Systems Modelling - Lab Report Example

MATLAB inbuilt function ss() is then called and given the matrices as input arguments so as to generate the state space model. The ss() function takes state matrices as inputs and generates the state space model. The program begins with the definition of variables to be used i.e. mass, spring constant and the damping coefficient. Next, the transfer function is defined and it poles computed. The function pole() returns a pair of poles that have real and complex parts. To establish stability, the sign of the real parts are checked.

Negative real part means the pole is in the LHS of the S-Plane thus implying that the system is stable. Otherwise the system will be unstable if poles have positive real parts. The program then illustrates the response of the system to initial conditions. Step input could also be used to demonstrate response of the system. From Figure 1.2 the conjugate pair has negative real parts. According to our criterion, the system is stable. The 2 poles are on the left hand side of the S-Plane indicating that the system is stable. Figure 1.3 demonstrates how the MSD system responds to initial condition of x0 = [1, 0].

From the graph it can be seen that the system undergoes a transient state and attains steady state as time increases. The code above has been modified in order to allow stability of the system to be determined for different values of gain. Gain is usually multiplied by the numerator of the transfer function, therefore the program will prompt the user to enter a value for the gain then the program computes the stability. First the constants are defined. The user is then prompted to enter the gain for the system which forms the numerator of the transfer function.

The denominator is defined as a vector containing mass, spring and dumping constants. The transfer function is then determined and its poles computed. The sign of the poles is checked and stability determined. When run with 8 as the gain, the system shows it is stable.