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Suppose that you want to optimize the control parameters in the Simulink^{®} model optsim.mdloptsim.mdl. (This model can be found in the optim/optimdemos folder. Note that Simulink must be installed on your system to load this model.) The model includes a nonlinear process plant modeled as a Simulink block diagram.
The plant is an under-damped third-order model with actuator limits. The actuator limits are a saturation limit and a slew rate limit. The actuator saturation limit cuts off input values greater than 2 units or less than –2 units. The slew rate limit of the actuator is 0.8 units/sec. The closed-loop response of the system to a step input is shown in Closed-Loop Response. You can see this response by opening the model (type optsimoptsim at the command line or click the model name), and selecting Run from the Simulation menu. The response plots to the scope.
The problem is to design a feedback control loop that tracks a unit step input to the system. The closed-loop plant is entered in terms of the blocks where the plant and actuator have been placed in a hierarchical Subsystem block. A Scope block displays output trajectories during the design process.
One way to solve this problem is to minimize the error between the output and the input signal. The variables are the parameters of the Proportional Integral Derivative (PID) controller. If you only need to minimize the error at one time unit, it would be a single objective function. But the goal is to minimize the error for all time steps from 0 to 100, thus producing a multiobjective function (one function for each time step).
The routine lsqnonlin is used to perform a least-squares fit on the tracking of the output. The tracking is performed via the function tracklsq, which returns the error signal yout, the output computed by calling sim, minus the input signal 1. The code for tracklsq is contained in the file runtracklsq.m, shown below.
The function runtracklsq sets up all the needed values and then calls lsqnonlin with the objective function tracklsq, which is nested inside runtracklsq. The variable options passed to lsqnonlin defines the criteria and display characteristics. In this case you ask for output, use the 'levenberg-marquardt' algorithm, and give termination tolerances for the step and objective function on the order of 0.001.
To run the simulation in the model optsim, the variables Kp, Ki, Kd, a1, and a2 (a1 and a2 are variables in the Plant block) must all be defined. Kp, Ki, and Kd are the variables to be optimized. The function tracklsq is nested inside runtracklsq so that the variables a1 and a2 are shared between the two functions. The variables a1 and a2 are initialized in runtracklsq.
The objective function tracklsq runs the simulation. The simulation can be run either in the base workspace or the current workspace, that is, the workspace of the function calling sim, which in this case is the workspace of tracklsq. In this example, the SrcWorkspace option is set to 'Current' to tell sim to run the simulation in the current workspace. The simulation is performed to 100 seconds.
When the simulation is completed, the myobj object is created in the current workspace (that is, the workspace of tracklsq). The Outport block in the block diagram model puts the yout field of the object into the current workspace at the end of the simulation.
The following is the code for runtracklsq:
function [Kp,Ki,Kd] = runtracklsq % RUNTRACKLSQ demonstrates using LSQNONLIN with Simulink. optsim % Load the model pid0 = [0.63 0.0504 1.9688]; % Set initial values a1 = 3; a2 = 43; % Initialize model plant variables options = optimoptions(@lsqnonlin,'Algorithm','levenberg-marquardt',... 'Display','off','TolX',0.001,'TolFun',0.001); pid = lsqnonlin(@tracklsq, pid0, [], [], options); Kp = pid(1); Ki = pid(2); Kd = pid(3); function F = tracklsq(pid) % Track the output of optsim to a signal of 1 % Variables a1 and a2 are needed by the model optsim. % They are shared with RUNTRACKLSQ so do not need to be % redefined here. Kp = pid(1); Ki = pid(2); Kd = pid(3); % Set sim options and compute function value myobj = sim('optsim','SrcWorkspace','Current', ... 'StopTime','100'); F = myobj.get('yout') - 1; end end
Copy the code for runtracklsq to a file named runtracklsq.m, placed in a folder on your MATLAB^{®} path.
When you run runtracklsq, the optimization gives the solution for the proportional, integral, and derivative (Kp, Ki, Kd) gains of the controller:
[Kp, Ki, Kd] = runtracklsq Done initializing optsim. Kp = 2.9633 Ki = 0.1436 Kd = 13.1386
Here is the resulting closed-loop step response.
Note The call to sim results in a call to one of the Simulink ordinary differential equation (ODE) solvers. A choice must be made about the type of solver to use. From the optimization point of view, a fixed-step solver is the best choice if that is sufficient to solve the ODE. However, in the case of a stiff system, a variable-step method might be required to solve the ODE. The numerical solution produced by a variable-step solver, however, is not a smooth function of parameters, because of step-size control mechanisms. This lack of smoothness can prevent the optimization routine from converging. The lack of smoothness is not introduced when a fixed-step solver is used. (For a further explanation, see [53].) Simulink Design Optimization™ software is recommended for solving multiobjective optimization problems in conjunction with Simulink variable-step solvers. It provides a special numeric gradient computation that works with Simulink and avoids introducing a problem of lack of smoothness. |