Power based control of physical systems: two case studies

Transcription

1 Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 8 Power based control of physical systems: two case studies Eloísa García Canseco Dimitri Jeltsema Jacquelien MA Scherpen Romeo Ortega Faculty of Mathematics and Natural Sciences, ITM, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands ( egarciacanseco{jmascherpen}@rugnl) Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 68 CD Delft, The Netherlands ( djeltsema@tudelftnl) Laboratoire des Signaux et Systèmes, SUPELEC, Plateau de Moulon, 99, Gif-sur-Yvette, cedex, France (Tel: ; romeoortega@lsssupelecfr) Abstract: Itis well known that energy balancing control is stymied by the presence of pervasive dissipation To overcome this problem in electrical circuits, the alternative paradigm of power shaping control was introduced in (Ortega et al, 3) where, as suggested by its name, stabilization is achieved shaping a function akin to power instead of the energy function In a previous work (García-Canseco et al, 6) we have extended this technique to general nonlinear systems The method relies on the solution of a PDE, which identifies the open loop storage function Despite the intrinsic difficulty of solving PDEs, we show through some physical examples, that the power shaping methodology yields storage functions corresponding to the power of the system To motivate the application of this control technique beyond the realm of electrical circuits, we illustrate the procedure with two case studies: a micro electromechanical system and a two tank system Copyright c 8 IFAC Keywords: Passivity based control, nonlinear control, stability, nonlinear systems INTRODUCTION The main idea behind passivity based control, is to shape the open loop storage function of thesystem, suchthat the closed loop system energyfunction has a minimum at the desired equilibrium point Within the so called energy balancing control methodology (Ortega et al,, ; van der Schaft, ), the closed loop energy function is the difference between the total (open loop) energy function of the system and the energy supplied by the controller Hence the name energy balancing Unfortunately, as shown in (Ortega et al, ), energy balancing control is stymied by the existence of pervasive dissipation a term which refers to the existence of resistive elements whose power dissipation does not vanish at the desired equilibrium point This is indeed the case in regulation of mechanical systems where the extracted power is the product of force and velocity and we want to drive the velocity to zero Unfortunately, it is no longer the case for most electrical or electromechanical systems where power involves the product of voltages and currents and the latter may be nonzero for nonzero equilibria Several control methodologies have been developed to overcome the so called dissipation obstacle, such as interconnection and damping assignment passivity based control (IDA PBC) (Ortega et al, ), where the stabilization problem is accomplished by endowing the closed loop system with adesiredport Hamiltonian structure In (Maschke et al, ), the authors derive a constructive procedure to generate new storage functions for nonzero equilibria in the presence of pervasive dissipation, by modifying the interconnection structure of the closed loop for port Hamiltonian systems with constant input control Additionally, (Jeltsema et al, 4) propose an alternative definition of the supply energy for port Hamiltonian systems when the damping is pervasive, and the energy balancing property is obtained via a swap of the damping terms In (Ortega et al, 7), some extensions of the control by interconnection methodology have been recently introduced to circumvent the dissipation obstacle In this paper, we concentrate on the paradigm of power shaping control, as originally introduced in (Ortega et al, 3) to overcome the dissipation obstacle in nonlinear RLC circuits As suggested by its name, stabilization is achieved by shaping the power instead of the energy as is done in the aforementioned methodologies The present work is a sequel of our previous developments (García-Canseco et al, 6), where we have extended the power shaping methodology to general nonlinear systems, and we have applied it to the stabilization problem of the benchmark tunnel diode circuit To encourage the /8/$ 8 IFAC /876-5-KR-3873

2 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 application of power shaping control beyond the realm of electrical circuits, we present two case studies that include the set point regulation problem of a micro electromechanical system and a two tank system Notation: All vectors defined in the paper are column vectors, including the gradient of a scalar function that we denote by the operator = ( / x) Differentiation of functions with scalar arguments is denoted by ( ) POWER SHAPING CONTROL The main resultof (García-Canseco et al, 6), which we state without proof, is contained in the following proposition Proposition Consider the general nonlinear system ẋ = f(x) + g(x)u, (a) y = h(x), (b) where x R n, and u,y R m are the input and output vectors, respectively Assume A There exist a matrix Q : R n R n n, Q(x), that i) solves the partial differential equation (Q(x)f(x)) = [ (Q(x)f(x))], () ii) and verifies Q(x) + Q (x) A There exist a scalar function P a : R n R verifying iii) g (x)q (x) P a (x) =, where g (x) is a full rank left annihilator of g(x), and iv) x = arg minp d (x), where P d (x) := [Q(x)f(x)] dx + P a (x) (3) Under these conditions, the control law u = [ g (x)q (x)q(x)g(x) ] g (x)q (x) P a (x) (4) ensures x is a (locally) stable equilibrium with Lyapunov function P d (x) Assume, in addition, A3 x is an isolated minimum of P d (x) and the largest invariant set contained in the set {x R n P d (x) [ Q (x) + Q (x) ] P d (x) = } equals {x } Then, the equilibrium x is (locally) asymptotically stable and an estimate of its domain of attraction is given by the largest bounded level set {x R n P d (x) c} Remark Invoking Poincare s lemma, we observe that () is equivalent to the existence of a potential function P : R n R such that Q(x)f(x) = P(x) (5) Substituting () in (5) and taking into account the full rank property of Q(x) in A, we get Q(x)ẋ = P(x) + G(x)u, (6) where G(x) :=Q(x)g(x) In the context of RLC circuits, the form of (6) is due to Brayton and Moser (964), and That is, g (x)g(x) =, and rank(g (x)) = n m is precisely the starting point of power shaping control (Ortega et al, 3) In the same context, Q : R n R n n represents a full rank matrix containing the incremental inductance and capacitance matrices and P : R n R is the circuit s mixed potential function, which has units of power, see (Ortega et al, 3; Jeltsema, 5) for further details Apracticaladvantage ofthebrayton Moser equations is that they naturally describe the dynamics of the system in terms of easily measurable quantities, that is, the inductor currents and capacitor voltages, instead of fluxes and charges that are normally used as canonical coordinates in port-hamiltonian systems See for instance (Jeltsema and Scherpen, 4, 7b), where some results in power converters have been derived using this framework Remark Assumption A of Proposition involves the solution ofthepde () subject tothesign constraint ii) which may be difficult to satisfy In (Ortega et al, 3), a more constructive procedure is proposed to, starting from apair {Q,P } describingthe dynamics (6), explicitly generate alternative pairs { Q, P } that also describe the dynamics, ie, Q(x)ẋ = P(x) + G(x)u, (7) where G(x) = Q(x)g(x) For ease of reference in the sequel, we repeat here this result adapting the notation to the present context Proposition (Ortega et al, 3) Let Q(x) be an invertible matrix solution of () and define the full rank matrix [ Q(x) := [Q(x)f(x)]M(x) + [M(x)Q(x)f(x)] + λi ] Q(x), where λ R and M : R n R n n, with M = M, can be arbitrarily chosen Then, the system (6) is equivalently described by (7), with P(x) := λ [Q(x)f(x)] dx + f (x)q (x)m(x)q(x)f(x) Remark 3 Clearly, the power shaping stage of the procedure after transforming () into the form (6) coincides with theone proposedin (Ortegaet al, ) for energy shaping using interconnection and damping assignment passivity based control (IDA PBC) Additional remarks on the relation between these techniques may be found in (Jeltsema, 5; Blankenstein, 5) and in the recent work (Ortega et al, 7) Indeed, for port Hamiltonian (ph) systems (van der Schaft, ) ẋ = [J(x) R(x)] H(x) + g(x)u, (8a) y = g (x) H(x), (8b) with full rank matrix J(x) R(x), a trivial solution of () is obtained by setting Q(x) = [J(x) R(x)] However, in such case the associated potential function is not modified and remains the total stored energy instead of power as is desired 557

3 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 The purpose of the next two sections is to illustrate the application of the power shaping methodology of Proposition using two well-known examples 3 CASE STUDY I: A MICRO ELECTROMECHANICAL SYSTEM 3 The Model Consider the micro electromechanical system depicted in Figure The dynamical equations of motion are given by (Maithripala et al, 5) (see also (van der Schaft, )) ẋ = x m ẋ = k(x x nom ) b m x Aǫ (9a) (9b) ẋ 3 = x RAǫ + R u, (9c) where the state vector x = [x x ] consist of the air gap x (with x nom the nominal value or zero voltage gap), the momentum x, and the charge of the device The plate area, the mass of the plate and the permittivity in the gap are represented by A, m, and ǫ, respectively The spring and friction coefficients are given respectively by the positive constants k and b The electrical input resistance is denoted by R and u represents the input voltage which is the control action As pointed out in (Maithripala et al, 5), x is usually not available for measurement The assignable equilibria of the system are determined by x = [x x 3] where (x 3) = kaǫ(x x nom ) The corresponding constant control is given by 3 Controller Design u = x x 3 Aǫ The control objective is to stabilize any constant desired air gap position x Following the power shaping procedure outlined in Section, we have the following result u + R Fig Model of an electrostatic microactuator b m k x x nom Proposition 3 The dynamics of the micro electromechani- -cal system (9), in closed loop with the controller u = α (x x x 3) α ( x 3) + u, () has a locally asymptotically stable equilibrium point x with Lyapunov function P d (x) = kx m (x x nom ) + x x 3 Aǫm + bx m + x x 3 RA ǫ + λ ( k(x x nom ) + x m + x x ) 3 (u ) Aǫ α RAǫ + α (x x x 3 + α 3 ( x 3) ) u, RAǫ α () provided α satisfies and α > k (x 3 ) (b + λm) (x 3) + kaǫα 3 RA ǫ (x 3 ), Aǫ(α 3 + 3x x nom )α > 3x x nom α = α α 3, α 3 = λraǫ, λ {( k b, ) ( x RAǫ, )} Proof Observe that the micro electromechanical system (9) in ph form (8), with J R = b, g =, /R /R and the energy function H(x) = k (x x nom ) + x m + x x 3 Aǫ () Since the matrix J R is full rank, a trivial solution of the PDE () is given by b Q = [J R] = (3) R Hence, (9) can be written in the form (6) as Qẋ = P(x) + Gu, where G = [J R] g Although (3) is negative semidefinite, the potential function still equals the original energy function given in (), that is, P(x) = H(x), and not a power function as desired Toproceed withthepower shaping methodology, we apply Proposition to look for another pair { Q, P } that alternatively describes the dynamics (9), with P a power like function It turns out that with the choice λ > and λ {( k b, ) ( x RAǫ, )}, M = b /R equations (9) can be rewritten as (7), with k λb λ x 3 Aǫ Q(x) = λ x m 3 x, Aǫ Aǫ λr, 558

4 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 and G = RAǫ x RAǫ λ Notice that Q(x) is locally negative semi definite, ie, Q(x) + Q (x) for all x R n such that x x is sufficiently small Moreover, the new mixed potential function P(x) = kx m (x x nom ) + x x 3 Aǫm + bx m + x x 3 RA + λh(x) (4) ǫ is a power like function Indeed, since x /m =: v m (velocity of the mass), x 3/(Aǫ) =: f e (force of electrical origin), k(x x nom ) =: f k (force of the spring), x /(Aǫ) := u c (voltage across the capacitor), and /λ =: τ has units of seconds, thus (4) can be recast into P( ) = b v m + f k v m + f e v m + u c R + H τ Clearly, the first term represents the mechanical resistive content, whereas the second and third term exhibit the product force velocity, and the fourth term represents the electrical resistive co content The last term is elucidated recalling that energy per second equals power Furthermore, the selection [ ] x λraǫ + λx + λ G RAǫ x 3 (x) = yields that condition iii) of Proposition, G (x) P a =, becomes the following PDEs (x +λraǫ) P a x +(λx +λ RA ) P a x + P a =, P a x =, whose solution has the form P a (x) = Ψ( (x + λraǫ)) The function Ψ( ) must be chosen so that P d (x) = P(x)+ P a (x) satisfies the equilibrium assignment and stability conditions of Proposition, that is, P d (x) should verify P d (x ) = and P d (x ) A suitable selection of Ψ( ) is given by Ψ(z(x, )) = κ(z z ) + µ(z z ), where z = (x +α 3 ), z = x 3(x +α 3 ), and α 3 = λraǫ The equilibrium is assigned with µ = u RAǫ Now, the Hessian of P d (x) is calculated as P d (x) x=x = ( ) λk + κ + RA ǫ (x k 3 ) m k m x 3 (x + α 3)(κRA ǫ + ) x 3 RA ǫ maǫ x 3 (x + α 3)(κRA ǫ + ) RA ǫ b + λm x 3 m maǫ (x ) RA ǫ + λx Aǫ + κ(x + α 3) Some computations show that P d (x ) if and only if κ > satisfies the following inequalities κ > RAǫk (x 3 ) (b + λm) (x 3) + kaǫα 3 Aǫ(x 3 ), RA ǫ (α 3 + 3x x nom )κ > 3x x nom Finally, by using (4) and setting α = RAǫκ, α = α α 3, yields the control law () Notice further that controller () does not depend on the unmeasurable coordinate x 33 Simulation Results Figures, 3, and 4 depict the level curves of the function P d (), where for simplicity, the model parameters R, m, k, b, A and ǫ have been set to one, and x nom = 4 The gains were selected as λ =, α = 3 Observe that x = [ 63] is a local minimum point of P d () Figure 5 show the closed loop behavior of the air gap position to a step change in x from to The initial conditions are x() = [ 5] As can be seen, the control law () depictedin Figure 6, effectively stabilizes the system x x Fig Micro electromechanical system: level curves in the plane (x,x ) for = x 3 = 63 x x Fig 3 Micro electromechanical system: level curves in the plane (x, ) for x = x = 559

5 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, x 7 x3 6 λ 5 4 u pump x x Fig 4 Micro electromechanical system: level curves inthe plane (x, ) for x = x = x time [sec] Fig 5 Micro electromechanical system: Air gap x u Fig 7 Two tank system ẋ = a gx + a gx + γ u (5a) A A A ẋ = a gx + γ u, (5b) A A where the state variables x > and x > represent the water level in the lower and upper tank, respectively The system parameters are all positive constants, where g is the gravitational constant and, A i and a i, with i =,, are the cross sections of the tanks and the outlet holes, respectively The valve parameter is the constant γ [,], with γ = if the valve is fully open, ie, all the water is directed to the upper tank, and γ = if the valve is closed The assignable equilibrium points of the system are determined by x a = a (γ ) x, with the corresponding constant control u = a gx, γ for all γ [,), and by x =, u = a gx for γ = 4 Controller Design The control objective is to stabilize a given equilibrium point x = [x x ] Following the power-shaping procedure outlined in Section, we have the following result time [sec] Fig 6 Micro electromechanical system: control law u () 4 CASE STUDY II: TWO TANK SYSTEM 4 The Model Consider the two tank system depicted in Figure 7 Using Torricelli s law, the dynamics of the system can be written as (Johnsen and Allgower, 6) Proposition 4 Consider the two tank system (5) in closed loop with the linear state feedback controller u = k (x x ) k (x x ) + u (6) If the tuning parameters k and k satisfy k >, k > ( γ)a k, (7) 4A then x is a globally asymptotically stable equilibrium of the closed loop system with Lyapunov function P d (x) = a k g 3A + a k g (8) 3( γ)a + A [k (x x ) + k (x x ) u ] + (u ) A 56

6 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 Proof Fixing the matrix Q constant, ie, Q = {q ij }, with i,j =,, a suitable solution to the PDE () yields q <, q = A q, q =, q < (9) A Hence, Q is invertible and under Assumption A4 verifies Q + Q if and only if q > A q 4A To simplify the computations, let q = k, q = A k A ( γ), where k and k are positive constants Consequently, the condition to make the symmetric part of the matrix Q negative semi definite becomes (7) Moreover, the mixed potential function P(x) = a k g 3A + a k g 3( γ)a, () can be seen as a power like function Indeed, by Torricelli s law, we know that the terms gx and gx have the units of velocity, hence we define v := gx and v := gx Furthermore, by fixing the units of k and k to kg/s so that the terms k x =: f and k x =: f have units of force, and defining the unitless constants β = a a 3A, β = 3( γ)a, the mixed potential function () can be recast into P( ) = β f v + β f v, which clearly exhibits the products force velocity Furthermore, by choosing [ g = γ ] γ, A A condition iii) of Proposition becomes P a P a = () k x k x The solution of () yields P a (x) = Ψ( k k x + x ), where Ψ( ) is an arbitrary differentiable function that must be chosen so that P d (x) = P(x) + P a (x) has a minimum at x Computing P d (x) from (3), we obtain P d (x) = a k g 3A + a ( ) k g 3( γ)a +Ψ k x + x, k which should satisfy P d (x ) = and P d (x ) As in the previous example, one possibility is to select a quadratic function of the form Ψ(z(x)) = κ (z z ) + µ(z z ), where z = k k x + x, z = z(x ), κ >, and µ are scalars Some simple calculations show that the minimum is assigned, ie, P d (x ) =, if we set µ = ku A The Hessian P d is calculated as k a g + κk κk P d = A x k k κk k a g k A ( γ) + κ x, which is positive definite for all positive x Setting κ = k A yields the Lyapunov function (8), which has a unique minimum at x Finally, from (4) we obtain the simple linear state feedback (6), which asymptotically stabilizes the equilibrium point x, provided (7) holds Remark 4 The controller (6) was also derived using the IDA PBC methodology in (Johnsen and Allgower, 6) We refer to the aforementioned work for simulations and experimental results 5 CONCLUSIONS AND FUTURE WORK Two case studies illustrating the power shaping methodology of general nonlinear systems, as recently proposed by (García-Canseco et al, 6), are presented Among the issues that remain open and are currently being explored are the solvability of the PDE () for a general class of systems and other applications of power shaping, for instance, to mechanical systems Although some modeling issues based on the Brayton Moser equations have been considered in (Jeltsema and Scherpen, 7a), many control issues using this power based framework still remain open REFERENCES G Blankenstein Power balancing for a new class of nonlinear systems and stabilization of RLC circuits International Journal of Control, 78(3):59 7, 5 RK Brayton and JK Moser A theory of nonlinear networks-i Quart of App Math, : 33, April 964 E García-Canseco, R Ortega, J M A Scherpen, and D Jeltsema Power shaping control of nonlinear systems: A benchmark example In 3rd Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nagoya, Japan, July 9-6 D Jeltsema Modeling and control of nonlinear networks: a power based perspective PhD thesis, Delft University of Technology, The Netherlands, May 5 D Jeltsema and JMA Scherpen A power based description of standar mechanical systems Systems & Control Letters, 56(5): , May 7a D Jeltsema and JMA Scherpen Tuning of passivity preservingcontrollers forswitched mode power converters IEEE Trans on Automatic Control, 49(8): , August 4 D Jeltsema and JMA Scherpen A power based perspective inmodeling andcontrol ofswitchedpower converters (past and present) IEEE Industrial Electronics Magazine, ():7,54, 7b D Jeltsema, R Ortega, and JMA Scherpen An energy balancing perspective of interconnection and damping assignment control of nonlinear systems Automatica, 4(9): , 4 J Johnsen and F Allgower Interconnection and damping assignment passivity based control of a four tank system In 3rd IFACWorkshop on Lagrangian and Hamiltonian Methods in Nonlinear Control, Nagoya, Japan, July 9-, 6 D H S Maithripala, J M Berg, and W P Dayawansa Control of an electrostatic mems using static and dynamic output feedback ASME Journal of Dynamical Systems Measurement and Control, 7:443 45, 5 B Maschke, R Ortega, and AJ van der Schaft Energy based lyapunov functions for forced Hamiltonian sys- 56

7 7th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-, 8 tems withdissipation IEEE Trans on Automatic Control, 45(8):498 5, R Ortega, AJ van der Schaft, I Mareels, and B Maschke Putting energy back in control IEEE Control Systems Magazine, ():8 33, R Ortega, AJ van der Schaft, BM Maschke, and G Escobar Interconnection and damping assignment passivity based control of port controlled Hamiltonian systems Automatica, 38(4): , R Ortega, D Jeltsema, and JMA Scherpen Power shaping: a new paradigm for stabilization of nonlinear RLC circuits IEEE Trans on Automatic Control, 48 ():76 767, October 3 R Ortega, A van der Schaft, F Castaños, and A Astolfi Control by state modulated interconnection of port hamiltonian systems In 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 7), Pretoria, South Africa, August -4, 7 AJ van der Schaft L Gain and passivity techniques in nonlinear control Springer Verlag, 56