Page 109 - Handbook of Biomechatronics

Model-Based Control of Biomechatronic Systems 105

Linear State Estimation
In this section, we introduce an optimal state observer called the Kalman
filter (KF). A KF is a data processing algorithm that estimates the current
value of the state variables of interest using the available information.
A KF incorporates all the available measurements to estimate the current
state variables by considering the system and measurement device dynamics,
the statistical significance of the measurement and system noise, and the
available information about the system’s initial condition. Here, consistent
with continuous LQ control, a continuous KF is introduced. Consider a lin-
ear time-varying continuous-time system:

_ xtðÞ ¼ AtðÞxtðÞ + BtðÞutðÞ + wtðÞ
(9)
ytðÞ ¼ CtðÞxtðÞ + vtðÞ
Here, y(t) is the measurement variable, and C is a continuous time-varying
matrix; w(t) and v(t) are Gaussian white noise with zero mean value and Q
and R are covariance matrices that represent process noise and sensor noise,
respectively. The process noise represents the uncertainty in the system
model, and sensor noise is usually used to show uncertainty in the measure-
ments. Q(t) and R(t) are symmetric and nonnegative definite matrices in
which each element represents the covariance of the corresponding mea-
surement or system noise. The initial state x(t 0 ) is also assumed to be Gauss-
ian random variable with a mean value of x 0 and a covariance P e0 . A KF is an
optimal state observer in which the state estimation ^xtðÞ is computed in a
way that the expected value of the estimation error squared is minimized
T
ð
(i.e., Ex t ðÞ ^xt ðÞð Þ xt ðÞ ^xt ðÞÞ ). The continuous-time KF observer is
in the following form:

^ x tðÞ ¼ A^xtðÞ + Bu tðÞ + LtðÞ y C^xtðÞÞ
ð
(10)
ðÞg
f
^ x 0ðÞ ¼ Ex t 0
where L(t) is often called the Kalman gain from:
T
LtðÞ ¼ P e tðÞC R 1
T T T 1 (11)
_ P e ¼ AP e + P e A + B w QB P e C R CP e
w
which solves forward in time with the boundary condition P e (t 0 )¼P e0 .
Based on the separation principle (Kirk, 2013), the optimal control input
can be determined by feeding the estimated states instead of the measure-
ments into Eq. (6). Then, the optimal feedback control becomes:

utðÞ ¼ KtðÞ^xtðÞ (12)