Page 110 - Handbook of Biomechatronics

106 Naser Mehrabi and John McPhee

Here, K(t) is the same gain array obtained from optimal control feedback in
Eq. (7). With the substitution of the control law with the observer Eq. (10),
the controller equations take the following form:

^ x _ tðÞ ¼ AtðÞ BtðÞKtðÞ LtðÞCtðÞ½ Š^xtðÞ + LtðÞytðÞ utðÞ ¼ KtðÞ^xtðÞ (13)

KFs are used to estimate the system state variables from indirect and noisy
measurements that are common in mechatronic systems (e.g., force sensors).
LQRs in conjunction with KF can be used to implement a biomechatronic
system device control logic while minimizing a cost function (e.g., electrical
energy consumption). As an example, this method can increase the battery
life of untethered biomechatronic devices or just simply decrease the device
energy consumption.

2.2.2 Nonlinear Control Theory
Nonlinear control theory covers a larger class of systems and can be used for
a wider range of real-life problems. Nonlinear systems do not obey the
superposition principle, and the equations of motion are governed by
nonlinear differential-algebraic equations (DAEs). A nonlinear system
can be linearized (approximated with a linear system) by use of Taylor
series expansion or perturbation methods around an operating point,
and then a linear control theory can be applied to design a controller
for the nonlinear system. However, the linear model is only valid if the
model varies in the sufficiently small range about the operating point,
while nonlinear controllers can incorporate nonlinear models to guarantee
performance under nonlinear phenomena (e.g., limit cycles, multiple
equilibria). In this section, we focus on the NMPC method that has
attracted attention both in industry and academia in recent years. NMPC
has been widely used in the chemical industry, where a lower sampling rate
is required, but recently it has been applied in other industries such as auto-
motive and assistive devices.
A NMPC can be considered as the general form of the LQ control
method in which the controller uses a nonlinear model and can account
for constraints on inputs and states. Moreover, the NMPC is not required
to have a quadratic performance criterion. The NMPC includes both
feedforward and feedback control schemes. The NMPC uses a control-
oriented model (COM) representing the physical system to predict the opti-
mal dynamics in a finite time interval ahead of current time called the
prediction horizon, and feedback information to correct the prediction errors.