athF ollo wing or NonMinimum Phase Systems Remo es erf ormance Limitations A
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athF ollo wing or NonMinimum Phase Systems Remo es erf ormance Limitations A

Pedro Aguiar Jo ao Hespanha and Petar ok oto vi Abstract highlight an essential differ ence between pathf ollo wing and efer encetracking or nonminimum phase systems It is wellkno wn that in the efer encetracking or nonminimum phase systems ther exi

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athF ollo wing or NonMinimum Phase Systems Remo es erf ormance Limitations A

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ath-F ollo wing or Non-Minimum Phase Systems Remo es erf ormance Limitations A. Pedro Aguiar Jo ao Hespanha, and Petar ok oto vi Abstract highlight an essential differ ence between path-f ollo wing and efer ence-tracking or non-minimum phase systems. It is well-kno wn that in the efer ence-tracking, or non-minimum phase systems, ther exists fundamental perf ormance limitation in terms of lo wer bound on the -norm of the tracking err or en when the contr ol eff ort is fr ee. sho that this is not the case or the less stringent path-f ollo wing pr oblem, wher the contr ol objecti

is to or ce the output to ollo geometric path without timing law assigned to it. Furthermor e, the same is true en when an additional desir ed speed assignment is imposed. Index erms Non-minimum phase systems, path-f ollo wing, cheap- contr ol, efer ence-tracking Obstacles to achie ving perfect tracking with feedback systems ha been well understood and quantied with classical Bode inte grals and as the limits of cheap optimal control performance [1][5]. In the absence of unstable zero dynamics non-minimum phase zer os perfect tracking of an reference signal is possible, that is, the

-norm of the tracking error can be made arbitrarily small. ith unstable zero dynamics this is no longer possible, because an amount of output ener gy must be used for stabilization. or linear systems the tracking error increases as the signal frequencies approach those of the unstable zeros [4], [6]. The limitations introduced by unstable zero dynamics are struc- tural. The cannot be oided without changing the system structure or re-formulating the tracking problem. One such reformulation is to select ne output for which the zero dynamics are stable and perfect tracking is possible [7], [8].

As discussed in [8], the ne output should be physically meaningful and allo good approximation to the original tracking task. Another reformulation of the tracking problem in estigated in this note is to di vide it into tw tasks: geometric path follo wing and speed assignment along the path. As sho wn in [9], this tw o-task formulation is suitable for man applications. It also of fers the e xibility to use the timing la as an additional control ariable and thus change the input/output structure of the system. In particular this e xibility can be emplo yed for rescaling of the

xosystem eigen alues in the internal model approach. The main contrib ution of this note is to present conditions under which for linear non-minimum phase systems the tw o-task path follo wing problem can be solv ed with arbitrarily small -norm of the path follo wing error In Section II we formulate the path-follo wing problems. After brief re vie of reference-tracking in Section III, Section IV presents the main results of the note sho wing that the well-kno wn performance limitations imposed on reference-tracking by non-minimum phase zeros do not appear in the less stringent path- follo wing

problem. An xample in Section illustrates the results. Concluding remarks are gi en in Section VI. The ork of A. Pedro Aguiar as supported in part by Pos-Doc Fello w- ship PRAXIS XXI from the Portuguese oundation of Science and echnol- ogy The ork of J. Hespanha and ok oto vi were supported by the National Science oundation under Grants ECS-0242798 and ECS-0228846, respecti ely The authors are with the Center for Control Engineering and Computation, Uni ersity of California, Santa Barbara, CA 93106-9560 USA (e-mail: aguiar@ece.ucsb .edu; hespanha@ece.ucsb .edu; petar@ece.ucsb .edu). Notation:

ranspose is denoted by comple conjugate by and comple conjugate transpose by The comple plane is denoted by where := Re := Re and := Re( Re is the real part and Im is the imaginary part. The Laplace transform of is is the space of square inte grable ector functions er with norm dt ath-follo wing problems are primarily concerned with the design of control la ws that dri an object (robot arm, mobile robot, ship, aircraft, etc.) to reach and follo geometric path secondary goal is to force the object mo ving along the path to satisfy some additional dynamic specication. common approach to

the path- follo wing problem is to parameterize the geometric path by path variable and then select timing law for [9][13]. Extending the approach of [10], path-follo wing controller as proposed in [9] for class of uncertain strict feedback nonlinear systems. frame ork for path-follo wing as method to oid some limitations in reference-tracking as described in [13]. The idea is to use as an additional control input to stabilize the unstable zero-dynamics while the original control ariables eep the system on the path. In this note, we consider geometric path-follo wing for non- minimum phase

systems and paths formed by linear combination of sinusoids in When the control ef fort is free, we sho that ther is no limitation upon the ac hie vable path err or performance The same result holds ven when we impose an additional speed assignment for the path variable This essential dif ference between reference-tracking and path- follo wing sho ws that is not appropriate to re-formulate path- follo wing problem as reference-tracking by making In man applications this ould introduce limits on the achie able performance that are not inherent to the original problem. no proceed with the

problem formulation. or the linear time- in ariant system Ax (1a) (1b) where is the state, and the control, the output is required to reach and follo geometric path := =1 (2) where is the scalar path parameter are real numbers, and are non-zero comple ectors. The geometric path can be generated by an xosystem of the form d (3a) Qw (3b) where is the xogenous state and or an timing la the path-following err or is dened as := )) (4) The tw path-follo wing problems to be solv ed are as follo ws.
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−1.5 −1 −0.5 0.5 1.5 −1.5 −1 −0.5 0.5 1.5

[m] [m] (a) 1.5 1 0.5 0.5 1.5 1.5 1 0.5 0.5 1.5 [m] [m] (b) Fig. 1. rajectory of the ehicle: 1(a) reference-tracking, 1(b) path-follo wing. Geometric path-f ollo wing: or desired path design controller that achie es: i) boundedness: the state is uniformly bounded for all and ery initial condition ), := ii) err or con ver ence: the path-follo wing error con er ges to zero as and iii) forwar motion: for all where is positi constant. Speed-assigned path-f ollo wing: In addition to geometric path- follo wing, gi en desired speed it is required that either as or for and some As illustrated by

Skjetne et al. [9], these path-follo wing problems pro vide natural settings for man engineering applications, including situations when the timing la is determined by human oper ator From theoretical standpoint our main interest is to determine whether the freedom to select timing la can be used to achie an arbitrarily small -norm of the path-follo wing error that is, whether in := dt (5) can be made arbitrarily small. Before we address this question in Section IV, let us briey recall that for the standard reference-tracking the answer is, in general, ne gati e. The standard linear

reference-tracking problem is to design feedback controller for (1) such that the closed-loop state is bounded, and for an reference signal the output asymptotically approaches or generated by kno wn xosystem, Qw (6) the so-called gulator or servomec hanism pr oblem as sho wn in [1], [14][16], to be solv able if and only if A; is stabilizable, is detectable, the number of inputs is at least as lar ge as the number of outputs and the zeros of A; do not coincide with the eigen alues of In that case, the internal model approach of Francis [15], [17] uses matrices and that satisfy Q; to design

the reference-tracking controller ( ) where is such that is Hurwitz. ith this controller the transients := and := con er ge to zero and are go erned by An important issue in reference-tracking problems is whether the -norm of the tracking error can be made arbitrarily small, that is, dt can be satised for an or this to be the case, the zeros of A; must be in the open left half-plane The non-minimum phase zeros, that is the zeros in impose fundamental limitation on the attainable tracking performance This is re ealed by the act that the limit as of the optimal alue of the cost

functional min dt (7) is strictly positi e, [1]. Qiu and Da vison [4] sho wed that for sin cos the best attainable performance is col trace =1 where and are the non-minimum zeros of A; ith the same formula holds for const
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or step reference signals and multi ariable linear systems, Chen, Qiu, and ok er [18] sho that the ef fect of nonminimum phase zeros is determined not only by the zero locations, ut also the mutual orientation between zero and input signal directions. or more general reference signals, Su, Qiu, and Chen [6] gi xplicit formulas which sho the dependence of on

the non-minimum phase zeros and their frequenc y-dependent directional information. Seron et al. [5] re-interpreted the Qiu-Da vison formula [4] and generalized it to class of nonlinear systems. The sho wed that the best attainable alue of is equal to the lo west control ef fort, measured by the -norm, needed to stabilize the zero dynamics dri en by the system output It is its role as stabilizing control input that pre ents the output from perfect tracking. Extensions to non-right-in ertible systems are gi en in [19], [20]. Control ener gy constraints as another source of fundamental

performance limitations ha been in estigated in [21]. no sho that the attainable performance for the path- follo wing problems is not limited by non-minimum phase zeros. As in [4], [6], [19], [20], we assume that initially the system is at rest, that is, we let Theor em 1: If A; is stabilizable, then for the geometric path- follo wing problem and an gi en positi constant there xist constant matrices and and timing la such that the feedback la Lw )) (8) achie es (5) Pr oof: Using (8) and the timing la (9) where is constant to be selected, we can vie the path- follo wing problem as the

reference-tracking of the signal )) =1 (10a) =1 (10b) where := From the xosystem (3) and (9) we obtain the set of equations Ax (11a) (11b) Qw (11c) that describe re gulator problem to which the results of [15], [22], [23] are applicable. The constant should be selected such that the zeros of (1) do not coincide with the eigen alues of pro that for an we can select and such that (5) holds, we apply the results in [4], [6] to (10b) (11) and obtain trace trace =1 =1 (12) where are the zeros of A; in and := col (Re Im Re Im Re Im (13) This is sho wn in Appendix and will also be useful in the

proof of Theorem 2. The result follo ws from (12) because for an there is suf ciently lar ge such that (5) holds. Remark 1: stress that the stabilizability of A; is the only condition (necessary and suf cient) for the solv ability of the geometric path-follo wing problem using (8) Ne xt we sho that an arbitrarily small -norm of the path- follo wing error is attainable en when the speed assignment is specied beforehand. Theor em 2: Let be specied so that the eigen alues of do not coincide with the zeros of (1), and assume that A; is stabilizable. Then, for the

speed-assigned path-follo wing problem, (5) can be satised for an with timing la and controller of the form (8) ut with time-v arying piece wise-constant matrices and Pr oof: construct path-follo wing controller that satises (5) we start with the follo wing structure (14a) (14b) where for each positi constant the matrices and satisfy (15a) Q; (15b) and := is the piece wise constant switching signal Each is chosen such that the matrix is Hurwitz. The xistence of matrices and presumes that will be chosen so that the eigen alues of do not coincide with the zeros of (1). observ

that (14) is speed-assignment path-follo wing controller for which con er ges to in nite time. no pro that an gi en performance specication (5) can be satised by appropriate selection of the nite sequence together with the parameters used in the feedback con- troller (14) start by selecting suf ciently lar ge such that dening := =1 =1 (16) where are the zeros of A; in and is dened by (13), is upper -bounded by sho in Appendix that is upper -bounded by =1 =1 in in in =1 in (17) where is positi constant, in := with the constant matrices of

appropriate dimensions, and the transient state errors in con er ge to zero as ast as
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sho that each term of (17) can be upper -bounded by so that The rst term is smaller than by construction. pro that the second term in (17) is also smaller than we select the parameters to satisfy ; (18) where := (4 )) and := Then, it follo ws that the second term in (17) can be upper -bounded by =1 The abo selection for the is made under the constraint that the eigen alues of do not coincide with the zeros of (1). This can al ays be satised by appropriately adjusting

Finally for gi en nite each of the last tw terms in (17) can be made smaller than by choosing suf ciently lar ge. ehicle with mass mo ving in the plane, on top of which lies mass is modeled by u; (19) where diag diag is the force, and are the positions of the ehicle and the mass respecti ely The mass is carried by the ehicle via the viscous friction force while the gra vity force dri es the mass ay from the position because the top of the ehicle is not at. The system is non-minimum phase with tw zeros in located at mg (2 The path-follo wing task is to mo the ehicle

along circular path with the desired steady-state elocity The path is centered at the origin with radius The ehicle starts with zero initial conditions and the model parameters are diag (15 10) diag (1 1) or the sak of comparison, we rst recast this problem as reference-tracking problem by creating the reference signal col cos )) sin( )) Fig. 1(a) displays the simulation results obtained with the control gain computed by solving the optimal cheap control problem (7) with 001 As xpected, the con er gence to the desired trajectory is achie ed with signicant transient error: 23 In

contrast, Fig. 1(b) sho ws the simulation results obtained with the path-follo wing controller described in the proof of Theorem 2. Starting with 10 the alues of were selected to decrease by in 10 seconds to 18 at time 180 The alues of the control gains are all equal to the alue of used in the rst xperiment. As it can be seen, the con er gence of the ehicle to the path is much smoother and the transient error is reduced to 06 ha re visited the classical issue of performance limitations in reference-tracking for linear non-minimum phase systems. ha demonstrated that the performance

limitations can be oided by reformulating the problem as path-follo wing, where the path ariable is treated as an additional control ariable. This conceptual result may be of practical signicance, because the path-follo wing formulation is con enient for man applications. Design of path- follo wing controllers for non-minimum phase systems and the study of performance under control ener gy constraints are topics for future research. deri (12) and inequality (17) we will need the follo wing results: transfer matrix in is said to be inner if in is stable with all the zeros located in and

in in transfer matrix out is said to be outer if it has full ro rank in the open right half comple plane. Ev ery transfer matrix can be actored as in out such that in is inner and out is outer [4, Lemma 2]. minimal realization A; of stable transfer matrix is called balanced ealization if the solutions to the yapuno equations AP QA are diagonal and equal. A. Derivation of (12) This deri ation closely follo ws [4]. Since the transients satisfy u; (20a) x; (20b) x: (20c) need to compute the limit lim of the cheap optimal control problem min dt; for (1) with zero initial conditions. The optimal

control la is where and is the unique, positi semidenite solution of the algebraic Riccati equation It is well kno wn [1] that lim xists, and is independent of which stabilizable and detectable state-space realizations the plant is used. As in [4] and without loss of generality we let A; be consistent with the inner -outer actorization in out where is the transfer matrix of system (1), in is square stable inner transfer matrix and out is minimum phase right- in ertible transfer matrix. Let in in in in be balanced realization of in and out out out out be an stabiliz- able and detectable

realization of out The partitioning of as col in out gi es in in out out in out out in in out in out
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and from Theorem in [4] it follo ws that lim and hence lim in in in in since in in The Laplace transform of in and are in sI in in out in out Therefore, in sI in in in (21) Since the substitution of from (10a) gi es in =1 (22a) =1 Re Re Im Im (22b) The computation of in in results in where is sho wn at the bottom of the page, and is gi en in (13). Finally the formula (12) is obtained by observing that trace =1 trace is inner matrix, and [4]. B. Derivation of (17) rst

compute := dt; with for all and note that =0 (23) As in the deri ation of formula (12) we get in in (24) where := is partitioned as col in out and that is, is the steady-state of when for all or is gi en by (12) replacing by and satises where is dened in (16) or we substitute in (24) and get in in in in in in in in in in in in in in in ith substituted in (2) and using (22a) we obtain in in =1 =1 Re Re Re Im Im Im where := and := Since has no poles on the imaginary axis, Re and Im are continuously dif ferentiable functions of and by the mean alue theorem, it follo ws that the

equation sho wn at the top of the ne xt page holds, where `;k `;k are constant matrices and the ector is dened in (13). Therefore, in in (25) where := ; ; ;n ;n ; ; ;n ;n Then (17) follo ws from (23) (25) and the act that there xists suf ciently lar ge such that trace for an := Re Im Re Im Re Im Re Im
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