-7); see also (2.9-9). In general, a target whose dynamics are described exactly by a dth-degree polynomial given by (5.9-1) can also have its target dynamics expressed by (2.4-1), which we repeat here for convenience: X nþ1 ¼ ÈX n where the state vector X n is now defined by (5.4-1) with m replaced by d and the transition matrix is a generalized form of (5.4-7). Note that in this text d represents the true degree of the target dynamics while m is the degree used by 252 Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright # 1998 John Wiley & Sons, Inc. ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic) the tracking filter to approximate the target dynamics. For the nonlinear dynamics model case, discussed briefly in Section 5.11 when considering the tracking of a satellite, d is the degree of the polynomial that approximates the elliptical motion of the satellite to negligible error. We shall now give three ways to derive the transition matrix of a target whose dynamics are described by an arbitrary degree polynomial. In the process we give three different methods for describing the target dynamics for a target whose motion is given by a polynomial. 8.1.2 Linear Constant-Coefficient Differential Equation Assume that the target dynamics is described exactly by the dth-degree polynomial given by (5.9-1). Then its dth derivative equals a constant, that is, D d xðtÞ¼const ð8:1-1Þ while its ðd þ 1Þth derivative equals zero, that is, D dþ1 xðtÞ¼0 ð8:1-2Þ As a result the class of all targets described by polynomials of degree d are also described by the simple linear constant-coefficient differential equation given by (8.1-2). Given (8.1-1) or (8.1-2) it is a straightforward manner to obtain the target dynamics model form given by (1.1-1) or (2.4-1) to (2.4-1b) for the case where d ¼ 1. Specifically, from (8.1-1) it follows that for this d ¼ 1 case DxðtÞ¼ _ xðtÞ¼const ð8:1-3Þ Thus _ x nþ1 ¼ _ x n ð8:1-4Þ Integrating this last equation yields x nþ1 ¼ x n þ T _ x n ð8:1-5Þ Equations (8.1-4) and (8.1-5) are the target dynamics equations for the constant-velocity target given by (1.1-1). Putting the above two equations in matrix form yields (2.4-1) with the transition matrix È given by (2.4-1b), the desired result. In a similar manner, starting with (8.1-1), one can derive the form of the target dynamics for d ¼ 2 given by (1.3-1) with, in turn, È given by (5.4-7). Thus for a target whose dynamics are given by a polynomial of degree d, it is possible to obtain from the differential equation form for the target dynamics given by (8.1-1) or (8.1-2), the transition matrix È by integration. TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME 253 8.1.3 Constant-Coefficient Linear Differential Vector Equation for State Vector X(t) A second method for obtaining the transition matrix È will now be developed. As indicated above, in general, a target for which D d xðtÞ¼const ð8:1-6Þ can be expressed by X nþ1 ¼ ÈX n ð8:1-7Þ Assume a target described exactly by a polynomial of degree 2, that is, d ¼ 2. Its continuous state vector can be written as XðtÞ¼ xðtÞ _ xðtÞ  xðtÞ 2 4 3 5 ¼ xðtÞ DxðtÞ D 2 xðtÞ 2 4 3 5 ð8:1-8Þ It is easily seen that this state vector satisfies the following constant-coefficient linear differential vector equation: DxðtÞ D 2 xðtÞ D 3 xðtÞ 2 4 3 5 ¼ 010 001 000 2 4 3 5 xðtÞ DxðtÞ D 2 xðtÞ 2 4 3 5 ð8:1-9Þ or d dt XðtÞ¼AXðtÞð8:1-10Þ where A ¼ 010 001 000 2 4 3 5 ð8:1-10aÞ The constant-coefficient linear differential vector equation given by (8.1-9), or more generally by (8.1-10), is a very useful form that is often used in the literature to describe the target dynamics of a time-invariant linear system. As shown in the next section, it applies to a more general class of target dynamics models than given by the polynomial trajectory. Let us proceed, however, for the time being assuming that the target trajectory is described exactly by a polynomial. We shall now show that the transition matrix È can be obtained from the matrix A of (8.1-10). 254 GENERAL FORM FOR LINEAR TIME-INVARIANT SYSTEM First express Xðt þ &Þ in a vector Taylor expansion as Xðt þ &Þ¼XðtÞþ&DXðtÞþ & 2 2! D 2 XðtÞÁÁÁ ¼ X 1 ¼0 &   ! D n XðtÞð8:1-11Þ From (8.1-10) D  XðtÞ¼A  XðtÞð8:1-12Þ Therefore (8.1-11) becomes Xðt þ &Þ¼ X 1 ¼0 ð&AÞ   ! "# XðtÞð8:1-13Þ We know from simple algebra that e x ¼ X 1 ¼0 x   ! ð8:1-14Þ Comparing (8.1-14) with (8.1-13), one would expect that X 1 ¼0 ð&AÞ   ! ¼ expð&AÞ¼Gð&AÞð8:1-15Þ Although A is now a matrix, (8.1-15) indeed does hold with exp ¼ e being to a matrix power being defined by (8.1-15). Moreover, the exponent function GðAÞ has the properties one expects for an exponential. These are [5, p. 95] Gð& 1 AÞGð& 2 AÞ¼G½ð& 1 þ & 2 ÞAð8:1-16Þ ½Gð& 1 AÞ k ¼ Gðk& 1 AÞð8:1-17Þ d d& Gð&AÞ¼Gð&AÞA ð8:1-18Þ We can thus rewrite (8.1-13) as Xðt þ &Þ¼expð&AÞXðtÞð8:1-19Þ Comparing (8.1-19) with (8.1-7), we see immediately that the transition matrix is Èð&Þ¼ expð&AÞð8:1-20Þ TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME 255 for the target whose dynamics are described by the constant-coefficient linear vector differential equation given by (8.1-10). Substituting (8.1-20) into (8.1-19) yields Xðt n þ &Þ¼Èð&ÞXðt n Þð8:1-21Þ Also from (8.1-15), and (8.1-20) it follows Èð&Þ¼I þ &A þ & 2 2! A 2 þ & 3 3! A 3 þÁÁÁ ð8:1-22Þ From (8.1-17) it follows that ðexp &AÞ k ¼ exp k&A ð8:1-23Þ Therefore ½Èð&Þ k ¼ Èðk&Þð8:1-24Þ By way of example, assume a target having a polynomial trajectory of degree d ¼ 2. From (8.1-10a) we have A. Substituting this value for A into (8.1-22) and letting & ¼ T yields (5.4-7), the transition matrix for the constant-accelerating target as desired. 8.1.4 Constant-Coefficient Linear Differential Vector Equation for Transition Matrix È A third useful alternate way for obtaining È is now developed [5. pp. 96–97]. First, from (8.1-21) we have Xð&Þ¼Èð&ÞXð0Þð8:1-25Þ Differentiating with respect to & yields d d& Èð&Þ  Xð0Þ¼ d d& Xð&Þð8:1-26Þ The differentiation of a matrix by & consists of differentiating each element of the matrix with respect to &. Applying (8.1-10) and (8.1-25) to (8.1-26) yields d d& Èð&Þ  Xð0Þ¼AXð&Þ ¼ AÈð&ÞXð0Þð8:1-27Þ 256 GENERAL FORM FOR LINEAR TIME-INVARIANT SYSTEM Thus d d& Èð&Þ¼AÈð&Þð8:1-28Þ On comparing (8.1-28) with (8.1-10) we see that the state vector XðtÞ and the transition matrix Èð&Þ both satisfy the same linear, time-invariant differential vector equation. Moreover, given this differential equation, it is possible to obtain Èð&Þ by numerically integrating it. This provides a third method for obtaining Èð&Þ. Define the matrix inverse of È by É, that is, Éð&Þ¼½Èð&Þ À1 ð8:1-29Þ The inverse É satisfies the associated differential equation [5, p. 97] d d& Éð&Þ¼ÀÉð&ÞA ð8:1-30Þ Thus Éð&Þ can be obtained by numerically integrating the above equation. To show that (8.1-30) is true, we first verify that the solution to (8.1-30) is Éð&Þ¼Éð0ÞexpðÀ&AÞð8:1-31Þ This we do by differentiating the above to obtain d d& Éð&Þ¼ÀÉð0Þ½expðÀ&AÞA ¼ÀÉð&ÞA ð8:1-32Þ Thus (8.1-31) satisfies (8.1-30), as we wished to show. For Éð0Þ let us choose Éð0Þ¼I ð8:1-33Þ This yields for Éð&Þ the following: Éð&Þ¼expðÀ&ÞA ð8:1-34Þ It now only remains to show that the above is the inverse of È. To do this, we use (8.1-16), which yields expð&AÞexpðÀ&AÞ¼expð0Þ ¼ I ð8:1-35Þ This completes our proof that È À1 ¼ É and É satisfies (8.1-30). TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME 257 For a target whose trajectory is given by a polynomial, it does not make sense to use the three ways given in this section to obtain È. The È can easily be obtained by using the straightforward method illustrated in Section 2.4; see (2.4-1), (2.4-1a), and (2.4-1b) and (1.3-1) in Section 1.3. However, as shall be seen later, for more complicated target models, use of the method involving the integration of the differential equation given by (8.1-28) represents the preferred method. In the next section we show that (8.1-10) applies to a more general class of targets than given by a polynomial trajectory. 8.2 MORE GENERAL MODEL CONSISTING OF THE SUM OF THE PRODUCT OF POLYNOMIALS AND EXPONENTIALS In the preceeding section we showed that the whole class of target dynamics consisting of polynomials of degree d are generated by the differential equation given by (8.1-2). In this section we consider the target whose trajectory is described by the sum of the product of polynomials and exponentials as given by xðtÞ¼ X k j¼0 p j ðtÞe jt ð8:2-1Þ where p j ðtÞ is a polynomial whose degree shall be specified shortly. The above xðtÞ is the solution of the more general [than (8.1-2)] linear, constant-coefficient differential vector equation given by [5, pp. 92–94] ðD dþ1 þ  d D d þÁÁÁþ 1 D þ  0 ÞxðtÞ¼0 ð8:2-2Þ We see that (8.1-2) is the special case of (8.2-2) for which  0 ¼  1 ¼ÁÁÁ¼  d ¼ 0. The  j of (8.2-1) are the k distinct roots of the characteristic equation  dþ1 þ  d  d þÁÁÁþ 1  þ  0 ¼ 0 ð8:2-3Þ The degree of p j ðtÞ is 1 less than the multiplicity of the root  j of the characteristic equation. By way of example let d ¼ 2. Then ðD 3 þ  2 D 2 þ  1 D þ  0 ÞxðtÞ¼0 ð8:2-4Þ Let the state vector XðtÞ for this process defined by (8.1-8). Then it follows directly from (8.2-4) that d dt XðtÞ¼ _ x  x _x 0 @ 1 A t ¼ 010 001 À 0 À 1 À 2 0 @ 1 A x _ x  x 0 @ 1 A t ð8:2-5Þ 258 GENERAL FORM FOR LINEAR TIME-INVARIANT SYSTEM or d dt XðtÞ¼AXðtÞð8:2-6Þ where A  010 001 À 0 À 1 À 2 0 @ 1 A ð8:2-6aÞ This gives us a more general form for A than obtained for targets following exactly a polynomial trajectory as given in Section 8.1; see (8.1-10a). The matrix A above can be made even more general. To do this, let ^ XðtÞ¼GXðtÞð8:2-7Þ where G is an arbitrary constant 3  3 nonsingular matrix. Applying (8.2-7) to (8.2-6) yields d dt G À1 ^ XðtÞ¼AG À1 ^ XðtÞð8:2-8Þ Because G is a constant, the above becomes G À1 d dt ^ XðtÞ¼AG À1 ^ XðtÞð8:2-9Þ or d dt ^ XðtÞ¼GAG À1 ^ XðtÞð8:2-10Þ or finally d dt ^ XðtÞ¼B ^ XðtÞð8:2-11Þ where B ¼ GAG À1 ð8:2-11aÞ Because G is arbitrary, B is arbitrary, but constant. Thus, (8.2-6) applies where A can be an arbitrary matrix and not just (8.2-6a). GENERAL MODEL CONSISTING OF THE SUM OF THE PRODUCT 259 . degree of the target dynamics while m is the degree used by 252 Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright # 1998 John Wiley & Sons,. the tracking filter to approximate the target dynamics. For the nonlinear dynamics model case, discussed briefly in Section 5.11 when considering the tracking