Reaching Conditions in Output Feedback Variable Structure Control

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

REACHING CONDITIONS IN OUTPUT FEEDBACK VARIABLE STRUCTURE CONTROL S.V. Yallapragada and 8.S. Heck School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

Abstract. Two methods are developed to design a variable structure control that satisfies the reaching conditions using only static output feedback. One of the developed methods can be used when there are known bounds of operation for the system. The other method is based on an efficient numerical optimization procedure. It is shown how the resulting control laws can be modified to be robust in the presence of parameter uncertainty or a disturbance. Keywords: variable structure control, output feedback, reaching conditions

I. INTRODUCTION

is of full rank. The VSC control is based on output feedback, so the ith component can be written as:

Variable Structure Control (VSC) is a robust non linear control strategy employing feedback of a discontinuous signal. Most of the previous work in VSC required either full state or estimated state feedback which resulted in an overly complicated implementation. It has been shown recently that static output feedback can be used in VSC (Heck and Ferri, 1989; Yallapragada and Heck, 1991; EIKhazala and DeCarlo, 1992; White, 1990). However, one major obstacle with output feedback is satisfying the "reaching conditions", i.e., conditions which guarantee that trajectories in the state space are directed towards a predefined switching surface.

(2)

The switching function is defined as s = [Sl, ... ,SmJT = Gy and the switching surface is s = o. The focus of this paper is to develop design techniques for choosing the control to satisfy existence and reaching conditions. The existence condition is that trajectories are directed toward the switching surface from the local region surrounding the surface. This corresponds to the local stability of the surface s = O. The reaching condition is that trajectories are directed toward the switching surface from any point in the state space. This corresponds to the global stability of the switching surface. Therefore, if sTs is the Lyapunov function, a suitable control u = f(y) must be chosen to guarantee that sTs < O.

This paper presents two methods for designing output feedback controllers that satisfy the reaching conditions. These methods are limited to the control of linear systems. One of the developed methods can be used when there are known bounds of operation for the system. The other method is based on an efficient numerical optimization procedure.

3. DESIGN METHODS

2. PROBLEM FORMUlATION

Two methods are proposed to design the control so that the reaching condition, sTs < 0, is satisfied.

The system to be controlled is assumed to be linear and time invariant:

x ==

Ax + Bu

Y

Cx

==

3.1. Method 1

(1)

The diagonalization method for state feedback (Utkin, 1978) can be modified for the output feedback case. To simplify the design process, construct a fictitious control u * which is related to

where xeR n, ueR m and yeRP and the product CB 411

Proot

It will be shown that if si> 0 then Si < 0 and if siO for i= 1, ...,m. From (4), an expression for Si is found to be:

the actual control u by a nonsingular transformation: (3)

~ where it is assumed that GCB is invertible and Q is an arbitrary positive defmite diagonal matrix with elements qj on the diagonal. The resulting state equation can be found by substituting (3) into (1). The corresponding expression for S is:

s=

(8)

<\1\.

Substitute x = K- 19 into (8) to yield:

~

= (GCAK-lY)i +

<\1\.

n

=

(4)

GCAx + Qu·

= (GCAx)j +

The only information available for feedback is the output vector y defmed in (1). Suppose that the following assumption is satisfied.

E (GCAK-l)ry. + <\1\. ~ J

. 1 J=

If si> 0, then a suitable upper bound for Si is

Assumption 1: The bounds of certain states or linear combination of states are known such that (n-p) inequalities are available in addition to the p equations of the output vector. czl cz2

S

zl

S

S

Zz

S

Pl P2

ut+.

Thus, if is chosen as in (6), then Sj0.

3.2 Method 2 The following output feedback control law can be designed using an efficient numerical optimization method:

where CZj'S and Il j's are known bounds on these variables. Let Cc ;: [Zl"",zn_p]T and define an augmented output vector as 9 ;: [CT C?]Tx. Let

u

(5)

and that K defined in (5) is invertible, then the output feedback control law defined in (2) satisfies the reaching condition, sTs < 0, when the control components are chosen to satisfy

H

.

<

-EP (GCAK-1kYl j=l

sTs = xTCTGT(GCA - NC)x - czlsit

.E

JJ

max[(GCAK-l)ijCZj_p, (GCAK-l)ijPj_p] (6)

=

E Isi!

i=1

Let L(N) be defined as:

<\1\ n

(10) 0

m

Ish

and

.E

S

where

-

n

J=p+l

- cz(GCB)-1 SGN(s) (9)

where s=Gy and SGN(s) is a vector with components sgn(si) = 1 if Sj~O and sgn(si) = -1 if Si <0. The gain matrix N is chosen to satisfy the reaching condition given by

Theorem 1: Suppose that Assumption 1 is satisfied

<\1\

= -(GCB)-1 Ny

.-

> -

P -1 E (GCAK )i'Yl. 1 JJ

J=

C TG T(GCA-NC) + (GCA-NC»TGC (11) 2

(7)

min[(GCAK -1)ij CZj_p' (GCAK -1)ij Ilj-p]

The reaching condition in (10) can be restated as follows: let cz > 0 and

J=p+1

where the ij subscript denotes the element in the jth row and jth column.

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tained by using the Cutting Plane Method to minimize the weighted average of the norm of N and the maximum eigenvalue of L(N). Let 4>(N) be defined as the quantity to be minimized:

The matrix N can be found using a numerical optimization method which minimizes the maximum eigenvalue of L(N). Note that in general, there will be multiple eigenvalues at 0. This is due to the fact that the maximum rank of CT GT is m, so the maximum rank of L(N) is 2m which is usually less than its dimension of n. As a result, this problem requires an algorithm that can handle nonsmooth optimization. Common techniques such as the Cutting Plane Method or the Ellipsoid Method (Boyd and Barratt, 1990) can be utilized to solve for N. However, special features of this problem warrant a few modifications in the problem formulation. First, since N is a gain matrix, a straightforward minimization of the eigenvalue might yield too large of a gain. Furthermore, it is only required that the maximum eigenvalue be less than or equal to zero to satisfy the reaching condition. A numerical method is described below which is based on the Cutting Plane Method and addresses these considerations.

mp 4>(N) = max Aj{L(N)} + p

where p ~O is a constant and the summation of nk2 is the square of the Frobenius norm on N. Note that the function 4>(N) is convex so that a global minimum exists. The Cutting Plane Method utilizes the convexness; therefore, it is guaranteed to converge. The algorithm is run until the maximum eigenvalue of L(N) is less than or equal to zero or until it reaches a minimum:

Cutting Plane Algorithm: 1. 2. 3.

mp =

4J

+

L 1\4

(15)

k=l

To use the Cutting Plane Method, the quantity defined in (11) must be rewritten as: L(N)

2

L 1\

(12)

Initialize N. Compute the maximum eigenvalue of L(N); if it is equal to or less than zero, stop. Form the matrices H, band f where fT

=

[Ompxl

k=l

1]

T H=[v L 1V +2pn1' ... ,v TImpv +2prmp' -1] where the 4's are nxn real symmetric matrices and the nk's are the elements of N. Specifically, the value of nk in (12) is defined as the kth element of the vector formed by stacking the columns of the N matrix: [nu,.··, n m1; n12' ... ' nm2; ... ; nIp' ... ' nmp]T Let Lo be defined by

4J

= C TG TGCA + (GCA)TGC 2

b = [- VT4JV +

k=l

where v is the eigenvector associated with the largest eigenvalue of L(N). Note that H is 1x(mp+1), b is lxI, and f is (mp+1)x1.

(13) 4.

Solve the linear programming problem: min fTx x

Let 4 be defined by

CTGT~C + CTNJ'GC

pEn; 1

subject to Hx s b

(14)

2 5. where Nk is a mxp matrix with zeros everywhere except a "1" in one position. For k= 1, the 1,1 element of NI is 1. As k is increased, the 1 moves down the first column, then the second column and so on. For example, if m=2 and p=2, then the Nk's are defined below.

N,

=

N3

=

[~ ~}

N2

=

[~ ~].

N,

=

6.

[~ ~].

[~ ~l

Compute 4>(N) from equation (15). Stop if the max A{L(N)} s or if 4>lb(N)~4>(N) or if 4>lb(N) converges. Go to step 3. In the next iteration of step 3, augment the new Hand b row vectors to the last Hand b. Therefore, the row dimensions of Hand B increase with each iteration. (To increase computational efficiency, only the last mp to 2mp rows for which the Lagrange multiplier in the linear program was nonzero need to be included in H and b.)

°

Note that some of the terms in the matrices defined in step 3 are actually the subgradients of 4>, thus providing the descent direction. This algorithm can easily be programmed into Matlab with

An algorithm is described below that finds the elements of the matrix N. This algorithm is ob413

system given in (17) where the control components are chosen to satisfy:

the Optimization Toolbox (needed for the linear programming solution). Comments:

CJi~

H

< -

EP (GCAK - 1)ij)j

- I(GC)jh A -

j=l

1. If an N does not exist that results in L(N) being negative semi-definite, then the term SGN(s) can be replaced with s/lsI2. The resulting control law will satisfy the reaching condition in a region of the state space; the smaller the maximum eigenvalue of L(N) the larger the region. Consider, for example, the control law in (9) with the term s/lsI2. The following expression is obtained:

n

E

max[(GCAK -l)ij aj_p' (GCAK -l)ij Pj_p]

j=p+l and

n

Suppose that L(N) possesses only one positive eigenvalue, A1 > 0, with corresponding unit length eigenvector v. The worst situation is that of x=av where a is a scalar. The expression in (16) becomes

E

min[(GCAK -l)ij tlj_p' (GCAK -l)jj Pj_p]

j=p+l where (GC)j denotes the ith row of the matrix Gc. Note that 1-12 denotes the Euclidian norm of a vector. The proof for this theorem follows closely to the proof for Theorem 1.

To satisfy the reaching condition, sTs
The control law that utilizes the numerical method described in the last section can also be modified. Suppose a matrix N is found numerically that satisfies the condition in (10), then the following theorem can be used to modify the control law given in (9).

2. Other numerical methods for solving this problem are available, see for example, (Overton, 1988). The Cutting Plane Method has an advantage since the function to be minimized is convex, the algorithm converges to the global minimum. The convergence rate, however, is often better with other techniques.

Theorem 3: Suppose hex) = Be where e is bounded by le 12 ~ p(t,x). If the control is chosen as

u

The control laws developed in the last section can be easily modified for application to a system in the following form: + Bu + hex)

-(GCB)-lNy-(a+yp)(GCB)-lSGN(s)

where a~O and y~IGCBI2' and if «GCANC)TGC+ CTGT(GCA-NC» ~ 0, then the reaching condition, sT s < 0, is satisfied for the system in (17).

4. ROBUSTNESS PROPERTIES

x = Ax

=

Proof: The following expression is derived for sTs

(17)

sTs

= xTCTGT(GCA

- NC)x -

(a + yp)lsh + sTGCBe

where hex) represents a bounded uncertainty, unknown disturbance and/or nonlinearities. It can be shown easily that the sliding mode equation is invariant to hex) with either the control in (2) or in (9) if the standard matching condition is assumed, h(x)eR(B). The controls must be modified to guarantee that the reaching condition is satisfied in the presence of hex).

With

the

Is12~ Is11,

Theorem 2: Suppose that Assumption 1 is satisfied, K defined in (5) is invertible, and IhI2~A, then the output feedback control law defined in (2)-(3) satisfies the reaching condition, sTs < 0, for the

conditions in the theorem the following is obtained:

and

where IGCB 12 is calculated as the maximum singular value of GCB. Therefore, if y ~ IGCB 12' then the reaching condition, sTs < 0, is satisfied.

414

written as:

The above theorem requires that h(x)eR(B). This condition is removed in the following theorem.

u = [-0.0124 0.1522]y - 0.417sgn(Gy)

Theorem 4: Suppose Ih(x)l2 s 4. If the control is chosen as

u

= -(GCB)-lNy

If an uncertainty in the model or a disturbance is present as represented in (17), then the results of Theorem 2 can be used to modify this control law as follows:

- ex(GCB)-lSGN(s)

where ex ~ IGCI 24 and if «GCA-NC)TGC+CTGT(GCA-NC)) s 0, then the reaching condition sTs s 0 is satisfied.

Proot

u

=

[-0.0124 0.1522]y-(0.417+1.1024)sgn(Gy)

where IGCl 2 = 1.102 and 4 is a known upper bound on Ih(x) 12,

The following expression is derived for sTs:

The second design method can be demonstrated by considering the following example. Define A as: 0

1

0

0

1.54

0

-0.744 -0.032

-5.2

0

0.337

-1.12

0.039 -0.996 -0.0003 -2.117 0

0.02

0

-4

0

0

0

-20

0

0

-25

An upper bound is found to be: sTs s xTCTGT(GCA - NC)x exlsl} + IshlGChlhh

0

0

-0.154 -0.0042

0

0.249

0

With the conditions in the theorem and the fact that Isll ~ Is12, it is obvious that sTssO.

0.5

0

0

0

0

A

=

-0.277

1

-0.0002

-17.1

-0.178

-12.2

o

o

-6.67

0

C

o0

20

0 000 0

~]

-1

1

0

0

0

0

0

0

0

0

0

0

1

0

0

-5.2 0 0.337

-1.12

-0.0067 0.0167 0.0033 0] [ G - 0.0167 -0.0333 0 0.0333 The Cutting Plane algorithm was used to find the value for N used in the control law given in equation (9). For a weighting of p = lx10-7,

where zl =Xl' The control is chosen using the results of Theorem 1 where Q is chosen to be the identity matrix.

-0.0825Yl + 1.0153Y2 + 2.78

0 0

0

0 0.249

N

u *-

[~

0

0

0 -0.154 -0.0042 1.54 0 -0.744 -0.032

-0.35 s zl s 0.35

-0.0825Yl + 1.0153Y2 - 2.78

0

0

BT =

The states are angle of attack, pitch rate and elevator angle, B = [00 6.67]T and the outputs are pitch rate and elevator angle. It was found in (Heck and Ferri, 1989) that G = [-0.4635, 1] gives good perform ace of the sliding mode for this system. Two outputs are measured for the three state system; therefore, the bounds of one additional variable must be found. In this case, it is known that the angle of attack should never be larger than 0.35 rad (20°), i.e.:

u *+

0

0

5. NUMERICAL EXAMPLES The control law developed in Method 1 is demonstrated on an example of the longitudinal dynamics of an aircraft where:

-1

0

=

-3.469 7.565 0.616 -4.617] [ 14.513 -30.473 -1.612 22.387

which yielded a maximum eigenvalue for L(N) of 3.414xlO-4. Note that this is very small in magnitude, but it is positive. As p is decreased, the eigenvalue decreases asymptotically to zero; however, some of the gains in N go to infinity. It can be shown that the rank of L(N) is less than or equal to 2m, thus there are at least 3 zero eigenvalues for this example. An alternative is to modify the

The actual control can be found from the inverse of the transformation defined in (3) and can be 415

region of the state space. This region is very large for the numerical example given in this paper.

control law as suggested in Comment 1 of Section

3, i.e., replace SGN(s) with s/lsI 2, to get a large

region of the state space in which the reaching condition holds.

Both of the developed control laws can be modified to be robust in the presence of bounded uncertainty, disturbance or nonlinearities. Specifically, it is shown that the modified control laws still satisfy the reaching conditions when the uncertainty, disturbance or nonlinearity is bounded in magnitude.

Interestingly, it was found in this example that as p is decreased, the direction of the eigenvector associated with the largest eigenvalue of L(N) asymoptotically approaches the surface s =O. As a result, the components of x that are not on the surface are directed towards the surface. Moreover, as p increases, then N seems to approach a high gain term of the form G/e where e is small and decreases as p decreases. This correlates with the expression in (10), i.e., substitution of G/e for N in (10) yields:

ACKNOWLEDGEMENT The author'S gratefully acknowledge the support of the National Science Foundation. M.K.H. Fan is also thanked for his help on using the numerical optimization procedure.

sTs = xTCTGTGCAx-.!.xTCTGTGCx-Cllsll e

REFERENCES Boyd, S.P. and C.H. Barratt (1990). Linear Controller Design Limits of Performance, pp. 311-350, Prentice Hall, Englewood Cliffs, NJ.

Thus, if there is a component of x that is not in the null space of C, then the second term dominates the expression driving sTs negative. If x is entirely in the null space of C, then it is on the sliding surface.

DeCarlo, RA., S.H. Zak and G.P. Matthews (1988). Variable structure control of nonlinear multivariable systems: a tutorial. Proc. of the IEEE, 76, 212-232.

6. CONCLUSIONS Two methods have been developed to design output feedback control laws that satisfy the reaching conditions for a variable structure control. Output feedback requires a much simpler implementation than either full-state feedback or estimated state feedback, however, satisfying the reaching conditions had previously been a major obstacle in the use of output feedback in VSC. The first method developed is based on a modification of the diagonalization method used for state feedback. This method can be used when bounds of operation on some of the system variables are known. This method is equivalent to determining a region of attraction in the state space for stability of the sliding surface.

EI-Khazala, R. and RA. DeCarlo (1992), "Variable Structure Output Feedback Control," Proc. of the ACC, Chicago, IL, pp. 871-875. Heck, B.S. and AA. Ferri (1989). Application of output feedback for variable structure systems. AIAA J. of Guidance. Control and Dynamics. 12, 932-935. Overton, M.L. (1988). On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Analysis Applications, 9, 256-268. Utkin, V.I. (1978). Sliding Modes and Their Application in Variable Structure Systems, MlR Publishers, Moscow.

The second method is used to find a gain matrix N from a numerical optimization procedure. For the control law suggested in this method, a specific matrix that depends on N, i.e. L(N), must be negative semi-definite to satisfy the reaching condition. The numerical procedure minimizes the weighted average of the maximum eigenvalue of L(N) and the square of the Frobenius norm of N. In this manner, the reaching condition can be satisfied without requiring an unduly large gain N. It should be noted that special numerical optimization algorithms must be used on this problem since the function to be minimized is not smooth. In the cases where L(N) cannot be made negative semidefinite, a modification to the control is suggested that ensures that the reaching condition is met in a

White, BA. (1990). In: Deterministic Control of Uncertain Systems, (A.S.I. Zinober, Ed.) pp. 144-169, Peter Peregrinus Ltd., London. Yallapragada, S.V. and B.S. Heck (1991). Reaching conditions for variable structure control with output feedback. Proc. of the ACC, Boston, MA, 32-36.

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