Question

1. Consider the scalar discrete-time optimal control problem: $x_{k+1} = -x_k + u_k$ $x_0 = 1$ $\text{Min: } J = \frac{1}{2}x_N^2 + \frac{1}{2}\sum_{k=0}^{N-1} (x_k^2 + u_k^2)$ where, $N$ is a very large positive integer (a) Solve the Riccati equation sequentially (backwards) as presented in class (b) Solve the steady state Riccati equation. Verify that the result is the same as obtained above (c) Determine the total cost $J$ (d) Determine the value of $J$ if the sub-optimal control law $u_k = 0.5x_k$ is applied

          1. Consider the scalar discrete-time optimal control problem:
$x_{k+1} = -x_k + u_k$
$x_0 = 1$
$\text{Min: } J = \frac{1}{2}x_N^2 + \frac{1}{2}\sum_{k=0}^{N-1} (x_k^2 + u_k^2)$
where, $N$ is a very large positive integer
(a) Solve the Riccati equation sequentially (backwards) as presented
in class
(b) Solve the steady state Riccati equation. Verify that the result is
the same as obtained above
(c) Determine the total cost $J$
(d) Determine the value of $J$ if the sub-optimal control law $u_k = 0.5x_k$
is applied

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1. Consider the scalar discrete-time optimal control problem:
xk+1 = -xk + uk
x0 = 1
Min: J = (1)/(2)xN^2 + (1)/(2)∑k=0^N-1 (xk^2 + uk^2)
where, N is a very large positive integer
(a) Solve the Riccati equation sequentially (backwards) as presented
in class
(b) Solve the steady state Riccati equation. Verify that the result is
the same as obtained above
(c) Determine the total cost J
(d) Determine the value of J if the sub-optimal control law uk = 0.5xk
is applied

Added by Francisco N.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sachin Rao

Step 1

To solve the Riccati equation sequentially, we start from the final time step and work our way backwards. In this case, the final time step is N. The Riccati equation is given by: P(k) = A(k)' * P(k+1) * A(k) - A(k)' * P(k+1) * B(k) * (R(k) + B(k)' * P(k+1) * Show more…

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MATLAB: Consider the scalar discrete-time optimal control problem: Where N is a very large positive integer. 1. Solve the Riccati equation sequentially. 2. Solve the steady-state Riccati equation. Verify that the result is the same as above. 3. Determine the total cost J. 1. Consider the scalar discrete-time optimal control problem: Tk+1 = -Tk + uk, To = 1, N = 1, 2, Min: J = k=0. where N is a very large positive integer. (a) Solve the Riccati equation sequentially (backwards) as presented in class. (b) Solve the steady-state Riccati equation. Verify that the result is the same as obtained above. (c) Determine the total cost J. (d) Determine the value of J if the sub-optimal control law u = 0.5 is applied.

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