MPC & PID

MPC & PID :

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ContentsWhy GPC?Why PID?Why GPC & PID?MPICMethods of MPC & PID:Auto-tuned PID controller using GPCHierarchical structure controlProposed new cost function with PID parametersAn improved PID-GPC algorithm

Why GPC?iterative optimizationhas strong adaptability for uncertainties such as modeling errors and environment disturbanceprediction with perfect theory frameFor long time-delay, non-minimum phase and non-linear processesWhy GPC?The purpose of taking new measurements at each time step compensate for :unmeasured disturbancesmodel inaccuracythat causing the measured system output to be different from the one predicted by the modelWhy PID?simple control structures strong robustnesshigh reliabilityeasy to implementwidely used for most industrial control systems

Why PID & GPC?PID control cannot achieve desirable effects:the practical production processes are often nonlinear and time-varying

To take the advantages of both GPC and PID, it is feasible to combine them together to form up(PID-GPC), by adding proportion, integral and differential structure to the conventional cost function.MPICProposed by Towhidkhah 1996. (Grid Method)Kordari, 1380.(Gradient Method):Cost function:

Gradient method:

Auto-Tuned PID Controller Using GPCPIDcontrol gains are automatically tuned by using a MPC methodAuto-Tuned PID Controller Using a Model Predictive Control Method for the Steam Generator Water Level by Man Gyun NaIEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 2001,cited 34.

Cost function

CARIMA model

The optimal prediction is derived by solving a Diophantine equation, whose solution can be found by an efficient recursive algorithm.Diophantine equation:

There exist unique polynomials by By taking and

It was proven that stability could be guaranteed if the horizons and input-weighting factor were correctly chosen.(Clarke & Mohtadi,1989)

If a controlled process is a second-order linear system:

The first optimal control input is

A standard PID controller can be expressed as

Hierarchical structure controlIncorporating GPC into the PID controller because of: Once GPC are adopted, the existing equipment, especially hardware parts, has to be upgraded at a large cost. The engineering level and the complicated algorithm will prohibit the implementation of the advanced control method. The advanced tuning methods usually lack explicit specifications and the plant operators are unfamiliar with the parameters tuning.consists of two levels: basic level: PID controller , which is not likely to give satisfactory performance as operating conditions changeoptimization level

the closed loop performance equals that of the standard GPCthe practical controller still remains a PID structure to plant personnel

The drawback of the conventional PID controller is that it has three degrees of freedom in tuning, which is difficult for plant engineers to tune the parameters to meet different specifications.

PID controller in the basic level:

CARIMA model:

Cost function

the optimal parameter vector for minimization ofTo achieve an optimal control variable at time interval k, a second-order Taylor expansion is given by

SimulationHVAC: heating, ventilation and air-conditionit is common for a cooling coil unit to be controlled to maintain the supply air temperature Tao at a set point value

This is a non-linear model, and the model is linearized at three different operating condition

Stage1: a short rise time and no overshootStage2: a large oscillationStage3: a large oscillationbecause conventional PID controller is chosen based on the stage 1 operating condition, system performance is unsatisfied for different operating condition.

To verify the robustness to disturbance, a white noise with 0.1 dithers are introduced in the controlled loop. a satisfied control performance with small oscillation is obtained.

Proposed new cost function with PID parametersConventional cost function:

New cost function:

Prediction:

Optimal solution

An improved PID-GPC algorithmGUANG-YI CHEN, PING REN, HAI-LONG PEI,2008mathematical difficulties:requirement of the solutions of a set of Diophantine equations inversion computation of higher dimensional matrixincreasing the computation quantity in PID-GPC algorithm

To broaden the application of PID-GPC, improving thecontrol algorithm by: lightening computational burdensimplifying the complexity of online implementation

Improvement It is assumed that the control increment has little change out of control horizon, that is u(t+j)=0, (j>NU).making the control increment u(t+j) (j=0,1,2,,NU-1) tend to 0 in NU steps smoothly

is a matrix of the dimension NUNU, and its inverse operation is very complex. just a scalar. that is just picking the reciprocal of a number

simulation

with the same value of k ,the bigger the value of q,the faster of the systems output response during the initial time

with the same value of q ,the smaller the value of k, the faster of the systems output response during the initial time.

There is no overshoot in improved PID-GPC, although it has a relatively slower output response.

the control variance is significantly smaller in improved GPC-PIDThe improved algorithm has the advantages of good control performance, less on line computation, and good application prospect.

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