C.32.LCA:5.1 - Retain a qualified loop and change how it receives targets

Consider a constructed discrete model, in normalized units,

[ x_{k+1}=x_k+u_k,\qquad u_k=\tfrac12(r_k-x_k). ]

The actuator condition is (|u_k|\leq0.06), and the state must remain in ([-1,1]). The state is available without delay or error in this first case. The controller computation fits each update. These are premises of the example, not claims about a physical device.

For a fixed target, the error contracts by one half per update. Nevertheless, if (x_0=0.5) and a planner immediately applies (r_0=0.7), the commanded input is (0.1), outside the actuator limit. The separate tracking result did not qualify that reference change.

One alternative is to redesign the controller and planner together. Another candidate retains the controller and limits the applied target from the desired goal (g_k\in[-1,1]):

[ r_k=x_k+\operatorname{clip}(g_k-x_k,-0.12,0.12). ]

Here (\operatorname{clip}(a,l,h)=\min(\max(a,l),h)). Substitution gives [ u_k=\tfrac12\operatorname{clip}(g_k-x_k,-0.12,0.12). ] Thus the actuator condition holds. The next state lies between (x_k) and (g_k), so an initial state in ([-1,1]) stays there. This establishes the stated discrete constraints for this model. It does not establish intersample physical safety or a prescribed arrival time.

For the constant goal (0.7), the first three applied targets are (0.62,0.68,0.7); the corresponding inputs are (0.06,0.06,0.04), and the resulting states are (0.56,0.62,0.66). The limiter uses the loop’s remaining action capability and returns a different progress forecast to the planner. It is a simple state-dependent governor construction, not a reason to add a new tracking layer.

Now request (x_2\geq0.68) from (x_0=0.5). Under the actuator bound, even an arbitrary control law can increase the state by at most (2(0.06)=0.12), giving (x_2\leq0.62). That requirement is impossible under the stated model and limit. The next choice concerns the deadline, attained position or available actuation; faster planning alone cannot obtain it. The participant authorized to change the demanded result must receive this bound.

Change the governor’s observation condition instead. Suppose the governor receives only (\hat x_k), with a qualified current bound (|x_k-\hat x_k|\leq0.02), while the local tracker retains the original state-based feedback law. Limiting (|r_k-\hat x_k|) to (0.12) would allow (|r_k-x_k|=0.14) and input (0.07). A sufficient repair is to limit (|r_k-\hat x_k|\leq0.10) and require (r_k\in[-1,1]). Then (|r_k-x_k|\leq0.12), while the original convex update preserves the state interval. The set of targets satisfying both conditions must be nonempty; it is nonempty here when the state lies in the interval and the stated estimate bound holds.

The error bound must describe the current state. A bound for a stale measurement cannot be reused without accounting for intervening motion. If the tracker also loses its state measurement, its response equation and the convexity argument need reconsideration. Replacing it with a learned policy likewise reopens the calculation: the old response equation no longer supplies its proof.

This case shows the common method’s contribution: recover the violated join, compare a governor with joint redesign, construct the admissible interface, calculate its consequence for the goal, and reopen the affected condition when observation or policy changes.