The Structure of Optimal Controllers of Spatially-Invariant Distributed ...

The Structure of Optimal Controllers of Spatially-Invariant Distributed Parameter Systems Bassam Bamieh 

Electrical and Computer Engineering and Coordinated Science Laboratory University of Illinois at Urbana-Champaign Urbana, Illinois 61801. E-mail: [email protected]

Submitted to the 36'th CDC

Abstract

We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of platoons, ow control with large arrays of sensors/actuators such as in Micro Electro Mechanical Systems, and systems described by partial dierential equations with constant coecients and distributed controls and measurements. We consider a variety of optimality criteria, and show that optimal controllers inherit the spatial invariance structure of the plant. We comment on the signi cant implications of this fact to controller design and implementation for such distributed parameter systems.

1 Introduction We consider a special class of distributed parameter systems which are sometimes referred to as spatio-temporal systems [1]. Such systems arise in a variety of diverse problems such as the control of in nite strings of vehicles (recently known as Platoons) [2,3,4], ow control with arrays of sensors and actuators [5,7], and the distributed control of partial dierential equations with constant coecients. In this work, we are ultimately motivated by the control design problem for systems with large actuator/sensor arrays. Such systems are becoming increasingly important in the context of Micro Electro Mechanical Systems (MEMS). It is becoming technologically feasible to manufacture large arrays of micro sensors and actuators with integrated control circuitry. Such devices have already been used for some ow control applications [6,7]. However, the control theory and design algorithms for such systems are far from fully developed. In this paper we address the structural aspects of optimal controllers for such systems. We consider systems in which actuators and sensors form large arrays. These arrays are assumed to be either periodic, or large enough that one can approximate them by in nite 

This research is supported by the National Science Foundation under CAREER award ECS-96-24152

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arrays. We show that when the underlying dynamics of the system are spatially invariant, then optimal controllers will necessarily have a spatially invariant structure. The spatial invariance of controllers has signi cant implications for both the design techniques of optimal controllers as well as the actual implementation of the control algorithms in the actuator/sensor array. It turns out that optimal controllers have a certain degree of inherent decentralization. Another consequence of our main result is that one need only design the controller for a single actuator, and all other controllers are obtained by symmetry. The optimal problems we pose are those that capture global or macro-objectives for the actuator/sensor arrays. The invariance result shows how one can design each individual controller to meet those global objectives. We begin this paper by discussing several illustrative examples of control problems that share the invariance property. We will isolate the common features in those problems, and present a formalism that allows us to state the main result on optimal controller invariance in section 3. In the last section, we discuss the implications of this invariance to the design and implementation of such distributed controllers.

2 Illustrative Examples In this section we discuss the control problem for several examples of systems with the spatial invariance property. The essential feature of these examples will be the underlying geometry.

2.1 The Heat Equation with Distributed Control and Measurements

Consider the dynamics of heat transfer or diusion in an in nite homogeneous medium with distributed control:

@ (x; t) = c @ 2 (x; t) + u(x; t); (1) @t @x2 where  is the material's temperature, and u is the control input. If we assume that (x; t)

is a measured signal, then the plant in this example would have an in nite number of inputs, ux (t) := u(x; t), and an in nite number of outputs x (t) := (x; t) indexed by the spatial variable x. Due to the spatial homogeneity of the equation and the in niteness of the domain, the input-output mapping of this system can be written as

x =

Z1

?1

Gx? u d;

where the dependence of the above quantities on time has been suppressed. Gx? is the \spatial" impulse response of the heat equation, and is a function of (x ?  ) due to the underlying spatial invariance of the system. Thus if we regard ux as the x'th input, and x as the x'th output, we see that the systems input-output mapping has a Toeplitz structure. In this example, we will regard the spatial variable x as indexing the actuators and sensors. The above situation is an idealized problem. A more realistic set up would be to have individual heating elements embedded in a material, together with an embedded array of temperature sensors (see gure 1). In this arrangement, ui , yj denote the i'th and j 'th 2

u?2

?

y?1

6

yo

6

u?1

?

uo

?

y1

6

u1

?

y2

6

u2

?

Figure 1: Material with arrays of heaters and sensors control input and measurement output respectively. Let us denote by Gi;j the dynamical system mapping ui to yj . Because of linearity, we can write:

yi =

1 X

j =?1

Gi;j uj :

In this particular example, each Gi;j is a single-input-single-output (SISO) in nite dimensional system (the dynamics are given by the heat equation with boundary input and output). However, due to the geometrical symmetry in the problem, we see that Gi;j = Gi+k;j+k for any integer k. This indicates that the doubly in nite matrix representation of the system [Gi;j ] has a Toeplitz structure. If we denote G l := Gl;0 (i.e. the 0'th column of the matrix), we can rewrite the input-output map as

yi =

1 X G i?j uj ;

j =?1

(

)

which is a convolution over the index of the actuators.

2.2 Channel and Pipe Flow

Another example which exhibits spatial invariance is in the problem of stabilization of Poiseuille ow. This interesting example is taken from the work of [5]. In the aforementioned paper, the situation is that of a 2D channel with blowing/suction actuation at evenly spaced points along the walls of the channel. The sensors are also an array of evenly spaced shear sensors (though their spacing maybe dierent from that of the actuators). A version of this problem is illustrated in gure 2, where vertical arrows towards the channel indicate actuators, and vertical arrows away from the channel indicate some sensor outputs. The work in [5] proposes and demonstrates the following interesting idea: It is possible by the use of active feedback control of the blowing/suction actuators using the shear sensors information to stabilize planar Poiseuille ow. This is done in the context of linear stability theory; Poiseuille ow is a stationary solution to the nonlinear Navier-Stokes equations, the dynamics of small perturbations around this solution are governed by linear PDE's, and the task of the feedback controller is to stabilize or enlarge the region of attraction of this stationary solution. A variety of interesting techniques are used in[5] which we will not mention here. We will instead focus on the structural aspect of the plant to be stabilized. The formalism and results we develop in this paper are applicable to other types of actuators as well, such as micro- aps [6,7], or piezoelectric devices. 3

6 6 6 6 6 6 6 6 6 --- ? ? ? ? ? ? ? ? ? Steady -Flow --6 6 6 6 6 6 6 6 6 ? ? ? ? ? ? ? ? ? Figure 2: 2D Poiseuille ow with active boundary control We will assume here an in nite 2D channel (in the horizontal direction). Let us denote by u1i , and u0i the i'th actuators in the upper and lower boundary of the channel respectively. Similarly, yi1 , and yi0 will denote the sensors. It can be shown that the input-output dynamics of the \plant" in this example are given by:

yil =

1 X

j =?1

Gl(?i?0j) u0j +

1 X

j =?1

Gl(?i?1j) u1j ;

l = 0; 1:

(2)

In the above equation, the operations l ? 0 and l ? 1 are to be interpreted in the group f0; 1g (thus 0 ? 1 = 1). G0i?j represents the mapping from the j 'th actuator to the i'th sensor on the same side of the channel, while G1i?j represents the mapping from the j 'th actuator to the i'th sensor on opposite sides of the channel. The fact that the input-output mapping can be written in the form (2) follows from the inherent symmetry of the problem. The linearized input-output dynamics have up/down and lateral shift symmetry. This can also be veri ed by investigating the underlying PDE's of the linearization. Note that in this particular example, each Glk represents a SISO in nite dimensional system. A generalization of this example would be to 3D Poiseuille ow in an in nite pipe. If one considers the situation where the actuators/sensors are arranged in the regular repeated pattern shown in gure 3. We use the following indexing scheme; let uli be the control

Figure 3: Geometry of actuators/sensors for pipe ow signal of the actuator in the i'th lateral and l'th angular position. We index the sensor outputs similarly. Note that the lateral index i 2 ZZ := f. . . ; ?2; ?1; 0; 1; 2; 3; . . .g, while the angular index l 2 ZZn := f0; 1; 2; . . . ; n ? 1g, where n is the number of actuators in any one circular ring. 4

With this indexing scheme we see that the dynamics of the linearized ow will be spatially invariant with respect to lateral and angular shifts. This means that we can write the input-output dynamics of the plant as

yil =

nX ?1 X 1 k=0 j =?1

Gl(?i?kj) ukj ;

(3)

where we again interpret l ? k as addition in the group ZZn (the discrete circle with n elements). We note the simpli cation that this indexing scheme, and the invariance of the system's dynamics yield. In the absence of such invariance, the input-output relation would be written as 1 nX ?1 X k Gl;k yil = i;j uj ; k=0 j =?1

where Gl;k i;j represents all possible input-output mappings. Spatial invariance implies that this in nite matrix has a generalized Toeplitz structure with respect to each set of indices.

2.3 Control of Vehicular Strings

This problem initially attracted attention in the late 60's and early seventies. A particularly insightful approach was followed by [2,3], who treated the problem of longitudinal control of an in nite string of vehicles. Recently these problems have attracted attention in the context of Automated Highway Systems (AHS) [8], and the longitudinal control of Platoons of vehicles. The issue that is most relevant to longitudinal control is that of \string stability". Though authors dier on the de nition of this term, intuitively, a platoon has string stability if disturbances are not ampli ed as they propagate through the platoon. The approach we adopt here is that of [4], where string stability is insured by minimizing a performance objective that is global to the platoon. If one uses the sum of all disturbance errors throughout the platoon as a performance objective, this intuitively insures that disturbances diminish as they propagate. More precisely, for in nite platoons, disturbance eects must diminish, for otherwise the performance objective would be in nite. In the in nite platoon problem, the i'th control input ui is the i'th vehicle's throttle input. With each vehicle, one associates either two states for position and velocity [2] or three states to account for the engine's time constant [4]. The performance objective is the regulation of a xed slot length between adjacent vehicles. This regulation must be done in the presence of force disturbances, which model wind gusts and non-smooth road obstacles on each vehicle. Such a problem was set up as an H2 minimization problem in [4]. If one views this in the context of the standard problem of robust control (see gure 4), the generalized plant G is a bidiagonal Toeplitz matrix of transfer functions. If one performs an H2 design for the optimal controller, it turns out that for large platoons (more than 20 vehicles) the optimal controller also has an approximately Toeplitz structure. As the platoon becomes larger, the optimal controller becomes essentially Toeplitz [4]. This implies that the controller in each vehicle operates on measurements from other vehicles in a spatially invariant manner. This result is a special case of the general invariance principle we prove in this paper. 5

3 Formalism and Main Result In order to treat the above example problems and others in a uni ed manner, we will need a formalize their common feature. Spatial invariance in these problems becomes clear when one views the \plant" as mapping the actuator inputs to sensor outputs. For all the problems we considered we were able to index the inputs and outputs to express the dynamics in terms of a convolution with respect to a certain arrangement of the indices. To formalize the above notion, we will require that the signal indices form a group. This is a consequence of the fact that convolution representations require the existence of an underlying group. We will assume that all signals (e.g. u) can be represented as follows: ui (t) := u(i1 ;...;in) (t); where i is a \spatial" multi-index. The number of indices is typically the number of spatial variables in the problem. For example, in the platoon problem the index i represents the i'th vehicle, and the set of indices for an in nite platoon is ZZ . In the channel ow problem i = (l; j ), where l 2 f0; 1g, and j 2 ZZ . If one assumes a limit where one has many actuators and sensors laterally (i.e. continuous actuation and measurement), the corresponding index set would be f0; 1g  IR. In the pipe ow problem, i = (; j ), where  2 ZZn , and j 2 ZZ , i.e. the index set is ZZn  ZZ . The index set in the continuous version of this problem would be [0; 2 ]  IR. In general, signal indices will be either nite or in nite discrete or continuous variables. To formalize this, we make the following essential assumption: Assumption 1 The set of indices of all signals forms a group. Speci cally, for a multiindex i = (i1; . . . ; in), we require that for each m, im 2 Gm , where Gm is a group. We thus see that signals are then de ned over the group G := G1  . . .  Gn: Remark 1 Implicit in the assumption is that all signals are de ned over the same group. This assumption is ful lled even if there is no symmetry between actuators and sensors. If the array of sensors and actuators form a lattice (with possibly an unequal distribution of sensors and actuators in each fundamental cell), then the group can be taken to be the group of translations required to generate the lattice from a fundamental cell. In this situation we consider the system's dynamics within each cell as one MIMO \block". This reorganization is similar to the \lifting" technique for representing periodic systems. The second fundamental assumption we make is that the system's dynamics are spatially invariant. Let the plant input-output mapping be represented by: X X X yi = Gi;j uj ; , y(i1 ;...;in ) = ... G(i1 ;...;in);(j1 ;...;jn) u(j1 ;...;jn); (4) j 2G

j1 2G1

jn 2Gn

which is a general abstract representation of a linear plant. Assumption 2 The input-output mapping of the plant is spatially invariant with respect to the signal index. Speci cally G(i1 ;...;in);(j1 ;...;jn ) = G(0;...;0);(i1?j1 ;...;in?jn ) : 6

Spatial invariance of a given system and its actuator/sensor array is typically easy to ascertain. Generally, the physics and the symmetries of the problem allow one to establish this invariance. We remark here that spatial invariance of either the system or the controller will always be understood with respect to the given underlying group G . When a plant is spatially invariant, we can write the input-output relation of eq. (4) as a convolution: X yi = Gi?j uj j 2G

This general notation now encompasses all the examples covered in the previous section.

Remark 2 When one of the groups is continuous, the above convolution should be per-

formed over that continuous variable. A more precise notation would be

yi =

Z

G

Gi?j uj dj;

where dj is the Haar measure on the group G [9]. However, for simplicity we will use the summation notation with the understanding that our statements are applicable to the case of continuous groups as well.

The main features of the above assumptions are the regularity of the actuator/sensor array and the spatial invariance of the underlying dynamics. The in niteness of the domains is not an essential feature in most cases. Of course, any physical systems will have a nite number of actuators and sensors. The in niteness assumption should be regarded as an approximation to the situation with a nite spatial domain, but where the system's dynamics are at a smaller spatial scale than the size of the underlying domain. This is very similar to the use of in nite-horizon problems to deal with systems that will operate only for a nite amount of time. When the dynamics are much faster than the time horizon, then an in nite horizon gives a reasonable approximation to the problem.

The Optimal Control Problem

Thus far we have been considering the structure of the system and the signals. We now set up the control problem and the performance objectives. We will consider a set up in terms of the \standard problem" of robust control [10,11,12]. This set up is shown in gure 4. We note that in our context, all signals w; z; u; y are indexed over the same group. To illustrate this, we consider the platoon example. Disturbance signals enter into the dynamics of every vehicle, while the slot length between every two vehicles is to be regulated. Thus all signals are indexed over ZZ in the case of the platoon problem. The objective in this problem is to nd a stabilizing controller that minimizes a given input-output norm from w to z . Many optimal and robust control problems can be converted to this standard problem. The norms on the input and output signals can be taken as the usual Lp norms:

0 Z 1 1p 0 1 1p X X kwkp := @ jwi(t)jpdtA = @ kwkppA : R I i2G

i2G

7

z y

G

-

C

 

w u

Figure 4: The Standard Problem We note that taking such norms on the regulated variables z allows us to penalize global objectives, i.e. to design the controller so that some macro-objective of the overall array system is optimized. In the usual notation, we will refer to the closed loop system in gure 4 by F (G; C ). Note that if we represent the generalized plant G by a matrix of transfer functions, the number of inputs and outputs is given by the number of elements in the group G (typically in nite). The i'th row of the controller C represents the transfer function from all the sensors to the i'th actuator. For any given p, the input-output sensitivity of the system is given by the Lp induced norm:

kF (G; C )kp?i := supP kkwzkkp : p w2L If the controller is a stabilizing controller the above worst case gain will be nite.

Optimal Controller Structure

The central question of this paper is whether one can do better with spatially-varying controllers over spatially-invariant ones. For a variety of reasons, spatially-invariant controllers are much easier to design and implement for such distributed parameter systems, much more so than their spatially-varying counterparts. It turns out that when the dynamics of the plant are spatially invariant, and when the performance objectives are spatially invariant, then there is no performance gain in using spatially-varying controllers. To state this result precisely, let LSI and LSV be the classes of Linear Spatially Invariant and Linear Spatially Varying (not necessarily stable) systems respectively. let us de ne the following two problems:

si :=

inf kF (G; C )kp?i; stabilizing C

C 2 LSI

and

sv :=

inf kF (G; C )kp?i; stabilizing C

C 2 LSV

which are the best achievable performances with LSI and LSV controllers respectively. We now can state the main result of this paper: 8

Theorem 1 If the plant and performance objectives are spatially invariant, i.e. if the

generalized plant G is spatially invariant, then the best achievable performance can be approached with a spatially invariant controller. More precisely

si = sv : Note that the statement of the theorem holds for any Lp norm, 1  p  1. In fact, the statement of this theorem is true for practically any useful system norm (including the H2 norm, though it is not an induced norm). The proof of this theorem is based on arguments similar to those used to show that for time-invariant plants, one can not improve performance by using time-varying controllers [13,14]. There are certain dierences between that case and the present one, and the proof of the above theorem will be reported in a subsequent version of this paper.

4 The Structure of Spatially Invariant Controllers Given that we have established that the class of spatially invariant controllers is optimal, we now remark on the implementation and design advantages of such controllers. Spatial invariance implies that one need only to design the controller of a single actuator (or a single cell in general). To illustrate this, let us pick one particular actuator and index it as the \zero" actuator. We will refer to its control signal as u0. The controller of this actuator uses information from all sensors as follows:

u0 =

X

j 2G

Cj yj ;

where Cj represents the transfer function (or gain) from the j 'th sensor to the \zero" actuator. The control signal of any other actuator, say the i'th is

ui =

X

j 2G

Cj+i yj ;

which can be viewed as the spatial convolution of fC?j g with fyj g. Thus the controller is fully described by the sequence of transfer functions fCj g. Each element of this sequence, e.g. Cj , should be regarded as the controller transfer function between a sensor and an actuator which are a \distance" j apart (recall that j 2 G ). This last fact allows us to derive some conclusions about the communication requirements between sensors and actuators in such an array system. In conclusion, we will enumerate the two most important consequences of the spatial invariance of optimal controllers: 1. Consider plants that have the property that disturbances do not get ampli ed by arbitrarily large amounts in short periods of time. This property does not require the plant to be open loop stable, but is more akin to well posedness. For such plants and quadratic problems, one can show that optimal controller \gains" Cj decay to zero as j ?! 1 (or as any subindex of j , corresponding to the in nite subgroups of G , goes to in nity). 9

This fact can be interpreted to mean that optimal controllers have a degree of decentralization, that is, a controller's dependence on information from a given sensor decays as its distance from that sensor increases. In practice, this means that a given actuator will only need information from sensors in its neighborhood. The shape and size of this neighborhood will of course depend on the performance objective to be optimized. 2. When both the controller and the plant are spatially invariant, a great simpli cation of the optimal controller design problem occurs. In this paper, we have not addressed how one actually performs the optimal designs. The essential idea here is that one can now make use of harmonic analysis over the group G [9], by taking Fourier transforms over the group G . This results in a great simpli cation of certain design problems and allows for the use of multidimensional transfer function techniques to answer many dicult questions. The questions of controllability, observability, and optimal designs with quadratic objectives can all be constructively treated in the transform domain. This point of view has been implicit in the work of several authors [2,3,1,15,16] who addressed certain aspects of these problems. It turns out that this point of view is extremely useful for optimal control design problems as well, and we will report on progress in this direction elsewhere.

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[9] W. Rudin, Foruier Analysis on Groups. New York, NY: Interscience-Wiley, 1962. [10] M. A. Dahleh and I. J. Diaz-Bobillo, Control of Uncertain Systems: a Linear Programming Approach. Englewood Clis, NJ: Prentice-Hall, Inc., 1995. [11] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Prentice Hall, 1996. [12] M. Green and D. Limebeer, Linear Robust Control. Prentice Hall, 1995. [13] J. Shamma and M. Dahleh, \Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization," IEEE Transactions on Automatic Control, vol. AC-36, pp. 838{748, July 1991. [14] H. Chapellat and M. Dahleh, \Analysis of time-varying control strategies for optimal disturbance rejection and robustness," IEEE Transactions on Automatic Control, vol. AC-37, November 1992. [15] E. W. Kamen and P. P. Khargonekar, \On the control of linear systems whose coecients are functions of parameters," IEEE Transactions on Automatic Control, vol. AC29, no. 1, pp. 25{33, 1984. [16] P. P. Khargonekar and E. Sontag, \On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings," IEEE Transactions on Automatic Control, vol. AC-27, no. 3, pp. 627{638, 1982.

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