Optimal & Model Predicive Control

The ojective of this notebook is to understand the MPC in the overall framework of control design for a dynamic system.

This first notebook takes you from the very simple concept of a dynamical system on which one can act using actuators to achieve some goal (control objective) to the role of MPC design as a way to elaborate a selective dynamic inversion of the model (objective -> actions) based on a dynamical modeling of the system.

This inversion is done using optimization procedure that will be studied later on. The notebook also shows some examples of the penetration of MPC in the academic and industrial world. It also shows some example of use of the MPC control design in the Gipsa-lab (where the author works!).

The previous notebook showed that MPC control design involves three main keywords that we recall hereafter:

1. Model-Based Prediction
2. On-line Optimization
3. Real-time

In this notebook, we shall learn a little bit more about The first weyword, namely:Model-Based Prediction. It is shown how starting from a Ordinary Differential Equations (ODEs) representation of a dynamical system, one can design a prediction engine via three different ODE's integration options.

In the previous noteboook, we learned how a dynamic model (in ODE form) can be used to make prediction (simulate) the dynamical system for given initial state x0 and a given set of parameters that define the dynamical beaviour of the system. As it has been mentioned in the first note book, this sequence 𝑈 has to be chosen so that the resulting trajectory (given the initial state 𝑥0) achieves in the best possible way the control objectives. In other words, the sequence has to be chosen to be the sequence that minimizes some cost function (or maximizes some performance index), namley, it has to be the optimal choice that solves an optimization problem.

This is more formally presented in this notebook!

Once the optimal control problem is formalized, a first attempt to solve it using the scipy.optimize module is studied In order to show that it is not compatible with real-time computation of MPC. This will justify the need for the next notebook.

In the previous noteboook, we showed that using the general purposes nonlinear programming solvers in order to solve the constrained open-loop optimal control problems can lead to computation times that might be too large to be compatible with repetitive solution of these problem as it is required in the MPC implementation.

That is the reason why a dedicated solvers have been developed these last years in order to come out with NLP frameworks that might solve the underlying problem in faster way.

One of these framework has been developed by researchers from the control system community or associated to it. This is the Casadi framework.

In the previous noteboook, we showed that using the general purposes nonlinear programming solvers in order to solve the constrained open-loop optimal control problems can lead to computation times that might be too large to be compatible with repetitive solution of these problem as it is required in the MPC implementation.

That is the reason why a dedicated solvers have been developed these last years in order to come out with NLP frameworks that might solve the underlying problem in faster way.

One of these framework has been developed by researchers from the control system community or associated to it. This is the Casadi framework.

In the previous notebook entitled, it has been shown how to use the Casadi framework in order to formulate and solve the open-loop optimal control problem. The result of the solution step is hopefully the best sequence of actions, namely, the one that minimizes the cost function while respecting the constraints.

Now because of the presence of uncertainties and model mismatch, applying this sequence in open-loop (without feedback based on the measurement) might lead to undesired drift or even instability of the resulting behaviour.

That is the reason why the MPC implementation consists in applying the first control in the optimal sequence until the next sampling instant and at the new reached state, formulate the new problem and solve it to get the new best sequence of actions, apply the first control in this new sequence and so on.

This is what we are going to do in this notebook. More precisely:

1. We shall first reproduce a part of the content of the previous notebook in order to define the simulator and the open-loop optimal control solver.
2. From this solver we shall define a feedback function that implements the principle of MPC recalled above (that extracts the first control in the optimal sequence to apply it to the system).
3. Having the feedback function, we shall simulate the closed-loop under the MPC design under different assumption regarding the model mismatches and the uncertainties. While doing so, we shall compare the behavioir of the system under the open-loop and the closed-loop control in order to enforce the understanding of the role of the feedback in getting the desired behaviour in closed loop.

This notebook presents the most important checkpoint of this course since it is here where the complete solution of a state feedback design is done on the crane example. The system's equations are given as well some elementary functions that you have then to use in order to design the MPC state feedback.

The feedback is then applied to a detuned model of the system (different from the one used in the design of the MPC feedback) in order to check the robustness of the solution to model mismatches that are unavoidable in the real-life.

In the previous notebook, it has been shown how to design a General Nonlinear MPC control for any system that is given by general set of nonlinear ordinary differential equations and using a general expression of the constraints function.

It is a fact however that in many situations, people work with a linearized model around some operating point and would like to handle constraints that are affine in the state and the control vectors. Moreover, it is very often that the cost function is expressed via a quandratic form of the state and the control trajectories.

This notebook show via a simple example that when the problem to be solved is a Quadratic Programming problem, it is better to solve it using a dedicated QP solver rather that using a general purpose solver.

The previous notebook underlined the fact that knowing that the underlying optimal control problem is quadratic can help us accelerating the resolution of the optimization problem by a significant facor thanks to the far more efficient QP solvers.

In this notebook, we shall through the steps that enables to put the underlying problems into QP form. This is possible under the following conditions:

1. The model of the system is affine in the state and the control
2. The cost function is exressed as a quadratic form in the state and control trajectories.
3. The constaints are expressed in affine form in the same variable.

In this notebook, we restrict ourselves to the particular case where the constraints are box constraints (saturation) in the state and the control. It is left to the reader to generalize to the complete general case without difficulties.