Suppose that you want to optimize the control parameters in
the Simulink® model
(This model can be found in the
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
Plant with Actuator Saturation
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
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).
lsqnonlin is used to perform a least-squares
fit on the tracking of the output. The tracking is performed via the
tracklsq, which returns the error signal
the output computed by calling
minus the input signal
1. The code for
contained in the file
runtracklsq.m, shown below.
runtracklsq sets up all the
needed values and then calls
lsqnonlin with the
tracklsq, which is nested inside
options passed to
the criteria and display characteristics. In this case you ask for
output, use the
and give termination tolerances for the step and objective function
on the order of
To run the simulation in the model
variables in the Plant block) must all be defined.
Kd are the variables to be optimized. The function
runtracklsq so that the variables
shared between the two functions. The variables
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
which in this case is the workspace of
In this example, the
SrcWorkspace option is set
'Current' to tell
run the simulation in the current workspace. The simulation is performed
When the simulation is completed, the
is created in the current workspace (that is, the workspace of
The Outport block in the block diagram model puts the
of the object into the current workspace at the end of the simulation.
The following is the code for
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','StepTolerance',0.001,'OptimalityTolerance',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
placed in a folder on your MATLAB® path.
When you run
runtracklsq, the optimization
gives the solution for the proportional, integral, and derivative
gains of the controller:
[Kp, Ki, Kd] = runtracklsq Kp = 3.1330 Ki = 0.1465 Kd = 14.3918
Here is the resulting closed-loop step response.
Closed-Loop Response Using lsqnonlin
The call to
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 .)
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.