## Control method

Fukushima proposed a control method of solving optimal control problems including nonlinear systems. Although, in Fukushima's method, there are similarities with the classical optimal control theory, it introduces a new optimal control approach which is fundamentally based on the employment of the energy generation, storage and dissipation of the controlled system. The total system energy stored in the system boundary is the sum of each energy. The rate of change of the instantaneous energy yields the net power flow of the dynamical system. Hence, the general power balance equation for a controlled system can be represented as follows:

P = uTq + vTq - qT M(q)q - dT (q, ¿j)cj - eT (q, z)q = 0, (7)

where u is the input force vector, v is the input disturbance force vector, M(q) e R"xn is the symmetric positive-definite inertia matrix with n, the number of the DOF of the controlled system, d is combination of the damping force vector, and Coriolis and centrifugal force vector, e is the potential force vector, z is the input disturbance displacement vector, and q is the generalized coordinates vector. The power equation of the system has dynamic characteristics of the controlled system. It is unique to the system and has an important role in the design of the optimal control system.

In optimal control theory, it is aimed to obtain a control law which satisfies the given constraints and extremizes the performance measure. A performance measure is mostly a combination of some scalar functions. The functional below is the performance measure used in Fukushima's control method:

where g is the performance criterion (control performance) to be selected for the given control problem. Combination of these two terms is known as the performance measure for a general optimal control system. In (8), g might represent the linear combination of more than one performance criterion. The use of multiple performance criteria depends on the selection of performance measures for the control objective. In (8), the integrand of the last term represents the power delivered to actuators; here uT is the transpose of the input force vector, and r' is a weighting factor.

According to the fundamental theorem of the calculus of variations (Elsgolc, 1961), the necessary condition for minimizing the performance measure is that the first variation of the functional must be zero. In Fukushima's method, the scalar function, L, is composed of the power equation (7) and the differential of the performance measure (8):

L=K(uTq + vTq - qT M(q)q - dT (q,q)q - eT (q, z)q) + g(q, q, q) + r'uTq , (9)

where k is an undecided constant. The included power balance equation is zero because it satisfies the energy conservation law. As the performance measure is minimized by means of the calculus of variations, L can be also minimized. The Euler equation is a necessary condition for minimization. After applying the Euler equation, the optimal control law is obtained. 