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Select web site. Since that the transfer function represents the relationship between a single input and a single output SISO at a time.
To get the first transfer function for the output and an input of U s we need to eliminate X s from the above equations. Solving the first equation for X s. Then substitute the above into the second equation. Therefore we get this transfer function. Where From the transfer function above it can be seen that there is both a pole and a zero at the origin. These are canceled by. The transfer function with the cart position X s as the output can be derived in a similar manner to arrive at the following.
State Space Representation: The linearized equations of motion from above can also be represented in state-space form if they are rearranged into a series of first order differential equations. Since the equations are linear, they can then be put into the standard matrix form. System Analysis Main Problem and Design Requirements :- The open-loop transfer functions of the inverted pendulum system as the following.
Where: The above two transfer functions are valid only for small values of the angle phi where is the deviation of the pendulum from the vertically upward position. Although the pendulum's position is shown to increase past radians 15 revolutions , the model is only valid for small phi. Also observe that the cart's position moves very far to the right, though there is no requirement on cart position for an impulsive force input.
The poles of a system can also tell us about its time response. Since our system has two outputs and one input, it is described by two transfer functions. We will specifically examine the poles and zeros of the system using the Mat-Lab.
The parameter 'v' shown below returns the poles and zeros as column vectors rather than as cell arrays. It is apparent from the analysis above that some sort of control will need to be designed to improve the response of the system.
In the design process we will assume a single-input, single-output system, with attempt to control the pendulum's angle without regard for the cart's position, subject to the following Transfer Function. Where The controller will attempt to maintain the pendulum vertically upward when the cart is subjected to a 1-Nsec impulse. This type of situation is often referred to as a regulator problem.
The external force applied to the cart can be considered as an impulsive disturbance. The Block Diagram for this problem is depicted below. To ease the design, the Block Diagram will be arranged to the following form. The resulting transfer function for the closed-loop system from an input of force to an output of pendulum angle PHI is then determined to be the following. So we try to modify the response by increasing the proportional gain by increasing the Kp variable to see what effect it has on the response.
Specifically, the settling time of the response is determined to be 1. Since the steady-state error approaches zero in a sufficiently fast manner, no additional integral action is needed Setting the integral gain constant Ki to zero to observe that some integral control is needed. The peak response, however, is larger than the requirement of 0. The overshoot often can be reduced by increasing the amount of derivative control. Since all of the given design requirements have been met, no further iteration is needed.
Integrating the Cart Into The System: In the designed system the block representing the response of the cart's position was not included because that variable is not being controlled.
To see what is happening to the cart's position when the controller for the pendulum's angle PHI is in place. We need to consider the full system block diagram as shown in the following figure. The block C s is the controller designed for maintaining the pendulum vertical. The closed-loop transfer function T2 s from an input force applied to the cart to an output of cart position is, therefore, given by the following.
And the transfer function for Pcart s is defined as follows. State Space Control Design Objective: After finding the shortcomings of the PID Controller in controlling the system, a new topology of design is tried in order to sufficiently control all aspects of the system.
In this method, we will use the state-space method to design the digital controller. The design criteria for this system for a 0.
These tests for controllability and observability are identical to the situation of continuous control except that now the state space model is discrete. Control Design via Pole Placement: Next step is to assume that all four states are measurable and design the control gain matrix K, the Linear Quadratic Regulator LQR method was used to find the control gain matrix K that results in the optimal balance between system errors and control effort, To use this LQR method, we need to specify two parameters, the performance index matrix and the state-cost matrix Q.
The state-cost matrix Q has the following structure. The element in the 1, 1 position of represents the weight on the cart's position and the element in the 3, 3 position represents the weight on the pendulum's angle.
As you can see, this plot is not satisfactory. The pendulum and cart's overshoot appear fine, but their settling times need improvement and the cart's rise time needs to be reduced.
Increasing the 1, 1 and 3,3 elements makes the settling and rise times go down, and lowers the angle the pendulum moves. In other words, you are putting more weight on the errors at the cost of increased control effort u.
We can easily correct this by introducing a feed forward scaling factor. Precompensator Design:- Unlike other design methods, the full-state feedback system does not compare the output directly to the reference, rather, it compares the state vector multiplied by the control matrix Kx to the reference.
Thus, we should not expect the output to converge to the commanded reference. To obtain the desired output, we need to scale the reference input so that the output equals the reference. This can be easily done by introducing a feed forward scaling factor N. The basic full state-feedback schematic with scaling factor is shown above. Now we have designed a system that satisfies all of the design requirements, however, that the scaling factor N was designed based on a model of the system.
If the model is in error or there are unknown disturbances, then the steady-state error will no longer be driven to zero. Observer Design:- Observer is a technique for estimating the state of the system based on the measured outputs and a model of the plant, thus in this section we will design a full-order state observer to estimate all of the system's state variables, including those that are measured.
Designing the observer equates to finding the observer gain matrix L, To accomplish this, we need to first determine the closed-loop poles of the system without the observer the eigenvalues of A-BK. A common guideline is to make the estimator poles eigenvalues of A-LC 4- 10 times faster than the slowest controller pole eigenvalue of A- BK. Making the estimator poles too fast can be problematic if the measurement is corrupted by noise or there are errors in the sensor measurement in general.
Based on the poles found above, we will place the observer poles at [ These poles can be modified later, if necessary. This is because the observer poles are fast, and because the model we assumed for the observer is identical to the model of the actual plant including the same initial conditions.
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