hinffi

Full-information H-infinity synthesis

Syntax

[K,CL,gamma] = hinffi(P,ncont)

[K,CL,gamma] = hinffi(P,ncont,gamTry)

[K,CL,gamma] = hinffi(P,ncont,gamRange)

[K,CL,gamma] = hinffi(___,opts)

[K,CL,gamma,info] = hinffi(___)

Description

Full-information synthesis assumes the controller has access to both the state vector x and the disturbance signal w. Synthesis with hinffi is the dual of the full-control problem covered by hinffc. For the more general output-feedback case when only output measurements are available, use hinfsyn.

[K,CL,gamma] = hinffi(P,ncont) computes the H_∞-optimal control law

$$u=K\left[\begin{array}{c}x\\ w\end{array}\right]$$

for the plant P. The plant is described by the state-space equations:

$$\begin{array}{c}dx=Ax+B_{1}w+B_{2}u\\ z=C_{1}x+D_{11}w+D_{12}u.\end{array}$$

Here, w represents the disturbance inputs, and z represents the error outputs to be kept small.

ncont is the number of control inputs u, which must be the last inputs of P. The gain matrix K minimizes the H_∞ norm of the closed-loop transfer function CL from the disturbance signals w to the error signals z.

[K,CL,gamma] = hinffi(P,ncont,gamTry) calculates a gain matrix for the target performance level gamTry. Specifying gamTry can be useful when the optimal achievable performance is better than you need for your application. In that case, a less-than-optimal solution can have smaller gains and be more numerically well-conditioned. If gamTry is not achievable, hinffi returns [] for K and CL, and Inf for gamma.

[K,CL,gamma] = hinffi(P,ncont,gamRange) searches the range gamRange for the best achievable performance. Specify the range with a vector of the form [gmin,gmax]. Limiting the search range can speed up computation by reducing the number of iterations performed to test different performance levels.

[K,CL,gamma] = hinffi(___,opts) specifies additional computation options. To create opts, use hinfsynOptions. Specify opts after all other input arguments.

[K,CL,gamma,info] = hinffi(___) returns a structure containing additional information about the H_∞ synthesis computation. You can use this argument with any of the previous syntaxes.

Input Arguments

P — Plant

LTI model

Plant, specified as an LTI model such as a state-space (ss) model. If P is a generalized state-space model with uncertain or tunable control design blocks, then hinffi uses the nominal or current value of those elements.

Construct P so that it has the partitioned form

$$\begin{array}{c}dx=Ax+B_{1}w+B_{2}u\\ z=C_{1}x+D_{11}w+D_{12}u.\end{array}$$

Here, w represents the disturbance inputs, and z represents the error outputs to be kept small. The ncont control inputs u are the last inputs.

For information about conditions imposed on the plant matrices and how the software addresses them, see hinfsyn.

ncont — Number of controls

nonnegative integer

Number of control input signals in the plant, specified as a nonnegative integer. hinffi takes the last ncont plant inputs as the control u. The returned gain matrix K has ncont outputs.

gamTry — Target performance level

positive scalar

Target performance level, specified as a positive scalar. hinffi attempts to compute a gain matrix such that the H_∞ of the closed-loop system does not exceed gamTry. If this performance level is achievable, then the returned gain matrix has gamma ≤ gamTry. If gamTry is not achievable, hinffi returns an empty matrix.

gamRange — Performance range for search

[0,Inf] (default) | vector of form [gmin,gmax]

Performance range for search, specified as a vector of the form [gmin,gmax]. The hinffi command tests only performance levels within that range. It returns a gain matrix with performance:

• gamma ≤ gmin, when gmin is achievable.

• gmin < gamma < gmax, when gmax is achievable and but gmin is not.

• gamma = Inf when gmax is not achievable. In this case, hinffi returns [] for K and CL.

If you know a range of feasible performance levels, specifying this range can speed up computation by reducing the number of iterations performed by hinffi to test different performance levels.

opts — Options

hinfsynOptions object

Additional options for the computation, specified as an options object you create using hinfsynOptions. Available options include displaying algorithm progress at the command line, turning off automatic scaling and regularization, and specifying an optimization method. For more information, see hinfsynOptions.

Output Arguments

K — Gain matrix

matrix | []

Gain matrix, returned as a matrix or []. The gain-matrix dimensions are ncont-by-n_y where n_y is the number of states plus the number disturbance inputs of P (inputs not included in ncont).

If you supply gamTry or gamRange and the specified performance values are not achievable, then K = [].

CL — Closed-loop transfer function

ss model object | []

Closed-loop transfer function, returned as a state-space (ss) model object or []. The returned performance level gamma is the H_∞ norm of CL.

If you supply gamTry or gamRange and the specified performance levels are not achievable, then CL = [].

gamma — Closed-loop performance

nonnegative scalar | Inf

Closed-loop performance, returned as a nonnegative scalar value or Inf. This value is the H_∞ norm of CL. If you do not provide performance levels to test using gamTry or gamRange, then gamma is the best achievable performance level.

If you provide gamTry or gamRange, then gamma is the actual performance level achieved by the gain matrix computed for the best passing performance level that the function tries. If the specified performance levels are not achievable, then gamma = Inf.

info — Synthesis data

structure | []

Additional synthesis data, returned as a structure or [] (if the specified performance level is not achievable). info has the following fields

Field	Description
gamma	Performance level used to compute the gain matrix K, returned as a nonnegative scalar. Typically, hinffi tests multiple target performance levels and returns a gain matrix corresponding to the best passing performance level (see the Algorithms section of hinfsyn for details). The value info.gamma is an upper limit on the actual achieved performance returned as the output argument gamma.
X	Riccati solution X∞ for the performance level info.gamma, returned as a matrix. For more information, see the Algorithms section of hinfsyn.
Preg	Regularized plant used for hinffi computation, returned as a state-space (ss) model object. By default, hinffi automatically adds extra disturbances and errors to the plant to ensure that it meets certain conditions (see the Algorithms section of hinfsyn). The field info.Preg contains the resulting plant model.

Algorithms

For information about the algorithms used for H_∞ synthesis, see hinfsyn.

References

[1] Doyle, J.C., K. Glover, P. Khargonekar, and B. Francis. "State-space solutions to standard H₂ and H_∞ control problems." IEEE Transactions on Automatic Control, Vol 34, Number 8, , August 1989, pp. 831–847.

Version History

Introduced in R2018b