Page 107 - Handbook of Biomechatronics

Model-Based Control of Biomechatronic Systems 103

2.2 Model-Based Closed-Loop Control
As shown in Fig. 1B, in a closed-loop control system with a feedback con-
troller, the system outputs feedback to the controller to regulate the control
action. Feedback controllers based on system dynamics can be categorized
into linear and nonlinear feedback controllers.

2.2.1 Linear Control Theory
A linear system is a system whose dynamics obey the superposition principle
and whose equations of motion are composed of linear differential equa-
tions. Optimal and robust control theories of linear systems with quadratic
cost functions have been well developed over decades and have been used in
many practical applications (Kirk, 2013; Doyle et al., 2013). In this section,
the linear quadratic (LQ) optimal control theory is presented. Consider the
linear time-varying system with a state differential equation:

_ xtðÞ ¼ AtðÞxtðÞ + BtðÞutðÞ
(4)
ztðÞ ¼ C 1 tðÞxtðÞ + D 1 tðÞutðÞ
where x, z, and u are system state variables, controlled variables, and control
inputs; A, B, C 1 , and D 1 are the time-varying matrix functions of time; and
x 0 is the state initial condition.

Linear-Quadratic Control
The LQ control law is optimal concerning a quadratic integral performance
criterion, as shown below:
t
Z 1

T
T
T
J ¼ x t 1 ðÞP 1 xt 1 + z tðÞR 3 tðÞztðÞ + u tðÞR 2 tðÞutðÞ dt (5)
ðÞ
t 0
Here, R 3 (t) is a nonnegative-definite symmetric matrix that determines
the weighting of each element of the controlled variable z.The quantity
T
z (t)R 3 (t)z(t) shows the error of the controlled variable z with respect to zero
at time t. R 2 (t) is a positive-definite symmetric weighting matrix that is used to
reduce the control effort. If needed, a terminal state condition can be added to
the objective function with a nonnegative-definite symmetric matrix P 1 [see
the first term in Eq. 5] such that the state x(t) at the final time t 1 is as close
as possible to zero. The optimal feedback controller with respect to the per-
formance criterion shown in Eq. (5) is in the form of a linear full-state feed-
back controller (Kirk, 2013) as shown in Fig. 2, and the optimal control law is