ABSTRACT
What happens if we apply the optimal controls u(t), t0 ≤ t ≤ tf, computed in the previous chapter, directly to the greenhouse? Will the greenhouse climate behavior x(t), t0 ≤ t ≤ tf be optimal? In other words, will x(t) = x*(t), t0 ≤ t ≤ tf hold? The most simple and straightforward answer would be yes. Unfortunately, this answer is only correct if all elements of the optimal control problem formulation are perfect descriptions of reality. Most optimal control problem formulations like the ones for greenhouses are not. The initial condition,
x(t0) = x0, (4.1)
and the dynamic systems model,
x f x u d p( ) ( ( ), ( ), ( ), )t t t t= , (4.2)
are at best very good approximations of reality. When controlling a greenhouse the external input trajectory d(t), t0 ≤ t ≤ tf, needed to solve the optimal control problem, will not be perfect, either, because it involves weather predictions. The cost function
J t t t L t t t t t
( ( )) , ( ( ), ( ), ( ), ) f
u x x u d p= ( )( ) + ∫Φ f f d 0
, (4.3)
is a description of the control objectives and as such can be perfect. Because the dynamic model, the initial condition and the external input trajectory d(t), t0 ≤ t ≤ tf are at best very good approximations x(t) = x*(t), t0 ≤ t ≤ tf will not hold. Stated differently,
Δx(t) = x(t) – x*(t) ≠ 0, (4.4)
for almost any t0 ≤ t ≤ tf. The associated important question now is: How large are the state perturbations (deviations) Δx(t)? This clearly depends on the magnitude of the errors just mentioned and the way these propagate. Without going into details, practical applications require a feedback control mechanism to limit the state perturbations in Equation 4.4. Observe that in Equation 4.4 x(t) represents actual system behavior. To obtain information about actual system behavior measurements must be performed on the system. Variables that are measured are also called outputs of the system. In general they are functions of the current state, input, external input and the parameters of the system. This is described by,
y(t) = g(x(t), u(t), d(t), p). (4.5)
In Equation 4.5 y is a ny dimensional column vector, its components being the measured variables (outputs), and g is the associated vector function. For control purposes the most favorable situation is,
y(t) = x(t), (4.6)
meaning that all state variables (the complete state) are measured. When Equation 4.6 applies, control engineers call this “having full state information.” In most practical circumstances, however, instead of the special case Equation 4.6, the general description Equation 4.5 applies. In that case state estimators are applied to obtain estimates ˆ( )x t of x(t). These state estimators are designed using Equations 4.1, 4.2, and 4.5. State estimation will be briey treated in Section 4.2.
