### A: SLICOT Performance Results

#### Performance of SLICOT and MATLAB

The performance (in terms of efficiency, reliability, and accuracy) which can be obtained using SLICOT components has been investigated in comparison with equivalent computations performed by some MATLAB functions included in the MATLAB nucleus or in the MATLAB Control Toolbox. The results show that some SLICOT computations are several times faster than MATLAB computations, at comparable accuracy; moreover, less memory is required by SLICOT routines, because the problem structure is fully exploited whenever possible.

### B: SLICOT Applications

#### SLICOT drives tractors

In control engineering applications, the numerical solution of one of the fundamental matrix equations of linear control theory often causes a bottleneck for its successful realization. Here we will describe one such application, the automatic steering of a farm tractor, in which this situation occurs. By using the SLICOT subroutine SB02MD, the problem is resolved, i.e., the used subroutine yields a numerical solution with sufficient accuracy and in due time.

#### SLICOT controls microwaves

Recent technological advances in microwave household appliances have created the possibility of a more refined control of the heating process in combination ovens. This includes the control of both a microwave heating source and a forced convection (air) heating source. Optimal control of a combination oven will give a higher quality end-product with a more uniform temperature distribution and, hence, a better cook-quality. This short article reports on the application of the (recently available) SLICOT reduction routines for calculation of an optimal control history for a microwave combination oven. We present a simple, finite horizon linear quadratic regulator solution to the heating problem, which was calculated using the reduced set of model equations. The example involves the calculation of an optimal heating profile for a container of mashed potato. The original (high dimensional) system model was reduced using the SLICOT routine AB09CD, which includes optimal Hankel norm approximation with square-root balancing. This routine allows a substantial reduction of computation time for the LQ design. The reduction routines were found to be more efficient than MATLAB routines. Recent progress is reported.

#### SLICOT controls helicopters

This example shows the design of a flight controller for a Bell 205 helicopter using H-inf optimisation approach. The design of flight controllers for helicopters is a difficult task due to the instability of the helicopter, complexity of the interactions between rotor, fuselage, air-flow, power-plant, etc., and, in particular, the model uncertainties and disturbances. In the last few years, the Control Systems Research Group at the University of Leicester, UK, has been, in collaboration with the Defence Evaluation and Research Agency (DERA) Bedford, UK, and the National Research Council (NRC) of Canada, developing robust controllers for helicopters. The design example shown here is one of the exercises in that research.

The system considered is a Bell 201 A-1 helicopter. The nominal model, representing the dynamics at velocity of 60 knots at sea level, consists of 8 states: forward velocity, vertical velocity, pitch rate, lateral velocity, roll rate, yaw rate, pitch angle and roll angle; 4 control inputs: main rotor collective, longitudinal cyclic, lateral cyclic and tail rotor collective; and 6 controlled outputs: vertical velocity, pitch angle, roll angle, yaw rate, pitch rate and roll rate. The H-inf mixed-sensitivity optimisation approach is used, with appropriately chosen weighting functions. The recently developed SLICOT subroutines SB10FD, SB10PD, SB10QD and SB10RD are employed and a sub-optimal controller is obtained. The simulations included show the properties of the closed-loop system. (Note that the first 4 outputs are plotted in the step responses of the closed-loop system.) It should be stressed that together with the derived controller the condition numbers and accuracy estimates are also produced which would greatly help the user judge the legitimacy of the computations and reliability of the designed controller. More details can be found in: D.-W. Gu, P.Hr. Petkov and M.M. Konstantinov: "H-inf and H2 Optimisation Toolbox in SLICOT", SLICOT Working Note 1999-12, September 1999

#### SLICOT controls distillation columns

This example shows the controller design for a distillation column using H-inf optimisation approach. A distillation column for production of benzene is considered here. The model is implemented using SpeedUp (Aspen Technology Inc.) originally and improved with Aspen Custom Modeller. The production of Benzene is carried out via hydrodealkylation of toluene C_7H_8, involved an exothermic reaction and an equlibrium reaction. The nonlinear system has 82 states. It is linearised at a set point and then reduced to a model of 17 states. There are 13 possible actuators (inputs), among which the most effective 6 are selected for the control system design. The system has 5 measurable outputs to be controlled, namely, flash inlet temperature, production rate, product purity, hydrogen to aromatics ratio and flash vapour outlet pressure. The H-inf mixed-sensitivity optimisation approach is used in the design. Appropriately weighting functions are chosen. The synthesis of a robust controller employs the recently developed SLICOT subroutines SB10FD, SB10PD, SB10QD and SB10RD. A sub-optimal controller is obtained. The simulations show the properties of the closed-loop system. (Note that the first 4 outputs are plotted in the step responses of the closed-loop system.) It should be stressed that together with the derived controller the condition numbers and accuracy estimates are also produced which would greatly help the user judge the legitimacy of the computations and reliability of the designed controller. More details can be found in: D.-W. Gu, P.Hr. Petkov and M.M. Konstantinov: "H-inf and H2 Optimisation Toolbox in SLICOT". SLICOT Working Note 1999-12, September 1999

#### SLICOT reduces models: Linearized aircraft model of ATTAS.

This model describes the linearized rigid body dynamics of the DLR Advanced Technology Testing Aircraft System (ATTAS) during the landing approach. The nonlinear model of ATTAS used for linearization has been obtained using the object oriented modelling tool Dymola. Besides flight dynamics, this model includes actuators and sensors dynamics, as well as engine dynamics. Several low pass filters to eliminate structure induced dynamics in outputs are also included. The total order of the model is 55. The linearized model is non-minimal and has additionally an unstable spiral mode. Moreover, because of presence of position states, there are three pure integrators in the model and an additional one for the heading angle. Overall, there are 6 control inputs, 3 wind disturbance inputs, and 9 measurement outputs. This model serves basically for the evaluation of linear handling criteria in a multi-model based robust autopilot design.

Several low order models have been computed using the Balance & Truncate method. A 15-th order approximation has been computed which fits almost exactly the original 55 order model both in terms of step responses as well as of Nyquist frequency responses. Click here to see the good agreement obtained between the frequency responses of the original and reduced model for (2,2)-element of the corresponding transfer function matrix. Reduced models for longitudinal and lateral modes have been also computed. The reduced longitudinal ATTAS model has 7 states, 4 inputs and 4 outputs, while a reduced model for the lateral flight has 10 states, 2 inputs and 5 outputs. For more information see:

1. A. Varga: "Selection of Model Reduction Routines", SLICOT Working Note 1998-2, June 1998.

2. A. Varga: "Model reduction routines for SLICOT", NICONET Report 1999-8: June 1999.

#### SLICOT reduces models: CD-player finite element model

This is a 120-th order single-input single-output system which describes the dynamics between the lens actuator and radial arm position of a portable compact disc player. Due to physical constraints on the size of the systems's controller, a reduced model with order at most 15 is desired. In order to test the SLICOT software, three 10-th order models have been also determined using the three available methods. Click here to see the performance of different computed approximations on basis of Bode plots. All methods approximate satisfactorily the central peak at frequency about 120Hz, but have different approximation properties at low and high frequencies. Both SPA and HNA approximations seem to be inappropriate, although the stationary error for the SPA method is zero. The B&T method appears to provide a good 10-th order approximation. For more information see:

1. A. Varga: "Selection of Model Reduction Routines", SLICOT Working Note 1998-2, June 1998.

2. A. Varga: "Model reduction routines for SLICOT", NICONET Report 1999-8: June 1999.

#### SLICOT reduces models: GEC ALSTHOM gasifier model

A detailed nonlinear industrial gasifier model has been developed by GEC ALSTHOM, in October 1997, as a benchmark problem for simulation and robust control. The model includes all significant effects; e.g., drying of coal and limestone, pyrolysis and volatilisation of coal, the gasification process itself and elutriation of fines. This model has been validated using measured time histories from the British Coal CTDD experimental test facility and it was shown that the model predicts the main trends in fuel gas quality. Linearized models at 0%, 50% and 100% load are available to support a multi-model based robust controller design. The three linearized models have order 25 and are non-minimal. Numerical difficulties with respect to using these models have been reported. The apparent cause of difficulties is a poor scaling of the model.

The computed reduced order models of state dimension 16 cannot be practically distinguished from the original models on basis of time or frequency responses. No preliminary scaling of the original models was necesssary, since this feature is available by default in all model reduction routines of SLICOT. Several lower order approximations of orders 6, 8 and 12 have been also computed. The 12th order models represent very good approximations of the original models and can serve as bases for designing a unique robust controller ensuring satisfactory performance for all three models. A comparison on basis of the (3,5)-elements of the corresponding transfer-function matrices can be seen here. For more information see:

1. A. Varga: "Selection of Model Reduction Routines", SLICOT Working Note 1998-2, June 1998.

2. A. Varga: "Model reduction routines for SLICOT", NICONET Report 1999-8: June 1999.

Ad van den Boom, November 12, 1999, updated: This email address is being protected from spambots. You need JavaScript enabled to view it. June 15, 2006