User:Vkadakkal

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[edit] LQR Controller with Preview (i/p depends on prev. i/p)

xk + 1 = Axk + Buk + fk

Performance Measure
J = \frac{1}{2} x_N^TMx_N + \frac{1}{2} \Sigma_{k=0}^{N-1} x_k^TQx_k + u_k^TRu_k + u_{k-1}^TRu_{k-1}

Augment the system
\left[\begin{array}{c} x_{k+1}\\ v_{k+1} \end{array}\right] = \left[\begin{array}{c c} A & 0\\ 0 & 0 \end{array}\right] \left[\begin{array}{c} x_k\\ v_k \end{array}\right] + \left[\begin{array}{c} B\\ I \end{array}\right] u_k + f_{a_k}

x_{a_k} = \left[\begin{array}{c} x_k\\ v_k \end{array}\right]

Performance measure is now:
J = \frac{1}{2} x_{a_N}^TQ_{AN}x_{a_N} + \frac{1}{2} \Sigma_{k=0}^{N-1} x_{a_k}^TQ_Ax_ {a_k} + u_k^TMx_{a_k} + u_k^TR_Au_k

where

Q_A = \left[\begin{array}{c c} Q & 0\\ 0 & T \end{array}\right] ,

M = \left[\begin{array}{c c} 0 & -T \end{array}\right]

and

RA = R + T


Let g[x_{a_k}] be the cost for moving the state from x_{a_k} to

x_{a_N}


g[x_{a_k}] = min \left( \frac{1}{2} x_{ak}^TQ_Ax_k + u_k^TMx_{a_k} + \frac{1}{2} u_k^TR_Au_k + g[x_{k+1}] \right)

Assume it's solution to be
g[x_{a_k}] = \frac{1}{2}x_{a_k}^TW_{N-k}x_{a_k} + x_{a_k}^TV_{N-k} + Z_{N-k}

= min \left(\frac{1}{2} x_{a_k}^TQ_Ax_{a_k} + u_k^TMx_{a_k} + \frac{1}{2} u_k^TR_Au_k + \frac{1}{2}x_{a_{k+1}}^TW_{N-k-1}x_{a_{k+1}} + x_{a_{k+1}}^TV_{N-k-1} + Z_{N-k-1} \right)

= min \left(\frac{1}{2}x_{a_k}^T[Q_A + A_A^TW_{N-k-1}A_A]x_{a_k} + \frac{1}{2}u_k^T[R_A + B_A^TW_{N-k-1}B_A]u_k \right)+ ...
...\left( u_k^T(Mx_{a_k} + B_A^TW_{N-k-1}A_Ax_{a_k} + B_A^TV_{N-k-1} + B_A^TW_{N-k-1}f_{a_k}) + \right)...
...\left( x_{a_k}^T(A_A^TW_{N-k-1}f_{a_k} + A_A^TV_{N-k-1}) + f_{a_k}^TV_{N-k-1} + \frac{1}{2}f_{a_k}^TW_{N-k-1}f_{a_k} + Z_{N-k-1} \right)

Differentiating w.r.t uk and equating it to 0, the optimal control

u_k^* is found to be:

u_k^* = -[R_A + B_A^TW_{N-k-1}B_A]^{-1} \left[ (M + B_A^TW_{N-k-1}A_A)x_{a_k} + B_A^TV_{N-k-1} + B_A^TW_{N-k-1}f_{a_k} \right]

Let
u_k^* = -(U_v + K_x x_{a_k} + K_f f_{a_k})
where,
S = [R_A + B_A^TW_{N-k-1}B_A]^{-1}
U_v = SB_A^TV_{N-k-1}
K_x = S(M + B_A^TW_{N-k-1}A_A)
K_f = SB_A^TW_{N-k-1}

Feeding u_k^* back into the expression for g[x_{a_k}], we

get:

RHS = \frac{1}{2}x_{a_k}^T[Q_A + A_A^TW_{N-k-1}A_A]x_{a_k} + \frac{1}{2}u_k^TS^{-1}u_k - u_k^TS^{-1}u_k + x_{a_k}^T [A_A^TW_{N-k-1}f_{a_k} + A_A^TV_{N-k-1}] +...

f_{a_k}^TV + \frac{1}{2}f_k^TWf_k + Z

= \frac{1}{2}x_{a_k}^T[Q_A + A_A^TW_{N-k-1}A_A - K_x^TS^{-1}K_x]x_{a_k} + x_{a_k}^T [A_A^TW_{N-k-1}f_{a_k} + A_A^TV_{N-k-1} - \frac{1}{2}K_x^TS^{-1}(U_v + K_f f_{a_k})] +...

U_v^TS^{-1}(U_v + K_x x_{a_k} + K_f f_{a_k}) + other terms

Grouping terms in the format for g[xk] and comparing, we get:

W_{N-k} = Q_A + A_A^TW_{N-k-1}A_A + K_x^TS^{-1}K_x

V_{N-k} = A_A^TW_{N-k-1}f_{a_k} + A_A^TV_{N-k-1} - \frac{1}{2}K_x^TS^{-1}(U_v + K_f f_{a_k})

ZN − k = All Other terms...


VN − k and WN − k are solved iteratively. ZN − k is not needed for computation of control u_k^*, so is left

untouched!




[edit] LQR Controller with Preview

xk + 1 = Axk + Buk + fk

Performance Measure
J = \frac{1}{2} x_N^TMx_N + \frac{1}{2} \Sigma_{k=0}^{N-1} x_k^TQx_k + u_k^TRu_k

Let g[xk] be the cost for moving the state from xk to

xN


g[x_k] = min \left( \frac{1}{2} x_k^TQx_k + \frac{1}{2} u_k^TRu_k + g[x_{k+1}] \right)

Assume it's solution to be
g[x_k] = \frac{1}{2}x_k^TW_{N-k}x_k + x_k^TV_{N-k} + Z_{N-k}

= min \left(\frac{1}{2} x_k^TQx_k + \frac{1}{2} u_k^TRu_k + \frac{1}{2}x_{k+1}^TW_{N-k-1}x_{k+1} + x_{k+1}^TV_{N-k-1} + Z_{N-k-1} \right)

= min \left(\frac{1}{2}x_k^T[Q + A^TW_{N-k-1}A]x_k + \frac{1}{2}u_k^T[R + B^TW_{N-k-1}B]u_k \right)+ ...
...\left( u_k^T(B^TW_{N-k-1}Ax_k + B^TV_{N-k-1} + B^TW_{N-k-1}f_k) + x_k^T(A^TW_{N-k-1}f_k + A^TV_{N-k-1}) \right)+ ...
...\left( f_k^TV_{N-k-1} + \frac{1}{2}f_k^TW_{N-k-1}f_k + Z_{N-k-1} \right)

Differentiating w.r.t uk and equating it to 0, the optimal control

u_k^* is found to be:

u_k^* = -[R + B^TW_{N-k-1}B]^{-1}B^T \left[ V_{N-k-1} + W_{N-k-1}Ax_k + W_{N-k-1}f_k \right]

Let
u_k^* = -(U_v + K_x x_k + K_f f_k)
where,
S = [R + BTWN − k − 1B] − 1
Uv = SBTVN − k − 1
Kx = SBTWN − k − 1A
Kf = SBTWN − k − 1

Feeding u_k^* back into the expression for g[xk], we get:

RHS = \frac{1}{2}x_k^T[Q+A^TWA]x_k + \frac{1}{2}[U_v^T + x_k^TK_x^T + f_k^TK_f^T][R + B^TWB][U_v + K_x x_k + K_f f_k]+ ...

- [U_v^T + x_k^TK_x^T + f_k^TK_f^T][B^TWAx_k + B^TV + B^TWf_k] + ...
x_k^T[A^TWf_k + A^TV] + f_K^TV + \frac{1}{2}f_k^TWf_k + Z


Grouping terms in the format for g[xk] and comparing, we get:

W_{N-k} = Q + A^TW_{N-k-1}A + K_x^T[R+B^TW_{N-k-1}B]K_x - 2K_x^TB^TW_{N-k-1}A

But K_x^TB^TW_{N-k-1}A = K_x^TB^TS^{-1}(SW_{N-k-1}A) = K_x^TS^{-1}K_x

So
W_{N-k} = Q + A^TW_{N-k-1}A - K_x^TS^{-1}K_x

V_{N-k} = A^TW_{N-k-1}f_k + A^TV_{N-k-1} - K_x^TB^T(V_{N-k-1} + W_{N-k-1}f_k)

ZN − k = All Other terms...


VN − k and WN − k are solved iteratively. ZN − k is not needed for computation of control u_k^*, so is left

untouched!