1. Find the transfer function of the system considering the angle of the pendulum, 6 as the output.
2. Select all the constant values randomly = 80+RAND#, M= 20 + RAND#, = 1.5 + RAND#
3. Select the time domain specifications randomly. Mention all time domain restrictions you set for your project here?
Fig. 1. Inverted pendulum on a cart. Figure 1 shows the famous "inverted pendulum on a cart" or IPC. This system is the source of lots of challenging control problems, and is often used to demonstrate and test new ideas. The angle of the pendulum, e, and the position of the cart, x, are configuration variables. Define v- dx/dt, o-ddt. The force Jit) is the input. The nonlinear equations of motion for the base position and pendulum angle are as follows: (la) (1b) M(?)tL f(1) + (mpeg/ l.)sin@cosO-rnle? sin ? where lo-1+m and A( (MmmM+mPsinyk. The equilibrium conditions for ( 1 ) reduce to sin ?-0. The constant solution ?-[F 0 0 0)" satisfies ( 1 ) for any constant r when A,)-0 ? x ? The natural tendency of this system is for the pendulum fall down. It also has no preference for any particular x-location of the cart. The object of this project is to design a control law so that the force fn) keeps the pendulum upright at - (and the cart at some commanded position, x- Linearizing (1) about the desired operating point gives the following linear system: (2a) where Ao-A(O) Assume that we can measure both 0and x. So both 0and x may be thought of as system outputs. There is only one input, namely the force f. We haven't learned yet how to control two outputs with one input. We'll do it by handling one loop at a time. It can be a challenge to control two things (? and x) with only one input (r). We'll only control ? in this project. The force f(t) is the input. The angle of the pendulum, ? is the output. For this system set the owing control objectives in time domain. Settling time ts#x seconds (eg:-2 seconds), Rise time tr#x seconds (eg.:-5 seconds) You should add/modify above time domain specifications and create a unique problem for your project group. Accordingly answer section 3 of the report. Using root-locus method, design a controller C(s) (lead or PD controller) to achieve the control objectives. Let the system parameters have random values. Accordingly answer section 2 of the report. This way your project group will have a unique system. m-80-RAND# Ad-20 RAND# 1 = 1.5 + RAND#
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