findop

Compute operating condition from specifications

Since R2023b

Syntax

op = findop(sys,Name=Value)

op = findop(sys,t,Name=Value)

op = findop(sys,t,p,Name=Value)

Description

op = findop(sys,Name=Value) computes the operating condition op for the state-space model sys. Use the name-value arguments to specify known target values for the state x, input u, output y, and state derivative dx. Set unknown or free entries of x, u, y, and dx to NaN.

For mechss models, use q, dq, and d2q instead of x and dx to constrain degrees of freedom (DOFs) and their first and second derivatives.

op = findop(sys,t,Name=Value) computes the operating conditions for ltvss models at time t.

op = findop(sys,t,p,Name=Value) computes the operating conditions for lpvss models at time t and parameter values p.

Examples

Find Operating Point for MIMO State-Space Model

This example shows how to compute operating condition at various specifications for a state-space model.

Create a state-space model using these multi-input multi-output state matrices.

$$A=\left[\begin{array}{cc}-7& 0\\ 0& -10\end{array}\right]B=\left[\begin{array}{cc}5& 0\\ 0& 2\end{array}\right]C=\left[\begin{array}{cc}1& -4\\ -4& 0.5\end{array}\right]D=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$$

Specify the state-space matrices and create the MIMO state-space model.

A = [-7,0;0,-10];
B = [5,0;0,2];
C = [1,-4;-4,0.5];
D = [0,-2;2,0];
sys = ss(A,B,C,D);

Compute steady-state condition with fixed output y = [–1;1].

 op1 = findop(sys,y=[-1;1])

op1 = 
  OperatingPoint with properties:

            t: 0
            p: []
            x: [2x1 double]
            u: [2x1 double]
            w: [0x1 double]
           dx: [2x1 double]
            y: [2x1 double]
           dw: [0x1 double]
           rx: [2x1 double]
           ry: [2x1 double]
           rw: [0x1 double]
    Equations: 4
     Unknowns: 4
       Status: 'Well-posed problem. Successfully computed the unique solution.'

Compute steady-state condition for constant input u = [1;2].

 op2 = findop(sys,u=[1;2])

op2 = 
  OperatingPoint with properties:

            t: 0
            p: []
            x: [2x1 double]
            u: [2x1 double]
            w: [0x1 double]
           dx: [2x1 double]
            y: [2x1 double]
           dw: [0x1 double]
           rx: [2x1 double]
           ry: [2x1 double]
           rw: [0x1 double]
    Equations: 4
     Unknowns: 4
       Status: 'Well-posed problem. Successfully computed the unique solution.'

Compute steady-state condition with x(1) = 0 and x(2) unspecified.

op3 = findop(sys,x=[0;NaN])

op3 = 
  OperatingPoint with properties:

            t: 0
            p: []
            x: [2x1 double]
            u: [2x1 double]
            w: [0x1 double]
           dx: [2x1 double]
            y: [2x1 double]
           dw: [0x1 double]
           rx: [2x1 double]
           ry: [2x1 double]
           rw: [0x1 double]
    Equations: 4
     Unknowns: 5
       Status: 'Underdetermined problem. Returned solution with minimum norm.'

Compute unsteady initial condition with dx = [1;0] and y = [-1;1].

op4 = findop(sys,y=[-1;1],dx=[1;0])

op4 = 
  OperatingPoint with properties:

            t: 0
            p: []
            x: [2x1 double]
            u: [2x1 double]
            w: [0x1 double]
           dx: [2x1 double]
            y: [2x1 double]
           dw: [0x1 double]
           rx: [2x1 double]
           ry: [2x1 double]
           rw: [0x1 double]
    Equations: 4
     Unknowns: 4
       Status: 'Well-posed problem. Successfully computed the unique solution.'

Use dot notation to access the values in these structures. For example, op4.u gives the input values for which the system achieved the unsteady initial condition in the previous specification.

op4.u

ans = 2×1

   -0.4785
    0.1840

Find Operating Condition for Sparse Second-Order Model

For this example, consider the sparse matrices for the 3-D beam model subjected to an impulsive point load at its tip.

Extract the sparse matrices from sparseBeam.mat.

load('sparseBeam.mat','M','K','B','F','G','D');

Create the mechss model object by specifying [] for matrix C, since there is no damping.

sys = mechss(M,[],K,B,F,G,D)

Sparse continuous-time second-order model with 3 outputs, 1 inputs, and 3408 degrees of freedom.

Use "spy" and "showStateInfo" to inspect model structure. 
Type "help mechssOptions" for available solver options for this model.

Compute the operating point at the fixed input u = 2.

op = findop(sys,u=2)

op = 
  OperatingPoint2 with properties:

            t: 0
            p: []
            q: [3408x1 double]
           dq: [3408x1 double]
            u: 2
            w: [0x1 double]
          d2q: [3408x1 double]
            y: [3x1 double]
           dw: [0x1 double]
           rq: [3408x1 double]
           ry: [3x1 double]
           rw: [0x1 double]
    Equations: 3411
     Unknowns: 3411
       Status: 'Well-posed problem. Successfully computed the unique solution.'

The function returns a steady-state condition at this input value. For this problem, the solution is unique.

The problem becomes overconstrained if you specify additional specifications.

q = NaN(3408,1);
q(1:100) = randn(100,1);
op1 = findop(sys,q=q,u=2,y=[NaN,1,NaN])

op1 = 
  OperatingPoint2 with properties:

            t: 0
            p: []
            q: [3408x1 double]
           dq: [3408x1 double]
            u: 2
            w: [0x1 double]
          d2q: [3408x1 double]
            y: [3x1 double]
           dw: [0x1 double]
           rq: [3408x1 double]
           ry: [3x1 double]
           rw: [0x1 double]
    Equations: 3411
     Unknowns: 3310
       Status: 'Overconstrained problem, residuals may exist. Returned least-squares solution.'

In this case, the function returns the least-squares solution.

Compute Operating Point for Discrete-Time LPV Model

Create a discrete-time linear-parameter varying model.

The matrices and offsets are given by:

$\mathit{A}=\sin \left(0.1\mathit{p}\right)$, $\mathit{B}=1$, $\mathit{C}=1$, $\mathit{D}=0$

$${\mathit{y}}_0=0.1\sin \left(\mathit{k}\right)$$.

These matrices and offsets are defined in the lpvFcnDiscrete.m data function provided with this example.

Specify the properties and create the LPV model.

Ts = 0.01;
ParamNames = 'p';
DataFcn = @lpvFcnDiscrete;
lpvSys = lpvss(ParamNames,DataFcn,Ts)

Discrete-time state-space LPV model with 1 outputs, 1 inputs, 1 states, and 1 parameters.

View the data function.

type lpvFcnDiscrete.m

function [A,B,C,D,E,dx0,x0,u0,y0,Delays] = lpvFcnDiscrete(k,p)
A = sin(0.1*p);
B = 1;
C = 1;
D = 0;
E = [];
dx0 = [];
x0 = [];
u0 = [];
y0 = 0.1*sin(k);
Delays = [];

Compute the operating condition for the LPV model at t = 0.05 and p = 10. For discrete-time models, specify time as the integer sample index.

op = findop(lpvSys,5,10,x=0.5)

op = 
  OperatingPoint with properties:

            t: 5
            p: 10
            x: 0.5000
            u: 0.0793
            w: [0x1 double]
           dx: 0.5000
            y: 0.4041
           dw: [0x1 double]
           rx: 0
           ry: 0
           rw: [0x1 double]
    Equations: 2
     Unknowns: 2
       Status: 'Well-posed problem. Successfully computed the unique solution.'

Input Arguments

sys — Dynamic system

state-space model

Dynamic system, specified as a SISO or MIMO state-space model. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric state-space models, such as ss models.

• Generalized or uncertain LTI models such as genss or uss models. (Using uncertain models requires Robust Control Toolbox™ software.)

• Sparse state-space models such as sparss and mechss models.

• Identified LTI models, such as idss, or idgrey models. (Using identified models requires System Identification Toolbox™ software.)

• Linear time-varying (ltvss) and linear parameter-varying (lpvss) models.

findop does not support non state-space models such as tf, zpk, pid, and frd models.

t — Time value

scalar

Time value for evaluating operating point condition for linear time-varying and linear-parameter varying models, specified as a scalar.

For discrete-time systems, t is the integer sample index k.

p — Parameter values for LPV models

vector

Parameter values for evaluating operating point condition for linear parameter-varying models, specified as a vector of length equal to the number of parameters in the model.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: op = findop(sys, y=[1;NaN], u=[NaN;0]) computes a steady-state operating condition with u(2)=0 and y(1)=1.

First-Order Models Only

dx — State derivative values

0 (default) | vector

State derivative values for computing the operating point, specified as a vector of length equal to the number of states in sys.

In discrete time, dx specifies the state incremental change x[k+1] − x[k].

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

x — State values

NaN (default) | vector

State values for computing the operating point, specified as a vector of length equal to the number of states in sys.

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

mechss Models Only

q — Displacement values

NaN (default) | vector

Displacement values for computing the operating point, specified as a vector of length equal to the number of degrees of freedom in the mechss model sys.

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

dq — Displacement first derivative values

0 (default) | vector

Displacement first derivative values for computing the operating point, specified as a vector of length equal to the number of degrees of freedom in the mechss model sys.

In discrete time, dq specifies the incremental change q[k+1] − q[k].

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

d2q — Displacement second derivative values

0 (default) | vector

Displacement second derivative values for computing the operating point, specified as a vector of length equal to the number of degrees of freedom in the mechss model sys.

In discrete time, d2q specifies the incremental change q[k+2] + q[k] – 2q[k+1].

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

All Models

u — Input values

NaN (default) | vector

Input values for computing the operating point, specified as a vector of length equal to the number of inputs in sys.

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

y — Output values

NaN (default) | vector

Output values for computing the operating point, specified as a vector of length equal to the number of outputs in sys.

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

w — Internal signal values

NaN (default) | vector

Internal signal specification, specified as a vector of length equal to the number of internal delays in the model. Use this input only when the model has internal delays. For more information about internal delays, see Internal Delays.

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN.

dw — Internal signal mismatch

0 (default) | vector

Since R2024b

Internal signal mismatch specification dw = z – w, specified as a vector of length equal to the number of internal delays in the model. Here:

$$z=z_0+C_2\left(x(t)-x_0\right)+D_{21}\left(u(t)-u_0\right)+D_{22}\left(w(t)-w_0\right)$$

The algorithm uses the specified numeric values as known values. To set entries as unspecified or free, use NaN. To specify steady state z = w for the internal delays, set dw to 0 (default value). Use this input only when the model has internal delays. For more information about internal delays, see Internal Delays.

Output Arguments

op — Operating point

OperatingPoint object | OperatingPoint2 object

Computed operating condition, returned as an OperatingPoint object or an OperatingPoint2 objects.

• The function returns an OperatingPoint object for all models except mechss. The object has following properties:

  Property	Description
  t (since R2024b)	Time value at evaluated operating point condition for linear time-varying and linear-parameter varying models, returned as a scalar. For discrete-time systems, this value is the integer sample index k. For LTI models, the function returns t = 0.
  p (since R2024b)	Parameter values at evaluated operating point condition for linear parameter-varying models, returned as a vector of length equal to the number of parameters in the model. For LTI models, the function returns p = [].
  x	Computed value for state vector, returned as a column vector of length equal to the number of states.
  u	Computed value for input vector, returned as a column vector of length equal to the number of inputs.
  w	Computed value for internal delay signals (if any), returned as a column vector of length equal to the number of internal delays in the model.
  dx	Computed value for state derivative vector, returned as a column vector of length equal to the number of states. In discrete-time, this value is equal to x[k+1].
  y	Computed value for output vector, returned as a column vector of length equal to the number of outputs.
  dw (since R2024b)	Computed value for internal signal mismatch dw = z – w, returned as a column vector of length equal to the number of internal delays.
  rx	Relative residuals for state equations, returned as a column vector of length equal to the number of states.
  ry	Relative residuals for output equations, returned as a column vector of length equal to the number of outputs.
  rw (since R2024b)	Relative residuals for delay equations, returned as a column vector of length equal to the number of internal delays.
  Equations	Number of equations to solve to compute operating conditions.
  Unknowns	Number of unknowns in the equations.
  Status	Status report.

• The function returns an OperatingPoint2 object for mechss models. The object has following properties:

  Property	Description
  t (since R2024b)	For mechss models, the function returns t = 0.
  p (since R2024b)	For mechss models, the function returns p = [].
  q	Computed value for displacement vector, returned as a column vector of length equal to the number of degrees of freedom.
  u	Computed value for input vector, returned as a column vector of length equal to the number of inputs.
  w	Computed value for internal delay signals (if any), returned as a column vector of length equal to the number of internal delays in the model.
  dq	Computed value for first derivative of displacement vector, returned as a column vector of length equal to the number of degrees of freedom. In discrete-time, this value is equal to q[k+1].
  d2q	Computed value for second derivative of displacement vector, returned as a column vector of length equal to the number of degrees of freedom. In discrete-time, this value is equal to q[k+2].
  y	Computed value for output vector, returned as a column vector of length equal to the number of outputs.
  dw (since R2024b)	Computed value for internal signal mismatch dw = z – w, returned as a column vector of length equal to the number of internal delays.
  rq	Relative residuals for displacement equations, returned as a column vector of length equal to the number of degrees of freedom.
  ry	Relative residuals for output equations, returned as a column vector of length equal to the number of outputs.
  rw (since R2024b)	Relative residuals for delay equations, returned as a column vector of length equal to the number of internal delays.
  Equations	Number of equations to solve to compute operating conditions.
  Unknowns	Number of unknowns in the equations.
  Status	Status report.

Additionally,

• For overconstrained problems, findop returns the least-squares solution.

• For underdetermined problems, findop returns the solution with smallest norm.

Version History

Introduced in R2023b

R2024b: Improvements for specifying internal delay specifications and additional properties in output objects

You can now use the dw argument to specify the internal signal mismatch dw = z – w when computing an operating condition. To reflect this change, the output objects now contain new fields dw and rw, where dw is the mismatch value at the operating condition and rw is the relative residual for the delay equation dw + w – z = 0.

Additionally:

• The function now returns rx and ry as relative residuals for the solved equations. This makes it easier to determine if the software achieved the operating point specification.

• The output objects now contain new fields for specified time t and parameter p values for LTV and LPV models. For all other models, the function returns t = 0 and p = [].

R2024a: Output changed

The output op of findop has changed. The function now returns an object instead of a structure.

• For mechss models, op is a OperatingPoint2 object.

• For all other models, op is a OperatingPoint object.

Additionally, when computing operating point condition for discrete-time models, the function now returns a different values for dx (non-mechss models) and dq and d2q (mechss models) fields.

Field	Since R2024a	Before R2024a
dx	x[k+1]	x[k+1] –x[k]
dq	q[k+1]	q[k+1] –q[k]
d2q	q[k+2]	q[k+2] + q[k] – 2q[k+1]

This change provides better consistency and alignment with how the software handles state-space models with offsets. If your existing code uses these fields to initialize an operating condition, consider checking your results in R2024a.