二次规划下的状态估计MPC：解决约束问题与高纯度精馏柱控制

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"模型预测控制（Model Predictive Control, MPC）是一种先进的控制策略，它结合了状态估计（State Estimation）以处理复杂的工业过程控制问题。本文主要关注于带有约束条件的MPC，这是一种扩展了传统控制方法如Quadratic Dynamic Matrix Control (QDMC)的技术，它能够处理硬性约束在操纵变量和输出上的限制。作者N. Lawrence Ricker在1990年的《工业与化学工程研究》上发表了一篇名为“带有状态估计的模型预测控制器”的论文，探讨了如何在MPC框架内实现状态估计。这种技术允许控制器在满足系统动态模型的同时，考虑到实际操作中的实时信息不确定性。通过状态估计，控制器能够更准确地预测系统的未来行为，并据此做出最优决策。在处理具有挑战性的高纯度蒸馏塔控制系统时，这种技术显示出显著优势。之前由Skogestad和Morari研究的问题对MPC算法来说尤为棘手。论文展示了，即使在存在双组分混合物控制的情况下，通过整合一个额外的调整参数，结合状态估计的MPC可以提供与Skogestad和Morari设计的最佳p-最优控制器相当的稳健性能。这表明，状态估计对于MPC的有效应用至关重要，尤其是在那些需要处理非线性、集成环节或有复杂约束的工业流程中。此外，文中还提到了该方法在包括积分环节的过程中的应用，这意味着它不仅适用于连续过程，也适用于涉及累积效应的系统。这篇文章强调了状态估计在提升MPC控制效果和适应复杂工业环境中的重要作用，为优化过程控制提供了新的理论支持和实践指导。"

374

Ind. Eng. Chem. Res.

1990,29,

374-382

Model Predictive Control with State Estimation

Lawrence Ricker

Departmen,t

Chemical Engineering,

BF-

IO,

University

Washington, Seattle, Washington

98195

state-space formulation of the multivariable model-predictive controller

(MPC)

with provisions

for state estimation is developed. Hard constraints on the manipulated variables

and

outputs are

accommodated, as in Quadratic Dynamic Matrix Control (QDMC) and related algorithms. For

unconstrained problems, a low-order analytical form of the controller is obtained. The potential

benefits of

MPC

with

state

estimation are demonstrated for the case of dual-composition,

control

of the high-purity distillation column problem studied previously by Skogestad

and

Morari, which

is an especially challenging problem for MPC-type algorithms.

is shown

that

the use of the state

estimator with a single tuning parameter (beyond that required for standard

MPC)

provides robust

performance equivalent to the best p-optimal controller designed by Skogestad and Morari. The

method is also applied to

process that includes integration and thus is not asymptotically stable

in the open loop.

Over the past 10 years, there has been a growing interest

in the use of control structures in which a model is placed

in parallel with the plant. The advantages of this from

the point of view of theoretical analysis and controller

design have been explored thoroughly by Morari and co-

workers (e.g., Garcia and Morari, 1982, 1985; Morari and

Doyle, 1986, Morari and Zafiriou, 1989), Rouhani and

Mehra (19821, and others. Its advantages in industrial

applications (e.g., explicit handling of inequality con-

straints) have been discussed by Richalet and Rault (19781,

Cutler et al. (19831, Ricker et al. (19861, Prett and Garcia

(19881, and others. It is also beginning to be used as the

basis for adaptive control algorithms (e.g., Clarke et al.,

1987; De Keyser et al., 1988). In the present work, how-

ever, such controllers are assumed to be nonadaptive and

will be referred to by the generic term “model-predictive

control” (MPC).

disadvantage of the typical MPC formulation (DMC,

QDMC, etc.) is that

is designed for

step

changes in the

setpoints and the output disturbances (Prett and Garcia,

1988). This may be a good assumption in the case of the

setpoints but is rarely

in the case of the disturbances.

Performance then depends on the degree to which the real

disturbances are “steplike” and on how well shortcomings

in the design assumptions can be overcome by controller

tuning. In the extreme case of a ramp disturbance at the

output of a nonminimum-phase process, the performance

can be very poor (see the examples).

Zafiriou and Morari (1987) and Astrom and Wittenmark

(1984) have pointed out that the disadvantages cited above

can be overcome through the use of a “2-degree-of-

freedom” structure, Le.,

which the controller treats the

effects of disturbances (as seen at the plant output) dif-

ferently from the way it handles setpoint changes. Morari

and Zafiriou (1989) cover the design of robust %degree-

of-freedom controllers but do

not

show how inequality

constraints can be incorporated.

For problems without inequality constraints, the LQG

design procedure (in which state estimation

combined

with a linear state feedback controller) leads naturally

controller with a 2-degree-of-freedom structure (e.g.,

Astrom and Wittenmark, 1984). Prett and Garcia (1988)

have shown that for the unconstrained problem the DMC

disturbance assumptions are, in fact, equivalent to the use

of a specific state estimator gain matrix in an LQG con-

troller. Navratil et al. (1989) and Li et al. (1989) show that

state estimation can be incorporated in MPC but do not

consider constraints. Marquis and Broustail (1989) have

:il~~)

rlrwriht-rl

ombination

MPC and state estimation

which has been operating on industrial problems (with

constraints) for more than 6 years.

Other methods have been proposed to deal with the

problem of disturbance rejection in an IMC (or Smith

Predictor) structure. For example, Wellons and Edgar

(19851, Yuan and Seborg (19861, and Svoronos (1986) use

a form of observer to estimate the magnitude of a hypo-

thetical load disturbance. The estimated disturbance is

then incorporated into the prediction of future outputs.

The philosophy is similar

the state estimation approach,

but their work is intended for

SISO

systems without

constraints.

The main goal of this paper is

show that

can be

advantageous

use state-estimation techniques other than

the approach used in DMC. The next section gives the

details of the algorithm. The final section is a demon-

stration of its properties for two example problems.

Derivation

the Algorithm

Plant.

The states of a plant are never fully measurable.

In fact, one would not know the “parameters” or even the

“order” of the plant, since it is generally a time-varying,

nonlinear, distributed-parameter system. The values of

the signals leaving the controller are known, and one can

measure the plant outputs, but nothing else in the plant

is accessible. As shown in Figure

an unmeasured dis-

turbance corrupts the control signal before

reaches the

plant,

even the values of the plant inputs are never

known exactly. The distinction between the known in-

ternal model and the unknown plant is an essential ele-

ment of past research and

is reemphasized here.

Suppose that we approximate the plant by a discrete-

time, linear, time-invariant, state-space model:

(1)

(2)

where

is the current sampling period,

is a vector of

states,

is a vector of

p!an_t input?,

is a vector of

measured outputs, and

CP,

and

are constant ma-

trices of appropriate size. The plant inputs can be con-

sidered to be of various types, as shown in Figure

iiT(k)

[mT(k)

vT(k)

*T(k)

ZT(k)

eT(k)lT

(3)

where

m(k)

is a vector of

manipulated variables (i.e.,

control signals),

v(k)

is a vector of

measured disturb-

ances, and

*&),

Z(k),

and

B(k)

are vectors of unmeasured

disturbances of length

fiz,

and

respectively.

course,

*(k),

Z(k),

and

6(k)

could be combined into a single

1990

American

Chemical Society

f(k+l)

&f(k)

h(k)

Y(k)

Cf(k)

Dti(k)

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