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we have now looked at two components of a state variable compensator the state feedback and reference feed-forward in this video we look at the third and lowest component the state observer the full state variable compensator is shown in this diagram the plant is given by this block with input u and output Y we have already looked at the state feedback and reference feed-forward the state feedback requires the system States but the states of a real-world system are not known to us the task of the state observer is to reconstruct the States as accurately as possible given the information available to us this information includes the plant input the plant output Y and the plant model described by the system matrices a B C and D the reconstructed or estimated States are called X hat the structure that we choose for the observer is shown in this block diagram where we assume that the direct transmission term of the plant D is equal to 0 the inputs to the observer are the plant input U and the plant output Y and the output of the observer is the estimated States X hat the observer can be thought of as the combination of two mechanisms firstly the response of the plant to the input is simulated by using a plant model which is done in this part of the observer u is the plant input the estimated States x hat are the states of the simulated plant y hat is the output of the simulated plant and the system matrices a B and C come from the plant model only a simulation of the plant however would not accurately reconstruct the States since any modeling errors or disturbances that the plant experienced will cause the behavior of the simulated plant to deviate from that of the actual plant to minimize this deviation the second mechanism compares the simulated output with the actual output and the difference is multiplied with observer gain M which is a column vector the result is added to X hat dot and is supposed to correct any errors in the estimated states if the difference between the simulated and actual outputs is small this corrective action will be small but if the difference between the simulated and actual outputs is large this corrective action will be large the question we now address is how to design the observer game M we want to find an observer gain such that the simulation mechanism and the correction mechanism work together well to minimize the difference between the observer States and plant States we do this by describing the dynamics of the observer in terms of m and we then design M such that the observer has some required dynamics we start this process by writing down the equations describing the observer from the block diagram in equation 1 X hat dot is equal to a times X hat plus B times u plus m times y minus y hat and in equation 2 y hat is equal to C times X hat equations 1 and 2 from the previous page are rewritten here we want the estimated States to be as close as possible to the plant States and to this int we defined the difference between them or the state error as X tilde equal to the plant States X minus the estimated States X hat our goal is to describe the dynamics of the state error or informally how quickly the state error goes to 0 to do this we first take the derivative of the state error which is equal to X dot minus X hat dot within substitute ax plus bu for X dot from the state equation of the plant model and the right hand side of equation 1 for X hat in the next line we substitute C X for y from the output equation of the plant model and the right-hand side of equation 2 for X hat and then gather the terms containing a and the terms containing M we now recognize X minus X hat to be the state error X tilde and we can write this equation in terms of X tilde only after taking out the common factor X tilde we get X tilde dot is equal to a minus MC times X tilde this equation is a standard state equation with States X tilde and nu a matrix a minus MC we can therefore calculate the poles of the observer by setting up the characteristic equation of the observer as the determinant of Si minus the new a matrix equal to 0 and solving for s remember that these poles describe the dynamics of the state error of a difference between the estimated and actual States we can now design the observer gain M such that the observer poles are in desired locations to illustrate these concepts let's work through an example we use the same hanging pendulum model as previously with the state variable model of the plant and given by these equations we can now describe the observer dynamics by calculating the observer characteristic polynomial in terms of the observer gain in the characteristic polynomial is given by the determinant of Si minus a plus M times C which results in the determinant of this matrix after calculating the determinant we obtained this polynomial where the coefficients are written in terms of the elements of the observer gain vector M we described the desired observer dynamics by specifying the observer polls in this example we choose both observer polls to lie - 20 we can now calculate the desired characteristic polynomial as the product of s - each poll which results in this polynomial we now compare the coefficients of the observer characteristic polynomial with that of the desired observer characteristic polynomial and we get M 1 equal to 40 and 9.81 + M 2 equal to 400 the observer gain vector M is therefore given by 40 + 390 point 1 9 in this example it is possible to place the observer polls in any desired location we will address the question for which systems we can place the observer pulse anywhere at a later stage also note that the design of the observer gain follows a very similar process to the design of the state feedback gain we will exploit this fact at a later stage