Metzler Matrix Transform Determination using a Nonsmooth Optimization Technique with an Application to Interval Observers

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1 Downloae 4/7/18 to Reistribution subject to SIAM license or copyright; see Metzler Matrix Transform Determination using a Nonsmooth Optimization Technique with an Application to Interval Observers Abstract Emmanuel Chambon, Pierre Apkarian Laurent Burlion The paper eals with the esign of cooperative systems which formulates as computing a state coorinate transform such that the resulting ynamics are both stable an cooperative. The esign of cooperative systems is a key problem to etermine interval observers. Solutions are provie in the literature to transform any system into a cooperative system. A novel approach is propose which reformulates into a stabilization problem. A solution is foun using nonsmooth optimization techniques. 1 Introuction. Cooperative systems that is systems whose Jacobian matrix is Metzler (see Definition 2.1) have the interesting property to keep partial orering between two trajectories [15, 1]. Such a property makes them key caniates for use as interval observers where the goal is to enclose a given variable x(t) between two other timeepenent variables x(t) x(t) x(t) especially when information on any external isturbance is not available. Relate works inclue for example [9]. Many systems incluing the consiere generic launch vehicle moel are however not cooperative. A solution to this problem was propose in [13] where a time-varying change of coorinates is use to efine interval observers applie to linear time-invariant non-cooperative systems. It was inee shown in [12] that in some cases no timeinvariant change of coorinates can be foun to guarantee exponential stability of the obtaine interval observer. Despite its theoretical interest an the guarantees it offers, this approach may be har to implement in practice. More recent works like [14] an [6] interestingly propose techniques to fin a static state coorinate transform P an an observer gain L so that M = P(A LC)P 1 is Metzler even for a non-metzler matrix A. Methos base on the numerical resolution of a Sylvester equation were propose. In these methos This article contains figures in color for better unerstaning. Onera The French Aerospace Lab, BP7425, 2 avenue Éouar Belin, FR-3155 Toulouse Ceex 4, France. Corresponing author: Emmanuel.Chambon@onera.fr. a target matrix M must be provie which may reuce the set of acceptable solutions. A similar approach is use in [4] with the aim to overcome the mentione limitation of time-invariant change of coorinates. A specific interval observer structure making use of classical observers is evelope with an application to nonlinear systems affine in the unmeasure part of the state variables. It is however assume that the static matrix transform is known. In this paper an alternative technique to these methos is propose. A Metzler Matrix Transform synthesis metho is evelope where the computation of both P an L is performe simultaneously which results in M not being fixe a priori. As far as control theory is concerne, nonsmooth H synthesis optimization techniques as presente in [3, 2] offer the possibility to tune a structure controller against multiple control requirements. Recent avances [1] also make it possible to take multiple moels into account when robustness is at stake. Amissible requirements inclue close-loop poles location or noise rejection. In this article, this control approach will be use to specify estimation quality requirements on gain L which will be use within a Luenberger observer. It is also shown that the problem of fining a pair (P,L) such that M = P(A LC)P 1 is Hurwitz Metzler may be expresse as a control problem on which these techniques will be use. Computing a Metzler Matrix Transform is ifficult. After reformulation into a control problem so as to use nonsmooth H synthesis techniques, a numerical solution to this mathematical problem is propose. Definitions an notations are recalle in 2. The paper is structure as follows. In 3 (resp. 4) the consiere mathematical (resp. control) problem is expose. A solution to these problems is then propose in 5. The theory of interval observers an how they can be use to eterministically bracket an estimation error is presente in 6. Examples of applications are etaile in 7. In particular, the algorithm an interval observer approach are teste on a generic launch vehicle rigi moel in Copyright by SIAM.

2 2 Definitions an notations. The Laplace transform is enote s. Given two integers (i,j) symbol δ ij is efine by (4.3) min p { } max Twi z i (C(s,p)) 2/ i=1,...,n soft Downloae 4/7/18 to Reistribution subject to SIAM license or copyright; see (2.1) (i,j), δ ij = { 1 if i = j otherwise Unless mentione, i an j are integer variables satisfying 1 i,j n where n refers to a matrix imension. I stans for ientity matrix with aequate imension. For a given matrix A R n m, let enote A + = max(a,) an A = A + A where max is the element-wise operator. Definition 2.1. Let A = (a ij ) R n n. The matrix A is sai to be Metzler if (2.2) i j, a ij The matrix A is Hurwitz if all its eigenvalues real parts are strictly negative. Definition 2.2. For given matrices A an C, a Metzler Matrix Transform is a pair (P,L) with P R n n an L R n m s.t. P (A LC)P 1 is Metzler. 3 Problem statement. Let G = (A,B,C,D) a system with n states, m n measurements. It is suppose (A, C) is observable. The following problem is consiere for which an optimization-base solution is propose in 5. Problem 3.1. Fin a Metzler Matrix Transform(P, L) such that M = P (A LC)P 1 is Hurwitz Metzler. In this approach L is use to place poles so that a Hurwitz matrix is obtaine. It can be ientifie to an observer gain. In resulting new coorinates, G reformulates into G = ( M,PB,CP 1,D ). 4 Control problem (application specific). When system G or G expresse in new coorinates shoul be observe or stabilize, this becomes a control problem. Using nonsmooth H control formalism, this problem can be formulate as follows Problem 4.1. Let C(s,p) be a collection of LTI systems epening on tunable parameters p. Let n soft an n har two positive integers an w an z two vectors of R n soft+n har. For a given integer i n soft + n har, the transfer functions between input w i an output z i over the collection C is enote T wi z i (C(s,p)). The problem is to fin p satisfying subject to T wj z j (C(s,p)) 2/ 1 with j = 1,...,n har. To solve this control problem optimization-base techniques can be use like nonsmooth H synthesis as presente in [1]. These techniques were implemente into numerical solvers as presente in [8]. More specifically the systune routine from the Matlab c Robust Control Toolbox 212b an higher can be use. An optimal solution is foun when the algorithm manages to rive har constraints below unity while minimizing soft constraints (also calle objectives). In 5 it will be shown that problem 3.1 reformulates into an optimization problem which can be solve using these optimization-base techniques. 5 Propose solution. Before going into the etails of the metho, the reaer shoul be remine that the propose solution is: a numerical solution: it is obtaine using an optimization algorithm; A local optimal solution: because the problem is non-convex in (P, L) algorithm converges towars an optimal local solution. Multiple ranom restarts of the optimization algorithmcanbeusetofinanoptimalsolutiononalarger set of solutions. In case some targete properties of the solution are not satisfie, it may be necessary to weaken some of the formulate constraints. It may help the optimization algorithm to fin a better local solution. 5.1 Note on the existence of a trivial solution. Since iagonal matrices comply with efinition 2.1 there exists a trivial solution to problem 3.1. Proof. With (A, C) observable, choose L so that A LC eigenvalues are negative real numbers an choose P as the matrix of associate eigenvectors. Then M = P (A LC)P 1 is iagonal Hurwitz hence is a solution to problem 3.1. Note this trivial solution may not satisfy control requirements state in problem 4.1. Also this may lea to large gains L when using pole placement methos. 5.2 Note on Sylvester equation approach. To fin Metzler Matrix Transform (P,L) such that M = 26 Copyright by SIAM.

3 Downloae 4/7/18 to Reistribution subject to SIAM license or copyright; see P(A LC)P 1 is Metzler, one can solve the following Sylvester equation as suggeste in [14] (5.4) PA MP = QC where A is known, L has been obtaine solving a pole placement problem an (M,Q = PL) are arbitrarily chosen such that M an A LC have the same eigenvalues an M is Hurwitz Metzler. Note however that this equation has a unique solution if an only if A an M have istinct eigenvalues. Because the approach presente in the sequel oes not rely on an a priori choice of M or Q it offers more freeom in the resolution. On the other han, the contingency of a local vs. global optimization technique has to be accepte. 5.3 Metzler conitions stabilization problem. Let M = P (A LC)P 1 R n n where P an L are ecision variables. For M to be Metzler, the n(n 1) following inequalities must be satisfie (5.5) i j, M ij = [ P (A LC)P 1] ij = i P (A LC)P 1 j where i = (δ ik ) 1 k n is a column vector. For a given pair (i,j), with i j, the corresponing inequality can be seen as an anti-stabilization problem in the ecision variables P an L. 5.4 Control synthesis approach an moels. To fin a pair (P,L) satisfying both M = P(A LC)P 1 an control constraints expresse on G (eventually consiere in-line with other systems), any nonsmooth nonconvex optimization technique can be use. It is propose to use the approach presente in [1] which can hanle multiple moels to synthesize control laws accoring to specifie constraints. As far as the conitions in 5.3 are consiere, the constraints can be reformulate into the propose control framework. The following fictitious 1 systems are consiere (5.6) i j, G ij M = [ Mij (P,L) R 1 1 n 1 From a control point of view, ensuring i j, M ij is equivalent to stabilizing these systems G ij M. The following collections are consiere in the control synthesis 1 In the sense they have no input nor output. ] (C1){ n(n} 1) uniimensional fictitious systems G ij M on which a requirement on close-loop i,j poles location is expresse (minimum ecay an max frequency); (C2) (optional) { } n(n 1) uniimensional fictitious systems G ij M withstatematrixm ij (P,L) Mij max. i,j This helps to restrict the set of acceptable solutions when < Mij max < +. As such, upon multiple restarts, the optimization algorithm converges more easily ue to limitation of the initialising variables excursion aroun a smaller set of potential local optima; (C3) Original plant moel G = (A,B,C,D) in-line with aitional systems eventually epening on L an aitional variables to enforce system stabilization or estimation criteria; (C4) (optional) any other moel to enforce specific properties. 6 Application to Interval Observers. The algorithm was evelope towars a specific application to interval observers which is presente here. The theory presente in this section is inspire from the works in [13] an [14]. Since the propose algorithm also synthesizes an observer gain L, it is propose to bracket the estimation error using interval observers so as to frame the plant state x in return. 6.1 Plant moel. The following plant moel expresse in the state space is consiere. Using Popov- Belevitch-Hautus Lemma, observability of(a, C) can be checke to account for possibly unobservable eigenvalues. ẋ = Ax+B ( +B ) u u (6.7) (G) = Ax+B u y = Cx where x R n an y R m with m n. Initial conitions are given by a vector x. Input is an unknown isturbance but it is suppose there exists known bouns an such that t, (t) (t) (t). It is suppose a ynamic controller K(s) has been esigne ensuring system stability in these conitions. Stabilizing output-feeback control law u = K(s)y is then applie to the system. 6.2 Luenberger observer. The following Luenberger observer is consiere 27 Copyright by SIAM.

4 Downloae 4/7/18 to Reistribution subject to SIAM license or copyright; see x = A x+b u u+l(y C x) (6.8) (G obs ) ŷ = C x x() = x The objective is to ensure minimal estimation error. 6.3 Estimation error system Let e = x x the estimation error, it is inferre (6.9) ė = (A LC)e+B with e() = e = x x. The objective is to fin (e,e,e,e ) s.t. for e [e,e ] then t, e(t) e(t) e(t). The pair (e,e ) is suppose known from now on. 6.4 Interval observer. To use the results presente in [14], a Metzler Matrix Transform nees to be foun an the error estimation system expresse into the corresponing new coorinates. It is suppose synthesis in 5 has returne (P,L) s.t. M = P (A LC)P 1 is Hurwitz Metzler an estimation quality requirements on system (6.9) are satisfie. Using the change of coorinates e z = Pe, system (6.9) becomes (6.1) ė z = P (A LC)P 1 e z +PB = Me z +B where B = PB. Using [5, Lemma 1] an notations in 2, an interval observer for system (6.1) is given by (6.11) uner initial conitions (6.12) { ėz = Me z +B + B ė z = Me z +B + B e z () = P + e P e e z () = P + e P e This interval observer is compose of two autonomous systems (no epenency on system output y). Let T = P 1, the bouns on e are obtaine using [5, Lemma 1]: (6.13) { e = T + e z T e z e = T + e z T e z Then for an initial estimation error e [e,e ], a time-varying bracketing of the state x is obtaine as x+e x x+e. The system compose of autonomous systems (6.11) with output y e = (e,e) is calle (G e ). 7 Examples. The following examples are use to emonstrate the possibilities of the algorithm etaile here with an application to interval observers. Note that syntheses were performe using systune function from the Robust Control Toolbox 214b [11] correctly parametrize to benefit from the Parallel Computing Toolbox. Other implementations may be use. 7.1 Thir-orer system with unobservable moe. This example is inspire from the partial linear system example presente in [14]. It is given by 2 (7.14) A = , C = [ 1 ] 3 4 The matrix B is not recalle here since no isturbance input is consiere. The problem reuces to fining a suitable Metzler Matrix Transform with limite control constraints. Note however that the complex moe 4±j 3 is not observable. To solve this problem, collections (C1) an (C2) with n = 3 an (i,j), Mij max = 1 are use. Moreover using (C4), A LC eigenvalues real part are force to stay within [ 1, 3.1 3]. A solution is typically foun in less than 5 iterations an after 3 restarts. The following results have been compute (7.15) M = , P = , L = It is reaily verifie that M = P(A LC)P 1 is Hurwitz Metzler. Accoring to results in 6 one can note that the estimation error epens on the initial error but converges towars zero since no other isturbance is consiere. 7.2 Sixth-orer system with two complex moes. This example is inspire from the one presente in [13]. It is given by (7.16) A = , B = 5 4, C = [ 1 ], D = [ 1 ] 28 Copyright by SIAM.

5 Downloae 4/7/18 to Reistribution subject to SIAM license or copyright; see The first input correspons to a isturbance input on the measurement. Its effect on the estimation quality nees to be mitigate using an appropriate isturbance rejection requirement. Note that since y = Cx+, the estimation error system is efine by (7.17) ė = (A LC)e L rather than by (6.9). To solve this problem, a collection of synthesis moels compose of (C1) to (C4) with n = 6 an (i,j), Mij max = is use. Estimation quality is ensure through minimization of the H 2 - norm of the transfer from to e using moel (C3). Stability of the estimation error subsystem is ensure through (C4) where A LC eigenvalues real parts are forcetostayintheinterval [ 1, 3.1 3]. Asolution is typically foun in 897 iterations an after 4 restarts, see (7.18) on Fig. 1. A similar construction to the one presente in 6 can then be use to frame state x. 7.3 Fifth-orer rigi launcher longituinal moel. The approach in 6 is applie to a generic rigi launch vehicle longituinal moel with unknown win input. No uncertainty is consiere in this article. Like other examples, the metho propose in 5 is use to compute P an L. Simulations are then run using a simplifie win profile Moel. The moel (7.19) in Fig. 2 is consiere where the first input correspons to the unknown win input an D =. The secon input correspons to the thruster orientation input signal to which the stabilizing control law is applie. Note that n = Synthesis an results. To solve the problem, the collections of moels (C1) to (C4) are use where (i,j), Mij max = 5 an A LC eigenvalues real parts are force to lie in [ 1.1 2, 1.1 2]. Using moel (C3) a minimization constraint on the weighte H 2 norm from unknown input R to estimation error e R 5 is expresse. The optimization algorithm generally gives a solution in aroun 11 iterations an 6 restarts. Results are shown in (7.2) in Fig Simulation. Since the system is unstable, the simulation is performe after structure H synthesis of a ynamic stabilizing controller K(s) for this system. As this is not the main purpose of this work, it is not etaile here. Fig. 3 illustrates how the ifferent systems G, G obs an G e are relate. Note that system G e oes not interact with the stabilize plant moel. Using win profile shown in Fig. 4 with +/ 7 m/s uncertainty on win spee, the results shown in Fig. 5 u G e G K e e z y G obs Figure3: Simulationmoelwhere(G e )isacombination of two autonomous systems. an 6 are obtaine. As expecte it is interesting to note that espite state estimate is use to express bounaries on the state, it oes not necessarily lie in the resulting interval [ x+e, x+e] since e (resp. e) can be positive (resp. negative). 7.4 Execution summary. Executions were performe running Matlab c R214b with Parallel Computing Toolbox version 6.5 on a Winows c 7 64 bits station with 8 Go RAM an Intel c Xeon processor E5-167 v2 3 GHz (qua-core). A summary of these executions for each example is shown in Table 1. Note the number of iterations an time results are only inicative since they may vary from one run to another but they are representative of the complexity of the problem. 8 Conclusions. In this article an application of nonsmooth optimization techniques [1] to numerically solve a mathematical problem combine to a control problem has been pre- Win spee (m/s) Simulation time (s) Figure 4: Win spee profile (in blue) an known bouns an use in simulation. x 29 Copyright by SIAM.

6 Downloae 4/7/18 to Reistribution subject to SIAM license or copyright; see (7.18) M = , P = , L = Figure 1: A solution obtaine on example (7.16) (7.19) A = , B = , C = (7.2) M = , P = , L = Figure 2: Rigi launch vehicle longituinal moel with etectors ynamics (top, see 7.3) an obtaine Metzler Matrix Transform (bottom). Ex. Res. Dim. (n) # Moels Restarts Iter. Ref. (7.14) (7.15) [14] (7.16) (7.18) [13] (7.19) (7.2) Table 1: Metzler Matrix Transform synthesis typical execution summary. sente. To etermine a Metzler Matrix Transform the corresponing mathematical problem 3.1 was reformulate into a control problem 4.1. Combine with control requirements, a local optimal solution to this problem can be foun. This approach has been successfully applie to various examples incluing a rigi launch vehicle moel. In comparison with other methos, this approach allows for simultaneous tuning of P an L an oes not make assumption on the targete M matrix structure. Moreover, by benefiting from observer structures as propose in [14] an [4] implementing the resulting interval observer is straightforwar in practise. Further improvements will be eicate to reuce complexity since non-negligible computing resources are neee with multiple algorithm restarts before converging towars an acceptable solution especially in the case of higher-imensional problems. Other problems coul also be consiere like fining static P such that P [A(y) L(y)C]P 1 is Metzler where y R m is the known system output signal. The interest woul be to benefit from the interval observer structure propose in [4]. In this work knowing such P is consiere an hy- 21 Copyright by SIAM.

7 Downloae 4/7/18 to Reistribution subject to SIAM license or copyright; see excapt eψ evz.1 eẋcapt e ψ Figure 5: Simulation results on example (7.19): estimation error e = x x is represente in blue. The bouns obtaine using interval observer expression as in (6.11) are represente in re an green. xcapt ψ Time(s) 33.6 vz Time(s) ẋcapt Time(s) 1 3 Time(s).5 5 ψ Time(s) Figure 6: Simulation results on example (7.19) using win profile in Fig. 4: the state (blue), the state estimate (magenta) an the upper (resp. lower) boun (re, resp. green) are represente. Zoom is performe on otherwise inistinguishable curves. pothesis, which might be limiting. Also note that other systems like positive continuous-time linear systems [7] also have a Metzler state-space matrix which gives hints on a wier scope for this article. References [1] P. Apkarian, Tuning controllers against multiple esign requirements, in Proc. of the American Control Conference, Washington, USA, June 213, pp [2] P. Apkarian an D. Noll, Nonsmooth H synthesis, IEEE Transactions on Automatic Control, 51 (26), pp [3] J. V. Burke, D. Henrion, A. S. Lewis, an M. L. Overton, HIFOO a MATLAB package for fixeorer controller esign an H optimization, in Proc. of the 5th IFAC Symposium on Robust Control Design, Toulouse, France, Aug. 26. [4] T. N. Dinh, F. Mazenc, an S.-I. Niculescu, Interval observer compose of observers for nonlinear systems, in Proc. of the European Control Conference, Strasbourg, France, June 214, pp [5] D. Efimov, T. Raïssi, S. Chebotarev, an A. Zolghari, Interval state observer for nonlinear timevarying systems, Automatica, 49 (213), pp [6] D. Efimov, T. Raïssi, an A. Zolghari, Control of nonlinear an LPV systems: Interval observer-base framework, IEEE Transactions on Automatic Control, 58 (213), pp [7] L. Farina an S. Rinali, Positive Linear Systems: Theory an Applications, John Wiley & Sons, 2. [8] P. Gahinet an P. Apkarian, Frequency-omain tuning of fixe-structure control systems, in Proc. of the UKACC International Conference on Control, Sept. 212, pp [9] J. L. Gouzé, A. Rapaport, an M. Z. Haj- Saok, Interval observers for uncertain biological systems, Ecological Moelling, 133 (2), pp [1] L. Mailleret, Stabilisation globale es systèmes positifs mal connus - Applications en Biologie, PhD thesis, Université e Nice Sophia-Antipolis, Nice, May 24. [11] MATLAB, Robust Control Toolbox version 5.2 (R214b), The MathWorks Inc., Natick, Massachusetts, 214. [12] F. Mazenc an O. Bernar, Asymptotically stable interval observers for planar systems with complex poles, IEEE Transactions on Automatic Control, 55 (21), pp [13], Interval observers for linear time-invariant systems with isturbances, Automatica, 47 (211), pp [14] T. Raïssi, D. Efimov, an A. Zolghari, Interval state estimation for a class of nonlinear systems, IEEE Transactions on Automatic Control, 57 (212), pp [15] H. L. Smith, Monotone ynamical systems: an introuction to the theory of competitive an cooperative systems, Bulletin of the American Mathematical Society, 33 (1996), pp Copyright by SIAM.