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Absolute stabilization of Lur’e systems via dynamic output feedback
Recommended by Dr. A Zemouche
Accepted Manuscript
Absolute Stabilization of Lur’e Systems via Dynamic Output Feedback Fan Zhang, Harry L. Trentelman, Gang Feng, Jacquelien M.A. Scherpen PII: DOI: Reference:
S0947-3580(18)30062-1 https://doi.org/10.1016/j.ejcon.2018.09.015 EJCON 297
To appear in:
European Journal of Control
Received date: Revised date: Accepted date:
30 January 2018 2 August 2018 20 September 2018
Please cite this article as: Fan Zhang, Harry L. Trentelman, Gang Feng, Jacquelien M.A. Scherpen, Absolute Stabilization of Lur’e Systems via Dynamic Output Feedback, European Journal of Control (2018), doi: https://doi.org/10.1016/j.ejcon.2018.09.015
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Absolute Stabilization of Lur’e Systems via Dynamic Output Feedback
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Fan Zhanga , Harry L. Trentelmanb,∗, Gang Fengc , Jacquelien M.A. Scherpenb a School of Mathematics, Southeast University, Nanjing, China of Science and Engineering, University of Groningen, Groningen, The Netherlands c Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China
b Faculty
Abstract
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In this paper we study the problem of absolute stabilization under dynamic output feedback for Lur’e systems with sector-bounded unknown nonlinearities. In most of the literature, besides the incremental sector-boundedness condition, the Lur’e-type nonlinearity itself is assumed to be known exactly and used in the dynamic output feedback controller design. In the present paper only the sectorboundedness condition is employed, and exact knowledge of the nonlinearity will not be used in the controller design. More precisely, we we will only employ knowledge of the sector in which the unknown nonlinearity lies. Two different approaches will be presented for the dynamic controller design, both using linear matrix inequality techniques. Numerical simulations of a flexible joint robotic arm will illustrate the theoretical results obtained in this paper.
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Keywords: Lur’e system, absolute stabilization, dynamic output feedback, linear matrix inequality, flexible joint robotic arm
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1. Introduction
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In this paper we study the problem of absolute stabilization by dynamic output feedback for Lur’e systems. A Lur’e system is the interconnection of a linear input-state-output system and an unknown static nonlinearity through a negative feedback loop, commonly represented as ( x˙ = Ap x + Bp u + Ep d, (1) z = Cp x, d = −φ(z), y = Mp x, where x(t) ∈ Rn , u(t) ∈ Rm , z(t) ∈ Rp and y(t) ∈ Rq are the state, control input, loop injection and measurement output, respectively. The matrices Ap , ∗ Corresponding
author Email addresses: [email protected], [email protected] (Fan Zhang), [email protected] (Harry L. Trentelman), [email protected] (Gang Feng), [email protected] (Jacquelien M.A. Scherpen)
Preprint submitted to Journal of LATEX Templates
September 27, 2018
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N − d u
φ(z) x˙ = Ap x + Bp u + Ep d z = Cp x y = Mp x Fig. 1: Lur’e system
z y
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Bp , Cp , Ep and Mp are of compatible dimensions. Without loss of generality, Bp and Mp are assumed to be of full column rank and full row rank, respectively. The equation d = −φ(z) describes a static negative feedback loop, where φ(·) is an unknown nonlinear function, see Fig. 1. This nonlinearity is however
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assumed to be sector-bounded, in the sense that it satisfies the following sectorboundedness condition.
Definition 1. Let S1 , S2 ∈ Rp×p be real symmetric matrices such that S1 is positive semi-definite and S2 −S1 is positive definite, i.e. 0 ≤ S1 < S2 . Then the nonlinearity φ(·) is called sector-bounded within the sector [S1 , S2 ] if it satisfies (φ(z) − S1 z)T (φ(z) − S2 z) ≤ 0,
∀ z ∈ Rp .
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We will be dealing with output feedback controllers of the form ( v˙ = Ac v + Bc y , u = Cc v + Dc y
(2)
(3)
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where v(t) ∈ Rnc is the state of the dynamic controller, the positive integer nc and the matrices Ac ∈ Rnc ×nc , Bc ∈ Rnc ×q , Cc ∈ Rm×nc , Dc ∈ Rm×q are to be determined. The aim is to design a controller of the form (3) that absolutely stabilizes the Lur’e system (1), i.e. the closed-loop system is absolutely stable, in the sense defined as follows:
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Definition 2. Given the bounds S1 and S2 with 0 ≤ S1 < S2 , the feedback interconnection of the Lur’e system (1) and the controller (3) is called absolutely stable if it is globally asymptotically stable for every φ(·) within the sector [S1 , S2 ].
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Thus, given the Lur’e system, including the sector bounds, we want to find a single dynamic output feedback controller that makes the closed-loop system globally asymptotically stable for all φ(·)’s satisfying the sector-boundedness condition. Whereas, in the state feedback case, it is not an issue at all since the static controller design does not need the knowledge of the nonlinearity itself and thus works for all nonlinearities within the same sector. Stabilization of Lur’e systems through output feedback has been studied before. However, in the major part of the literature, the Lur’e-type nonlinearity is assumed to be known exactly, and this knowledge is used in the to-be-designed observer-based feedback controllers. In [1, 2] this led to a nonlinear observer, 2
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called ‘circle-criterion observer’, which used exact knowledge of the nonlinearity. Moreover, a slope-restrictedness condition is required, which is more conservative than the sector-boundedness condition (2). In [3], a circle-criterion observer was adapted in stabilization of Lur’e systems with saturating actuators. It also inspired the output feedback controller design for Lur’e systems with set-valued nonlinearities [4], and the H∞ observer design for Lipschitz and monotonic nonlinear systems [5]. However, the above design failed to achieve absolute stabilization. In order to remove the requirement on exact knowledge of the Lur’e-type nonlinearity, a nonlinear approximation was proposed in [6], which in fact still needs information on the nonlinearity itself. The same authors studied fixed-order dynamic output feedback control of discrete-time Lur’e systems in [7]. The results therein have been generalized to the continuous-time case in Section 2, where the order of the dynamic controller is not fixed. The problem formulation in the present paper differs from the above references: the Lur’e-type nonlinearity is not known, and can therefore not be used as part of the controller. The only information on the nonlinearity that we will use are the exact values of the matrices S1 and S2 defining the sector. We will present two different approaches to design the controller of the form (3). Numerical simulations for a flexible joint robotic arm will be given to validate the obtained theoretical results. 2. Design Approach I
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The interconnection of the Lur’e system (1) and the controller (3) is represented by x˙ Ap + Bp Dc Mp Bp Cc x E = − p φ(Cp x). (4) v˙ B c Mp Ac v 0
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Collect the matrices Ac , Bc , Cc and Dc describing the controller (3) into the matrix F ∈ R(n+nc )×(n+nc ) defined as Dc Cc F , . (5) B c Ac
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Henceforth, we will speak about the ‘controller’ (5). The following lemma gives a sufficient condition in terms of solvability of a matrix inequality under which the controller is absolutely stabilizing.
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Lemma Ap 0 1. Consider Bp 0 the Lur’e system(1) with the sector [S1 , S2 ]. Define A , Cp 0 ], E , Ep and M , Mp 0 . Let a controller , B , , C , [ 0 0 0 0 I 0 I F be given by (5). If there exists a positive definite matrix P ∈ R(n+nc )×(n+nc ) such that −P E+ P (A + BF M ) + (A + BF M )T P C T (S1 + S2 ) −C T (S1 S2 + S2 S1 )C (6) < 0, −E T P + (S1 + S2 )C
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then the controller (5) is absolutely stabilizing. 3
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Proof. Let Q > 0. Define V (x, v) , [ xv ] Q [ xv ]. Obviously, V (x, v) is positive definite and radially unbounded. The time derivative of V (x, v) along trajectories of (4) is given by T x x V˙ (x, v) =2 Q (A + BF M ) − Eφ(Cp x) v v T Q(A + BF M )+ x x −QE T = v (A + BF M ) Q v . T φ(Cp x) φ(Cp x) −E Q 0
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It is easily verified that the sector-boundedness condition (2) for (1) implies T 1 T − 21 CpT (S1 + S2 ) x x 2 Cp (S1 S2 + S2 S1 )Cp ≤ 0, φ(Cp x) φ(Cp x) − 12 (S1 + S2 )Cp I
which, in terms of the closed-loop system, yields T 1 T x x 1 T C (S S + S S )C − C (S + S ) 1 2 2 1 1 2 2 v 2 v ≤ 0. − 12 (S1 + S2 )C I φ(Cp x) φ(Cp x)
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Thus, we find that V˙ (x, v) is negative definite for all φ(·)’s within the sector [S1 , S2 ] if there exists a positive real number τ together with Q > 0 such that −QE+ Q(A + BF M ) + (A + BF M )T Q τ C T (S1 + S2 ) −τ C T (S1 S2 + S2 S1 )C (7) <0
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−E T Q + τ (S1 + S2 )C
−2τ I
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holds. Obviously, by defining P , τ1 Q, the matrix inequality (7) is equivalent to (6). This completes the proof. Using Lemma 1 one can verify whether a given F is absolutely stabilizing. Of course, the question is now under what conditions an F satisfying the condition (6) actually exists. This will be dealt with in the following. Before doing this, we review the concept of minimal left annihilator of a matrix.
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Definition 3. [8] Let M ∈ Rn×m with rank(M ) = r < n. We denote by M ⊥ any matrix in R(n−r)×n of full row rank such that M ⊥ M = 0. Any such matrix M ⊥ is called a minimal left annihilator of M .
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Note that a minimal left annihilator is only defined for matrices with linearly dependent rows. The set of all such matrices is given by M ⊥ = W U2T , where W is an arbitrary nonsingular matrix and U2 is obtained from the singular value T ⊥ 0 V V decomposition M = [ U1 U2 ] [ Σ is not 0 0 ] [ 1 2 ] . Thus, for a given M , M ⊥ unique. Throughout this paper, M will denote any choice from this set of matrices. The following lemma now gives necessary and sufficient conditions under which, for a given dimension nc , there exists a controller F ∈ R(n+nc )×(n+nc ) for which the matrix inequality (6) has a solution P > 0: 4
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−2(S2 − S1 )−2
CX
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Lemma 2. Let nc be a given nonnegative integer. There exist F and P > 0 such that (6) holds if and only if there exist positive definite matrices X ∈ R(n+nc )×(n+nc ) and Y ∈ R(n+nc )×(n+nc ) such that XY = I, A − 12 E(S1 + S2 )C X+ T ⊥T ⊥ XC T X A − 12 E(S1 + S2 )C B B < 0, (8) + 12 EE T 0 0 Y A − 12 E(S1 + S2 )C + T T ⊥ ⊥T 1 M A − 2 E(S1 + S2 )C Y Y E M T < 0. 1 T + 2 C (S2 − S1 )2 C 0 0 ET Y
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−2I
Furthermore, in terms of X and Y , the following P and F satisfy (6): P , X −1 and T T −1 F , −rB T Θ−1 M XΘ−1 , (10) x XM x XM
where r and Θx are determined as follows: choose a real number r such that
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1 1 Θx , rBB T − QX − EE T − XC T (S2 − S1 )2 CX > 0 2 2 T with QX , A − 21 E(S1 + S2 )C X + X A − 12 E(S1 + S2 )C .
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Remark 1. In order to enhance readability of this paper, we defer the proof of Lemma 2 to the Appendix. Several ideas in our proof are borrowed from [9], Section 2.3. For instance, the existence of r ∈ R such that Θx in (11) is positive definite follows by applying Finsler’s lemma (see e.g. [8]) to the LMI (8).
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Remark 2. We note that Lemma 2 also applies to the situation that nc = 0, in which we want to obtain a static output feedback controller. We also refer to [7] for the design of static output feedback in the context of Lur’e systems. In general, asking for solvability with nc = 0 is too restrictive, and it will turn out that we need nc > 0 for the LMI’s (8) and (9) to be solvable.
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Note that in Lemma 2, the controller dimension nc is assumed to be given, and the lemma gives necessary and sufficient conditions for solvability of (6) with that given dimension nc . But, in general finding the dimension of the controller is part of the design problem. Our following result addresses the problem of how to find an nc together with a suitable F ∈ R(n+nc )×(n+nc ) :
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Lemma 3. There exists a nonnegative integer nc , matrices X > 0 and Y > 0 of size (n + nc ) × (n + nc ) such that XY = I and the LMI’s (8), (9) hold if and only if there exist matrices Xp > 0, Yp > 0 of size n × n such that Xp CpT −2(S2 − S1 )−2
T ⊥ Qy + 12 CpT (S2 − S1 )2 Cp Mp EpT Yp 0
rank
I ≥ 0, Yp Xp I
I Yp
Yp Ep −2I
T ⊥T Mp < 0, 0
(12)
(13)
(14)
≤ n + nc ,
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Xp I
⊥T Bp < 0, 0
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⊥ Qx + 21 Ep EpT Bp 0 Cp Xp
(15)
T where Qx , Ap − 21 Ep (S1 + S2 )Cp Xp + Xp Ap − 21 Ep (S1 + S2 )Cp ,
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T 1 1 Qy , Yp Ap − Ep (S1 + S2 )Cp + Ap − Ep (S1 + S2 )Cp Yp . 2 2
Proof. (only if ) Assume there exists nonnegative integer nc , X > 0 and Y > 0 of sizeh (n + nci) × (n + nc )h such that i XY = I, (8) and (9) hold. Partition Xp Xpc T Xpc Xc
Yp Ypc T Ypc Yc
appropriately. Note that B ⊥ = [ Bp⊥ 0 ], ⊥ T ⊥ T⊥ ⊥ B 0 , M M T ⊥ = [ MpT ⊥ 0 ], [ B = M0 0I . In this way we obtain 0] = 0 0 I T = I and Xp Ypc +Xpc Yc = 0. (12) and (13). XY = I implies that Xp Yp +Xpc Ypc Thus
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and Y =
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X =
T Yp − Xp−1 = Ypc Yc−1 Ypc ≥ 0.
(16)
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Using the Schur complement lemma, (16) is equivalent to (14). In addition, h i Xp I rank = rank(Xp ) + rank Yp − Xp−1 I Yp T = n + rank Ypc Yc−1 Ypc ≤ n + nc .
So (15) holds. (if ) Let Ypc and Yc > 0 be any matrices satisfying (16) while Xp > 0 and Yp > 0 satisfy (12), (13), (14), and nc is chosen (15). It can be verified h to satisfy i Yp Ypc
that a matrix pair (X, Y ) such that Y = Y T Yc and X = Y −1 satisfy the pc conditions in Lemma 2. This completes the proof. Lemmas 1, 2 and 3 together give a design method for an absolutely stabilizing controller (3), summarized in the following theorem. 6
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Theorem 1. Consider the Lur’e system (1) with the sector [S1 , S2 ]. If there exist positive definite matrices Xp ∈ Rn×n and Yp ∈ Rn×n such that (12), (13) and (14) holds, then there exists an absolutely stabilizing controller (3). Its design can be performed as follows:
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1) Compute Xp > 0 and Yp > 0 such that (12), (13) and (14); h i Xp I 2) Choose the controller dimension as nc , rank − n, then define I Yp A, B, C, E and M as introduced in Lemma 1; 3) Choose Yc > 0 and Ypc satisfying (16) to obtain Y > 0 and define X , Y −1 ; 4) Compute r > 0 in (11) to get Θx > 0; 5) Compute F defined by (10) and partition it as
Dc
Cc Bc Ac
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3. Design Approach II
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For Case 1 of the simulation example in the subsequent Section 4 we have applied the design approach I to stabilize a flexible joint robotic arm modelled by the Lur’e system (1). It turns out that, although absolute stabilization is achieved, the transient performance of the closed-loop system is not satisfactory and a rather large control input is required. Therefore, in order to overcome the above drawbacks, in the next section we will present an alternative design approach.
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In this section we will take an alternative approach to designing an absolutely stabilizing dynamic controller for the Lur’e system (1). A main difference is that we will fix the controller state dimension nc to be equal to n, the state space dimension of the Lur’e system. We set the direct feedthrough matrix of the controller (3) equal to Dc = 0, and look at controllers of the form ( x ˆ˙ = Ac x ˆ + Bc y . (17) u = Cc x ˆ
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Then the interconnection of (1) and (17) is given by x˙ Ap Bp Cc x E = − p φ(Cp x). B c Mp Ac x ˆ 0 x ˆ˙
(18)
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The controller state x ˆ can be viewed as an estimate of x. By defining the error e(t) , x(t) − x ˆ(t), we also have x˙ Ap + B p C c −Bp Cc x E = − p φ(Cp x). (19) e˙ Ap − Bc Mp + Bp Cc − Ac −Bp Cc + Ac e Ep Clearly, for a given sector [S1 , S2 ], the closed-loop system (18) is absolutely stable if and only if the same holds for (19). The following theorem provides a solution to the design problem for Ac , Bc and Cc in (17). 7
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Theorem 2. Consider the Lur’e system (1) with the sector [S1 , S2 ]. Denote A¯p , Ap − 12 Ep (S1 + S2 )Cp and C¯p , (S2 − S1 )Cp . If there exist positive definite matrices Xp ∈ Rn×n and Yp ∈ Rn×n such that (20)
T ⊥T Mp < 0, 0
(21)
Xp C¯pT −I
T ⊥ Yp A¯p + A¯Tp Yp + 41 C¯pT C¯p Mp EpT Yp 0
Yp Ep −I
1 Yp − Xp−1 > 0, 4
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⊥T Bp < 0, 0
⊥ A¯p Xp + Xp A¯Tp + 41 Ep EpT Bp 0 C¯p Xp
(22)
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then there exists an absolutely stabilizing controller of the form (17). Moreover, if the positive definite matrices Xp and Yp satisfy the LMI’s (20), (21) and the matrix inequality (22), then suitable Ac , Bc and Cc are obtained as follows: Xp C¯pT < 0, −I
(23)
2) Choose a real number r2 such that Yp A¯p + A¯Tp Yp + 41 C¯pT C¯p − 2r2 MpT Mp EpT Yp
Yp Ep < 0, −I
(24)
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1) Choose a real number r1 such that A¯p Xp + Xp A¯T + 14 Ep EpT − 2r1 Bp BpT C¯p Xp
3) Define H , −r1 BpT Xp−1 , G , −r2 Yp−1 MpT ,
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T 1 RH , Xp−1 A¯p + Bp H + A¯p + Bp H Xp−1 + Xp−1 Ep EpT Xp−1 + C¯pT C¯p , 4
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T 1 RG , A¯p + GMp Yp−1 + Yp−1 A¯p + GMp + Yp−1 C¯pT C¯p Yp−1 + Ep EpT , 4
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4) Choose a real number k ∈ (0, 1) such that Yp RG Yp <
1 (1 − k)RH , 4
(25)
¯ , Zp−1 Yp G, 5) Define Zp , Yp − 14 Xp−1 , G 1 −1 1 −1 −1 ¯ T −1 T −1 T ¯ ¯ ¯ ∆1 , kZp Xp Ap + Ap Xp + Xp Ep Ep Xp + Cp Cp , 4 4 1 −1 ∆2 , − Zp (1 − k)H T BpT Xp−1 − kXp−1 Bp H , 4 8
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6) Choose
¯ Bc , −G, and Cc , H. ¯, Proof. In this proof, we denote C , [ Cp 0 ], E
Ep Ep
i
and
−Bp Cc . −Bp Cc + Ac
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Ap + Bp Cc A, Ap − Bc Mp + Bp Cc − Ac
h
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1 ¯ p + ∆ 1 + Bp H + ∆ 2 , Ac , A¯p + Ep EpT Xp−1 + GM 4
Similarly as in the proof of Lemma 1, it can be shown that if there exists a positive definite matrix P ∈ R2n×2n and a positive real number τ such that ¯ −P E+ T T P A + A P − τ C (S1 S2 + S2 S1 )C τ C T (S1 + S2 ) (26) < 0, T ¯ −E P + τ (S1 + S2 )C −2τ I
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then (19) is absolutely stable. Note that the unknown τ > 0 can be removed 1 from (26) through dividing (26) by 2τ and defining P¯ , 2τ P . We see that (26) has a solution pair (P, τ ) if and only if there is a positive definite solution P¯ to
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1 ¯E ¯ T P¯ < 0, P¯ A¯ + A¯T P¯ + C¯ T C¯ + P¯ E 4
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where C¯ , [ C¯p 0 ] and A¯p + Bp Cc A¯ , ¯ Ap − Bc Mp + Bp Cc − Ac
(27)
−Bp Cc . −Bp Cc + Ac
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Before moving on, by Finsler’s lemma, (20) and (21) imply that there exist real numbers r1 and r2 such that (23) and (24) hold, respectively. By taking Schur complements, (23) is equivalent to RH < 0, and similarly we have RG < 0. In addition, we choose P¯ to be 1 −1 Xp 0 4 ¯ . P , 0 Zp
By straightforward computation, the block (1, 1) of the left hand side of (27) turns out to be 14 RH . The blocks (2, 1) and (2, 2) can be computed to be equal to − 14 kRH and Yp RG Yp − 14 RH + 12 kRH , respectively. Thus the left hand side of (27) equals 1 − 41 kRH 4 RH , − 14 kRH Yp RG Yp − 41 RH + 12 kRH 9
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i.e.
1
4 (1
− k)RH 0
0 Yp RG Yp − 14 (1 − k)RH
1 1 + k 4 −1
−1 ⊗ RH . 1
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Obviously, the first term in the above is negative define, and the second term is 1 −1 ≥ 0 and RH < 0. Therefore, (26) as negative semi-define since k > 0, −1 1 well as (27) holds. This completes the proof. 4. A Simulation Example
0
2x3
φ(x3 )
ψ(x3 ) x 3
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In this section we will consider a flexible joint robotic arm model as an application example of Lur’e systems and show how to compute absolutely stabilizing dynamic output feedback controllers. The corresponding simulation results of the closed-loop control systems will be also given, along with some discussions. A type of flexible joint robotic arms can be modeled using the Lur’e system (1) [10], where x = [x1 x2 x3 x4 ]T , Ap = [0 1 0 0; −48.6 − 1.25 48.6 T T 0; 0 0 0 1; 19.5 0 − 16.17 0], Bp = [0 21.6 0 0] , Cp = [0 0 1 0], Ep = [0 0 0 3.33] and Mp = [ 10 01 00 00 ]. Its nonlinearity is represented by φ(x3 ) = x3 + sin x3 , which satisfies the sector-boundedness condition φ(x3 )(φ(x3 ) − 2x3 ) ≤ 0, see the red curve in Fig. 2. Note that different from [10] (see Page 727 therein), here the dynamics of x4 has been rewritten in order to render the new nonlinear part to lie inside the sector [0, 2].
Fig. 2: Two nonlinearities within sector [0, 2]
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In the following two cases, we will apply our first and second approaches, respectively. In the first case, we will compute a suitable controller of the form (3) by using Theorem 1. Case 1. From (12), we can compute Xp > 0 to be 0.4305 −0.6514 0.2940 −1.0916 −0.6514 10.2188 0.3089 −1.5696 . Xp = 0.2940 0.3089 0.6282 −0.8073 −1.0916 −1.5696 −0.8073 7.9034 10
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Here we enforce the left hand side of (14) to be strictly positive definite. Thus Yp > 0 determined by LMI (13) and the strict version of (14) can be computed to be 32.2408 0.0115 −0.0063 0.0089 0.0115 32.0406 −1.2093 −0.3112 Yp = −0.0063 −1.2093 34.7891 −1.7252. 0.0089 −0.3112 −1.7252 0.6567
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Hence the state dimension nc of a possible controller can be chosen as nc = 4 by (15). It is equal to the state space dimension of the flexible joint robotic arm. Without loss of generality, we choose Yc as Yc = I4 . It follows that 4.9737 0 0 0 −0.1380 5.6440 0 0 Ypc = 0.5414 −0.1516 5.6269 0 −0.1790 −0.0774 −0.2664 0.5398
from (16) by using Cholesky decomposition. Then, a suitable Y > 0 is obtained. Consequently, a suitable X > 0 is derived by X = Y −1 which is given in (28). Now a suitable real number r in (11) can be computed as r = 10866 and 0.2940 0.3089 0.6282 −0.8073 −1.9043 −1.7107 −3.7499 0.4358
−1.0916 −1.5696 −0.8073 7.9034 7.0643 9.3485 6.6483 −4.2659
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−0.6514 10.2188 0.3089 −1.5696 4.2019 −57.7501 −2.1563 0.8472
−2.5854 4.2019 −1.9043 7.0643 16.7348 −23.4576 12.5974 −3.8130
3.6364 −57.7501 −1.7107 9.3485 −23.4576 327.4086 12.1161 −5.0458
−1.9452 −2.1563 −3.7499 6.6483 12.5974 12.1161 23.8713 −3.5884
0.5892 0.8472 0.4358 −4.2659 −3.8130 −5.0458 −3.5884
(28)
3.3025
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0.4305 −0.6514 0.2940 −1.0916 X = −2.5854 3.6364 −1.9452 0.5892
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Θx > 0 is also known. So (10) provides a suitable F which can be appropriately partitioned to get −1.1022 2.0013 0.1776 −0.0569 −0.0285 0.2499 0.0212 −0.0074 Ac = 104 0.2491 −2.9095 −0.1279 −0.2999, 0.1392 −8.0287 −0.3002 −0.9877 −0.5235 1.1507 0.0023 0.1454 Bc = 105 0.0878 −1.6542, −0.0300 −4.5570 3 Cc = 10 0.3098 −5.8578 −0.2395 −0.6580 ,
and
Dc = 104 0.0806
−3.3280 .
Ignoring the physical quantities and units we take the initial state x(0) = T [1 2 3 4] . Setting the initial state of the dynamic controller (3) to be zero, the 11
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state trajectories of the closed-loop system (4) are plotted in Fig. 3. Clearly, our designed dynamic output feedback controller works. However, v(t) varies greatly compared to x(t) in transient state. This can be also seen from the plot of the control input given in Fig. 4, which is extremely large in transient state. Moreover, due to the very large controller matrices, it takes too much time to compute the state trajectories using Matlab and sometimes the computer will run out of memory.
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(3), we replace φ(x3 ) with the the dead-zone nonlinx3 < −2 − 2 ≤ x3 < 2 x3 ≥ 2
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output feedback controller earity 3 2 x3 + 3 ψ(x3 ) = 0 3 2 x3 − 3
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which is also sector-bounded within [0, 2], seethe orange curve in Fig. 2. Using c Cc the same initial states and the same F = D , the corresponding state Bc Ac trajectories are plotted in Fig. 5 along with the plot of the control input in Fig. 6. Obviously, the same F works against ψ(·) within the same sector while there exists the same issue discussed above. Case 2. In this case, we will compute a suitable dynamic output feedback controller of the form (17) by using Theorem 2. From (20), we can compute Xp > 0 to be 0.3030 −0.7051 0.1766 −0.9492 −0.7051 11.0552 0.3412 −1.2957 . Xp = 0.1766 0.3412 0.4949 −0.7042 −0.9492 −1.2957 −0.7042 7.6735
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controller works well. The required control input is not large either, see Fig. 8.
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Finally, we replace φ(·) with ψ(·) in (18). The corresponding state trajectories are plotted in Fig. 9 along with the plot of the control input in Fig. 10. Certainly, the feature of absolute stabilization holds.
Fig. 9: The state trajectories regarding ψ(·) in Case 2
Remark 3. If we compare the two design approaches I and II, from the simulation example we see that they perform quite differently. In particular, the obtained controllers using the two design approaches perform quite differently 19
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in transient state. More precisely, in order to drive the flexible joint robotic arm from the same initial states to the origin, the controller based on design approach I uses much more energy and its internal states vary greatly as compared to the states of the flexible joint robotic arm. A very powerful actuator would be needed, which could be expensive or does not even exist in practice. In the design approach I, all (linear) matrix inequalities are handled by algebraic techniques, while the choice of the dimension of the dynamic controller is flexible. In contrast, the design approach II is inspired by an H∞ optimal control algorithm for linear systems against external and unknown disturbance [11]. The Lur’e-type nonlinearity is viewed as a kind of state-dependent disturbance acting on the linear part of the Lur’e system. Thus the design approach II is based on H∞ -optimization. It is not unreasonable to believe that, in general, it achieves better transient performance than the first approach. Unfortunately, we have no rigorous analysis at the moment to provide comparisons between the two design approaches.
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We note that in dynamic output feedback control, often measurement noise is involved. In Case 1, measurement noise is amplified by the very large direct feedthrough matrix Dc and then enter the channels of the system states directly. Thus, even if the dynamic controller itself is not sensitive to measurement noise, the performance of the closed-loop system can be destroyed. In Case 2, we have no such issue thanks to the fact that Dc = 0. A simulation has also been performed considering an additive measurement noise, which is a Gaussian distributed random signal with mean zero and standard deviation 0.01. The simulation results are given below. Clearly, the dynamic controller obtained in Case 1 does not work at all in the presence of measurement noise, see Figs. 11 and 12. Both the state trajectories and the control input are expanding. In contrast, with the same measurement noise, the controller obtained in Case 2 still works and the effect of the measurement noise is not that prominent, see Fig. 13. Apart from smaller overshoot, the dynamic controller based on design approach II is not sensitive to measurement noise, see Fig. 14.
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5. Conclusions
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We have studied absolute stabilization of Lur’e systems under dynamic output feedback in the sense that the Lur’e-type nonlinearities are unknown. Only their sector bounds are assumed to be known. This setup is different from existing work on dynamic output feedback control for Lur’e systems, where the nonlinearities are always assumed to be known precisely and used to design the controllers. Two approaches are presented for the controller design. Our designed controllers can be computed using the Matlab LMI Control Toolbox. Their effectiveness has been validated through numerical simulation results on a Lur’e-type model of flexible joint robotic arms. For future research it is appealing to design a separate observer for Lur’e systems without using the Lur’e-type nonlinearities themselves. Such observer 21
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is indeed stronger than the circle-criterion observer, and would be applied to, for example, output feedback control and fault diagnosis of Lur’e systems. The robustness will be also considered against incomplete feedback information and model uncertainties in the future. Acknowledgements
This work was supported by the State Key Laboratory of Intelligent Control and Decision of Complex Systems, the Young Scientist Fund of the National Natural Science Foundation of China (Grant No. 61703099), the China Postdoctoral Science Foundation Funded Project (Grant No. 2017M621589), and grants from the Research Grants Council of Hong Kong (Grant No. CityU-11261516).
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References References
[1] M. Arcak, P. Kokotovi´c, Nonlinear observers: a circle criterion design and robustness analysis, Automatica 37 (12) (2001) 1923–1930.
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[2] M. Arcak, P. Kokotovi´c, Observer-based control of systems with sloperestricted nonlinearities, IEEE Transactions on Automatic Control 46 (7) (2001) 1146–1150.
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[3] J. da Silva Jr., E. Castelan, J. Corso, D. Eckhard, Dynamic output feedback stabilization for systems with sector-bounded nonlinearities and saturating actuators, Journal of the Franklin Institute 350 (3) (2013) 464–484.
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[4] J. de Bruin, A. Doris, N. van de Wouw, W. Heemels, H. Nijmeijer, Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities, Automatica 45 (2) (2009) 405–415.
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[5] A. Zemouche, R. Rajamani, G. Phanomchoeng, B. Boulkroune, H. Rafaralahy, Circle criterion-based H∞ observer design for Lipschitz and monotonic nonlinear systems - Enhanced LMI conditions and constructive discussions, Automatica 85 (2017) 412–425.
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[6] K.-K. Kim, R. Braatz, Observer-based output feedback control of discretetime Lur’e systems with sector-bounded slope-restricted nonlinearities, International Journal of Robust and Nonlinear Control 24 (16) (2014) 2458– 2472. [7] K.-K. Kim, R. Braatz, Robust static and fixed-order dynamic output feedback control of discrete-time parametric uncertain Lur´e systems: Sequential SDP relaxation approaches, Optimal Control Applications and Methods 38 (1) (2017) 36–58.
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[8] T. Iwasaki, R. Skelton, All controllers for the general H∞ control problem: LMI existence conditions and state space formulas, Automatica 30 (8) (1994) 1307–1317.
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[9] R. Skelton, T. Iwasaki, K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, CRC Press, 1997.
[10] R. Rajamani, Y. Cho, Existence and design of observers for nonlinear systems: Relation to distance to unobservability, International Journal of Control 69 (5) (1998) 717–731.
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[11] C. Scherer, The Riccati Inequality and State-Space H∞ -Optimal Control, Ph.D. thesis, University of Wuerzburg (June 1991).
Appendix
Proof of Lemma 2. The existence of F and to the existence of solutions F and X > 0 to BF M X + (BF M X)T XC T 1 T +Q + EE X 2 −2(S2 − S1 )
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T where QY , Y A − 21 E(S1 + S2 )C + A − 12 E(S1 + S2 )C Y , which then implies (9). (if ) By Finsler’s lemma [8], (8) implies that there exists a real number r such that −XC T rBB T − QX − 12 EE T > 0, −CX 2(S2 − S1 )−2 equivalently, Θx > 0. Similarly, (9) implies that there exists a positive definite matrix S > 0 such that T −1 M S M − QY − 21 C T (S2 − S1 )2 C −Y E > 0, −E T Y 2I 27
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equivalently, Θy , XM T S −1 M X − QX − 12 XC T (S2 − S1 )2 CX − 12 EE T > 0. Define
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or equivalently, using the matrix inversion lemma [9], −1 rI − r2 B T Θx + XM T Z −1 M X B > 0, equivalently, using the Schur complement lemma, rI rB T > 0, rB Θx + XM T Z −1 M X which is equivalent to
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Accepted Manuscript
Absolute Stabilization of Lur’e Systems via Dynamic Output Feedback Fan Zhang, Harry L. Trentelman, Gang Feng, Jacquelien M.A. Scherpen PII: DOI: Reference:
S0947-3580(18)30062-1 https://doi.org/10.1016/j.ejcon.2018.09.015 EJCON 297
To appear in:
European Journal of Control
Received date: Revised date: Accepted date:
30 January 2018 2 August 2018 20 September 2018
Please cite this article as: Fan Zhang, Harry L. Trentelman, Gang Feng, Jacquelien M.A. Scherpen, Absolute Stabilization of Lur’e Systems via Dynamic Output Feedback, European Journal of Control (2018), doi: https://doi.org/10.1016/j.ejcon.2018.09.015
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Absolute Stabilization of Lur’e Systems via Dynamic Output Feedback
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Fan Zhanga , Harry L. Trentelmanb,∗, Gang Fengc , Jacquelien M.A. Scherpenb a School of Mathematics, Southeast University, Nanjing, China of Science and Engineering, University of Groningen, Groningen, The Netherlands c Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China
b Faculty
Abstract
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In this paper we study the problem of absolute stabilization under dynamic output feedback for Lur’e systems with sector-bounded unknown nonlinearities. In most of the literature, besides the incremental sector-boundedness condition, the Lur’e-type nonlinearity itself is assumed to be known exactly and used in the dynamic output feedback controller design. In the present paper only the sectorboundedness condition is employed, and exact knowledge of the nonlinearity will not be used in the controller design. More precisely, we we will only employ knowledge of the sector in which the unknown nonlinearity lies. Two different approaches will be presented for the dynamic controller design, both using linear matrix inequality techniques. Numerical simulations of a flexible joint robotic arm will illustrate the theoretical results obtained in this paper.
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Keywords: Lur’e system, absolute stabilization, dynamic output feedback, linear matrix inequality, flexible joint robotic arm
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1. Introduction
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In this paper we study the problem of absolute stabilization by dynamic output feedback for Lur’e systems. A Lur’e system is the interconnection of a linear input-state-output system and an unknown static nonlinearity through a negative feedback loop, commonly represented as ( x˙ = Ap x + Bp u + Ep d, (1) z = Cp x, d = −φ(z), y = Mp x, where x(t) ∈ Rn , u(t) ∈ Rm , z(t) ∈ Rp and y(t) ∈ Rq are the state, control input, loop injection and measurement output, respectively. The matrices Ap , ∗ Corresponding
author Email addresses: [email protected], [email protected] (Fan Zhang), [email protected] (Harry L. Trentelman), [email protected] (Gang Feng), [email protected] (Jacquelien M.A. Scherpen)
Preprint submitted to Journal of LATEX Templates
September 27, 2018
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Bp , Cp , Ep and Mp are of compatible dimensions. Without loss of generality, Bp and Mp are assumed to be of full column rank and full row rank, respectively. The equation d = −φ(z) describes a static negative feedback loop, where φ(·) is an unknown nonlinear function, see Fig. 1. This nonlinearity is however
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assumed to be sector-bounded, in the sense that it satisfies the following sectorboundedness condition.
Definition 1. Let S1 , S2 ∈ Rp×p be real symmetric matrices such that S1 is positive semi-definite and S2 −S1 is positive definite, i.e. 0 ≤ S1 < S2 . Then the nonlinearity φ(·) is called sector-bounded within the sector [S1 , S2 ] if it satisfies (φ(z) − S1 z)T (φ(z) − S2 z) ≤ 0,
∀ z ∈ Rp .
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We will be dealing with output feedback controllers of the form ( v˙ = Ac v + Bc y , u = Cc v + Dc y
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where v(t) ∈ Rnc is the state of the dynamic controller, the positive integer nc and the matrices Ac ∈ Rnc ×nc , Bc ∈ Rnc ×q , Cc ∈ Rm×nc , Dc ∈ Rm×q are to be determined. The aim is to design a controller of the form (3) that absolutely stabilizes the Lur’e system (1), i.e. the closed-loop system is absolutely stable, in the sense defined as follows:
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Definition 2. Given the bounds S1 and S2 with 0 ≤ S1 < S2 , the feedback interconnection of the Lur’e system (1) and the controller (3) is called absolutely stable if it is globally asymptotically stable for every φ(·) within the sector [S1 , S2 ].
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Thus, given the Lur’e system, including the sector bounds, we want to find a single dynamic output feedback controller that makes the closed-loop system globally asymptotically stable for all φ(·)’s satisfying the sector-boundedness condition. Whereas, in the state feedback case, it is not an issue at all since the static controller design does not need the knowledge of the nonlinearity itself and thus works for all nonlinearities within the same sector. Stabilization of Lur’e systems through output feedback has been studied before. However, in the major part of the literature, the Lur’e-type nonlinearity is assumed to be known exactly, and this knowledge is used in the to-be-designed observer-based feedback controllers. In [1, 2] this led to a nonlinear observer, 2
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called ‘circle-criterion observer’, which used exact knowledge of the nonlinearity. Moreover, a slope-restrictedness condition is required, which is more conservative than the sector-boundedness condition (2). In [3], a circle-criterion observer was adapted in stabilization of Lur’e systems with saturating actuators. It also inspired the output feedback controller design for Lur’e systems with set-valued nonlinearities [4], and the H∞ observer design for Lipschitz and monotonic nonlinear systems [5]. However, the above design failed to achieve absolute stabilization. In order to remove the requirement on exact knowledge of the Lur’e-type nonlinearity, a nonlinear approximation was proposed in [6], which in fact still needs information on the nonlinearity itself. The same authors studied fixed-order dynamic output feedback control of discrete-time Lur’e systems in [7]. The results therein have been generalized to the continuous-time case in Section 2, where the order of the dynamic controller is not fixed. The problem formulation in the present paper differs from the above references: the Lur’e-type nonlinearity is not known, and can therefore not be used as part of the controller. The only information on the nonlinearity that we will use are the exact values of the matrices S1 and S2 defining the sector. We will present two different approaches to design the controller of the form (3). Numerical simulations for a flexible joint robotic arm will be given to validate the obtained theoretical results. 2. Design Approach I
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The interconnection of the Lur’e system (1) and the controller (3) is represented by x˙ Ap + Bp Dc Mp Bp Cc x E = − p φ(Cp x). (4) v˙ B c Mp Ac v 0
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Henceforth, we will speak about the ‘controller’ (5). The following lemma gives a sufficient condition in terms of solvability of a matrix inequality under which the controller is absolutely stabilizing.
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Lemma Ap 0 1. Consider Bp 0 the Lur’e system(1) with the sector [S1 , S2 ]. Define A , Cp 0 ], E , Ep and M , Mp 0 . Let a controller , B , , C , [ 0 0 0 0 I 0 I F be given by (5). If there exists a positive definite matrix P ∈ R(n+nc )×(n+nc ) such that −P E+ P (A + BF M ) + (A + BF M )T P C T (S1 + S2 ) −C T (S1 S2 + S2 S1 )C (6) < 0, −E T P + (S1 + S2 )C
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Proof. Let Q > 0. Define V (x, v) , [ xv ] Q [ xv ]. Obviously, V (x, v) is positive definite and radially unbounded. The time derivative of V (x, v) along trajectories of (4) is given by T x x V˙ (x, v) =2 Q (A + BF M ) − Eφ(Cp x) v v T Q(A + BF M )+ x x −QE T = v (A + BF M ) Q v . T φ(Cp x) φ(Cp x) −E Q 0
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It is easily verified that the sector-boundedness condition (2) for (1) implies T 1 T − 21 CpT (S1 + S2 ) x x 2 Cp (S1 S2 + S2 S1 )Cp ≤ 0, φ(Cp x) φ(Cp x) − 12 (S1 + S2 )Cp I
which, in terms of the closed-loop system, yields T 1 T x x 1 T C (S S + S S )C − C (S + S ) 1 2 2 1 1 2 2 v 2 v ≤ 0. − 12 (S1 + S2 )C I φ(Cp x) φ(Cp x)
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holds. Obviously, by defining P , τ1 Q, the matrix inequality (7) is equivalent to (6). This completes the proof. Using Lemma 1 one can verify whether a given F is absolutely stabilizing. Of course, the question is now under what conditions an F satisfying the condition (6) actually exists. This will be dealt with in the following. Before doing this, we review the concept of minimal left annihilator of a matrix.
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Definition 3. [8] Let M ∈ Rn×m with rank(M ) = r < n. We denote by M ⊥ any matrix in R(n−r)×n of full row rank such that M ⊥ M = 0. Any such matrix M ⊥ is called a minimal left annihilator of M .
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Note that a minimal left annihilator is only defined for matrices with linearly dependent rows. The set of all such matrices is given by M ⊥ = W U2T , where W is an arbitrary nonsingular matrix and U2 is obtained from the singular value T ⊥ 0 V V decomposition M = [ U1 U2 ] [ Σ is not 0 0 ] [ 1 2 ] . Thus, for a given M , M ⊥ unique. Throughout this paper, M will denote any choice from this set of matrices. The following lemma now gives necessary and sufficient conditions under which, for a given dimension nc , there exists a controller F ∈ R(n+nc )×(n+nc ) for which the matrix inequality (6) has a solution P > 0: 4
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−2(S2 − S1 )−2
CX
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Lemma 2. Let nc be a given nonnegative integer. There exist F and P > 0 such that (6) holds if and only if there exist positive definite matrices X ∈ R(n+nc )×(n+nc ) and Y ∈ R(n+nc )×(n+nc ) such that XY = I, A − 12 E(S1 + S2 )C X+ T ⊥T ⊥ XC T X A − 12 E(S1 + S2 )C B B < 0, (8) + 12 EE T 0 0 Y A − 12 E(S1 + S2 )C + T T ⊥ ⊥T 1 M A − 2 E(S1 + S2 )C Y Y E M T < 0. 1 T + 2 C (S2 − S1 )2 C 0 0 ET Y
(9)
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−2I
Furthermore, in terms of X and Y , the following P and F satisfy (6): P , X −1 and T T −1 F , −rB T Θ−1 M XΘ−1 , (10) x XM x XM
where r and Θx are determined as follows: choose a real number r such that
(11)
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1 1 Θx , rBB T − QX − EE T − XC T (S2 − S1 )2 CX > 0 2 2 T with QX , A − 21 E(S1 + S2 )C X + X A − 12 E(S1 + S2 )C .
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Remark 1. In order to enhance readability of this paper, we defer the proof of Lemma 2 to the Appendix. Several ideas in our proof are borrowed from [9], Section 2.3. For instance, the existence of r ∈ R such that Θx in (11) is positive definite follows by applying Finsler’s lemma (see e.g. [8]) to the LMI (8).
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Remark 2. We note that Lemma 2 also applies to the situation that nc = 0, in which we want to obtain a static output feedback controller. We also refer to [7] for the design of static output feedback in the context of Lur’e systems. In general, asking for solvability with nc = 0 is too restrictive, and it will turn out that we need nc > 0 for the LMI’s (8) and (9) to be solvable.
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Note that in Lemma 2, the controller dimension nc is assumed to be given, and the lemma gives necessary and sufficient conditions for solvability of (6) with that given dimension nc . But, in general finding the dimension of the controller is part of the design problem. Our following result addresses the problem of how to find an nc together with a suitable F ∈ R(n+nc )×(n+nc ) :
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Lemma 3. There exists a nonnegative integer nc , matrices X > 0 and Y > 0 of size (n + nc ) × (n + nc ) such that XY = I and the LMI’s (8), (9) hold if and only if there exist matrices Xp > 0, Yp > 0 of size n × n such that Xp CpT −2(S2 − S1 )−2
T ⊥ Qy + 12 CpT (S2 − S1 )2 Cp Mp EpT Yp 0
rank
I ≥ 0, Yp Xp I
I Yp
Yp Ep −2I
T ⊥T Mp < 0, 0
(12)
(13)
(14)
≤ n + nc ,
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Xp I
⊥T Bp < 0, 0
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⊥ Qx + 21 Ep EpT Bp 0 Cp Xp
(15)
T where Qx , Ap − 21 Ep (S1 + S2 )Cp Xp + Xp Ap − 21 Ep (S1 + S2 )Cp ,
M
T 1 1 Qy , Yp Ap − Ep (S1 + S2 )Cp + Ap − Ep (S1 + S2 )Cp Yp . 2 2
Proof. (only if ) Assume there exists nonnegative integer nc , X > 0 and Y > 0 of sizeh (n + nci) × (n + nc )h such that i XY = I, (8) and (9) hold. Partition Xp Xpc T Xpc Xc
Yp Ypc T Ypc Yc
appropriately. Note that B ⊥ = [ Bp⊥ 0 ], ⊥ T ⊥ T⊥ ⊥ B 0 , M M T ⊥ = [ MpT ⊥ 0 ], [ B = M0 0I . In this way we obtain 0] = 0 0 I T = I and Xp Ypc +Xpc Yc = 0. (12) and (13). XY = I implies that Xp Yp +Xpc Ypc Thus
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and Y =
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X =
T Yp − Xp−1 = Ypc Yc−1 Ypc ≥ 0.
(16)
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Using the Schur complement lemma, (16) is equivalent to (14). In addition, h i Xp I rank = rank(Xp ) + rank Yp − Xp−1 I Yp T = n + rank Ypc Yc−1 Ypc ≤ n + nc .
So (15) holds. (if ) Let Ypc and Yc > 0 be any matrices satisfying (16) while Xp > 0 and Yp > 0 satisfy (12), (13), (14), and nc is chosen (15). It can be verified h to satisfy i Yp Ypc
that a matrix pair (X, Y ) such that Y = Y T Yc and X = Y −1 satisfy the pc conditions in Lemma 2. This completes the proof. Lemmas 1, 2 and 3 together give a design method for an absolutely stabilizing controller (3), summarized in the following theorem. 6
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Theorem 1. Consider the Lur’e system (1) with the sector [S1 , S2 ]. If there exist positive definite matrices Xp ∈ Rn×n and Yp ∈ Rn×n such that (12), (13) and (14) holds, then there exists an absolutely stabilizing controller (3). Its design can be performed as follows:
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1) Compute Xp > 0 and Yp > 0 such that (12), (13) and (14); h i Xp I 2) Choose the controller dimension as nc , rank − n, then define I Yp A, B, C, E and M as introduced in Lemma 1; 3) Choose Yc > 0 and Ypc satisfying (16) to obtain Y > 0 and define X , Y −1 ; 4) Compute r > 0 in (11) to get Θx > 0; 5) Compute F defined by (10) and partition it as
Dc
Cc Bc Ac
.
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3. Design Approach II
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For Case 1 of the simulation example in the subsequent Section 4 we have applied the design approach I to stabilize a flexible joint robotic arm modelled by the Lur’e system (1). It turns out that, although absolute stabilization is achieved, the transient performance of the closed-loop system is not satisfactory and a rather large control input is required. Therefore, in order to overcome the above drawbacks, in the next section we will present an alternative design approach.
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In this section we will take an alternative approach to designing an absolutely stabilizing dynamic controller for the Lur’e system (1). A main difference is that we will fix the controller state dimension nc to be equal to n, the state space dimension of the Lur’e system. We set the direct feedthrough matrix of the controller (3) equal to Dc = 0, and look at controllers of the form ( x ˆ˙ = Ac x ˆ + Bc y . (17) u = Cc x ˆ
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Then the interconnection of (1) and (17) is given by x˙ Ap Bp Cc x E = − p φ(Cp x). B c Mp Ac x ˆ 0 x ˆ˙
(18)
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The controller state x ˆ can be viewed as an estimate of x. By defining the error e(t) , x(t) − x ˆ(t), we also have x˙ Ap + B p C c −Bp Cc x E = − p φ(Cp x). (19) e˙ Ap − Bc Mp + Bp Cc − Ac −Bp Cc + Ac e Ep Clearly, for a given sector [S1 , S2 ], the closed-loop system (18) is absolutely stable if and only if the same holds for (19). The following theorem provides a solution to the design problem for Ac , Bc and Cc in (17). 7
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Theorem 2. Consider the Lur’e system (1) with the sector [S1 , S2 ]. Denote A¯p , Ap − 12 Ep (S1 + S2 )Cp and C¯p , (S2 − S1 )Cp . If there exist positive definite matrices Xp ∈ Rn×n and Yp ∈ Rn×n such that (20)
T ⊥T Mp < 0, 0
(21)
Xp C¯pT −I
T ⊥ Yp A¯p + A¯Tp Yp + 41 C¯pT C¯p Mp EpT Yp 0
Yp Ep −I
1 Yp − Xp−1 > 0, 4
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⊥T Bp < 0, 0
⊥ A¯p Xp + Xp A¯Tp + 41 Ep EpT Bp 0 C¯p Xp
(22)
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then there exists an absolutely stabilizing controller of the form (17). Moreover, if the positive definite matrices Xp and Yp satisfy the LMI’s (20), (21) and the matrix inequality (22), then suitable Ac , Bc and Cc are obtained as follows: Xp C¯pT < 0, −I
(23)
2) Choose a real number r2 such that Yp A¯p + A¯Tp Yp + 41 C¯pT C¯p − 2r2 MpT Mp EpT Yp
Yp Ep < 0, −I
(24)
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1) Choose a real number r1 such that A¯p Xp + Xp A¯T + 14 Ep EpT − 2r1 Bp BpT C¯p Xp
3) Define H , −r1 BpT Xp−1 , G , −r2 Yp−1 MpT ,
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T 1 RH , Xp−1 A¯p + Bp H + A¯p + Bp H Xp−1 + Xp−1 Ep EpT Xp−1 + C¯pT C¯p , 4
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T 1 RG , A¯p + GMp Yp−1 + Yp−1 A¯p + GMp + Yp−1 C¯pT C¯p Yp−1 + Ep EpT , 4
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4) Choose a real number k ∈ (0, 1) such that Yp RG Yp <
1 (1 − k)RH , 4
(25)
¯ , Zp−1 Yp G, 5) Define Zp , Yp − 14 Xp−1 , G 1 −1 1 −1 −1 ¯ T −1 T −1 T ¯ ¯ ¯ ∆1 , kZp Xp Ap + Ap Xp + Xp Ep Ep Xp + Cp Cp , 4 4 1 −1 ∆2 , − Zp (1 − k)H T BpT Xp−1 − kXp−1 Bp H , 4 8
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6) Choose
¯ Bc , −G, and Cc , H. ¯, Proof. In this proof, we denote C , [ Cp 0 ], E
Ep Ep
i
and
−Bp Cc . −Bp Cc + Ac
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Ap + Bp Cc A, Ap − Bc Mp + Bp Cc − Ac
h
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1 ¯ p + ∆ 1 + Bp H + ∆ 2 , Ac , A¯p + Ep EpT Xp−1 + GM 4
Similarly as in the proof of Lemma 1, it can be shown that if there exists a positive definite matrix P ∈ R2n×2n and a positive real number τ such that ¯ −P E+ T T P A + A P − τ C (S1 S2 + S2 S1 )C τ C T (S1 + S2 ) (26) < 0, T ¯ −E P + τ (S1 + S2 )C −2τ I
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then (19) is absolutely stable. Note that the unknown τ > 0 can be removed 1 from (26) through dividing (26) by 2τ and defining P¯ , 2τ P . We see that (26) has a solution pair (P, τ ) if and only if there is a positive definite solution P¯ to
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1 ¯E ¯ T P¯ < 0, P¯ A¯ + A¯T P¯ + C¯ T C¯ + P¯ E 4
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where C¯ , [ C¯p 0 ] and A¯p + Bp Cc A¯ , ¯ Ap − Bc Mp + Bp Cc − Ac
(27)
−Bp Cc . −Bp Cc + Ac
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Before moving on, by Finsler’s lemma, (20) and (21) imply that there exist real numbers r1 and r2 such that (23) and (24) hold, respectively. By taking Schur complements, (23) is equivalent to RH < 0, and similarly we have RG < 0. In addition, we choose P¯ to be 1 −1 Xp 0 4 ¯ . P , 0 Zp
By straightforward computation, the block (1, 1) of the left hand side of (27) turns out to be 14 RH . The blocks (2, 1) and (2, 2) can be computed to be equal to − 14 kRH and Yp RG Yp − 14 RH + 12 kRH , respectively. Thus the left hand side of (27) equals 1 − 41 kRH 4 RH , − 14 kRH Yp RG Yp − 41 RH + 12 kRH 9
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i.e.
1
4 (1
− k)RH 0
0 Yp RG Yp − 14 (1 − k)RH
1 1 + k 4 −1
−1 ⊗ RH . 1
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Obviously, the first term in the above is negative define, and the second term is 1 −1 ≥ 0 and RH < 0. Therefore, (26) as negative semi-define since k > 0, −1 1 well as (27) holds. This completes the proof. 4. A Simulation Example
0
2x3
φ(x3 )
ψ(x3 ) x 3
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In this section we will consider a flexible joint robotic arm model as an application example of Lur’e systems and show how to compute absolutely stabilizing dynamic output feedback controllers. The corresponding simulation results of the closed-loop control systems will be also given, along with some discussions. A type of flexible joint robotic arms can be modeled using the Lur’e system (1) [10], where x = [x1 x2 x3 x4 ]T , Ap = [0 1 0 0; −48.6 − 1.25 48.6 T T 0; 0 0 0 1; 19.5 0 − 16.17 0], Bp = [0 21.6 0 0] , Cp = [0 0 1 0], Ep = [0 0 0 3.33] and Mp = [ 10 01 00 00 ]. Its nonlinearity is represented by φ(x3 ) = x3 + sin x3 , which satisfies the sector-boundedness condition φ(x3 )(φ(x3 ) − 2x3 ) ≤ 0, see the red curve in Fig. 2. Note that different from [10] (see Page 727 therein), here the dynamics of x4 has been rewritten in order to render the new nonlinear part to lie inside the sector [0, 2].
Fig. 2: Two nonlinearities within sector [0, 2]
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In the following two cases, we will apply our first and second approaches, respectively. In the first case, we will compute a suitable controller of the form (3) by using Theorem 1. Case 1. From (12), we can compute Xp > 0 to be 0.4305 −0.6514 0.2940 −1.0916 −0.6514 10.2188 0.3089 −1.5696 . Xp = 0.2940 0.3089 0.6282 −0.8073 −1.0916 −1.5696 −0.8073 7.9034 10
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Here we enforce the left hand side of (14) to be strictly positive definite. Thus Yp > 0 determined by LMI (13) and the strict version of (14) can be computed to be 32.2408 0.0115 −0.0063 0.0089 0.0115 32.0406 −1.2093 −0.3112 Yp = −0.0063 −1.2093 34.7891 −1.7252. 0.0089 −0.3112 −1.7252 0.6567
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Hence the state dimension nc of a possible controller can be chosen as nc = 4 by (15). It is equal to the state space dimension of the flexible joint robotic arm. Without loss of generality, we choose Yc as Yc = I4 . It follows that 4.9737 0 0 0 −0.1380 5.6440 0 0 Ypc = 0.5414 −0.1516 5.6269 0 −0.1790 −0.0774 −0.2664 0.5398
from (16) by using Cholesky decomposition. Then, a suitable Y > 0 is obtained. Consequently, a suitable X > 0 is derived by X = Y −1 which is given in (28). Now a suitable real number r in (11) can be computed as r = 10866 and 0.2940 0.3089 0.6282 −0.8073 −1.9043 −1.7107 −3.7499 0.4358
−1.0916 −1.5696 −0.8073 7.9034 7.0643 9.3485 6.6483 −4.2659
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−0.6514 10.2188 0.3089 −1.5696 4.2019 −57.7501 −2.1563 0.8472
−2.5854 4.2019 −1.9043 7.0643 16.7348 −23.4576 12.5974 −3.8130
3.6364 −57.7501 −1.7107 9.3485 −23.4576 327.4086 12.1161 −5.0458
−1.9452 −2.1563 −3.7499 6.6483 12.5974 12.1161 23.8713 −3.5884
0.5892 0.8472 0.4358 −4.2659 −3.8130 −5.0458 −3.5884
(28)
3.3025
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0.4305 −0.6514 0.2940 −1.0916 X = −2.5854 3.6364 −1.9452 0.5892
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Θx > 0 is also known. So (10) provides a suitable F which can be appropriately partitioned to get −1.1022 2.0013 0.1776 −0.0569 −0.0285 0.2499 0.0212 −0.0074 Ac = 104 0.2491 −2.9095 −0.1279 −0.2999, 0.1392 −8.0287 −0.3002 −0.9877 −0.5235 1.1507 0.0023 0.1454 Bc = 105 0.0878 −1.6542, −0.0300 −4.5570 3 Cc = 10 0.3098 −5.8578 −0.2395 −0.6580 ,
and
Dc = 104 0.0806
−3.3280 .
Ignoring the physical quantities and units we take the initial state x(0) = T [1 2 3 4] . Setting the initial state of the dynamic controller (3) to be zero, the 11
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x1 v1
0
100
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0 −100 20
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1
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1
−20 20
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state trajectories of the closed-loop system (4) are plotted in Fig. 3. Clearly, our designed dynamic output feedback controller works. However, v(t) varies greatly compared to x(t) in transient state. This can be also seen from the plot of the control input given in Fig. 4, which is extremely large in transient state. Moreover, due to the very large controller matrices, it takes too much time to compute the state trajectories using Matlab and sometimes the computer will run out of memory.
1
2
2
4
5
3
4
5
3
4
5
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1
6 x4 v4
2
3
time(s)
4
5
6
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0
6 x3 v3
0
−20
6
x2 v2
Fig. 3: The state trajectories regarding φ(·) in Case 1
In order to illustrate the feature of absolute stabilization for the dynamic
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1
x 10
0 4
0
−1
x 10
−1 −2
−2
u −3
−3
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−4
−4
−5 −5
−6
−6
0
0.005
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−7
0
1
2
0.01 3
time(s)
4
0.015 5
6
AC
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PT
−7
Fig. 4: The control input regarding φ(·) in Case 1
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(3), we replace φ(x3 ) with the the dead-zone nonlinx3 < −2 − 2 ≤ x3 < 2 x3 ≥ 2
,
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output feedback controller earity 3 2 x3 + 3 ψ(x3 ) = 0 3 2 x3 − 3
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which is also sector-bounded within [0, 2], seethe orange curve in Fig. 2. Using c Cc the same initial states and the same F = D , the corresponding state Bc Ac trajectories are plotted in Fig. 5 along with the plot of the control input in Fig. 6. Obviously, the same F works against ψ(·) within the same sector while there exists the same issue discussed above. Case 2. In this case, we will compute a suitable dynamic output feedback controller of the form (17) by using Theorem 2. From (20), we can compute Xp > 0 to be 0.3030 −0.7051 0.1766 −0.9492 −0.7051 11.0552 0.3412 −1.2957 . Xp = 0.1766 0.3412 0.4949 −0.7042 −0.9492 −1.2957 −0.7042 7.6735
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Then (22) becomes linear and together with LMI (21), a suitable Yp > 0 is computed as 3.7727 0.2374 −0.7035 0.3210 0.2374 1.0201 −0.0941 0.0119 Yp = −0.7035 −0.0941 1.7614 −0.1749. 0.3210 0.0119 −0.1749 0.1095
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Similarly, by LMI’s (23), (24) and (25), suitable real numbers r1 , r2 and k ∈ (0, 1) are computed to be r1 = 0.9299, r2 = 4632 and k = 0.9794, respectively. Now it is straightforward to get −4.8316 −0.0155 0.0371 −0.0126 −0.4889 −4.9996 0.1835 −0.0557 Ac = 103 −0.0475 −1.2683 0.0153 −0.0051, −0.5459 −7.9553 0.1742 −0.0697 4.7265 0.0000 0.0000 4.9183 Bc = 103 0.0000 1.2617, 0.0000 7.8693 and
Cc = −19.0775
−3.4955
5.7736
−2.4202 .
Using the same initial states as in Case 1, the state trajectories of the closedloop system (18) are plotted in Fig. 7. Clearly, this dynamic output feedback 14
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0 −20 50
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0 −20 20
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6
x3 v3
3
4
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6 x4 v4
3
time(s)
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6
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Fig. 5: The state trajectories regarding ψ(·) in Case 1
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u −3
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−5 −5
−6
−6
0
0.005
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−7
0
1
2
0.01 3
time(s)
4
0.015 5
6
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−7
Fig. 6: The control input regarding ψ(·) in Case 1
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x1 x ˆ1
0 −2 5
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1
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0 −5 20
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4
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x3 x ˆ3
3
4
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6 x4 x ˆ4
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time(s)
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6
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x2 x ˆ2
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Fig. 7: The state trajectories regarding φ(·) in Case 2
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4 2 0 −2 −4 −5
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u
0
−10
−10 −12
0
0.02
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−15
0
1
0.04 2
3
time(s)
0.06
0.08
4
5
0.1 6
AC
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−14
Fig. 8: The control input regarding φ(·) in Case 2
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controller works well. The required control input is not large either, see Fig. 8.
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x1 x ˆ1
0 −2 5
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0 −5 10
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6 x4 x ˆ4
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x3 x ˆ3
0
−10
6
x2 x ˆ2
0 −5
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Finally, we replace φ(·) with ψ(·) in (18). The corresponding state trajectories are plotted in Fig. 9 along with the plot of the control input in Fig. 10. Certainly, the feature of absolute stabilization holds.
Fig. 9: The state trajectories regarding ψ(·) in Case 2
Remark 3. If we compare the two design approaches I and II, from the simulation example we see that they perform quite differently. In particular, the obtained controllers using the two design approaches perform quite differently 19
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4 2 0 −2 −4 −5
−6 −8
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u
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−10
−10 −12
0
0.02
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−15
0
1
0.04 2
3
time(s)
0.06
0.08
4
5
0.1 6
AC
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−14
Fig. 10: The control input regarding ψ(·) in Case 2
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in transient state. More precisely, in order to drive the flexible joint robotic arm from the same initial states to the origin, the controller based on design approach I uses much more energy and its internal states vary greatly as compared to the states of the flexible joint robotic arm. A very powerful actuator would be needed, which could be expensive or does not even exist in practice. In the design approach I, all (linear) matrix inequalities are handled by algebraic techniques, while the choice of the dimension of the dynamic controller is flexible. In contrast, the design approach II is inspired by an H∞ optimal control algorithm for linear systems against external and unknown disturbance [11]. The Lur’e-type nonlinearity is viewed as a kind of state-dependent disturbance acting on the linear part of the Lur’e system. Thus the design approach II is based on H∞ -optimization. It is not unreasonable to believe that, in general, it achieves better transient performance than the first approach. Unfortunately, we have no rigorous analysis at the moment to provide comparisons between the two design approaches.
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We note that in dynamic output feedback control, often measurement noise is involved. In Case 1, measurement noise is amplified by the very large direct feedthrough matrix Dc and then enter the channels of the system states directly. Thus, even if the dynamic controller itself is not sensitive to measurement noise, the performance of the closed-loop system can be destroyed. In Case 2, we have no such issue thanks to the fact that Dc = 0. A simulation has also been performed considering an additive measurement noise, which is a Gaussian distributed random signal with mean zero and standard deviation 0.01. The simulation results are given below. Clearly, the dynamic controller obtained in Case 1 does not work at all in the presence of measurement noise, see Figs. 11 and 12. Both the state trajectories and the control input are expanding. In contrast, with the same measurement noise, the controller obtained in Case 2 still works and the effect of the measurement noise is not that prominent, see Fig. 13. Apart from smaller overshoot, the dynamic controller based on design approach II is not sensitive to measurement noise, see Fig. 14.
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5. Conclusions
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We have studied absolute stabilization of Lur’e systems under dynamic output feedback in the sense that the Lur’e-type nonlinearities are unknown. Only their sector bounds are assumed to be known. This setup is different from existing work on dynamic output feedback control for Lur’e systems, where the nonlinearities are always assumed to be known precisely and used to design the controllers. Two approaches are presented for the controller design. Our designed controllers can be computed using the Matlab LMI Control Toolbox. Their effectiveness has been validated through numerical simulation results on a Lur’e-type model of flexible joint robotic arms. For future research it is appealing to design a separate observer for Lur’e systems without using the Lur’e-type nonlinearities themselves. Such observer 21
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Fig. 12: The control input regarding φ(·) in Case 1 with measurement noise
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Fig. 13: The state trajectories regarding φ(·) in Case 2 with measurement noise
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is indeed stronger than the circle-criterion observer, and would be applied to, for example, output feedback control and fault diagnosis of Lur’e systems. The robustness will be also considered against incomplete feedback information and model uncertainties in the future. Acknowledgements
This work was supported by the State Key Laboratory of Intelligent Control and Decision of Complex Systems, the Young Scientist Fund of the National Natural Science Foundation of China (Grant No. 61703099), the China Postdoctoral Science Foundation Funded Project (Grant No. 2017M621589), and grants from the Research Grants Council of Hong Kong (Grant No. CityU-11261516).
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References References
[1] M. Arcak, P. Kokotovi´c, Nonlinear observers: a circle criterion design and robustness analysis, Automatica 37 (12) (2001) 1923–1930.
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[2] M. Arcak, P. Kokotovi´c, Observer-based control of systems with sloperestricted nonlinearities, IEEE Transactions on Automatic Control 46 (7) (2001) 1146–1150.
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[3] J. da Silva Jr., E. Castelan, J. Corso, D. Eckhard, Dynamic output feedback stabilization for systems with sector-bounded nonlinearities and saturating actuators, Journal of the Franklin Institute 350 (3) (2013) 464–484.
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[4] J. de Bruin, A. Doris, N. van de Wouw, W. Heemels, H. Nijmeijer, Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities, Automatica 45 (2) (2009) 405–415.
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[5] A. Zemouche, R. Rajamani, G. Phanomchoeng, B. Boulkroune, H. Rafaralahy, Circle criterion-based H∞ observer design for Lipschitz and monotonic nonlinear systems - Enhanced LMI conditions and constructive discussions, Automatica 85 (2017) 412–425.
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[6] K.-K. Kim, R. Braatz, Observer-based output feedback control of discretetime Lur’e systems with sector-bounded slope-restricted nonlinearities, International Journal of Robust and Nonlinear Control 24 (16) (2014) 2458– 2472. [7] K.-K. Kim, R. Braatz, Robust static and fixed-order dynamic output feedback control of discrete-time parametric uncertain Lur´e systems: Sequential SDP relaxation approaches, Optimal Control Applications and Methods 38 (1) (2017) 36–58.
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[8] T. Iwasaki, R. Skelton, All controllers for the general H∞ control problem: LMI existence conditions and state space formulas, Automatica 30 (8) (1994) 1307–1317.
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[9] R. Skelton, T. Iwasaki, K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, CRC Press, 1997.
[10] R. Rajamani, Y. Cho, Existence and design of observers for nonlinear systems: Relation to distance to unobservability, International Journal of Control 69 (5) (1998) 717–731.
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[11] C. Scherer, The Riccati Inequality and State-Space H∞ -Optimal Control, Ph.D. thesis, University of Wuerzburg (June 1991).
Appendix
Proof of Lemma 2. The existence of F and to the existence of solutions F and X > 0 to BF M X + (BF M X)T XC T 1 T +Q + EE X 2 −2(S2 − S1 )
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CX
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P > 0 satisfying (6) is equivalent
< 0.
(29)
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This can be seen by taking X = P −1 and using appropriate Schur complements. (only if ) Let X be a positive definite solution to (29). Define Y = X −1 . ⊥ ⊥ Then Y > 0 as well, and XY = I. We have [ B = B0 0I . Obviously, (8) 0] holds. (29) implies that Y BF M + (Y BF M )T + Q + 1 C T (S − S )2 C Y E 2 1 Y < 0, 2 ET Y
−2I
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T where QY , Y A − 21 E(S1 + S2 )C + A − 12 E(S1 + S2 )C Y , which then implies (9). (if ) By Finsler’s lemma [8], (8) implies that there exists a real number r such that −XC T rBB T − QX − 12 EE T > 0, −CX 2(S2 − S1 )−2 equivalently, Θx > 0. Similarly, (9) implies that there exists a positive definite matrix S > 0 such that T −1 M S M − QY − 21 C T (S2 − S1 )2 C −Y E > 0, −E T Y 2I 27
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equivalently, Θy , XM T S −1 M X − QX − 12 XC T (S2 − S1 )2 CX − 12 EE T > 0. Define
T T r2 B T Θ−1 M XΘ−1 x XM x XM
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M XΘ−1 x B,
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Ξ ,rI − r2 B T Θ−1 x B+
T where M XΘ−1 is positive definite since Θx > 0, X > 0 and M has full x XM row rank. Obviously, Ξ > 0 if and only if there exists a positive definite matrix Z > 0 such that
rI − r2 B T Θ−1 x B+
T T M XΘ−1 r2 B T Θ−1 x XM x XM + Z
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M XΘ−1 x B > 0,
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or equivalently, using the matrix inversion lemma [9], −1 rI − r2 B T Θx + XM T Z −1 M X B > 0, equivalently, using the Schur complement lemma, rI rB T > 0, rB Θx + XM T Z −1 M X which is equivalent to
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Θx + XM T Z −1 M X − rBB T 1 1 = − QX − EE T − XC T (S2 − S1 )2 CX + XM T Z −1 M X > 0. 2 2
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If we take Z = S, the above inequality is exactly Θy > 0. This thus shows that Ξ > 0. Now, clearly, h i T T −1 F + rB T Θ−1 M XΘ−1 M XΘ−1 x XM x XM x X h iT T T −1 M T F + rB T Θ−1 M XΘ−1 =0<Ξ x XM x XM
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T T for the particular choice F , −rB T Θ−1 M XΘ−1 x XM x XM inequality holds if and only if T rB T + F M X Θ−1 rB T + F M X < rI, x
which in turn is equivalent to T Θx rB T + F M X > 0, rB T + F M X rI
and to T 1 rB T + F M X rB T + F M X < Θx = r 28
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1 1 rBB T − QX − EE T − XC T (S2 − S1 )2 CX. 2 2
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is always positive semi-definite. This
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which yields (29) since completes the proof.
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It then follows that
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