The Kalman filter design in Problem 11.7-1 resulted in the steady-state filter equations (a) Consider the plant

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The Kalman filter design in Problem 11.7-1 resulted in the steady-state filter equationsq(k)= a(k) + 0.7105[y(k) - q(k)] q(k+1)= 0.8q(k) + 0.2u(k)

(a) Consider the plant and filter to be open-loop; that is, the state estimate is not fed back for control purposes.
Suppose that the input u(k) is constant at a value of 10. Find the steady-state values of the plant state x(k) and the plant output y(k), if the random inputs w(k) and v(k) are zero.

(b) For the conditions specified in part (a), find the steady-state value of q(k), the plant state estimate.

(c) The steady-state Kalman filter has the property that Q(z)>U(z) = X(z)>U(z). Does this property verify your results in parts (a) and (b)?

(d) The Kalman filter requires that the average value of w(k) be zero. Suppose that w(k) is constant with a value of 5. Repeat parts (a) and (b), and calculate the percent error in the state estimate.

Problem 11.7-1

Suppose that a plant is described byx(k+ 1) = 0.8x(k) + 0.2u(k) + w(k), y(k)= x(k)+v(k) where w(k) and v(k) are random and uncorrelated, with

(a) Design a Kalman filter for this system. Continue the gain calculations until the gain is approximately constant. Use M(0) = 2.

(b) In part (a), we specified M(0) = 2. What are we stating about our estimate of the state x(0) ?

(c) Write the difference equations for the steady-state Kalman filter, as in part (a).

(d) Suppose that an LQ design is performed for this plant, with the resulting gain K = 0.2197. Find the control-estimator transfer function (see Fig. 9-8) for this IH-LQG design.

(e) Find the closed-loop system characteristic equation for part (d).

(f) Find the closed-loop system time constants.

(g) Suppose that the state x(k) is estimated to be 90.1 by the Kalman filter at a certain time kT. Give the three-sigma range about the value 90.1 that will almost certainly contain the true value of x(k) at that time kT.

(h) Find the system phase and gain margins.

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Related Book For  answer-question

Digital Control System Analysis And Design

ISBN: 9780132938310

4th Edition

Authors: Charles Phillips, H. Nagle, Aranya Chakrabortty

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