Matrix exponentials

The exponential matrix e in equation (8.46) is ealled the state-transition matrix < t) and represents the natural response of the system. Henee... 

Consider again the CSTR where the consecutive reactions A B C take place. The required discrete nonlinear control law is found by obtaining a discrete version of the immersion developed in Example 3 and its associated exponential holder which has the form He, and in this particular example is a 1 X 5 dimension vector, where and H are presented in matrices (38). The exponential holder can be calculated by using the definition of an exponential matrix,... 

Now express the exponential matrix as a linear combination of the characteristic matrices of A, the fj(A ) s which satisfy the relations... 

With the exponential matrix defined in the usual fashion,... 

The exponential matrix of a matrix can be obtained as follows > exponential(B,t) ... 

The exponential matrix method can be used to determine the temperatures in a... 

Next, find the exponential matrix for the coefficient matrix A ... 

Higher order linear ODEs can also be solved by changing them into a system of first order ODEs and using the exponential matrix approach discussed earlier. The most general form of a linear ODE of n order is[l]... 

Again, the solution is obtained by finding the exponential matrix and the non-homogeneous part (see equation 2.12). The procedure used to solve higher order linear ODEs can be summarized as follows ... 

In the previous sections we solved linear ODEs using exponential matrix (section 2.1.2 - 2.1.4) and the Laplace transform technique (section 2.1.5). Alternatively, Maple s dsolve command can be used to solve linear ODEs. However, the analytical solution obtained from the dsolve command may not be in a simplified form. 

We observe that solutions obtained for T2(t) and T3(t) using the dsolve command are long and messy compared to the solution obtained using the exponential matrix approach (Example 2.3). When more than three differential equations are to be solved it is recommended that the exponential matrix method be used. As an exercise, readers can verily that the solution obtained using the dsolve command is equivalent to the solution obtained in example 2.3. 

Higher order ODEs (of order n) were converted to a system of n coupled linear first order ODEs in section 2.1.4. This system was then solved using the exponential matrix developed earlier. This approach yields analytical solutions for linear ODEs of any order. In section 2.1.5, the given system of coupled linear ODEs was converted to Laplace domain. The resulting linear system of algebraic equations was then solved for the solution in the Laplace domain. The solution obtained in the Laplace domain was then converted to the time domain. 

Maple s dsolve command was used to solve linear ODEs in section 2.1.6. In our opinion, exponential matrix method is the best method to arrive at an elegant analytical solution. The Laplace transform technique illustrated in section 2.1.5 could be used for integro-differential equations. Maple s dsolve command has to be used if the exponential matrix method fails. 

Plot your concentration profiles for k2 = 1/8 and k3 = 1/2. All other parameters are same as that of example 2.4. What do you observe Consider heating of a fluid stream by steam coils in a series of tanks (see example 2.3). Write down the differential equations describing the evolution of temperature in a system of four such tanks in series. Find the evolution of temperature with time in each tank using the exponential matrix method. Plot your temperature profiles for the parameter values [a. Pi = [0.1, 0.005] and [0.1, 0.01]. Find the time taken by the last tank to reach 99% of its steady state value. How does this time compare with the time for a 3-tank system and a 2-tank system when the total weight of all the tanks remains the same All other parameters are same as that of example 2.3. 

Exponential Matrix Method for Linear Boundary Value Problems... 

The procedure for solving linear initial value problems using the matrizant is the same as that in section 2.1.2 except that instead of finding the exponential matrix, the matrizant is found. 

An analytical solution can be obtained using the exponential matrix method described in section 3.1.2 ... 

Find the exponential matrix (exp(At)) by using Maple s exponential(A,t) command and store it in mat. 

Now the exponential matrix in the second term is represented as an infinite series ... 

Since D is a diagonal matrix, the exponential matrix of D is easily obtained as ... 

The following procedure in Maple can be used to obtain exponential matrix for any matrix with distinct eigenvalues. 

When the above procedure was used to calculate the exponential matrix for example 7.1, the time taken for N=40 interior node points was less than 30 seconds. For the same number of node points, Maple takes more than 5 minutes to calculate the exponential matrix in a 2.6Ghz processor with 2 GB RAM. The 3-D plot obtained for example 7.1 with N=40 node points is given below ... 

It is recommended that one check the exponential matrix obtained using this expedited procedure with the exponential matrix obtained using Maple s exponential matrix command for at least two values of interior node points (for e.g., N = 2, 4 etc). Once it is verified that the above procedure works for a particular problem, one can use the procedure for obtaining the exponential matrix efficiently for high values of N. 

In section 5.1.7, a procedure to expedite the calculation of exponential matrix was developed. This procedure is valid as long as all the eigenvalues are distinct. 

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