MIMO H Controller Design Based on Genetic Algoritms. Application to a Flight Control System

MANUEL J. LOPEZ, LUIS GARCIA

Dpto. de Ingeniería de Sistemas y Automática

Universidad de Cádiz

11510 Puerto Real, Cádiz

SPAIN

Abstract: In this paper we propose a method for tuning output feedback multivariable H controller based on standard theory and on genetic algorithms. Problem design considers time domain specifications such as overshoot, rising time and stationary error for MIMO systems, as well as robustness properties based on multiplicative stability margins for plant model uncertainties. Genetic algorithms are employed to satisfy time domain design specifications, which are not considered in an explicit way in the standard H control theory. The result is a design method based on robust control theory (H) and on a robust optimisation technique (genetic algorithms), which is applied to a flight control system.

Key-Words: H control , robust control, optimisation problem, genetic algorithms, flight control system.

1 Introduction

During last twenty years significant advances in robust control methods has been achieved [4], [7], [12], such us: LQG/LTR, , synthesis GPC, IMC and eigenstructure assignment among other ones. At the present time there is a solid base of analytical algorithms to solve these problems. Nevertheless, the design parameters selection is difficult in order to achieve engineering specifications. Several optimisation techniques, denominated as “intelligent” (genetic algorithms, neural networks, simulated annealing, tabu search), can be used for obtaining controller parameters directly, or in order to select design parameters [8], [10]. In this paper the second option has been implemented based on genetic algorithms (GAs).

Conventional algorithms for function optimisation are generally limited to convex regular functions. However, many functions are multi-modal, discontinuous, and non-differentiable. Stochastic sampling methods have been used to optimise these functions. Whereas traditional search techniques use characteristics of the problem to determine the next sampling point (e.g., gradients, Hessians, linearity, and continuity), stochastic search techniques make no such assumptions. Instead, the next sample points are determined based on stochastic sampling and decision rules. Due to it, there has been widespread interest from the control community in applying the Genetic Algorithm (GA) to problems in control system engineering. The GA is robust, global and generally more straightforward to apply in situations where there is little or no a priori knowledge about the process to be controlled [10], [8].

Control engineers need systematic design procedures for design high performance feedback control system. No matter how powerful the design methodology is, a typical application requires several iterations. For that it is essential that reliable computer aided design (CAD) software be used in the design procedure. If a suitable combination of control systems design methodology and CAD software is obtained, the practitioner control engineer will have an easy-to-use design method.

A systematic and useful approach for designing robust controllers is H control. This technique is applied to SISO (single-input single-output) and to MIMO (multi-input multi-output) systems. Due to it, all the control gains are computed simultaneously, and the trial and error one-loop-at-a-time design disappears. Instead, the fundamental engineering decision is the selection of a suitable design parameters set in order to satisfy design specifications.

In H linear theory, robustness (with respect to plant model uncertainty) and performance (specifications for setpoint tracking, disturbance attenuation) are considered in the frequency domain, by means of weighting transfer functions. Nevertheless, temporary response specifications (such as overshoot, rise time and settle time) are not considered “a priori” or explicitly, so that this one must be taken into account after the controller has been obtained in a trial and error procedure. In this paper we propose a method for tuning H controller based on genetic algorithms (GA). Time responses specifications as well as robustness properties are considered for a flight control system, using H theory for solving an H optimisation problem and genetic algorithms for tuning H controller with time domain specifications (overshoot, statioray error, and rise time).

The paper is organized as follows: In section two genetic algorithm optimisation is described, H control technology is presented in section three, and our method based on genetic algorithm for H design is described in section four, where this method is applied to a flight control system obtaining simulation results. Finally, conclusions are resumed in section five.

2. Genetic Algorithm Optimisation

Genetic algorithms (GA) have been used to solve difficult problems with objective functions that do not possess well properties such as continuity, differentiability, satisfaction of the Lipschitz Condition, etc., [1], [5]. These algorithms maintain and manipulate a population of solutions and implement the principle of survival of the fittest in their search to produce better and better approximations to a solution. This provides an implicit as well as explicit parallelism that allows for the exploitation of several promising areas of the solution space at the same time. The implicit parallelism is due to the schema theory developed by Holland, while the explicit parallelism arises from the manipulation of a population of points.

The power of the Genetic Algorithms (GA) comes from the mechanism of evolution, which allows searching through a huge number of possibilities for solutions. The simplicity of the representations and operations in the GA is another feature to make them so popular. Bit strings are used in Holland’s work to encode candidate solutions to the problems which are computational analogy to “chromosomes” of a species in nature, while each bit in the string is an analogue to the “gene”. A fitness function is evaluated on these chromosomes, then genetic operators transform the parent chromosomes to their offspring according to their fitness rating. Commonly used genetic operators are composed of selection, crossover and mutation [3], [5].

The implementation of GA involves some preparatory stages. Having decoded the chromosome representation (genotype) into the decision variable domain (phenotype), it is possible to assess the performance, or fitness, of individual members of a population. This is done through an objective function that characterises an individual’s performance in the problem domain. During the reproduction phase, each individual is assigned a fitness value derived from its raw performance measure given by objective function.

Once the individuals have been assigned a fitness value, they can be chosen from population, with a probability according to their relative fitness, and recombined to produce the next generation. Genetic operators manipulate the genes. The recombination operator is used to exchange genetic information between pairs of individuals. The crossover operation is applied with a probability px when the pairs are chosen for breeding. Mutation causes the individual genetic representation to be changed according to some probabilistic rule.

Mutation is generally considered to be a background operator that ensures that the probability of searching a particular subspace of the problem space is never zero. This has the effect of tending to inhibit the possibility of converging to a local optimum [1], [6], [10].

A Genetic Algorithm Toolbox for the MATLAB package provide a uniform and familiar environment on which to build GA for control design [1]. This toolbox makes GA accessible to the control engineer within the framework of an existing Computer Aided Control System Design (CACSD) package.

Fig. 1. Generalised plant with weighting transfer functions

2. The H Control Problem

In control literature, a standard feedback control system configuration consists of the plant (G), controller (Gr), reference signal (r), measurement noise, and disturbances (acting at the plant input, di, and acting at the plant output, do). The signals to evaluate the performance of the system are the control signal (u), the error signal (e) and the output of the plant (y). All signals are multivariable in general, and nominal mathematical models for G and Gr are considered LTI (Linear Time Invariant).

To take into account the performance specifications and stability robustness, the standard closed loop block diagram can be transformed into the block diagram of Fig. 1, where weights depending on frequency are added. The relation between w (contains all external inputs) and z (vector of all the signals required to characterize the behaviour of the closed-loop system) is z = Tzw w, where

z = [WS e WSK u WT y ]T

and from Fig. 1 it can be obtained that Tzw is a transfer function matriz

Tzw = [Hij], i = 1, 2, 3; j = 1, 2, 3, 4

and its elements are given by

H11 = WSSWr, H12 = -WSSWn

H13 = -WSSWdo, H14 = -WSSGWdi

H21 = WSKGrSWr, H22 = -WSKGrSWn

H23 = -WSKGrSWr, H24 = -WSKSiWdi

H31 = WTTWr, H32 = -WTTWn

H33 = WTSWdo, H34 = -WTSGWdi

where S, Si and T are respectively: output sensitivity function S=(I+GGr)-1, input sensitivity function Si=(I+GrG)-1 and complementary sensitivity function T=(I+GGr)-1GGr.

3. HController Based on Genetic Algorithms


In conventional applications of Hcontrol theory, controller is designed for robustness and performance specifications expressed in the frequency domain; but the typical indicators of the time response (overshoot, rise time, etc.) are not considered “a priori”. In practice, it is difficult to obtain adequate time responses using this approach. In this paper we present a method to satisfy time responses specifications as well as robustness properties. For that, H control theory and genetic algorithms (H-GA) are employed.

We have implemented the following algorithm for the H controller design:

(a)The target close loop time domain specifications (for example: rise time, overshoot, settle time, stationary error) are established.

(b)The structure of weighting matrices is selected (design parameters).

(c)The parameters values intervals are set.

(d)The H problem is solved aided with GA for obtaining design specifications (the target feedback loop).

(e)Performance and robustness stability indicators are calculated in order to evaluate the control system.

4. Flight Control System Application

We shall illustrate the H-GA technique on a lateral aircraft control augmented system (CAS). As test system the F-16 aircraft model has been used [11], [9]. Computations and simulations have been implemented using MATLAB environment, Simulink, Robust Control Toolbox, LMI Toolbox from Mathworks© and the GA toolbox referenced before [1]. In simulations a non-linear model of the F-16 and a digital controller are used in order to carry out realistic tests.

The tracking control system shown in Fig. 2 is meant to provide coordinated turns by causing the bank (roll) angle to follow a desired command while maintaining the sideslip angle at zero. It is a two-channel or MIMO system.

Fig. 2. Aircraft control system for turn coordination.

The Flight Control System (FCS) should hold at the commanded value of and at the commanded value of , which is equal to zero. The tracking error is

(1)

The non-linear F-16 model is linearized at nominal flight condition (true airspeed VT=502 ft/s, sea level altitude, dynamic pressure q=300 lb/ft2). The basic lateral states are four: sideslip , bank angle , roll rate p, and yaw rate r. Additional states and are introduced by the aileron a rudder actuators, both are modelled as (2)

At high frequency, the aircraft model is accurate up to a frequency of 2 rad/s, after the uncertainty grows without bound at the rate of 20 dB/decade. The uncertainty in the plant model could be due to actuator modelling inaccuracies, aircraft flexible modes, time delays, and other effects non considered (Etkin and Reid, 1996). At low frequency closed-loop system should be robust to wind gust disturbances. This information is used in order to obtain bounds on weighting functions singular values.

Due to the fact that the loop gain of the plant model has neither integrator behaviour nor large singular values at low frequency, the mathematical model of the plant dynamics is modified (augmented plant):

  • By integrator in each control channel, so that the closed-loop steady-state error will be zero.
  • By precompensator Pre, for balancing the singular values at low frequency. With this, the speed of the responses will be similar in all channels.

The resulting augmented plant is shown in Fig. 2. In this case, the entire state vector, including aircraft states and integrators is

(3)

The non-linear mathematical model of the aircraft used in simulations are given in [11], [9].

Due to knowledge based on experimental data, the structured selected for the weighting matrices W(s) are given by

W(s) = diag( bj / (s+aj) ), j=1, 2 (4)

where the parameters of the matrices are obtained by minimisation through GAs with objective function

where Mp is the overshoot for step change in the setpoint, tr is the rising time, ess is the steady-state error and wi (i=1,2,3) are weighting factors (subscript “o” means specified values or target specifications).

In order to implement our design methodology the following steps are considered:

step 1: The plant model is scaled with a precompensator for balancing the singular values at low frequency.

step 2: The plant model is augmented with integrators, so that the closed-loop steady-state error tends to zero.

step 3: Matrices structure and range of the design parameters are selected as are given before in (4).

step 4: Parameters of the weighting transfer functions are adjusted by means of GA optimisation.

step 5: Performance and robustness indicators for the linear model are analysed.

step 6: Simulations tests with the non-linear model of the plant are implemented, for nominal conditions and non-nominal conditions.

We apply this H-GA optimisation procedure with time response specifications Mpo= 4% and tro= 0.2 s respectively; and it is obtained Mp = 3.8%, tr= 0.20 seconds, which are near enough to target specifications.

As complementary indicators based on the temporary response are calculated:

where is the sampling time interval.

Useful robustness indicators based on frequency domain are computed too, using the sensitivity and complementary sensitivity functions of the closed loop system, at the plant input (Si,Ti) and at the plant output (So,To). These indicators are obtained as follows:

  • Multiplicative Stability Margin (MSM) at the plant input and at the output, for diagonal-structured multiplicative uncertainty:

= 79.6%

= 85.8%

where is the structured singular value.

Fig.3. Bank or roll step response. Linear and non-linear comparison.

  • MSM for simultaneous diagonal-structured multiplicative uncertainty:

= 34.5%,

where the interconnection matrix, M, is given by

( 4)

From the analysis with the linear model or the plant we have obtained that performance and robustness indicators are satisfactory, due to the fact that the time response indicators are nearby to specifications and multiplicative stability margins (for structured uncertainty) are greater than 75 %. [LGG1]

The next step in the analysis and design procedure is to test the controller with the non-linear model of the plant. In this model it is consider the non-linear differential equations that describe the complex dynamics of the F-16 aircraft [9], [11], and the non-linear behaviour of the actuators, with saturation in position and speed (aileron: 21.5º, 80.0º/sec.; and rudder: 30.0º, 120.0º/sec.).

Although during the design process the plant dynamics and controller have been considered as continuous time systems, for simulation tests a discrete time implementation of the controller is used.

In Fig. 3 the time responses for a step change in set-point are shown, for both linear and non-linear models. As it can be seen very similar behaviours are obtained. Fig. 4 shows actuators signals for the same set-point change. In Fig. 5 roll angle and sideslip angle are given for a disturbance acting at the plant input. Disturbance (pulse in aileron channel) and actuator signals are given too. Fig. 6 presents the controlled variables and manipulated variables for the case of disturbance acting at the plant input, but in this case for rudder channel. As we can see in these Fig. 5 and 6, disturbances rejection objective is achieved.

Fig. 4. Actuators activity for a step change in bank angle setpoint.

Fig. 5. Ailerons disturbance effects.

In order to test the control system robustness for flight conditions different to nominal case, closed loop simulations are made with a fixed controller. This can be seen in Fig. 7, where satisfactory temporary responses for step change in set-point are given. The non-nominal conditions correspond to changes in the true airspeed: a) V=402 feet/sec, and b) V=602 feet/sec. ( 20 % variations with respect to nominal condition).

For greater variations in flight conditions a new controller must be designed for good performance. Our next work treats with an adaptive controller version of the H-GA proposed in this paper.

Fig. 6. Rudder disturbance effects.

Fig. 7. Time responses for different flight conditions.

5 Concluding Remarks

In this paper a new method based on genetic algorithms (GA) for H controller design has been presented. The controller is designed for robustness and time domain specifications requirements. Suitable time responses as well as robustness [LGG2]properties are obtained using an optimisation procedure, where genetic algorithms are employed to satisfy design specifications for a flight control system.

This work is a first step for future works. An expert system based on design specifications and plant knowledge will be incorporated to select the design parameters and for controller adaptation to different flight conditions.

The contribution shows a methodology to combine a robust control technique (H) with a robust optimisation method (GA).

Acknowledgement

This work was supported in part by [a3]the C.I.C.Y.T project DPI2002-02995.

References:

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[2] Etkin, B., L.D. Reid (1996). Dynamics of flight. Stability and Control, Willey&Son.

[3] Goldberg, D. (1989). Genetic Algorithms in Search, Optimisation, and Machine Learning, Addison-Wesley.

[4] Grimble, M.J. (1994). Robust Industrial Control: Optimal Approach for Polynomial Systems, Prentice Hall.

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