^{1}

^{2}

^{1}

^{2}

A formulation of an LPV control problem with regional pole placement constraints is presented,
which is suitable for the application of a Full Block S-Procedure. It is demonstrated that improved
bounds can be obtained on the induced

An interesting technique that has allowed linear parameter varying (LPV) control synthesis algorithms to obtain less conservative performance bounds is given the name of Full Block S-Procedure (FBSP). See [

Subsequent work presented in [

Another reference that is relevant from the practical point of view in LPV control is given by [

It must be recalled that synthesis conditions like the ones presented in [

It is rather frequent, in the application of LPV methods, that feasible problems with acceptable

The aspect of transient response of systems has not been dealt with in the FBSP framework of [

The paper is organized as follows. Section

In this section the synthesis method of LPV control with FBMs, PDLFs, and regional pole placement constraints is presented.

The set

The following lemma is crucial in order to use FBMs for LPV control.

Let

See [

Condition (

In order to state the control problem, consider an LFT parameter-dependent plant:

The synthesis method used in this paper is based upon the results in [

A subset

These regions make up a dense subset in the set of regions of the complex plane, symmetric with respect to the real axis. This makes them appealing for specifying pole placement design objectives.

Let

Internal stability is enforced.

The poles of each closed-loop LTI system, resulting from all constant parameter trajectories in

See [

For the optimization problem to be convex, this method seeks a unique closed loop Lyapunov Matrix

Next, to proceed towards the derivation of synthesis conditions, a dependence of the so-called

with

In order to apply the FBSP on LMIs (

Let

a pair of symmetric positive definite

a quadruple

a real positive performance index

See [

The inequalities of (

The application of Lemma

The computation of the controller's state space matrices is carried out following the algorithm prescribed in [

As observed in [

The example we consider is a sounding rocket (see Figure

Vehicle diagram.

Position of frame

A simple model of aerodynamic drag and lift forces taken from [

The actuator considered for the rocket is a nozzle gimbal which allows small rotations around the

Linearization of (

An evaluation of the Jacobian matrices of (

It also shows that the system can be decomposed in four decoupled systems corresponding to the

The aerodynamic forces are included in the error dynamics in a first-order approximation for zero angle of attack nominal trajectories as follows:

These terms end up inside the

As a consequence of decoupling, the control problem is reduced to synthesizing one controller per subsystem. This simplifies the statement since systems

For each subsystem an augmented plant as the one in (

For each subsystem the augmented plant is the result of the block interconnection of Figure

Block interconnection making up the augmented plant (left) and Inclusion of low-pass transfer function model of actuator dynamics (right).

For the sake of clarity as notation regards, note that the

System

with

Systems

with

System

The synthesis procedure was carried out on all subsystems. The pole placement region used was

Vehicle's details.

Name | Symbol | Value |
---|---|---|

Initial/final Mass | ||

Initial/final Inertia | ^{2} | |

Initial/final Inertia | ^{2} | |

Aerodynamic Derivative | ||

Nominal thrust | ||

Distance from CM to nozzle | ||

Distance from CM to CP | ||

Maximum Dynamic Pressure |

This choice was made, in order to have the same kind of affine parameter dependence of the original plant in the controller. Once the controllers for each subsystem were synthesized, they were appended to make up the complete LPV controller.

A heuristic approach to address the “

A considerable increase in the number of decision variables can take place. For this application, with the pole placement region being a half plane, the number of decision variables of the optimization problem for each lateral controller goes from 73 with the algorithm of [

With respect to previous work [

To stress the fact that the method is not only valid but also applicable, real-time numerical simulations were carried out. The control law was implemented in a computer based upon an Atmel TSC695E SPARC7 class microprocessor operating at 20 MHz. This 32-bit microcontroller has been available in commercial space systems for more than a decade, setting a

To evaluate the response of the system to disturbance signals the

The simulations carried out show responses with the initial conditions deviated from the nominal ones as follows:

with

All quantities in bold face should nominally be zero, except for

In Figure

Response to initial conditions. Error State Variables versus time (sec). (a), (b), and (c): position components (m). (d), (e), and (f): velocity components (m/sec)). (g), (h), and (i): quaternion components (deg). (j), (k), and (l): angular velocity components (rad/sec).

Response to initial conditions. Column 1: error Control Forces

In this work, the use of FBMs was extended to LPV synthesis with regional pole placement constraints. The usefulness of the method was tested on an application example with HIL simulations. The design of an LPV controller for a 6 DOF vehicle with pole placement constraints shows an adequate response without degrading the

This research was partially supported by the Universidad Nacional de Quilmes, Argentina, through Grant PUNQ 0530/07. The setup for simulation was assembled at the laboratories of CONAE, the Argentine Space Agency. The second author has been supported by CONICET and a PRH Grant from the Ministry of Science and Technology of Argentina. The authors wish to acknowledge the contribution of Dr. Ke Dong to this work, who gently sent the source code of the software developed for the example in [