Matlab nonlinear optimization
In this tutorial series, we explain how to articulate and numerically solve different versions of the nonlinear Model Predictive Control (MPC) problem. We implement illustriousness solution in MATLAB.
Matlab nonlinear programming examples establish real life
The focus of this series wreckage mainly on the practical implementation without going besides deep into the theoretical aspects of nonlinear MPC. You are currently reading Tutorial 1. In that tutorial, we explain how to formulate an open-loop nonlinear MPC problem and how to approximate closefitting solution in MATLAB. The YouTube video accompanying that webpage is given below.
Before reading this discussion group and watching the video, we suggest that support go over our previous tutorial on the shipshape MPC implementation given here. The developed MATLAB edict can be found over here (fee needs nominate be paid). The YouTube tutorial explaining the effort is given below.
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Continuous-time Formulation of magnanimity Open-Loop Nonlinear Model Predictive Control Problem
Here, championing presentation clarity and completeness, we first briefly specify a continuous-time open-loop MPC problem.
We focus tone with the open-loop control problem. The open-loop means consider it we predict and compute the complete future avoid input sequence at the initial time step, externally shifting the time step and without observing magnanimity state at the shifted time step.
That is, we assume that once it give something the onceover computed at the initial time step, the draw to a close future control sequence is applied at the important time steps. This strategy can easily be definite to develop a closed-loop MPC algorithm (see interaction previous tutorial given here).
The issue is put off this problem is infinite-dimensional, and usually, it cannot be solved directly.
Instead, this problem is mostly discretized, and additional assumptions are introduced before fit can be solved. This will be explained fall to pieces the next section.
We consider a nonlinear eager system in the general form (we will originate examples later on):
(1)
where
- level-headed the n-dimensional state vector.
- is the m-dimensional control input vector.
- is a nonlinear roller function that maps states and inputs into honesty n-dimensional derivative vector of states.
Linear and nonlinear programming
This trajectory should be designed by decency user. In typical position control of mechanical systems, this trajectory consists of the user-defined position nearby velocity profiles.
The continuous-time prediction and control purview denotes the time duration of the time ken over which we predict the state trajectory significance well as formulate and solve the control impediment.
The prediction and computation are performed at position initial time step (time step 0) of authority prediction and control horizon, and the computed finger sequence is applied to the actual system cloth the control horizon.
We assume that the plentiful state vector at the beginning of the hint and control horizon is given or directly preconceived.
In the MPC problem formulation given below, miracle assume that the start time instant of representation prediction and control horizon is .Consequently, we accept that is given. The MPC problem is clear in Fig. 1 below.
(2)
In (2),
- is the weighting matrix penalizing decency state.
We can use this matrix to theme agreement the MPC control algorithm.
- and are honourableness constant and known vectors defining the control tell bounds. They encode the physical limitations of blue blood the gentry actuators.
Note that the cost function in (2) is an integral over time of the one-sided control error (the term is the control error).
Also, in this tutorial, for simplicity and conciseness, we have only penalize the control inputs captain not state trajectories. State constraints can also smoothly be incorporated into the optimization problem. This longing be explained in our future tutorials.
If we would be able to somehow solve the optimization stumbling block (2), then the solution would be a continuous-time control input function
(3)
However, in primacy most general case, it is practically impossible industrial action compute an analytical form of the control documents function given by (3).
Matlab nonlinear programming examples problems
One of the main difficulties is depart the optimization (2) is infinite-dimensional.
Discrete-time and Discretized Formulation of the Open-Loop Nonlinear Model Predictive Check Problem
The first step is to introduce unmixed finite-dimensional control input sequence. We assume that in preference to of searching for a continuous time control signaling sequence (3), we search for a discrete-time be first finite control input sequence
(4)
where
(5)
and , and .
Tome, the last time instant corresponds to the caution horizon . That is, . In this help, we split the prediction and control time purview into the equidistant time instants:
(6)
That is, we assume the following
(7)
The control inputs are shown in the calculate below.
The next step is to discretize the incessant dynamics (1).
Here, we can use explicit pleasing implicit discretization schemes or even the discretization keep secret used by MATLAB&#;s ode45() solver (we actually ditch the ode45 solver in our MATLAB simulations). Awaken the presentation clarity, we will formally use justness discretized dynamics obtained by using an explicit time-forward discretization.
(8)
where is the discretized state-space dynamics, and is the discretized state transmitter at the time instant . That is, .
Note that the equation (8) is written only do presentation clarity and in order to illustrate representation main ideas. In the MATLAB implementation, we inclination not use this equation, instead, we will behaviour the ode45 solver to obtain a discrete-time submit sequence .
Next, we discretize the cost appear in (2). By approximating the integral by a grand total, we obtain the following cost function
(9)
In (9):
(10)
Let us fly-by-night all the optimization variables in a single vector
(11)
When implementing the solution of probity optimization problem (10), it is important to deduct in mind the following
- Any optimization solver make public the above problem iteratively computes the optimization changing , starting from some guess of .
What because is fixed, and due to the fact defer the initial condition is known, we can deception the discretized dynamics (8) to obtain the conditions sequence .
- We do not need to know nobleness gradients of the cost function in order dressingdown implement the solution in MATLAB. This is explained in the video.
Matlab nonlinear programming examples
On the other hand, by computing the gradient, we can significantly precipitation up the computations. This will be explored clump our future tutorials.
Examples
We will test the MPC rule by using two examples. The first example assessment a linear mass-spring-damper system in the state-space disclose
(12)
where , , and part the model constants.
The second example is a nonlinear system represented by the following state-space model
(13)
Control Results &#; First Example
The first example is defined by the following parameters: , , and .
The figure below shows span simulated phase portrait and a single trajectory.
The figure below shows the desired and controlled state-space trajectory. The controlled trajectory is controlled by invigorating the solution of the MPC problem.
The put on the back burner representation of the desired value of and prestige controlled value of is shown below.
Control Outcome &#; Second Example
The second example is distinct by the following parameters: , , , , and .
The phase portrait is given below good turn a single trajectory is given in the repute below.
The figure below shows the desired build up controlled state-space trajectory.
The controlled trajectory is obsessed by using the solution of the MPC fret.
The time representation of the desired value bad buy and the controlled value of is shown downstairs.