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Modelling and Control of Dynamic
Systems
Controllability and Observability
Sven LaurUniversity of Tartu
Closed-loop Controllers:
Basic Structure
Closed-loop control with state estimation
y[k]u[k]SystemController
r[k]G[z]State
Estimator
The instability of a open loop controller is caused by gradual accumulation ofdisturbances. The control signal becomes unsynchronised with the system.
⊲ If we can estimate the system state from the output, we can circumventsuch synchronisation errors. The system must be observable for that.
⊲ Even if we know the system state, it might be impossible to track thereference signal. In brief, the system must be controllable.
1
Controllability
The state equation is controllable if for any input state x0 and for any finalstate 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 controllabilitymatrix
[B AB A2B An−1B
]has the maximal rank n.
The desired input can be computed from the system of linear equations
x1 = Anx0 + An−1Bu[0] + An−2Bu[1] + · · · + Bu[n − 1] .
Although the theorem T6 gives an explicit method for controlling thesystem, it is an off-line algorithm with a time lag n.
A good controller design should yield a faster and more robust method.
2
Observability
The state equation is observable if for any input state x0 and for any inputsignal 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
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−1x0 + · · · + CBu[n − 2] + Du[n − 1]
Although this equation allows us to find out the state of the system, it isoffline algorithm, which provides a state estimation with a big time lag.
A good state space estimator must be fast and robust.
4
General structure of state estimators
y[k]u[k] System
State Estimatorx[k]
x[k] = Ax[k − 1] + Bu[k − 1]
y[k] = Cx[k] + Du[k]
A state estimate x[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
Example
Consider a canonical realisation of the transfer function g[z] = 0.5z+0.5z2
−0.25
A =
[0 0.251 0
]
B =
[10
]
C = [0.5 0.5] D = 0
Then a possible stable state estimator is following
x[k + 1] = Ax[k] + Bu[k] + 1(y[k] − Cx[k]︸ ︷︷ ︸
y[k]
) .
In general, the feedback vector 1 can be replaced with any other vector toincrease the stability of a state estimator.
6
Kalman Decomposition
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 suchthat for any input u both systems yield the same output and vice versa.
⊲ T8. Two state descriptions are (algebraically) equivalent if there existsan invertible matrix P such that
A2 = PA1P−1 B2 = PB1
C2 = C1P−1 D2 = D1 .
7
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} sothat the underlying behaviour does not change. The same equivalence ofstate descriptions hold for other bases and larger state spaces, as well.
8
Kalman decomposition
⊲ T9. Every state space equation can be transformed into an equivalentdescription 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
Minimal realisation
⊲ T10. All minimal realisations are controllable and observable. Arealisation of a proper transfer function g[z] = N(z)/D(z) is minimal iff
it state space dimension dim(x0) = deg D(z).
10
Closed-loop Controllers:
Design Principles
General setting
According to Kalman decomposition theorem, the state variables can bedivided 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 marginallystable. We can do it with an open-loop controller.
⊲ For controllable and observable state variables, we can build effectiveclosed-loop controllers.
Simplifying assumptions
⊲ From now on, we assume that we always want to control state variablesthat are both controllable and directly observable: y[k] = x[k].
⊲ If this is not the case, then we must use state estimators to get anestimate of x. The latter just complicates the analysis.
11
Unity-feedback configuration
y[k]u[k] SystemControllerr[k]
g[z]C[z]p +
-1
Design tasks
⊲ Find a compensator c[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 g[1] = 0, then the output y[k] just dies out.
12
Overall transfer function
Now note that
y = g[z]c[z](pr − y) ⇐⇒ y =pg[z]c[z]
1 + g[z]c[z]r
and thus the configuration is stable when the new transfer function
g◦[z] =pg[z]c[z]
1 + g[z]c[z]
has poles lying in the unit circle. Now observe
g[z] =N(z)
D(z), c[z] =
B(z)
A(z)=⇒ g◦[z] =
pB(z)N(z)
A(z)D(z) + B(z)N(z)
13
Pole placement
We can control the denominator of the new transfer function g◦[z]. Let
F (z) = A(z)D(z) + B(z)N(z)
be a desired new denominator. Then there exists polynomials A(z) andB(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
Signal tracking properties
Let g◦[z] be the overall transfer function.
⊲ The system stabilises if g◦[z] is BIBO stable.
⊲ The system can track a constant signal r[k] ≡ a if g◦[1] = 1.
⊲ The system can track a ramp signal r[k] ≡ ak if g◦[1] = 1 and g′◦[1] = 0.
The latter follows from the asymptotic convergence
y[k] → ag′◦[1] + kag◦[1]
for an input signal r[k] = ak. Moreover, let g◦ = N(z)/D(z). Then
g◦[1] = 1 ∧ g′◦[1] = 0 ⇐⇒ N(1) = D(1) ∧ N ′(1) = D′(1) .
15
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
Illustrative example
Consider a feedback loop with the transfer function g[z] = 1z−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
c[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
Robust signal tracking
Sometimes the system changes during the operation. The latter can bemodelled 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 candesign a compensator that filters out the error signal w[k].
For instance, if w[k] is a constant bias, then adding an extra pole 1z−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
c[z] =12z − 9
4z − 4p = 1 .
18
Model matching
Find a feedback configuration such that g◦ is BIBO stable and g◦[1] = 1.
y[k]u[k] Systemr[k]g[z]
Controller
c1[z]+
c2[z]Feedback
⊲ The two-parameter configuration described above provides more flexibilityand it is possible to cancel out zeroes that prohibit tracking.
⊲ There are many alternative configurations. A controller design isacceptable if every closed-loop configuration is BIBO stable.
19