![]() ![]() We will use the MATLAB command ctrb to generate the controllability matrix and the MATLAB command rank to test the rank of the matrix. Since our controllability matrix is 4x4, the rank of the matrix must be 4. Powers of the matrix will not increase the rank of the controllability matrix since these additional terms will just be linear combinations of Adding additional terms to the controllability matrix with higher The number corresponds to the number of state variables of the system. Rank where the rank of a matrix is the number of linearly independent rows (or columns). For the system to be completely state controllable, the controllability matrix must have Satisfaction of this property means that we can drive the state of the system anywhere we like in finite time (under the Zero (consult your textbook for more details).īefore we design our controller, we will first verify that the system is controllable. Another option is to use the lqr command which returns the optimal controller gain assuming a linear plant, quadratic cost function, and reference equal to If you know the desired closed-loop pole locations, you can use the MATLAB commands place or acker. can measure) all four of the state variables. The next step in the design process is to find the vector of state-feedback control gains assuming that we have access (i.e. This should confirm your intuition that the system is unstable Sys_ss = ss(A,B,C,D, 'statename',states, 'inputname',inputs, 'outputname',outputs) Īs you can see, there is one right-half plane pole at 5.5651. P = I*(M+m)+M*m*l^2 %denominator for the A and B matrices After execution in the MATLAB command window, the output will list the open-loop poles (eigenvalues of ) as shown below. Enter theįollowing lines of code into an m-file. The first step in designing a full-state feedback controller is to determine the open-loop poles of the system. To view the system's open-loop response please refer to the Inverted Pendulum: System Analysis page. vertical) and theĬart should move to its new commanded position. Reference is given to the system, the pendulum should be displaced, but eventually return to zero (i.e. We want to design a controller so that when a step ![]() The output contains both the position of the cart and the angle of the pendulum. The 4 states represent the position and velocity of the cart and theĪngle and angular velocity of the pendulum. In this problem, represents the step command of the cart's position. Note that here we feedback all of the system's states, rather than using the system's outputs The schematic of this type of control system is shown below where is a matrix of control gains. This problem can be solved using full-state feedback. Well suited to the control of multiple outputs as we have here. Pendulum vertical while controlling the cart's position to move 0.2 meters to the right. In this example, we are attempting to keep the ![]() We did not attempt to control the cart's position. In the other examples we were attemping to keep the pendulum vertical in response to an impulsive disturbanceįorce applied to the cart. Steady-state error of less than 2% for andĪs you may have noticed if you went through some of the other inverted pendulum examples, the design criteria for this exampleĪre different.Pendulum angle never more than 20 degrees (0.35 radians) from the vertical.Settling time for and of less than 5 seconds. ![]()
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