MPC with Hard constraints

One of the advantages of using MPC over other controllers is it allows operation closer to constraints compared with conventional controls, which leads to more protable operation. Often these constraints are associated with direct costs, fre-quently energy costs. For instance, in a manufacturing unit the power consumption must be kept as minimum as possible with same level of production, this is a con-straint on manufacturing process. Concon-straints can be present in both input as well as output. Most commonly the input constraints on the control signals, that is to the process or manipulated variables and rate constraints are hard constraints.

This may be because of various reasons like saturation, physical limitations etc., These constraints can never be violated.

For example in Equation (2.6), the inequality constraints

"Ωθ≤ω"

is called hard constraints as the condition needs to be strictly satised when cal-culating the optimization solution. The best example of hard constraints in real time is the high and low limit of the manipulated variables which cannot be varied beyond the limits because of the physical restrictions like vibrations etc.,

Clarke (1988) has proposed a generalized model predictive controller with hard input constraints. The controller is based on the minimization of long-range cost function. The model used for controller is CARIMA model. The closed loop performance of the controller is investigated using simulation.

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Iino et al. (1993) have proposed a new method by modifying the Generalized predictive control. Firstly, a Kalman lter based predictor is introduced in order to improve the robustness of the predictor against noises. Secondly, a time-dependent weighting factor is introduced into the MPC's quadratic type cost function, in order to improve the transient response characteristics. Thirdly, a parameter tuning method is proposed that adjusts the weighting factors in the cost function considering robust stability of the control system. Finally, the proposed MPC method with and without constraint conditions that are the upper/lower limits and rate limits for both manipulation variables and process control variables, is formulated. The controller is tested in an ethylene plant's dynamic simulator. The models are obtained by simple step tests in the plant. ARMA types of models are used for prediction.

A design method of LQ optimal control law is considered for constrained continuous-time systems by Kojima and Morari (2004). Here the control laws are obtained based on quadratic programming. The control law converges to exact solutions by introducing singular value decomposition for nite-time horizon linear systems.

By employing the control problem to a double integrator with constraints it is clar-ied that the receding horizon control is equivalent to that of the state feedback control where the gain is calculated by a piecewise ane state functions.

Guzman et al. (2009) have provided a solution for output tracking problem for uncertain systems subject to input saturation. In order to tackle constraints and modeling errors an external supervisory control method is proposed. Thus a cas-cade loop with any type of inner control and a GPC for outer loop is considered.

A robust constrained Linear Matrix Inequality (LMI) based approach is developed as a solution to control such systems. The existing control loop is rst converted into state space representation and LMI is used to provide state space feed back for the inner loop controller. The controller is then tested in an integrator plant with delay with a inner loop PI controller. The inner control loop is studied with PI controller in the presence of uncertainties and it is found that stability prob-lems occur. Then the controller is included with the GPC for controlling the inner loop considering input saturation and it is found that the performance results also

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ensuring constrained robust stability.

A survey of some of the reported work on the model predictive controllers with hard constraints relevant to the present work is given in Table 2.2

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Table 2.2 Reported work on design and implementation of MPC with hard constraints

S.No Author Problem Comments

1 Clarke and

Tsang (1988)

Generalised Predictive Control with hard input

constraints. Minimize long range cost function. Modeling the GPC - CARIMA model.

Hard constrained MPCs may sometime lead to input saturation that is not desirable.

2 Iino et al., (1993)

A new input/output MPC with frequency domain technique and its application to ethylene plant. MPC with hard input and output constraints Techniques - DMC and MAC. ARMA type of model for prediction.

Hard constrained MPCs may sometime lead to input saturation that is not desirable.

3. Kojima and

Morari (2004)

Design method of LQ optimal control law for constrained continuous time domain

Systems. Control laws based on QP. Control problem a double integrator with constraints.

Receding horizon principle used.

Results verify the control equivalent to state feed back with gain calculated by piecewise affine state functions 4 Guzman et al.,

(2009)

A robust constrained reference governor approach using linear matrix inequalities . Two different loop controllers.

An outer Loop MPC control with Linear Matrix Inequalities and local control loop for maintaining the variables.

Solve a set of constraints described by LMI and BMI complex to be extended for constraints other than input

constraints.