achapt 15 - STATE ESTIMATION 15 STATE ESTIMATION State...

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STATE ESTIMATION 15 STATE ESTIMATION State estimation is a method of estimating some of the states in a system based on measurements of other states and knowledge of how the system behaves. The estimates of some of the states together with the measurements of the other states provide complete state information. Additionally, the estimates are made in real time. Therefore, the estimated states can be used in a feedback controller as though they are actual measurements. State estimation is also useful in fault detection. In fault detection, the estimated state is compared with the measurement to confirm that they agree. When they don’t agree then a failure has been detected. This chapter first examines state estimation of single degree-of-freedom systems by looking at the equations in the configuration space. Next, the problem is examined in the state space. Finally, the state estimation problem is formulated for the two degree-of-freedom system. The formulation for the two-degree-of-freedom system is the same as for an n degree-of-freedom system, so the results in that section have broad applicability. MAE 461: DYNAMICS AND CONTROLS
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STATE ESTIMATION 1. The Configuration Space A single degree-of-freedom system has two states – a displacement and a velocity. Consider two different possibilities: 1) a measurement of position is available and we want to estimate velocity, and 2) a measurement of velocity is available and we want to estimate position. Estimate of Position Figure 15 – 1 shows a representative single degree- of-freedom system. Figure 15 – 1 Single Degree-of-Freedom System The circle with the v inside it represents a velocity sensor. The question arises how to obtain a real- time estimate of x. In the configuration space the equation of motion is (15 – 1) f kx x m = + Equation (15 – 1) represents the actual system. It is actually not available except for the purposes of the analysis conducted here. We also assume that we know the system, so we can construct a model of it. The equation of motion of the actual model of the system is (15 – 2) f x k x m = + ˆ ˆ + p MAE 461: DYNAMICS AND CONTROLS
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STATE ESTIMATION This equation, unlike the actual equation of motion, is available to us. It is coded in a computer and solved numerically by the computer to yield real-time estimates of x and x . Equation (15 – 2) is called an observer . Comparing Eq. (15 – 1) and Eq. (15 – 2), we have set m , k , and f , in both to be the same because they are known. On the other hand, just because the systems are the same, doesn’t mean that the positions are the same, so we denoted the estimate of x by x ˆ . Also notice that a forcing function p has been added to the model on the right side of the equation. The purpose of p , as you’ll see shortly, is to drive the error to zero; the error is defined as (15 – 3) x x e ˆ = Let’s now use Eq. (15 – 1) to understand how the error is effected by p. Subtract Eq. (15 – 2) from Eq. (15 – 1) to get (15 – 4) p ke e m = + Equation (15 – 4) is called the error equation . It governs the error between x
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This note was uploaded on 09/17/2009 for the course MAE 469 taught by Professor Silverberg during the Fall '08 term at N.C. State.

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achapt 15 - STATE ESTIMATION 15 STATE ESTIMATION State...

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