Certainty equivalence property (CEP) is another name for model-predictive control (MPC). Based on CEP, it does not differ whether you include additive noise into calculation or not. It means it does not matter whether you consider the system as xdot=ax+bu+ w, or without additive disturbance variable as: xdot=ax+bu, anyway the optimal control-law is indifferent for the two linear systems, regardless of including disturbance variable (w), or neglecting it. The property has been proven in the following paper for the linear case (LQG):
Article The Certainty Equivalence Property in Stochastic Control Theory
Moreover, as I've tried it myself, even for nonlinear case, the CEP strongly holds, as long as noise is additive (xdot=f(x,u)+w), not multiplicative as (xdot=(f(x,u)+w*x).
Both optimal regulator, and MPC, give similar control-law (or gains) for the stochastic system (xdot=f(x,u)+w), with or without disturbance variable w, but not for the stochastic system (xdot=f(x,u)+w*x). In the case of (xdot=f(x,u)+w*x), ignoring the multiplicative disturbance is not allowed, and if multiplicative term (w*x) is neglected, the optimal control-law found, would be dissimilar for the two-type of systems, and might not be robust with respect to multiplicative disturbance. This property is simply the certainty equivalence property (CEP), or through another terminology, the model-predictive control (MPC). Notice that, it seems MPC is based on discrete optimization for minimum cost, hence MPC has some other variant setup and independent format from optimal-control, but they are analogous as optimal output-feedback control is usually analytic and continuous, while MPC seems as discrete version of that, based on optimization.
MPC is resourceful when analyzing disturbance is not simple, hence CEP is used to authorize ignorance of disturbance variable.
Dear Saeb,
-The optimal control is an open loop control while MPC is a closed loop control.
-When the optimal control is calculated a sequence of input signals that guide this system, If we apply this input sequence to the real system, there is a gap between the calculated behavior and the actual behavior due to the mismatch system models and disturbances. As for the MPC control, the correction is integrated because you recalculate your optimal sequence after having periodically reset the initial conditions of the MPC problem to the actual state of the system.
Best regards
Dear Mohamed-Mourad Lafifi.
Thanks for your comment. I should notify that, optimal control is not always open-loop. It could have feedback feature as dynamic programming and quadratic regulator. Even the numerical optimal control techniques, could have feedback feature as the control input depends on state as u(t)=k(x(t)).
Regards.
Dear Saeb,
As Dr Mohamed-Mourad Lafifi said, optimal control is an open loop control and optimal state gain calculations is performed offline. َSo this calculations is based on the plant model and the initial state value. Also note that u(t)=k(x(t)) in the optimal control is a numerical trick for reduction of online computations and is a tools to reducing memory usage of the optimal control algorithm. If there is any mismatch in the state estimation or plant model, the controller response is not optimal and system may lead to the instability.
Best Regards
Dear Mr. Majid Shariat. Thank you for the wise comment. Regards.
The main difference is that is that LQR is solved either over an infinite horizon or a fixed finite horizon whole MPC is solved over a receding horizon. Hence it can handle plant variations and hard constraints better.
https://en.wikipedia.org/wiki/Model_predictive_control
read the section : MPC vs. LQR in the URL.
A serious drawback of eMPC is exponential growth of the total number of the control regions with respect to some key parameters of the controlled system, e.g., the number of states, thus dramatically increasing controller memory requirements and making the first step of PWA evaluation, i.e. searching for the current control region, computationally expensive.
The drawback of the approach however is that the size of the problem grows exponentially with the number of uncertainties and the prediction horizon
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