'Optimisation' is used in a wide variety of applications apart from optimal control. In optimal control, sometimes the controller achieving the optimum has a closed-form (as a function of the past state - as in LQR); sometimes it needs to be evaluated numerically at each step (as in MPC). Numerical optimisation techniques used in MPC (and elsewhere) are described, for example in Boyd and Vandenberghe's book on convex optimisation.

The main difference is in the goals:

In "optimization", the main goal is to achieve the "best" value of a cost function.

In MPC, the main goal is to achieve stability, robust stability etc. Technically, this is done through the application of a part of the optimal control.

MPC is a simplified version of optimal control which is for dynamical systems. Optimization is for static system problem. However, MPC use optimization method to find the control at each time step.

I think that the answer by Éva Gyurkovics is useful. Roughly, one could say that  MPC is a control method based on repeated optimal control, see also the book by Rawlings and Mayne. But let me explain this step by step.

First consider the following optimization problem

minu(.) J(x0,u(.)) = \int_{0}^{T} F(x(\tau), u(\tau))d\tau

subject to

dx/dt = f(x,u), x(0) = x0

where J(x0,u(.)) is the objective functional to be minimized. More precisely, this problem is called optimal control problem (OCP), since a control system appears a constraint of the optimization. The constant T, which appears as upper bound in the integral, is called optimization horizon. Note that an optimal solution to OCP(x0) is an input trajectory u(t,x0) and the corresponding state trajectory x(t, x0, u(.,x0)), both of them are defined for all t \in [0,T]. Clearly, the solution to the OCP depends on the initial condition x0 of the ODE. Thus, we write OCP(x0)  and the optimal input is written as u(t,x0). For sake of simplicity, I have not listed any state or input constraints. I also wrote the OCP in continuous time, discrete time formulations look similar. I will come back to this later. 

Now, you should note that the term optimal control refers to the mathematical theory that deals with problems like OCP(x0). And model predictive control (MPC) is a control strategy that uses the following simple idea:

1. At a sampling instant tk obtain the state of the system to be controlled, i.e., x(tk).

2. Solve OCP(x(tk)) over the horizon [tk, tk+T].

3. Apply u(t,x(tk))  for t \in [tk, tk+\delta). SET k --> k+1. GOTO 1.

Thus,  MPC can be understood as repeated optimal control. In essence, in MPC one controls a system via inputs obtained from solving OCP(x(tk)). However, this does not imply that the input sequence obtained via an MPC strategy is optimal.  What does this mean?

Consider a specific OCP:

OCP1(x0) defined over a long horizon T1

Instead of solving the OCP1(x0) over the complete horizon from 0 to T1, one could apply an MPC strategy and solve a sequence of OCPs over a shorter horizon T2 < T1 in an MPC manner. This means, first solve OCP1(x0) defined over the short horizon T2, second store the first part of this solution, lets say from 0 to time \delta

Optimization is a very general concept and yes MPC is an optimization based approach. You optimize a cost function subject to some constraints. In MPC you consider a finite prediction horizon and solve an optimization problem subject to the system model (in basic case) and physical (and other) constraints.  

You may consider some constraints for insuring stability or relaxing some of constraints for achieving feasibility.  

The short answer is that you set up an optimization problem based on your control objective and solve it to find the best input(s). 

Hi, real-time optimal control is an open-loop control, and MPC is a closed-loop control. If I added the initial states corrections into real-time optimal control, is real-time optimal control model same as the MPC? That is to say, can the optimal objectives set the same between real-time control and MPC?

They both are closed-loop control. MPC is an approximation of Optimal Control over infinite time horizon with a control over a finite time horizon closed loop control. The former is not real-time feasible in general while the latter is.

Dr. Hossein Sartipizadeh is correct in his views.

I think these two expressions are totally 2 separate concepts. Optimization is a huge topic, but MPC is a kind of control model. When we need to solve a MPC problem, we can use optimization method or other mathematical tools.

Predictive Control fed by an Indirect Converter with and without energy storage using PSO Optimization

Model predictive control is one of the optimal control strategies which capture the dynamics of the system over a time horizon. It is basicaly a control system designing approach. Where as optimization is a very generic tool. In simple words, optimization techniques are used to find best possible control settings in MPC, but not the other way round. MPC is a way of modeling dynamical system, whereas optimization is a tool to find the best possible settings not only in it but also in other scenarios where dynamic behaviors are absent.

I think that, the main difference between optimization and MPC is in the main goals. Firstly, optimization is a generic approach which the main idea is to find the "best" value of a cost function (optimal value ). On the other hand, MPC is one of the optimal control approaches ( MPC is a control strategie based on repeated optimal control ) which take the system dynamics over a horizon subject to constraints and the main goal is to achieve stability, robust stability etc.

the main components in an MPC are:

• The plant model
• An objective function
• A state estimator
• An algorithm for solving constrained optimization problems

The following actions take place on a cyclic basis and are repeated with equidistant intervals, of which the sampling time is chosen with respect to the time scale of the controlled process and optimality considerations.

so generally, the optimal problem is a part of MPC.

The recursive least squares algorithm (RLS) allows for (real-time) dynamical application of least squares regression to time series. While investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS algorithm with forgetting factor (RLS-FF) to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, while giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter (pilot plant). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...