trim

Find trim point of dynamic system

Syntax

[x,u,y,dx] = trim('sys') [x,u,y,dx] = trim('sys',x0,u0,y0) [x,u,y,dx] = trim('sys',x0,u0,y0,ix,iu,iy) [x,u,y,dx] = trim('sys',x0,u0,y0,ix,iu,iy,dx0,idx) [x,u,y,dx,options] = trim('sys',x0,u0,y0,ix,iu,iy,dx0,idx,options) [x,u,y,dx,options] = trim('sys',x0,u0,y0,ix,iu,iy,dx0,idx,options,t)

Description

Note

trim provides only basic trimming functionality. For full trimming functionality, use Simulink^® Control Design™ software. For more information, see Compute Steady-State Operating Points (Simulink Control Design).

A trim point, also known as an equilibrium point, is a point in the parameter space of a dynamic system at which the system is in a steady state. For example, a trim point of an aircraft is a setting of its controls that causes the aircraft to fly straight and level. Mathematically, a trim point is a point where the system's state derivatives equal zero. trim starts from an initial point and searches, using a sequential quadratic programming algorithm, until it finds the nearest trim point. You must supply the initial point implicitly or explicitly. If trim cannot find a trim point, it returns the point encountered in its search where the state derivatives are closest to zero in a min-max sense; that is, it returns the point that minimizes the maximum deviation from zero of the derivatives. trim can find trim points that meet specific input, output, or state conditions, and it can find points where a system is changing in a specified manner, that is, points where the system's state derivatives equal specific nonzero values.

[x,u,y,dx] = trim('sys') finds the equilibrium point of the model 'sys', nearest to the system's initial state, x0. Specifically, trim finds the equilibrium point that minimizes the maximum absolute value of [x-x0,u,y]. If trim cannot find an equilibrium point near the system's initial state, it returns the point at which the system is nearest to equilibrium. Specifically, it returns the point that minimizes abs(dx) where dx represents the derivative of the system. You can obtain x0 using this command.

[sizes,x0,xstr] = sys([],[],[],0)

[x,u,y,dx] = trim('sys',x0,u0,y0) finds the trim point nearest to x0, u0, y0, that is, the point that minimizes the maximum value of

abs([x-x0; u-u0; y-y0])

Caution

When you use the trim function to find a trim point near a specified initial operating point,

The trim function returns only a local value when you use the function to find a trim point near a specified initial operating point. Other, more suitable trim points might exist. To find the most suitable trim point for a particular application, as a best practice, try a number of initial guesses for the initial state, input, and output values.

[x,u,y,dx] = trim('sys',x0,u0,y0,ix,iu,iy) finds the trim point closest to x0, u0, y0 that satisfies a specified set of state, input, and/or output conditions. The integer vectors ix, iu, and iy select the values in x0, u0, and y0 that must be satisfied. If trim cannot find an equilibrium point that satisfies the specified set of conditions exactly, it returns the nearest point that satisfies the conditions, namely,

abs([x(ix)-x0(ix); u(iu)-u0(iu); y(iy)-y0(iy)])

[x,u,y,dx] = trim('sys',x0,u0,y0,ix,iu,iy,dx0,idx) finds specific nonequilibrium points, that is, points at which the system's state derivatives have some specified nonzero value. Here, dx0 specifies the state derivative values at the search's starting point and idx selects the values in dx0 that the search must satisfy exactly.

[x,u,y,dx,options] = trim('sys',x0,u0,y0,ix,iu,iy,dx0,idx,options) specifies an array of optimization parameters that trim passes to the optimization function that it uses to find trim points. The optimization function, in turn, uses this array to control the optimization process and to return information about the process. trim returns the options array at the end of the search process. By exposing the underlying optimization process in this way, trim allows you to monitor and fine-tune the search for trim points.

The following table describes how each element affects the search for a trim point. Array elements 1, 2, 3, 4, and 10 are particularly useful for finding trim points.

No.	Default	Description
1	0	Specifies display options. 0 specifies no display; 1 specifies tabular output; -1 suppresses warning messages.
2	10^–4	Precision the computed trim point must attain to terminate the search.
3	10^–4	Precision the trim search goal function must attain to terminate the search.
4	10^–6	Precision the state derivatives must attain to terminate the search.
5	N/A	Not used.
6	N/A	Not used.
7	N/A	Used internally.
8	N/A	Returns the value of the trim search goal function (λ in goal attainment).
9	N/A	Not used.
10	N/A	Returns the number of iterations used to find a trim point.
11	N/A	Returns the number of function gradient evaluations.
12	0	Not used.
13	0	Number of equality constraints.
14	100*(Number of variables)	Maximum number of function evaluations to use to find a trim point.
15	N/A	Not used.
16	10^–8	Used internally.
17	0.1	Used internally.
18	N/A	Returns the step length.

[x,u,y,dx,options] = trim('sys',x0,u0,y0,ix,iu,iy,dx0,idx,options,t) sets the time to t if the system is dependent on time.

Note

If you fix any of the state, input, or output values, the trim function uses the unspecified free variables to derive the solution that satisfies the specified constraints.

Examples

collapse all

Find Steady-State Points

The Simulink trim function uses a model to determine steady-state points of a dynamic system that satisfy specified input, output, and state conditions. Consider, for example, this model, named ex_lmod.

The block diagram of the model named ex_lmod.

You can use the trim function to find the values of the input and the states that result in a value of 1 for both output values.

First, create variables named x and u to store values that represent initial guesses of the state variable values and input values, respectively. Then, create a variable named y to store the desired output value.

x = [0; 0; 0];
u = 0;
y = [1; 1];

Create index variables to specify which variables are fixed and which can vary. To specify that the state and input values can vary, assign an empty matrix ([]) to the index variables ix and iu. To specify that the output values are fixed, specify the value of the index variable iy as [1;2].

ix = [];      % Don't fix any of the states
iu = [];      % Don't fix the input
iy = [1;2];   % Fix both output 1 and output 2

Call the trim function to find the values of states and inputs that result in the desired output value. The values the trim function returns might differ due to rounding error.

[x,u,y,dx] = trim('lmod',x,u,y,ix,iu,iy)

Equilibrium point problems might not have a solution. In that case, the trim function first tries setting the derivatives to 0 and then returns a solution that minimizes the maximum deviation from the desired result.

Find Equilibrium Points

Consider a linear state-space system modeled using a State-Space block. The system equations have the form:

$\begin{array}{l} \dot{x} = A x + B u \\ y = C x + D u \end{array}$

Suppose a model named sys contains a State-Space block that has these values for the A, B, C, and D matrices.

A = [-0.09 -0.01;  1   0];
B = [ 0    -7;     0  -2];
C = [ 0     2;     1  -5];
D = [-3     0;     1   0];

To find an equilibrium point in the model sys, call the trim function and specify only the name of the model as an input argument.

[x,u,y,dx,options] = trim('sys')

To find an equilibrium point near an operating point defined by state and input values, call the trim function and specify additional input arguments that contain the state and input values to define the operating point.

x0 = [1;1];
u0 = [1;1];
[x,u,y,dx,options] = trim('sys', x0, u0);

x =
    1.0e-13 *
   -0.5160
   -0.5169
u =
    0.3333
    0.0000
y =
   -1.0000
    0.3333
dx =
    1.0e-12 *
    0.1979
    0.0035

Check the options return argument to see the number of iterations required to identify the equilibrium point near the specified operating point.

options(10)

ans = 
      25

To find an equilibrium point at which both output values are fixed, specify additional input arguments to indicate the target output value and that the output values are fixed.

y = [1;1];
iy = [1;2];
[x,u,y,dx] = trim('sys', [], [], y, [], [], iy)

x =
    0.0009
   -0.3075
u =
   -0.5383
    0.0004
y =
    1.0000
    1.0000
dx =
    1.0e-15 *
   -0.0170
    0.1483

To find an equilibrium point with specified derivative values at which both outputs are fixed, specify additional input arguments for the derivative values, the target output value, and that the output values are fixed.

y = [1;1];
iy = [1;2];
dx = [0;1];
idx = [1;2];
[x,u,y,dx,options] = trim('sys',[],[],y,[],[],iy,dx,idx)

Check the options return argument to see the number of iterations required to identify the equilibrium point near the specified operating point.

options(10)

ans = 
      13

Algorithms

The trim function uses a sequential quadratic programming algorithm to find trim points. For a description of this algorithm, see Sequential Quadratic Programming (SQP) (Optimization Toolbox).

Version History

Introduced before R2006a