Strong tracking filter based adaptive generic model control

doi:10.1016/S0959-1524(98)00052-3

Journal of Process Control

Volume 9, Issue 4, August 1999, Pages 337-350

https://doi.org/10.1016/S0959-1524(98)00052-3 Get rights and content

Abstract

Generic Model Control (GMC) is a control algorithm capable of using nonlinear process model directly. Parameters in GMC controllers are easily tuned, and measurable disturbances can be compensated effectively. However, the existence of large modeling errors and unmeasurable disturbances will make the performance of GMC deteriorate. In this paper, based on the theory of Strong Tracking Filter (STF), a new approach to Adaptive Generic Model Control (AGMC) is proposed. Two AGMC schemes are developed. The first is a parameter-estimation-based AGMC. After introducing a new concept of Input Equivalent Disturbance (IED), another AGMC scheme called IED-estimation-based AGMC is further proposed. The unmeasurable disturbance and structural process/model mismatches can be effectively overcome by the second AGMC scheme. The laboratory experimental results on a three-tank-system demonstrate the effectiveness of the proposed AGMC approach.

Introduction

The inherent nonlinearities of many chemical processes have limited the application of multivariable linear model-predictive controllers. Since 1988, there has been a growing interest in the use of Generic Model Control (GMC), when Lee and Sullivan[1] introduced the new control structure in which a nonlinear process model can be incorporated in a control strategy. This class of controller has been shown to have certain robustness for a wide range of process nonlinearity against process/model mismatches. Zhou and Lee[2] analyzed the robust stability of GMC under the condition that the explicit control law is available. In a case study, Lee and Newell[3] have demonstrated that the performance of GMC is superior over both PI control and Dynamic Matrix Control strategies for a forced circulation evaporator. To make the GMC algorithm more practical to be applied into industry, Brown et al.[4] presented a strategy to handle the constraints within GMC by defining GMC specification curves for both constraint variables and controlled variables. Lee and Zhou[5] designed a new multivariable dead-time compensator under the GMC framework by decomposing an n×n MIMO dead-time problem into a series of n SISO feedback controllers, each with a feedforward compensation term.

GMC has been applied experimentally to a diverse range of processes, such as bench-scale yeast fermentation[6], the binary distillation7, 8, and a continuous metallurgical process[9]. These publications highlighted the simplicity of the GMC technique in treating process nonlinearities and the economic incentives for using GMC instead of standard linear controllers. Furthermore, an industrially relevant procedure of tuning GMC controllers based upon Auto Tune Variation (ATV) technique was developed[10] and the effect of process/model mismatches on the effectiveness of ATV tuning procedure was evaluated.

All model-based controllers rely on the assumption that the process model effectively characterizes the dominant features of the process dynamics. As the degree of process/model mismatch increases, the closed-loop performance will degrade away from a pre-specified trajectory. The true process may differ from the process model in two ways: the first, termed parametric mismatch, where the structure of the process model is the same as the true process, but with different parameters; the second, termed structure mismatch, where the structure of the process model differs from the true process. Several methods under the framework of GMC have tried to deal with the impact of process/model mismatches. Lundberg and Bezanson[11] have shown that GMC lacks robustness for critical and overdamped closed-loop specifications. Therefore they proposed an enhanced robust GMC by using derivative feedback to compensate for process/model mismatch. Yamuna and Gangiah (1991)[12] proposed adaptive GMC and adaptive robust GMC to compensate for parametric and structural mismatches. Dunia et al. &/it;(1997)[13] implemented the Sliding Mode Control (SMC) for the GMC reference trajectory, and showed that SMC allows one to incorporate the effect of the uncertainty bounds in the controller structure, making GMC robust to processes with bounded uncertainties. Lee et al.[14] presented a process/model mismatch compensation algorithm to compensate for model errors and update the model parameters at steady-state. Because the steady-state is very difficult to determine, Generic Model Adaptive Control (GMAC), a more practical method of updating the critical parameters regularly, was further presented by Signal and Lee (1992)[15]. The updating of biased model parameters are based on the minimization of the difference of the process measurements and the prediction of the reference trajectory. It is sensitive to the modeling error. When the process outputs are biased from the reference trajectory owing tounknown disturbances or constraints, which is the common situation in practice, the GMAC approach has difficulty tracking the time-varying parameters. Therefore, the control performance of GMAC will deteriorate. This statement will be shown by a simulation example later.

In this paper, a new approach to Adaptive Generic Model Control (AGMC) is proposed in order to further improve the performance of GMAC which was developed by Signal and Lee (1992)[15]. Two AGMC schemes are developed.

The first scheme is based on estimation of time-varying parameters on-line in the imbedded nonlinear model of GMC. We propose to use a new filter, called a Strong Tracking Filter (STF)[16], to estimate time-varying parameters on-line. The negative influence of time-varying parameters on the control performance is effectively overcome, by updating them in every control period. This constitutes a parameter-estimation-based AGMC scheme.

The second scheme is based on the implementation of a new concept of Input Equivalent Disturbance (IED). When there are structural process/model mismatches, GMC cannot reject the influence of such disturbances quickly. A new concept of Input Equivalent Disturbance (IED) is introduced in this paper, and the process/model mismatches are lumped into IED. This constitutes an IED-estimation-based AGMC scheme.

The paper is organized as follows: Section 2is devoted to a review of concepts in GMC. The outline of Strong Tracking Filter is presented in Section 3. After introducing a new concept of Input Equivalent Disturbance (IED), Section 4presents the main idea of Adaptive GMC with STF, and gives the detailed theoretical development. In Section 5, the experimental results are given to show the effectiveness of the proposed approach. Concluding remarks are provided in the last section.

Section snippets

The basic principle of generic model control (GMC)

Consider the following MIMO nonlinear process[1]: $x ̇ t = f x,d + g x,d · u t y t = h x$

where the state vector $x t ∈ R^{n}$ , the input vector $u t ∈ R^{m}$ , the output vector $y t ∈ R^{m}$ , and the disturbance vector $d ∈ R^{P} · f ·, g ·, h ·$ are nonlinear functions of x,d, respectively, which have the sufficient order of continuous partial derivative with regard to x in the region R₀ in the x space.

For the completeness of this paper, we induce the concept of relative order of nonlinear system. First we introduce an operator of system (1): $L_{f}^{k} h_{i} x =$

Outline of strong tracking filter

Consider the following discrete nonlinear system: $I : x k+1 = f u k, x k +Γ k ν k y k+1 = h x k+1 + e k+1$

where the state vector $x k ∈ R^{n}$ , the input vector $u k ∈ R^{m}$ , the output vector $y k ∈ R^{m}$ , and the nonlinear functions f:R^m×Rⁿ→Rⁿ,h:Rⁿ→R^m, have continuous partial derivative with regard to x; process noise $ν k ∈ R^{q}$ and measurement noise $e k ∈ R^{m}$ are gaussian white noises with covariance $Q k$ and $R k$ , respectively. $Γ k$ is a known matrix with appropriate dimension.

There exists the following Strong Tracking Filter (STF)[16]: $x ̂ k+1∣k+1 = x ̂$

Parameter-estimation-based AGMC scheme

Consider the following discrete nonlinear time-varying process: ${II : x k+1 = f x k,θ k d (k + g x k,θ k, d k · u k +Γ k v k y k+1 = h x k+1,θ k+1, d k+1 + e k+1$

where $θ k ∈ R^{l}$ is a vector of unknown time-varying process parameter, $d k ∈ R^{p}$ is the unmeasurable process disturbance, and the definition of the other variables are as same as the above section. Assume the system is locally observable, and the unknown parameter $θ k$ and disturbance $d k$ can be identified.

In order to estimate the time-varying process parameter $θ k$ , unmeasurable

Experimental studies

The experimental equipment is a three-tank-system called DTS200, manufactured and provided by Amira Automation Company in German.

Fig. 3 shows the layout of the setup. This setup consists of three plexiglas cylinders T1, T3 and T2 with the equivalent cross section A. These are connected serially with each other by cylindrical pipes with the cross section S_n. Located at T2 is the single outflow valve. It has a circular cross section S_n. The outflowing liquid (usually distilled water) is collected

Conclusions

Generic Model Control is a nonlinear process model based control algorithm. Because the GMC algorithm uses the nonlinear model directly in the control law, to a certain degree, the control performance relies on the model parameters. Especially, when the model parameters change abruptly, the control performance deteriorates greatly. The simulation and experimental results show that the new approaches of Adaptive Generic Model Control which are developed in this paper can overcome the influence

Acknowledgements

Supported by the National Natural Science Foundation, the National “863” Plan and the State Education Ministry of China.

Appendix A

Here provide the formulae of the gain of STF $K k+1$ : $K (k+1 )= P (k+1∣k ) H^{T} (x ̂ (k+1∣k ) ) [H (x ̂ (k+1∣k ) ) P (k+1∣k) H^{T} (x ̂ (k+1∣k))+ R (k ) ]^{−1}$ $P k+1∣k = LMD k+1 F u k, x ̂ k∣k · P k∣k F^{T} u k, x ̂ k∣k +Γ k Q k Γ^{T} k$ $P k+1∣k+1 = I − K k+1 H x ̂ k+1∣k · P k+1∣k$ $γ k+1 =y k+1 − h x ̂ k+1∣k$

where $F u k, x ̂ k∣k = ∂ f u k, x k ∂ x ∣_{x=x̂k∣k}$ $H x ̂ k+1∣k = ∂ h x k+1 ∂ x ∣_{x} = x ̂ k+1∣k$ $LMD k+1 = diag λ_{1} k+1,λ_{2} k+1,…λ_{n} k+1$ $λ_{i} = α_{i} η k+1; α_{i} η k+1 >1 {1;} α_{i} η k+1 ⩽1$ $η k+1 = tr N k+1 ∑ i=1 n α_{i} M_{ii}$

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Strong tracking filter based adaptive generic model control

Abstract

Introduction

Section snippets

The basic principle of generic model control (GMC)

Outline of strong tracking filter

Parameter-estimation-based AGMC scheme

Experimental studies

Conclusions

Acknowledgements

Comput. Chem. Engng

Comp. Chem. Engng

Chem. Engng. Sci

Robust stability analysis of generic model control

Chem. Eng. Comm

Generic model control—a case study

Canad. J. Chem. Engng

A constrained nonlinear multivariable control algorithm

Trans. IChemE

A new multivariable deadtime control algorithm

Chem Eng. Comm