Page 310 - The Mechatronics Handbook

0066_frame_C14.fm Page 24 Wednesday, January 9, 2002 1:50 PM

one finds

x A 0 B 0
x ˙ Σ t() = A Σ x Σ + B Σ u + N Σ r, y = Hx, x Σ = , A Σ = , B Σ = , N Σ =
x ref – H 0 0 I

Minimizing the quadratic performance functional

J = 1 ∫ t f ( x Σ Qx Σ + u Gu) t
T
T
d
--
2 t 0
one finds the control law using the first-order necessary condition for optimality. In particular, we have
∂V
u = – G B Σ T ∂V – G – 1 B T --------
-------- =
–
1
∂x Σ 0 ∂x Σ
Here, Q is the positive semi-definite constant-coefficient matrix, and G is the positive weighting constant-
coefficient matrix.
The solution of the Hamilton–Jacobi equation


∂V 1 T  ∂V  T 1 ∂V  T – 1 T ∂V
– ------- = --x Σ Qx Σ + -------- Ax Σ – -- -------- B Σ G B Σ --------

∂t 2  ∂x Σ  2 ∂x Σ  ∂x Σ
1 T
is satisfied by the quadratic return function V = -- x Σ Kx Σ . Here, K is the symmetric matrix, which must
2
be found by solving the nonlinear differential equation
T
1
T
T
–
T
˙
– K = Q + A Σ K + K A Σ – K B Σ G B Σ K, Kt f () = K f
The controller is given as
u = – G B Σ Kx Σ = – G – 1 B T Kx Σ
T
1
–
0
From x ˙ ref t() = et(), one has
x ref t() = ∫ et() t
d
Therefore, we obtain the integral control law

ut() = – G – 1 B T K xt()
0 et() dt

In this control algorithm, the error vector is used in addition to the state feedback.
As was illustrated, the bounds are imposed on the control, and u min ≤ u ≤ u max . Therefore, the bounded
controllers must be designed. Using the nonquadratic performance functional [9]


t f
∫
T
J =   ∫ x Σ Qx Σ + G tan u u d t
1
–
d
t 0 
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