TuningGoal.StepTracking

Step response requirement for control system tuning

Description

Use TuningGoal.StepTracking to specify a target step response from specified inputs to specified outputs of a control system. Use this tuning goal with control system tuning commands such as systune or looptune.

Creation

Syntax

Req = TuningGoal.StepTracking(inputname,outputname,refsys)

Req = TuningGoal.StepTracking(inputname,outputname,tau)

Req = TuningGoal.StepTracking(inputname,outputname,tau,overshoot)

Description

Req = TuningGoal.StepTracking(inputname,outputname,refsys) creates a tuning goal that constrains the step response between the specified signal locations to match the step response of a stable reference system, refsys. The constraint is satisfied when the relative difference between the tuned and target responses falls within a tolerance specified by the RelGap property of the tuning goal (see Properties). inputname and outputname can describe a SISO or MIMO response of your control system. For MIMO responses, the number of inputs must equal the number of outputs.

Req = TuningGoal.StepTracking(inputname,outputname,tau) specifies the desired step response as a first-order response with time constant tau:

$$\text{Req}\text{.ReferenceModel}=\frac{1/\text{tau}}{s+1/\text{tau}}.$$

Req = TuningGoal.StepTracking(inputname,outputname,tau,overshoot) specifies the desired step response as a second-order response with natural period tau, natural frequency 1/tau, and percent overshoot overshoot:

$$\text{Req}\text{.ReferenceModel}=\frac{{\left(1/\text{tau}\right)}^2}{s^2+2\left(\text{zeta}/\text{tau}\right)s+{\left(1/\text{tau}\right)}^2}.$$

The damping is given by zeta = cos(atan2(pi,-log(overshoot/100))).

Input Arguments

inputname — Input signals for tuning goal

character vector | cell array of character vectors

Input signals for the tuning goal, specified as a character vector or, for multiple-input tuning goals, a cell array of character vectors.

• If you are using the tuning goal to tune a Simulink^® model of a control system, then inputname can include:

  • Any model input.

  • Any linear analysis point marked in the model.

  • Any linear analysis point in an slTuner (Simulink Control Design) interface associated with the Simulink model. Use addPoint (Simulink Control Design) to add analysis points to the slTuner interface. Use getPoints (Simulink Control Design) to get the list of analysis points available in an slTuner interface to your model.

  For example, suppose that the slTuner interface contains analysis points u1 and u2. Use 'u1' to designate that point as an input signal when creating tuning goals. Use {'u1','u2'} to designate a two-channel input.

• If you are using the tuning goal to tune a generalized state-space (genss) model of a control system, then inputname can include:

  • Any input of the genss model

  • Any AnalysisPoint location in the control system model

  For example, if you are tuning a control system model, T, then inputname can be any input name in T.InputName. Also, if T contains an AnalysisPoint block with a location named AP_u, then inputname can include 'AP_u'. Use getPoints to get a list of analysis points available in a genss model.

  If inputname is an AnalysisPoint location of a generalized model, the input signal for the tuning goal is the implied input associated with the AnalysisPoint block:

  

For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design.

outputname — Output signals for tuning goal

character vector | cell array of character vectors

Output signals for the tuning goal, specified as a character vector or, for multiple-output tuning goals, a cell array of character vectors.

• If you are using the tuning goal to tune a Simulink model of a control system, then outputname can include:

  • Any model output.

  • Any linear analysis point marked in the model.

  • Any linear analysis point in an slTuner (Simulink Control Design) interface associated with the Simulink model. Use addPoint (Simulink Control Design) to add analysis points to the slTuner interface. Use getPoints (Simulink Control Design) to get the list of analysis points available in an slTuner interface to your model.

  For example, suppose that the slTuner interface contains analysis points y1 and y2. Use 'y1' to designate that point as an output signal when creating tuning goals. Use {'y1','y2'} to designate a two-channel output.

• If you are using the tuning goal to tune a generalized state-space (genss) model of a control system, then outputname can include:

  • Any output of the genss model

  • Any AnalysisPoint location in the control system model

  For example, if you are tuning a control system model, T, then outputname can be any output name in T.OutputName. Also, if T contains an AnalysisPoint block with a location named AP_u, then outputname can include 'AP_u'. Use getPoints to get a list of analysis points available in a genss model.

  If outputname is an AnalysisPoint location of a generalized model, the output signal for the tuning goal is the implied output associated with the AnalysisPoint block:

  

For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design.

refsys — Reference system for target step response

tf model object | zpk model object | ss model object

Reference system for target step response, specified as a dynamic system model, such as a tf, zpk, or ss model. refsys must be stable and must have DC gain of 1 (zero steady-state error).

refsys can be continuous or discrete. If refsys is discrete, it can include time delays which are treated as poles at z = 0.

refsys can be MIMO, provided that it is square and that its DC singular value (sigma) is 1. If refsys is a MIMO model, then its number of inputs and outputs must match the dimensions of inputname and outputname.

For best results, refsys should also include intrinsic system characteristics such as non-minimum-phase zeros (undershoot).

tau — Time constant or natural period of target step response

positive scalar

Time constant or natural period of target step response, specified as a positive scalar.

If you use the syntax Req = TuningGoal.StepTracking(inputname,outputname,tau) to specify a first-order target response, then tau is the time constant of the response decay. In that case, the target is the step response of the system given by:

$$\text{Req}\text{.ReferenceModel}=\frac{1/\text{tau}}{s+1/\text{tau}}.$$

If you use the syntax Req = TuningGoal.StepTracking(inputname,outputname,tau,overshoot) to specify a second-order target response, then tau is the inverse of the natural frequency of the response. In that case, the target is the step response of the system given by:

$$\text{Req}\text{.ReferenceModel}=\frac{{\left(1/\text{tau}\right)}^2}{s^2+2\left(\text{zeta}/\text{tau}\right)s+{\left(1/\text{tau}\right)}^2}.$$

The damping of the system is given by zeta = cos(atan2(pi,-log(overshoot/100))).

overshoot — Percent overshoot of target step response

scalar in the range (0, 100)

Percent overshoot of target step response, specified as a scalar value in the range (0,100).

Properties

ReferenceModel — Reference system for target step response

SISO or MIMO ss model object

Reference system for target step response, specified as a SISO or MIMO state-space (ss) model. When you use the tuning goal to tune a control system, the step response from inputname to outputname is tuned to match this target response to within the tolerance specified by the RelGap property.

If you use the refsys input argument to create the tuning goal, then the value of ReferenceModel is ss(refsys).

If you use the tau or tau and overshoot input arguments, thenReferenceModel is a state-space representation of the corresponding first-order or second-order transfer function.

ReferenceModel must be stable and have unit DC gain (zero steady-state error). For best results, ReferenceModel should also include intrinsic system characteristics such as non-minimum-phase zeros (undershoot).

RelGap — Maximum relative matching error

positive scalar

Maximum relative matching error, specified as a positive scalar value. This property specifies the matching tolerance as the maximum relative gap between the target and actual step responses. The relative gap is defined as:

$$\text{gap}=\frac{{\Vert y\left(t\right)-y_{ref}\left(t\right)\Vert }_2}{{\Vert 1-y_{ref}\left(t\right)\Vert }_2}.$$

y(t) – y_ref(t) is the response mismatch, and 1 – y_ref(t) is the step-tracking error of the target model. $${\Vert \cdot \Vert }_2$$ denotes the signal energy (2-norm).

Increase the value of RelGap to loosen the matching tolerance.

InputScaling — Reference signal scaling

[] (default) | vector of positive real elements

Reference signal scaling, specified as a vector of positive real elements.

For a MIMO tracking requirement, when the choice of units results in a mix of small and large signals in different channels of the response, use this property to specify the relative amplitude of each entry in the vector-valued step input. This information is used to scale the off-diagonal terms in the transfer function from reference to tracking error. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal.

For example, suppose that Req is a tuning goal that signals {'y1','y2'} track reference signals {'r1','r2'}. Suppose further that you require the outputs to track the references with less than 10% cross-coupling. If r1 and r2 have comparable amplitudes, then it is sufficient to keep the gains from r1 to y2 and r2 and y1 below 0.1. However, if r1 is 100 times larger than r2, the gain from r1 to y2 must be less than 0.001 to ensure that r1 changes y2 by less than 10% of the r2 target. To ensure this result, set the InputScaling property as follows.

Req.InputScaling = [100,1];

This tells the software to take into account that the first reference signal is 100 times greater than the second reference signal.

The default value, [] , means no scaling.

Input — Input signal names

cell array of character vectors

Input signal names, specified as a cell array of character vectors that identify the inputs of the transfer function that the tuning goal constrains. The initial value of the Input property is set by the inputname input argument when you construct the tuning goal.

Output — Output signal names

cell array of character vectors

Output signal names, specified as a cell array of character vectors that identify the outputs of the transfer function that the tuning goal constrains. The initial value of the Output property is set by the outputname input argument when you construct the tuning goal.

Models — Models to which tuning goal applies

NaN (default) | vector of scalars

Models to which the tuning goal applies, specified as a vector of indices.

Use the Models property when tuning an array of control system models with systune, to enforce a tuning goal for a subset of models in the array. For example, suppose you want to apply the tuning goal, Req, to the second, third, and fourth models in a model array passed to systune. To restrict enforcement of the tuning goal, use the following command:

Req.Models = 2:4;

When Models = NaN, the tuning goal applies to all models.

Openings — Feedback loops to open when evaluating the tuning goal

{} (default) | cell array of character vectors

Feedback loops to open when evaluating the tuning goal, specified as a cell array of character vectors that identify loop-opening locations. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify.

If you are using the tuning goal to tune a Simulink model of a control system, then Openings can include any linear analysis point marked in the model, or any linear analysis point in an slTuner (Simulink Control Design) interface associated with the Simulink model. Use addPoint (Simulink Control Design) to add analysis points and loop openings to the slTuner interface. Use getPoints (Simulink Control Design) to get the list of analysis points available in an slTuner interface to your model.

If you are using the tuning goal to tune a generalized state-space (genss) model of a control system, then Openings can include any AnalysisPoint location in the control system model. Use getPoints to get the list of analysis points available in the genss model.

For example, if Openings = {'u1','u2'}, then the tuning goal is evaluated with loops open at analysis points u1 and u2.

Name — Name of tuning goal

[] (default) | character vector

Name of the tuning goal, specified as a character vector.

For example, if Req is a tuning goal:

Req.Name = 'LoopReq';

Examples

Step Response Requirement with Specified Tolerance

Create a requirement for the step response from a signal named 'r' to a signal named 'y'. Constrain the step response to match the transfer function H = 10/(s+10), but allow 20% relative variation between the target and the tuned responses.

H = tf(10,[1 10]);
Req = TuningGoal.StepResp('r','y',H);

By default, this requirement allows a relative gap of 0.1 between the target and tuned responses. To change the relative gap to 20%, set the RelGap property of the requirement.

Req.RelGap = 0.2;

Examine the requirement.

viewGoal(Req);

The dashed line shows the target step response specified by this requirement. You can use this requirement to tune a control system model, T, that contains valid input and output locations named 'r' and 'y'. If you do so, the command viewGoal(Req,T) plots the achieved step response from 'r' to 'y' for comparison to the target response.

First-Order Step Response with Known Time Constant

Create a requirement that specifies a first-order step response with time constant of 5 seconds. Create the requirement for the step response from a signal named 'r' to a signal named 'y'.

Req = TuningGoal.StepResp('r','y',5);

When you use this requirement to tune a control system model, T, the time constant 5 is taken to be expressed in the prevailing units of the control system. For example, if T is a genss model and the property T.TimeUnit is 'seconds', then this requirement specifies a target time constant of 5 seconds for the response from the input 'r' to the output 'y' of 'T'.

The specified time constant is converted into a reference state-space model stored in the ReferenceModel property of the requirement.

refsys = tf(Req.ReferenceModel)

refsys =
 
    0.2
  -------
  s + 0.2
 
Continuous-time transfer function.

As expected, refsys is a first-order model.

Examine the requirement. The viewGoal command displays the target response, which is the step response of the reference model.

viewGoal(Req);

The dashed line shows the target step response specified by this requirement, a first-order response with a time constant of five seconds.

Second-Order Step Response with Known Natural Period and Overshoot

Create a requirement that specifies a second-order step response with a natural period of 5 seconds, and a 10% overshoot. Create the requirement for the step response from a signal named 'r' to a signal named 'y'.

Req = TuningGoal.StepResp('r','y',5,10);

When you use this requirement to tune a control system model, T, the natural period 5 is taken to be expressed in the prevailing units of the control system. For example, if T is a genss model and the property T.TimeUnit is 'seconds', then this requirement specifies a target natural period of 5 seconds for the response from the input 'r' to the output 'y' of 'T'.

The specified parameters of the response are converted into a reference state-space model stored in the ReferenceModel property of the requirement.

refsys = tf(Req.ReferenceModel)

refsys =
 
          0.04
  ---------------------
  s^2 + 0.2365 s + 0.04
 
Continuous-time transfer function.

As expected, refsys is a second-order model.

Examine the requirement. The viewGoal command displays the target response, which is the step response of the reference model.

viewGoal(Req);

The dashed line shows the target step response specified by this requirement, a second-order response with 10% overshoot and a natural period of five seconds.

Tracking Goal with Limited Model Application and Additional Loop Openings

Create a tuning goal that specifies a first-order step response with time constant of 5 seconds. Set the Models and Openings properties to further configure the tuning goal’s applicability.

Req = TuningGoal.StepTracking('r','y',5);
Req.Models = [2 3];
Req.Openings = 'OuterLoop'

When tuning a control system that has an input 'r', an output 'y', and an analysis-point location 'OuterLoop', you can use Req as an input to looptune or systune. Setting the Openings property specifies that the step response from 'r' to 'y' is measured with the loop opened at 'OuterLoop'. When tuning an array of control system models, setting the Models property restricts how the tuning goal is applied. In this example, the tuning goal applies only to the second and third models in an array.

Tips

• This tuning goal imposes an implicit stability constraint on the closed-loop transfer function from Input to Output, evaluated with loops opened at the points identified in Openings. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The MinDecay and MaxRadius options of systuneOptions control the bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, use systuneOptions to change these defaults.

Algorithms

When you tune a control system using a TuningGoal, the software converts the tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint.

For TuningGoal.StepTracking, f(x) is given by:

$$f\left(x\right)=\frac{{\Vert \frac{1}{s}\left(T\left(s,x\right)-T_{ref}\left(s\right)\right)\Vert }_2}{\text{RelGap}{\Vert \frac{1}{s}\left(T_{ref}\left(s\right)-I\right)\Vert }_2}.$$

T(s,x) is the closed-loop transfer function from Input to Output with parameter values x, and T_ref(s) is the reference model specified in the ReferenceModel property. $${\Vert \cdot \Vert }_2$$ denotes the H₂ norm (see norm).

Version History

Introduced in R2016a

R2016a: Functionality moved from Robust Control Toolbox

Prior to R2016a, this functionality required a Robust Control Toolbox™ license.