Reachability and Controllability
• Reachability. The reachability problem is to “find the set of all the final states x(t1) reachable starting from a given initial state x(t0)”:
– A state x(t1) of a dynamic system is reachable from the state x(t0) in the time interval [t0, t1] if it exists an input function u(·) ∈ U such that x(t1) = ψ(t0, t1, x(t0), u(·)).
– Let X+(t0, t1, x(t0)) denote the “set of all the final states x(t1) reachable at time t1 starting from the initial state x(t0)”.
• Controllability. The controllability problem is “to find the set of all the initial states x(t0) controllable to a given final state x(t1):
– A state x(t0) of a dynamic system is controllable to state x(t1) in the time interval [t0, t1] if it exists an input function u(·) ∈ U such that x(t1) = ψ(t0, t1, x(t0), u(·)).
– Let X−(t0, t1, x(t1)) the “set of all the initial states x(t0) controllable to the final state x(t1) at time t1”.
-
t0 t1
x(t1)
X+(t0, t1, x(t0)) X−(t0, t1, x(t1))
x(t0)
9 z
• For time-invariant systems one can use t0 = 0 and t1 = t:
X+(t0, t1, x(t0)) ⇒ X+(t, x(0)), X−(t0, t1, x(t1)) ⇒ X−(t, x(t))
• For discrete-time systems it is t → k:
X+(t, x(0)) ⇒ X+(k, x(0)), X−(t, x(t)) ⇒ X−(k, x(k))
Discrete time-invariant linear systems
let us consider the following discrete time-invariant linear system:
x(k + 1) = Ax(k) + Bu(k)
Reachability
• The set X+(k) of all the states reachable from the origin in k steps is equal to the set of all the states x(k) obtained starting from the initial condition x(0) = 0 and considering only the forced evolution of the system:
x(k) =
k−1
X
j=0
A(k−j−1)Bu(j) = [B AB . . . Ak−1B]
u(k − 1) u(k − 2)
...
u(0)
and varying the input u(0), u(1), . . . , u(k − 1) in all the possible ways.
• Definition. Reachability matrix in k steps:
R+(k) = [B AB . . . A△ k−1B]
• The set X+(k) of all the states reachable from the origin in k steps is a vectorial space which is equal to the image of matrix R+(k):
X+(k) = Im[R+(k)]
• The subspaces X+(k) reachable in 1, 2, . . . , k steps satisfy the following chain of inclusions (n is the dimension of the state space):
X+(1) ⊆ X+(2) ⊆ . . . ⊆ X+(n) = X+(n + 1) = . . .
• The maximum reachable subspace X+(n) is obtained, at the most, in n steps.
R+ △= R+(k)
k=n = R+(n) = [B AB . . . An−1B]
• The subspace X+ of all the state reachable from the origin in a time interval however long is equal to the image of matrix R+:
X+ = Im[B AB . . . An−1B] = ImR+
• Definition. A system is reachable if the subspace X+ of all the reachable states from the origin is equal to the whole state space X:
X+ = X
• Necessary and sufficient condition for a system to be reachable is:
rank(R+) = n
• For discrete, time-invariant linear systems the set X+(k, x0) has the struc- ture of a “linear variety”:
X+(k, x0) = Akx0 + ImR+(k)
• Graphical representation:
X+(k, x0)
R+(k) Akx0
x1
x2
Controllability
• A state x0 = x(0) is controllable to zero in k steps if it exists an input sequence u(0), u(1), . . . , u(k − 1) which brings the initial state x0 to a final state equal to the origin x(k) = 0 in the time interval [0, k]:
0 = x(k) = Akx0 +
k−1
X
j=0
A(k−j−1)Bu(j) that is:
−Akx0 =
k−1
X
j=0
A(k−j−1)Bu(j) ⇒ Akx0 ∈ X+(k) = ImR+ if the state “−Akx(0)” is reachable from the origin in k steps.
• Property. A system is controllable if and only if the following relation holds:
ImAn ⊆ X+(n) = ImR+ where R+ is the reachability matrix of the system.
• For discrete linear systems the reachability and controllability properties are NOT equivalent:
1) The reachability implies the controllability.
reachability ⇒ controllability
In fact the reachability implies X+ = X from which it follows that:
ImAn ⊆ X+ = X, that is the system is surely controllable.
2) The controllability does not imply the reachability:
controllability 6⇒ reachability In fact if, for example A = 0 and rank(B) < n, then:
rank([B AB . . . An−1B]) = rank([B 0 . . . 0] < n that is the system is not reachable even if it is controllable.
• If A is a full rank matrix, then the reachability and the controllability imply one another.
Invariant time-continuous linear systems
Let us consider the following invariant time-continuous linear system:
˙x = Ax(t) + Bu(t)
Reachability:
• A state x(t) is reachable at time t starting from zero if it exists an input function u(·) such that:
x(t) = Z t
0
eA(t−τ)Bu(τ )dτ
• Let X+(t) be the set of all the states reachable from the origin x(0) = 0 in the time interval [0, t] and let X+ denote be the set of all the states reachable from the origin x(0) = 0 in the time interval [0, ∞].
• Let Rt denote the linear function Rt : U → X defined as follows:
Rt : u(·) → x(t) = Z t
0
eA(t−τ)Bu(τ )dτ
• The set U is infinite dimensional. The states x(t) reachable at time t are all the states which belongs to the image of the linear function Rt:
X+(t) = {x : x ∈ ImRt}
• The set X+(t), being the image of a linear function, is a vectorial subspace of the state space X.
• Property. For each t > 0, the reachable subspace X+(t) is the image of the reachability matrix R+:
X+(t) = X+ = ImR+
• For continuous-time systems, the reachable subspace does NOT depend on the length of the time interval [0, t].
• The smaller is the time interval [0, t] the larger is the control action u(t).
Controllability
. For invariant time-continuous linear systems, the con- trollable subspace X− does not depend on the amplitude of the time interval [0, t] and it is equal to the reachable subspace X+.————–
Example. Let us consider the following electrical network:
IL
J(t) V
L
R
C VC
R
The dynamic equations of the systems are:
Ld IL
dt = VC + R(J − IL) − R IL Cd VC
dt = J − IL
V = VC + R(J − IL)
where ILis the current which flows in the inductance, VC is the voltage across the capacitor, J is the input current and V is the output voltage. In matrix form, the system dynamics can be represented as follows:
˙x =
" −2R
L 1 L
−1 C 0
# x+
" R
L 1 C
# J V = −R 1 x + [ R ]J
x=
IL
VC
The reachability matrix of the system is R+ =
" R
L 1
LC − 2RL22 1
C −LCR
#
, det R+ = 1 LC
R2 L − 1
C
The system is reachable only if R+ is a full rank matrix. The system is NOT completely reachable if:
R2 = L
C ↔ R C = L
R
that is if the inductance time constant LR is equal to the capacitor time constant R C. In this case the two system eigenvalues are coincident: λ1,2 = −√LC1 .
˙x(t) =
1 0 0 1 1 0 0 1 1
x(t)+
1
−1 0
u(t)
Reachability matrix R+ and computation of the subspace X+:
R+ = b Ab A2b =
1 1 1
−1 0 1
0 −1 −1
, X+ = ImR+ = Im
1 0 0 0 1 0 0 0 1
. The system is reachable.
————–
Example. Compute the reachability matrix R+ of the following system:
˙x(t) =
1 −1 0 0 1 −1
0 0 1
x(t) +
1 1 0
u(t) Reachability matrix R+ and computation of the subspace X+:
R+ = b Ab A2b =
1 0 −1 1 1 1 0 0 0
, X+ = ImR+ = Im
1 0 1 1 0 0
. The system is NOT completely reachable.
————–
Example. Compute the reachability matrix R+ of the following system:
˙x(t) =
1 0 0 0 1 −1
−1 0 1
x(t)+
0 0 1 1 0 1
u(t)
Reachability matrix R+ and computation of the subspace X+:
R+ = B AB A2B =
0 0 | 0 0 | 0 0 1 1 | 1 0 | 1 −1 0 1 | 0 1 | 0 1
, X+ = ImR+ = Im
0 0 1 1 0 1
. The system is NOT completely reachable
Equivalent systems
• Property. Discrete or continuous-time linear systems which are algebrai- cally equivalent have the same reachability properties.
• Let T be a full rank transformation matrix which links two algebraically equivalent time-invariant linear systems S = (A, B, C, D) and ¯S = (A, B, C, D):
x = Tx → ( A = T−1AT, B = T−1B
C = CT, D = D
The reachability subspaces in k steps of the two systems S and ¯S, that is X+(k) and X+(k), are linked by the following relation:
X+(k) = Im[B . . . Ak−1B] = Im(T−1[B . . . Ak−1B]) = T−1X+(k)
• The subspace X+ is invariant with respect to a state space transformation:
x = Tx → X+ = TX+
• Let R+ and R+ be the reachability matrices of the two systems. The following relation holds:
R+ = T−1R+ ⇔ R+ = TR+
• If the two systems have only one input, R+ and R+ are squared full rank matrices which satisfy the following relation:
T = R+(R+)−1
