mpcaugss (Model Predictive Control Toolbox)

Model Predictive Control Toolbox

mpcaugss

Differences the states of a system and augments them with the output variables. Mainly used as a utility function for setting up model predictive controllers.

Syntax

[phia,gama,ca] = mpcaugss(phi,gam,c) 
[phia,gama,ca,da,na] = mpcaugss(phi,gam,c,d)

Description

Consider the process shown in the above block diagram. A state-space representation is:

where x is a vector of n state variables, u is a vector of nu manipulated variables, d is a vector of nd measured disturbances, w is a vector of nw unmeasured disturbances, y is a vector of ny plant outputs, z is measurement noise, and , u, d, w, etc., are constant matrices of appropriate size. The variable = Cx(k) represents the plant output before the addition of the direct contribution of the inputs [Duu(k) + Ddv(k) + Dww(k)] and the measurement noise [z(k)]. (The variable is the output before addition of the measurement noise). Define:

u(k) = u(k) - u(k - 1) x(k) = x(k) - x(k - 1)

etc. Then equations 4.28 and 4.29 can be converted to the form

xa(k+1) = axa(k) + ua u(k) + da (k) + wa w(k)

y(k) = Caxa(k) + Duu(k) + Ddd(k) + Dww(k) + z(k)

where, by definition,

The mpcaugss function takes the matrices , (= [u d w]), C as input, and creates the augmented matrices a, a, Ca and Da in the form shown above. The D input matrix is optional. If you include it, mpcaugss assumes it has the form D = [Du Dd Dw]. If you omit it, the default is zero. Note that all MPC design routines require Du = Dd = 0.

The last output variable, na, is the order of the augmented system, i.e.,
na = n + ny. It is optional.

Example
The following system has 2 states, 3 inputs, and 2 outputs.

phi=diag([0.8, -0.2]);
gam=[1 -1 0;0 2 -0.5];
c=[0.4 0;0 1.5];

Here is the augmentation command, followed by the calculated results:

[phia,gama,ca,da,na]=mpcaugss(phi,gam,c)
phia =
    0.8000         0         0         0
         0   -0.2000         0         0
    0.3200         0    1.0000         0
         0   -0.3000         0    1.0000
gama =
    1.0000   -1.0000         0
         0    2.0000   -0.5000
    0.4000   -0.4000         0
         0    3.0000   -0.7500
ca =
     0     0     1     0
     0     0     0     1
da =
     0     0     0
     0     0     0
na =
     4

mod2step, step format mpccl