control of nonlinear systems using high-gain observers
CONTROL OF NONLINEAR SYSTEMS USING HIGH-GAIN OBSERVERS∗ Leonid B. Freidovich Department of Applied Physics and Electronics Umea˚ University ˚ SWEDEN SE-901 87 Umea, [email protected]
∗
Seminar talk at TFE, Umea˚ University
April 3, 2006 ↓
1
Outline [part I: introduction]
1. Why to estimate derivatives and how
2. What is peaking phenomenon and how to overcome it
3. How to design a high-gain observer
2
Why and how to estimate derivatives? Consider the normalized mass-spring system:
x ¨ + x = u. Simple state feedback control law
√ u = − 2 x˙ is not realizable if the only output is y = x. However, following the Separation Principle, would it be acceptable to take
√ u = − 2 vˆ, where vˆ is an estimate for y˙ ?
←
3
Besides HGO, the most frequently used formulae to estimate derivatives are the following. • Euler’s difference formula :
y(t) − y(t − ε) vˆ(t) = , ε with y(t) ≡ 0 for −ε ≤ t < 0 and ε > 0 being sufficiently small, • Sliding-mode-based observer (Levant’s formula):
p d x ˆ + α sign(ˆ x − y) = −λ |ˆ x − y| sign(ˆ x − y), dt with α = 8 and λ = 6 for the case when |v(t)| ≤ 2.
vˆ = x ˆ˙
4
• ’Dirty’ derivative formula:
vˆ =
s εs + 1
y,
with vˆ(0) = 0 and ε > 0 being sufficiently small.
4
• ’Dirty’ derivative formula:
vˆ =
s εs + 1
y,
with vˆ(0) = 0 and ε > 0 being sufficiently small. In the case of negligible measurement noise, should we take ε > 0 as small as possible in order to recover the trajectories under the state feedback? Let us simulate the system with ’dirty’ derivative estimate
√ x ¨ + x = − 2 vˆ + d1(t),
ε vˆ˙ + vˆ = y˙ + d2(t),
where d1(t) and d2(t) are impulses at t = 5 and t = 10 , correspondingly, for the two cases: ε = 0.01 and ε = 0.001. Simulation results: contrary to the intuition, decreasing ε is useless!
5 1
state feedback observer with eps=0.01 observer with eps=0.001
x(t)
0.5 0 −0.5
0
5
10
15
−3
4
x 10
x(t)
3 2 1 0 14.91
14.92
14.93
14.94
14.95
t
14.96
14.97
14.98
14.99
15
Figure 1: Should we make ε > 0 smaller? – No!
6
Explanation of the problem for the Euler’s formula:
y(t) vˆ(t) = ε
∀t: 0≤t 0 smaller? – YES, if saturation is used!
10
←
High-gain observer design A Luenberg observer (full order observer) for the system
x˙ = v,
v˙ = f (x, v, u),
y=x
is
x ˆ˙ = vˆ + h1 (y − x ˆ) ,
vˆ˙ = fˆ(ˆ x, vˆ, u) + h2 (y − x ˆ) ,
where fˆ(·) is a nominal model of f (·), bounded with respect to x ˆ and h1, h2 are the observer gains to be chosen. Let as introduce a change of coordinates
x ˜=x−x ˆ,
v˜ = v − vˆ,
δ(·) = f (·) − fˆ(·).
11
To make the observer fast we place the poles of the error dynamics
¨˜ + h1 x x ˜˙ + h2 x ˜ = δ(·) 1 into − by choosing ε 2 1 1 h1 = 2 , h2 = . ε ε Note: in the case when δ(·) ≡ 0 , we have
vˆ =
s (1 + ε s)2
y.
In distinction to the ’dirty’ derivative formula, this is a low pass filter.
12
Defining the scaled estimation error
η1 (y − x ˆ)/ε η= = η2 v − vˆ
we obtain
−2 εη˙ = −1
1 0 η+ ε 0 δ(·)
x ˜(0) , η2(0) = v ˜(0) , and with η1(0) = ε a0 t vˆ(t) = v(t) − η2(t) = v(t) + exp − + . . . . ε ε
13
Summary and remarks (supported by theory and/or based on the experience): • HGO is a fast nonlinear (or linear) full order observer with high observer gain chosen via pole placement, • to protect the system from the destabilizing effect of peaking, HGOs, as well as any continuous differentiating schemes, have to be followed by saturation, • under certain regularity assumptions, it can be shown that making the main observer parameter (ε) sufficiently small would result in: ? better robustness with respect to uncertainty of the model, ? arbitrary closeness of the trajectories under state and output feedback controllers, provided the sensor noise is negligible, ? degradation of the performance in the case when the sensor noise is significant, since the cut-off frequency of the observer is inversely proportional to ε,
14
• nonlinear HGO allows to exploit the partial knowledge of the model and is the most reliable way to estimate derivatives in the situation when ε could not be taken too small, • in most cases, it is better to assign the eigenvalues of the observer to be small real numbers of the order of −1/ε, • the value of ε, as well as the saturation levels, should be considered as tuning parameters, since the theoretical estimates are usually too conservative. • generalization of the design procedure and of the available (based on the singular perturbation technique) proofs for the higher order and multi-output cases are not complicated in the case when there are no switching or other discontinuities in the controller and the plant dynamics.
∗
Seminar talk at TFE, Umea˚ University
April 3, 2006 ↓
1
Outline [part I: introduction]
1. Why to estimate derivatives and how
2. What is peaking phenomenon and how to overcome it
3. How to design a high-gain observer
2
Why and how to estimate derivatives? Consider the normalized mass-spring system:
x ¨ + x = u. Simple state feedback control law
√ u = − 2 x˙ is not realizable if the only output is y = x. However, following the Separation Principle, would it be acceptable to take
√ u = − 2 vˆ, where vˆ is an estimate for y˙ ?
←
3
Besides HGO, the most frequently used formulae to estimate derivatives are the following. • Euler’s difference formula :
y(t) − y(t − ε) vˆ(t) = , ε with y(t) ≡ 0 for −ε ≤ t < 0 and ε > 0 being sufficiently small, • Sliding-mode-based observer (Levant’s formula):
p d x ˆ + α sign(ˆ x − y) = −λ |ˆ x − y| sign(ˆ x − y), dt with α = 8 and λ = 6 for the case when |v(t)| ≤ 2.
vˆ = x ˆ˙
4
• ’Dirty’ derivative formula:
vˆ =
s εs + 1
y,
with vˆ(0) = 0 and ε > 0 being sufficiently small.
4
• ’Dirty’ derivative formula:
vˆ =
s εs + 1
y,
with vˆ(0) = 0 and ε > 0 being sufficiently small. In the case of negligible measurement noise, should we take ε > 0 as small as possible in order to recover the trajectories under the state feedback? Let us simulate the system with ’dirty’ derivative estimate
√ x ¨ + x = − 2 vˆ + d1(t),
ε vˆ˙ + vˆ = y˙ + d2(t),
where d1(t) and d2(t) are impulses at t = 5 and t = 10 , correspondingly, for the two cases: ε = 0.01 and ε = 0.001. Simulation results: contrary to the intuition, decreasing ε is useless!
5 1
state feedback observer with eps=0.01 observer with eps=0.001
x(t)
0.5 0 −0.5
0
5
10
15
−3
4
x 10
x(t)
3 2 1 0 14.91
14.92
14.93
14.94
14.95
t
14.96
14.97
14.98
14.99
15
Figure 1: Should we make ε > 0 smaller? – No!
6
Explanation of the problem for the Euler’s formula:
y(t) vˆ(t) = ε
∀t: 0≤t 0 smaller? – YES, if saturation is used!
10
←
High-gain observer design A Luenberg observer (full order observer) for the system
x˙ = v,
v˙ = f (x, v, u),
y=x
is
x ˆ˙ = vˆ + h1 (y − x ˆ) ,
vˆ˙ = fˆ(ˆ x, vˆ, u) + h2 (y − x ˆ) ,
where fˆ(·) is a nominal model of f (·), bounded with respect to x ˆ and h1, h2 are the observer gains to be chosen. Let as introduce a change of coordinates
x ˜=x−x ˆ,
v˜ = v − vˆ,
δ(·) = f (·) − fˆ(·).
11
To make the observer fast we place the poles of the error dynamics
¨˜ + h1 x x ˜˙ + h2 x ˜ = δ(·) 1 into − by choosing ε 2 1 1 h1 = 2 , h2 = . ε ε Note: in the case when δ(·) ≡ 0 , we have
vˆ =
s (1 + ε s)2
y.
In distinction to the ’dirty’ derivative formula, this is a low pass filter.
12
Defining the scaled estimation error
η1 (y − x ˆ)/ε η= = η2 v − vˆ
we obtain
−2 εη˙ = −1
1 0 η+ ε 0 δ(·)
x ˜(0) , η2(0) = v ˜(0) , and with η1(0) = ε a0 t vˆ(t) = v(t) − η2(t) = v(t) + exp − + . . . . ε ε
13
Summary and remarks (supported by theory and/or based on the experience): • HGO is a fast nonlinear (or linear) full order observer with high observer gain chosen via pole placement, • to protect the system from the destabilizing effect of peaking, HGOs, as well as any continuous differentiating schemes, have to be followed by saturation, • under certain regularity assumptions, it can be shown that making the main observer parameter (ε) sufficiently small would result in: ? better robustness with respect to uncertainty of the model, ? arbitrary closeness of the trajectories under state and output feedback controllers, provided the sensor noise is negligible, ? degradation of the performance in the case when the sensor noise is significant, since the cut-off frequency of the observer is inversely proportional to ε,
14
• nonlinear HGO allows to exploit the partial knowledge of the model and is the most reliable way to estimate derivatives in the situation when ε could not be taken too small, • in most cases, it is better to assign the eigenvalues of the observer to be small real numbers of the order of −1/ε, • the value of ε, as well as the saturation levels, should be considered as tuning parameters, since the theoretical estimates are usually too conservative. • generalization of the design procedure and of the available (based on the singular perturbation technique) proofs for the higher order and multi-output cases are not complicated in the case when there are no switching or other discontinuities in the controller and the plant dynamics.
