Passive control. Carles Batlle. II EURON/GEOPLEX Summer School on Modeling and Control of Complex Dynamical Systems Bertinoro, Italy, July

Transcription

1 Passive control theory I Carles Batlle II EURON/GEOPLEX Summer School on Modeling and Control of Complex Dynamical Systems Bertinoro, Italy, July

2 Contents of this lecture Change of paradigm in control: from signal based to energy based Fits well with the PHDS modeling approach Energy balance control Control as interconnection Casimir functions and the dissipation obstacle

3 References Ortega, R., A.van der Schaft, I. Mareels, and B. Maschke, Putting energy back in control. IEEE Control Systems Magazine 21, pp , 21. Ortega, R., A.van der Schaft, and B. Maschke, Interconnection and damping assignment passivity-based control of portcontrolled Hamiltonian systems. Automatica 38, pp , 22. Rodríguez, H., and R. Ortega, Stabilization of electromechanical systems via interconnection and damping assignment. Int. Journal of Robust and Nonlinear Control 13, pp , 23.

4 Energy based control Control problems have been approached traditionally adopting a signal-processing viewpoint. Very useful for linear time-invariant systems, where signals can be discriminated via filtering. For nonlinear systems, frequency mixing makes things harder: very involved computations very complex and energy comsuming controls are needed to suppress the large set of undesirable signals

5 Most of the problem stems from not using any information about the physical structure of the system. We have learnt from the previous lectures that energy plays an essential role in the description of physical systems. energy based control the controller should shape the energy of the system, and even change how energy flows inside the system.

6 Passivity and energy balance control Consider a system with states x R n,inputsu R m and outputs y R m The map u 7 y is passive if there exists a state function H(x), bounded from below, and a nonnegative function d(t) suchthat u T (s)y(s) ds {z } energy supplied to the system = H(x(t)) H(x()) + {z } d(t) {z} stored energy dissipated

7 k q λ p m F (t) u = F, y = v = p/m q = p m ṗ = F (t) kq λ p m u(s)y(s) ds = F (s)v(s) ds = µ ṗ(s)+kq(s)+ λ m p(s) p(s) m ds = µ 1 2m p2 (s)+ 1 2 kq2 (s) s=t s= + λ m 2 p 2 (s) ds H(x) = 1 2m p kq2 = H(x(t)) H(x()) + λ p 2 (s) ds m 2 d(s)

8 Consider an explicit PHDS ẋ = (J(x) R(x)) x H(x)+g(x)u y = g T (x) x H(x) J T = J R T = R u T y dτ = = = u T g T x H dτ = (gu) T x H dτ ẋt +( x H) T J +( x H) T R x H dτ τ H dτ + = H(x(t)) H(x()) + ( x H) T R x H dτ ( x H) T R x H dτ PHDS are passive for u 7 y and storage function H does not depend on the PHDS being explicit!

9 If x is a global minimum of H(x) andd(t) >, and we set u =,H(x(t)) will decrease in time and the system will reach x asymptotically. Therateofconvergencecanbeincreased if we actually extract energy from the system with u = K di y with K T di = K di > If d only, then invariant sets have to be examined and LaSalle theorem has to be invoked. However, the minimum of the energy of the system is not a very interesting point from an engineering perspective. Physics As engineers, we are not really concerned in knowing what Nature does, but rather in forcing Nature to do what we want.

10 Key idea of passivity based control (PBC) use feedback u(t) = β(x(t)) + v(t) so that the closed-loop system is again a passive system, with energy function H d, with respect to v 7 y, and such that H d has the global minimum at the desired point. If β T (x(s))y(s) ds = H a (x(t)) Why should this be a state function? then the closed loop system is passive (with input v) and has energy function H d (x) =H(x)+H a (x) prove that

11 If the preceding assumptions hold, then H d (x(t)) {z } closed-loop energy = H(x(t)) β T (x(s))y(s) ds {z } stored energy {z } supplied energy This is called the Energy Balance Equation (EBE) For ẋ = f + gu, y = h systems, the EBE is equivalent to the PDE β T (x)h(x) = µ T Ha x (x) (f(x)+g(x)β(x))

12 As an example, consider the RLC circuit + V L C R µ x ẋ = 2 /L x 1 /C x 2 R/L + µ q x = state φ u = V control y = i = φ L output µ V 1 The map V 7 i is passive with energy function H(x) = 1 2C x L x2 2 and dissipation d(t) = R t R L 2 φ 2 (s) ds If we set V =, the natural equilibria is (, ) For V = V,theforced equilibria are (x 1, ), with x 1 = CV

13 The PDE for energy balance control to be solved is x 2 L µ H a x1 x 1 C + R L x Ha 2 β(x) = x 2 x 2 L β(x) Do we really have to solve this? The natural energy already has a minimum at the desired forced equilibria x 2 = we only need to shape the energy with respect to x 1 we can take H a as a function of x 1 only β(x 1 )= H a x 1 (x 1 ) No equation at all: it defines the control!

14 The only remaining task is to choose H a so that H d has the minimum at (x 1, ). H a (x 1 )= 1 µ 1 x 2 1 2C a C + 1 x 1 C x 1 a C a is a design parameter to be tuned for performance It can be seen that the resulting H d has a minimum at (x 1, ) iff C a > C u = H a (x 1 )= x µ x 1 C a C + 1 x 1 C a

15 Consider now this slightly different circuit + V L R C ẋ 1 = 1 RC x L x 2, ẋ 2 = 1 C x 1 + V (t) Only the dissipation structure has changed, but the admissible equilibria are of the form x 1 = CV, x 2 = L R V The power delivered by the source, Vx 2 /L, is nonzero at any equilibrium point except for the trivial one thesourcehastoprovide an infinite amount of energy to keep any nontrivial equilibrium point this is the infamous dissipation obstacle, which will reappear later

16 Control as interconnection We would like to have a physical interpretation of the preceding ideas Plant and controller as two physical systems exchanging energy over a network u plant Σ y network u c y c The interconnection is power continuous if controller Σ c u T c (t)y c (t)+u T (t)y(t) = t

17 As an example, consider the standard feedback interconnection u plant Σ y y c controller Σ c u c u c = y u = y c It is clearly power continuous u c y c + uy = yy c +( y c )y =

18 Assume we have one of such interconnections, and add extra inputs u u + v, u c u c + v c v u y c plant Σ controller Σ c y u c v c Then Let Σ and Σ c have state variables x and ξ, and let the maps u 7 y and u c 7 y c be passive, with energy functions H(x) andh c (ξ), respectively. the map (v, v c ) 7 (y, y c ) is passive for the interconnected system, with energy function H d (x, ξ) =H(x)+H c (ξ). power continuous interconnection of passive systems yields passive systems

19 We have a passive system with state variables (x, ξ) and energy function H d (x, ξ) =H(x)+H c (ξ). We would like to get an energy function H d in terms of x only, so that we can set the minimum at the desired point. Inordertodothis,wemustrestrict thedynamicstoasubmanifold of the (x, ξ) space parameterized by x, sothat Ω K = {(x, ξ) ; ξ = F (x)+k} level constant à µ F ξ! T is dinamically invariant ẋ =. x ξ=f (x)+k ThiscanbeformulatedbetterinthePHDSframework

20 Casimir functions and the dissipation obstacle Suppose both the plant and the controller are PHDS Σ : Σ c : ½ ẋ = (J(x) R(x)) H x (x)+g(x)u y = g T (x) H x (x) ( ξ = (Jc (ξ) R c (ξ)) H c ξ (ξ)+g c(ξ)u c y c = gc T (ξ) H c ξ (ξ) For simplicity, take the standard feedback interconnection u = y c, u c = y µ ẋ ξ = µ J(x) R(x) g(x)g T c (ξ) g c (ξ)g T (x) J c (ξ) R c (ξ) µ x H d ξ H d with H d (x, ξ) =H(x)+H c (ξ)

21 Letuslookforinvariantmanifoldsoftheform Condition Ċ k =yields C K (x, ξ) =F (x) ξ + K ³ F x T I µ µ J R ggc T x H d g c g T J c R c ξ H d = since H d = H + H a and we want total freedom in choosing H a, we cannot count on the gradient of H d,so ³ F x T I µ J R ggc T g c g T J c R c = This is a PDE for F (x)

22 Functions C K (x, ξ) =F (x) ξ + K such that F satisfies the above PDE on C K = are called Casimir functions. They are invariants associated to the structure of the system (J, R, g, J c,r c,g c ), independently of the Hamiltonian function It can be shown that the PDE for F has solution iff, onc K =, 1. ( x F ) T J x F = J c 2. R x F = 3. R c = 4. ( x F ) T J = g c g T No dissipation in the variables on which the Casimir depends (itmustbeinvariant!) R x F = is the dissipation obstacle again.

23 If a Casimir can be found, the closed loop dynamics on C K is given by ẋ =(J(x) R(x)) H d x with H d (x) =H(x)+H c (F (x)+k) while the ξ evolve as their x parametrization dictates. No dissipation obstacle appears in mechanical systems, since dissipation only exists for velocities and these already have the energy minimum at the desired regulation point (v = ). The situation is different in other domains. For the first of the two simple RLC circuits considered previously, dissipation appears in a coordinate, x 2,which already has the minimum at the desired point. For the second one, the minimum of the energy has to be moved for both coordinates, and hence the dissipation obstacle is unavoidable.