State space eigenvalues matlab how to#

The schematic of a full-state feedback control system is shown below. This proves that our discrete system is both completely state controllable and completely state observable. Adding the following commands to your m-file and running in the MATLAB command window

The function rank can give you the rank of each matrix. Since the number of state variables in our system is 4, the rank of both matrices must be 4. These tests for controllability and observability are identical to the situation of continuous control except that now the In the same token, for the system to beĬompletely state observable, the observability matrix The rank of the matrix is the number of linearly independent rows (or columns). For the system to be completely state controllable, The next step is to check the controllability and the observability of the system. Now we have obtained the discrete state-space model of the form: Sys_ss = ss(A,B,C,D, 'statename',states, 'inputname',inputs, 'outputname',outputs) P = I*(M+m)+M*m*l^2 %denominator for the A and B matrices Running this m-file in the MATLAB command window gives you the following four matrices representing the discrete time state-space Enter the following commands into an m-file. For further details, refer to the Introduction: Digital Controller Design page. The discretization method we will use is the zero-order hold ( 'zoh'). Least 30 times larger than the closed-loop bandwidth frequency which can be determined from the closed-loop Bode plot.Īssuming that the closed-loop bandwidth frequencies are around 1 rad/sec for both the cart and the pendulum, let the sampling A good rule of thumb is that the sampling frequency be at One measure of a system's 'speed' is its closed-loop bandwidth. In choosing a sample time, note that it is desired that the sampling frequency be fast compared to the dynamics of the system.

State space eigenvalues matlab how to#

You should already be familiar with how to construct a state-space system from, ,, and matrices. This function requires that we specify three arguments: a continuous system model, the sampling time ( Ts in sec/sample), and the 'method'. We will accomplish this employing the MATLAB function c2d.

Our first step in designing a digital controller is to convert the above continuous state-space equations to a discrete form.

Steady-state error of less than 2% for and.

Pendulum angle never more than 20 degrees (0.35 radians) from the vertical.

Settling time for and theta of less than 5 seconds.

The design criteria for this system for a 0.2-m step in desired cart position are as follows: Where: (M) mass of the cart 0.5 kg (m) mass of the pendulum 0.2 kg (b) coefficient of friction for cart 0.1 N/m/sec (l) length to pendulum center of mass 0.3 m (I) mass moment of inertia of the pendulum 0.006 kg.m^2 (F) force applied to the cart (x) cart position coordinate (theta) pendulum angle from vertical (down)įor this problem the outputs are the cart's displacement ( in meters) and the pendulum angle ( in radians) where represents the deviation of the pedulum's position from equilibrium, that is, = +. If you refer to the Inverted Pendulum: System Modeling page, the linearized state-space equations were derived as: In this digital control version of the inverted pendulum problem, we will use the state-space method to design the digitalĬontroller.