CAD.5:4.6 - Establish control under that information

First test whether the information permits a common action. In the following calculation, the selected (u) is executed as represented by the plant model; any bounded execution discrepancy must already be included in its disturbance or augmented state. For a required next-state set (R_{\mathrm{next}}), the robust admissible inputs are

\mathcal U(X_k)= \{u\in U:A X_k\oplus\{Bu\}\oplus W\subseteq R_{\mathrm{next}}\}.

The quantifiers are “choose one (u), then allow every (x\in X_k) and (w\in W),” not “for every possible state, choose a different action.” For a halfspace (h^\top x\le b) of (R_{\mathrm{next}}), the test is

\sup_{x\in X_k}h^\top Ax+h^\top Bu+\sup_{w\in W}h^\top w\le b.

This provides a direct calculation for polytopes and support-function representations. When the test fails on an outer approximation, refine the approximation or change the design before claiming impossibility. A negative result for one chosen controller is weaker still.

One feasible step does not establish a continuing guarantee. A sufficient construction for a specified feedback (u=K\hat x+g), with constant (g) in the selected operating mode, uses (F=A+BK):

x_{k+1}=F x_k+B g+w_k-BK\eta_k.

Find a set (R), containing the allowed initial states and contained in the required state set, for which

F R\oplus\{Bg\}\oplus W\oplus(-BK)V_\eta\subseteq R, \qquad K(R\oplus(-V_\eta))\oplus\{g\}\subseteq U.

Together with a valid uniform error bound, these conditions prove state and input admissibility by induction. For a stable (F), one constructive candidate is the reachable sum of the forcing set (D_f={Bg}\oplus W\oplus(-BK)V_\eta):

R_\infty=\bigoplus_{i=0}^{\infty}F^i D_f.

Use a finite outer approximation with a bounded tail, and enlarge it for the specified initial set if necessary; verify the inclusions after enlargement. For example, if an induced norm satisfies (|F|\le q_F<1) and (D_f) lies in a ball of radius (d_f), the ball of radius (d_f/(1-q_F)), enlarged to contain the initial ball, is invariant. A Euclidean norm larger than one does not prove that no other norm or set works.

These are sufficient conditions, not a universal synthesis algorithm. A stable hidden mode may be handled without measurement; an unstable or tightly constrained uncertain direction may exhaust the input or state margin. Failure can mean excessive approximation, a poor given feedback, an infeasible information arrangement, or an infeasible demand. Distinguish them.

A decisive impossibility witness consists of two admissible histories producing the same available packets and internal information but requiring disjoint actions, or an unavoidable uncertainty width larger than the permitted state interval. Such a witness rules out all causal controllers under those observation conditions, not merely the proposed feedback. Section :5 supplies both a positive construction and such a negative result.