Limit-cycles prevention via multiple H∞ constraints with an application to anti-windup design Jean-Marc Biannic ONERA-DCSD, Toulouse France http://www.onera.fr/staff/jean-marc-biannic/ http://jm.biannic.free.fr/
NOLCOS 2013, Toulouse France.
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
INTRODUCTION (1/2) Stability analysis of nonlinear control systems can often be treated as a problem of investigating the existence of sustained oscillations known as limit-cycles,
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
INTRODUCTION (1/2) Stability analysis of nonlinear control systems can often be treated as a problem of investigating the existence of sustained oscillations known as limit-cycles, Although the detection of such oscillatory behavior cannot be regarded as a rigorous stability test, the knowledge of their existence and characteristics (magnitude and frequency) is often crucial to predict wether the plant will become unstable or not,
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
INTRODUCTION (1/2) Stability analysis of nonlinear control systems can often be treated as a problem of investigating the existence of sustained oscillations known as limit-cycles, Although the detection of such oscillatory behavior cannot be regarded as a rigorous stability test, the knowledge of their existence and characteristics (magnitude and frequency) is often crucial to predict wether the plant will become unstable or not, A variety of techniques have then been developed in the past decades to detect and avoid limit-cycles among which describing function (DF) methods are still very popular today thanks to their close connections with linear frequency-domain techniques. These techniques offer possibilities for systematic control systems design in the presence of input saturations which are often present in practice.
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
INTRODUCTION (1/2)
Inspired by such results, the central contribution of this paper is based on these classical frequency-domain conditions to check the existence of limit-cycles in magnitude and rate limited control systems. The main result consists of a new characterization of the above conditions via multiple H∞ constraints from which a new anti-windup design approach can be derived, The talk is organized as follows : Describing functions and necessary condition of oscillation Avoiding limit cycles Application to anti-windup design Illustration
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
DESCRIBING FUNCTIONS Considered class of nonlinear systems : z
w
Φ +
z
v
Ψ −
−
+
v
G(s)
G(s) −
MRL
(a)
(b)
Φ w
+
z
M(s) (c)
where the nonlinear operators Ψ and Φ are described by : +/− Lm z
+/− Lr +
z
+
v
v
w
1/s −
−
MRL Φ Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
DESCRIBING FUNCTIONS In the context of sinusoidal-input describing function (SIDF) analysis, the input signal z(t) is assumed to be z(t) = x sin ωt. The output v (t) = Ψ(x sin ωt) is then a periodic signal and the describing function gain is defined as the fundamental of the Fourier series of v (t) divided (k) by the input magnitude x. Denoted NΨ , the resulting complex-valued th gain reads (for the k harmonic) : (k) NΨ (x, ω)
jω = πx
Z
2π ω
Ψ(x sin ωt)e −jkωt dt
(1)
0
and the existence of limit-cycle oscillations (xc sin ωc t) in the nonlinear closed-loop plant can be investigated through the resolution of the harmonic balance equation : (1)
(1)
1 + G (jωc )NΨ (xc , ωc ) = 0 ⇔ M(jωc ) = 1/NΦ (xc , ωc )
Jean-Marc Biannic
(2)
Limit-cycles prevention via multiple H∞ constraints
DESCRIBING FUNCTIONS The resolution of the above equation is usually performed graphically in the Nyquist plane by looking for intersections between the Nyquist plot (1) of M(s) and the critical loci of the nonlinearity 1/NΦ (x, ω). 2 0
t=3
t=3.5
ï2 ï4
Imaginary axis
t=5 ï6
t=4
ï8 ï10
t=4.5
ï12 ï14 ï16 ï18
t=6 0
2
4
6
8
10 Real axis
12
14
16
18
20
(1)
Figure : Critical loci 1/NΦ (x, ω) in the Nyquist plane (Lr = 1, Lm = 0.25). Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
DESCRIBING FUNCTIONS AND LIMIT CYCLES The resolution of the above equation is usually performed graphically in the Nyquist plane by looking for intersections between the Nyquist plot (1) of M(s) and the critical loci of the nonlinearity 1/NΦ (x, ω). 2 0
t=3
t=3.5
ï2 ï4
Imaginary axis
t=5 ï6
t=4
Nyquist locus
ï8 ï10
t=4.5
ï12 ï14 ï16 ï18
t=6 0
2
4
6
8
10 Real axis
12
14
16
18
20
(1)
Figure : Critical loci 1/NΦ (x, ω) and portion of Nyquist plot for M(s) Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
AVOIDING LIMIT-CYCLES To avoid limit-cycles, no intersection should appear : (1)
∀ω ≥ 0, ∀x ≥ 0, M(jω) 6= 1/NΦ (x, ω)
(3)
(1)
Since |1/NΦ (x, ω)| > 1, ∀ω ≥ 0, ∀x ≥ 0, a standard but conservative constraint to enforce (3) is : ∀ω ≥ 0, |M(jω)| < 1 ⇔ kM(s)k∞ < 1
(4)
It can be relaxed as follows :
kM(s)k∞ < c1 ∀ω ≥ 0, |M(jω) − α| > ρ
Jean-Marc Biannic
c1 > 1 α > c1 with ρ≥α−1
(5)
Limit-cycles prevention via multiple H∞ constraints
AVOIDING LIMIT-CYCLES 4
2
−2
−4
−6
−8
−10 −6
Allowed region associated to the small gain criterion Allowed region characterized by two H
−4
−2
0
8
Imaginary axis
0
constraints
2
4
6
8
10
12
Real axis
Figure : Comparison of constraints (4) and (5) Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
AVOIDING LIMIT-CYCLES With the following lemma : Lemma Given α > 0 and a stable LTI model M(s), such that kM(s)k∞ < α, then the linear feedback interconnection T (s) = (I − α−1 M(s))−1 is well-posed and for any positive real c, the H∞ constraint : kT (s)k∞ = k(I − α−1 M(s))−1 k∞ < c
(6)
guarantees that the Nyquist plot of M(s) remains outside the disk D(α, α/c) with center α and radius ρ = α/c. conditions (5) take the form of a multi-channel H∞ problem : 1 < c1 < α kM(s)k∞ < c1 with α c2 ≤ α−1 kT (s)k∞ < c2 Jean-Marc Biannic
(7)
Limit-cycles prevention via multiple H∞ constraints
AVOIDING LIMIT-CYCLES Remark It is not always possible in practice to avoid limit-cycles, especially for open-loop unstable plants. In this case, the critical loci and the nyquist plot will generally intersect near 1 which cannot be excluded by the "red circle". In other words, c2 cannot be minimized to satisfy the α . constraint c2 ≤ α−1 However, by exploiting more precisely the geometry of the critical loci, a region R can be defined such that : ∀ω ≥ 0, M(jω) ∈ / R ⇒ xc > xL or ωc > ωL
(8)
Interestingly, the exclusion constraint M(jω) ∈ / R can also be enforced by the H∞ conditions (7) for an appropriate choice of c1 , c2 and α.
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
APPLICATION TO ANTI-WINDUP DESIGN The anti-windup general structure J(s)
ξ z
Φ
w
w −
z
Φ
v
K(s)
G(s) +
L(s)
J(s)
(a)
(b)
M(s) = Tw →z (s) = L1 (s) + L2 (s)J(s)
(9)
Resolution via mutli-objective H∞ design ˆ = arg min max (µkT1 (s)k∞ , kT2 (s)k∞ ) J(s)
(10)
J(s)
T1 (s) = (L1 (s) + L2 (s)J(s)) −1 T2 (s) = I − α−1 (L1 (s) + L2 (s)J(s)) Jean-Marc Biannic
(11)
Limit-cycles prevention via multiple H∞ constraints
ILLUSTRATION Consider the following second-order open loop model : G (s) =
1 s(s + 0.1)
Next define a stabilizing PID controller for this plant : Z t ˙ u(t) = (θc (τ ) − θ(τ ))dτ − 2.5θ(t) − 2.4θ(t)
(12)
(13)
0
The nominal closed-loop (without saturations) reads : Tθc →θ (s) =
(s +
Jean-Marc Biannic
1)(s 2
1 + 1.5s + 1)
(14)
Limit-cycles prevention via multiple H∞ constraints
ILLUSTRATION : LIMIT-CYCLE Introduce magnitude (Lm = 0.25) and rate (Lr = 1) limitations on the control signals. Limit-cycle oscillations appear for θc = 2.114 : 3
T=11.28 s
Free and constrained controller outputs
2
1
0
ï1 0.5 0
ï2
ï0.5 10.6 10.8 11 11.2 11.4 11.6 ï3
0
5
10
15
Jean-Marc Biannic
20 Time (sec)
25
30
35
40
Limit-cycles prevention via multiple H∞ constraints
ILLUSTRATION : LIMIT-CYCLE Limit-cycle is confirmed by Nyquist-plane analysis : 0.05 t=0.55
Limit cycle risk
Imaginary axis
0
ï0.05
t=1.6
ï0.1 t=1.8 t=0.75 ï0.15 t=2
ï0.2
1
1.5
2
2.5
Real axis
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
ILLUSTRATION : ANTI-WINDUP DESIGN
The PID control law is modified as follows : Z t ˙ u(t) = (θc (τ ) − θ(τ ) + ξ(τ ))dτ − 2.5θ(t) − 2.4θ(t)
(15)
0
From this structure, the linear interconnection L(s) is obtained as : L(s) = [L1 (s) L2 (s)] =
[1 + 2.5s + 2.4s 2 − s(s + 0.1)] (1 + s)(1 + 1.5s + s 2 )
(16)
and the multi-objective design problem is solved via nonsmooth optimization for α = 1.5 and µ ≥ 1. This parameter is increased until µkT1 (s)k∞ = kT2 (s)k∞ = 3.6. Then, one obtains Jˆ = 1.53.
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
ILLUSTRATION : RESULTS
0.1 0 ï0.1
Modified nyquist plot (J=1.53)
Imaginary axis
ï0.2
Critical locii
ï0.3 ï0.4
||T2(s)||' = c2
ï0.5 ï0.6
Initial nyquist plot (J=0)
ï0.7
||T (s)|| = c 1 ' 1
ï0.8 ï0.9
0
0.5
1
1.5
2
2.5
Real axis
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
ILLUSTRATION : RESULTS
3
Controlled outputs
2.5
J=0
2 1.5
J=1.53
1 0.5 0
0
5
0
5
10
15
10
15
Constrained control signals
0.4
0.2
0
−0.2
−0.4
Time (sec)
Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
CONCLUSIONS AND FUTURE WORK Based on the well-known describing function approach, we have shown that limit cycles can be avoided as soon as H∞ constraints are simultaneously satisfied by appropriately chosen linear interconnections, With this result, a new anti-windup design strategy can be derived for possibly high-order systems, The proposed framework is flexible and can be easily enriched to take into account additional H∞ constraints. Future work will be devoted to extensions of the anti-windup design approach to systems involving multiple and possibly more general nonlinearities. A specific attention will also be devoted to the tuning aspects so that the design procedure can be automatized and made available to non expert users. Jean-Marc Biannic
Limit-cycles prevention via multiple H∞ constraints
