Dynamical systems with active singularities: Input/state/output ...

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Automatica 44 (2008) 1741–1752 www.elsevier.com/locate/automatica

Dynamical systems with active singularities: Input/state/output modeling and controlI Joseph Bentsman a,∗ , Boris M. Miller b,c , Evgeny Ya. Rubinovich d a Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA b Institute of Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia c School of Mathematical Sciences, Monash University, Clayton, 3800, Victoria, Australia d Institute of Control Sciences, 65 Profsoyuznaya Street, Moscow, 117997, Russia

Received 15 March 2006; received in revised form 5 November 2007; accepted 8 November 2007 Available online 25 March 2008

Abstract Controller synthesis setting for systems with active, or controlled, singularities is extended to the case of incomplete information. The general physically-based description of mechanical systems with controlled singularities and sensing in the singular phase that also admits control over observations is developed. An example is presented that demonstrates the computation of a singular phase sensor-based optimal control law. It is found that optimal control of this class of systems should, in general, be sought in the class of complex signals termed temporal multi-impacts — very short duration isolated sets of temporally sequenced impulses. The system subject to these impacts is shown to exhibit a novel mode of controlled dynamic behavior — the interlaced singular phase. c 2008 Elsevier Ltd. All rights reserved.

Keywords: Mechanical systems; Controlled singularities; Control over observations; Temporal multi-impact; Constraints; Optimal control

1. Introduction Rapid progress in fast sensing and actuation compatible with the impact-type transitions, or singular phases, of a number of technologically significant systems has a potential of providing a qualitative jump in their performance. These transitions can be either naturally present or created through actuation (Brogliato & Zavala Rio, 2000; Tornambe, 1999). The resulting systems dynamics, such as that of power microgrids (Mazumder, Tahir, & Kamisetty, 2005), mobile sensor networks (Hristu-Varsakelis & Levine, 2005), and robotic manipulators (Grizzle, Abba, & Plestan, 2001), becomes, however, rather nontrivial (Ronsse, Levefre, & Sepulchre, 2007), and impact games, such as pingpong, can help in providing a conceptual clarity to the behavior I Revised and extended version of a paper presented at the 16th World Congress of IFAC, Prague, Czech Republic, July 3–8, 2005. This paper was recommended for publication in revised form by Associate Editor Bernard Brogliato under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +1 217 244 1076: fax: +1 217 244 6534. E-mail addresses: [email protected] (J. Bentsman), [email protected] (B.M. Miller), [email protected] (E.Ya. Rubinovich).

c 2008 Elsevier Ltd. All rights reserved. 0005-1098/$ - see front matter doi:10.1016/j.automatica.2007.11.026

of these systems. In these games, the player can be viewed as generating a constraint (Pagilla & Tomizuka, 1995; Spong, 2001) and controlling its properties during the very short duration engagement phase with the ball. This player action gives rise to a concept of active, or controlled, constraints capable of radically changing the attainability set of the postimpact system state. The engagement phase of the system with such constraint can then be termed an active singularity, and the system motion in the domain of constraint violation - the singular motion phase. Based on these concepts, a new class of systems, dynamical systems with active, or controlled, singularities, and the corresponding rigorous technique for modeling and controller synthesis for this class of systems, has been recently introduced and developed in Bentsman and Miller (2007). Extending the framework of Bentsman and Miller (2007), the present work gives rise to a class of systems that admit an incomplete observation in the singular motion phase. The optimal control of the latter system class is carried out on an example (cf. Section 2) and shown to bring out the novel dynamic mode of system interaction with the constraints, termed temporal multi-impact, characterized

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by very short duration isolated sets of temporally sequenced control signals, referred to as temporal multi-impulses. Through an application example of the latter new class of control signals, this mode is shown to vastly increase the attainability set of the post-multi-impact system states in comparison to the singleimpact mode considered in Bentsman and Miller (2007). To adequately represent temporal multi-impact, a new mode of system behavior – the interlaced singular phase – exhibited by the system under a temporal multi-impulse, is introduced, and the framework of Bentsman and Miller (2007) is generalized (cf. Sections 3 and 4) to encompass this mode, providing a modeling and controller synthesis setting sufficiently rich for accommodating the output feedback optimal controller synthesis for systems with active singularities.

Fig. 1. The constraint-free motion phase.

2. Optimal multi-impulse control based on the input/ state/output model: A motivating example In Bentsman and Miller (2007), an example of modeling and control of an elastic ball/racket system was considered, with the racket rotation in the ball/racket engagement, or singular, phase inducing a ball bounce-off angle increment with respect to its incidence angle. The singular phase dynamics was characterized by a smooth controller action within a single impact structure. The example detailed the general framework introduced therein; however, it was limited to the full state access assumption and lacked optimization in controller synthesis. These gaps are bridged in the example presented next: namely, the physically-based and limit modeling, and controller synthesis setting are augmented by output sensing characterized by its own dynamics, and the control objective of stopping the ball is addressed through optimal control law calculation. These two features are found to (i) make attainability set of a single impact insufficient to encompass the zero terminal ball velocity point, (ii) give rise to a new mode of system interaction with the constraint in the singular phase — temporal multi-impact, described in Section 1, and (iii) introduce the corresponding sequence of subphases, termed the interlaced singular phase, that partitions a singular phase into subintervals, each associated with one of the two alternating forces – one force acting during the ball/racket engagement and the other – outside of the latter to quickly bring the racket back into contact with the ball. In the limit, as elasticity tends to zero, the entire partitioned singular phase still shrinks into a single point, just as the singular phase in Bentsman and Miller (2007). It is only through such fine structuring of the singular phase that the ball zero velocity can be attained. Some of the analytical steps in the example presentation are omitted, as they will be given in the general form in subsequent sections. 2.1. Physically-based system representation Consider a ball of the unit mass colliding in a free fall with a racket of mass M moving along the vertical axis with a constant speed, as shown in Fig. 1. This system has the state vector X = (x p , xv , X p , X v ), where x p , xv and X p , X v are the positions

Fig. 2. The singular motion phase: motion in the inhibited domain.

and the velocities of the ball and the racket, respectively. The area free of constraint is described by the inequality G(X ) = x p − X p > 0.

(1)

In this area, the system motion is said to be in the constraintfree phase and the equations of motion have the form x˙ p (t) = xv (t), x˙v (t) = −g,

X˙ p (t) = X v (t), X˙ v (t) = 0,

(2)

where g is the acceleration due to gravity. In the inhibited area, depicted in Fig. 2, the motion is said to be in the singular phase and is governed by the equations x˙ p (t) = xv (t),

X˙ p (t) = X v (t),

x˙v (t) = −g − µFv0s (X (t), µ),   X˙ v (t) = M −1 F(t, µ) + µFv0s (X (t), µ) .

(3)

In (3) µFv0s (X (t), µ) is a viscoelastic force during the contact of the ball and the racket described by Fv0s (X (t), µ) = x p (t) − X p (t) + 2κµ−1/2 (xv (t) − X v (t)), (4) where µ > 0 is the elasticity coefficient and 0 ≤ κ ≤ 1 is the damping, and F(t, µ) could be interpreted as an external impulsive control force acting on the racket in the singular and the constraint-free phases and driven in these phases by the short duration control signals w1 and w2 , respectively, interpreted, for example, as computational commands. These two distinct manifestations of this force, further referred to as F s and F r s , respectively, are described in detail in Section 2.5. Define the dependence of w1 and w2 on t and µ as  √ wi ( µ(t − τ )), t ≥ τ, µ wi (t) = i = 1, 2. (5) 0, otherwise,

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To retain the notational simplicity of Eq. (3) in the subsequent derivations, the variable γ , with γ0 = w0 , and the related variable ζ = M −1 γ , with ζ0 = M −1 γ0 , will be used in this section to denote w1 or w2 , and M −1 w1 or M −1 w2 , respectively, and specialized to any of the latter signals as needed. Then, the force F(t, µ) admits the representation √ √ F(t, µ) = µγ ( µ(t − τ )), t ≥ τ, (6) µ

µ

where τ is the first impact time for w1 or exit time for w2 . The control signals wi are assumed to satisfy the constraint |wi (·)| ≤ w0 < ∞,

i = 1, 2.

(7)

Thus, (3)–(7) represents an active, or controlled, singularity. Suppose, further on, that the pressure on the racket during the contact phase is accessible through the sensor output signal ξ(t) equal to the viscoelastic force acting on the racket: ξ(t) = x p (t) − X p (t) + 2κµ−1/2 (xv (t) − X v (t)).

(8)

Eqs. (1)–(8), further referred to as the physically-based, or the original, system, describe the continuous motion of the ball/racket system for µ < ∞. 2.2. Problem statement Although the original system incorporates active singularity in a physically meaningful manner, it is not well suited for controller synthesis in the singular phase for arbitrary µ. Indeed, in practice parameter µ varies in a broad range, depending on the material, and as µ in (1)–(8) increases, the right hand side in (3) becomes arbitrarily large, leading to numerical ill-conditioning of the standard computational optimal control procedures. Therefore, it is of interest to obtain a representation that is free from µ for computation purposes, but admits derivation of control law that, when reparametrized by µ, captures the entire family of optimal or near optimal control solutions for the original system. As indicated in the Introduction, this representation is shown in Bentsman and Miller (2007) to be given by the so-called controlled infinitesimal dynamics equation. In general case this equation yields a solution optimal only in the limit as µ → ∞ in the auxiliary time and suboptimal for finite µ in the actual time, i.e. time t of the original system, with optimality quality improving as µ increases. However, for a large class of systems, including the one at hand, the µ-reparametrization turns out to provide a family of control laws optimal in the actual time for arbitrary µ. The equation of controlled infinitesimal dynamics also permits obtaining the equation, referred to as the limit system, that describes the motion of the original system, and, specifically, the velocity jumps, as µ → ∞. This system is a numerically well-conditioned original system approximation for large µ. Thus, the problem falls into three objectives: (i) the controller synthesis setting objective — development of an analytical setting that (a) permits computation of the optimal control law by standard controller synthesis techniques in auxiliary time, and (b) once this law is obtained, admits its reparametrization by µ for application to the original system,

(ii) the control law calculation objective — calculation of an impulsive optimal control law which minimizes the velocity of the ball bounce after the impact in the limit system, and, finally, (iii) the limit modeling objective — derivation of the limit system corresponding to the original one. These objectives are addressed below for system (1)–(8). 2.3. Controller synthesis setting: Infinitesimal dynamics equation To develop controller synthesis setting, introduce the spacetime transformation √ s = µ(t − τ ), t ≥ τ, √ y µp (s) = x p (τ ) + µ[x p (τ + µ−1/2 s) − x p (τ )], √ Y pµ (s) = X p (τ ) + µ[X p (τ + µ−1/2 s) − X p (τ )], (9) yvµ (s) = xv (τ + µ−1/2 s), Yvµ (s) = X v (τ + µ−1/2 s), ηµ (s) = ξ(τ + µ−1/2 s) where s represents the auxiliary time variable. This transformation extends the one given in Bentsman and Miller (2007) to accommodate sensor dynamics. Substituting (9) into (1)–(8) and using variable γ to denote w1 or w2 yields the following µ µ µ µ system for new variables (y p , yv , Y p , Yv , ηµ ), describing the motion in the enlarged space-time scale: y˙ µp (s) = yvµ (s), Y˙ pµ (s) = Yvµ (s), g y˙vµ (s) = − √ − y µp (s) + Y pµ (s) − 2κ yvµ (s) + 2κYvµ (s), µ Y˙vµ (s) = M −1 (γ (s) + y µp (s) − Y pµ (s) + 2κ(yvµ (s) − Yvµ (s))), µ

η (s) =

y µp (s) − Y pµ (s) + 2κ

yvµ (s) − Yvµ (s)

(10)


,

with the dots denoting d/ds and the initial conditions given by y µp (0) = Y pµ (0) = 0, Yvµ (0)

=

X vµ (τ −),

yvµ (0) = xv (τ −), µ

η (0) = 2κ

xvµ (τ −) −

(11) X vµ (τ −)




.

In the limit as µ → ∞, (10) and (11) yield the system of controlled infinitesimal dynamics for the variables (y p , yv , Y p , Yv , η) given by y˙ p (s) = yv (s),

Y˙ p (s) = Yv (s),

y˙v (s) = −y p (s) + Y p (s) − 2κ yv (s) + 2κYv (s), Y˙v (s) = M −1 (γ (s) + y p (s) − Y p (s)

(12)

+ 2κ(yv (s) − Yv (s))), η(s) = y p (s) − Y p (s) + 2κ (yv (s) − Yv (s)) , with the initial conditions: y p (0) = Y p (0) = 0, Yv (0) = X v (τ −),

yv (0) = xv (τ −), η(0) = 2κ (xv (τ −) − X v (τ −)) .

For convenience of controller synthesis, introduce relative coordinates q(s) = (q p (s), qv (s)) as q p (s) = y p (s) − Y p (s),

qv (s) = yv (s) − Yv (s).

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and the resulting optimal control law is given by ζ˜ (s) = −ζ0 sign (ψv (s)), ζ0 , M −1 γ0 . Further on, the adjoint system is given by

Then, (12) takes the form q˙ p (s) = qv (s), q˙v (s) = −ζ (s) − aq p (s) − 2aκqv (s),

(13)

y˙v (s) = −q p (s) − 2κqv (s) = −η(s), where a = 1 + M −1 , q p (0) = 0, and qv (0) = xv (τ −) − X v (τ −) < 0. Transformation (9) and controlled infinitesimal dynamics Eq. (13) satisfy objective (i). The next two subsections present optimal control law calculation for system (13) in auxiliary time s.

2.4.1. Problem structure Suppose the viscoelastic force is characterized by the socalled restitution, or repulsiveness, property (Bentsman & Miller, 2007), i.e. guarantees the repulsion of the ball from the inhibited domain in a finite time without an external force. This means that the rebound conditions that take place at the instant s ∗ of exit from the constraint are given by q˙ p (s ∗ ) = qv (s ∗ ) > 0.

In this case, system (13) admits an explicit solution  sin(ωs) −λs q p (s) = e qv (0) ω  Z s 0 0 sin ω(s − s ) − ζ (s 0 )ds 0 , eλs ω 0

(14)

where subscript 0 in yv0 and s0∗ corresponds to the zero control value. In view of yv (0) < 0, this implies that if M > exp(λπ ω−1 ),

(b) yv0 (s0∗ ) < 0,

if M < exp(λπ ω−1 ),

(c) yv0 (s0∗ ) = 0,

if M = exp(λπ ω−1 ).

In the case (c), the ball stops without any external force. However, in the cases (a) and (b), the racket needs to be forced to attain the reduced ball velocity at instant s ∗ . In the latter two cases, it is natural to take yv2 (s ∗ ) → min

i.e. ψ y = C y = const,

and the terminal transversality conditions at s = s ∗ take the form 2yv δyv + ψ p δq p + ψv δqv + ψ y δyv − Hδs = 0, 2yv (s ∗ ) + C y = 0,

ψv (s ∗ ) = 0,

(16)

as the performance criterion. Denoting the adjoint variables corresponding to q p , qv , and yv by ψ p , ψv , and ψ y , respectively, and applying the Pontrjagin’s maximum principle to the optimal control problem (13)–(16), Hamiltonian H = H(q p , qv , yv , ψ p , ψv , ψ y , ζ ) takes the form

H = ψ p qv − ψv ζ − aψv + ψ y q p + 2κqv → max,

H(s ∗ ) = 0.

Then, H(s ∗ ) = 0 implies ψ p (s ∗ ) − 2κC y = 0,

(18)

and (17) and (18) imply ψ˙ v (s ∗ ) = ψv (s ∗ ) = 0.

(19)

Eq. (17) for the adjoint variable ψv (s) that determines the sign of the optimal control signal has zero terminal conditions and may be easily integrated backward in time. Indeed, suppose C y = −2yv (s ∗ ) 6= 0. Then, defining ϑ = s ∗ − s and ϕv (ϑ) = C y−1 ψv (s ∗ − ϑ), it follows from (17) and (19) that ϕ¨v (ϑ) + 2aκ ϕ˙v (ϑ) + aϕv (ϑ) = −1,

(15)

where λ = κa, ω2 = a − λ2 > 0, and, hence, the restitution condition takes the form a −1/2 > κ. Setting, without loss of generality, Yv (0) = 0, suppose, first, that ζ (s) ≡ 0. Then, (13)–(15) yield    yv0 (s0∗ ) = a −1 yv (0) M −1 − exp −λπ ω−1 ,

(a) yv0 (s0∗ ) > 0,

ψ˙ y (s) = 0,

(17)

where δq p = 0 due to condition (14). This yields

2.4. Optimal control law computation: Open-loop case

q p (s ∗ ) = 0,

ψ˙ p (s) = aψv (s) + ψ y (s), ψ˙ v (s) = −ψ p (s) + 2aκψv (s) + 2κψ y (s),

ϕv (0) = 0.

The latter equation admits an explicit solution h   i ϕv (ϑ) = a −1 e−λϑ cos ωϑ + λω−1 sin ωϑ − 1 . It is easily seen that ϕv (ϑ) < 0 for ϑ > 0, so that the optimal control does not change sign. Thus, at the instant s ∗ two events are possible: (i) 0 < yv (s ∗ ) < yv0 (s0∗ ) < |yv (0)|, implying C y < 0 and, further on, ψv (s) = C y ϕv (s ∗ − s) > 0, which yields the optimal control ζ˜ (s) ≡ −ζ0 . The racket in this case is subject to the external control pulse −w0 , 0 ≤ s < s ∗ producing impulse of motion −w0 s ∗ in the negative direction of the coordinate axis. However, the ball rebounds from the racket in the positive direction. The external force pulse in this case is too small to stop the ball. Since the racket remains behind the ball, the damping problem for the case (i) is completed, with zero velocity of the ball not attainable. (ii) yv (0) < yv0 (s0∗ ) < yv (s ∗ ) < 0, implying C y > 0 and, further on, ψv (s) = C y ϕv (s ∗ −s) < 0, which yields the optimal control ζ˜ (s) ≡ ζ0 . The racket in this case is subject to the external control pulse w0 , 0 ≤ s < s ∗ in the positive direction of the coordinate axis, and the ball retains the negative motion direction. In this case, the external pulse is too small to stop the ball as well, but the racket is now located in front of the ball, permitting a continuation of the damping problem solution. 2.4.2. Open-loop multi-impact control In both cases, the terminal time s ∗ is found by setting the right hand side of (15) to zero, with ζ (s 0 ) = −ζ0 or +ζ0 ,

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Fig. 3. Damped oscillations of the relative coordinates q p and qv in the inhibited domain.

respectively. This gives   ∗ eλs = cos ωs ∗ + ω−1 λ ∓ ζ0−1 aqv (0) sin ωs ∗ .

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Fig. 4. Variation of the coordinates q p , qv and yv in the inhibited and admissible domains under the three-impulse control in the case of a light racket.

(20)

The ball velocity increment is calculated as 1yv (s ) = yv (s ) − yv (0) = − ∗

∗

s∗

Z

q p (s)ds.

(21)

0

Analogously, the racket velocity increment is calculated as 1Yv (s ∗ ) = Yv (s ∗ ) − Yv (0) = 1yv (s ∗ ) − 1qv (s ∗ ) Z s∗ Z s∗ = −M −1 q p (s)ds + ζ (s)ds. 0

(22)

0

Continuing the case (ii) solution, it is easily seen that the application of the constant control ζ (s) = ζ0 after instant s ∗ yields the next collision after a time interval T = 2qv (s ∗ )ζ0−1 with qv (s ∗ + T ) = −qv (s ∗ ) < 0 and q p (s ∗ + T ) = 0 under the external force pulse w0 , s ∗ ≤ s < s ∗ +T applied to the racket. It is seen that the phase vector (q p , qv , yv ) components values at s ∗ +T can be considered as new initial data followed by the case (ii). This process can be repeated several times until the value of the initial relative velocity gets into the interval qˆv ≤ qv (0) ≤ 0, where qˆv is some threshold velocity. When the latter occurs, the ball becomes stuck to the racket. More precisely, it starts exhibiting damped oscillatory motion, but inside an inhibited domain. Physically it means that an inertia force acting on the ball due to the racket acceleration is too large to permit the ball leave the racket. Fig. 3 illustrates this case for the following initial data: a = 2,

κ = 0.25,

qv (0) = −0.0822,

ζ0 = 0.3,

q p (0) = 0,

(23)

yv (0) = −0.3329,

Yv (0) = −0.2507. In the original time t, as µ → ∞, T → 0, so that the entire ball-stopping sequence shrinks into a single point.

Fig. 5. The three-impulse control formed by pulses F1s , F1r s , and (F2s , F3s ). The singular and the intersingular motion phases are shown by the grey and the hatched areas, respectively.

Sequences with such characteristics, termed multi-impulses, are considered in more detail next. 2.5. Optimal multi-impact control law computation with output feedback based decision structure In the numerical example considered, the mass of the ball was equal to that of the racket (a = 2). Now, keeping the same κ = 0.25, and q p (0) = 0 as in (23), consider the case of a light racket (M = 1/3 and, hence, a = 4) moving towards the ball at a higher initial relative velocity qv (0) = −3.2302, with yv (0) = −1.4802 and, hence, Yv (0) = 1.75. Figs. 4 and 5 illustrate this case. In the first interval [0, s1 ] the external control signal is ζ1 (s) = M −1 w11 (s) = ζ0 = 0.3 where the second index in w1 j stands for the first continuous segment of w1 (s). This signal generates the force F1s = µ1/2 w0 applied to the racket. In this interval the ball moves inside the racket. At time s1

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the ball and the racket disengage and during the time interval [s1 , s2 ] they move separately: the ball — with a constant speed y(s) = y(s1 ) = const in the constraint-free domain and the racket - with a linear increasing speed Y (s) = y(s1 ) − q(s) under a constant external control action of the same magnitude and direction ζ2 (s) = M −1 w21 (s) = ζ0 = 0.3. The latter action generates the same force F = µ1/2 w0 as before, but applied to the racket outside of the contact phase. Using the notation introduced in Section 2.1, this force will be denoted as F1r s , and the motion phase corresponding to it will be referred to as the intersingular. At time s2 the racket collides with the ball once again. During the time interval [s2 , s4 ] the ball moves inside the racket, first under the control action ζ3 (s) = M −1 w12 (s) = ζ0 = 0.3 and the corresponding force F2s = µ1/2 w0 on [s2 , s3 ], and then, after the optimal control sign change at time s3 , under the control action ζ4 (s) = M −1 w13 (s) = −ζ0 = −0.3 and the corresponding force F3s = −µ1/2 w0 . Finally, at time s4 the ball stops. The signal ζ2 = M −1 w21 and the force F1r s are seen to be the elements of the optimal signal and force sequences {ζ1 , ζ2 , ζ3 , ζ4 } and {F1s , F1r s , F2s , F3s }, respectively. The latter sequence is depicted in Fig. 5 for µ = 9. These sequences, further referred to as the temporal multi-impulse, induce contact and non-contact phases on the time interval cumulatively going to zero as µ → ∞. Since the corresponding motion is characterized by contact interruption, yet its total duration tends to zero as µ → ∞, it is further referred to as the interlaced singular phase. Now, let us address the original task of using only the sensor output to solve the damping problem. Since the initial relative velocity qv (0) < 0 is unknown, it is also not known which of the cases, (i) or (ii), takes place from the beginning. For this reason, the first control pulse should be negative, i.e. in the direction of the initial velocity of the ball (ζ1 (s) ≡ −ζ0 ). Further, the controller begins integrating the sensor output signal η(s) up to the time moment s ∗ detected by the sensor. On the basis of (13)–(15), this gives s∗

Z J= 0

η(s)ds =

s∗

Z

q p (s)ds.

(24)

0

This integral is easily calculated and permits deducing the initial relative velocity qv (0) from (24) by an explicit, but rather cumbersome formula. On the basis of qv (0) and (15) the value of qv (s ∗ ) is calculated as   sin ωs ∗ 1 ∗ ∗ . qv (s ) = λs ∗ qv (0) cos ωs − (λqv (0) − ζ0 ) e ω By definition, the absolute ball velocity is yv (s ∗ ) = qv (s ∗ ) + Yv (s ∗ ). The velocity of the racket Yv (s) is known to the controller. If yv (s ∗ ) ≥ 0, the control process is completed. If yv (s ∗ ) < 0 the control process is continued by the algorithm described above until the ball stops. From the solution of the limit problem in the singular phase depicted in Figs. 4 and 5 it follows that (i) starting from an instant τ (corresponding to s = 0) the controller forms, on the basis of the sensor output signal, a finite number of force pulses applied to the racket, with the first one directed along the initial velocity of

the ball; (ii) the resulting control signal is the multi-impulse; (iii) the single-impact mode considered in Bentsman and Miller (2007) makes the ball stopping in the latter case of light racket impossible. Consequently, the multi-impact mode is seen to vastly increase the attainability set of the post-multiimpact system state, encompassing the zero velocity point, in comparison to the single-impact mode considered in Bentsman and Miller (2007). 2.6. Limit system representation In the original coordinates the limit trajectories of the ball and the racket are described by equations: x˙¯ p (t) = x¯v (t), x˙¯ v (t) = −g − 1yv (s ∗ )δ(t − τ ), X˙¯ p (t) = X¯ v (t), X˙¯ v (t) = 1Yv (s ∗ )δ(t − τ ), where 1yv (s ∗ ) and 1Yv (s ∗ ) are calculated through (21) and (22), respectively. Here δ(t) is a Dirac δ-function. 2.7. Control law implementation in the original system Now, let us discuss the case of finite µ. Suppose first that the gravity force is absent. This corresponds to the motion along the horizontal line. Space–time transformation (9) leads to the system (13) directly, without going to the limit. This means that for any finite µ it is possible to carry out an inverse space–time transformation and obtain the exact optimal solution of the original problem. The value µ−1/2 plays in this transformation the role of a scaling factor. In particular, real time t = sµ−1/2 . The external force impulses of motion applied to the racket coincide in dimension and magnitude in the original and the auxiliary problems, and the graphs of the absolute velocity xv (t) of the ball, the main variable of interest, are identical to those of yv (s) in Fig. 4 with auxiliary time replaced by the original one according to the first line in (9). If, on the other hand, the gravity force is not equal to zero, then it will give an additional force impulse gsT µ−1/2 , where sT is the terminal time in the auxiliary problem. Then, the ball can not be stopped at time sT µ−1/2 , but the order of its velocity deviation from zero stays vanishingly small and equal to µ−1/2 . 3. General problem statement To present the novel dynamical features of the example considered in a broader context, the general framework for modeling and controller synthesis of dynamical systems with active singularities under incomplete information is developed next. Only objectives (i) and (iii) of the example problem statement – the controller synthesis setting development and the limit model derivation – are addressed. Objective (ii) – optimal control law computation – will be considered in a separate publication. Let the controlled dynamical system be described by the state vector x(t) = (x p (t), xv (t)), x p (t) ∈ R n , xv (t) ∈ R n , where vectors x p and xv are referred to as the sets of generalized positions and generalized velocities, respectively, and t ∈ [0, T ], where T is sufficiently large.

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Suppose that system motion includes interaction with some elastic constraint. Let the elastic deformation of the constraint be parametrized by some coefficient µ > 0, so that for finite µ the constraint would admit a system motion, although inhibited, within the domain occupied by it. Let the constraintfree domain be given by {(x p , t) : G(x p , t) > 0},

(25)

where G : R n × [0, T ] → R is a sufficiently smooth function. Following Section 2, the system motions in the domain occupied by the constraint and in the constraint-free domain will be referred to as the singular and the constraintfree motion phases, respectively. 3.1. Motion in the constraint-free and the singular phases Generalizing representation of Bentsman and Miller (2007) in the context of the example of Section 2, let the system motion be described by x˙ p (t) = F pr (x p (t), xv (t), t), x˙v (t) = Fvr (x p (t), xv (t), u(t), t) µ

+ µFvs (x p (t), xv (t), w1 (ξ, t), t, µ)

(26)

µ

+ µFvr s (x p (t), xv (t), w2 (ξ, t), t, µ), where u(t) ∈ U ⊂ R r is a control variable (a measurable function) in the constraint-free phase, U is a compact set, F pr (x p , xv , u, t) and Fvr (x p , xv , u, t) are the generalized forces µ in the constraint-free phase, w1 (ξ, t) is a control signal in µ s the singular phase, µFv (x p , xv , w1 (ξ, t), t, µ) is a generalized controlled force arising from a contact with the constraint in µ the inhibited area, µFvr s (x p , xv , w2 (ξ, t), t, µ) is an additional generalized controlled force in the constraint-free phase µ governed by a control signal w2 (ξ, t) (a measurable function), and ξ is the sensor output signal. µ The first of the latter two forces, µFvs (x p , xv , w1 (ξ, t),t,µ), is characterized by µ

Fvs (x p , xv , w1 , t, µ) = 0, if (1) G(x p , t) > 0 or (2) G(x p , t) = 0 and d G(x p , t) = G 0x p (x p , t)F pr (x p , xv , t) + G 0t (x p , t) = 0 dt F pr

further in Section 4, and satisfies the condition µ

Fvr s (x p , xv , w2 , t, µ) = 0,

if G(x p , t) < 0.

(28)

The introduction of this force lays the groundwork for addressing optimal control problems with complex multiimpact structure, such as that encountered in Section 2. Once this structure is in place, whether or not the multi-impact will appear depends on the specific features of the problem at hand. Let in the singular phase, when G(x p (t), t) ≤ 0, components of the state vector (x p (t), xv (t)) be unobservable directly, and it be possible to observe only signal ξ(t) ∈ R k . Then, the control variables in the singular phase can be taken to be continuous functionals of the sensor output signal ξ(t) and measurable in time. Example (3)–(6) of Section 2 is easily recast into (26) as follows. First, x p and xv of (26) are defined as two-vectors [x p1 , x p2 ]T and [xv1 , xv2 ]T given by [x p , X p ]T and [xv , X v ]T , respectively. The functions F pr , Fvr , µFvs , and µFvr s in the rhs of (26) are then simply traced to be given in terms of functions µFv0s , F s , and F r s defined in (3)–(6) by F pr = [xv1 (t), xv2 (t)]T , Fvr = [−g, 0]T ,   −µFv0s (x p , xv , µ),  µFvs =  M −1 F s −1 0s + µM Fv (x p , xv , µ),   −µFv0s (x p , xv , µ), √ 1/2 −1 = µ M w1 (ξ, µ(t − τ )) , (29) +µM −1 Fv0s (x p , xv , µ),     0, 0, rs √ µFv = = . (30) M −1 F r s µ1/2 M −1 w2 (ξ, µ(t − τ )). r s of For example, as seen from (30), the second component Fv2 r s Fv takes the form √ rs Fv2 (31) = µ−1/2 M −1 w2 (ξ, µ(t − τ )).

The generalized forces µFvs and µFvr s are also seen not to retain the physical dimension of force that characterizes F s and F r s . The time τ in (29) and (30) indicates the start of the appropriate phase. The dependence of w1 and w2 on ξ indicates that these control signals are in general functionals of ξ , as illustrated in Section 2.5. 3.2. Sensor equations and admissible control in the singular phase

(27) where

G 0x p

and

G 0t

denote partial derivatives with respect to

x p and t, respectively, and dtd | F pr G(x p , t) denotes the time derivative of G(x p , t) along the trajectories of x˙ p (t) = F pr (x p (t), xv (t), t). Noting that G(x p , t) does not depend on xv , the last expression in (27) is seen to represent the time derivative of G(x p , t) along the trajectories of the entire system (26). µ The force µFvr s (x p , xv , w2 (ξ, t), t, µ), the last of the forces in (26), characterizes an external impulsive action on the system in the constrained-free domain during the so-called intersingular motion introduced in Section 2 and formally defined

To admit control of the sensing environment, let the sensor output signal ξ(t) satisfy the equation ξ˙ (t) = µH (x p (t), xv (t), α µ (ξ, t), t, µ),

(32)

where α µ (ξ, t) is a control signal and H (x p , xv , α µ , t, µ) = 0

if G(x p , t) > 0.

Let the motion in the singular phase begin at τ , where τ is the first instant when d G(x p (τ ), τ ) = 0 and G(x p (τ ), τ ) < 0. (33) dt F pr

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Denoting by γ any of the control signals w1 , w2 , α, define its dependence on t and µ in the singular (interlaced singular) phase as  √ γ (ξ, µ(t − τ )), t ≥ τ, γ µ (ξ, t) = (34) 0, otherwise. Let the following Lipschitz condition take place |γ (ξ 0 , t) − γ (ξ 00 , t)| ≤ Lkξ 0 − ξ 00 kt ,

L = const,

(35)

where kξ kt = ess sup |ξ(s)| = min{λ : |ξ(s)| ≤ λ, τ ≤s≤t

t

Z a.s. s ∈ [τ, t]},

or

kξ kt =

τ

|ξ(s)|2 ds

1/2

Fig. 6. A single multi-impulse case. Prelimit and limit trajectories. Crosshatched region corresponds to the intersingular motion.

.

µ

Definition 1. Admissible control w1 (ξ, t) in a singular phase is a restricted measurable by t functional, where dependence on τ, t, µ, ξ is given by (34) and (35) and a restriction has the form µ w1 (ξ, t) ∈ W1 ⊂ R r1 . Here W1 is a compact set including zero µ element. Admissible controls w2 (ξ, t) and α µ (ξ, t) are defined analogously. It is assumed that the right hand sides of (26)–(32) are sufficiently smooth to guarantee unique solution of (26)–(32) for any admissible controls. 3.3. Controller synthesis setting and limit modeling objectives As indicated in Sections 1 and 2, Eqs. (26) and (32) are not directly suitable for the controller synthesis and modeling due to their unbounded right hand sides (rhs). This problem is addressed by the following specific objectives: Controller synthesis setting objective: provide an analytical setting that permits reduction of an ill-posed problem of µ synthesis of the singular phase control signals w1 (ξ, t), µ w2 (ξ, t), and α µ (ξ, t) in (26)–(32) to a well-posed two-step approximation procedure: (a) synthesis of bounded singular phase control signals w1 (η, s), w2 (η, s), and α(η, s) in the µ auxiliary fictitious time s, and (b) calculation of w1 (ξ, t), µ w2 (ξ, t), and α µ (ξ, t) implementable in the original system (26)–(32) using signals synthesized in (a). Tasks (a) and (b), that can be viewed as the direct and the converse ones, respectively, are addressed in the next section by Theorems 1 and 3. Limit modeling objective: obtain a model that generates a discontinuous (limit) motion controlled by w1 (η, ·), w2 (η, ·), and α(η, ·) representing a consistent approximation of motion µ µ of (25)-(26) controlled by w1 (ξ, t), w2 (ξ, t), and α µ (ξ, t). This objective is addressed in the next section by Corollary 1 and Theorem 2. 4. Singular motion phase under multi-impact µ

In the case of Fvr s (x p , xv , w2 (ξ, t), t, µ) ≡ 0, the multiimpacts, such as the ball-stopping sequence shown in Fig. 5

in the example of Section 2, reduce to single impacts, since the constraint-free motion between two adjacent singular phases can be no longer shaped on the time scale compatible with these phases. As a result, optimal control solutions for systems considered in the present work that can take the form of coordinated time-sequences of impulses can no longer be generated. This motivates focusing on the case µ Fvr s (x p , xv , w2 (ξ, t), t, µ) 6≡ 0. 4.1. Controlled infinitesimal dynamics equations under multiimpact In order to consider the case of Fvr s 6≡ 0 introduce the following definitions. Definition 2. Constrained-free system motion between two sequentially occurring singular phases will be called the intersingular motion if its duration goes to zero as µ → ∞. Definition 3. An arbitrary finite connected sequence of alternating singular and inter-singular motions, which starts and ends with singular phase will be referred to as the interlaced singular phase of the system motion. µ

Definition 4. A pair of admissible controls w1 (ξ, t) and µ µ w2 (ξ, t) in (26) such that w1 (ξ, t) = 0, G(x p (t), t) > 0 and µ w2 (ξ, t) = 0, G(x p (t), t) < 0 is said to be a temporal multiimpulse control if they exist only on the disjoint finite subset of the time subintervals within the time interval of an isolated interlaced singular phase. Definitions 2–4 characterize the temporal multi-impact mode of system interaction with the constraint. The case of a µ single multi-impulse is depicted in Fig. 6 where x¯ p (t) and x p (t) denote prelimit and limit generalized positions, respectively, µ µ∗ and τi and τi denote, respectively, the times of entering and exiting the constraint. As it is seen from these definitions and Fig. 6, this mode is comprised by the time subintervals partitioning the time interval of the interlaced singular phase such that in each subinterval, alternating, either G(x p (t), t) > 0 µ µ and w2 (ξ, t) is invoked or G(x p (t), t) < 0 and w1 (ξ, t) is invoked. In the rest of the paper, the qualifier “temporal” will

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be mostly omitted with no loss of clarity, since the paper does not consider spatially distributed simultaneous impacts. According to (33), the singular motion phase begins at the first time τ that the system engages the constraint. Therefore, for a finite value of µ there exists a non-zero time interval of the constraint violation. Then, applying transformation (9), with (x, X ) and (y, Y ) represented by the single variables x and y, respectively, i.e. with the third and the fifth lines deleted, to µ µ system (26) and (32), the new variables {y p (s), yv (s), ηµ (s)} are straightforwardly shown to satisfy the multiscale equation  µ y p (s) − x p (τ ) µ r µ −1/2 y˙ p (s) = F p + x p (τ ), yv (s), τ + µ s , µ1/2  µ √ s y p (s) − x p (τ ) µ y˙v (s) = µFv + x p (τ ), µ1/2  µ µ −1/2 × yv (s), w1 (η , s), τ + µ s, µ 

µ

y p (s) − x p (τ ) + x p (τ ), µ1/2  µ −1/2 µ s, µ × yv (s), w2 (η , s), τ + µ

+

√

µ Fvr s



η˙ µ (s) =

√ µH



Fvr



(36)

µ

µ y p (s) − x p (τ ) µ1/2



µ↑∞

 yp − x p µ −1/2 + x , y , w (η , s), τ + µ s, µ p v 1 µ1/2

= F¯vs (y p , yv , w1 (η, s), x p , τ ),  1/2 r s y p − x p lim µ Fv + x p , yv , w2 (ηµ , s), τ µ↑∞ µ1/2  −1/2 + µ s, µ = F¯vr s (y p , yv , w2 (η, s), x p , τ ), lim µ

1/2

µ↑∞

 H

(38)

 yp − x p µ −1/2 + x p , yv , α(η , s), τ + µ s, µ µ1/2

= H¯ (yv , α(η, s), x p , τ ), where convergence is uniform in any bounded vicinity of (y p , yv , η, s).  Now, fix some admissible controls u, w1 , w2 , α on [0, T ] and define recursively a finite sequence of instants

j = 1, . . . , N , as follows. First step. Consider the system for Fvs 6≡ 0 and Fvr s = 0 given by y˙ p (s) = F pr (x p (τ ), yv (s), τ ), y˙v (s) = F¯vs (y p (s), yv (s), w1 (η, s), x p (τ ), τ ),

(39)

η(s) ˙ = H¯ (yv , α(η, s), x p (τ ), τ ) with y p (0) = x p (τ ), yv (0) = xv (τ −), η(0) = ξ(τ ). System (39) is supposed to have the unique solution on some interval [0, s1∗ (τ ) + ε], where ε > 0 and

+ x p (τ ), yvµ (s),

 × α(ηµ , s), τ + µ−1/2 s, µ ,

s1∗ (τ )

with the initial conditions given by µ µ y p (0) = x p (τ ), yv (0) = xv (τ −), ηµ (0) = ξ(τ ). Let us now formulate the theorems that incorporate the interlaced singular phase into the system motion. We begin with the theorem that describes the limit behavior as µ → ∞ of new variables introduced through (9) and satisfying (36), developing first the necessary background. Assumption 1. Suppose that Fvs (analogously Fvr s and H ) satisfies the Lipschitz condition in the following form: there exist L > 0, µ0 > 0 such that for any (x p , x 0p , xv , xv0 ), t ∈ [0, T ], w1 ∈ W1 , and µ ≥ µ0 kFvs (x p , xv , w1 , t, µ) − Fvs (x 0p , xv0 , w1 , t, µ)k ≤ L{kx p − x 0p k + µ−1/2 kxv − xv0 k}.

lim µ1/2 Fvs

0 = s1 (τ ) < s1∗ (τ ) < · · · < s j (τ ) < s ∗j (τ ) < · · · ,

y p (s) − x p (τ ) + x p (τ ), µ1/2  µ −1/2 −1/2 × yv (s), u(τ + µ s), τ + µ s ,

+µ

−1/2

exists

(37)

( = inf

s>0

G 0t |(x p (τ ),τ ) s + G 0x p |(x p (τ ),τ ) × (y p (s) − x p (τ )) = 0,

)

G 0t |(x p (τ ),τ ) + G 0x p |(x p (τ ),τ ) × F pr (x p (τ ), yv (s), τ ) > 0

.

(40) Along with (39) consider its counterpart for Fvs = 0 and Fvr s 6≡ 0 given by y˙ p (s) = F pr (x p (τ ), yv (s), τ ), y˙v (s) = F¯vr s (y p (s), yv (s), w2 (η, s), x p (τ ), τ ),

(41)

η(s) ˙ = 0, which is supposed to have the unique solution on an interval [s1∗ (τ ), s2 (τ ) + ε1 ], where ε1 > 0 and y p (s1∗ (τ )), yv (s1∗ (τ )), η(s1∗ (τ )) coincide with the terminal values of solutions of (39) at instant s1∗ (τ ), i.e. a solution of the system (41) is a continuous extension of the solution of the system (39). Define s2 (τ )

Assumption 2. Along with Assumption 1 assume that: (1) for any admissible controls w1 , w2 , α and for any (x p , τ ) such that G(x p , τ ) = 0 and dtd | F pr G(x p (τ ), τ ) < 0 there

=

inf

s>s1∗ (τ )

( 0 ) G t |(x p (τ ),τ ) s + G 0x p |(x p (τ ),τ ) × (y p (s) − x p (τ )) = 0, G 0t |(x p (τ ),τ ) + G 0x p |(x p (τ ),τ ) × F pr (x p (τ ), yv (s), τ ) < 0

.

(42)

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∗ sµ1 (τ ) (

= inf

s>0

G(x p (τ + µ−1/2 s), τ + µ−1/2 s) = 0,

)

G 0t |(x p (τ +µ−1/2 s),τ +µ−1/2 s) + G 0x p |(x p (τ +µ−1/2 s),τ +µ−1/2 s) F pr (x p (τ + µ−1/2 s), xv (τ + µ−1/2 s), τ + µ−1/2 s) > 0 Box I.

sµ2 (τ ) ( =

inf

G(x µp (τ + µ−1/2 s), τ + µ−1/2 s) = 0,

)

G 0t |(x µp (τ +µ−1/2 s),τ +µ−1/2 s) + G 0x p |(x µp (τ +µ−1/2 s),τ +µ−1/2 s) F pr (x µp (τ + µ−1/2 s), xvµ (τ + µ−1/2 s), τ + µ−1/2 s) < 0

∗ (τ ) s>sµ1

Box II.

∗ sµ2 (τ )

( =

inf

s>sµ2 (τ )

G(x µp (τ + µ−1/2 s), τ + µ−1/2 s) = 0,

)

G 0t |(x µp (τ +µ−1/2 s),τ +µ−1/2 s) + G 0x p |(x µp (τ +µ−1/2 s),τ +µ−1/2 s) F pr (x µp (τ + µ−1/2 s), xvµ (τ + µ−1/2 s), τ + µ−1/2 s) > 0 Box III.

Second step. Taking the terminal values y p (s2 (τ )), yv (s2 (τ )), η(s2 (τ )) of solutions of (41) as initial conditions for the system (39), assume that the latter has a unique solution on an interval [s2 (τ ), s2∗ (τ ) + ε2 ], where ε2 > 0 and s2∗ (τ ) =

inf

s>s2 (τ )

( 0 ) G t |(x p (τ ),τ ) s + G 0x p |(x p (τ ),τ ) × (y p (s) − x p (τ )) = 0, G 0t |(x p (τ ),τ ) + G 0x p |(x p (τ ),τ ) × F pr (x p (τ ), yv (s), τ ) > 0

,

(43) and so on. Set ε0 = min j {ε, ε j } and denote by (y p (s), yv (s), η(s)) defined above solution of the systems (39) and (41) for s ∈ [0, s N∗ (τ ) + ε0 ]. Next, for sufficiently large µ > µ0 consider the systems (26) and (36) under fixed control signals defined above. Denoting solutions of those systems by µ µ µ µ (x p (t), xv (t)) and (y p (s), yv (s), ηµ (s)), respectively, define recursively a finite sequence of instants of system trajectory intersections with the constraint boundary

sµj (τ ) → s j (τ ),

∗ sµj (τ ) → s ∗j (τ ).

(45)

The proof of this theorem follows from extending the arguments of Theorem 1 of Bentsman and Miller (2007) to the formalism (36)–(43), Boxes I, II and III. Assertion (45) is proved inductively. Conditions (40) and Box I mean that the “forces” F¯vs and Fvs have the property to repulse the system from the inhibited domain under any admissible control signals. Remark 1. Eqs. (39) and (41), referred to as the controlled infinitesimal dynamics equations, represent dynamics of the interlaced singular phase in the limit as µ ↑ ∞ in the extended µ µ time s. Unlike (26), where w1 (ξ, t) and w2 (ξ, t) appear in the unbounded rhs terms, (39) and (41) have bounded rhs and are, therefore, amenable to synthesis of the singular phase control signals w1 (η, s) and w2 (η, s) using standard optimal control methods. 4.2. Limit system representation under multi-impact

∗ ∗ 0 = sµ1 (τ ) < sµ1 (τ ) < · · · < sµj (τ ) < sµj (τ ) < · · · ,

j = 1, . . . , N , as follows. ∗ (τ ) and s (τ ) in Boxes I and II. First step. Define sµ1 µ2 Second step. Define sµ∗ 2 (τ ) in Box III and so on. ∗ (τ )], j = 1, . . . , N , correHere the intervals [sµj (τ ), sµj spond to the singular phases of motion and the intervals ∗ (τ ), s (sµj µ, j+1 (τ )), j = 1, . . . , N − 1, correspond to the intersingular ones. Theorem 1. Let Assumptions 1 and 2 hold. Then, if µ → ∞, (y µp (s), yvµ (s), ηµ (s)) → (y p (s), yv (s), η(s))

uniformly on [0, s N∗ (τ ) + ε0 ], and for j = 1, . . . , N

(44)

The multi-impact limit representation of the original system is given next. Corollary 1. For sufficiently small ε > 0 on the interval [0, τ +ε), solution of the original system (26) converges to some discontinuous functions (x¯ p (t), x¯v (t)), such that x¯ p (t) = x p (t),

x¯v (t) = xv (t),

t < τ,

and

∗ x¯ p (τ ) = lim x p (τ + µ−1/2 sµN (τ )) = x p (τ ), µ↑∞

∗ x¯v (τ ) = lim xv (τ + µ−1/2 sµN (τ )) = yv (s N∗ (τ )). µ↑∞

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J. Bentsman et al. / Automatica 44 (2008) 1741–1752

The functions (x¯ p (t), x¯v (t)) can be interpreted as the generalized solution of the original system (26). Let us now use Corollary 1 to formulate a theorem describing evolution of the variables (x¯ p (t), x¯v (t)). µ

µ

Theorem 2. Let (x p (t), xv (t)) denote the ordinary solution of the original system (26) where a superscript µ is used to indicate dependence of this solution on parameter µ. Let Ψv (·) be a v-component of the shift operator along the paths of system (39) and (41) obtained by integrating (39) and (41) for s ∈ [0, s N∗ (τ )], so that
yv (s N∗ (τ )) = yv (0) + Ψv y p (0), yv (0), w1τ (·), w2τ (·), τ . where w1τ (·) = {w1 (η, s) : 0 ≤ s ≤ s N∗ (τ )} and w2τ (·) = {w2 (η, s) : 0 ≤ s ≤ s N∗ (τ )} (analogous notation is used for control signal α). Then, the generalized solution (x¯ p (t), x¯v (t)) of the original system (26) is a pointwise limit of its ordinary solution as µ → ∞, and satisfies on an interval [0, τ + ε] the system of generalized differential equations x˙¯ p (t) = F pr (x¯ p (t), x¯v (t), t), x˙¯ v (t) = Fvr (x¯ p (t), x¯v (t), u(t), t)

(46)

+ Ψv x¯ p (τ ), x¯v (τ −), w1τ (·), w2τ (·), τ δ(t − τ ),


with x¯ p (0) = x p (0), x¯v (0) = xv (0), x¯ p (τ ) = x p (τ ), x¯v (τ −) = xv (τ −). The proof of this theorem is similar to that of Theorem 2 of Bentsman and Miller (2007) and is omitted. Since (46) encompasses limit motions corresponding to both regular and singular original system motion phases, it is referred to as the full limit system. 4.3. Control law implementation under multi-impact The next assertion is, in a certain sense, a converse of Theorem 2, demonstrating how for given controls u(t), w1τ (·), w2τ (·), ατ (·) that generate a solution (x¯ p (t), x¯v (t)) of the system (46) to construct controls that produce a solution µ µ (x p (t), xv (t)) of the original system (26) that pointwise converges to (x¯ p (t), x¯v (t)) as µ → ∞. Introduce a finite series corresponding to a multi-impulse as µ

µ∗

µ

µ∗

µ

µ∗

τ µ = τ1 < τ1 < τ2 < τ2 < · · · < τ N < τ N = τ µ∗ , where µ

τ j = τ µ + µ−1/2 sµj (τ µ ),

µ∗

τj

∗ = τ µ + µ−1/2 sµj (τ µ ),

j = 1, . . . , N . Theorem 3. Let (x¯ p (t), x¯v (t)), t ∈ [0, T ], be a solution of the system (46) under some admissible controls u, w1 , w2 and α. Then, if µ → ∞, the corresponding sequence µ µ of ordinary solutions (x p (t), xv (t)) of the system (26) with the same control signals u(t), t ∈ [0, T ], and γ µ (ξ, t) = √ γ (ξ, µ(t − τ µ )), t ∈ [τ µ , τ µ∗ ], converges everywhere on [0, T ], except, possibly, at the point τ , to the general solution (x¯ p (t), x¯v (t)) of the system (46). Here γ is any of the controls

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w1 , w2 and α, which, generally, admit an extension in the neighborhood of the point τ µ∗ by the rule  u(t), t ∈ [0, τ ], u µ (t) = any admissible, t > τ,  √ γ (ξ, µ(t − τ )), t ∈ [τ, τ + tµ∗ ], µ γ (ξ, t) = any admissible, otherwise, where γ stands for either of w1 , w2 , or α, and ( ) ∗ s N∗ (τ ) sµN (τ ) ∗ tµ = min √ , √ . µ µ The proof of this theorem is similar to that of Theorem 4 of Bentsman and Miller (2007) and is omitted. This type of control is shown in the example of collision damping by a single multi-impulse in Section 2.5. 5. Conclusions Extending the framework of Bentsman and Miller (2007), this work presents the first input/state/output model of a dynamical system with active singularity that incorporates both sensing and actuation and admits observations control. New concepts, the interlaced singular phase and the multi-impulse control, are found to be necessary for accommodating solutions of the optimal control problems in the latter system class. An example is given that on one hand motivates the development of the framework proposed and on the other – demonstrates its use for the design of the observations–based multi-impulse optimal control law for this class of systems. Acknowledgements This work was supported by the US National Science Foundation under grants CMS-0324630 and ECS-0501407, and Russian Foundation for Basic Research under grant 07-0800739-a. Anonymous reviewers are gratefully acknowledged. References Bentsman, J., & Miller, B. M. (2007). Dynamical systems with active singularities of elastic type: a controller synthesis framework. IEEE Transactions on Automatic Control, 52, 1–18. Brogliato, B., & Zavala Rio, A. (2000). On the control of complementaryslackness juggling mechanical systems. IEEE Transactions on Automatic Control, 45, 235–246. Grizzle, J. W., Abba, G., & Plestan, F. (2001). Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transactions on Automatic Control, 46, 51–64. Hristu-Varsakelis, D., & Levine, W. S. (2005). Handbook of Networked and Embedded Control Systems. Boston: Birkhouser. Mazumder, S. K., Tahir, M., & Kamisetty, S. L. (2005). Wireless pwm control of a parallel dc-dc buck converter. IEEE Transactions on Power Electronics, 20, 1280–1286. Pagilla, P. R., & Tomizuka, M. (1995). Control of mechanical systems subject to unilateral constraints. In: Proceedings of the 34th IEEE conference on decision and control (pp. 4311–4316). Ronsse, R., Levefre, P., & Sepulchre, R. (2007). Rhythmic feedback control of a blind planar juggler. IEEE Transactions on Robotics, 23, 790–802. Spong, M. (2001). Impact controllability of an air hockey puck. Systems and Control Letters, 42, 333–345. Tornambe, A. (1999). Modeling and control of impact mechanical systems: theory and experimental results. IEEE Transactions on Automatic Control, 44, 294–309.

Author's personal copy

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J. Bentsman et al. / Automatica 44 (2008) 1741–1752

Joseph Bentsman (S’83-M’84) received the Electrical Engineering diploma from Byelorussian Polytechnic Institute, Minsk, Belarus, in 1979, and the Ph.D. degree in electrical engineering from the Illinois Institute of Technology, Chicago, IL, in 1984. From 1975 to 1980 he worked as an Engineer at the Design Bureau of Broaching Machine Tools, Minsk, Belarus. In 1985, he was a Lecturer and a Postdoctoral Research Fellow in the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI. At present, he is an Associate Professor at the Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign. His current research interests include control of distributed parameter systems, nonsmooth and impulsive control, nonlinear dynamics, wavelet methods, sensor networks, and robust predictive and self-tuning control. He has authored over 100 journal and conference papers in these areas. Dr. Bentsman is a recipient of the 1989 National Science Foundation Presidential Young Investigator Award in Dynamic Systems and Control. He served as an associate editor of the ASME Journal of Dynamic Systems, Measurement and Control and is currently on the editorial board of Nonlinear Phenomena in Complex Systems. He also currently serves as an invited editor for the special issue on adaptive identification and control of distributed parameter systems in the International Journal of Adaptive Control and Signal Processing. Boris M. Miller graduated from Moscow Institute of Physics and Technology (MIPT), Russia, and received the Ph.D. degree in cybernetics from MIPT, in 1978. He received the Degree of Dr. Sc. (Mathematics) from the Institute of Applied Mathematics, Moscow, Russia, in 1991. He is a Professor of Department of Control Sciences of MIPT and Head of Laboratory of the Institute for Information Transmission Problems, Russian Academy of Sciences. Currently, he is with the School of Mathematical Sciences of Monash

University, Victoria, Australia. His main scientific interests are in optimal control and stochastic systems theory. He wrote with E. Rubinovich the monograph “Impulsive Control in Continuous and Discrete-Continuous Systems”, Dordrecht: Kluwer; NY 2003 and the monograph “Optimization of Dynamic Systems with Impulse Controls” (in Russian), Fizmatlit, Moscow, 2005. He is also the co-author of the textbook written with A. Pankov “Theory of Random Processes (in Problems and Examples)” (in Russian), Fizmatlit, Moscow, 2003. Evgeny Ya. Rubinovich was born in Moscow, Russia on December 05, 1946. He received the M.Sc. degree in Electrical Engineering and Control Sciences, 1971 from Moscow Institute of Physics and Technology, Russia, the Ph.D. and Dr.Sc. (Engineering) degrees from the Institute of Control Sciences of Russian Academy of Sciences, all in Control Sciences, in 1979 and 1992, respectively. From 1971 to 1979, from 1979 to 1992, from 1992 to 2004 and from 2004 to 2006 he was Associate Researcher, Senior Researcher, Leading Researcher and a head of Sector, respectively, at the Laboratory “Processes of Control and Management Based on Incomplete Data” of the Institute of Control Sciences, Moscow, Russia. Since 2006 he has been a Chief Researcher at the Institute of Control Sciences. He served as an Associate Professor of the Department of Control Sciences at Moscow Institute of Physics and Technology from 1986 to 1995. Since 1995 he has been a Full Professor of Moscow Institute of Physics and Technology. He has published over sixty articles in various fields, including optimal control theory, differential games, stochastic control theory and co-authored (with B.M.Miller) books “Impulsive control in continuous and discretecontinuous systems (Foundations of the hybrid systems theory)”, 457p. Kluwer, Academic/Plenum Publishers, 2003 and “Optimization of dynamic systems with impulsive controls”, 430p. Nauka, Moscow, 2005 (in Russian).