Command vector

In reverse-time, starting with P(A ) = 0 at NT = 20 seconds, compute the state feedback gain matrix K(kT) and Riccati matrix P(kT) using equations (9.29) and (9.30). Aiso in reverse time, use the desired state vector r(/c7 ) to drive the tracking equation (9.53) with the boundary condition s(N) = 0 and hence compute the command vector y kT). 

In forward-time, use the command vector vf/cT) and state vector xf/cT) to caicuiate Uopif/cT) in equation (9.55) and hence, using the piant state transition equation (9.56) caicuiate the state trajectories. 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

P is a vector of inputs and T a vector of target (desired) values. The command newff creates the feed-forward network, defines the activation functions and the training method. The default is Fevenberg-Marquardt back-propagation training since it is fast, but it does require a lot of memory. The train command trains the network, and in this case, the network is trained for 50 epochs. The results before and after training are plotted. 

MATLAB is most at home dealing with arrays, which we will refer to as matrices and vectors. They are all created by enclosing a set of numbers in brackets, [ ]. First, we define a row vector by entering in the MATLAB Command Window ... 

The function logspace () generates a vector with numbers nicely spaced on the logarithmic scale. Its use is optional. The default of the function gives 50 points and is usually adequate. For a smoother curve, use more points. For example, this command will use 150 points logspace (-1,1,150). 

Computer Methods These methods are easily programmed in a spreadsheet program such as Microsoft Excel. In MATLAB, the trapezoid rule can be calculated by using the command trapz(x,y), where x is a vector of x values and y is a vector of values /(%.). Alternatively, use the commands... 

In Matlab the plus (+) and minus (-) operators cannot be directly applied to equations that involve vectors or matrices of different dimensions. In order to perform the same operation as in the former Excel example, column vector b ,3 must be replicated three times to match the dimensions of A. For this, the Matlab command repmat can be used. 

Alternatively, Matlab s built-in function norm can be used to determine normalisation coefficients and perform the same task. An example for column-wise normalisation of a matrix X with orthogonal columns is given below. It is worthwhile to compare X with equation (2.15) the subspace command can be used to determine the angle between the vectors (in rad) and reconfirm orthogonality. ... 

We create a range of temperatures and volumes. The command logspace (-1.2,0,50) creates a vector of 50 logarithmically spaced values between 10 12 and 10°=1 the command meshgrid produces the matrices V and T that contain all the pressures and temperatures required for the grid of values needed for the plot. 

The command produces the vector c spec with all species concentrations. 

Note that entry-wise defined matrices and vectors are delimited by square brackets [ and ] in MATLAB commands. 

Note further that the entries of one row of a matrix (or of a row vector) can be entered into MATLAB s workspace either separated by a blank space or by a comma, while a semicolon indicates the start of a new row. A comma or a blank at the end of a command line will cause screen display of the object that has just been defined, while a semicolon after a command will not. 

The commas after our MATLAB commands above are necessary as delimiters to be able to place several commands onto one line of code. They create screen output, whereas semicolons would have suppressed it. The final entry result = [size(A ), length(A )] on the command line creates the vector (4, 3, 4), called result by us. It contains the number of rows (4) of AT, followed by AT,s number of columns (3) and its length or maximal dimension (4). 

Note that we have to place zeros in the positions of A that correspond to unused variables in any equation in (1.1). Below are the corresponding MATLAB commands that let us find the solution vector... 

Our two command lines below first generate the coefficient matrix A and the right hand side vector b for (1.1), followed by the MATLAB backslash linear equations solver that computes the solution vector x. This is followed by a simple verification of the error inherent in the residual vector A-x — b for our numerical solution x. This error is nearly zero since in MATLAB the number -1.3323e-15 describes the real number —1.3323 10-15. 

To illustrate we first verify the identical behavior of the MATLAB QR based polynomial-root finder roots and MATLAB s QR based matrix eigenvalue finder eig for p s companion matrix P = C(p) First we define p by its coefficient vector in MATLAB s workspace, then we invoke the MATLAB polynomial-root finder roots, followed by its matrix eigenvalue finder eig on the companion matrix ofp. Finally we display the companion matrix P of p. As an example we use p(x) = x3 — 2.x2 + 4 here and represent p by its coefficient vector [1 -2 0 4] in the following line of MATLAB commands. 

MATLAB allows the user to choose the IVP integrator, such as ode23 or ode45 etc, and to select a stiff or nonstiff integrator, each as warranted by the specific problem. Moreover, each of the MATLAB s ODE solvers ode... allows us to specify certain options , as done in fluidbed.m in the fourth MATLAB command line options = odeset( RelTol ,10 -6, AbsTol ,10 -8, Vectorized , on ) for example. 

Organizing J in a three-dimensional array is elegant, but it does not fit well into the standard routines of MATLAB for matrix manipulation. There is no command for the calculation of the pseudoinverse J+ of such a three-dimensional array. There are several ways around this problem one of them is discussed in the following. The matrices R(k) and R(k + 5k) as well as each matrix < RIdk, are vectorized, i.e., unfolded into long column vectors r(k) and r(k + 5k). The nk vectorized partial derivatives then form the columns of the matricized Jacobian J. The structure of the resulting analogue to Equation 7.13 can be represented graphically in Equation 7.17. 

There are three Tektronix terminals that are emulated, the 4010, 4014 and 4027. The 4010 is a high resolution terminal with cross hair cursors for graphics input, the 4014 adds a hardware line patterns, choice of character sizes and user definable character sets with local storage for commonly used structures. The 4027 is a color raster terminal with 64 colors, 16 character fonts, as well as polygon and vector commands. 

These files store the workspace or variables created during a session. All matrices, vectors and scalars with unique names are saved. Many chemometricians exchange data in this format. The command save places all this information into a file called matlab.mat in the current working directory. Alternatively, you can use the Save Workspace item on die File menu. Normally you wish to save the information as a named file, in which case you enter the filename after the save command. The following code saves the results of a session as a file called mydata in the directory c results, the first line being dependent on the current working directory and requires you to have created this first ... 

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