Model-Predictive-Control

This repo includes the MATLAB codes for a DMC and an EPFC controller.

DMC

A DMC controller is implemented to control a linear system. The linear system is actually the linearized version of a nonlinear system, which as an example is considered to be a cart with a nonlinear spring, as that used for the EPFC controller in here. A very simple explanation of DMC is included in this file.

The functions and scripts are as follows:

DMC functions

• simulate_linear_system and [simulate_nonlinear_system(DMC/simulate_nonlinear_system) simulate the systems by running the whole simulation, calling the MPC controller at each timestep to get the control input, applying the control input to the systems and moving the dynamics forward.
• dmc_linear determines the control input for the linear system using the DMC logic
• dmc_nonlinear is the same as dmc_linear with the difference that it moves the nonlinear system's dynamics forward to calculate Ypast.
• update_linear_state moves the linear system's dynamics forward one step. You can insert your linear system dynamics here.
• update_nonlinear_state moves the nonlinear system's dynamics forward one step. You can insert your nonlinear system dynamics here. A considerably smaller step size (compared to control sample time) should be considered when simulating the nonlinear system itself. The substeps parameter can be tuned for that (keep it at least at 10 for a realistic simulation).
• plot_simulation_results, plot_comparison_results, plot_comparison_results_for_Q and plot_comparison_results_for_alpha are used for plotting the results.
• plot_static_gain is used for analyzing the nonlinear system's static gain at a given point

DMC scripts

• plot_step_response is used to analyzing the linearized system's step response to obtain a reliable model horizon N
• general runs the simulation with the set parameters and saves the results. Parameters include:
  • Ts sampling time
  • tf final simulation time
  • N model horizon
  • u0_nonlinear the nonlinear system's operating point control input
  • u0_linear initial value of the linearized system's control input
  • bias and span used for generating the reference signal
  • P prediction horizon
  • M control horizon
  • alpha filter coefficient for filtering the desired reference
  • Q weight of each future output sample (set not as a matrix, but as a vector. This will be later used to construct a diagonal Q)
  • R weight of each future input sample (set not as a matrix, but as a vector. This will be later used to construct a diagonal Q)
  • N_model not actually used. Set it equal to N
  • x0 initial condition of the system
  • is_programmed if set to true, the reference will be considered programmed, meaning that the controller is aware of the future values of the reference signal.
  • is_open_loop if set to true, the disturbance D term will not be
  • noise_power power of white noise on the output (in dB)
  • dist_amp the amplitude of disturbance on the output. The disturbance is considered to be a pulse signal.
  • dist_start_time the time when the disturbance is applied
• compare_Ms compares different values of M.
• compare_Ns compares different values of N.
• compare_Ps compares different values of P.
• compare_Qs compares different values of Q.
• compare_Rs compares different values of R.
• compare_alphas compares different values of alpha.
• [pulse](DMC/pulse0 pulse reference signal
• sinusoid sinusoid reference signal

EPFC

An extended PFC controller is implemented to control a nonlinear system. The exemplar system considered is a cart with a nonlinear spring attached to it, as shown in the image:

The system state space representation is as follows:

x1_dot = x2
x2_dot = -0.33 * exp(-x1) * x1 - 1.1 * x2 + u
y = x1

Any other system can be used instead. It's simply a matter of system definition in one function, which will be shown later. A change of variables is considered as:

v = u + x1

This change of variables allows the new system to have a one-to-one relationship between v and y. If a system does not need a change of variables, line 162 of run_simulation can simply be changed to:

u = v;

A very simple explanation of EPFC is included in this file.

The functions and scripts are as follows:

EPFC functions

• run_simulation runs the simulation given linearization method: set to 'perturbation' or 'jacobian'. Jacobian uses the provided Jacobian of the system (which should be provided in linearize_dynamics). The perturbation method uses the predictive model and applies two inputs. One where the input is kept unchanged as it is at this sample time, and one with a small change in the control input value. By dividing the change of the output in the two cases by the change of the input, a linearized model is achieved.
• linearize_dynamics provides the jacobian for the 'jacobian' lineariztion method. You should set this function according to your system's dynamics.
• get_step_response_nonlinear provides the outputs for a given constant input. It is used in 'perturbation' linearization technique.
• update_nonlinear_state moves the nonlinear system's dynamics forward one step. You can insert your system dynamics here. A considerably smaller step size (compared to control sample time) should be considered when simulating the nonlinear system itself. The substeps parameter can be tuned for that (keep it at least at 10 for a realistic simulation).
• update_nonlinear_state_actual is used when model mismatch is considered. You can skip setting up this function if your model is exact. If not, use the model at hand in update_nonlinear_state and in update_nonlinear_state_actual write the actual system model (unknown).
• plot_simulation_results and plot_comparison_results are used for plotting and saving the outputs. The outputs are saved in a folder called simulation_results in downloads. If no such folder exists, one will be created.

EPFC scripts

• plot_static_gain is used for analyzing the nonlinear system static gain and consider a change of variables if necessary
• main runs the simulation with the set parameters and saves the results. Parameters include:
  • Ts sampling time
  • tf final simulation time
  • N-model not actually used in PFC, but set it to something larger than all the mus
  • mu output coincidence points
  • m input coincidence points (degrees of freedom)
  • q weight of each output coincidence point (same size as mu)
  • r weight of each input coincidence point (same size as m)
  • psi filter coefficients for filtering the desired reference (same size as 'mu')
  • u_min, u_max control input saturation lower and higher limits. Note that no saturation is actually applied to the input, but these are passed to the MPC controller which then calculates the control input as to not violate these constraints.
  • x0 initial condition of the system
  • u0 initial control input
  • beta, Ni and TOL are used when finding D_nonlinar.
  • method determines the linearization method: 'jacobian' or 'perturbation'
  • is_programmed_ref: if set to true, the reference will be considered programmed, meaning that the controller is aware of the future values of the reference signal.
  • noise_power power of white noise on the output
  • dist_amp the amplitude of disturbance on the output. The disturbance is considered to be a pulse signal.
  • dist_time the time when the disturbance is applied
  • dist_duration the duration of the disturbance
• one_input_one_output simulates the system for one input and one output coincidence points.
• one_input_three_outputs simulates the system for one input and three output coincidence points.
• one_input_three_outputs simulates the system for three input and three output coincidence points.
• compare_coincidence_points compares the three cases above.
• compare_input_coincidence_points compare different sets of input coincidence points.
• compare_output_coincidence_points compare different sets of output coincidence points.
• compare_constrained_vs_unconstrained compares the controller performance in presence and absence of input constraints
• compare_linearization_method compares 'perturbation' and 'jacobian' linearization methods.
• compare_q_values compares different values of q.
• compare_r_values compares different values of r.
• compare-psi_values compares different values of psi.
• compare_nominal_vs-uncertainty compares the controller performance in presence and absence of uncertainty in the model.
• compare_programmed compares programmed vs unprogrammed reference signal.
• noise_and_disturbance is just main with more noise and disturbance to see their effects.
• initial_condition is just main with different initial conditions to see their effects.