2.1. A Simple Differential Equation Model

First an impetus is provided for solving differential eq uations in an approach unique to control analysis. The mass balance of a well-mixed tank can be written (see Review Problems) as

where C is the concentration of a component, C _in is the inlet concentration, C ₀ is the initial concentration, and ? is the space time. In classical control problems, we invariably rearrange the equation as

(2.1)

and further redefine variables C' = C - C ₀ and = C _in - C ₀. ^[1] We designate C' and as deviation variables - they denote how a quantity deviates from the original value at t = 0. ^[2] Because C ₀ is a constant, we can rewrite Eq. (2.1) as

(2.2)

Note that the equation now has a zero initial condition. For reference, the solution to Eq. (2.2) is ^[3]

(2.3)

If is zero, we have the trivial solution C' = 0. It is obvious from Eq. (2.2) immediately. For a more interesting situation in which C' is nonzero or for C to deviate from the initial C ₀, must be nonzero, or in other words, C _in is different from C ₀. In the terminology of differential equations, the right-hand side (RHS) is called the forcing function. In control, it is called the input. Not only is