Passivity and Dissipativity

Transcription

1 Passivity and Dissipativity 1 I. INTRODUCTION Example I.1. Consider again the mass spring system in Figure 1 and that we can exert a force u and measure the velocity y. Setting the mass m = 1, the state equations then become ẋ1 x = 2, ẋ 2 F x 1 ρx 2 + u y = x 2. How can we make a statement about the stability of the system? We consider the energy of the system: The derivative of V x is given by V x = Assume that initially t =, x 1 = x 2 =. Then V t = t V τdτ = t x1 F sds x2 2. V x = ρx x 2 u = ρx yu. ρx yu t dτ yτuτdτ t yτ uτ dτ. We can see that if u, y are bounded, then V t is bounded. Due to the properties of V x, we can then bound the state. In this way we can state that if the measured input and output are bounded, then the state is bounded. This may be taken to imply stability of the system, in some sense. In general, we consider the state space system Σ : ẋ = fx, u y = hx, u u U = R m y Y = R p where x = x 1,..., x n are local coordinates for an n-dimensional state space manifold X, together with a function s : U Y R called the supply rate. Fig. 1. A mass spring system with additional damping.

2 2 Fig. 2. Two interconnected systems. Definition I.1 Storage function. A state space system Σ is said to be dissipative with respect to the supply rate s if there exists a function S : X R +, called the storage function, such that for all x X, all t 1 > t, and all input functions u t1 Sxt 1 Sxt + sut, ytdt t where xt = x, and xt 1 is the state of system Σ at time t 1 resulting from initial condition x and input function u. This inequality is called the dissipation inequality. It expresses the concept that the stored energy Sxt 1 of Σ at any time t 1 is at most equal to the sum of the stored energy Sxt present at the time t and the total energy t 1 t sut, ytdt which is supplied externally during the time interval [t, t 1 ]. Hence, there can be no internal creation of energy; only internal dissipation of energy is possible. Two special supply rates are worthy of special mention. These are dealt with in the following sections. II. PASSIVE SYSTEMS Definition II.1 Passive system. A state space system Σ is called passive if it is dissipative with respect to the supply rate su, y = u y. Σ is strictly input passive if there exists δ > such that Σ is dissipative with respect to su, y = u y δ u 2. Σ is strictly output passive if there exists ɛ > such that Σ is dissipative with respect to su, y = u y δ y 2. Remark II.1. Some authors work with a weaker concept. A state space system Σ is called cyclo-passive if it is passive, but with the storage function being allowed to be negative, not strictly positive. ctd: We can reinterpret the result from above: Sx = x1 F sds x2 2. We can see that the system is dissipative with respect to the supply rate su, y = u y. Remark II.2 Linear systems. Consider a linear SISO system Gs. Then Gs is passive if and only if Gjω lies entirely in the RHP and Gs is stable. Such systems are called strictly positive real systems. Note that such systems are stabilized by any negative feedback. Remark II.3 Interconnections. Passive systems are additionally interesting as interconnections of passive systems are also passive. Here interconnections may be series, parallel or feedback connections of the passive systems. See Figure 2. Theorem II.1. The feedback connection of two passive systems is passive.

3 3 Proof: Let V 1 x and V 2 x be the storage functions for Σ 1 and Σ 2 respectively, so that e i y i V i, i = 1, 2. Where Then it follows that y 1 = Σ 1 e 1 = Σ 1 u 1 y 2 y 2 = Σ 2 e 2 = Σ 1 u 2 + y 1 e 1 y 1 + e 2 y 2 = u 1 y 2 y 1 + u 2 + y 1 y 2 = u 1 y 1 + u 2 y 2 V 1 + V 2 Taking V 1 + V 2 as the storage function for the feedback connection we obtain u y V. III. FINITE L 2 -GAIN SYSTEMS Theorem III.1. Let the system Σ be dissipative with respect to su, y = 1 2 function S : R n R +.. Then the system has L 2 -gain γ. 1 2 t1 t γ 2 u 2 2 y 2 2dt Sxt 1 Sxt Sxt y 2 2dt γ 2 u 2 2dt + 2Sx γ 2 + u 2 2 y 2 2 with storage Remark III.1. This result lays the grounds for the development of nonlinear H theory, see [4]. IV. TESTING FOR DISSIPATIVITY Of course, an important question is how we may decide if Σ is dissipative with respect to a given supply rate s. The following theorem gives a theoretical answer. Theorem IV.1. Consider the system with supply rate s. Then it is dissipative with respect to s if and only if the function S a x = sup u,t is uniformly bounded S a x <, x R n. su, ydt, x = x Further, if S a x is uniformly bounded, then it is a storage function, and for all other storage functions S a x Sx. Proof: Suppose S a if finite. Clearly S a. Compare now S a xt with S a xt 1 t 1 t sut, ytdt, for a given u : [t, t 1 ] R m and resulting state xt 1. Since S a is given as the supremum over all u it immediately follow that t1 S a xt S a xt 1 sut, ytdt t

4 4 and thus S a is a storage function, proving that the system Σ is dissipative to the supply rate s. Suppose conversely that Σ is dissipative. Then there exists S such that for all u which shows that Sx + Sx sup u,t proving finiteness of S a, as well as S a x Sx. sut, ytdt SxT sut, ytdt = S a x Remark IV.1. Note that in linking dissipativity with the existence of the function S a, we have removed attention from the dissipation inequality, to existence of the solution to an optimization problem. Remark IV.2. The quantity S a can be interpreted as the maximal energy which can be extracted from the system Σ starting at an initial condition x. The function S a is therefore called the available storage. The above theorem states that Σ is dissipative if and only if this available maximally extractable energy is finite for every initial condition. Note that If the system is reachable from some initial condition x, then we only have to check this property for x. Lemma IV.1. Assume that Σ is reachable from x X. Then Σ is dissipative if and only if S a x <. Consider the dissipation inequality in the limit where t 2 t 1. Then it may be seen that satisfaction of the dissipation inequality is equivalent to fulfilling the partial differential equation Sxfx, u su, hx, u, x R n, u R m. This differential equation is called the differential dissipation inequality. V. HAMILTONIAN SYSTEMS A particular class of systems that is worthy of study are Hamiltonian systems. Here we consider three types: Hamiltonian systems, Mechanical Hamiltonian systems and generalized Hamiltonian systems, or Port Hamiltonian systems. Hamiltonian systems : q = H q, p p ṗ = H q, p + Bqu q y = B q H q, p. p In general, for mechanical systems the Hamiltonian is the system energy, and is given by the sum of the kinetic and potential energies of the system Hp, q = 1 2 p Mqp + P q Mq = M q >, P q >

5 5 Generalized Port Hamiltonian Systems ẋ = Jx H x + gxu y = g x H x where Jx = J x. Note that all such systems are strictly passive, meaning that the change in energy storage is equal to the energy transport at the system boundary, and the Hamiltonian gives a storage function Ḣ = u y. Including the effect of damping Rx = R x, the Port-Hamiltonian system becomes ẋ = Jx Rx H x + gxu y = g x H x The system thus satisfies a generalized energy conservation law For a Port-Hamiltonian system with damping Ḣ = u y dx Stored Energy = Supplied Energy + Dissipation Ḣ = u y H x Rx H x } {{ } dx The problem of designing a feedback controller for a Hamiltonian system can be considered as that of designing the energy function of the system. This gives rise to the idea of Passivity Based Control PBC. VI. STABILIZATION OF NONLINEAR SYSTEMS VIA DISSIPATIVITY The following Lemma establishes the link to Lyapunov stability theory Lemma VI.1. Let S be a continuously differentiable storage function for the system Σ and assume that the supply rate s satifies s, y, y R n. Let x = be a minimum of Sx. Then x = is local asymptotically stable for the unforced system u = and V x = Sx S is a local Lyapunov function. This holds for passive and L 2 -systems. More generally nonlinear systems of the form Σ : ẋ = fx + gxu y = hx + jxu are called locally zero state detectable ZSD if there exists a neighborhood D of O such that for all xt D ut =, hxt = t = lim xt =. t Respectively, zero state observable ZSO if it follows that xt = for all t >. This allows the first stabilization result

6 6 Theorem VI.1. Consider that the system Σ is passive and ZSD. Let φy be any smooth function such that φ = and y φy >, y. Let the storage function Sx > be well defined. Then the control law u = φy asymptotically stabilizes the equilibrium point x =. Furthermore, if the system is globally ZSD, then x = is globally asymptotically stable. Proof: From the dissipation equation it follows that Sxt must be strictly decreasing. Thus xt must converge to a local minimum. As Ṡ, hx. As the system is ZSD this implies xt converges to, globally if the system is globally ZSD. It follows that for all passive systems, the output feedback u = y will stabilize the system. Passive systems are thus very attractive from a stabilization point of view. Furthermore, the control problem becomes the problems of energy shaping, i.e. it becomes the problem of finding a controller that forms the energy of the system. Consider the system ẋ = fx + gxu Σ : y = hx The state feedback u SF : R n R m is said to be a Passivity Based Controller PBC if there exist functions H d : R n R and h d : R n R m such that u = u SF + v renders the closed loop system ẋ = f d x + gxv Σ d : = fx + gxu SF x + gxv y d = h d x passive with storage function H d x. That is, if it satisfies The new power balance is then where Ḣ d y d v. Ḣ d = y d v d d y d = u d = g H d d d = d x fx + gxu SF x Note that the energy, the dissipation and also the outputs have been modified according to the idea of the outputs and inputs being natural complements to each other. Theorem VI.2. The state feedback u SF : R n R m is a PBC for the system Σ with energy balance Ḣ = y u dx if and only if there exist functions H a : R n R and d a : R n R with d a x dx such that h xu SF x = H a x fx + gxu SF x d a x. An alternative approach is control by interconnection. Here we consider two subsystems, the plant Σ and controller Σ c which are connected by an interconnection Σ I, see Figure 3 The principle is to add the energies of the plant and controller to get the desired behavior. Example of control by interconnection: James Watts centrifugal governor, see Figure 4: The interconnection is power preserving if Simplest example: Classical feedback interconnection: u = u c y u + y c u c =. 1 1 y y c

7 7 Fig. 3. Fig. 4. James Watt s centrifugal governor. Theorem VI.3. An interconnection of passive systems Σ, Σ c, with storage functions Sx, resp. S c z and a power preserving interconnection is itself passive, with energy function Sx + S c z. Problem: Although the S c z is free, it is not clear how to affect the state x. A basic control by interconnection can be defined by a nonlinear integrator. Consider the strictly passive controller given by ż = u c Σ c : y = H c z z With the interconnection matrix u 1 y v = + 1 so that Ḣ + Ḣc v y. Then we get the following result. Theorem VI.4. Assume there exists a vector function C : R n R m such that H x J + R gx g C = x u c y c

8 8 Then for all functions φ : R n R m, defining the shaped energy storage function W : R n R m R by W x, z = Hx + H c z + φgx z the interconnected system is passive and satisfies Ẇ v y. For further approaches to dissipative and passivity based control see [1] [4]. REFERENCES [1] David Hill, Jun Zhoa, Robert Gregg, and Romeo Ortega. 2 years of Passivity based control: Theory and applications. CDC Workshop Shanghai, 29. [2] Hassan K Khalil and JW Grizzle. Nonlinear systems, volume 3. Prentice hall Upper Saddle River, 22. [3] Rogelio Lozano, B Maschke, B Brogliato, and O Egeland. Dissipative systems analysis and control: theory and applications. Springer- Verlag New York, Inc., 2. [4] Arjan van der Schaft. L2-Gain and Passivity in Nonlinear Control. Springer-Verlag New York, Inc., 1999.