Offset Free Model Predictive Control

Transcription

1 Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec , 27 Offset Free Model Predictive Control Francesco Borrelli, Manfred Morari. Abstract This work addresses the problem of offset-free Model Predictive Control (MPC) when tracking a constant reference. It builds on the work in 1, 2 and summarizes in a compact and intuitive manner the conditions that need to be satisfied to obtain offset-free MPC by using the arguments of the internal model principle. I. INTRODUCTION The main concept of MPC is to use a model of the plant to predict the future evolution of the system 3 7. At each time step t a certain performance index is optimized over a sequence of future input moves subject to operating constraints. The first of such optimal moves is the control action applied to the plant at time t. At time t 1, a new optimization is solved over a shifted prediction horizon. In order to obtain offset-free control with MPC, the system model is augumented with a disturbance model which is used to estimate and predict the mismatch between measured and predicted outputs. The state and disturbance estimates are used to initialize the MPC control problem. The MPC control algorithms presented in 1, 2, 8 13 guarantee offsetfree control around a neighborhood of the steady-state and they differ in the MPC problem setup, the type of disturbance model used and the assumptions which guarantee offset-free control. This work addresses the problem of offset-free Model Predictive Control (MPC) when tracking a constant reference. It builds on the work in 1, 2 and summarizes in a compact and intuitive manner the conditions that need to be satisfied to obtain offset-free MPC by using the arguments of the internal model principle. We distinguish between z R p c controlled outputs, y m R p measured outputs and d R n d disturbances and provide (i) a simple proof of zero steady state offset when n d p p c, (ii) a simple algorithm for computing the space spanned by the offset when n d p c < p and (iii) insights on one/infinity norm, δu formulation and explicit MPC parameter storage. II. PROBLEM FORMULATION Consider the discrete-time time-invariant system { x(t 1) f(x(t),u(t)) y m (t) g(x(t)) Corresponding author. F. Borrelli is with the Dipartimento di Ingegneria, Università degli Studi del Sannio, 821 Benevento, Italy, francesco.borrelli@unisannio.it M. Morari is with the Automatic Control Laboratory, ETH, CH-892 Zurich, Switzerland, morari@control.ee.ethz.ch (1) with the constraints Ex(t) Lu(t) M (2) In (1) (2), x(t) R n, u(t) R m, and y m (t) R p are the state, input, and output vector respectively, The matrices E, L and M define, state, input and output constraints. The objective is to design an MPC controller 3, 4 based on a linear system model in order to have y m (t) track y ref, where y ref R p is an asymptotically constant reference signal. Moreover, we require zero steady-state tracking error, i.e., (y m (t) y ref ) for t. III. INPUT STEADY-STATE (s ) FORMULATION. A. The Plant Model Consider the following LTI system model { x(t 1) Ax(t) Bu(t) y(t) Cx(t), and assume that the pair (A,B) is controllable, and the pair (C,A) is observable. B. The Observer The plant model (3) is augmented with a disturbance model in order to capture the mismatch between (1) and (3) in steady state. Several disturbance model have been presented in the literature 1, 2, In this note we follow the paper 2 and use the form: x(t 1) Ax(t) Bu(t) B d d(t) d(t 1) d(t) y(t) Cx(t) C d d(t) with d(t) R n d. Later we will focus on specific versions of the model (4). The observer estimates both states and disturbances based on this augmented model. Conditions for the observability of (4) are given in the following proposition. Proposition 1: 2, 8, 14, 15 The augmented system (4) is observable if and only if (C,A) is observable and d full column rank. (5) C C d Proof: From the Hautus observability condition system (4) is observable iff A λi C B d I λi C d (3) (4) full row rank λ (6) Again from the Hautus condition, the first set of rows is linearly independent iff (C,A) is observable. The second set /7/$ IEEE. 1245

2 46th IEEE CDC, New Orleans, USA, Dec , 27 of rows is linearly independent from the first n rows except possibly for λ 1. Thus, for the augmented system the Hautus condition needs to be checked for λ 1 only, where it becomes (5). Remark 1: Note that for condition (5) to be satisfied the number of disturbances in d needs to be smaller or equal to the number of available measurements in y, n d p. Condition (5) can be nicely interpreted. It requires that the model of the disturbance effect on the output d y must not have a zero at (1,). Alternatively we can look at the steady state of system (4) d xs (7) C C d where we have denoted the steady state values with a subscript s and have omitted the forcing term u for simplicity. We note that from observability condition (5) for system (4) equation (7) is required to have a unique solution, which means, that we must be able to deduce a unique value for the disturbance d s from a measurement of y s in steady state. The state and disturbance estimator is designed based on the augmented model as follows: ˆx(t 1) ˆd(t 1) A Bd I K1 d s ˆx(t) ˆd(t) y s B u(t) ( y m (t) Cˆx(t) C d ˆd(t)) (8) where K 1 and are chosen so that the estimator is stable. Proposition 2: Choose n d p. The steady state of the observer (8) satisfies: C ˆxs y m,s C d ˆds where y m,s and are the steady state output and input of the system (1), ˆx s and ˆd s are state and disturbance estimates from the observer (8) at steady state, respectively. Proof: From (8) it follows ˆx(t 1) ˆd(t 1) A K1 C B d K 1 C d B C u(t) I C d K1 y m (t) ˆx(t) ˆd(t) (1) By design the observer (1) is asymptotically stable. Specifically it has no poles at (1,) and therefore (9) ( ) A I K1 C B d K 1 C d (11) C C d or ( ) ( ) I A I K1C B d K 1C d C C d (12) Condition (12) implies that is a nonsingular matrix. From (8) we note that the disturbance estimate ˆd converges only if ( y m,s Cˆx s C d ˆds )). As is nonsingular this implies that at steady state, the observer estimates (8) satisfy y m,s Cˆx s C d ˆds (13) 1246 where y m,s is the steady state output of the system (1) and ˆx s, ˆds are state and disturbance estimates from the observer (8) at steady state, respectively. Equation (9) follows directly from (13) and (8). Remark 2: The lower part of (9), i.e. (13), implies that the observer tracks the measurement without error in steady state y m,s Cˆx s C d ˆds C. The MPC design For offset-free tracking at steady state we want y m,s y ref. The observer condition (9) suggests that in steady state MPC should satisfy xs C y ref C d ˆds (14) where x s is the controller state in steady state. For to exist for any ˆd s and y ref the matrix must C be of full row rank which implies m p. The MPC is designed as follows min Ut N 1 k Q(y k y ref ) 2 R(u k,t,t ) 2 subj. to Ex k Lu k,t M, k,...,n 1 x k1 Ax k B d d k Bu k,t, k y k Cx k C d d k, k d k1 d k, x ˆx(t), d ˆd(t), with,t given by: xs,t C,t B d ˆd(t) y ref C d ˆd(t) (15) (16) where Mx 2 x Mx and Q,R,N selected for the nominal closed loop system to be stable. Let Ut {u,t,...,u N 1,t } be the optimal solution of (15)-(16) at time t. Then, the first sample of Ut is applied to system (1) u(t) u,t. (17) Denote by c (ˆx(t), ˆd(t),y ref ) u,t(ˆx(t), ˆd(t),y ref ) the control law when the estimated state and disturbance are ˆx(t) and ˆd(t), respectively, then the closed loop system obtained by controlling (1) with the MPC (15)-(16)-(17) and the observer (8) is: x(t 1) f(x(t),c (ˆx(t), ˆd(t),y ref )) ˆx(t 1) (A K 1 C)ˆx(t) (B d K 1 C d ) ˆd(t) Bc (ˆx(t), ˆd(t),y ref ) K 1 y m (t) ˆd(t 1) ( C)ˆx(t) (I C d ) ˆd(t) y m (t) (18) Theorem 1: Assume that the MPC problem (15)-(16) is feasible for all t and unconstrained for t j with j N. Assume that the closed-loop system (18) is stable. Then the measurement y m (t) y ref asymptotically. Proof: At steady state ˆx(t) ˆx s, ˆd(t) ˆds x s,t x s, u(t) c (ˆx s, ˆd s,y ref ) and,t,c. Note that the

3 46th IEEE CDC, New Orleans, USA, Dec , 27 steady state controller input (computed and implemented) might be different from the steady state desired input,c. The asymptotic values ˆx s, x s, and,c satisfy the observer conditions (9) C ˆxs y m,s C d ˆds and the controller requirement (16) xs B d ˆds C,c y ref C d ˆds (19) (2) Define δx ˆx s x s, δu,c, and the offset ǫ y m,s y ref. By subtracting (19) from (2) we obtain: (A I)δx Bδu Cδx ǫ (21) Next we prove that δx and thus ǫ. Consider the MPC problem (15)-(16) and the following change of variables δx k x k x s,t, δu k,t u k,t,t. Notice that y k y ref Cx k C d d k y ref Cδx k Cx s,t C d d k y ref Cδx k from condition (16) with d k ˆd(t). Similarly, one can show that δx k1 Aδx k Bδu k,t. Then, the MPC problem (15) becomes: min δu,t,...,δu N 1,t N 1 k Q(Cδx k ) 2 Rδu k,t 2 subj. to Eδx k Lδu k,t M 2, k N 1 δx k1 Aδx k Bδu k,t, k δx δx(t), δx(t) ˆx(t) x s,t. (22) Denote by K MPC the unconstrained MPC controller (22), i.e., δu k,t K MPC δx(t). At steady state δu k,t,c δu and δx(t) ˆx s x s δx. Therefore, at steady state, δu K MPC δx. From (21) (A I BK MPC )δx (23) which implies δx since (A IBK MPC ) is nonsingular (K MPC is stabilizing). Remark 3: Theorem 1 was proven in 2 by using a different approach. Remark 4: The proof of Theorem 1 assumes only that the models used for the control design (3) and the observer design (4) are identical in steady state in the sense that they give rise to the same relation y y(u,d). It does not make any assumptions about the behavior of the real plant (1), i.e. the model-plant mismatch, with the exception that the closed-loop system (18) must be stable. The models used in the controller and the observer could even be different as long as they satisfy the same steady state relation. Remark 5: If condition (16) does not specify,t uniquely, it is customary to ermine,t through an optimization problem, for example, minimizing the magnitude of,t subject to the constraint (16) 2. Remark 6: Changing the first term in the objective function of (15) to Q x (x k x s,t (y ref, ˆd(t))) 2 to penalize deviations from the target state rather than the target reference y ref, does not change the properties of the control law. Remark 7: If the 2-norm in the objective function of (15) is replaced with a 1 or norm ( N 1 k Q(y k y ref ) p R(u k,t,t ) p, p 1 or p ), then Theorem 1 continues to hold. In fact, the unconstrained MPC controlled K MPC (of the δ formulation in (22)) is piecewise linear around the origin 16. In particular, around the origin, u (k) K MPC (x(k)) is a continuous piecewise linear function of the state x: K MPC (x) F i x if H i x K i, i 1,...,N r, (24) where H i and K i in equation (24) are the matrices describing the i-th polyhedron CR i {x R n H i x K i } inside which the feedback optimal control law u (k) has the linear form F i x(k). The polyhedra CR i, i 1,...,N r are a partition of the set of feasible states of problem (15) and they all contain the origin. For Theorem 1 to hold, it is sufficient to require that all the linear feedback laws F i x(k) for i 1,...,N r are stabilizing. D. Special Disturbance Models The following proposition follows directly from Proposition 1. Proposition 3: The augmented system (4) with n d p and C d I is observable if and only if (C,A) is observable and d (A I B C I d C). (25) Remark 8: We note here clearly how the observability requirement restricts the choice of the disturbance model. If the plant has no integrators, then (A I) and we can choose B d. If the plant has integrators then B d has to be chosen specifically to make (A I B d C). We can understand the meaning of this requirement when we look at the steady state of system (4) d xs (26) C I where we have denoted the steady state values with a subscript s and have omitted the forcing term u for simplicity. We note that observability of system (4) requires equation (26) to have a unique solution, which means, that we must be able to deduce a unique value for the disturbance d s from a measurement of y s in steady state. Proposition 4: The augmented system (4) with n d p and C d is observable if and only if (C,A) is observable and d C A I B d C C(A I B d C) 1 B d. (27) d s y s 1247

4 46th IEEE CDC, New Orleans, USA, Dec , 27 IV. s FORMULATION: MEASURED OUTPUTS CONTROLLED OUTPUTS This section deals with the case when the number of measured variables is greater than the number of measured variables which one desires to track (next defined as controlled outputs ). We will distinguish between the dimension p of the measured outputs y m and the dimension p c of the controlled outputs z. The case when n d p > p c is identical to the previous section and guarantees zero steady state offset but requires a number of additional disturbances n d greater than the number of controlled outputs p c. Next we study the case when the number of additional disturbances n d is equal to the number of controller outputs, i.e., n d p c < p. System (3) is augmented with the controlled output equation z(t) Hy(t), (28) with H R pc p and then the MPC (15) is modified as follows min Ut N 1 k Q(z k y ref ) 2 R(u k,t,t ) 2 subj. to Ex k Lu k,t M, k,...,n 1 x k1 Ax k B d d k Bu k,t, k z k HCx k HC d d k, k d k1 d k, x ˆx(t), d ˆd(t), with y ref R pc and,t given by: xs,t B d ˆd(t) HC,t y ref HC d ˆd(t) (29) (3) We use the same observer (8) and, since n d p, Proposition 2 is replaced by the following proposition. Proposition 5: The steady state of the observer (8) satisfies: A I K1 C B C ˆxs K1 y m,s (B d K 1 C d ) ˆd s y m,s C d ˆds (31) where y m,s and are the steady state output and input of the system (1), ˆx s and ˆd s are state and disturbance estimates from the observer (8) at steady state, respectively. The controller (29)-(3) steady state satisfies: HC xs,c y ref HC d ˆds (32) where x s and,c are the controller desired state and input at steady state. Similarly to the proof of Theorem 1 it can be shown that at steady state, the MPC controller (29)-(3) satisfies:,c K MPC (ˆx s x s ), (33) where K MPC is the unconstrained MPC controller (29)-(3). By combining the equations (31), (32), (33) and using the offset equation ǫ Hy m y ref we obtain that observer and controller steady state values satisfy the following equation: with M 1 equal to 1248 M 1 ˆx s x s,c ˆd s y m,s I ǫ (34) A I K 1C B B d K 1C d K 1 C C d B d HC HC d K MPC I K MPC I (35) Rewrite (34) in a compact way as follows: M 1 v M 2 ǫ (36) with v ˆx s,, x s,,c, ˆds, y m,s. The set of all possible offsets ǫ belongs to the projection of the subspace K {(v,ǫ) M 1 v M 2 ǫ } on the offset space, this is denoted by Π ǫ (K). The following Algorithm 4.1 describes a standard procedure to ermine the dimension n d n d of the minimal subspace H Π ǫ (K) where the offset ǫ lies, and when n d < n d it finds the hyperplane equations defining H. Algorithm 4.1: Step 1. Let K {(v,ǫ) M 1 vm 2 ǫ } be the subspace containing v and ǫ; Step 2. if rank(m 1 ) rank(m 1 M 2 ) then Π ǫ (K) is full dimensional, i.e., H R n d ; Step 3. if rank(m 1 M 2 (:,1 : j 1) M 2 (:,j 1 : n d )) rank(m 1 M 2 ) for all j 1...,n d then we have zero steady state offset and H. Step 4. else let {u 1,...,u k } be a basis of the kernel of M 1 then H {ǫ Zǫ }, where Z u T 1... u T k M 2 We emphasize that the synthesis of MPC controllers with zero steady state offset is not an easy problem. In fact, H is a function of K 1,, H and K MPC. Remark 9: The case when the observer is chosen as ˆx(t 1) ˆd(t 1) K1 A Bd I ˆx(t) ˆd(t) B u(t) ( Hy m(t) HCˆx(t) HC d ˆd(t)) (37) falls in the class of systems studied in Section III-C. Therefore, the conditions presented in this section are relevant only if all the measurements y m are used for the observer design and p c < p. Remark 1: By defining ǫ s y m,s Cˆx s C d ˆds,

5 46th IEEE CDC, New Orleans, USA, Dec , 27 equation (34) can be rewritten as follows ˆx s N 1 x s,c ˆd s I ǫ (38) ǫ s with N 1 B d K 1 B d HC HC d K MPC I K MPC I (39) By using direct substitution, from equation (38) the following equation can be derived: H(I C(I A BK MPC) 1 (A BK MPC)) I ǫ s ǫ (4) Equation (4) is the main equation used in 2. Clearly Algorithm 4.1can be applied to equation (4) as well. V. DELTA INPUT (δu) FORMULATION. A. The Plant Model: x(t 1) Ax(t) Bu(t) u(t) u(t 1) δu(t) y(t) Cx(t) (41) System (41) is controllable if (A,B) is controllable. We use the same observer as before, but pose the MPC controller as follows. min δu,t,...,δu N 1,t Q(y k y ref ) 2 Rδu k,t 2 subj. to Ex k Lu k,t M, k,...,n 1 x k1 Ax k B d d k Bu k,t, k y k Cx k C d d k, k d k1 d k u k,t u k 1,t δu k,t, k u 1,t x ˆx(t), d ˆd(t) (42) where Q,R,N are selected for the nominal closed loop system to be stable. The advantage of this formulation is that we do not need to compute a target at each time step and that we penalize the optimization variable δu k,t directly rather than the deviation of u k,t from the target. The target depends on the the estimated disturbance ˆd s which captures many effects and may fluctuate wildly giving rise to erratic control performance. Therefore the δu formulation if often preferred in practice. A disadvantage of the proposed δu formulation is that it may give undesirable performance 1249 when there is an excess of manipulated variables in u. Specifically, in steady state condition (14) holds ˆxs B d ˆds (43) C y ref C d ˆds }{{} H When n m rank(h) then the null space of H defines a hyperplane in the space x(t),u(t) of dimension nm rank(h). Points of this hyperplane are system equilibria ((A I)x Bu ) which cannot be observed (Cx ). The MPC will converge to some non-unique point on this hyperplane. VI. EXPLICIT CONTROLLER In the last few years there has been growing interest to apply MPC to systems where the computational resources are insufficient to solve the optimization problem (15,16) on-line in real time. Methods have been developed 17 2 to solve (15), (16) explicitly to obtain a state feedback control law u(t) c (ˆx(t), ˆd(t),y ref ) in the form of a look-up table. The applicability of these methods is limited by the complexity of the control law c ( ) which is greatly affected by the number of parameters, e.g. elements in the vectors ˆx(t), ˆd(t) and y ref. Thus it is of interest to examine the proposed control formulations and disturbance models from this perspective. Examining (15), (16) we note that the control law depends on ˆx(t), ˆd(t) and yref, i.e. on nn d p n2p parameters as we have assumed n d p. This is surprising as we expect that we need to add only p integrators to a system to control p outputs without offset. In the special case that the system itself has no integrators we are allowed to choose B d in the disturbance model. Then the problem (15), (16) becomes N 1 min Ut k Q(Cx k (y ref C d ˆd(t))) 2 R(u k,t,t (y ref C d ˆd(t))) 2 subj. to Ex k Lu k,t M, k,...,n 1 x k1 Ax k Bu k,t, k x ˆx(t), (44) with,t given by: ˆxs,t C,t y ref C d ˆd(t) (45) where the control law now depends only on the n p parameters in ˆx(t) and (y ref C d ˆd(t)). Similarly, if B d and p c < p (see Section IV) one can choose n d p to obtain zero-steady state offset and do not pay any price in terms of MPC complexity. In fact, the MPC control law defined by (29)-(3) depends only on the np c parameters in ˆx(t) and (y ref HC d ˆd(t)) VII. ACKNOWLEDGEMENTS We would like to thank Franta Kraus and Urban Mäder for their helpful comments.

6 46th IEEE CDC, New Orleans, USA, Dec , 27 REFERENCES 1 G. Pannocchia, Robust disturbance modeling for model predictive control with application to multivariable ill-conditioned processes, J. Process Control, vol. 13, no. 8, pp , G. Pannocchia and J. B. Rawlings, Disturbance models for offset-free MPC control, AIChE Journal, vol. 49, no. 2, pp , C. Garcia, D. Prett, and M. Morari, Model predictive control: Theory and practice-a survey, Automatica, vol. 25, pp , D. Mayne, J. Rawlings, C. Rao, and P. Scokaert, Constrained model predictive control: Stability and optimality, Automatica, vol. 36, no. 6, pp , June 2. 5 M. Morari and J. Lee, Model predictive control: past, present and future, Computers & Chemical Engineering, vol. 23, no. 4 5, pp , D. Hrovat, MPC-based idle speed control for IC engine, in Proceedings FISITA 1996, Prague, CZ, F. Borrelli, A. Bemporad, M. Fodor, and D. Hrovat, An MPC/hybrid system approach to traction control, IEEE Trans. Control Systems Technology, vol. 14, no. 3, pp , May T. A. Badgwell and K. R. Muske, Disturbance model design for linear model predictive control, Proc. American Contr. Conf., pp , T. Meadowcroft, G. Stephanopoulos, and C. Brosilow, The Modular Multivariable Controller: 1: Steady-state properties, AIChE Journal, vol. 38, no. 8, pp , Y. Liu and C. B. Brosilow, Simulation of large scale dynamic systems I. modular integration methods, Journal of Computational Chemistry, vol. 11, no. 3, pp , E. Zafiriou and M. Morari, A general controller synthesis methodology based on the IMC structure and the H 2 -, H - and µ-optimal control theories, Journal of Computational Chemistry, vol. 12, no. 7, pp , J. M. Maciejowski, The implicit daisy-chaining property of constrained predictive control, Appl. Math. and Comp. Sci., vol. 8, no. 4, pp , L. Magni, G. D. Nicolao, and R. Scattolini, Output feedback and tracking of nonlinear systems with model predictive control, Automatica, vol. 37, no. 1, pp , M. Morari and G. Stephanopoulos, Minimizing unobservability in inferential control schemes, Int. J. Control, vol. 31, pp , , Studies in the synthesis of control structures for chemical processes; Part III: Optimal selection of secondary measurements within the framework of state estimation in the presence of persistent unknown disturbances, AIChE J., vol. 26, pp , A. Bemporad, F. Borrelli, and M. Morari, Model Predictive Control Based on Linear Programming - The Explicit Solution, IEEE Transactions on Automatic Control, vol. 47, no. 12, pp , Dec M. Kvasnica, P. Grieder, and M. Baotić, Multi-Parametric Toolbox (MPT), F. Borrelli, Constrained Optimal Control of Linear & Hybrid Systems. Springer Verlag, 23, vol A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos, The Explicit Linear Quadratic Regulator for Constrained Systems, Automatica, vol. 38, no. 1, pp. 3 2, Jan F. Borrelli, M. Baotic, A. Bemporad, and M. Morari, Dynamic programming for constrained optimal control of discrete-time hybrid systems, Automatica, vol. 41, pp , January