Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007 FrB03.5 Offset-Free Reference Tracking for Predictive Controllers Urban Maeder, and Manfred Morari Automatic Control Laboratory, ETH Zurich, Physikstrasse 3, ETL K13.1, CH – 8092 Zurich, Switzerland maeder | morari @control.ee.ethz.ch Abstract— We discuss the offset-free reference tracking problem for linear constrained systems in the presence of disturbances and plant-model mismatch. Introducing disturbance models, we show how to construct a controller / observer combination, such that zero offset is achieved. Contrary to other approaches, the plant model is augmented only by as many disturbance states as there are tracked variables, thus yielding the controller with minimal complexity according to the internal model principle. when explicit Model Predictive Control is used [2], where the complexity increases quickly with the number of state variables. II. P RELIMINARIES Consider the feedback system in Figure 1. Let the nominal model be defined by x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) I. I NTRODUCTION In Model Predictive Control (MPC), a model of the plant is used to predict the future evolution of the system state [6], [7]. At each time step an optimization problem is solved over the sequence of future input moves, possibly subject to constraints. The first optimal control move obtained by the optimization is then applied to the plant. At the next time step, the prediction horizon of the optimization problem is shifted forward in time and the procedure is repeated. By the internal model principle [4], there have to be as many integrators in the controller as there are variables for offset-free control. Most traditional methods for LQG-style state-feedback controllers suggest adding the integral of the tracking error as additional state variable [3], [5]. Such a scheme is inherently offset-free, but limited in the choice of the disturbance model. In a constrained setup, these methods suffer from windup effects [1], [8]. While MPC directly considers constraints, applying these methods to MPC has some disadvantages. For instance, the choice of an invariant terminal set which guarantees feasibility is not clear. Some traditional methods do not allow freedom in designing the disturbance model, since they only consider the tracking error. For MPC, a two-degreeof-freedom method, where reference and output signals are considered independently, is therefore preferable. For these reasons, interest in disturbance model and observer based approaches has increased [10], [9]. Both teams of authors independently derived conditions for offset-free reference tracking. They both prove that if the plant is augmented with as many disturbance states as there are measured variables, there will be no offset. In the case where only a subset of the measured variables is actually to be controlled with zero offset, this method yields too complex models. By the internal model principle, it should be possible to add only as many disturbance states as there are outputs to control with zero offset. Adding more variables than necessary will lead to a unnecessarily complex optimization problem in MPC. This effect is particularly bothersome 1-4244-1498-9/07/$25.00 ©2007 IEEE. (1) where x(k) ∈ Rnx , u(k) ∈ Rnu and y(k) ∈ Rny . We assume (A, B) stabilizable and (C, A) detectable with C full row rank. The controlled variables are defined as z(k) = Hy(k) (2) with z(k) ∈ Rnz , H full row rank and nu ≥ nz . Input and states are constrained x(k) ∈ X , u(k) ∈ U, (3) where Y and U are convex sets with the origin in their interior. The goal is to asymptotically eliminate the control error given a constant reference signal r∞ , that is z(k) → r∞ , k → ∞, (4) in the presence of constant, non-decaying disturbances and plant-model mismatch, assuming stability of the closed loop. The receding-horizon optimal control problem with quadratic performance cost is considered ∗ JN (x(k)) := min u0 ,...,uN −1 N −1 X (uTi Rui + xTi Qxi ) i=0 + xTN QN xN s. t. xi ∈ X, ui−1 ∈ U, ∀ i ∈ {1, . . . , N }, xN ∈ T , xi+1 = Axi + Bui , R ≻ 0, Q 0, QN 0. (5a) (5b) (5c) (5d) (5e) The optimization problem is defined by the system model (A, B), the prediction horizon N , the cost matrices Q and R, the terminal weight QN , the terminal set T and the current state x0 . The problem is repeatedly solved at every time step k for x0 = x(k) the current the state vector. The optimal input u(k) = u∗o is applied to the plant, the rest of the optimal open-loop control sequence u∗1 . . . u∗N −1 is discarded. We note that the resulting control law is linear in some region 5252 46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 u y P H FrB03.5 Assume that both the system and controller converge to steady-state and the reference signal is given by r(k) = rss . If ∃E ∈ Rnx̄ ×nz and E full column rank such that z E T (Ak − I) T E Bk E T Br K r Fig. 1. Control System Structure containing the origin, where none of the constraints are active. The terminal weight and the terminal set can be chosen such that the nominal closed-loop system is guaranteed to be stable [6]. For example, if QN is the solution to the discrete algebraic Riccati equation 0 = Q + AT QN A − QN − (AT QN B)(B T QN B + R)−1 (B T QN A), (6) (7) If T is such that it is invariant and feasible under this local controller, then both closed-loop stability and feasibility hold. For reference tracking, it is clear that zero offset can only be achieved when the controller is locally unconstrained, around the requested set point. Thus, we will assume there exists a set T which contains the requested set point and which is rendered invariant by the linear controller Kx . Also, Kx must be feasible within T . The system state must enter the set T , otherwise the reference cannot be feasibly attained. In the following, it will be assumed the state enters T and the local control law is linear. The analysis of offset can thus be performed for the linear control law Kx . The state vector is assumed not to be directly measurable; Hence, a linear state observer is employed. x̂(k + 1|k) = (A − LC)x̂(k|k − 1) + Bu(k) + Ly(k). (8) The control signal is u(k) = −Kx x̂(k|k − 1). (9) Introducing the variables x̄(k) = x̂(k|k−1), Ak = A−LC − BK, Bk = L and Ck = −Kx , the controller dynamics are defined by x̄(k + 1) = u(k) = Ak x̄(k) + Bk y(k) + Br r(k) Ck x̄(k) + Dr r(k) (11a) = H, = −I, (11b) (11c) then there is zero offset in the controlled variable z(k), i.e. z(k) → zss = rss for k → ∞. Proof. Let uss , yss , x̄ss and rss be the signals at steady-state and the controller’s state vector, respectively. For the system to be in steady-state, (Ak − I)x̄ss + Bk yss + Br rss = 0 must hold. Left multiplying by E T yields E T Bk yss +E T Br rss = 0 and by (11b) and (11c) Hyss = rss and hence zss = rss . It is observed that E spans the left eigenspace of the integrator modes (λ = 1) of Ak . Hence, Ak must contain at least nz integrator modes for E to exist. Equations (11b) and (11c) impose conditions on the input directions of the integrators of K in the signals y(k) and r(k) [11]. III. D ISTURBANCE M ODEL then the local unconstrained control law is u(k) = −Kx x(k), Kx = (B T QN B + R)−1 (B T PN A). = 0, (10) with x̄(k) ∈ Rn̄x , u(k) ∈ Rnu and y(k) ∈ Rny We first state a fundamental, sufficient condition for zero offset. Theorem 1 (Zero Offset): Consider the feedback system shown in Figure 1, let the controller K be given by (10). To account for the disturbances and plant-model mismatch, we augment the plant model by a disturbance with integrator B A Bd , Bm = Am = , Cm = [C Cd ] , (12) 0 I 0 with Bd and Cd of appropriate dimension. This model assumes the disturbance is constant over time. This is a common assumption in practice, when offset-free control at steady-state is needed. The choice of the matrices Bd and Cd may be motivated by a real, physical disturbance acting on the system, but they may also be chosen freely, as design parameters. The augmented model (Cm , Am ) is assumed to be detectable. A. Equivalence of State- / Output Disturbance Models In the following, the equivalence of state and output disturbance models will be established. A similarity transform is introduced I −T12 T = . (13) 0 I Thus, the transformed system matrices are A (I − A)T12 − Bd −1 T Am T = 0 I Cm T −1 = [C (14) CT12 + Cd ] . It is easy to see that any disturbance model with Cd 6= 0 and Bd = 0 (output disturbance) can be transformed into a pure state disturbance model by solving CT12 = −Cd . (15) for T12 . Since C has full row rank, a solution always exists. 5253 46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Conversely, a state disturbance model (Bd 6= 0, Cd = 0) can be transformed into a pure output disturbance model by solving (I − A)T12 = Bd (16) for T12 . A solution exists iff span(Bd ) ⊆ span(I − A), which is trivially true for plants with no eigenvalues at 1. For simplicity, we will assume Cd = 0 from now on. Since output disturbance models can always be transformed to state disturbance models by similarity transform, this is completely general. The proofs and methods are also straightforward to extend to the case Cd 6= 0. IV. C ONTROLLER In this section, we develop a general method for tracking constant reference signals in the presence of constant disturbances. The approach is well known and essentially similar to [9] and [10]. The idea is to first compute the steady-state target xss (rss , dss ) and uss (rss , dss ) for a given disturbance and reference, and then devise a stabilizing controller of the form u(k) = uss − k(x̂(k|k − 1) − xss ). (17) The target variables xss and uss are determined by solving a steady-state equation I −A HC −B 0 xss uss = Bd 0 dss + 0 I A solution exists for any rss , dss if I − A −B rank = nx + nz . HC 0 rss . (18) (19) Assume nu ≥ nz . Then, the rank condition (19) can be stated equivalently in the frequency domain. Let G(z) = HC(zI − A)−1 B. = = Xd dss + Xr rss , Ud dss + Ur rss . (21a) (21b) The MPC problem (5) is modified ∗ JN (x(k)) := min u0 ,...,uN −1 N −1 X ((ui − uss )T R(ui − uss ) i=0 + (xi − xss )T Q(xi − xss ) (22a) T + (xN − xss ) QN (xN − xss ) s. t. xN ∈ T (xss , uss ), (22b) (5b), (5d), (5e). (22c) The terminal set T now depends on the target variables. If (22) is feasible, then the optimal control law is linear in x(k) in some region around the target. Let this unconstrained controller be given by u(k) = uss − Kx (x̂(k|k − 1) − xss ). Introducing the feedback gain matrices Kd = −Kx Xd − Ud , (23) Kr = −Kx Xr − Ur , (24) Equation (23) is rewritten as linear feedback law of the ˆ − augmented state estimate x̄(k) = [x̂(k|k − 1)T d(k|k T T 1) ] , where dss is replaced by the current estimate of the disturbance. Similarly, rss is replaced by the current reference value. The feedback law is u(k) = −[Kx Kd ]x̄(k) − Kr r(k). (25) This shows the degrees of freedom of the control structure in both the reference and control error. V. E STIMATOR A state estimator is constructed for the augmented model (Am , Cm ). The estimator gain matrix is Lx . (26) L= Ld The controller dynamics can be written as x̄(k + 1) = u(k) = Ak x̄(k) + Bk y(k) + Br r(k) Ck x̄(k) + Dr r(k) (27) Inserting the augmented model description (12), the control law (25) and the estimator gain (26) yields the controller dynamics A − BKx − Lx C Bd − BKd , Ak = −Ld C I BKr Lx (28) , , Br = Bk = 0 Ld Ck = −[Kx Kd ], Dr = −[Kr ]. A. Conditions (20) Then, (19) holds if and only if z = 1 is not a transmission zero for G(z), i.e. the steady-state gain matrix G(1) has full row rank [11]. Let the solution to (18) exist and be given by xss uss FrB03.5 For brevity, the following notation is introduced Φ = I − A + BKx , Ψ = Φ + Lx C. (29) Consider the nominal system (12) subject to the control law (25). The steady-state gains are easily computed and they are given by zss = HCΦ−1 (Bd − BKd )dss for the disturbance and zss = HCΦ−1 BKr rss for the reference. Nominal tracking (disturbances are measurable, no plantmodel mismatch) can thus be achieved by choosing Kd and Kr correctly. For offset-free control, the estimator has to be considered additionally. In the following, conditions are given for the matrices Kx , Kd , Kr , Lx and Ld such that (11a-c) hold for the dynamic system (28). These results are mostly equivalent to the conditions given in [9] and [10]. Lemma 1 (Conditions for zero offset): Consider the controller defined by (27) and (28). Assume the closed loop reaches steady state, the reference r(k) = rss constant and the following holds ker(Ld ) ⊆ ker H(I + CΦ−1 Lx ) (30a) HCΦ−1 BKr = I (30b) HCΦ−1 (Bd − BKd ) = 0. (30c) 5254 46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 Then, there is zero offset in the controlled variable, i.e. z(k) → zss = rss . Proof. Equation (30a) can be rewritten as ker(Ld C) ⊆ ker(HCΦ−1 Ψ). If it holds, there exist ∆ ∈ Rnz ×nd which chooses a linear combination of the rows of Ld C such that ∆Ld C = HCΦ−1 Ψ. (31) Consider E T = [−HCΦ−1 ∆] as candidate left eigenspace of Ak to the eigenvalue λ = 1. By using (30), we check if the conditions of Theorem 1 hold. i) Using (11a) yields E T (Ak − I) = HCΦ−1 Ψ − ∆Ld C − HCΦ−1 (Bd − BKd ) . The first element is zero by using (31), the second element vanishes by (30c), which proves the presence of the integrator modes. ii) Checking for the integrator’s input directions using (11b), we check if E T Bk = H and E T Br = −H. For the former, we get E T Bk = −HCΦ−1 Lx + ∆Ld , and (31) −HCΦ−1 Lx C + ∆Ld C −1 = HC FrB03.5 the Hautus observability condition for the system (Am − [LTx 0]T Cm , Ĉ) at λ = 1 Ĉ 0 rank I − A + Lx C −Bd < nx + nd , (35) 0 0 it is clear that detectability does not hold. Therefore, L̂d cannot stabilize all nd integrator modes, nor can Ld if it is rank deficient. Lemma 3 (Estimator): Consider the controller defined by (27) and (28). Assume A − Lx C is stable and nd = nz . Consider the estimator gain Lx + L̄x H̄ L̄ = (36) L̄d H̄ where H̄ = H(I + CΦ−1 Lx ). Assume L̄x and L̄d can be chosen such that Am − L̄Cm stable. Then, (30a) holds for the estimator gain L̄. Proof. Inserting (36) into (31) yields ∆L¯d H̄C = HCΦ−1 (I − A + BKx + (Lx + L̄x H̄)C). For the theorem to hold, ∆ must exist. Substituting H̄ yields −1 −HCΦ Lx C + HCΦ Ψ = HC HC Φ−1 (−Lx C + Ψ) = HC {z } | ∆L̄d H(I + CΦ−1 Lx )C HCΦ−1 Ψ + L̄x H(I + CΦ−1 Lx )C I For the reference direction, we immediately get E T Br = −HCΦ−1 BKr = −I by (30b). Remark. When Kd and Kr are computed as in (21) and (24), it is easy to check that (30b) and (30c) hold. The construction of the estimator gains Lx and Ld given Kx such that (30a) also holds will be discussed next. B. Properties In the following, the problem of constructing the estimator gains such that (30a) holds is discussed. In principle, given Kx , and an Lx which stabilizes A−Lx C, the set of candidate Ld is defined by ∆Ld = H(I + CΦ−1 Lx ). = (32) In order to derive a construction method for Ld , we will state some properties first. Lemma 2 (Rank of Ld ): Consider the the linear system defined by the matrices A Bd , Cm = [C 0] , (33) Am = 0 Ind ×nd where A ∈ Rnx ×nx , Cm ∈ Rny ×nx and Bd ∈ Rnx ×nd . Assume (Am , Cm ) detectable. Consider the class of estimator T gains L = LTx LTd such that Am − LCm is stable. Then rank(Ld ) = nd . (34) Proof. Choose an arbitrary Lx such that A− Lx Cm is stable. Suppose there exists a Ld such that rank(Ld ) < nd and Am −LCm is stable. Since rank(Ld C) < nd , we can choose Ĉ, L̂d such that L̂d Ĉ = Ld C and rank(Ĉ) < nd . Applying ∆L̄d HCΦ−1 Ψ = ∆L̄d = HCΦ−1 (I + L̄x HCΦ−1 )Ψ I + HCΦ−1 L̄x . From the last equation, it is immediately clear that ∆ exists, since L̄d has full rank by Lemma 2. C. Construction Method The previous sections suggest a direct construction method for L. Algorithm 1: Consider the linear system as in (1) and (12). Assume Cd = 0, (Cm , Am ) detectable and nd = nz . Suppose Kd and Kr are known and (30c), (30b) hold. 1) Compute Lx such that A − Lx C is stable and (H̄Cm , Ā) detectable, where H̄ = H(I + C(I − A + BKx )−1 Lx ) and Ā = Am − [LTx 0]T Cm . 2) Compute L̄ such that Ā − L̄H̄Cm is stable. 3) The final estimator gain Lx + L̄x H̄ L2 = (37) L̄d H̄ stabilizes Am −L2 Cm and - by Lemma 3 - (30a) holds. A better way to construct the estimator gain is described in the following, modified algorithm, which allows to move the poles associated to the integrator modes independently from the other modes. Algorithm 2: 1) Compute Lx such that A − Lx C is stable and (H̄Cm , Ā) detectable, where H̄ = H(I + C(I − A + BKx )−1 Lx ) and Ā = Am − [LTx 0]T Cm . 2) Apply the linear transform I −(I − A + Lx C)−1 Bd T = (38) 0 I 5255 46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 which brings the system to block-diagonal form A − Lx C 0 Āt = T ĀT −1 = (39) 0 I C̄t = H̄Cm T −1 (40) −1 = H̄ C C(I − A + Lx C) Bd . (41) 3) Compute L̄d such that (I − L̄d H̄C(I − A + Lx C)−1 Bd ) (42) is stable. 4) Recover the estimator gain for the original system 0 Lx −1 (43) +T L2 = L̄d H̄ 0 Algorithm 2 does not move the eigenvalues already established by A − Lx C. Hence, the integrator eigenvalues can be placed independently. D. Summary An algorithm was proposed for constructing the estimator such that offset-free control is achieved. The approach uses a minimal order disturbance model. For every controlled variable, one disturbance state is added, whereas existing methods suggest adding as many disturbance states as there are measured variables. In MPC, this improvement leads to simpler optimization problems and thus a better performance of the optimization algorithm. Assuming stabilizability and detectability of the system, necessary conditions for the existence of a tracking controller are that the plant has no transmission zero at 1, that is I − A −B rank = nx + nz , (44) HC 0 and that the chosen disturbance model is detectable. Algorithm 1 and 2 require detectability of (Ā, H̄C) for the construction of the estimator. It is an open question whether there always exists an estimator gain Lx for a given Kx such that detectability holds. Given the local linear control law Kx , the resulting estimator will remove offset of the closed loop in the controlled variables. Algorithm 2 describes a method to construct the estimator in two steps. The closedloop estimator poles for the disturbances can be moved independently of the poles of A − Lx C. VI. E XAMPLE As an example, we consider the double integrator 1 1 0.5 A= , B= , C = I. 0 1 1 (46) The actuator is constrained by |u| ≤ 0.2. The dynamics of the physical plant are slightly different and given by 1 1 Ar = . (48) −0.1 0.9 Furthermore, the system shall be subject to disturbances. The disturbance model is given by 1 Bd = . (49) .5 Solving the target problem for this disturbance model, we get Xd = [0 −3/4]T , Ud = −1/2, Xr = [1 0]T and Ur = 0. For the cost function, we choose Q = I, R = I, (47) (50) which yields the LQ feedback gains Kx = [0.435 1.03], Kd = 1.271 and Kr = −0.435. Closed loop performance is analyzed for two estimators, defined by the gain matrices L1 and L2 . L1 is the standard Kalman filter gain, while L2 is computed employing Algorithm 2. 1.26 0.87 L1 = 0.29 0.68 , 0.33 0.13 1.51 1.25 L2 = 0.38 0.83 . 0.47 0.37 (51) It is easy to check that L1 does not satisfy (30a), while L2 does. To account for the input constraint, we set up the MPC problem, slightly modifying (5) ∗ JN (x(k)) := min u0 ,...,uN −1 N −1 X (ui − ut )T R(ui − ut ) i=0 + (xi − xt )T Q(xi − xt ) + (xN − xt )T QN (xNp − xt ) s. t. |ui−1 | < 0.2, ∀ i ∈ {1, . . . , N }, xi+1 = Axi + Bui + Bd d0 (52) xt = Xd d0 + Xr r0 , ut = Ud d0 + Ur r0 . Qn is set to the solution to the DARE which was obtained while computing Kx . To close the control loop, we set x0 = ˆ x̂(k|k − 1), d0 = d(k|k − 1) and r0 = r(k) and solve (52) for every time step. The results are plotted in Figures 2 and 3 for a prediction horizon of N = 1. For the standard Kalman estimator gain L1 , the closed loop shows offset. The second plot shows no offset, but virtually the same performance. This suggests the proposed method does fit nicely into the classical framework of controller - observer design. VII. C ONCLUSION (45) The goal is to track x1 , that is H = [1 0]. FrB03.5 We have seen that robust reference tracking is not straightforward in Model Predictive Control. First we noted the classic integral control schemes proposed in the literature [3], [5] cannot be used for MPC straightforwardly. A twodegree-of-freedom approach employing a disturbance model which may reflect knowledge of the physical structure of the system is better suited. 5256 46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 FrB03.5 x1(k) 2 x (k) 2 x (k) 2 1.5 1 x (k) 2 1.5 r(k) 1 1 0.5 0.5 0 r(k) 0 0 5 10 15 20 time 25 30 35 40 0 1 5 10 15 20 time 25 30 35 40 1 u(k) u(k) 0.5 0.5 0 0 0 5 10 15 20 time 25 30 35 1 40 0 x (k) 5 10 15 20 time 25 30 35 x (k) 1 1 1 x (k) x (k) 2 0 40 2 0 d(k) −1 d(k) −1 −2 0 5 10 15 20 time 25 30 35 1 −2 40 u(k) 0 5 10 15 20 time 25 30 35 40 u(k) 1 0.8 0.6 0.5 0.4 0.2 0 0 0 5 10 15 20 time 25 30 35 40 0 Fig. 2. Reference step (top plots) and disturbance steps (bottom plots) for L1 estimator 5 10 15 20 time 25 30 35 40 Fig. 3. Reference step (top plots) and disturbance steps (bottom plots) for L2 estimator R EFERENCES However, the disturbance modeling approach is difficult: The integrators which are required for robustness do not appear automatically in the controller; they are tightly coupled to the observer used. The solution proposed in the literature was to add as many disturbance states to the model as there are measured variables. This may lead to an unnecessary increase in the dimension of the state space. In this paper, we investigated the nature of the problem further. We showed that the number of disturbance states can in fact be reduced to the number of controlled variables. The construction method for the observer gain starts with a linear state-feedback controller and then constructs an observer gain which yields zero offset. It is simple and based on the standard state-space algorithms. 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