QP Matrices

This section describes the matrices associated with the model predictive control optimization problem described in Optimization Problem.

• Prediction
• Optimization Variables
• Cost Function
• Constraints

Prediction

Assume for simplicity that the disturbance model in Equation 2-1 and Equation 2-2 is a unit gain (i.e., d(k)=nd(k) is a white Gaussian noise). For simplicity, denote by

Then, the prediction model given by

• 

• .

Consider for simplicity the prediction of the future trajectories of the model performed at time k=0. We set n_d(i)=0 for all prediction instants i, and obtain

which gives

• 

where

• 

• 

• 

Optimization Variables

Let m be the number of free control moves and denote by z= [z₀; ...; z_m-1]. Then,

(2-7)

where J_M depends on the choice of blocking moves. Together with the slack variable , vectors z₀, ..., z_m-1 constitute the free optimization variables of the optimization problem (in case of systems with a single manipulated variables, z₀, ..., z_m-1 are scalars).

Figure 2-3: Blocking Moves: Inputs and Input Iincrements for moves=[2 3 2]

Consider for instance the blocking moves depicted in Figure 2-3, which corresponds to the choice moves=[2 3 2], or, equivalently, u(0)=u(1),  u(2)=u(3)=u(4),  u(5)=u(6), u(0)=z0,  u(2)=z1,  u(5)=z2, u(1)= u(3)= u(4)= u(6)=0.

Then, the corresponding matrix J_M is

• 

Cost Function

Standard Form

The function to be optimized is

where

(2-8)

Finally, after substituting u(k), u(k), y(k), J(z) can be rewritten as

(2-9)

Note    In order to keep the QP problem always strictly convex, if the condition number of the Hessian matrix KU is larger than 1012, the quantity 10*sqrt(eps) is added on each diagonal term. This may only occur when all input rates are not weighted (Wu=0) (see Weights).

Alternative Cost Function

If the alternative cost function shown in Equation 2-5 is being used, Equation 2-8 is replaced by the following:

(2-10)

where the block-diagonal matrices repeat p times, i.e., once for each step in the prediction horizon.

You also have the option to use a combination of the standard and alternative forms. See Weights for more details.

Constraints

Let us now consider the limits on inputs, input increments, and outputs along with the constraint 0.

• 

Note    Upper and lower bounds that are not finite are removed, as well as the input and input-increment bounds over blocked moves.

Similarly to what was done for the cost function, we can substitute u(k), u(k), y(k), and obtain

(2-11)

where matrices M_z,M,M_lim,M_v,M_u,M_x are obtained from the upper and lower bounds and ECR values.

Function mpc_buildmat constructs the QP problem matrices.

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State Estimation

Model Predictive Control Computation

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