3. Closed-loop control with state estimation
y[k]
u[k]
System
Controller
r[k]
Ĝ[z]
State
Estimator
The instability of a open loop controller is caused by gradual accumulation of
disturbances. The control signal becomes unsynchronised with the system.
⊲ If we can estimate the system state from the output, we can circumvent
such synchronisation errors. The system must be observable for that.
⊲ Even if we know the system state, it might be impossible to track the
reference signal. In brief, the system must be controllable.
1
4. Controllability
The state equation is controllable if for any input state x0 and for any final
state x1 there exists an input u that transfers x0 to x1 in a finite time.
⊲ T6. The n dimensional pair (A, B) is controllable iff the controllability
matrix
B AB A2
B An−1
B
has the maximal rank n.
The desired input can be computed from the system of linear equations
x1 = An
x0 + An−1
Bu[0] + An−2
Bu[1] + · · · + Bu[n − 1] .
Although the theorem T6 gives an explicit method for controlling the
system, it is an off-line algorithm with a time lag n.
A good controller design should yield a faster and more robust method.
2
5. Observability
The state equation is observable if for any input state x0 and for any input
signal u, finite the output y sequence determines uniquely x0.
⊲ T7. The pair (A, C) is observable iff the observability matrix
C
CA
. . .
CAn−1
has the maximal rank n.
3
6. Offline state estimation algorithm
Again, the input state x0 can be computed from a system of linear equations
y[0] = Cx0 + Du[0]
y[1] = CAx0 + CBu[0] + Du[1]
· · ·
y[n − 1] = CAn−1
x0 + · · · + CBu[n − 2] + Du[n − 1]
Although this equation allows us to find out the state of the system, it is
offline algorithm, which provides a state estimation with a big time lag.
A good state space estimator must be fast and robust.
4
7. General structure of state estimators
y[k]
u[k] System
State Estimator
x̂[k]
x[k] = Ax[k − 1] + Bu[k − 1]
y[k] = Cx[k] + Du[k]
A state estimate x̂[k] is updated according to u[k] and y[k]:
⊲ Update rules are based on linear operations.
⊲ The state estimator must converge quickly to the true value x[k]
⊲ The state estimator must tolerate noise in the inputs u[k] and y[k].
5
8. Example
Consider a canonical realisation of the transfer function ĝ[z] = 0.5z+0.5
z2−0.25
A =
0 0.25
1 0
B =
1
0
C = [0.5 0.5] D = 0
Then a possible stable state estimator is following
x̂[k + 1] = Ax̂[k] + Bu[k] + 1(y[k] − Cx̂[k]
| {z }
ŷ[k]
) .
In general, the feedback vector 1 can be replaced with any other vector to
increase the stability of a state estimator.
6
10. Equivalent state equations
Two different state descriptions of linear systems
(
x[k + 1] = A1x[k] + B1u[k]
y[k] = C1x[k] + D1u[k]
(
x[k + 1] = A2x[k] + B2u[k]
y[k] = C2x[k] + D2u[k]
are equivalent if for any initial state x0 there exists an initial state x0 such
that for any input u both systems yield the same output and vice versa.
⊲ T8. Two state descriptions are (algebraically) equivalent if there exists
an invertible matrix P such that
A2 = P A1P −1
B2 = P B1
C2 = C1P −1
D2 = D1 .
7
11. Linear state space transformations
The basis e1 = (0, 1) and e2 = (1, 0) is a canonical base in R2
. However a
basis a1 = (1, 1) and a2 = (1, −1) is also a basis.
Now any state x = x1e1 + x2e2 can be represented as x = x1a1 + x2a2
and vice versa. The latter is known as a basis transformation:
x1 =
x1 + x2
2
x2 =
x1 − x2
2
(
x1 = x1 + x2
x2 = x1 − x2
For obvious reasons, we can do all computations wrt the basis {a1, a2} so
that the underlying behaviour does not change. The same equivalence of
state descriptions hold for other bases and larger state spaces, as well.
8
12. Kalman decomposition
⊲ T9. Every state space equation can be transformed into an equivalent
description to a canonical form
xco[k + 1]
xco[k + 1]
xco[k + 1]
xco[k + 1]
=
Aco 0 A13 0
A21 Aco A23 A24
0 0 Aco 0
0 0 A43 Aco
xco[k]
xco[k]
xco[k]
xco[k]
+
Bco
Bco
0
0
u[k]
y[k] = [Cco 0 Cco 0] x[k] + Du[k]
where
⋄ xco is controllable and observable
⋄ xco is controllable but not observable
⋄ xco is observable but not controllable
⋄ xco is neither controllable nor observable
9
13. Minimal realisation
⊲ T10. All minimal realisations are controllable and observable. A
realisation of a proper transfer function ĝ[z] = N(z)/D(z) is minimal iff
it state space dimension dim(x0) = deg D(z).
10
15. General setting
According to Kalman decomposition theorem, the state variables can be
divided into four classes depending on controllability and observability.
⊲ We cannot do anything with non-controllable state variables.
⊲ Non-observable variables can be controlled only if they are marginally
stable. We can do it with an open-loop controller.
⊲ For controllable and observable state variables, we can build effective
closed-loop controllers.
Simplifying assumptions
⊲ From now on, we assume that we always want to control state variables
that are both controllable and directly observable: y[k] = x[k].
⊲ If this is not the case, then we must use state estimators to get an
estimate of x. The latter just complicates the analysis.
11
16. Unity-feedback configuration
y[k]
u[k] System
Controller
r[k]
ĝ[z]
Ĉ[z]
p +
-1
Design tasks
⊲ Find a compensator ĉ[z] such that system becomes stable.
⊲ Find a proper value of p such that system starts to track reference signal.
It is sometimes impossible to find p such that y[k] ≈ r[k].
⊲ The latter is impossible if ĝ[1] = 0, then the output y[k] just dies out.
12
17. Overall transfer function
Now note that
ŷ = ĝ[z]ĉ[z](pr̂ − ŷ) ⇐⇒ ŷ =
pĝ[z]ĉ[z]
1 + ĝ[z]ĉ[z]
r̂
and thus the configuration is stable when the new transfer function
ĝ◦[z] =
pĝ[z]ĉ[z]
1 + ĝ[z]ĉ[z]
has poles lying in the unit circle. Now observe
ĝ[z] =
N(z)
D(z)
, ĉ[z] =
B(z)
A(z)
=⇒ ĝ◦[z] =
pB(z)N(z)
A(z)D(z) + B(z)N(z)
13
18. Pole placement
We can control the denominator of the new transfer function ĝ◦[z]. Let
F(z) = A(z)D(z) + B(z)N(z)
be a desired new denominator. Then there exists polynomials A(z) and
B(z) for every polynomial F(z) provided that D(z) and N(z) are coprime.
The degrees of the polynomials satisfy deg B(z) ≥ deg F(z) − deg D(z).
ℑ(s)
ℜ(s)
BIBO
stable
ℜ(z)
ℑ(z)
BIBO
stable
14
19. Signal tracking properties
Let ĝ◦[z] be the overall transfer function.
⊲ The system stabilises if ĝ◦[z] is BIBO stable.
⊲ The system can track a constant signal r[k] ≡ a if ĝ◦[1] = 1.
⊲ The system can track a ramp signal r[k] ≡ ak if ĝ◦[1] = 1 and ĝ′
◦[1] = 0.
The latter follows from the asymptotic convergence
y[k] → aĝ′
◦[1] + kaĝ◦[1]
for an input signal r[k] = ak. Moreover, let ĝ◦ = N(z)/D(z). Then
ĝ◦[1] = 1 ∧ ĝ′
◦[1] = 0 ⇐⇒ N(1) = D(1) ∧ N′
(1) = D′
(1) .
15
20. Trade-offs in pole placement
A pole placement is a trade-off between three design criteria:
⊲ response time
⊲ overshoot ratio
⊲ maximal strength of the input signal
There is no general recipe for pole placement. Rules of thumb are given in
⊲ C.-T. Chen. Linear System Theory and Design. page 238.
16
21. Illustrative example
Consider a feedback loop with the transfer function g[z] = 1
z−2. Then we
can try many different pole placements
F(z) ∈
z − 0.5, z + 0.5, z2
− 0.25, z2
+ z + 0.25, z2
− z + 0.25
The corresponding compensators are
ĉ[z] ∈
3
2
,
5
2
,
15
4z + 8
,
24
4z + 12
,
9
4z + 4
,
p ∈
1
3
,
3
5
,
1
5
, ,
9
25
,
1
9
.
They can be found systematically by solving a system of linear equations.
17
22. Robust signal tracking
Sometimes the system changes during the operation. The latter can be
modelled as an additional additive term w[k] in the input signal.
If we know the poles of reference signal r[k] and w[k] ahead, then we can
design a compensator that filters out the error signal w[k].
For instance, if w[k] is a constant bias, then adding an extra pole 1
z−1
cancels out the effect of bias. See pages 277–283 for further examples.
In our example, the robust compensator for F[z] = z2
− 0.25 is
ĉ[z] =
12z − 9
4z − 4
p = 1 .
18
23. Model matching
Find a feedback configuration such that ĝ◦ is BIBO stable and ĝ◦[1] = 1.
y[k]
u[k] System
r[k]
ĝ[z]
Controller
ĉ1[z]
+
ĉ2[z]
Feedback
⊲ The two-parameter configuration described above provides more flexibility
and it is possible to cancel out zeroes that prohibit tracking.
⊲ There are many alternative configurations. A controller design is
acceptable if every closed-loop configuration is BIBO stable.
19
