Model Predictive Control for Hammerstein Systems with Unknown

Article pubs.acs.org/IECR

Model Predictive Control for Hammerstein Systems with Unknown Input Nonlinearities Haokun Wang, Jun Zhao, Zuhua Xu,* and Zhijiang Shao National Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China ABSTRACT: In this study, a model predictive control scheme for Hammerstein systems with unknown nonlinearities is presented. The Hammerstein model is transformed into a linear model with one known input and another unknown input, and then a minimum-variance unbiased (MVU) filter is introduced to estimate both the state and the unknown input. This enables the use of linear techniques for controlling these systems. The nonlinear dynamics are treated as unknown input and can be estimated online by the MVU filter. The proposed approach is therefore applicable for handling model mismatch and timevarying operation conditions. The proposed approach can achieve offset-free control in the presence of unknown nonlinearities and/or asymptotically constant unmeasured disturbances. Simulation results demonstrate the potential of the proposed approach for application to the control of linear systems with unknown input nonlinearities.

1. INTRODUCTION Input nonlinearities are ubiquitous in process industries and may arise from the inherent nonlinear characteristics of the actuators as well as from time-varying operation conditions such as flow fluctuation from upstream units and feed composition variations. Such nonlinear characteristics can severely degrade control system performance and in some cases drive the system to instability. Thus, it is important to account for input nonlinearities in the control system design process. A class of linear system with input nonlinearities can be represented by the Hammerstein model, in which a memoryless nonlinear function is in series with a linear dynamic model. The identification of the Hammerstein system involves estimating both the nonlinear and the linear parts. Both parts can be identified from prespecified tests separately or simultaneously.1−4 Hammerstein systems have been useful in modeling many industrial plants, such as pH neutralization processes,5−9 distillation columns,4,7,10 heat exchangers,4,6,10 and solid oxide fuel cells.11 The control problem of Hammerstein systems has drawn considerable attention over the past few decades. Model predictive control (MPC) has been the most successful advanced control technique applied in the process industries.12,13 Nonlinear models can be directly integrated into the MPC algorithms, because the predictive models in MPC are not limited to linear models. As shown by Sentoni et al.,4 the Hammerstein model can be used as the predictive model in the MPC framework. This approach leads to a nonconvex optimization problem, which is not attractive in practice, because the algorithm is computationally expensive and may get stuck in a local minimum.14 This drawback can be partially alleviated by exploiting the block structure for sensitivity information.15 A common way of handling Hammerstein systems is the nonlinearity inversion based approach.5,7,8 In this approach, a linear controller is first designed for the linear dynamic model, and then the nonlinearity is removed by an inversion of the © 2014 American Chemical Society

nonlinear function. Since there are two steps in this approach, we call it the two-step approach in this study. The two-step approach is widely used, because it is relatively easy to implement in practice. This approach is often sensitive to model mismatch, especially the nonlinear model mismatch, because the nonlinearity is removed off-line. Thus, this approach is applicable for systems with perfectly known input nonlinearities. A robust controller can also be adopted in designing a Hammerstein control system. In this approach, the nonlinearity is transformed into a polytopic description. The remaining uncertain linear model is used in an MPC algorithm in which the optimization problem entails minimizing a linear objective function subject to linear matrix inequalities, and the online optimization problem remains convex.14,16 But it should be noted that the conservativeness and the online computational costs need to be carefully considered with respect to practical implementations. Although successfully implemented in many applications, existing approaches are limited to cases where an accurate nonlinear model is available. However, an accurate model of the nonlinearities is difficult to obtain in practice, because the nonlinearities often exhibit time-varying features or characteristics. The nonlinear part of the Hammerstein model is often used to represent nonlinear behavior that caused by load changes, unmeasured disturbances, and nonlinear characteristics of actuators. Thus, such nonlinearities are often unclear or unknown to practitioners. The control performance will degrade without accurate information about the nonlinearities, because existing approaches are heavily dependent on model accuracy, especially the accuracy of the nonlinear part. It is Received: Revised: Accepted: Published: 7714

December 17, 2013 April 7, 2014 April 9, 2014 April 9, 2014 dx.doi.org/10.1021/ie404276h | Ind. Eng. Chem. Res. 2014, 53, 7714−7722

Industrial & Engineering Chemistry Research

Article

This paper aims to design an MPC for the Hammerstein model (1) with unknown nonlinearities. The two-step approach cannot be used, because the nonlinearity is unknown, and cannot be removed by an inversion. To deal with the unknown nonlinearities, the Hammerstein model 1 is transformed into a linear model with input uncertainties. Then, the optimal filtering problem for the transformed model is addressed. 2.1. Model Transformation. At a given working condition u0, the Taylor series linearization of f(uk) gives

therefore more practical to account for unknown input nonlinearities in designing a control system. This study aims to design an MPC for Hammerstein systems with unknown nonlinearities. The motivation for this study comes from applications in the control of linear systems with unclear input nonlinearities, which are widespread in process industries and are not directly solvable with existing results. This current study combines and extends the study of ours,17 transforms the Hammerstein model into a linear model with one known input and another unknown input, and then introduces a minimum variance unbiased (MVU) filter18,19 to estimate both the state and the unknown input. This enables the use of linear MPC techniques for controlling these systems. Prior knowledge of the nonlinearities is not required because the nonlinear dynamics are treated as unknown input, which can be detected online by an MVU filter. The proposed approach is therefore applicable for handling model mismatch and time-varying operation conditions. Offset-free control can be achieved in the presence of unknown nonlinearities and/or asymptotically constant unmeasured disturbances. The MVU filter has quick estimation behavior, which enables the controller to quickly reject unmeasured disturbances. The rest of this paper is organized as follows. In Section 2, we first show that a Hammerstein system can be transformed into a linear system with input uncertainties and then discuss the optimal filtering problem for the transformed system. Section 3 addresses the design of an MVU filter and analyzes the estimation performance. The offset-free control problem is discussed in Section 4. Illustrative examples are provided in Section 5 to demonstrate the effectiveness of the proposed method. A brief conclusion is given in Section 6.

nk = f (u0) + = αuk + ok

(2)

where α = ∂f/∂u|u0 is the first derivative of f(uk), ϵk is the summary of the higher order derivative terms, and ok = f(u0) − ∂f/∂u|u0 u0 + ϵk. Then, eq 2 can be rewritten as

nk = uk + dk

(3)

where dk = (α − I )uk + ok

(4)

The main idea behind this transformation is to represent the nonlinearities as input uncertainties. From the above transformation, dk is expressed as a function of u0, uk, and ok. The magnitude of dk therefore depends on the operation conditions and the order of ok. If f(uk) is exactly known and u0 is given, dk can be calculated from eq 4, and the nonlinearities can be represented by dk with specified accuracy. In fact, nk can be obtained by a linear controller for the linear model in a similar way to that in the two-step approach,7,8 and then uk can be calculated by uk = nk − dk. Thus, the inversion step in the twostep approach can be replaced by the calculation of dk. However, as already stated, the nonlinear function is often unclear or unknown to practitioners. Thus, any approaches that depend on accurate information about the nonlinearities are not applicable for cases with unknown nonlinearities. If f(uk) is unknown, both α and ok become unknown, and the value of dk is therefore unknown. From transformation 3, nk is divided into two parts, the known part uk and the unknown part dk. The Hammerstein model 1 can then be represented by the following linear system with unknown input:

2. PROBLEM FORMULATION In this study, the following Hammerstein system with unknown nonlinearities is considered: ⎧ xk + 1 = Axk + Bnk + wk ⎪ ⎪ ⎨ yk = Cxk + vk ⎪ ⎪ n = f (u ) ⎩ k k

∂f |u (uk − u0) + ϵk ∂u 0

(1)

where x ∈ nx, u ∈ nu, and y ∈ ny are the state, input, and output vectors, respectively. A, B, and C are known matrices with appropriate dimensions. n is the input to the LTI block, and f(uk) is the unknown static nonlinearity function. wk and vk are process noise and measurement noise with known covariances Qk = (wkwTk ) and Rk = (vkvTk ). Typically, the measurement noise covariance Rk can be derived from the properties of the sensors that are used. The dynamics of the process noise wk can be negated by assuming that Qk ≈ 0. Although both Qk and Rk are assumed to be known, they also can be treated as tuning parameters to obtain a trade-off between effectiveness in obtaining offset-free control and low sensitivity to noises. Throughout the study, we assume that the following two conditions are satisfied. Assumption 1. Consider the Hammerstein model (1), for any given y and u, there exists a unique solution x* such that x* = Ax* + Bf(u) and y = Cx* hold. Assumption 2. Consider the Hammerstein model (1), for all given output y there exists a unique solution x* and u* such that x* = Ax* + Bf(u*) and y = Cx* hold. When assumptions 1 and 2 hold, the input multiplicity problem is excluded, and the design process can be simplified.

xk + 1 = Axk + Buk + Bdk + wk

(5)

yk = Cxk + vk

(6)

where dk can be seen as input uncertainties, unmeasured disturbances, or unknown input disturbances. For the rest of the paper, both “measurements” and “outputs” can interchangeably mean yk, and the term “unknown input” denotes unmeasured disturbance or unknown input disturbance. The equivalence between the original system and the transformed system 5−6 is shown in Figure 1. It should be noted that for any given pair (nk,uk), there always exists unique dk such that eq 3 holds. Therefore, the existence of this transformation can be guaranteed as long as assumptions (1) and (2) hold. This transformation does not require prior knowledge of the nonlinear function f(uk), and all the nonlinear dynamics are included in dk. This transformation enables us to design controllers for system (1) in a linear framework. The transformation does not depend on the nonlinear model and the operation conditions; thus, it is possible to deal online with model mismatch and time-varying 7715

dx.doi.org/10.1021/ie404276h | Ind. Eng. Chem. Res. 2014, 53, 7714−7722


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arbitrary statistics. This type of filter is exactly what is needed for system 5−6. These filters have been discussed by Kitanidis18 and Gillijns and De Moor.19 A complete treatise on the design of an MVU filter is beyond the scope of this study; here the main results proposed by Gillijns and De Moor19 are briefly recounted. 3.1. MVU Filter Design. Throughout the study, we assume that (C,A) is observable, (A,Q1/2) is stabilizable, and the initial state x0 is of the mean x̂0 and covariance P0 and is independent of vk for all k. The following recursive filter is given for system 5−6: xk̂ | k − 1 = Axk̂ − 1 + Buk − 1

(7)

dk̂ − 1 = Mk(yk − Cxk̂ | k − 1)

(8)

xk̂ * = xk̂ | k − 1 + Bdk̂ − 1

(9)

xk̂ = xk̂ * + Kk(yk − Cxk̂ *)

Figure 1. Equivalence between the Hammerstein model and linear models with unknown input.

(10)

Defining ỹk = yk − Cx̂k|k−1 yields ỹk = CBdk−1 + ek, where ek = C(Ax̃k−1 + wk−1) + vk and x̃k−1 = xk−1 − x̂k−1. Then we can obtain d̂k−1 = Mk(CBdk−1 + ek). To obtain an unbiased estimate of dk−1, the condition

operation conditions. The remaining problem is how to obtain the dk value, and this problem will be discussed in the following section. 2.2. Optimal Filtering for the Transformed System. After transformation 3, a proper filter must be found for system 5−6. The filter should be able to estimate both xk and dk for designing the controllers. On the other hand, the estimate of dk should be close to the actual result, because a reliable estimate enables analysis of the system’s nonlinearities. As previously stated, it is difficult to model dk, because dk is a function of u0, uk, and ok, and the nonlinear function f(uk) is often unclear or unknown. A simple solution for the estimation problem is augmented Kalman filtering,20,21 where the system state is augmented to include the unknown input as an additional component of the state and then apply the Kalman filter to the augmented system. In these filters, the unknown input is assumed to be a stochastic process with known statistics. However, it is clear that this assumption is not valid, because dk is used to represent the input uncertainties as shown in eq 3 and cannot be described by a general stochastic process. Thus, optimal filtering cannot be guaranteed if traditional Kalman filters are used directly, and the true value of dk cannot be obtained. Fortunately, linear MVU filtering techniques can be used to solve the above-mentioned estimation problems.18,19 dk is allowed to have arbitrary statistics in the design of an MVU filter. This is very important because the nature of dk is unknown here. After the dk value is obtained, the control problem can be solved in the linear framework. In this study, an MVU filter is introduced into the framework of the MPC, which can provide optimal filtering in the presence of unknown input with arbitrary statistics. The benefits of this filter will be shown in the next section.

MkCB = I

(11)

must be satisfied, condition

19

which is equivalent to the following rank

rank(CB) = nu

(12)

The estimate of dk−1 can be obtained by using the GaussMarkov theorem.22 And Kk can be determined by minimizing the error variance subject to the unbiasedness condition 11. An MVU filter with control input is given as follows:19 ⎧ xk̂ | k − 1 = Axk̂ − 1 + Buk − 1 ⎪ ⎪ Mk = (BT CTR̃ k−1CB)−1BT CTR̃ k−1 ⎪ ⎪ d ̂ = M (y − Cx ̂ k k k | k − 1) ⎪ k−1 ⎨ ⎪ xk̂ * = xk̂ | k − 1 + Bdk̂ − 1 ⎪ ⎪ Kk = (Pk*CT + Sk*)αkT (αkR̃ k*αkT )−1αk ⎪ ⎪ xk̂ = xk̂ * + Kk(y − Cxk̂ *) ⎩ k

(13)

where αk is an arbitrary matrix which must be chosen such that αkR̃ *k αTk has full rank. ⎧ R̃ = C(AP AT + Q )CT + R k−1 k k−1 ⎪ k ⎪ T ⎪ Pk | k − 1 = APk − 1A + Q k − 1 ⎪ ⎪ Sk* = −BMkR k ⎨ ⎪ * T ⎪ Pk = (I − BMkC)Pk | k − 1(I − BMkC) ⎪ + BMkR kMkT BT ⎪ ⎪ R̃ * = CP*CT + R + CS * + (S *)T CT ⎩ k k k k k

3. MINIMUM-VARIANCE UNBIASED FILTER DESIGN Because it is difficult to specify a model for dk, optimal filtering cannot be guaranteed when using traditional Kalman filters. In this section, linear MVU filtering techniques18,19 will be introduced, and the estimation performance will be analyzed. The resulting filter can simultaneously estimate both the state and the unknown input, even when the unknown input has

Remark 1. In fact, eq 7 is an open loop state estimator, and the corresponding estimated output is Cx̂k|k−1. From eq 8, the estimate of dk−1 requires ỹk, which is the error between the measured output yk and the estimated output Cx̂k|k−1. The gain matrix Mk is obtained by Gauss-Markov theorem (weighted 7716



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least-squares).22 This means unmeasured disturbances and unmodeled dynamics are lumped into dk−1 in the least-squares sense. □ Remark 2. Stability issues of the above MVU filter have been addressed by Fang and de Callafon.23 Similar to the Kalman filter, the MVU filter is asymptotically stable if unbiasedness condition 12 is satisfied, (C,A) is observable, (A,Q1/2) is stabilizable, and |λi[(I − KC)(I − BMC)A]| < 1, where λi(X) is the ith eigenvalue of X. □ 3.2. Offset-Free Output Estimation. First, the estimated output is defined as ŷk = Cx̂k. From eqs 6, 9, and 10, the corresponding estimated error can be expressed as

state targets xs and us for the regulator. xs and us can be determined from the following quadratic program:24 min(us − ut )T R s(us − ut ) xs , us

s.t.

⎡ I − A −B ⎤⎡ xs ⎤ ⎡ Bd ̂ ⎤ ⎥⎢ ⎥ = ⎢ k ⎥ ⎢⎣ C 0 ⎦⎣ us ⎦ ⎣ r ⎦ ⎡ xmin ⎤ ⎡ xs ⎤ ⎡ xmax ⎤ ⎢u ⎥ ≤ ⎢u ⎥ ≤ ⎢u ⎥ ⎣ min ⎦ ⎣ s ⎦ ⎣ max ⎦

(14)

where ymin ≤ r ≤ ymax, ut is the desired value of the input at steady state, and Rs is a positive definite weighting matrix for the deviation of the input vector from the target input ut. If there are insufficient degrees of freedom, the following quadratic program can be used to track the output target in a least-squares sense.24

ξk = yk − yk̂ = C(I − KkC)(Axk̃ − 1 + Bdk̃ − 1 + wk − 1) − (I − CKk)vk

where d̃k−1 = dk−1 − d̂k−1. Since x̂k−1 and d̂k−1 are unbiased estimates of xk−1 and dk−1, we can obtain  (ξk) = 0 ((*) =0 is the expectation value of (*)). This means a zero offset output estimation can be obtained for every sample time. This property is important for analyzing the offset-free control performance in the next section.

min(r − Cxs)T Q s(r − Cxs) xs , us

s.t.

4. MODEL PREDICTIVE CONTROL In this section, MPC for the Hammerstein model with unknown nonlinearities will be discussed. It has been shown that system (1) can be transformed into a linear system with unknown input, and both the state and the unknown input can be estimated by the MVU filter. This enables one to control such systems using a linear framework. Next, we briefly review the main results of a linear MPC and demonstrate that the proposed approach can achieve offset-free control in the presence of unknown nonlinearities and/or unmeasured disturbances. 4.1. Linear Model Predictive Control. There are three main parts in current formulations of an MPC: a state estimator, a dynamic constrained regulator, and a target calculator. A basic block diagram for a linear MPC scheme is shown in Figure 2 where r and ut are the set-points for the

⎡ xs ⎤ [ I − A −B ]⎢ ⎥ = Bd dk̂ ⎣ us ⎦ ⎡ xmin ⎤ ⎡ xs ⎤ ⎡ xmax ⎤ ⎢u ⎥ ≤ ⎢u ⎥ ≤ ⎢u ⎥ ⎣ min ⎦ ⎣ s ⎦ ⎣ max ⎦

̂ where r is the set-points for the output, ymin ≤ r ≤ ymax, d̂i = dk−1 (i = k, k + 1, ..., k + N − 1), and Qs is a positive definite weighting matrix for the output tracking error. 4.1.2. Dynamic Regulation. The regulation problem can be formulated by the following quadratic programming with constraints and penalties (Wx ≥ 0, Wu ≥ 0, and WΔu ≥ 0): k+N−1

min ΔuN

∑

2 2 2 ∥xî − xs∥W + ∥ui − us∥W + ∥Δui∥W x u Δu

i=k

s.t.

xî + 1 = Axî + Bui + Bdî umin ≤ ui ≤ umax Δumin ≤ Δui ≤ Δumax Δui = ui − ui − 1

(15)

where uN = [Δuk, ···, Δuk+N−1] and the estimated input d̂i is ̂ (i assumed to be a constant in the prediction horizon dî = dk−1 = k, k + 1, ..., k + N − 1). Here, the control horizon and the prediction horizon are assumed to be the same number N. 4.2. Offset-Free Nominal Tracking. Now an MVU filter is introduced into the design of the linear MPC for the Hammerstein systems with unknown nonlinearities. The offset-free tracking problem will be addressed in the proposed framework. Before the discussion, two more assumptions are required. Assumption 3. The target problem 14 has a unique feasible solution, and the regulation problem 15 is feasible for all k. Assumption 4. The reference is asymptotically constant (rk → r∞), and the closed-loop system reaches a steady state (xk → x∞, nk → n∞, uk → u∞, and yk → y∞). It is well-known that offset-free control cannot be achieved with an unmeasured persistent disturbance. Here only the deterministic case (Qk ≈ 0 and Rk ≈ 0) is considered. A very simple proof for offset-free control for the Hammerstein model is given in the following theorem.

Figure 2. Block diagram of an MPC for Hammerstein systems with unknown nonlinearities.

output and the input, respectively, xs and us are the calculated state and input, x̂ and d̂ are the estimated state and unknown input, respectively. Note that the measured disturbance is excluded, because it is easy to include feedforward algorithms in the MPC framework by treating the measured disturbance as input to the predictor. This will not affect our main results. 4.1.1. Target Calculation. In order to guarantee zero offset steady-state, a target calculator is used to calculate the steady7717



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Theorem 1. Consider Hammerstein model 1, the MVU filter 13, and MPC 14 and 15, if assumptions 1−4 are satisfied and unbiasedness condition 12 holds, then offset-free nominal tracking (y∞ → r∞) can be achieved. Proof: If assumptions 1 and 2 hold, the Hammerstein model 1 can be uniquely transformed into a linear system with unknown input as shown before. If uk → u∞ and nk → n∞, from 3 we know that dk → d∞, where d∞ is the steady state value of dk. Since unbiasedness condition 12 holds, unbiased estimation of dk can be obtained from the MVU filter 14, and we can then obtain d̂k → d∞. Then the problem becomes a standard offsetfree control problem, which has been fully discussed by Muske and Badgwell25 (Theorem 4) and Maeder et al.26 (Theorem 1), and the proof is omitted for brevity. □ Remark 3. As pointed out by Pannocchia and Bemporad,27 a critical problem for achieving offset-free control is to design an observer that gives a zero offset output estimation. Morari and Maeder28 also showed that a zero offset output estimation and offset-free nominal tracking imply an offset-free MPC. We have shown that the MVU filter gives a zero offset output estimation for every time k. This enables the proposed method to more quickly reject unmeasured disturbances. □ Remark 4. Although dk is assumed originating from the nonlinear part of the system in the above discussions, unmeasured disturbances and unmodeled dynamics can also be lumped into the term dk as shown in Remark 1. For example, we can assume that unmeasured disturbance dink is originated from external unknown sources or model mismatch, and suppose that dink has the same dimension as dk, dink + dk can then be treated as the unknown input. If dink → din∞, offset-free control can also be achieved. The proof is similar to that of Theorem 1. □

Figure 3. Static nonlinearities and their inverses.

⎧ ⎡ 0.8187 ⎪ ⎢ ⎪ ⎢0 = A ⎪ ⎢0 ⎪ ⎢ ⎣0 ⎪ ⎪ ⎡ 0.9063 ⎪ ⎨ ⎢ ⎪ ⎢0 ⎪ B = ⎢0 ⎪ ⎢ ⎣0 ⎪ ⎪ ⎡ 0.2000 ⎪ = C ⎢ ⎪ ⎣ 0.4000 ⎩

0 0.8465 0 0

0 0 0.0191 0.3487

⎤ 0 ⎥ 0 ⎥ − 0.2325⎥ ⎥ 0.8328 ⎦

⎤ ⎥ 0.4606 ⎥ 0.3487 ⎥ ⎥ 0.2508 ⎦ 0

⎤ 0.6667 0 0 ⎥ 0 0.6667 0.3333⎦

5. ILLUSTRATIVE EXAMPLES In the following tests, MPC parameters are chosen as Wx = I, Wu = I, and WΔu = I, and the prediction horizon and the control horizon are assumed to be the same number 20. The filter is initialized as Qk ≈ 0, Rk ≈ 0, and P0 = 0. It is easy to design an MPC for the above Hammerstein system using the two-step approach if the nonlinear functions are perfectly known. In this section, the proposed approach will be compared with the twostep approach addressed by Patwardhan et al.8 with known nonlinear functions. Actually, the two approaches cannot be compared with each other if the nonlinear functions are unknown, because the two-step approach cannot be used in this case. Here we simply want to show the limitations of the twostep approach, even though the nonlinearities are known. The nonlinear functions are assumed to be unknown when using the proposed approach. First, we will show the performance of the proposed approach under the nominal model case. Then, the proposed approach will be tested in the presence of model mismatch and time-varying operation conditions. 5.1. Nominal Performance. First, nominal performance of the proposed approach under unmeasured disturbances will be examined. Both the nonlinear and linear parts will not change, and both deterministic and stochastic disturbances are considered in the following tests. 5.1.1. Nominal Tracking Performance. At time 0−50 and 100−150, two step changes in the two references are applied (as shown in Figure 4). Here, we assume that the nonlinear

In this section, several examples are provided to illustrate the effectiveness of the proposed approach. First, the system will be tested under the nominal case. And then model mismatch and time-varying cases are considered. A two-input−two-output Hammerstein model is given by the following equations: ⎡ 1 ⎤ 2 ⎥ n ⎡ y1 ⎤ ⎢ 5s + 1 + 6 s 1 ⎥⎡ 1 ⎤ ⎢ ⎥=⎢ ⎥⎢⎣ n2 ⎥⎦ 2s + 1 ⎣ y2 ⎦ ⎢ 2 ⎢ ⎥ ⎣ 5s + 1 (3s + 1)(s + 2) ⎦

where the input nonlinearities are given as ⎧ ⎪ n1 = ⎪ ⎨ ⎪ ⎪ n2 = ⎩

e 2u1 − 1 e 2u1 + 1 e u2 − 1 e u2 + 1

The nonlinear function is smooth saturation, which is often encountered in practice. The shapes of the static nonlinearities and their inverses are shown in Figure 3. A minimal state-space realization of the linear dynamic model (discrete time with a sampling time of 1 s) gives 7718



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function is unknown and the linear model is known, and the nominal tracking performance is illustrated in Figure 4.

Figure 6. Offset-free control of the Hammerstein system in the presence of unmeasured disturbances.

5.1.3. Stochastic Disturbances Rejection. The offset-free control problem has been discussed under deterministic disturbances. Now, stochastic disturbances are considered, because stochastic disturbances are often encountered in practical applications. Stochastic measurement noises with unknown statistics are considered in this part (Figure 7). In

Figure 4. Nominal tracking performance of the proposed approach.

Figure 4 shows that the proposed approach can achieve offset-free nominal tracking. In fact, this result is guaranteed by Theorem 1. 5.1.2. Deterministic Disturbances Rejection. Next, disturbance rejection performance will be examined in the presence of deterministic unmeasured disturbances. At time 0−50 and 100−150 two unmeasured disturbances (step signals) are added to both the inputs as shown in Figure 5. The disturbance rejection performance is demonstrated in Figure 6.

Figure 7. Unmeasured stochastic disturbances.

fact, the two unmeasured disturbances are combinations of two signals: white noises with zero means and sine signals. Because the statistics of measurement noises are unknown, we also assume that Qk ≈ 0 and Rk ≈ 0 in this test. We can also tune the two parameters to obtain a trade-off between effectiveness in obtaining offset-free control and low sensitivity to noises. Here, 2 200 the ISE = ∑200 i=1 (r −yi) and IAE = ∑i=1 |r −yi| indices are used to evaluate the control performance. The two approaches are compared under the same parameters as before. Disturbances are added to the output, and the corresponding performance is shown in Table 1. Clearly, the proposed approach performs better than the two-step approach from both of the indices in this test.

Figure 5. Unmeasured input disturbances.

From Figure 6 we can see that the proposed approach achieves offset-free tracking. This is mainly because the MVU filter can estimate the disturbances, and the MPC can then reject the disturbances according to the estimated results. As stated in Remark 4, if unmeasured disturbances are treated as part of the unknown input, offset-free tracking can also be guaranteed by Theorem 1. 7719



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Table 1. Rejection Performance under Stochastic Output Disturbances ISE

IAE

approaches

y1

y2

y1

y2

two-step proposed

0.854 0.442

11.034 7.321

0.950 0.883

11.275 10.984

5.2. Model Mismatch Case. Model mismatch is considered in this subsection. First, the nonlinear part is mismatched, and then, the linear model mismatch is considered. The design of the two-step approach is based on the original Hammerstein model and that the original nonlinear functions are known and removed by inverses. 5.2.1. Nonlinear Part. In this test, the linear dynamic model will not change; only the nonlinear functions are mismatched. The linear function n1 = 0.2u1 and the nonlinear function n2 = (e3u2 − 1)/(e3u2 + 1) are used to replace the original nonlinear functions. The original and the mismatched nonlinear functions are plotted in Figure 8.

Figure 9. Tracking performance under nonlinear model mismatch.

Figure 10. Disturbance rejection performance under nonlinear model mismatch.

Figure 8. Model mismatch of the nonlinear functions (Solid line: real nonlinearities, dashed line: mismatched nonlinearities).

In the design of the proposed approach, the nonlinear functions are assumed to be unknown (this assumption is more practical). The tracking performance of the two approaches is compared in Figure 9. Figure 9 shows that the two-step approach is sensitive to the mismatched models, because the nonlinearity is removed off-line through an inversion calculation. The proposed approach gives better performance, because accurate information on the nonlinear functions is not required. Then, an unmeasured disturbance (step signal with a value of 0.3) is added to the second input at the beginning of this test. The disturbance rejection performance of both the two-step approach and the proposed approach is given in Figure 10. It is clear that the proposed approach performs much better, because nonlinear model mismatch is treated as the unknown input, and it can be estimated online by the MVU filter. The proposed MPC quickly reacts to the disturbance, as if it knows where the disturbance is located. For example, only the second control input varies in the rejection process. This is because the disturbance is added into the second input of the system, and

the MPC just only needs to tune the second control input to reject the disturbance according to the estimated results. In the meantime, the first control input does not require significant changes. This result can be seen from the subfigure in the lower left corner of Figure 10. 5.2.2. Linear Part. In this test, the nonlinear functions will not change, only the linear model is mismatched. The following linear model is used to replace the original model: ⎡ 1.5 3 ⎤ ⎡ y1 ⎤ ⎢ 6s + 1 5s + 1 ⎥⎡ n1 ⎤ ⎥⎢ ⎥ ⎢ ⎥=⎢ 1 ⎥⎢ n 2 ⎥ ⎢ y2 ⎥ ⎢ 2.5 ⎣ ⎦ ⎢ ⎥⎣ ⎦ ⎢⎣ 4s + 1 2s + 1 ⎥⎦

Tracking performance of the two approaches are plotted in Figure 11. It is shown that there is no significant difference between the two approaches under the linear mismatch case. As previously stated, unmodeled dynamics can be lumped into the disturbance term (Remark 1) by the proposed approach. 7720



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The following two subfigures show that the two-step approach is more sensitive to the time-varying models. Then, disturbance rejection performance is examined in this case. An unknown disturbance (step signal with value of 0.4) is added into the first input at the start of this test, and the resulting disturbance rejection performance is shown in Figure 13.

Figure 11. Tracking performance under linear model mismatch.

Therefore, the proposed approach can handle linear model mismatch. In the above tests, both the set-point tracking performance and the disturbance rejection performance under model mismatch are examined. Generally speaking, the proposed approach performs better than the two-step approach, especially the performance of disturbance rejection. 5.3. Time-Varying Case. Here, time-varying nonlinear models are considered. In practice, the nonlinear characteristics may vary over time because of the effects of aging on the actuators. In this test, the two nonlinear functions are assumed varying as n1 = (1 + 0.2 sin 0.1t)(e2u1 − 1)/(e2u1 + 1) and n2 = (1−0.001t)(eu2 − 1)/(eu2 + 1). The tracking performance of the two approaches is compared in Figure 12. Because nonlinearities of the system can be estimated by the MVU filter online, the tracking performance of the proposed approach is not affected by the time-varying models. From the top two subfigures of Figure 12, we can see that the proposed approach tracks the references more quickly than the two-step approach.

Figure 13. Disturbance rejection performance under time-varying nonlinear functions.

From Figure 13, we can see that the disturbance rejection performance of the two-step approach is more sensitive to the time-varying models. The proposed approach gives better performance in this case. This is because the time-varying dynamics are treated as unknown input and can be estimated online by the MVU filter.

6. CONCLUSIONS The use of MPC for Hammerstein systems with unknown nonlinearities is discussed in this study. A Hammerstein model can be transformed into a linear model with unknown input. As distinct from existing approaches, the proposed method does not require an accurate model of the nonlinearities. The proposed method can achieve offset-free control. A direct implication of this study is that an industrial practitioner can treat the Hammerstein system as a linear system with unknown input but then use an MVU filter to estimate both the state and the unknown input. Although MPC techniques are used in this study, the proposed approach is not solely applicable to MPCs. In fact, most of the linear controllers can be adopted as long as the designer uses the proposed transformation and the MVU filter. Tracking performance using this approach is often comparable with that obtained by using an accurate nonlinear model, and the proposed approach performs better in rejecting unmeasured disturbances. As a result, modeling effort can be directed for identifying the linear dynamics rather than attempting to identify the accurate nonlinear model, which is not easy to obtain in practice.

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AUTHOR INFORMATION

Corresponding Author

*(Z.X.) E-mail: [email protected]. Phone: +86-57187953068. Fax: +86-571-87953353.

Figure 12. Tracking performance under time-varying nonlinear functions. 7721



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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (No. 61273145, 61273146), the 863 Program of China (No. 2014AA041802), and the 973 Program of China (No. 2012CB720503). Part of this work was presented at IFAC LSS 2013.

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Model Predictive Control for Hammerstein Systems with Unknown

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