MMP.9:5.1 - Replace many response components by two evolving quantities

Suppose a dimensionless model has an observed x and n hidden response components:

x_dot=-x+sum_i y_i+u(t), y_i_dot=c_i*x-2*y_i,

where the nonnegative c_i sum to one. Initial values and a prescribed input u(t) are supplied. The question asks for x over a finite horizon; each y_i separately is irrelevant to that result.

Retaining x alone leaves the unclosed contribution sum_i y_i. Solve each hidden equation and sum:

sum_i y_i(t)=exp(-2*t)*sum_i y_i(0)+integral_0^t exp(-2*(t-s))*x(s) ds.

All hidden contributions have the same decay kernel. Define v=sum_i y_i. Differentiating the sum gives the two-variable model:

x_dot=-x+v+u(t), v_dot=x-2*v, v(0)=sum_i y_i(0).

The derived equations and initial sum preserve x for every supplied input for which these linear equations have their solution. For n>1 this uses two evolving quantities instead of n+1. The required hidden initial information is their sum. Arbitrarily setting v(0)=0 would already change x_dot(0) when the actual sum is nonzero.

The initial sum requires n terms once. Each subsequent evaluation of the reduced right-hand side uses x, v and u(t); it no longer recomputes n component contributions. This saves repeated arithmetic when those components would otherwise be advanced separately.

Now one component has decay rate 3 instead of 2. Summing produces v_dot=x-2*v-y_1; the old two-variable model has lost a contribution. Retain y_1 separately and update it by y_1_dot=c_1*x-3*y_1, or separate the two decay groups. The change determines which added state is needed. The same grouping can combine any components that share their response kernel; different kernels remain distinct until another justified approximation combines them.