Optim al decentralized control of large scale system s

Xiao Qi SHI,Daniel E.DAVISON ,Raymond KWONG ,Edward J.DAVISON

1.System s Control Group,Department of Electrical and Computer Engineering,University of Toronto,Ontario,Canada,M 5S 3G4;

2.Department of Electrical and Computer Engineering,University of Waterloo,Ontario,Canada,N2L 3G1

1 Introduction

Large flexible space structures(LFSS)are now becoming a way of life in the space industry,and tw o control problem s which often occur are attitude control and shape control.The behavior of an LFSS is usually modelled via finite element methods by a set of differential equations whose order may be quite high(sayn＞100),and in this case the practice of controlling only a subset of elastic body modes can lead to the“spillover problem”[1]in which stabilizing the subset of elastic modes may cause instability due to excitation of any uncontrolled elastic m odes.

Early approaches for LFSS control generally have been directed towards centralized control,e.g.,using model reduction methods[2],modal control methods[3,4],output feedback control[1,5],adaptive control techniques[6],and disp lacement feedback[7].An overview of the literature is given in[8].However,more recent research is focused on decentralized control,e.g.,[9-11],and in the survey paper[12],w here it is naturally applicable to LFSS system s.This paper is directed towards decentralized control of LFSS.

In our formulation,we choose a stabilizing controller structure in which there are parameters that can be optimized.optimization of these parameters is then carried out by minim izing a quadratic performance criterion.We demonstrate the effectiveness of the optimized decentralized controller by applying it to an unstable LFSS.We show that the controller not only gives excellent tracking and disturbance rejection performance,but it also possesses a high degree of robustness as well as fail-safe properties com pared to the standard centralized LQR-observer controller.

The paper is organized as follows.The plant model,which is based on a standard model of an LFSS,is described in the next section,as is the problem statem ent.Section III presents themain results of this work,namely the proposal of a low-order controller that solves the decentralized robust servomechanism problem.Section 3 gives the proposed controller structure and outlines the optimization approach used to solve for the three controller gains.Extensive simulations of the controller performance,when applied to an LFSS structure taken from the literature,are then given in Section 4.The numerical example is continued in Sections 5 and 6,where the robustness and fail-safe properties of the proposed scheme are studied.

2 Plant model and problem statement

2.1 Plant model

The behavior of an LFSS[11,13]may be described by the linear time-invariant(LTI)system

w hereMandKare the“inertia”and“stiffness”matrices respectively,q∈Rnis a vector,a vector of control inputs and ω is a vector of unmeasurable constant disturbances.Heredepends on the type and location of the control actuators and similarlydepends on the type of disturbance which occurs in the LFSS.It will be assumed that the control inputsare provided for by point-force actuators or by torques.

The well-known point transformationq=Tρ can now be used to diagonalizeMandKin(1)to obtain

where Ω2is a diagonal matrix,and in this model,the outputs to be regulated are given by

It is assumed that the outputyis measurable and that the system is square with the number of inputs and outputs denoted by ν.The resulting equations of motion of the LFSS are

where.The state vectorx∈R2nis given by

andIn the above expressions,Cdepends on the type and location of outputsyto be regulated,and matricesEandFdepend on the types of disturbances associated with the outputy.

It now will be assumed that the sensors and actuators are mutually dual(i.e.,collocated),which im p lies by[11]that(4)may be written as

for a suitable choice ofBandu.

In addition,it will be assumed that the sensors and actuators are located at ν control stations,such that only the outputs of a given control station may be used in controlling the inputs of that station,i.e.,it will be assumed that a decentralized controller is used,which im plies that the model(5),in turn,can be written asLFSS model:

wherex∈ R2nis the state,are the outputs to be regulated,ui∈ Rmi,i=1,2,...,ν are the control inputs,ω is an unmeasurable constant disturbance,andare constant set points.In this paper,we will take for simplicity thatmi=1 for alli=1,2,...,ν.

De finition 1In(5),assume that there arerigid body modes[11]present and let the rigid body modes and elastic body modes be ordered so that

whereis a diagonal matrix of strictly negative elements corresponding to the elastic modes of the LFSS.Then the model

is called arigid body modelof the LFSS.

Finally,to simplify notation,(6)is compactly rewritten as

where

2.2 Problem statement and existence conditions

Given the plant(8),it is desired to construct an optimal decentralized controller for the system so that the closed-loop system solves the decentralized robust servomechanism problem described in[14],i.e.,it satisfies the following properties:

1)All eigenvalues of the closed-loop system lie in the open left-half plane.

2)Asymptotic tracking and regulation occurs for all constant set pointsallconstant unmeasurable disturbances ω,and all initial conditions of the system.

3)Property 2)holds for all perturbations that do not cause the resulting closed-loop system to become unstable.

If possible,it is also desired that perfect control[15]occurs,i.e.,the controllerhas the property thatthe tracking error can be made arbitrarily small with no peaking occurring.

The following are results obtained from[14]and[15]:

Proposition 1Given system(6),with equal number of inputs and outputs,there exists a solution to the decentralized robust servomechanism problem[14]for constant set points and constant unmeasurable disturbances if and only if the following two conditions are satisfied:

i)The plant(8)has no unstable decentralized fixed m odes[14],i.e.,the plant can be stabilized using decentralized control.

ii)The plant(8)has no transmission zeros at 0,[16],i.e.,

Proposition 2Assuming conditions i)and ii)in Proposition 1 both hold,a necessary condition for a decentralized controller to achieve perfect control is that the plant(6)is minim um phase.

3 Controller structure and parameter optimization

3.1 Controller structure

Consider now the open-loop LTI system defined in(6)and assume that the existence conditions of Proposition 1 are satisfied.The control objective is to solve the decentralized robust servomechanism problem(DRSP)[14]for system(8)and this will be done by implementing the following 3-term(PID)controller:

where βp,βd,and βIare positive scalar gains yet to be determined.

To sim p lify the notation,let(9)be rewritten as

w hereu∈ Rν,y∈ Rν,η∈ Rν,ande∈ Rν.

The following proposition guarantees that a controller of the form(10)solves the DRSP.

Proposition 3Assum e the conditions of Proposition 1 hold.Then there exist values of the parameters βp,βd,and βIsuch that controller(10)solves the DRSP.

The proof of Proposition 3 follow s directly from Theorem 3 of[11].We now wish to optimize the choice of the controller parameters βp,βd,and βI.

3.2 Parameter optimization

The three scalar parameters βp, βd,and βIwill be obtained by solving a parameter optimization problem as was done in the centralized control design of[17],except now the controller design is decentralized.We introduce the performance index

w heree=y-yrefis the error signal and ?＞0 is a tuning parameter.For the problem at hand,the goal is to minimize(with respect to the three parameters βp,βd,and βI)performance index(11)subject to the constraint

obtained from(8).

where

It can be readily verified that the controller given in(13)has the same form as(10).Next,substituteeand˙ufrom(13)into(11)to obtain

or

where Γ＞0 is the solution to the Lyapunov equation

Assume the controller is initialized so thatu(0)=0,and choose the performance index:

This performance index measures the “average cost”of(11)for all tracking signals.Likewise the alternative performance index

measures the “average cost”of(11)for all disturbance signals.In all cases,we apply parameter optimization to determine βp,βd,and βI.We can summarize the above development as a theorem.

Theorem 1Controller parameters βp,βd,and βIthat optimize(11),when used in the controller given by(13)and(14),produce a control law that solves the DRSP.

ProofProposition 3 guarantees that there exist values for βp,βd,and βIwhich solve the DRSP using controller(10).By construction,the controller defined by(13)and(14)is the same as(10).Standard results from quadratic optimal control show that the optimized values of βp, βd,and βImust also solve the DRSP,while also minimizing the cost criterion(11). □

In view of Proposition 2,we can expect to improve the response of the closed-loop system by decreasing ? in(11).Further discussion on the choice of ? is provided in Section 4.

3.3 Practical aspects of numerical optimization

The Nelder-Mead parameter optimization method was used to minimize the performance index(18),over the parameters βp,βd,and βI.To carry out the parame-ter optimization,one must have an initial starting point.This was selected by initially choosing a simple controllerui=-kiηi,i=1,2,...,ν,which stabilizes the rigid body m odes;in particular,the choice ofki=10-2,i=1,2,...,ν was made for the numerical exam p le in the next section.It also follows from[11]and[18]that there exists a feasible starting point for the optimization problem given by βp=0, βd=0,and with βIsufficiently sm all.The initial choice of βp=10-8,βd=10-4,and βI=10-6was made in what follow s.Other initial starting points could also be used.

4 LFSSnumericalexam p le and simulations

4.1 Plant model and controller

The numerical example chosen to illustrate the type of results that can be obtained was taken from Hablani[13],who considered an unstable LFSS structure which has a“flexible pancake”structure with 5 collocated sensors and actuators(see Fig.1).The system has 24 states with 5 inputs and 5 outputs with the disturbance matrixEbeing a 24×5 dense matrix and the disturbance matrixFa dense 5×5 matrix,and it is desired to carry out“shape control”in the presence of these disturbances.

Fig.1 Large flexible space structure[13].

The exam p le of Hablani’s LFSS originally had ordern=100 states with 5 inputs and 5 outputs,but for simplicity,the order was reduced ton=24 with 5 inputs and 5 outputs.A list of the open-loop eigenvalues of the system is given in Table 1.

For this system,it can be verified that the plant is minim um phase and the conditions in Proposition 1 and Proposition 2 are all satisfied.Hence,there exists a solution to the control problem,so that in principle one can obtain a “good”decentralized controller by choosing ? smallenough in the parametero ptimization problem.O f course,engineering constraintssuch assignal saturation will be a limiting factor in the final controller obtained.

To find a controller for this system,w e chose the parameter ?=10-9in(11),and minimized the performance index(18).Table 2 shows the rate of progress of the parameter optimization obtained for this example.Other controllers that were obtained for different values of ? are given in Table 3.It may be seen that excellent control has been obtained in all cases and that the controllers are simple to implement.Note that as ?→0,the system becomes arbitrarily fast.However,the control parameters become larger as ?→ 0,and so a trade off in the choice of ? must be made.

Table 1 List of open-loop eigenvalues.

Table 2 Rate of progress of parameter convergence for ?=10-9 and starting point βp=10-8,βd=10-4,βI=10-6.

Table 3 optimized βp,βd,βI and associated 5%settling time,t s,for different ? values.

The final optimal controller obtained has ?=10-9and is given as follows:

A list of the closed-loop eigenvalues of the system using the controller(20)is given in Table 4.

Table 4 List of closed-loop eigenvalues.

4.2 Simulation results

The controller(obtained with ?=10-9)is able to track the reference signals as well as reject theEandFdisturbance signals in all cases,and the resultant transient behavior is smooth.Representative simulation results are given in Figs.2-7.In term s of the notation used in the figures,we define the set point signal vectorsyref1=[1 0 0 0 0]T,...,yref5=[0 0 0 0 1]T,and the disturbance term sEandFcontain a random 24×5 matrix and a random 5×5 matrix,whereEandFconsist of vectors[E1E2E3E4E5]and vectors[F1F2F3F4F5],respectively.

Figs.2 and 3 show simulation results of the nominal system for unit step tracking.Perfect steady-state tracking is obtained and the response is smooth with settling time near 120 seconds.Note that to produce a smooth output response,the control signals(see Fig.3)are oscillatory.

Figs.4 and 5 show the nominal behavior of the system for the case of rejecting theEdisturbances w henE=B.It is observed that excellent disturbance rejection is obtained.Finally,Figs.6 and 7 show the nominal behavior of the system for the case of rejecting theFdisturbances.Again,excellent control is obtained.

Fig.2 Output response for tracking steps y ref1,...,y ref5.

Fig.4 Output response for rejecting E disturbances;the column denoted E1 is associated with the step disturbance ω=[1 0 0 0 0]T in the plant(5),the column denoted E2 is associated with the step disturbance ω=[0 1 0 0 0]T in the plant(5),etc.The matrix F is set to zero in these simulations.

Fig.5 Control signals for the output responses in Fig.4.

Fig.6 Outputresponse for rejecting F disturbances;the column denoted F1 isassociated with the step disturbanceω=[1 0 0 0 0]T in the plant(5),the column denoted F2 is associated with the step disturbance ω=[0 1 0 0 0]T in the plant(5),etc.The matrix E is set to zero in these simulations.

Fig.7 Control signals for the output responses in Fig.6.

5 Measure of robustness of decentralized controller

In a decentralized controller design,there always will be some uncertainty in the mathematical model of the plant.For example,there will always be high-frequency effects that have been ignored.It would be useful to obtain some measure of how sensitive the controlled system is to high-frequency m odes of the plant.This can be done by finding the so calledreal stability radiusΔD[19,20]of the controlled plant.

Consider the asymptotic stable system˙x=Ax+Bu,y=Cxthat is subject to the following perturbation

whereis assumed to be asymptotically stable but unknown,andare unknown,andis a scalar.

Then in the limit asby singular perturbation analysis,the above system sim plifies to

where.Thus we can now consider ΔD to be an uncertain matrix,and we can determine the real stability radius denoted by rstab for system(22)from[19,20]which has the property that it is the largest bound such that the perturbed closed-loop system is stable for allIn particular,the real stability radius rstab is obtained by finding the largest value of the norm ΔD such that the perturbed closed-loop system is stable.For the LFSS exam p le,the perturbed closed-loop system is given by

and the real stability radius for(23)can be directly obtained from[19,20].The matrix ΔD in(23)is 5 × 5.It is to be noted that the stability radius rstab computed in[19,20]is exact.In this case,a summary of the real stability radius obtained for the proposed optimal three term controller for various values of ? is given in Table 5;a comparison is made with the real stability radius of the standard centralized LQR-observer controller,using the same performance index obtained in[11].

Table 5 Comparison of robustness of proposed controller with LQR-observer controller.

From Table 5,it can be seen that,w hen the proposed controller is used,the system remains stable for all ΔD matrices which have norm less than 2.542×10-3.Moreover,there exists some ΔD with norm equal to 2.542×10-3for which the perturbed closed-loop system will be unstable.

Note that the proposed controller has a real stability radius of 2.542 × 10-3for all values of ?,as compared to the real stability radius for the LQR-observer controller,which degrades as ? decreases;the radius is only 4.231×10-8for the LQR-observer controller when ?=10-14,which is some five orders of magnitude worse than that of the proposed controller.

Thus,for this exam p le it is seen that the proposed decentralized controller has very strong robustness prop ertiesunlike the standard centralized LQR-observer controller which becomes fragile as one attempts to increase the performance of the system to obtain a faster response.

6 Fail-safe properties of the decentralized controller

In the control of modern industrial system s,it is always a concern as to what will happen if a sensor and/or actuator fails.It is highly desirable that such a failure should not result in the failed system having poles with positive real parts,which would be catastrophic,but should result only in some relatively mild deterioration of performance.

Sensor and actuator failure are studied in[10],where it is shown that if a single sensor or single actuator fails,then the unfailed sensors and actuators will be unaffected,but the resultant closed-loop system will have a pole at the origin.This implies that the resultant failed system may drift over time.Thus,in practice it is important that if a sensor or actuator fails,then the corresponding actuator or sensor should be disconnected.(How ever,if the system is in a noise-free enviroment,it maynotbenecessary to carryoutsuch adisconnection.)We now present a series of simulation results to illustrate the fail-safe properties of the proposed controller for the case of a com plete failure of sensor 3 and/or actuator 3.

First,as show n in Tables 6 and 7,we confirm that the closed-loop system for the proposed controller has only one pole at the origin with the remaining poles all stable for a single actuator or a sensor failure.In contrast,the centralized LQR-observer controller results in one pole at zero and 12 poles with positive real parts,which is a severe instability.Consequently,the proposed decentralized controller has excellent fail-safe properties compared to the standard LQR-observer controller,which has no fail-safe protection.

Table 6 List of closed-loop eigenvalues for the LQR-observer controller with failure of sensor 3 or actuator 3.

Table 7 List of closed-loop eigenvalues for the proposed controller with failure of sensor 3 or actuator 3.

Figs.8-23 show simulation results to illustrate how the controller performance is affected by a sensor and/or actuator failure for channel 3.Figs.8 and 9 show the tracking behavior for set point signalsyrefi,i=1,2,...,5 in the case of a joint sensor and actuator failure.In this case,the response of the outputs are very similar to Fig.1 except,unsurprisingly,for the response of channel 3.Figs.10-11 and Figs.12-13 show the behavior of the system under a joint sensor and actuator failure for the case of rejecting theEdisturbances andFdisturbances,respectively.Note that the overall impact of the failure on disturbance-rejection behavior is very mild.

In the situation where actuator 3 fails,but sensor 3 continues to function,Figs.14 and 15 show the impact on tracking performance,Figs.16 and 17 show the impact on rejectingEdisturbances,and Figs.18-19 show the impact on rejectingFdisturbances.The simulation results show that tracking performance is only mildly affected(other than that of channel 3);the impact of the actuator failure onEdisturbance rejection is more severe(in particular,the outputy3in Fig.16 becom es unbounded),but still remarkably mild com pared to the impact typically exhibited by controllers under actuator failure.

Finally,Figs.20-23 show what happens in the situation where sensor 3 fails,but actuator 3 continues to operate.Figs.20 and 21 show the impact on tracking performance.In this case the actuator signalu3becomes unbounded due to the integral control term,but in practice,the actuator signal would saturate and become a constant due to its physical limit.Figs.22 and 23 show the impact on disturbance rejection forEdisturbances;the impact is very mild.

Fig.8 Output response for tracking steps y ref1,...,y ref5 with failures of both sensor 3 and actuator 3.

Fig.9 Control signals for the output responses in Fig.8.

Fig.10 Output response for rejecting E disturbances with failures of both sensor 3 and actuator 3.

Fig.11 Control signals for the output responses in Fig.10.

Fig.12 Output response for rejecting F disturbances with failures of both sensor 3 and actuator 3.

Fig.13 Control signals for the output responses in Fig.12.

Fig.14 Output response for tracking steps y ref1,...,y ref5 with failure of actuator 3.

Fig.15 Control signals for the output responses in Fig.14.

Fig.16 Output response for rejecting E disturbances with failure of actuator 3.

Fig.17 Control signals for the output responses in Fig.16.

Fig.18 Output response for rejecting F disturbances with failure of actuator 3.

Fig.19 Control signals for the output responses in Fig.18.

Fig.20 Output response for tracking steps y ref1,...,y ref5 with failure of sensor 3.

Fig.21 Control signals for the output responses in Fig.20.

Fig.22 Output response for rejecting E disturbances with failure of sensor 3.

Fig.23 Control signals for the output responses in Fig.22.

7 Conclusions

This paper has proposed a new decentralized controller design method for large-scale system s such as those arising in large flexible space structures.The proposed design achieves excellent performance using a low-order controller.In fact,it is somewhat surprising that the multivariable proposed decentralized controller,which has only 3 scalar parameters,can perform better than the standard centralized LQR-observer.In particular,in this study,the LFSS plant model originally proposed in[13]had 5 inputs and 5 outputs and consisted of 100 state variables.For simplicity,this plant model w as reduced to a 24th system,and in this case,if a LQR observer controller is used,it results in a 29th order controller,as com pared to the 5th order proposed decentralized controller.However,if the original 100th order plant model was used instead of the 24th order model,the centralized LQR-observer controller would then have a controller of order 105,as com pared to the present controller,which would still be only 5th order.It is also of interest to note that the centralized LQR observer controller for the 24th order model has a real stability radius some five orders of magnitude w orse than the proposed decentralized controller,when cheap control with ?=10-14is used.The proposed decentralized controller also has the significant advantage of having good fail-safe properties when sensor and/or actuator failure occurs.

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