AWBT Objectives

7 .3.4 Extended Kalman Filter

7.4 AWBT Objectives

\Ve now turn our attention to the considerations which guide the selection of these design parameters. In this section we introduce the AWBT design objectives in a qualit~.tive way. In the following sections we will develop quantitative analysis mea- sures for each objective. These analysis tools allow us to assess A WBT performance quantitatively for any given H₁ and H_{2 .}

7.4.1 Stability

Our first concern must be that the closed loop system remain stable when limitations and substitutions occur. It is well-known that introduction of a limitation or sub- stitution into a stable linear closed loop system can cause instability. In the case of limit.at.ions, typical instability mechanisms are that the plant input remains against

its constraint indefinitely or limit cycles across the linear regime. In the case of sub- stitutions instability appears as cycling between operating modes. It is precisely these stability problems which have motivated all of the AWBT analysis results available in the literature ( e.g., [40,48,46,47,90,20]).

In Section 7.7.1 we derive certain necessary conditions for internal stability of the AWBT compensated system shown in Figure 3. These conditions are complemented by easily computed sufficient conditions (Section 7.7.2) which guarantee stability for all nonlinearities, N, within given mnir. sPctor honncis. The 11Pvelopment of these bounds for common limitation and substitution mechanisms is outlined in Section 7 .6.

In addition to nominal stability results, we obtain sufficient conditions for robust stability with respect to uncertainties in the linear plant model, G( s ).

7.4.2 Mode Switching Performance

The performance objective of an AWBT design is to allow the system to transition smoothly to and from constraints and between operating modes. The problem of smooth transitions can be considered as a controller state initialization problem. In general a limitation or substitution of the output of a particular controller can be considered as a switch from open loop operation to clo5ed loop operation, i.e., a controller whose output is limited or switched out has no incremental effect on the true plant input - the system is effectively open loop. When the limitation is removed or the controller switched back in, linear closed loop operation is initiated. Since limitations and substitutions can occur at essentially arbitrary times, it is important that the controller be properly "initialized" at all times so that the transition is effected smoothly.

Proper initialization of the controller requires that its state be correctly updated even when it is "off-line" or open loop. Since K(s) is designed based on the assumption that u.'

=

u. we cannot expect that the controller state will be updated correctly when u' =j:. u, i.e., when the controller thinks it is driving the plant but it is actually not. For linear designs in which the state of K ( s) has a direct physical interpretation it is often

clear how the state update should be modified during limitations and substitutions.

For example the extended Kalman filter implementation was developed to insure that the controller states remain valid estimates of the plant states independent of limitations or substitutions. Similarly, PI anti-reset windup is based on maintaining

llie proper value of the integrated error, the single controller state, during limitations and substitutions.

In the general case the design of I< ( s) will not provide a physical interpretation of the controller states. K(s) is simply provided in the form of Laplace transform or a set of state space equations. As a result we cannot avoid state positioning errors due to limitations and substitutions. In this case we seek to minimize the impact of these errors. This can be achieved by finding a

f< (

s) for which the current and future controller output, u, is relatively independent of past controller inputs, y, and the current (possibly incorrect) controller state. This independence is a function of the dynamic memory of

k (

s ). For example, a pure integrator has infinite memory;

any past input, resulting in a state positioning error, will effect the controller output for all future times. Such controllers are highly sensitive to state positioning errors resulting from limitations and substitutions. On the other hand, a purely proportional controller, with no states, is memoryless. Past inputs have no effect on current and future outputs. These controllers are insensitive to limitation and substitutions. In Section 7.8 we develop a quantitative measure of dynamic memory and show how it can be used to analyze AWBT performance.

7.4.3 Recovery of Linear Performance

Assuming that a perfect estimate of the plant input is available ( ^'Um = u'), the admissibility requirements insure that when N

=

I the closed loop performance of the AWBT compensated system is identical to that of the idealized linear design. When this assumption is not satisfied, however, we do not have any guarantee regarding the linear (N

= /)

performance of the AWBT compensated system.

In general, for the AWBT compensated system we have

while for the idealized linear design

(7.,56)

Since a perfect estimate of the actual plant input, at all frequencies, is never a realistic assumption, we must insure that when N

=

I the AWBT implementation meets the performance specifications given for the linear design. In Section 7.9 we outline an analysis which will allow us to determine if these specifications are met for a particular AWBT design. In addition we investigate the degree of deterioration in linear performance we can expect as the dynamic memory of

f< (

s) is reduced to zero.

7.4.4 Directional Sensitivity

The switching performanr,e ohjed.ive ann linP,'H pPrformance objective consider the open loop (N

=

⁰⁾ and closed loop (N

= /)

situations. In the case of plant input substitutions these are the only situations which are realized. In the case of limita- tions, however, the plant input is modified rather than replaced. For multiple input plants, a limitation, acting on only some of the inputs, can change the direction of the plant input, i.e., the relative magnitudes of the elements in the plant input, u'(s), are different than in the controller output, u(s). This important effect, as originally pointed out by Doyle et al. [351, can Ci:I.U::se ::iiguifici:l.ut perfunui:l.uce <leteriuraLiuu.

From the perspective of linear theory this effect may be regarded as a plant input perturbation with diagonal structure. In the linear case ( where the perturbation is unknown but bounded LTI operator) it is well-known that ill-conditioned plants coupled with inverse based controllers result in closed loop systems which are very sensitive to diagonal input uncertainty [87]. In Section 7.10 we outline an extension of the stability analysis methods to handle robust performance, i.e., to determine

what level of performance can be guaranteed for all nonlinear perturbations within given bounds. This result will allow us to determine whether or not the system is

"directionally sensitive," i.e., whether or not modification of the plant input direction causes severe performance deterioration.