Nonlinear Model Predictive Control for Periodic Systems using LMIs

Transcription

1 Marcus Reble Christoph Böhm Fran Allgöwer Nonlinear Model Predictive Control for Periodic Systems using LMIs Stuttgart, June 29 Institute for Systems Theory and Automatic Control (IST), University of Stuttgart, Germany, Pfaffenwaldring 9, 7 Stuttgart/ Germany {reble,cboehm,allgower}@ist.uni-stuttgart.de Abstract The problem of stabilizing constrained nonlinear discrete-time periodic systems using model predictive control (MPC) is considered in this paper. The results are based on a recently developed MPC scheme for linear periodic systems which is extended to the nonlinear case by using differential inclusion. Alternatively, it can be viewed as an extension of a well-nown robust MPC scheme exploiting the periodicity of the system and therefore reducing conservatism. At each time instant, a new periodic linear state feedbac law is obtained based on the repeated solution of a convex optimization problem involving a set of linear matrix inequalities (LMIs). The approach can therefore tae advantage of existing efficient and reliable algorithms for LMIs. A numerical example demonstrates the effectiveness of the proposed scheme. Keywords Model Predictive Control Periodic systems Convex optimization Preprint Series Issue No Stuttgart Research Centre for Simulation Technology (SRC SimTech) SimTech Cluster of Excellence Pfaffenwaldring 7a 769 Stuttgart publications@simtech.uni-stuttgart.de

2 2 M. Reble et al. 1 Introduction Periodic dynamics can be found in many real systems and have received significant attention in publications related to control over the last decades. Examples of periodic systems are given by compressors of jet engines 12, chemical processes, or helicopter rotors 2. Nontechnical systems with periodic dynamics are quite frequently encountered in economics 9 and biology. Various important results for linear periodic systems have been published. Output feedbac stabilization via a Riccati equation approach is addressed in 8. Conditions based on Linear Matrix Inequalities (LMIs) for state feedbac and output feedbac of unconstrained linear periodic systems are given in 11. The issue of robustness is considered in 4. MPC schemes for linear periodic systems are suggested in e.g. 1, 13. However, only few publications refer to nonlinear periodic systems. A recent example is the adaptive control scheme proposed in 1. In this paper, we propose a scheme for nonlinear periodic systems based on Model Predictive Control (MPC). MPC has become a popular control design method both in academic research 16 and for industrial application 17. The primary advantage is its capability to tae input and state constraints into account while in many cases its main disadvantage is the prohibitive computational demand of the required online solution of a finite horizon optimal control problem. Much effort has been made to find problem formulations that allow a reduction of the online computations. A combination of the recent results for linear periodic systems with well nown results for robust constrained nonlinear MPC based differential inclusion 14 allow the formulation of the design of an MPC controller for nonlinear periodic systems in terms of a convex optimization problem involving LMIs. Thus, efficient solvers for convex optimization problems can be applied and mae this approach computationally attractive 7. In principle, the approach in 14 is applicable to the system class considered in this paper. However, it is more conservative. Hence, feasibility problems might occur for cases in which our scheme provides a feasible controller as shown in this paper. The remainder of the paper is organized as follows: In Section 2, the considered control problem and preliminary results are presented. The main result, a stabilizing MPC controller design based on LMIs, is presented in Section 3. The effectiveness of the proposed scheme and advantages to other approaches such as 14 are illustrated via a numerical example in Section 4. Section gives a summary and concluding remars. 2 Problem Setup Consider the nonlinear discrete-time N-periodic system x +1 = f (x, u ) (1) with initial condition x = x. In the equation above, x R n denotes the state of the system, u R m the input variable, the time variable and f +N (x, u) = f (x, u) is a periodic function with time period N assumed to be continuously differentiable with respect to x and u. Furthermore, we assume the state of the system and the input to be bounded by constraints of the form c p x + d p u 1, p = 1,... M (2) at each time instant, in which c p R 1 n and d p R 1 m. Before we present our main result, namely a stabilizing MPC scheme using LMIs, we want to summarize some well-nown facts, which help with understanding the resulting equations. 2.1 Linear Differential Inclusion If the state and input of the system are assumed to be bounded for all time instants, there exist periodic matrices A (j) +N = A(j) and B (j) +N = B(j), j = 1,...,L, such that for all x and u satisfying the constraints (2) and all times it holds that { } Co A (1) B (1),...,A(L) B (L) (3) f x f u

3 Nonlinear Model Predictive Control for Periodic Systems using LMIs 3 in which Co{ } denotes the convex hull of a set. The original nonlinear system (1) is equivalent to a linear time-varying system whose state-space matrices A, B lie in the convex hull given in Equation (3), cf. 14, 1. The problem of stabilizing nonlinear periodic systems can therefore be replaced by finding a controller for the periodic linear differential inclusion (LDI) given by (3). A controller apparently stabilizes the original nonlinear system, if it stabilizes each possible realization of a linear time-varying system x +1 = A x + B u, (4) { } in which A B Co A (1) B (1),..., A(L) B (L) or formally with α (j) and L j=1 α (j) = 1. A B = L j=1 α (j) A(j) B (j) () It is quite obvious that some amount of conservatism is inherent in the approach presented since it essentially guarantees robustness with respect to a larger class of possible system dynamics. However, the formulation of the controller design as an LMI presented in the following maes the approach computationally attractive. 2.2 Model Predictive Control (MPC) In MPC, it is attempted to solve an optimal control problem repeatedly online at each time instant based on the prediction of the future behavior of the system. For this, a prediction model is used. Let x +i denote the prediction of the state at time + i calculated at time based on measurement of x = x. Similarly, u +i is the control input at time + i calculated at time. Using this notation, the objective function to be minimized at each time instant is J () = x T +i Qx +i + u T +i Ru +i (6) i= with Q R n n and R R m m each symmetric and positive definite. In both and 14, an upper bound on the objective function (6) is minimized instead of minimizing the objective function itself. The latter would require the calculation of inifinitely many control inputs or state feedbac matrices for a nonlinear and in general nonconvex problem and is therefore inappropriate for practical applications. Thus, in the following section the simplifications made are explained in more detail. 2.3 Upper Bound on Cost Function Following the lines of, a time-varying linear state feedbac law u +i = K +i x +i (7) is considered, in which K +i R m n. The state feedbac gain is N-periodic, i.e. K +i+n = K +i, and will be recalculated at each time instant. Note that, e.g. if necessary from a computational point of view, it is also possible to calculate K i only after one period N as in, cf. Figure 1. In that case during one period, i.e. rn, rn + N 1, r N, the feedbac gain is not recalculated as done in classical MPC, in which the control law is usually recomputed at each time instant. Thus, it is less computational demanding. However, the control scheme is still capable of attenuating disturbances because a feedbac control law is obtained rather than a sequence of open loop control inputs.

4 4 M. Reble et al. K K K +N +N K +N+1 +N+1 K +i K +N+i +N +1 +N 1 +N +N+1 Fig. 1 Recalculation at each sampling instant (top) and after each period (bottom). A quadratic form V (x) = x T Px can be found via LMI conditions such that J () V (x ), i.e. V is an upper bound on the optimal cost function, 14. The goal of the controller design is then to find a state feedbac law which minimizes that upper bound. If the optimization problem is initially feasible, it is feasible for all times and guarantees asymptotic stability in the sense of convergence to the origin. Note that two simplifications are made in calculating the upper bound V (x ) which introduce conservatism in the estimate of the infinite cost. First, the nonlinear system dynamics (1) are replaced by the LDI (4) along the lines of 14. This introduces some conservatism because the upper bound V is essentially an upper bound for all possible linear systems x +1 = A (j) x +B (j) u, j = 1,...,L, and the upper bound on the optimal cost function is assumed to be quadratic. Note that the optimal cost function is not necessarily quadratic due to the constraints. Second, only a finite number of feedbac matrices is calculated according to the period N by assuming a periodic linear state feedbac. This type of feedbac is obviously not capable of using the input to full capacity within the constraints as is shown in Section 4 via a numerical example. 2.4 Stability of Periodic systems The periodic Lyapunov lemma 3, 6, 11 gives a valuable result concerning stability of periodic systems. A linear periodic system x +1 = A x +B u with A +N = A K, B +N = B, is asymptotically stable for controller u = K x if and only if there exists an N-periodic matrix P +N = P > such that (A + B K ) T P +1 (A + B K ) P <, forall =,...,N 1. With W = P 1 > this is equivalent to the LMI condition W W A T. A W W +1 3 Main Result For stabilizing the nonlinear periodic system, we propose an MPC scheme which minimizes an upper bound on the cost (6) at each time instant. The resulting optimization problem is convex and can be rewritten in terms of LMIs. Therefore, the presented control scheme can be seen as a natural combination of the robust MPC scheme in 14 with the results for linear periodic systems based on Theorem 2 in. First we state conditions for the calculation of a periodic state feedbac matrix K +i+n = K +i and an upper bound on the cost function (6) using LMIs.

5 Nonlinear Model Predictive Control for Periodic Systems using LMIs Proposition 1 Assume there exist matrices Λ +i > and Γ +i and a positive scalar α such that the LMIs Λ +i (j),t +i Λ +i Q 1 2 Γ+i T R 1 2 (j) +i Λ +i+1, (8) Q 1 2 Λ +i α I R 1 2Γ +i α I 1 c p Λ +i + d p Γ +i (c p Λ +i + d p Γ +i ) T, (9) Λ +i Λ +N =Λ, (1) i =,...,N 1, j = 1,...,L, p = 1,...,M, with (j) +i = A(j) +i Λ +i + B (j) +i Γ +i are satisfied at time instant and the initial condition x lies in the ellipsoid given by x T P x α. Then x T P x with P +i = Λ 1 +i α is an upper bound on the cost function (6). Furthermore u +i = K +i x +i with N-periodic feedbac matrix K +i = Γ +i Λ 1 +i, (11) stabilizes the origin in the sense of convergence to the origin and the states and inputs satisfy the constraints (2) at all times. Proof The proof is divided in two parts. Upper bound on the cost function and convergence: Rewriting the terms in LMI (8) by using A, B, P and K and applying the Schur complement yields (A (j) +i +B(j) +i K +i ) T P +i+1 (A (j) +i +B(j) +i K +i ) P +i + Q + K T +i RK +i, (12) for i =,...,, j = 1,...,L. If the calculated periodic state feedbac matrix K +i is applied for all times i >, the positive definiteness of matrices Q and R implies x T +i (A (j) +i +B(j) +i K +i ) T P +i+1 (A (j) +i +B(j) +i K +i )x +i < x T +i P +i x +i. Using () and some simple algebra, one obtains x T +i+1 P +i+1 x +i+1 < x T +i P +i x +i (13) for each possible system of form (4). Since P +N = P, i.e. the matrix P is periodic with period N, the periodic Lyapunov lemma 3, 6 directly shows asymptotic stability for those closed-loop systems. Furthermore, Equation (3) then immediately guarantees stability of the nonlinear system (1) when applying the calculated periodic control for all times. Note that stability of the receding horizon strategy, in which the control law is recalculated at each time instant, is not guaranteed by this argument and will be shown in Theorem 2. Summing (12) from i = to i and noting that x +i for i yields x T P x x T +i (Q + KT +i RK +i )x +i i= J (). (14) This shows that x T P x is an upper bound on the cost function (6). Constraints: The initial condition is assumed to satisfy x T P x α. This and (13) yield Inequality (9) ensures that all ellipsoids x T +i P +i x +i α, i >. (1) E,i = {x : x T P +i x α } lie entirely in the constraint set defined by (2). Further details can be found in, e.g.,, 7.

6 6 M. Reble et al. Note that in Proposition 1 a static feedbac was considered in the sense that the periodic feedbac matrices K +i are only calculated once. In principle, the controller in Proposition 1 is suitable for the stabilization of the considered class of systems. However, a better performance can be achieved if the matrices K +i are recalculated online. In particular, the input can be better used to full capacity within the constraints. Thus, we propose an MPC scheme which minimizes the upper bound on the infinite horizon at each time instant. Our main result is presented in the following theorem. Theorem 2 The model predictive controller for the nonlinear periodic system (1) which minimizes the upper bound x T P x on the cost (6) calculated at time for measured state x is given by u +i = K +i +i x +i with K +i = Γ +i Λ 1 +i, (16) in which Λ +i > and Γ +i are solutions of the following optimization problem min α,λ +i,γ +i α, (17) subject to Λ +i (j),t +i Λ +i Q 1 2 Γ+i T R 1 2 (j) +i Λ +i+1 Q 1 2 Λ+i α I R 1 2Γ +i α I 1 x T, (18) x Λ, (19) 1 c p Λ +i + d p Γ +i (c p Λ +i + d p Γ +i ) T, (2) Λ +i Λ +N =Λ, (21) i =,...,N 1, j = 1,...,L, p = 1,...,M, in which (j) +i = A(j) +i Λ +i + B (j) +i Γ +i and P +i = Λ 1 +i α. The optimization problem is feasible for all time instants if it is initially feasible for =. The states and inputs satisfy the constraints (2) at all times and the closed loop is asymptotically stable in the sense of convergence to the origin. Proof The proof is divided in three parts. Minimization of an upper bound: In Proposition 1 it was shown that x T P x is an upper bound on the cost function (6). LMI (18) can be rewritten using the Schur complement as x T P x α. This and (17) show that an upper bound x T P x of the cost function J in (6) is minimized at each recalculation time. Feasibility: Suppose there is a feasible solution to the minimization problem at time. Then this solution is also a feasible solution at time + 1, but maybe not optimal. In order to see this note that neither (19) nor (2) depend explicitly on the initial state of the system and are therefore also satisfied at time +1. Only inequality (18) depends on the state and is equivalent to x T P x α. Then (13) guarantees x T +1 P +1 x +1 < x T P x α (22) for each possible system of form (4). We assume no mismatch between model and plant, in the sense that (3) holds, i. e. the nonlinear system can be represented by some linear time-varying system with state-space matrices in the convex hull (3). Hence the actual state x +1 at the next time instant will clearly satisfy (22), i. e. x T +1 P +1 x +1 α.

7 Nonlinear Model Predictive Control for Periodic Systems using LMIs x1 x2 u Fig. 2 Simulation results of the example system controlled by the proposed controller (blac lines) and by the periodic feedbac matrix calculated at time instant = (gray lines) for u max = 6. 6 Thus, (18) is satisfied for state x +1 and Λ +1. Hence, the in the previous time step calculated state feedbac matrix K +1 is also a feasible state feedbac matrix at time + 1. By induction, it follows that the optimization problem is feasible for all time instants if it is initially feasible. Stability: In the last part, we show that the MPC scheme, in which the state feedbac is recalculated at each time instant, is asymptotically stable. It was shown that previously calculated solutions are feasible, but not necessarily optimal, thus it follows from (22) x T +1 P +1 +1x +1 x T +1 P +1 x +1 (23a) < x T P x α. (23b) The matrix P +i +i does not explicitly depend on the absolute time +i, but only depends on x +i and mod( + i, N). Hence, we can write P +i +i = P(x +i, + i) with P(x, + N) = P(x, ) for all x and. The periodic nonlinear Lyapunov function V (x) = x T P(x, )x has period N, i.e. V +N (x) = V (x). Furthermore, it is positive definite because a positive definite P +r +r exists for all r N. Note that in general, V cannot be written in a explicit closed form since P is the solution of the convex optimization problem given in the theorem. From (23), it follows V +i+n (x +i+n ) V +i (x +i ) < x +i Qx +i, i =,...,N 1. This guarantees asymptotic stability in the sense of convergence to the origin via the periodic Lyapunov lemma 3, 6 for each possible system of form (4). This and differential inclusion (3) directly imply the stability of the nonlinear system (1), which concludes the proof. Note that the presented scheme is similar to the approach presented by Kothare in 14. The scheme presented therein is in principle suitable for stabilizing periodic systems as in the problem formulation. However, the controller would have to be robust with respect to all possible dynamics occuring within one period. This obviously maes this approach very conservative. The most crucial innovation in our scheme is that a periodic feedbac matrix K +i is considered. In particular, we want to direct the reader s attention to the fact that two distinct terms Λ +i and Λ +i+1 appear in Equation (19). Note that each solution of LMIs as in 14 also solves the LMIs in Theorem 2 with the additional constraint of constant matrices P and Λ as opposed to periodic matrices in this wor. Due to the explicit consideration of the periodic dynamics as in Theorem 2, it is possible to significantly reduce conservatism. As explained before, in principle it is also possible to recalculate the feedbac matrix K only after a period, cf. Figure 1. This decreases the computational demand and can still guarantee a certain amount of disturbance rejection due to the calculation of a feedbac matrix as opposed to an open loop input. However, recalculation after each time instant reduces conservatism and improves controller performance.

8 8 M. Reble et al x1 x2 u Fig. 3 Simulation results of the example system controlled by the proposed controller (blac lines) and by a robust controller which does not tae into account the periodic dynamics of the system (gray lines) for u max = x1 x2 u Fig. 4 Simulation results of the example system controlled by the proposed controller updated at each time step (blac lines) and updated only after each period (gray lines) for u max = Simulation Results An exemplary open-loop unstable nonlinear periodic system with period N = 3 is considered in this section. It is given by the difference equation x +1 = f (x, u ) = G x + ϕ(x,2 ) + H u (24) in which the matrices G and H.9.4 G = G 2 = , G 1 =.6.6, H =, cos( 2π 3 ), and the nonlinearity ϕ(x,2 ) = sin( π 2 x,2) T. The initial condition is chosen as x = 1 1 T. The derivative of the nonlinearity ϕ can be bounded, therefore it is possible to find matrices A (j) and B (j) satisfying (3). The resulting matrices are necessary

9 Nonlinear Model Predictive Control for Periodic Systems using LMIs 9 for the formulation of LMI (19), and are given by A (1) = A (1) 1 = A (1) 2 =.9.4, A (2) , A (1) = 1 =, A (2) 2 =.9.4, , , and B (j) = H, j = 1, 2, respectively. The weighting matrices for the MPC scheme are chosen as Q = 1 1 and R = 1. (2) Furthermore, the state is constrained by x 2 x 2,max = 1 and the input constraint u u max has to hold for all time instants. As was seen in the proof of Theorem 2, the constraints are guaranteed to be satisfied by the proposed scheme, if it is feasible, due to LMI (2). 4.1 Comparison to a static periodic controller First, we consider the proposed predictive control scheme approach for u max = 6. It is compared to a controller using the first periodic feedbac calculated at = without any recalculation, i.e. the sequence K, K 1, K 2, K 3 = K,... is applied to the system as considered in Proposition 1. This feedbac law also stabilizes the system as it was shown to be a feasible, but not necessarily optimal, solution of LMIs (18)-(2) for all times if initially feasible. This periodic feedbac is closely connected to the LMI-based controller proposed in 11 for linear discrete-time periodic systems. The state and input trajectories for both controllers are displayed in Figure 2. It is not surprising that the predictive control scheme shows a faster convergence since it can better use the input to full capacity within the constraints than a static periodic linear feedbac due to the recalculation at each time instant. The closer the states are to the origin, the more aggressive a feedbac gain can be without violating the input constraints. This demonstrates the advantages compared to a static controller as in Comparison to a robust MPC scheme Second, we want to compare the presented scheme to the robust model predictive controller 14 which does not explicitly consider the periodic dynamics of the system. This controller considers all possible A (j), for all = 1, 2, 3, j = 1, 2 at each time instant. Thus, the controller is robust with respect to the time-varying dynamics, which are considered as uncertainties. For u max = 6, the resulting LMIs have no feasible solution in contrast to the LMIs for the proposed scheme, which directly incorporates the periodic dynamics. It can be expected that for many periodic systems the approach using 14 leads to feasibility problems whereas which our scheme provides a suitable solution. The LMIs for such a robust controller are significantly more liely to be infeasible than those for the proposed scheme and that the result is much more conservative. If the constraint is relaxed to u max = 1, a robust controller can be calculated. Figure 3 shows the state trajectories and the input for both controllers. As expected, the proposed controller shows a better performance due to the less conservative formulation even in the case of significantly relaxed constraints. It should be stressed again that for tighter constraints the robust MPC scheme in 14 does not yield a stabilizing controller in contrast to the scheme proposed in this wor.

10 1 M. Reble et al. 4.3 Comparison to recalculation after one period Figure 4 compares the presented scheme with an update of feedbac matrix K after each time step to an update after only one period. The update at each time step leads to a slightly faster convergence and a better exploitation of the input constraints. Furthermore, it is clear that the disturbance rejection is better for a more frequent update. However, the total computation time for the simulation with update at each time step is 34.3 seconds, and for an update after each period it is 1.9 seconds. Conclusions In this paper a new model predictive control scheme using linear matrix inequalities for the stabilization of constrained nonlinear periodic time-varying systems is presented. The control law is calculated such that an upper bound on the infinite horizon cost function is minimized at each time instant. Major advantages are on the one hand computational attractivity of the formulation as a convex optimization problem, for which efficient solvers are available, on the other hand the explicit use of information about the periodic dynamics, which helps reducing the conservatism in comparison to existing schemes. Although a certain amount of conservatism is inherent in the proposed scheme due to the use of differential inclusion, the effectiveness compared to other possible control designs was demonstrated via a numerical example. References 1. K. Abidi and J.-X. Xu. A discrete-time periodic adaptive control approach for time-varying parameters with nown periodicity. IEEE Trans. Autom. Control, 3(2):7 81, P. Arcara, S. Bittanti, and M. Lovera. Periodic control of helicopter rotors for attenuation of vibrations in forward flight. IEEE Transactions on Control Systems Technology, 8(6), S. Bittanti, P. Bolzern, and P. Colaneri. Stability analysis of linear periodic systems via the Lyapunov equation. In Proc. 9th IFAC World Congress, Budapest, Hungary, pages , S. Bittanti and P. Colaneri. Stabilization of periodic systems: overview and advances. In Proc. Int. Worshop on Periodic Control Systems, pages , Como, Italy, 21.. C. Böhm, T. Raff, M. Reble, and F. Allgöwer. LMI-based model predictive control for linear discrete-time periodic systems. In International Worshop on Assessment and Future Directions of Nonlinear Model Predictive Control, Pavia, Italy, P. Bolzern and P. Colaneri. The periodic Lyapunov equation. SIAM Journal on Matrix Analysis and Applications, 9:499 12, S. Boyd, L. El Ghaoui, E. Feron, and V. Balarishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, USA, P. Colaneri, C. de Souza, and V. Kucera. Output stabilizability of periodic systems: necessary and sufficient conditions. In Proc. American Control Conference, volume, pages , Philadelphia, USA, T.P. de Lima. Studying a basic price equation as a periodic system. In Proc. Int. Worshop on Periodic Control Systems, pages 7 62, Como, Italy, G. De Nicolao. Cyclomonotonicity, Riccati equations and periodic receding horizon control. Automatica, 3(9): , C.E. De Souza and A. Trofino. An LMI approach to stabilization of linear discrete-time periodic systems. Int. J. Control, 73(8):696 73, P. Falugi, L. Giarré, L. Chisci, and G. Zappa. LPV predictive control of the stall and surge for jet engine. In Proc. Int. Worshop on Periodic Control Systems, pages 33 38, Como, Italy, K.B. Kim, J.W. Lee, and W.H. Kwon. Intervalwise receding horizon H tracing control for discrete linear periodic systems. IEEE Trans. Autom. Control, 4(4):747 72, M.V. Kothare, V. Balarishnan, and M. Morari. Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(1): , R.W. Liu. Convergent systems. IEEE Trans. Autom. Control, 13(4): , D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scoaert. Constrained model predictive control: stability and optimality. Automatica, 26(6): , 2.

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