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Find trim point of dynamic system
[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)
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 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-0). 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])
[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.
Consider a linear state-space system modeled using a State-Space block
$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$
The A, B, C, and D matrices to enter at the command line or in the block parameters dialog are:.
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 this model called sys, use:
[x,u,y,dx,options] = trim('sys') x = 0 0 u = 0 0 y = 0 0 dx = 0 0
The number of iterations taken is:
options(10) ans = 7
To find an equilibrium point near x = [1;1], u = [1;1], enter
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
The number of iterations taken is
options(10) ans = 25
To find an equilibrium point with the outputs fixed to 1, use:
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 the outputs fixed to 1 and the derivatives set to 0 and 1, use
y = [1;1]; iy = [1;2]; dx = [0;1]; idx = [1;2]; [x,u,y,dx,options] = trim('sys',[],[],y,[],[],iy,dx,idx) x = 0.9752 -0.0827 u = -0.3884 -0.0124 y = 1.0000 1.0000 dx = 0.0000 1.0000
The number of iterations taken is
options(10) ans = 13
The trim point found by trim starting from any given initial point is only a local value. Other, more suitable trim points may exist. Thus, if you want to find the most suitable trim point for a particular application, it is important to try a number of initial guesses for x, u, and y.