J.C. Willems, Dissipative dynamical systems Part I: General theory; Part II: Linear systems with quadratic supply rates, Archive for Rational Mechanics and Analysis, volume 45, pages 321-393, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....case, if u 2 L ; U ) then it suces to integrate [ u ] once, then apply C D, and nally di erentiate once in the distribution sense. 12 3 Scattering Passive and Conservative Systems The following de nition is a slightly modi ed version of the de nitions in the two classical papers [31, 32] by Willems (although we use a slightly di erent terminology: our passive is the same as Willems dissipative) De nition 3.1. Let J be a bounded self adjoint operator on . A system node S on (U; X;Y ) is J passive if, for all t 0, the solution (x; y) in Lemma 2.3 satis ....

J. C. Willems. Dissipative dynamical systems Part I: General theory. Arch. Rational Mech. Anal., 45:321-351, 1972.

....[8, 9] 11] 17] 19, 20] 21] 22, 23, 24, 25, 28] 29, 30] 33] 35, 36, 37, 38, 39] 40] 41] and [42] and the references therein) 3. Scattering Passive and Conservative Systems. The following de nition is a slightly modi ed version of the de nitions in the two classical papers [43, 44] by Willems (although we use a slightly di erent terminology: our passive is the same as Willems dissipative) Definition 3.1. Let J be a bounded self adjoint operator on [ U ] A system node S on (U; X;Y ) is J passive if, for all t 0, the solution (x; y) in Lemma 2.2 satis ....

J. C. Willems, Dissipative dynamical systems Part I: General theory, Arch. Rational Mech. Anal., 45 (1972), pp. 321-351.

.... : ES) For more details, explanations and examples we refer the reader to [3] and [7, 8, 9, 10] and the references therein) 3 Passive and Conservative Scattering and Impedance Systems The following de nitions are slightly modi ed versions of the de nitions in the two classical papers [16, 17] by Willems (although we use a slightly di erent terminology: our passive is the same as Willems dissipative, and we use Willems storage function as the norm in the state space) De nition 3.1. A system node S is scattering passive if, for all t 0, the solution (x; y) in Lemma 2.2 satis ....

J. C. Willems. Dissipative dynamical systems Part I: General theory. Arch. Rational Mech. Anal., 45:321-351, 1972.

....classes of flow control laws. The passivity concept is motivated by physical systems that conserve or dissipate energy, for example, passive circuits [12] and mechanical structures [13] Passivity provides a useful tool in nonlinear stability analysis and control design for feedback systems [14] [15]. The main result is the celebrated passivity theorem which states that the negative feedback connection of two passive systems is passive. The storage functions (generalization of energy in physical systems) for the subsystems in the feedback interconnection can be combined and used in the ....

J. Willems, "Dissipative dynamical systems, part I: General theory, part II: Linear systems with quadratic supply rate," Arch. Rational Mech. Anal., vol. 45, pp. 321--393, 1972.

....a passive relationship with its physical environment. A dynamic system with input u 6 g and output y 6 Y is passive with respect to the supply rate s: g x y R if, for any u: g and any t O, f s(u(r) y(r) dr c 2, for sone c 6 R which may depend on the system s initial condition [4]. Mechanical systems are subject to two types of inputs: control inputs T (torques or forces generated by actuators) and environment forces F (disturbances, contact forces) While it is well known that mechanical systems are passive with respect to the supply rate defined by Ttotl where Trot ....

J.C. Willems, "Dissipative dynamical systems, part i: General theory," Archive for Rational Mechan- ics and Analysis, vol. 45, pp. 321-351, 1972.

....control problem for G if and only if the H1 control problem for G is solvable. oe oe oe oe p = F (p; u; y) u u p y Figure 3.1: Central controller K . The non negative function e : R n Theta Xe R defined by the simple formula e(x; p) Gammap(x) W (p) 3:11) is a storage function [15] for the closed loop system. This follows because rxe(x;p) Gammar xp(x) and rpe(x;p) GammaE x rpW (p) where Ex is the evaluation operator hEx ; fi = f(x) which impies that e(x; p) satisfies the PDE sup y frxe Delta (A B1(y Gamma C2 ) B2u ) hrp e; F (p; u ; y)i 1 2 ....

J.C. Willems, Dissipative Dynamical Systems Part I: General Theory, Arch. Rational Mech. Anal., 45 (1972) 321---351.

....[1,4,5] There are two commonly used approaches for providing solutions to nonlinear H1 control and filtering problems. One is based on the dissipativity theory and the differential game theory. Another is based on the nonlinear version of the classical Bounded Real Lemma as developed by Willems [6] and Hill and Moylan [7] However, the nonlinear H1 filters proposed so far are mainly limited to time invariant systems. Therefore they can not be applied to general time varying systems on the infinite horizon since one of two Riccati differential equations required to solve the problem can not ....

J. C. Willems, "Dissipative dynamical systems Part I General theory", Arch. Rational Mech. Anal.,vol. 45, pp. 321-351, 1972.

....invariant set S as t 1. Sucient conditions for the existence of a continuous state feedback law are given, based on a new theorem. 1 Introduction The standard formulation of local state feedback H1 control is mainly based on the theory of dissipative systems rst introduced by Willems [Wil72]. In this paper we shall take o from this problem, which we will approach by the theory of di erential games as outlined in the papers by Isidori [Isi92] and Isidori and Astol [IA92b, IA92a] but we allow for non zero initial conditions following the approach of van der Schaft [vdS92b] ....

Jan C. Willems. Dissipative dynamical systems part I: General theory. Arch. Rational Mech. Anal., 45:321-351, 1972. 10

....After this paper was accepted for publication we became aware of [18] where GAS of a PID controller with a saturated proportional gain and velocity measurement is established. 4. It is now well known [3] that the robot total energy function T (q; q) Ug (q) qualifies as a storage function [19] for the supply rate w(u; q) u T q. From this property output strict passifiability 4 of the map u 7 q inmediately follows taking u = GammaK D q u1 , with u1 the input that shapes the potential energy. In output feedback problems we require a passifiability property with q, instead ....

J. C. Willems, "Dissipative dynamical systems. Part I: General theory," Arch. Rat. Mech. and Analysis, vol. 45, no. 5, 1972.

....the plant s model accurately; this uncertainty can lead to instability or in the best case, to a steady state error. An alternative approach is the so called passivity based control. This technique applies to a certain class of systems which are dissipative with respect to a storage function [2]. Passive systems constitute a particular case of dissipative systems for which the storage function happens to be an energy function. Hence, the rationale behind the passivitybased approach is physical: roughly speaking, a passive system is a system from which one cannot pull out more energy than ....

J. C. Willems, "Dissipative dynamical systems. Part I: General theory," Arch. Rat. Mech. and Analysis, vol. 45, no. 5, pp. 321-351, 1972.

....p2 with storage function T c ( q c ; q c ) V c (q c ; q p2 ) These properties follow, of course, from the passivity of EL systems established in proposition 2.1. With this perspective we see the close connection between the developments above and the theory of dissipative systems a la [10] [21]. Actually, we can replace condition A.2 above by the (stronger) assumption of zero state detectability from q p2 . Remark 3.3. It is clear that the kinetic energy of the controller plays no role on the stabilization task. In fact, as will be illustrated below, it can be trivially chosen as T c ....

Willems, J. C., "Dissipative Dynamical Systems. Part I: General Theory", Arch. Rat. Mech. and Analysis, vol. 45, no. 5, 1972. 11

....game, see [2] there are two opposing players: the disturbance input and the control input. Extensions to nonlinear problems are also possible, by interpreting the H 1 operator norm in terms of the L 2 Gamma Gain of the system, thus making the theory of dissipative systems applicable, see [3,5]) However, in the case of output feedback minimax dynamic games, several issues associated with the controller design are not so well developed and understood. This is due, in part, to the fundamental difficulty that for general nonlinear output feedback minimax games one has to introduce an ....

J. Willems, "Dissipative dynamical systems part I: General theory," Arch. Rational Mech. Anal., vol. 45, pp. 321--351, 1972.

....GAS is shown as in the proof of Theorem 2.33 in Sepulchre, Jankovic Kokotovic (1997) because, under Assumption 2, Sigma n is GAS when v = 0 and G 1 is stable. The dissipation inequality (30) corresponds to the property of dissipativity with suply rate (Qv 2 2Svw Rw 2 ) introduced by Willems (1972) and Hill Moylan (1976) In the form of inequality (21) it is used by Megretski Rantzer (1997) in stability analysis with integral quadratic constraints. Theorem 3.1 shows that GAS is preserved for stable G 1 of the form (18) whose Nyquist plot is in the disk shown in Figure 5. We are ....

Willems, J. (1972), `Dissipative dynamical systems. Part I: General theory. Part II: Linear systems with quadratic suply rates', Arch. Rational Mechanics and Analisys vol. 45(Nr. 5), pp. 321--393.

....the multivalued hysteresis operator. The definitions of finite gain and stability as they apply to relations will therefore be needed. These, along with the more standard definitions listed above, can all be found in [14] The concept of dissipativity was first introduced in 1971 by Willems[15], and has its roots in electrical and mechanical systems. Since then it has been widely studied, including works by Moylan and Hill[16] and Vidyasagar[17] and has proved a valuable tool in analyzing the stability of non linear systems. The extension to relations of many of the dissipativity tools ....

J.C. Willems, "Dissipative dynamical systems, Part I: General theory," Archives for Rational Mechanics & Analysis, vol. 45, pp. 321--351, 1972.

....problems[6, 9, e.g. In this paper we describe the Preisach model and show that it can be formulated as a classical dynamical system. Using a purely abstract analysis of the Preisach operator, it is shown that these systems are dissipative, in a generalization of the sense first described in [13]. An initial difficulty in studying hysteresis is to arrive at a precise definition. The word comes from the Greek word meaning to lag behind . It is tempting to characterize hysteresis in terms of this lag, or perhaps by the fact that hysteretic systems Motion u y Gamma y u Gamma Temp ....

....section is concerned with developing a state space representation for Preisach models for which 2 M p . By placing the model in a state space framework, more general stability techniques such as Lyapunov and dissipativity theory may be applied. The dynamical system framework used is that of [13], where a complete definition can be found. The system is defined through the input, output and state spaces U ; Y and X , as well as the state transition operator OE and the read out operator r. OE : IR 2 Theta X Theta U 7 X must satisfy the standard axioms: consistency: OE(t o ; t o ; x o ....

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J.C. Willems, "Dissipative Dynamical Systems, Part I: General Theory", Archives for Rational Mechanics and Analysis, Vol. 45, pp. 321-351, 1972.

....the positive definite function V3 (ae) ln(1 ae T ae) Remark 1: According to [11] the systems (1) 3) and (2) are passive with corresponding storage functions V1 , V2 and V3 , respectively. Moreover, the proof of Proposition 1 shows that the systems (1) 3) and (2) are, in fact, lossless [12]. The passivity of system (1) is a well known fact and has been used repeatedly in the past. The passivity of system (3) or of the system (2) however, is neither as a well known nor as a frequently used result. Passivity in terms of the Euler parameter vector and the Euler rotation vector has ....

J. C. Willems, "Dissipative dynamical systems. Part I: General theory," Archive for Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321--351, 1972.

....Z t 2 t 1 jr(w( z( j d 1 ; 2.3) and I r(w( z( d 0 : 2.4) The supply rate is chosen based on the system characteristics and properties. In fact, it usually comprises a quadratic function, because such a form leads to Riccati type stability conditions. Definition (Willems [47, 48]) A system M of the form in Eqn. 2.1) or Eqn. 2.2) with states denoted by x 2 R n , is said to be passive with respect to the supply rate r(w; z) if there exists a positive definite function VM : R n R , called a storage function, that satisfies the dissipation inequality VM (x(t 2 ) ....

....the case of two or more interconnected dissipative systems, the storage functions may be combined to form a Lyapunov function for the interconnected system. The stability of interconnected systems is based on the fact that the interconnection of two dissipative systems is stable. Lemma (Willems [47, 48]) Consider two dynamic systems M 1 and M 2 with state space representation as in Eqn. 2.1) or Eqn. 2.2) and input output pairs (w 1 ; z 1 ) and (w 2 ; z 2 ) respectively. For the system interconnection illustrated in Fig. 2 1, with w 1 = z 2 M 1 M 2 oe z 2 w 1 w 2 z 1 Figure 2 1: ....

Willems, J. C., "Dissipative Dynamical Systems Part I: General Theory," Archive Rational Mechanics Analysis, vol. 45, pp. 321--351, 1972.

....that interact with humans: Robotic exercise machines and robotic rehabilitation devices interact with humans closely. By ensuring that the controlled system is passive, safety can be enhanced. 3. Augmented Mechanical System The state space formulation of the passivity property of a dynamic system [5] involves the definition of a storage function, a positive valued function on the state space. A dynamic system is passive w.r.t. to the supply rate s : U Theta Y , if a storage function W : X exists s.t. for any initial state x0 2 X , and any input u : U , and any t 0, Z t ....

Willems, J. C., "Dissipative Dynamical Systems, Part I: General Theory", Archive for Rational Mechanics

....the behavior of the user. For our purpose, we specify safety by requiring that the controlled exercise machine appear like a passive device to the user. Viewing the user s force F (t) and the velocity x(t) as the input and output of the controlled exercise machine system, define the supply rate [11] to be F (t) x(t) which is the power input into the system. The exercise machine is said to be passive w.r.t. the supply rate F (t) x(t) if Z t 0 F ( x( d Gammac 2 ; 7) for all t 0, and any human force F ( not necessarily satisfying (4) and some c 2 . When (7) is satisfied, the ....

J. C. Willems, "Dissipative dynamical systems, part i: General theory," Archive for Rational Mechanics and Analysis, pp. 321-- 351, 1972.

....equation. Extending these results to nonlinear systems has been a topic of recent interest, such as in the papers [3] 7] 8] 10] In this paper, we are studying the case of single input linear systems with input saturation. By applying the concept of dissipativity introduced by Willems in [11], we are able to reduce the problem to finding a (piecewise) smooth candidate function whose derivatives satisfy three differential inequalities. For a subclass of closed loop systems, a technique is presented to calculate a valid candidate function. This technique is generalisable to more general ....

....on d : H ( sup 2R 8 : H( 2 dV dx fi fi fi fi fi f( Gamma K( Gammafl 2 k k 2 kg( Gamma K( k 2 o : Then k Upsilonwk 2 2 fl 2 kwk 2 2 for all w 2 W 2 is implied by H ( 0 for all 2 d . Proof: This theorem is originally due to Willems ([11], 12] however later work (as in James [5] has studied the consequences of relaxing the smoothness conditions. 2 Lemma 2.9: For a system of the form given in Definition 2.1, the induced norm fl over any nontrivial set W 2 is greater than or equal to the H1 norm of the linearised system at ....

J.C. Willems, "Dissipative Dynamical Systems, Part I: General Theory," Arch. Rational Mech. Anal., 45, pp. 321--351, 1971

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J.C. Willems, Dissipative dynamical systems Part I: General theory; Part II: Linear systems with quadratic supply rates, Archive for Rational Mechanics and Analysis, volume 45, pages 321-393, 1972.

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J.C. Willems, Dissipative dynamical systems - Part I: General theory, Part II: Linear systems with quadratic supply rates, Archive for Rational Mechanics and Analysis, volume 45, pages 321-351 and 352-393, 1972.

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J.C. Willems, Dissipative Dynamical Systems Part I: General Theory, Arch. Rational Mech. Anal., 45 (1972) 321---351.

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References 20 Willems, J.C. (1972). Dissipative dynamical systems part I: general theory, Arch. Rat. Mech. Anal. 45, 321-351.

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Willems, J. C., "Dissipative Dynamical Systems, Part I: General Theory", Archive for Rational Mechanics and Analysis, pp 321-351, 1972.

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