Page 795 - Mechanical Engineers' Handbook (Volume 2)

786 Control System Design Using State-Space Methods

lines are violated, the state-space-based controller design procedure can be modified appro-
priately. 36
A recent example of a MIMO controller design method that has evolved from a com-
bination of frequency-domain methods and state-space methods is the linear-quadratic-
Gaussian method with loop-transfer-recovery (LQG/LTR), developed by Athans. 37 The
procedure relies on the fact that results and requirements relating to control system robustness
to modeling errors are best presented in the frequency domain. The powerful controller and
estimator structures resulting from LQR formulations of the control and state estimation
problem are used. These structures are useful in this design method because their robustness
14
and performance have been well studied, using frequency-domain measures. The resulting
method relies upon designer expertise in formulating good performance specifications at the
outset. The design method therefore avoids the main weakness of state-space methods—
namely, the weak connection between performance measures used by these methods and
performance measures of engineering significance. The computation of the controller is,
however, straightforward in this method, since the controller structures are derived from
well-established state-space methods and are well supported by commercial CACSD pack-
ages. 4–6
The first step in the design method is the definition of the design plant model. This
model includes not only the nominal model of the system to be controlled but also scaling
of the variables and augmentation of the dynamics, such as the inclusion of integrators
dictated by control objectives. The number of control inputs r and the number of outputs p
are assumed to be equal in the following development. The model is also linear and time
invariant and strictly proper, that is, the transmission matrix D in the system equations (6)
and (7) in Chapter 17 is zero. The transfer function matrix of the system is

1
H(s) C(sI A) B (120)
n
Modeling inaccuracy is treated as follows. The actual transfer function matrix is given by
H (s) [I E(s)]H(s) (121)
n
A
where E(s) characterizes the modeling error. The maximum singular value max of the matrix
E( j ) is assumed to be bounded by a known bound e ( ):
m
max [E(i )] e ( ) (122)
m
The second step in the design procedure is the specification of a target feedback loop
that has satisfactory robustness, stability, and performance specifications. The target feedback
loop is shown in Fig. 14. It is obviously not directly implementable since the control inputs
u do not appear in the system. The matrix L is a constant matrix and is chosen as described
later. It is the designer’s task to experiment with different choices of L and evaluate whether

Figure 14 Target feedback loop block diagram.