2013 21st Mediterranean Conference on Control & Automation (MED) Platanias-Chania, Crete, Greece, June 25-28, 2013
Simple Output Controller for Nonlinear Systems with Multisinusoidal Disturbance Anton A. Pyrkin, Member, IEEE, Alexey A. Bobtsov, Senior member, IEEE, Sergey A. Kolyubin, Graduate Student Member, IEEE Abstract— The problem of control design for a class of nonlinear system with multisinusoidal disturbance is considered. It is assumed that the linear part is unknown and minimum phase. The nonlinear part is known inaccurately, it is irreducible to an input of the linear block, and generally does not satisfy sector restrictions. An output controller ensuring semiglobal stability is designed. Controller was supplemented with the cancellation scheme of a multisinusoidal external disturbance that improves performance without its significant complication.
I. INTRODUCTION From practical point of view it is important not only to solve the control problem itself, but also one have to be sure that the new proposed approach is implementable. Controllers of simple structure with clear parameters’ adjustment schemes are highly attractive for engineers in different areas including aerospace, robotics, and power systems [15], [22], [28], [34], [43], [45]. Particularly, output adaptive control methods in demand for a plenty of practical applications, where plant state measurement is hard or even impossible to realize. This paper is focused on the recent advantages in the development of adaptive output control approach intensively using system passification principle named by the authors as “consecutive compensator”, that was considered in the number of previous works [12]–[16]. One of the classical problems of control theory is the problem of stabilization of nonlinear systems. As it was surveyed by Kokotovic and Arcak [30], there had been already a significant number of constructive results. Today the line of investigation mainly concerns the tasks of analysis and design of control for the systems, in which nonlinearity is irreducible to an input [30]. Some of the publications devoted to the given themes were presented by Kokotovic’s group, in particular, papers [2]–[4]. In [2]–[4] conditions of existence of a static feedback, which ensures asymptotic stability of A. Pyrkin and S. Kolyubin are with the Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, NO-7491, Norway. A. Pyrkin, A. Bobtsov, and S. Kolyubin are with the Department of Control Systems and Informatics, Saint Petersburg National Research University of Information Technologies Mechanics and Optics, Kronverkskiy av. 49, Saint Petersburg, 197101, Russia. A. Bobtsov is with the Laboratory “Control of Complex Systems”, Institute for Problems of Mechanical Engineering, Bolshoy pr. V.O. 61, St.Petersburg, 199178, Russia This work was supported the Federal Target Program “Scientific and Educational Personnel of Innovative Russia” for 2009-2013 (Agreements 14.B37.21.1480 and 14.B37.21.0871). E-mail:
[email protected],
[email protected],
[email protected] 978-1-4799-0997-1/13/$31.00 ©2013 IEEE
the system, are offered. In these papers authors consider the following nonlinear system: χ˙ = Aχ χ + Gχ ϕ(y) + Bχ u,
y = Hχ χ,
(1)
where χ ∈ Rn is a measured vector of state variables; u ∈ Rl is a control; y ∈ Rq is an output; Aχ , Bχ , Gχ , Hχ are known matrices; ϕ(y) is a known nonlinearity. Disadvantages of methods considered in [2]–[4] are the following: design of a control law requires measurements of state variables and also full knowledge of the system parameters and the nonlinearity structure. One of the important problems in the field of nonlinear control is stabilization by output feedback (see for example [1], [5]–[12], [14], [19], [23]–[26], [29], [33], [35]–[37], [40]–[42], [44], [45]). An interesting result for output feedback control of such class of nonlinear systems was presented by Lin’s group in the papers [40], [40], [41]. In the work [40] authors consider a class of SISO time-varying systems of the following kind: χ˙ i = χi+1 + ϕi (t, χ, u), χ˙ n = u + ϕn (t, χ, u), y = χ1 , where function ϕi (t, χ, u) ≤ θ (|χ1 | + . . . + |χn |) and positive number θ is known. In comparison with results obtained by Kokotovic’s group, the approach of Lin’s group ensures an output stabilization of nonlinear systems without measurement of state variable χ(t). But in the papers [2]–[4] nonlinear function ϕ(y) may be not necessarily globally Lipschitz (for example ϕ(y) = −y 3 ). Extending the output feedback design introduced in [40], authors of the paper [42] explicitly construct a linear time-varying output feedback control law, which globally regulates the states of the systems without information about the growth rate (i.e. θ is unknown). One more approach was presented in the papers [14], [17], [39] based on passification [20], [21]. In comparison with results obtained by Kokotovic group, the approach in [14] ensures an output stabilization of nonlinear systems without measurement of the state variable χ(t) for the case when ϕ(y) is unknown nonlinearity. In comparison with results obtained by Lin group, the approach in [14] ensures output stabilization of nonlinear systems for the general case (in papers [40], [40], [41] authors consider nonlinear system in triangular canonical form). And one more approach was presented in the paper [12]. The problem was solved for
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the case of unknown parameters of the linear block and unknown nonlinear part. Only the output variable was used for measurements. The disadvantage of the approach [12] is the following: transfer function from an input u to an output y has a relative degree ρ = 1 and its numerator is Hurwitz. Developing results, presented in [2]–[4], [12], [14], [40], [42], in this work we assume that only an output variable of the system is measured (as in [12], [14], [40], [42]), parameters of the linear block are unknown (unlike [2]– [4], [40], [42]), and its transfer function is minimum phase (unlike [12]). We also assume that nonlinear part of the system is known inaccurately, irreducible to an input of the linear block and does not satisfy sector restrictions (unlike [14], [40], [42]). The control objective is to design the simple output regulator, ensuring semiglobal stability. We also assume that nonlinear part of the system is known inaccurately, irreducible to an input of the linear block and does not satisfy sector restrictions (unlike [14], [40]). The control objective is to design the simple output regulator, ensuring semiglobal stability for nonlinear disturbed system. In this paper we assume that we know the frequencies of the disturbance to simplify the analysis of the stabilization scheme, but it is possible to apply known results [18], [38] to design the adaptive estimation scheme for unknown frequencies. II. PROBLEM FORMULATION Consider the following nonlinear system y(t) =
c(p) e(p) b(p) u(t) + f (t) + δ(t), a(p) a(p) a(p)
(2)
where p = d/dt denotes differential operator; output y(t) is measured, but its derivatives are not measured; b(p) = bm pm + · · · + b1 p + b0 , c(p) = cr pr + · · · + c1 p + c0 , e(p) = eg pg + · · · + e1 p + e0 , and a(p) = pn + · · · + a1 p + a0 are monic coprime polynomials with unknown coefficients; b(p) number r, g ≤ n − 1; transfer function a(p) has relative degree ρ = n − m; unknown function f (t) = ϕ(y, t) is such that: s |ϕ(y, t)| ≤ C0 |y(t)| for all y(t), (3)
III. CONTROL DESIGN Let us choose the control law u(t) as follows u(t) = −(µ + κ)
δ(t) = A0 +
Ai sin(ωi t + ϕi )
(4)
H(p) =
α(p)b(p)(p + 1)2l+1 a(p)γ(p) + µα(p)b(p)(p + 1)2l+1
is SPR (more detailed explanation on choice of α(p) and µ > 0 can be found in [14]), κ > 0 is chosen to compensate the unknown nonlinearity ϕ. The function yˆ(t) is calculated according to the following algorithm ξ˙1 = σξ2 , ˙ ξ2 = σξ3 , (6) ... ˙ ξρ−1 = σ (−k1 ξ1 − . . . − kρ−1 ξρ−1 + k1 y) , yˆ = ξ1 ,
(7)
where number σ > µ + κ (see proof of the theorem 1, inequality (32)) and parameters ki are calculated for the system (6) to be exponentially stable. It is obvious, that the control (5)–(7) is technically possible as contains known or measurable signals. Substituting (5) into equation (2), we obtain α(p)(p + 1)2l+1 b(p) − (µ + κ)ˆ y (t) y(t) = a(p) γ(p) c(p) e(p) + ϕ(y, t) + δ(t) a(p) a(p) b(p) α(p)(p + 1)2l+1 (µ + κ) [y(t) − ε(t)] =− a(p) γ(p) c(p) e(p) + ϕ(y, t) + δ(t), (8) a(p) a(p) where the error ε(t) = y(t) − yˆ(t). After simple transformations, for model (8) we have a(p)γ(p) + µα(p)b(p)(p + 1)2l+1 y(t) = b(p)α(p)(p + 1)2l+1 [(µ + κ)ε(t) − κy(t)] + c(p)γ(p)ϕ(y, t) + e(p)γ(p)δ(t)
(9)
and y(t) = W (p) [−κy(t) + (µ + κ)ε(t)] c(p)γ(p) + ϕ(y, t) a(p)γ(p) + µα(p)b(p)(p + 1)2l+1 e(p)γ(p) + δ(t), (10) a(p)γ(p) + µα(p)b(p)(p + 1)2l+1
i=1
is the multiharmonic disturbance with known frequencies ωi and unknown amplitudes Ai , phases ϕi , A0 is unknown constant shift. The purpose of control is to ensure semiglobal stability of equilibrium position y = 0 in the system (2) under the following assumptions: Assumption 1: Polynomial b(p) is Hurwitz and the parameter b0 > 0. Assumption 2: The frequencies ωi , ∀i and the relative degree ρ are known.
(5)
where γ(p) = p(p2 + ω12 )(p2 + ω22 ) · . . . · (p2 + ωl2 ), while α(p) is Hurwitz polynomial of (ρ − 1) degree, constant µ > 0 are chosen such way that transfer function
where number C0 is unknown and number s > 1 is unknown, and l X
α(p)(p + 1)2l+1 yˆ(t), γ(p)
2l+1
b(p)α(p)(p+1) where transfer function W (p) = a(p)γ(p)+µα(p)b(p)(p+1) 2l+1 is SPR. Since all eigenvalues of γ(p) equal to all frequencies of δ(t) it is easy to show that the last term of (10) is zero because γ(p)δ(t) = 0.
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Then let us present model (10) in the form x(t) ˙ = Ax(t) + b(−κy(t) + (µ + κ)ε(t)) + qϕ(y, t), (11) y(t) = cT x(t),
Substituting in (22) equations (13), (20) and taking into account inequalities 2xT (t)P bhT η(t) ≤ δxT (t)P bbT P x(t)
(12)
+ δ −1 η T (t)hhT η(t),
where x ∈ Rn is a state vector of system (11); A, b, q and c are appropriate matrix and vectors of transition from model (10) to model (11), (12). Since transfer function W (p) is SPR then AT P + P A = −R,
P b = c,
T
yˆ(t) = h ξ(t),
+ δ −1 [ϕ(y, t)]2 , T
0 1 0 0 0 1 0 0 0 where Γ = .. .. .. . . . −k −k −k 1 2 3 and hT = 1 0 0 . . . 0 . Consider vector
... ... ... .. . ...
T
T
T
T
T
(24) T
2η (t)N hc bh η(t) ≤ η (t)N hc bb ch N η(t)
(13)
(15) 0 0 0 0 0 , d = 0 , .. .. . . k1 −kρ−1
(23)
T
2x (t)P qϕ(y, t) ≤ δx (t)P qq P x(t)
+ η T (t)hhT η(t),
where R = RT and parameters of matrix R depend on µ and do not depend on κ. Let us rewrite model (6), (7) in the form ˙ = σ(Γξ(t) + dy(t)), ξ(t) (14) T
T
T
T
2η (t)N hc Ax(t) ≤ δ
−1
T
(25) T
T
T
η(t)N hc AA ch N η (t)
+ δxT (t)x(t), T
T
T
T
(26) T
T
2η (t)N hc qϕ(y, t) ≤ κη (t)N hc qq ch N η(t) + κ−1 [ϕ(y, t)]2 ,
(27)
−2κη T (t)N hcT by(t) ≤ δ −1 κη T (t)N hcT bbT chT N η(t) + δκxT (t)P bbT P x(t),
(28)
we obtain V˙ (t) ≤ −xT (t)Rx(t) − ση T (t)M η(t) − κ2 y(t)
η(t) = hy(t) − ξ(t),
− κxT (t)P bbT P x(t) + δ(µ + κ)xT (t)P bbT P x(t)
(16)
+ δ −1 (µ + κ)η T (t)hhT η(t) + δxT (t)P qq T P x(t)
then by force of vector h structure the error ε(t) will become T
+ δ −1 [ϕ(y, t)]2 + (µ + κ)η T (t)N hcT bbT chT N η(t)
T
ε(t) = y(t) − yˆ(t) = h hy(t) − h ξ(t) T
T
= h (hy(t) − ξ(t)) = h η(t).
+ (µ + κ)η T (t)hhT η(t) (17)
+ δ −1 η(t)N hcT AAT chT N η T (t)
For derivative of η(t) we obtain
+ δxT (t)x(t) + κη T (t)N hcT qq T chT N η(t)
η(t) ˙ = hy(t) ˙ − σ(Γ(hy(t) − η(t)) + dy(t)) = hy(t) ˙ + σΓη(t) − σ(d + Γh)y(t).
+ κ−1 [ϕ(y, t)]2 + δ −1 κη T (t)N hcT bbT chT N η(t) (18)
Since d = −Γh (can be checked by substitution), then η(t) ˙ = hy(t) ˙ + σΓη(t), ε(t) = hT η(t),
(19)
+ δκxT (t)P bbT P x(t). where the number δ > 0. Let the number 0 < δ < 0.5 be such that
where matrix Γ is Hurwitz by force of calculated parameters ki of system (6) and ΓT N + N Γ = −M,
V (t) = xT (t)P x(t) + η T (t)N η(t).
−R + δI + (δµ + 2δκ − κ)P bbT P +δP qq T P ≤ −Q1 < 0.
(20)
where N = N T > 0, M = M T > 0. Theorem 1: For any µ > 0 and any X > 0 there exist numbers κ > 0 and σ > µ + κ such that the system (11), (12), (19) is asymptotically stable for all initial condition x0 satisfying kx0 k ≤ X. Proof: Consider the Lyapunov function
(29)
(30)
Substituting (30) into the inequality (29), we obtain V˙ (t) ≤ −xT (t)Q1 x(t) − ση T (t)M η(t) − κ2 y(t)
(21)
Differentiating (21) yields V˙ (t) = xT (t) AT P + P A x(t) − 2κxT (t)P by(t)
+ δ −1 (µ + κ)η T (t)hhT η(t) + δ −1 [ϕ(y, t)]2 + (µ + κ)η T (t)N hcT bbT chT N η(t) + (µ + κ)η T (t)hhT η(t) + δ −1 η(t)N hcT AAT chT N η T (t) + κη T (t)N hcT qq T chT N η(t) + κ−1 [ϕ(y, t)]2 + δ −1 κη T (t)N hcT bbT chT N η(t). (31)
+ 2(µ + κ)xT (t)P bhT η(t) + 2xT (t)P qϕ(y, t) + η T (t)σ(ΓT N + N Γ)η(t) + 2η T (t)N hcT Ax(t) + 2(µ + κ)η T (t)N hcT bhT η(t)
Let number σ be such that the following ratio is executed
+ 2η T (t)N hcT qϕ(y, t)
−σM + δ −1 (µ + κ)hhT + (µ + κ)N hcT bbT chT N
T
T
− 2κη (t)N hc by(t).
(22) 1089
+κN hcT qq T chT N + δ −1 κN hcT bbT chT N ≤ −Q2 . (32)
Substituting (32) into the inequality (31), we have V˙ (t) ≤ −xT (t)Q1 x(t) − η T (t)Q2 η(t) − κ2 y(t) + δ −1 + κ−1 [ϕ(y, t)]2 ≤ −λ0 V (t) − κ2 y(t) + δ −1 + κ−1 C02 y 2s−1 (t)y(t) 2 ≤ −λ0 V (t) − κ2 y(t) + ψ0 C04 δ −1 + κ−1 y 2 (t) + ψ0−1 y 4s−2 (t),
(33)
where from (3) [ϕ(y, t)]2 ≤ C02 |y(t)|2s , numbers λ0 > 0 and ψ0 > 0. Let number κ be such that 2 (34) κ ≥ ψ0 C04 κ−1 + δ −1 . Then we have V˙ (t) ≤ −λ0 V (t) + ψ0−1 y 4s−2 (t) ≤ −λ0 V (t) + ψ0−1 λ1 xT (t)P x(t) ≤ ≤
2s−1
−λ0 V (t) + ψ0−1 λ1 V 2s−1 (t) −V (t) λ0 + ψ0−1 λ1 V 2s−2 (t) ,
where number λ1 > 0 is such that 2s−1 4s−2 λ1 xT (t)P x(t) ≥ cT x(t) = y 4s−2 (t).
(35)
(36)
for any t ≥ t0 . From (37) asymptotic stability of the system (11), (12), (19) follows. IV. EXAMPLE Consider the model of the heart muscle contraction oscillation [27] without disturbance (38)
χ˙ 2 = −0.1(χ1 + 1)(χ1 − 1)(χ1 + 3)(χ1 − 2.2)χ2 − χ1 + u, y = χ1 .
(39) (40)
Transforming the model (38)–(40) to the form (2) we obtain p 1 u(t) + 2 ϕ(y), +1 p +1 ϕ(y) = −0.1 0.2y 5 + 0.2y 4 −7.6y 3 /3 − 0.4y 2 + 6.6y . y(t) =
p2
(41)
(42)
Choose the control according to equation (5)–(7) taking the fractional term in (5) is equal one. u(t) = −α(p)(µ + κ)ˆ y (t) = −(p + 1)(µ + κ)ˆ y (t), ˙ξ1 = σ(−k1 ξ1 + k1 y) = σ(−ξ1 + y), yˆ = ξ1 ,
(p + 1)5 , + 4)(p2 + 1) and Fig. 1c diagram (b) shows that the output variable tends to zero as required. p(p2
2s−2 Choosing the number ψ0 > λ−1 (t0 ) for inequal0 λ1 V ity (35) we obtain V 2s−2 (t)
