Controller design for probabilistic performance objectives
3.1 Controller design
represents the performance of the composite structure and actuator system. Hence, (3.1)
where
(3.2)
J(¢*, 8)
=
min J(¢, 8), t/JEif!J(¢, 8) := P(FI¢, 8, 'D)
may be determined by including the controller parameters in the evaluation of (2.21).
In general, the optimal solution¢* requires a nonlinear optimization over if!, which can be conducted using a variety of existing numerical methods (see Pierre 1986 or Press et al. 1992, for example). The solution techniques that are used for the applications in this thesis are the unconstrained optimization algorithms that exist in MATLAB (The MathWorks, Inc. 1994a), such as the Neider and Mead (Press et al. 1992) nonlinear simplex algorithm (MATLAB function fmins()) for the multi-variable optimizations and a combined golden-section-search and parabolic interpolation (Press et al. 1992) for single-variable optimizations (MATLAB function fmin()).
The optimization performed for (3.1) is subject to constraints on the controller, such as a limit on the available actuator effort or a penalty on the expected energy used by the controller. These constraints can either be imposed as boundaries on the possible controller set or as a penalty function incorporated into the design ob- jective of (3.1). The latter approach allows for the use of unconstrained optimization methods and is adopted for the controller optimizations performed in this thesis.
One comment on the controller class is that any parameterized controller could be considered for the control design problem, such as an output-feedback controller with a state estimator or a nonlinear controller. A practical limit exists on the controller class that depends on the speed of the computer and number of uncertain parameters used to describe the system. In addition, the efficiency of the opti- mization routines that are used for both the search for the "design point" in the asymptotic approximation to (2.21) as well as the controller optimization for (3.1)
influence the feasible problem size. Note that for the probabilistic robust control design, the integration to determine "total performance" from Section 2.5 must be performed for each function evaluation in the optimization. Hence, the solution time has an upper bound that grows exponentially with the product of the number of uncertain parameters and the number of parameters in the control law.
3.1.2 More linear system representation
The generalized equations of motion for a linear system given by (2.1) can be aug- mented such that the inputs are the exogenous disturbance, sensor noise, and mod- eling error, which are all grouped into w, and the control input, u,
(3.3)
X
~
Ax+ [
Bu Bw] {:}{:} ~ [~:] x+ [:,. D:]{ J
which is presented in block-diagram form in Figure 3.1. Here, the system output has been partitioned into the measured outputs y and the performance variables z. In (3.3), the assumption of no direct feed-through of w --+ z and u --+ y has been made (i.e., Dzw and Dyu are zero, as shown). This is not a very restrictive assumption, as the controller takes the measured output y as input, so feedback of
u to the controller can be performed internally for the controller. In addition, direct feed-through of the disturbance w to z would cause the variance of the performance variables to be unbounded. If the performance variables are to include the modeling error by modeling the prediction error as a Gaussian process, their influence should be filtered using a frequency weighting function that rolls off at higher frequencies (Doyle et al. 1992).
For output-feedback control, u is a function of the measured output variable y and the disturbance input w. If the control law is linear, the controller can be
0 Dyw
Dzu 0
{:}
Figure 3.1 Linear system representation with control input u, distur- bance w, measured outputs y, and performance variables z.
represented by
(3.4)
where Xc is the controller state, which contains the "memory" of the controller1. In block-diagram form, this linear control system is shown in Figure 3.2.
u y
Figure 3.2 Representation of linear control system.
These two systems can be inter-connected to form the closed-loop system, where the measured outputs from the structural system become the inputs to the con- troller, and the outputs from the controller are input to the structure. This closed- loop system is shown in Figure 3.3, where the direction of the input-output arrows on the controller block have been reversed from the sense of Figure 3.2 to simplify the diagram.
The equations of motion for the closed loop system can be obtained for the
1 A controller with state dimension of zero is called "memory-less."
Gc=[*]
Cc Dey u
[ A
Bu Bw
]
G= Cy 0 Dyw
Cz Dzu 0
z w
Figure 3.3 Closed-loop system.
system interconnection of Figure 3.3 from solving (3.3) and (3.4), and are given by
(3.5)
{
~
} =[A+
BuDcCy BuCcl { x }+
[Bw+
BuDcDyw] wXc BcCy Ac Xc BcDyw
z
~ [c, + n,.n,c, n,.c,] { :J + [n,.n,n""]
wIn order to have finite variance of the response, either DzuDcDyw
=
0 or w must be band-limited. For the examples worked out later in this chapter, the Gaussian input w will be passed through a low-pass linear filter. The state equations in (3.5) can easily be augmented to include this filter, but that is not presented here.3.1.3 Performance measures
The reliability-based controller design seeks to minimize the failure probability of the composite structure/actuator system. As mentioned previously in Chapter 2, structural failure occurs when the inter-story drift in any story exceeds a specified fraction of the story height. In addition, the probability of actuator failure is incor- porated into the performance objective, and the actuator is considered to have failed when the required actuator effort exceeds its allowable maximum. The controller
design procedure then searches over the set of possible controller parameter values to minimize this composite failure probability.
The failure probability can be computed on the basis of a single model of the system, termed the nominal performance, or for a whole set of possible models, termed the robust performance. The nominal performance is computed on the basis of (2.7), and the robust performance is determined by evaluating the total failure probability for the system, (2.21).
In addition, the 1i2-performance of the system is considered for the control design. The design seeks to minimize the 1i2-norm of the system, either for a nominal system, or for an uncertain one using (2.22) to compute the expected 1i2 performance.