【正文】 2 are the angles of the lower pendulum and the upper pendulum from the vertical line, mi denote the mass, lithe distance between the center of the hinge its center of gravity, Ji is moment of inertia and ci is the friction coefficient for rotation between the pendulum and hinge. L is the length (between the hinges) of the lower pendulum. M, F are the mass and friction coefficient of the cart and g is the acceleration of gravity.Then the following relations exist (summing the kinetic, potential and dissipation energies) gives the mathematical model for the double inverted pendulum system,where In the neighborhood x = 0, the following linear model is derived. Thus a mathematical model of the double inverted pendulum has beenderived.Parameters have been given by Furuta et al , we will take those numerical values in order to simulate the double pendulum system. LQR SteadyState Optimal ControlThe previous section developed the optimal control gain that minimized the cost in . We saw that the result was a timevarying gain, K(k), but that there would usually be a portion of the solution that produced a constant gain, ,which would be much easier to implement in a control system. In fact, for the infinite time problem, called the regular case, the constantgain solution is the optimum. We call this solution the linear quadratic regulator, or LQR, because it applies to linear systems, the cost is quadratic, and it applies to the regulator case. This section will discuss how one finds the LQR solution and various properties of the solution.One obvious method to pute the value of K during the early, constant portion of a problem is to pute S backward in time until it reaches a steady value, ,Then use to pute . This has been done in some software packages and gives reliable answers. Its drawback is that it requires substantially more putation than the alternate methods.Another method is to look for steadystate solution of the Riccati equation. In steady state, S(k) bees equal to S(k+1) (we’ll call them both ) and the Riccati reduces to=Which is usually referred to as the algebraic Riccati equation. Because of the quadratic appearance of , there is more than one solution, and one needs to know that S must be positive definite to select the correct one. The fact that S is positive definite follows by inspection of Eq.() and that J must be positive. For extremely simple problem for , but in most cases this is impossible, and a numerical solution required. Most software packages use variation on a method called eigenvector。t and flat surface of Z of zero, extremity directly the occurrence contact, the weakness is a design hard tangibly get improvement the system function and flat surface of Z last zero, relation between the extremity variety.Everyone know that in control system being design, for consecution the control object of the system, beg a control regulation of system very easily. But when the sample period choose bigger, will continue the control regulation that system design applies to the longlost control system realization directly, the dynamic state of the system responds to and will grow worse to even appear the unsteady phenomenon. We according to shut the wreath consecution system and is longlost with it to turn of shut certain etc. price between the wreath system relation, the control regulation of the continuous system e out by design begs the process of one more dragon system of longlost control system regulation to be called the again design of the numeral control and ．KuoCombine to longlost control system of appearance feedback be apart from to look like the expression type. According to various standard or design that speak of above method, overview numerical control system of design again the problem.The pendulum system we will take the double inverted pendulum as described in [3] with the assumption. The double inverted pendulum moves on a horizontal rail is schematically shown in Figure 1. The system is posed of:A double inverted pendulum consisting of two aluminum rods which are connected by a hinge, and the lower rod connected to a cart by a second hinge. The motion of the rods is smooth and is restricted to the vertical plane containing the double pendulum.A cart moves on a horizontal rail.The cartdriving system consisting of a motor, a pulley and belt transmission system using a timing belt and power amplifier.Some assumptions were given by Furute et al (1978) in the mathematical model of the system.Each pendulum is rigid body.The length of the belt does not change during the experiment.The driving force to the cart is directly applied to the cart without delay and is proportional to the input to the amplifier.The friction force against the motion of the cart, and frictional force generated at the connecting hinges is proportional to the difference the angular velocities of the upper and lower pendulums.The above assumptions were given by Furute et al. A mathematical model can be derived on the Lagrange equation according to the above assumptions. But the kinetic energy, potential energy and dissipation energy will be given here for our case of the motion of the double inverted pendulum on a horizontal rail, assuming that subscripts 1 and 2 denote the lower and upper pendulums respectively.(1) For the Lower Pendulum: Kinetic EnergyPotential EnergyDissipation Energy (2) For the upper pendulum: Kinetic EnergyPotential EnergyDissipation Energy(3) For the cart: Kinetic EnergyPotential EnergyDissipation Energywhere r is the distance of the cart from the reference position, 181。)附錄B Digital redesigning and Mathematical Model In control system have continuous control object, because the digital puter leads to go into, the numeral controls the analysis of the system, designing and carrying out the methods to all produce some new topics that need the continue the con