Practical Robust Linear Model Predictive Control for Offset-free Output Regulation Yang Su, [email protected] Abstract For a variety of control problems in process industry and others, the control objective is to achieve offset-free regulation and sound robustness/disturbance rejection. Model predictive control(MPC) is often adopted as a general and powerful solution. Meanwhile in academic, MPC has been extensively investigated. Various algorithms, focused on different aspects such as stability, feasibility, robustness, reference tracking, etc, have been proposed. However, it is difficult to apply some theoretical results into practical problems with offsetfree requirement. In this paper, MPC based on the error integration approach[1], explicitly including an error integrator into the model, is revisited. It is shown that it is straightforward to apply various MPC algorithms into such scheme. In particular, a H∞ γ-suboptimal MPC is presented as an example. Issues, including robustness, transient response and setpoint feasibility, are discussed. The purpose of this paper is to illustrate the potential of error integrator approach, as an alternate of the commonly used disturbance estimation approach[2], for practical robust MPC design.

Keywords: Model predictive control; H∞ γ-suboptimal MPC; Offset-free tracking; Integrator.

1

Introduction

In process control industry, offset free output regulation is the basic requirement for controller design. For small scale problems, such as single input single output systems, PID controllers and lead/lag controllers are popular choices with the integral control is used for steady error elimination. For large 1

scale problems or systems requiring higher level control performance, Model Predictive Control(MPC), as the most promising advanced control methodology, is widely adopted due to its optimality and generality. As for the general controller design, there are only a few methods to design offset-free MPC, including the disturbance estimation approach[2] as the most popular one, the ∆u formulation[3], and the error integration approach[1][4][5]. All these methods augment the original system models with some forms of ‘integrator’. In addition to the basic requirement, control challenges often include satisfactory transient response and/or good robustness/disturbance rejection. In fact, extensive efforts have been devoted to MPC on topics, such as closed loop stability, recursive feasibility and robustness against various uncertainties, under a simplified framework–the state feedback control for regulation problem. This is rational since offset-free tracking problem is often formulated as a regulation problem, regardless which method is adopted. However, although great advancement in MPC for regulation problem(MPC regulator), it is difficult to apply such approaches into practical offset-free MPC problem, if the disturbance estimation approach is adopted. As will be explained later, it is due to the fact that at each sampling instant k, new steady state xs and control us are calculated based on the instant estimation of disturbance d. Therefore there is no explicit system evolution prediction equation, where the difficulty of applying MPC regulator comes in. Correspondingly, there is little result in literature on robust offset-free MPC algorithms, even for linear systems[6]. Another method, error integral control, is a well acknowledged solution for removal of steady error. MPC, designed for composite model augmented with error integrator, are presented in [1][4][7][8]. Due to the error integrator in the feedback loop, the offset-free property is naturally assured if the system stabilizes at a steady state; otherwise the integrator will integrate the error to infinity. A nice property with such scheme is that steady state and control input for a given setpoint needs not to be recalculated at every sampling instant k, therefore allowing an analytical expression of future system prediction. This property enables the application of advanced MPC regulator approaches in the literature to practical offset-free MPC design. In particular, the robust MPC approach is emphasised in this paper. A H∞ γ-suboptimal MPC is presented, about which the author thinks it may connect the modern control theory and MPC, and provides a practical MPC for the task of robust regulation/disturbance rejection, which may be the pri2

mary control objective for certain applications. Other aspects, such as step response and setpoint feasibility, are also briefly discussed. The rest of the paper is organized as follow. A comparison of disturbance estimation approach and error integration approach is presented in section 2; an offset-free H∞ γ-suboptimal MPC control and the discussion on related aspects are presented in section 3; one simulation example is given in section 4; the paper concludes in section 5.

2

Offset-free MPC Algorithms Comparison

Here a comparison between disturbance estimation approach and error integration approach is presented. As presented in [2], the model used in disturbance estimation approach is given by    xk+1 = Axk + Buk + Bd dk dk+1 = dk (1)   yk = Cxk + Cd dk

The state and disturbance can be estimated via state observer and is denoted by xˆk and dˆk . The prediction of system evolution is given by   xk + Buk + Bd dˆk   xˆk+1 = Aˆ (2) dˆk+1 = dˆk    yˆ = C xˆ + C dˆ k k d k

Based on estimated disturbance dˆk , the steady state and control input xs,k , us,k , required for the given setpoint, are calculated. Then the control problem is converted to a regulation problem to drive the estimated state to xs,k . It is noted that the stabilizing MPC control can be used in this step, which can be used to ensure the feasibility and stability for the dynamic (2). However, at the next time k + 1, the new state and disturbance estimation xˆk+1 , dˆk+1 are obtained. It is often dˆk+1 6= dˆk and therefore xs,k 6= xs,k+1 , which means the MPC problem solved in the two steps are different. This fact undermines the MPC design properties, such as recursive feasibility and stability. Briefly speaking, it is difficult to predict the actual system evolution using disturbance estimation approach. It is even impossible for nonlinear MPC, in which there is no analytical expression for xs,k , us,k . Therefore, it is difficult 3

to apply various MPC regulator approaches to practical offset-free tracking scenario. Next, it is shown that there is no such problem in error integration approach. This approach simply augments the system model with new state variable that integrates the tracking error ek as ξk+1 = ξk + ek . Thereafter MPC is designed for the augmented model. It was shown that if the model satisfies certain conditions and the output converge to a steady value, it is offset-free. What is more important is that the system evolution prediction, including the uncertainty evolution, can be obtained analytically via recursive iteration of the composite system model. This advantage enables straightforward application of various MPC regulator approaches to offsetfree tracking problem. In particular, the application of robust MPC design is also possible. In the next section, a H∞ γ-suboptimal offset-free MPC is presented as a example, which may be considered as a connection between MPC and modern robust control theory. The model used in [1] is for robust regulation to setpoint r = 0. For non-zero r, the only difference is to regulate the state to an non-zero steady state. In addition, implementable algorithms for piecewise constant setpoint tracking are proposed in [9][10]. Therefore in the following part, the model adopted will include the non-zero setpoint r for completeness. However, it should be emphasized that the presented MPC is mainly for robust regulation and disturbance rejection, but not the transient response for step setpoint change. The reason will be explained in due course.

3

Offset-free H∞ γ-suboptimal MPC Design

The system is described as ( xk+1 = Axk + Buk + D1 wk yk = Cxk + D2 wk ,

(3)

where x ∈ Rn , u ∈ Rm , and y ∈ Rp are state, control input, output variables, respectively. The external disturbance variable w ∈ Rw is used to describe the model/plant mismatch. The system is controllable, observable and subjected to constraints as (x, u) ∈ C, w ∈ W, where C, W are polytopic sets. For simplicity, it is assumed that x is measurable and the tracking variable is y(a more general formulation is to control a linear transform Hy). Otherwise, x can be estimated by a state observer and the MPC is designed similarly. 4

3.1

MPC Algorithm

The model used for MPC design is given as follow: ( xk+1 = Axk + Buk + D1 wk ξk+1 = −Cxk + ξk + rk − D2 wk

(4)

where ξk is the error integrator that integrates the tracking error ek = rk −yk . Denote the composite state variable as zk = [xTk ξkT ]T and the compact model as

where A =



zk+1 = Azk + Buk + Rrk + Dwk ,        D1 0 B A 0 . and D = ,R = ,B = −D2 I 0 −C I

(5)

Assumption 1. [5]The open loop transfer matrix C(zI − A)−1 B posses no zeros at z = 1. The nominal steady state for a given rk is chosen as zs = [xTs 0T ] and us which satisfy xs = Axs + Bus , ys = Cxs = rk

(6)

Define the difference between the state/input and their targets as x˜k = xk −xs and u˜k = uk − us . Let z˜k = [˜ xTk ξkT ]T , whose dynamic is given by z˜k+1 = A˜ zk + B˜ uk + Dwk

(7)

Assumption 1 assures that (A, B) in (5) is controllable, therefore MPC can be designed to robustly stabilize z˜k as in [1]. In recognition of recent advancement in Robust MPC design, the H∞ γ-suboptimal MPC based on disturbance affine parameterization [11] is adopted as below. The predicted control is parameterized as u˜i|k = vi|k +

i−1 X

Mi,j wj|k , 0 ≤ i ≤ N − 1.

(8)

j=0

Denote M, v and w as  0 ... ...  M1,0 0 . ..  M= .. . . .. ..  . MN −1,0 . . . MN −1,N −2

0 0 .. . 0



    , v =   5

v0|k .. . vN −1|k





  , w = 

w0|k .. . wN −1|k



  . (9)

Define the variable of interests as ζi|k = Cζ z˜i|k + Dζ u˜i|k . The finite horizon cost function is given by JN =

N −1 X

T (|ζi|k |2 − γ 2 |wi|k |2 ) + z˜N ˜N |k |k P z

(10)

i=0

To achieve a prescribed level of disturbance rejection, the MPC is formulated as a min-max problem P(˜ zk , xs , us ): min max JN

(11)

(7 − 8), 0 ≤ i ≤ N − 1, (xi|k , ui|k ) = (˜ xi|k + xs , u˜i|k + us ) ∈ C, ∀wi|k ∈ W, 0 ≤ i ≤ N − 1, (˜ zN |k , xs , us ) ∈ Ω.

(12)

M,v

w

subject to:

As in standard stabilizing MPC design, a H∞ terminal control u˜ = K˜ z and T terminal cost function z˜ P z˜ are assumed, which can be obtained via linear robust control[12][13]. The terminal closed loop system is z˜+ = (A + BK)˜ z. The terminal region Ω, in which it is valid, can be chosen as the maximal robust positive invariant set[14] for the dynamic  + z + Dw,   z˜ = (A + BK)˜ + xs = xs (13)   + us = us

subject to the constraints

(˜ x + xs , K˜ z + us ) ∈ C, w ∈ W.

(14)

The solution of P(˜ zk , xs , us ) is denoted as M∗ and v∗ . The control input is ∗ calculated as uk = us + v0|k . Remark 1. It is easy to see that there is redundancy for defining the steady state and input in (13) due to the coupling relation (6). It is possible to parameterize xs , us as xs = Mx θ and us = Mu θ, and then θ can be used instead in MPC formulation[9]. It provides a compact design. Here xs , us are adopted for presentation simplicity. 6

Theorem 1. Let Assumption 1 hold and the optimization problem P(˜ zk , xs , us ) is feasible at k = 0. The MPC is recursive feasible for constant setpoint and the response satisfies ∞ X

2

|ζk | ≤ β(˜ z0 ) + γ

k=0

2

∞ X

|wk |2 .

(15)

k=0

If the output variable converges to a steady value, there is no offset error. Proof. Since the MPC tracking problem is completely converted to a state regulation problem, the proofs in [11] and [1] for recursive feasibility, offsetfree property and stability are straightforward applicable to this case and thus omitted here. Remark 2. The adopted model for offset free design, based on internal model principle, is common in servo control theory[15]. It is easy to be extended for more general systems and control objectives. MPC with error integrator for nonlinear systems is studied in [4]. For other types of setpoint signals, such as ramp, sinusoid, periodic and others, it is possible to replace the integrator by servo dynamic blocks, which contain the corresponding unstable dynamic, and to design MPC based on the modified model[16]. Remark 3. As shown above, the H∞ γ-suboptimal MPC [11] can be applied to offset-free tracking problem without extra difficulty. This is duo to use of the composite model, which includes the error integrator. In fact, other MPC approaches, for example Tube MPC[17], Output MPC[18], Explicit MPC[19], Dynamic Programming[20], etc., can also be applied, depending the complexity and requirement. Of course, if wk is ignored in the model, the MPC approaches based on nominal models are also applicable. Remark 4. The state dimension is increased by ny , due to the addition of integrator, therefore the computation may be more complex than the MPC using disturbance estimation approach.

3.2

Transient Response

The proposed controller can robustly track piecewise constant reference signal. However, it may not be able to render satisfactory transient response for setpoint step change. It is simply due to the fact that the MPC is designed to 7

steer the integrator to 0, which implies that the tracking error has to change sign before it settles down. Therefore overshot is likely to be observed on output variable. That is why in section 2, it is stated that the primary objective of the proposed MPC design is on disturbance rejection, which is a more important requirement for process control applications. For cases requiring frequent setpoint change and high performance transient response, such as mechanical motion control, new systematic design is required. For example, the transient performance may be enhanced by employing the 2 degree-offreedom MPC approach[21] or by minimizing the distance between predicted trajectory and reference trajectory but not only a reference point, which is not within the scope of the paper. In addition, the overshot is possible to be reduced by simply adding a setpoint filter[22], which smooths the setpoint, before it is forwarded to MPC control loop. This method is simple, but of course not systematic. Remark 5. To achieve economic optimality, Self-optimizing control[23], which can assure optimal operation by controlling certain variables at constant, is widely used in industry. Since it is a purely robust regulation problem, the proposed MPC is a suitable choice to work with self-optimizing control. Remark 6. Reference Governor(RG)[10], which also considers constraints, is an alternate to MPC. It assumes a good linear control is available and RG is designed for the closed loop system. The state for the linear closed loop system includes original state and also error integrator. Therefore these two approaches share similarity. RG is more suitable for transient response. The MPC using error integrator is more general, easily applicable to nonlinear systems and closely related to extensive MPC literature.

3.3

Setpoint Feasibility

Sometimes the required setpoint decided by the commander or upper level controller may not be feasible for the dynamic MPC controller, especially for the terminal constraint MPC which may becomes infeasible. Here we denote the given setpoint as rg . Following the usual practice, an artificial setpoint, which is feasible for the current state and as close as possible to rg , can be selected by optimization problem Ps (zk , rg ): min

M,v,xs ,us ,˜ x0|k

|Cxs − rg |2 8

(16)

subject to: (12) , xs = Axs + Bus and xk = x˜0|k + xs .

(17)

The solution is denoted as x˜∗0|k , x∗s (k), u∗s (k). The objective function may take other form f (xs , us , rg ). The artificial setpoint rk = Cx∗s (k) is then used in MPC problem P(˜ x∗0|k , x∗s (k), u∗s (k)). Other advanced approaches, such as [9], propose to incorporate the selection of feasible setpoint into the MPC optimization problem. However, it is not applicable, at least if without any modification, to the case here. Theorem 2. The artificial setpoint rk will converge to rg or the closest feasible one. Proof. The definition of some common sets are given first. The set of feasible steady state and setpoint is Ox = {xs |(˜ z , xs , us ) ∈ Ω, xs = Axs + Bus } and Or = {Cxs |xs ∈ Ox }; the minimal disturbance invariant set[24] for dynamic (7) is θ; the terminal invariant set for each steady state xs is O(xs ) = {˜ z |(˜ z , xs , us ) ∈ Ω, xs = Axs + Bus }; the minimal disturbance invariant set around each steady state xs is Θ(xs ) = [xTs 0T ]T ⊕ θ. As in Theorem 1, the closed loop system is recursive feasible, so rk will always be a feasible setpoint for zk+1 . The function |Cx∗s (k)−rg |2 is therefore a decreasing function with time. Let assume that x∗s (k) converges to xˆs . Define x¯s = arg minxs ∈Ox |Cxs − rg |2 . Next, it is shown that xˆs = x¯s . It is noted that if xˆs is used in the H∞ γ-suboptimal MPC problem for a sufficient long period, the MPC will be equivalent to the terminal linear controller uk = u∗s + K(zk − [ˆ xTs 0T ]T ) and zk will converge to the set Θ(ˆ xs ). λ xs − xˆs ) and 0 ≤ λ ≤ 1. Since θ is a subset of O(xs )(θ ⊂ Let xs = xˆs + λ(¯ O(xs )) for ∀xs ∈ Ox , there exists a sufficiently small λ such that zk ∈ O(xλs ), ∀zk ∈ Θ(ˆ xs ). Therefore xλs is a feasible steady state for zk . λ |Cxs − rg |2 = |C xˆs − rg + λC(¯ xs − xˆs )|2 = |C xˆs − rg |2 − λ2 |¯ xs − xˆs |2 + 2λ(C x¯s − rg )T (C x¯s − C xˆs ) < |C xˆs − rg |2 − λ2 |¯ xs − xˆs |2 , which implies xˆs will not be the solution to Ps (zk , rg ) and contradicts the assumption. Therefore rk will converge to C x¯s or rk = rg if rg ∈ Or .

9

1 output y

ampliltude

0.5

0

−0.5

0

5

10

15

20 k

25

30

35

40

Figure 1: Output response

4

Numerical Example

To show the results graphically, a simple first order system, augmented with the error integrator, is considered as xk+1 = 2xk + 2uk + wk1 ξk+1 = −yk + ξk + rk ,

(18)

where the output variable yk = xk + wk2 . The constraints are |u| ≤ 1, |w 1 | ≤ 0.2 and |w 2 | ≤ 0.02. Other parameters used in the design are listed as: prediction horizon N = 5; the variable of interest ζk = [x ξ u]T ; setpoint r = 0; the initial state x0 = 1 and ξ0 = 0; l2 gain γ = 3 and the terminal H∞ -γ suboptimal control gain is K = [−1.3471 0.3550]. During the period of k ≤ 20, wk1 = 0.1 and wk2 = 0; for k ≥ 20, wk1 and wk2 are random noise. The output trajectory and input trajectory are shown in Fig. 1 & 2, respectively. The closed loop system is stable: at steady state, there is no offset; for time varying disturbance, the disturbance attenuation level satisfies the design requirement.

10

0.4 control input u 0.2 0

amplitude

−0.2 −0.4 −0.6 −0.8 −1 −1.2

0

5

10

15

20 k

25

30

35

40

Figure 2: Control input

5

Conclusions

Offset-free MPC design with the widely used error integrator scheme is revisited in this paper. The advantages and disadvantages of this scheme are highlighted and analysed. In particular, its potential in robust MPC design is emphasized with one H∞ γ-suboptimal MPC controller presented, which may link Robust MPC and linear H∞ control. Although most of the technical parts are directly migrated from MPC for regulation problem, it provides a practically feasible solution to implement robust MPC in real environment.

Acknowledgements The author is grateful to Assoc. Prof. Tan Kok Kiong from National University of Singapore for his guidance on servo mpc design.

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