Model Predictive Control System Design and Implementation Using ...

1.5 Predictive Control of MIMO Systems 23
where I q×q is the identity matrix with dimensions q × q, which is the number
of outputs; and o m is a q × n 1 zero matrix. In (1.38), A m , B m and C m have
dimension n 1 × n 1 , n 1 × m and q × n 1 , respectively.
For notational simplicity, we denote (1.38) by
x(k +1)=Ax(k)+BΔu(k)+B ɛ ɛ(k)
y(k) =Cx(k), (1.39)
where A, B and C are matrices corresponding to the forms given in (1.38).
In the following, the dimensionality of the augmented state-space equation is
taken to be n (= n 1 + q).
There are two points that are worth investigating here. The first is related
to the eigenvalues of the augmented design model. The second point is related
to the realization of the state-space model. Both points will help us understand
the model.
Eigenvalues of the Augmented Model
Note that the characteristic polynomial equation of the augmented model is
]
λI − Am o
ρ(λ) =det[ T m
−C m A m (λ − 1)I q×q
=(λ − 1) q det(λI − A m ) = 0 (1.40)
where we used the property that the determinant of a block lower triangular
matrix equals the product of the determinants of the matrices on the diagonal.
Hence, the eigenvalues of the augmented model are the union of the eigenvalues
of the plant model and the q eigenvalues, λ = 1. This means that there are
q integrators embedded into the augmented design model. This is the means
we use to obtain integral action for the MPC systems.
Controllability and Observability of the Augmented Model
Because the original plant model is augmented with integrators and the MPC
design is performed on the basis of the augmented state-space model, it is
important for control system design that the augmented model does not become
uncontrollable or unobservable, particularly with respect to the unstable
dynamics of the system. Controllability is a pre-requisite for the predictive
control system to achieve the desired closed-loop control performance and observability
is a pre-requisite for a successful design of an observer. However,
the conditions may be relaxed to the requirement of stabilizability and detectability,
if only closed-loop stability is of concern. 1 In this book, unless it is
1 A system is stabilizable if its uncontrollable modes, if any, are stable. Its controllable
modes may be stable or unstable. A system is detectable, if its unobservable
modes, if any, are stable. Its observable modes may be stable or unstable. Stable
modes here means that the corresponding eigenvalues are strictly inside the unit
circle.