Documentation |
Package: TuningGoal
Noise amplification constraint for control system tuning
Use the TuningGoal.Variance object to specify a tuning requirement that limits the noise amplification from specified inputs to outputs. The noise amplification is defined as either:
The square root of the output variance, for a unit-variance white-noise input
The root-mean-square of the output, for a unit-variance white-noise input
The H_{2} norm of the transfer function from the specified inputs to outputs, which equals the total energy of the impulse response
These definitions are different interpretations of the same quantity. TuningGoal.Variance imposes the same limit on these quantities.
You can use the TuningGoal.Variance requirement for control system tuning with tuning commands, such as systune or looptune. Specifying this requirement allows you to tune the system response to white-noise inputs. For stochastic inputs with a nonuniform spectrum (colored noise), use TuningGoal.WeightedVariance instead.
After you create a requirement object, you can further configure the tuning requirement by setting Properties of the object.
Req = TuningGoal.Variance(inputname,outputname,maxamp) creates a tuning requirement. This tuning requirement limits the noise amplification of the transfer function from inputname to outputname to the scalar value maxamp.
When you tune a control system in discrete time, this requirement assumes that the physical plant and noise process are continuous. To ensure that continuous-time and discrete-time tuning give consistent results, maxamp is interpreted as a constraint on the continuous-time H_{2} norm. If the plant and noise processes are truly discrete and you want to constrain the discrete-time H_{2} norm instead, multiply maxamp by $$\sqrt{{T}_{s}}$$. T_{s} is the sampling time of the model you are tuning.
inputname |
Input signals for the requirement, specified as a string or as a cell array of strings, for multiple-input requirements. If you are using the requirement to tune a Simulink^{®} model of a control system, then inputname can include:
If you are using the requirement to tune a generalized state-space (genss) model of a control system, then inputname can include:
For example, if you are tuning a control system model, T, then inputname can be a string contained 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 requirement is the implied input associated with the AnalysisPoint block:
For more information about analysis points in control system models, see Managing Signals in Control System Analysis and Design. |
outputname |
Output signals for the requirement, specified as a string or as a cell array of strings, for multiple-output requirements. If you are using the requirement to tune a Simulink model of a control system, then outputname can include:
If you are using the requirement to tune a generalized state-space (genss) model of a control system, then outputname can include:
For example, if you are tuning a control system model, T, then inputname can be a string contained in T.OutputName. Also, if T contains an AnalysisPoint block with a location named AP_y, then inputname can include 'AP_y'. 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 requirement is the implied output associated with the AnalysisPoint block:
For more information about analysis points in control system models, see Managing Signals in Control System Analysis and Design. |
maxamp |
Maximum noise amplification from inputname to outputname, specified as a positive scalar value. This value specifies the maximum value of the output variance at the signals specified in outputname, for unit-variance white noise signal at inputname. This value corresponds to the maximum H_{2} norm from inputname to outputname. When you tune a control system in discrete time, this requirement assumes that the physical plant and noise process are continuous, and interprets maxamp as a bound on the continuous-time H_{2} norm. This ensures that continuous-time and discrete-time tuning give consistent results. If the plant and noise processes are truly discrete, and you want to bound the discrete-time H_{2} norm instead, specify the value maxamp/$$\sqrt{{T}_{s}}$$. T_{s} is the sampling time of the model you are tuning. |
MaxAmplification |
Maximum noise amplification, specified as a positive scalar value. This property specifies the maximum value of the output variance at the signals specified in Output, for unit-variance white noise signal at Input. This value corresponds to the maximum H_{2} norm from Input to Output. The initial value of MaxAmplification is set by the maxamp input argument when you construct the requirement. |
InputScaling |
Input signal scaling, specified as a vector of positive real values. Use this property to specify the relative amplitude of each entry in vector-valued input signals when the choice of units results in a mix of small and large signals. This information is used to scale the closed-loop transfer function from Input to Output when the tuning requirement is evaluated. Suppose T(s) is the closed-loop transfer function from Input to Output. The requirement is evaluated for the scaled transfer function D_{o}^{–1}T(s)D_{i}. The diagonal matrices D_{o} and D_{i} have the OutputScaling and InputScaling values on the diagonal, respectively. The default value, [] , means no scaling. Default: [] |
OutputScaling |
Output signal scaling, specified as a vector of positive real values. Use this property to specify the relative amplitude of each entry in vector-valued output signals when the choice of units results in a mix of small and large signals. This information is used to scale the closed-loop transfer function from Input to Output when the tuning requirement is evaluated. Suppose T(s) is the closed-loop transfer function from Input to Output. The requirement is evaluated for the scaled transfer function D_{o}^{–1}T(s)D_{i}. The diagonal matrices D_{o} and D_{i} have the OutputScaling and InputScaling values on the diagonal, respectively. The default value, [] , means no scaling. Default: [] |
Input |
Input signal names, specified as a cell array of strings. These strings specify the names of the inputs of the transfer function that the tuning requirement constrains. The initial value of the Input property is set by the inputname input argument when you construct the requirement object. |
Output |
Output signal names, specified as a cell array of strings. These strings specify the names of the outputs of the transfer function that the tuning requirement constrains. The initial value of the Output property is set by the outputname input argument when you construct the requirement object. |
Models |
Models to which the tuning requirement 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 requirement for a subset of models in the array. For example, suppose you want to apply the tuning requirement, Req, to the second, third, and fourth models in a model array passed to systune. To restrict enforcement of the requirement, use the following command: Req.Models = 2:4; When Models = NaN, the tuning requirement applies to all models. Default: NaN |
Openings |
Feedback loops to open when evaluating the requirement, specified as a cell array of strings that identify loop-opening locations. The tuning requirement is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. If you are using the requirement 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 interface associated with the Simulink model. Use addPoint to add analysis points and loop openings to the slTuner interface. Use getPoints to get the list of analysis points available in an slTuner interface to your model. If you are using the requirement 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. Default: {} |
Name |
Name of the requirement object, specified as a string. For example, if Req is a requirement: Req.Name = 'LoopReq'; Default: [] |
When you use this requirement to tune a continuous-time control system, systune attempts to enforce zero feedthrough (D = 0) on the transfer that the requirement constrains. Zero feedthrough is imposed because the H_{2} norm, and therefore the value of the tuning goal (see Algorithms), is infinite for continuous-time systems with nonzero feedthrough.
systune enforces zero feedthrough by fixing to zero all tunable parameters that contribute to the feedthrough term. systune returns an error when fixing these tunable parameters is insufficient to enforce zero feedthrough. In such cases, you must modify the requirement or the control structure, or manually fix some tunable parameters of your system to values that eliminate the feedthrough term.
When the constrained transfer function has several tunable blocks in series, the software's approach of zeroing all parameters that contribute to the overall feedthrough might be conservative. In that case, it is sufficient to zero the feedthrough term of one of the blocks. If you want to control which block has feedthrough fixed to zero, you can manually fix the feedthrough of the tuned block of your choice.
To fix parameters of tunable blocks to specified values, use the Value and Free properties of the block parametrization. For example, consider a tuned state-space block:
C = ltiblock.ss('C',1,2,3);
To enforce zero feedthrough on this block, set its D matrix value to zero, and fix the parameter.
C.d.Value = 0; C.d.Free = false;
For more information on fixing parameter values, see the Control Design Block reference pages, such as ltiblock.ss.
When you tune a control system using a TuningGoal object to specify a tuning requirement, the software converts the requirement into a normalized scalar value f(x). The vector 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 requirement is a hard constraint.
For the TuningGoal.Variance requirement, f(x) is given by:
$$f\left(x\right)={\Vert \frac{1}{\text{MaxAmplification}}T\left(s,x\right)\Vert}_{2}.$$
T(s,x) is the closed-loop transfer function from Input to Output. $${\Vert \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\Vert}_{2}$$ denotes the H_{2} norm (see norm).
For tuning discrete-time control systems, f(x) is given by:
$$f\left(x\right)={\Vert \frac{1}{\text{MaxAmplification}\sqrt{{T}_{s}}}T\left(z,x\right)\Vert}_{2}.$$
T_{s} is the sampling time of the discrete-time transfer function T(z,x).