3.2 Finite Dimensional Linear Time-invariant (LTI) Systems
The most simple and yet important system class is the class of finite di- mensional linear time-invariant systems (abbreviated as FDLTI, or simply LTI systems). These systems and the analysis and control techniques based thereon are the subject of standard introductory courses on systems and control based on a state-space approach. The majority of the well-known and popular analysis and control techniques, such as controllability and ob- servability analysis, Kalman filtering or LQR (Linear Quadratic Regulator) theory, were developed and investigated for such systems and later on have been extended to more complicated system classes.
Therefore, the LTI case will be used as a reference case throughout the book. Most of the new techniques and approaches will build on these used for LTI systems.
3.2.1 The General Form of State-space Models
For continuous time LTI systems the general form of state-space models is as follows:
˙
x(t) =Ax(t) +Bu(t) (state equation)
y(t) =Cx(t) +Du(t) (output equation) (3.1)
with given initial conditionx(t0) =x(0) and
x(t)∈Rn, y(t)∈Rp, u(t)∈Rr
being the state, output and input vectors of finite dimensional spaces and A∈Rn×n, B∈Rn×r, C ∈Rp×n, D∈Rp×r
being matrices with constant (time-independent) elements.
It is important to note the above state-space model applies also for MIMO systems whenp >1, r >1.
Most often we suppress t denoting the time dependence from the state, input and output signals and write simplyxinstead ofx(t).
Definition 3.2.1 (LTI state-space representation)
The state-space representation (SSR) of LTI systems is the quadruplet of constant matrices(A, B, C, D)in Equation (3.1). The dimension of an SSR is the dimension of the state vector:dimx(t) =n. The state-space X is the set of all states:
x(t)∈ X, dimX =n
Example 3.2.1 (LTI state-space model equation)
Let us consider the following state-space model:
˙ x=
"
−Vvcc −cP cU AρcVc
U A cP cρcVc
U A
cP hρhVh −vVhh −cP hU AρhVh
# x+
vc
Vc 0 0 vVhh
u (3.2)
y=x (3.3)
withx, u, y ∈ R2being two-dimensional vectors. If all the model parameters
vc, Vc, U, A, cP c, ρc, vh, Vh, cP h, ρh
are constants, then the above model is indeed linear and time- invariant, with the state equation (3.2) and output equation (3.3) and with the special output matrices (C=I, D= 0 ).
Note that the above LTI state-space model describes a heat ex- changer cell, the process system description of which is introduced in detail later in Subsection 4.4.2.
3.2.2 Linear Transformation of States
State-space models are not unique: if we have a state-space representation (A, B, C, D) for an LTI system, then we can easily find infinitely many other ones with the same dimension.
Definition 3.2.2 (Equivalent state-space models)
Two state-space representations are equivalent if they have the same input–
output description.
Equivalent state-space models of LTI systems can be generated by applying coordinate transformation on the state-space. Transforming the coordinates in the state-space is often very useful in order to highlight some properties of interest (e.g.reachability, observability,etc.), or to show how certain control problems can be solved.
In the case of LTI state-space models, equivalent state-space representa- tions can be obtained bylinear coordinate transformation, which relates two possible equivalent state-space models
˙
x(t) =Ax(t) +Bu(t), x(t) =˙ Ax(t) +Bu(t)
y(t) =Cx(t) +Du(t), y(t) =Cx(t) +Du(t) (3.4) related by the transformation
T ∈Rn×n, det T 6= 0
3.2 Finite Dimensional Linear Time-invariant (LTI) Systems 27
i.e.
x=T x ⇒ x=T−1x (3.5)
Observe that we do not transform the input and output signals because we want to preserve the input–output behavior of the state representations to have equivalent state-space models.
If we transform the first set of state equations in Equation (3.4) using the transformation matrixT, we get
dimX = dimX =n (3.6)
and
T−1x˙ =AT−1x+Bu so finally
x˙ =T AT−1x+T Bu, y=CT−1x+Du (3.7) In this way, we can define infinitely many state variables for the same system, and the realization matrices are related by
A=T AT−1, B=T B, C=CT−1, D=D (3.8)
3.2.3 Special Realization Forms of LTI Systems
Realizations of special forms play an important role in investigating the dy- namic properties of an LTI system.
Diagonal Form Realization. Diagonal form realization is characterized by a diagonal state matrixAthat plays a role in analyzing asymptotic stability.
It is important to note that there are systems which cannot be transformed into diagonal forms (it is known from linear algebra that the necessary and sufficient condition for diagonalizing ann×nquadratic matrix is that it has nlinearly independent eigenvectors).
Definition 3.2.3 (Diagonal form realization of LTI systems)
A diagonal form realization is a realization with the matrices(A, B, C)in the following special form:
A=
λ1. . . 0 . . . . . . . . . . . . . . . 0 . . . λn
, B=
b1
. . . bn
, C=
c1. . . cn
Note that theλi parameters present in the state matrix Aof a diagonal form realization are the eigenvalues of the matrix and the poles of the system.
A suitable transformation matrixT that brings a given realization (A, B, C) into its diagonal form can be constructed from the eigenvectors of the state matrixA.
Controller Form Realization. This realization plays a role in assessing the joint controllability and observability of an LTI system because it is always controllable.
Definition 3.2.4 (Controller form realization of LTI systems) The controller form realization of an LTI system is given by the state-space model
˙
x(t) =Acx(t) +Bcu(t)
y(t) =Ccx(t) (3.9)
with the matrices of the following special form:
Ac=
−a1−a2. . . −an
1 0 . . . 0 . . . . . . . . . . . . 0 0 . .1 0
with the coefficients of the polynomiala(s) =sn+a1sn−1+...+an−1s+an
and
Bc=
1 0 . . . 0
Cc =
b1b2. . . bn
with the coefficients of the polynomialb(s) =b1sn−1+...+bn−1s+bn where the above polynomials appear in the transfer functionH(s) = b(s)a(s) (see later in Definition 5.1.2 in Subsection 5.1.2).