Unidad Orgánica: Oficina General de Planeamiento y Presupuesto Oficina de Presupuesto
I SEM III TRIM IV TRIM ANUAL I SEM III TRIM IV TRIM ANUAL CORRECTIVOS
From a conceptual point of the view, the input parametrization problem, as stated in the previous section, is the following. Let a model be given of the system that is to be controlled. The input of the system is given by u(t) ∈ IRnu and the state by x(t) ∈ IRn.
The inputs and states are constrained by Kuu(t) < ku and Kxx(t) < kx respectively. At this point the system can be either linear or nonlinear but it is assumed to be discrete time. Let the on-line optimization problem for receding horizon control be given by
inf
U∈UJ (U, x0) (5.1)
where U is the input trajectory which is an element of some setU which is totally specified by the input and state constraints imposed on the system, J (·) is a cost function that reflects the desired performance and x0 is some variable that is a function of the mea- surement with which the optimization problem is initialized at each time instant t. The variable x0 can be either a measured state (full information state feedback) or a predicted state (partial information state feedback) or the measured output (general dynamic out- put feedback).
The input parametrization problem is to find a suitable subspace Ur with a lower di- mension than the full space U such that the solution to the reduced order optimization problem
inf U∈U∩Ur
J (U, x0) (5.2)
provides good control for all possible measurements x0.
A possible way to quantify good control is by making the reduced problem deviate as little as possible from the original problem. A possible way to choose a space Ur of dimension
p is then by solving
min Up,dim(Up)≤p
where V (·) reflects the deviation of the reduced order problem to the full order problem. Possible cost functions are
• minimize the worst-case deviation in value of the cost function V1(Up) = sup
x0∈X
/
min
U∈U∩Up
J (U, x0) − minU∈UJ (U, x0)
0
(5.4)
where the measurement x0 varies over some set X .
• minimize the worst-case deviation between the optimal input trajectories V2(Up) = sup
x0∈X
11
11arg min
U∈U∩Up
J (U, x0)−arg minU∈UJ (U, x0)1111 γ
(5.5)
where this deviation is measured in some norm · γ.
• minimize the worst-case difference between the first samples of the optimal input
trajectories
V3(Up) = sup x0∈X
[Inu 0 · · · 0](Ur∗(Up, x0) − U∗(x0)) γ (5.6) with U∗(Up, x0) = arg min
U∈U∩Up
J (U, x0) and U∗(x0) = arg min
U∈UJ (U, x0). This function seems suitable because only the first sample of the input trajectory is actually applied in a receding horizon strategy.
The optimization problems discussed above are related to n-width problems [119] where an optimal n-dimensional subspace is calculated by an optimization of a cost function with a free variable that is a set of a certain dimension. Unfortunately, the problems stated above are as such intractable. This is mainly because the cost functions must be evaluated for all possible sets of active constraints which is combinatorial in size. Therefore, only in specific cases a solution can be found.
Let the applied cost function be given by the quadratic function
J (u(·)) =
P−1 t=0
{xT(t)Q
1x(t) + uT(t)Q2u(t)} + xT(P )Q0x(P ). (5.7)
which is equivalent to the one used in the LQR problem (2.4.1) and the MPC problem (2.19). In terms of the stacked input vector UT = [uT(0)· · · uT(P − 1)] this cost function is given by
J (U ) = UTHU + 2xT0gTU (5.8) as explained in section 2.4.4.
The first case in which a solution to the input parametrization problem (5.3) can be found, is the unconstrained case. In this case the optimal control profiles are generated by a time-varying state feedback which can be calculated a priori. This controller is given in (2.16) where use is made of the Riccati difference equation (2.17). Note that the
control problem that is solved is an open loop control problem, while the solution is a closed loop controller. The only variable that cannot be computed a priori is the state
x0 with which the optimization problem is initialized at each time instant t. This simple observation leads to the following lemma for the subspace which contains all solutions to the discrete-time LQR problem.
Lemma 5.2.1 Consider the quadratic cost function given by (5.7) with finite horizon P
and let there be no constraints. Then, the subspace Ur = im{H−1g}
with H ∈ IRP nu×P nu, g ∈ IRP nu×n given in (5.8) is a solution of (5.3) with any of the cost
functions (5.4), (5.5) or (5.6) and X = IRn.
Proof: The lemma follows from the fact that the unconstrained solution to (5.8) is given
by U (x0) = −H−1gx0. ✷
Apparently if one wants to find the unconstrained optimum for all possible initializations
x0, a search over an n-dimensional subspace is sufficient instead of a P nu-dimensional one.
Lemma 5.2.1 holds for a finite horizon criterion but a similar simple result also holds for the infinite horizon case.
Lemma 5.2.2 Consider the quadratic cost function given by (2.19) with infinite horizon
P =∞ and let there be no constraints. Then, the subspace
Ur = im{ F F (A− BF ) F (A− BF )2 .. . } ⊂ ln 2[0,∞), (5.9)
with {A, B} state space matrices given in (2.20) and F the LQ-optimal state feedback given by F = (BTXB + Q
2)−1BTXA with X the unique nonnegative definite solution of
the Algebraic Riccati Equation
X = AT[X − XB(BTXB + Q2)−1BTX]A + Q1,
is a solution of (5.3) with any of the cost functions (5.4), (5.5) or (5.6) and X = IRn.
Proof: The unconstrained solution to the problem (2.24) with infinite horizons is the LQ
optimal control profile given by
u(t, x0) = −F (A − BF )tx0, t = 0, 1, 2, . . .
The lemma above indicates that an efficient input parametrization for infinite horizon model predictive control is generated by a dynamical system {F, A − BF }. With this parametrization the infinite dimensional optimization problem is reduced to a finite di- mensional optimization problem with a number of free variables that is equal to the model order.