A Volterra Method for Nonlinear Control Design

Copyright © IFAC l'onlinear Control Systems Design. Capri. italY 1989

A VOLTERRA METHOD FOR NONLINEAR CONTROL DESIGN S. A. AI-Baiyat* and M. K. Sain** *Electrical Engineering, King Fahd University of Petroleum (3 Minerals , Dhahmn 31261 , Saudi Arabia **Electrical alld Computer Engzlleerillg, L'lliversity of Sot re Dame, ;\'otre Dame, IN 46556, USA

Abstract. In Al-Baiyat and Sain (1986). the Volterra model has been used for the design of a nonlinear feedback control system for a given multiple-inputs multiple-outputs nonlinear system. The approach extended the ideas of Peczkowski. Sain. and Leake (1979). which deal with the simultaneous design for a specific output response and a reasonable control signal. This paper continues that study. which involves negative unity output feedback around linear analytic plants to achieve partial feedpack linearization. A tensor representation of the plant kernels as well as explicit realizations of the compensator is given. for multiple control inputs. An example is given. Keywords:

Nonlinear control systems; multivariable control systems; transfer functions; decoupling; control system synthesis; Volterra; tensors.

INTRODUCTION

(M,T). for a given plant p. such that T = PM.

Contributions to the theory of input-output models for nonlinear systems. from a Volterra-type viewpoint. have been made by Brockett (1976). Brockett and Gilbert (1976). Sandberg (1982). Fliess and colleagues (1983). and Lesiak and Krener (1978). In the case of multiple input. multiple output. time-invariant systems such a representation is

> TJ >

> Tj > 0

(2)

Consider next the question of synthesizing the pair (M.T) in a specific feedback structure. Define the operator G: R + U, which describes the mapping of errOrS to the plant control action. Then the problem of negative unity output feedback synthesis is to find a controller G which synthesizes the pair (M,T) according to the structure of Fig. 1.

)1

T (1)

)1

M where Pj. j = 1.2 •••• is called the jth Volterra kernel. Recently. AI-Baiyat and Sain (1986) have used the Volterra model for the design of a nonlinear output feedback control system for a given linear analytic plant. The approach was based upon nonlinear feedback synthesis as introduced for a single point of operation by Peczkowski. Sain. and Leake (1979) and extended later to general operating regions by Peczkowski and Sain (1981; 1982; 1985a.b). In 1986. AI-Baiyat and Sain contributed an extension to the 1979 work. Using operators based upon Volterra convolutions. a method of design was outlined. proceeding recursively from the first order. 1979. methods to second order. third and higher orders. This paper continues the study by giving tensor representations of multiple-input plant kernels as well as explicit realizations for negative unity output feedback controllers around linear analytic plants to achieve partial feedback linearization.

I--E~

flGI

u

I-h I -I

>

P

y

Fig. 1. The approach of AI-Baiyat and Sain (1986) used the multidimensional Laplace transform. An interesting special case is the following. Suppose we want to design such that the higher order kernels of the overall system are zero up to degree d, that is (3)

PROBLEM FORMULATION

This means we want to use the feedback design to linearize the system up to degree d. If the first-order plant is invertible. and if the firstorder loop is well posed. it was shown that the kernels of the controller G are given by

Let R, U. and Y denote the spaces of requests to the system. controls to the plant. and responses from the plant, respectively. With these spaces defined, let P: U + Y denote a nonlinear plant. If the feedback action of the controller is well defined, then there will be an operator M: R + U generating control actions from requests and anoth operator T R + Y describing plant responses to requests. A general design problem is to find

Gl(S) - Pl-1(S)Tl(S)(I-Tl(S»-I.

(4)

G2(SI.S2) = -Pl-l(SI+S2)P2(SI. S 2) (S)

125

S. A. AI-Baiyat and M. K. Sain

126

(6) and in general:

i-j+2-k 1

i

I

I

j =2

(1978), and Fliess (1983). Brockett used the Carleman Linearization idea to obtain a bilinear model that approximates the linear analytic system. Once a bilinear model is obtained then it is straightforward in principle to find a Volterra representation for the new model. Brockett's work was restricted to single input, single output linear analytic systems. Here, this work is extended to the class of multi-input multi-output linear analytic systems, by utilizing tensor product. So the goal is to approximate the system given in (8) by a bilinear system. This process begins by the expansion of (8) around the origin using a truncated power series, N

k2=1

x

=

I

N

A1kXk +

k=l

SkI + k

j-1

(12.a)

(12.b)

+k

j-1

+1,"" Si»}'

>1

i

where xk = X 0 x 0 0 x (k times). In order to find a bilinear model we need to find dxi/dt, i ~ 2. In these differential equations we will also truncate any term beyond N deg. We can develop the following differential equation for ~, j > 1

( 7)

The goal of this paper is to realize the higher order controllers giv e n in Equations (5-7), for the class of multi-input, multi-output linear analytic plants

x-

D1kXk 0 u

+1 + ••• +Si)

(Gk (Sl, ... ,Sk ) 0 Gk (Sk +l, ... ,sk +k ) 0 ... 0 1 1 2 1 1 2 Gi-k - ... -k. (Sk + 1 J-1 1

I

k=O

N-j+1

N-j+1

I

.!L [xj] dt

Ajk Xk +j - 1 +

k=l

I

Djk Xk + j - 1 0 u,

k=O

xj (0)

j = 1, ••• ,N

0,

(13)

in which x(O) = 0, 0 f(O)

f(x) + g(x)u

(8.a)

0,

Ajk = A1k e In e ••• e In + In e A1k e ••• e In y

= h(x)

(8.b) (14)

+ ••• + In e In e ••• e A1k where f, and the m columns g l, ••• ,gm of the matrix g(x) are analytic functions, in the neighborhood of the origin, from X + X, a nd h: X + Y is analytic also.

and e In + In e D1k e ••• 0 In

D1k e In e VOLTERRA KERNELS FOR LINEAR ANALYTIC SYSTEMS HAVI NG MULTIP LE INPUTS

+ ... +

In this section, we devel op a Volterra kernel r e presentation for linear ana l ytic systems having multiple inputs. We start by treating the bilinear systems. Here bilinea r systems are described by a state space description of the form

In e ••• e D1k e In + In e ••• e In e D1k'

(15)

In equations (14) and (15) there are j terms with (j-1) products. In (15) the terms with the bar are defined by -1

x(t)

Ax(t) + Dx(t) e u(t) + Bu(t),

(9.a)

In e D1k e ••• 0 In - (In e D1k e ••• e In)Qjk

y(t)

Cx(t)

(9.b)

where Q is an invertible map on Xk+1 0 U e xj-2 into Xk+j-1 e U, the powers of X being tensor powers. Then we have a bilinear system

x(O) = 0,

where x(t) ERn, u(t) E Rm, y(t) E RP, A E Rnxn, D £ Rnxmn, B E Rn xm, a nd C £ RPxn. We use the Pea no-Baker formula in conjunc tion with the tensor product.

1- xl -I

I

PI ( t

, T) =

Ce A ( t

x2

1

With some manipulation it can be shown (Al-Baiyat, 1986) that the Volterra kernels of this system are given in triangular form by

Pj(t,'l, ... ,T j ) = Ce D([e

A21

A2(N-1 )

x2

0

0

A3(N-1)

x3

I

0

0

~1

xN

J

\- Dll D12 B]

I

e lm) .. , ] e lm)] e lm), (11)

Next, the Volterra series for analytic systems is s t udied. sentation of such s ystems ha s Brockett (1976), Rugh (1981),

the class of linear Th e Volterra rep rebeen investigated by Lesiak and Krener

xl

0

D([

A(Tj_1- Tj )

-

A1N

+

I

A(T1- T2) D([e

All A12

I II =

xN A(t-T1)

I

( 10)

-T ) B ,

j-

(16)

!I

D20 D21 0

i

I

0

DIN

-1111

D2(N- 1)

D30

D3(N-2)

0

DN1

!I

xl e u

I

1

1-

D10

I

0

x2 0 u

I

x3 e u

1 1 1+

I

J

-I

xN e u

0

-I

u.

I

J

0

(17.a)

A Volterra Method for Nonlinear Control Design The output equation for this system is given as (17.b)

REALIZATION : SECOND ORDER In this and the next section we will investigate the realization problem of multi-input. multi-output Volterra systems. Our approach is a generalization of Schetzen (1965) by utilizing tensor product. We begin our treatment by considering the class of second order homogeneous systems that can be synthesized by using three linear systems connected as in Fig. 2.

u

_ _S_21_ _ =

~

127

our goal is to construct a jth order basic system consisting of (2j-l) linear systems and at most (j-l) multipliers. tensor product. To present this extension. let us first recall the following result (Rugh. 1981). Suppose ft(tl ••••• tt) can be written as a convolution of the form

(23) then. in the transform domain (23) will be

Now let us construct a third order homogeneous system as in Fig. 3. From the reasoning above. the kernel of the resulting third order homogeneous system can be written by inspection as

zll ® z22

Fig. 2. Let us denote the kernel of the second order system by h2(Tl.T2) arid the kernels of the linear systems 511. 522. and 521 by hll(t). h22(t). and h21(t) respectively. Now yet) is the output of the linear system 521 whose input is z2(t). Hence we can write

So if a given transfer function of a third order homogeneous system can be decomposed as in formula (25) we can synthesize such a system by five linear systems connected as in Fig. 3. The realization problem of higher order systems can be tackled in the same way. connected as in Fig. 4. The kernel of the jth order system will be Hj(SI ••••• Sj)

=

Hjl(SI+ ••• +S j)

([HU-l)l(SI+·· .+Sj-l)( [ ••• ([H21(SI+S2)(Hll(SI) ® H22(S2»] 0 ••• )]0 (18)

HU-1)2(Sj-l»]

0

(26)

Hj2(Sj».

So again if a given transfer function of a jth order homogeneous system can be decomposed as in equation (26). then we can realize such a system by (2j-l) linear systems connected as in Fig. 4. in which AN APPLICATION OF TENSOR REALIZATION (20) Our goal is to use the structure of Fig. 2 in the realization problem. So it is more convenient to work with the transform of the kernel h2(Tl.T2). To do that let us take the transform of (20)

In the preceding section a realization scheme for a jth order homogeneous Volterra system was developed. The scheme is very useful in completing our design process. since the resulting controllers from our design method. equations (4-6). have a structure similar to that of equation (26). In this section we show how the realization scheme is utilized in realizing our controllers. Specifically. we will show in detail

I-h

r---------~) I 532

(21)

e

By making the following change of variables tl - Tl -a. and t2 = T2 -a. and using the fact that hll(t) and h22(t) are causal functions we will have

I I

' Ii i

i

r

I

i

1

I

I I

522

f-----l

i l l i

!Z22

~,

l

In this section. the realization method that we have given for second order Volterra systems in the preceding section is extended to the realization of higher order systems. In brief.

z32

I

I

I I z21 I i-~'-~\ j - j y L>i 511 0 521 ' 0 I 531 H I

REALIZATION : HIGHER ORDER

1

1--1

I I

u

I

-- I

( l

If the transfer function of a given second order system can be decomposed in the form of equation (22). then that system can be realized by three linear systems connected as in Fig. 2.

1

i__ ! \

\

I1_ _1

Fig. 3

\ 1_ _ 1 z21 ® z32

S. A. AI-Baiyat and M. K. Sain

128

I----------------~ -- - I

SU-2)2

---

h

I----=--_---~ !~h ~l S22 Z22 I Z32

h

-U--'---1

41

'"

I~hI 1_ _ 1

I 1

z(j-2)2

~1-:-~I-:-~I_S_(_j-_2_)I_ I -I --

- - Zll

Z21

Zll ® Z22

Z31

Z(j-2)1 Z(j-2)1 ® Zj2

Z21 ® Z32 Fig. 4.

the decomposition of the second and the third order controllers, obtained from partial feedback linearization, according to formula (26). We will also discuss higher order controllers. But, before that let us give the kernels of the plant in the transform domain, since they will be useful in the realization process. By taking the transform of (11) it can be shown (Al-Baiyat, 1986) that the jth order kernel is given by Pj(SI, ••• ,Sj)

=

given in (6). write

To simplify our analysis let us

(36) To show the decomposability of the controller let us start with

C«SI+··.+Sj)I-A)-1 (37 )

O([«SI+.··+Sj-l)I-A)-IO ([ ••• O([(SII-A)-IB] ® lm)·.·] ® lm)] ® lm)·

(27)

Again from the assumption that the plant is linear analytic (37) becomes

Special cases of the jth order kernel are Al(sl,S2,s3) = - {C«SI+S2+S3)I-A)-IB} - 1 (28 ) C«SI+S2+S3)I-A)-IO P2(SI ,S2) ~ C«SI+S2)I-A)-IO( [(sll-A)-IB]

® lm), ([(SII-A)-IBGl(SI)] ® G2(S2,S3».

(38)

(29) P3(SI,s2,s3)=C«SI+s2+s 3)I-A)-IO([«SI+S2)I-A)-IO ([(SII-A)-IB] ® lm)] ® lm).

(30)

Let us show the decomposability of a second order controller resulting from partial feedback linearization. Suppose that the plant is a linear analytic system; hence, substituting for P2(sl,S2) in (5) gives G2(SI,S2) = - {C«SI+S2)I-A)-IB}-IC«sl+S2)I-A)-IO ([(SII-A)-IB]11 Im)(Gl(SI) ® Gl(S2».

We let 31 F31(SI+S2+S3) = - {C«SI+S2+S3)I-A)-IB}-1 C«SI+S2+S3)I-A)-IO

(39)

31 F32(sl) = (SII-A)-IBGl(SI)·

(40)

and

Hence, (38) becomes 31 31 Al(sl,S2,s3) = F31(sl+S2+s3)(F32(sl) ® G2(s2,s3»,

(31) (41)

But this can be written as which is in the form of Fig. 4. into A2(SI,S2,S3), which is

Next let us look

A2(SI,S2,S3) = - Pl- l (SI+s2+s 3)P2(SI+S2,S3) (42)

Let F21(SI+s2)= -{C«SI+S2)I-A)-IB}-IC«SI+S2)I-A)-IO, (33) (34 )

By using the appropriate expressions for Pl(sl+S2+s3) and P2(SI+S2,S3) we get A2(sl,S2,s3)

= -

{C«SI+S2+S3)I-A)-IB}-1

C«SI+S2+S3)I-A)-IO

Clearly, we have (35) which can be realized by three linear systems connected as in Fig. 2. Next, let us investigate the decomposability of the third order controller

([«SI+S2)I-A)-IB]

®

I)(G2(sl,S2) ® Gl(s3», (43)

- {C«SI+S2+S3)I-A)-IB}-IC«SI+S2+S3)I-A)-IO ([«SI+S2)I-A)-IBG2(SI,S2)] ® Gl(S3».

(44)

A Volterra Method for Nonlincar Control Design By using (35) for G2(SI,S2) we can write «SI+S2)I-A)- IBG 2(SI,S2) = «SI+S2)I-A)-IBF21(SI+S2)(Fll(SI) ® Gl(S2»' (45)

129

design and a linear design to the desired shows that our design has an improvement over the linear design. However when we increase the step requests so that we are out of the linear region our design shows much better performance than the linear. See Fig. 6.

Now we make the following definitions CONCLUSION

32 F31(SI+s2+s3) = - {C«SI+s2+s3)I-A)-IB}-1 C«sl+S2+S3)I-A)-ID.

(48)

32 32 F31(SI+S2+S3)([F21(SI+S2)

This paper continues the designing process outlined in AI-Baiyat and Sain (1986) by explicitly studying the problem of designing a negative unity output feedback control system for the class of linear analytic plants such that partial feedback linearization is achieved. As part of the study we have developed a method for the calculation of Volterra kernels of the plants with multiple input. The method utilizes the tensor product. We also gave a general scheme for realizing multiinput, multi-output systems. This scheme was used to realize our controllers. An example was worked out to illustrate the design method. REFERENCES

(49) So again A2(sl,s2,s3) is realizable since it has the same form as (25). Finally by following the same logic in realizing Al(SI,S2,S3) and A2(SI,S2,S3) it is a simple matter to show that A3(sl,S2,S3) is also realizable. Hence the third order controller is realizable. By following the same type of analysis it can be shown that higher order controllers are realizable. The reason for that is the fact that the ith order controller, see equation (7), depends on lower order controllers up to the (i-I) order. Hence, if the lower order controllers are realizable, then the jth order controller is also realizable. This is because from equation (7) we have a tensor product of realizable controllers and the result of that product passes through the jth kernel of the plant, which we know is its realization. Literature discussion appears in the Appendix.

Consider the nonlinear plant

o

Xl = sin x2 + u2 cos x2,

Xl(O)

X2

X2(0) - 0

- sin Xl - sin x2 + ul cos Xl,

and Yl = Xl, Y2 - x2; design a nonlinear feedback control system such that T (s) .36 1. 1 (s+.4)(s+.9)

The first step in our design method is to calculate the kernels of the plant in the transform domain. This can be done for this particular plant, by using equations (17, 28-30). Having done that, we can now find the first, second, and third order controllers by using equations (4-6). Due to the space limitation we have not shown the calculation of the plant kernels and the controllers; also we have not shown the realization of the controllers. The interested reader can find those calculations in (Al-Baiyat, 1986). In order to check our design a computer simulation for the closed loop system has been done. Figure 5 shows responses for the requests RI = .5, and R2 = .6. In this figure a comparison between our

AI-Baiyat, S.A. (1986). Nonlinear Feedback Synthesis: A Volterra Approach. Ph.D. Dissertation, University of Notre Dame, Notre Dame, Indiana. AI-Baiyat, S.A., and M.K. Sain (1986). Control Design with Transfer Functions Associated to Higher Order Volterra Kernels. In Proceedings Twenty-Fifth IEEE Conference ~ Decision and Control, IEEE, New York, pp. 1306-1311. Brockett, R.W. (1976). Volterra Series and Geometric Control Theory. Automatica, 12, 167176 (addendum with E. Gilbert, 12, 635). Fliess, M., M. Lamnabi, and F. Lamnabi-Lagarrigue (1983). An Algebraic Approach to Nonlinear Functional Expansions. IEEE Transactions on Automatic Control, 30, 554-570. -Frazho, A. (l~A Shift Operator Approach to Bilinear System Theory. SIAM J. Control and Optimization, ~, 640-658-.----Gilbert, E.G. (1978). Bilinear and 2-Power Input-Output Maps: Finite Dimensional Realizations and the Role of Functional Series. IEEE Transactions on Automatic Control, 23, 418-425. --Gilbert, E.G. (1983). Minimal-Order Realizations for Continous-Time Two-Power Input-Output Maps. IEEE Transactions on Automatic Control, ~,

452-464.

-------

Harper, T. and W.J. Rugh (1976). Structural Features of Factorable Volterra Systems. IEEE Transactions ~ Automatic Control, l!, 822----832. Lesiak, C., and A.J. Krener (1978). The Existence and Uniqueness of Volterra Series for Nonlinear Systems. IEEE Transactions on Automatic Control, 23, 1090-1095. Mitz~ ~.J:-Clancy, and W.J. Rugh (1979). On Transfer Function Representations for Homogeneous Nonlinear Systems. IEEE Transactions on Automatic Control, 24, 242-249. Peczkowski, J.L., and M.K.-Sain (1981). Scheduled Nonlinear Control Design for a Turbojet Engine. In Proceedings IEEE International Symposium ~ Circuits and Systems, IEEE, New York, pp. 248-251. Peczkowski, J.L., and M.K. Sain (1985). Synthesis of System Responses: A Nonlinear Multivariable Control Design Approach. In Proceedings American Control Conference, IEEE, New York, pp. 1322-~ Peczkowski, J.L., M.K. Sain, and R.J. Leake (1979). Multivariable Synthesis with Inverses. In Proceedings Eighteenth Joint Automatic Control Conference, IEEE, New York, pp. 375-380. Rugh, W.J. (1981). Nonlinear System Theory: The

130

S. A. AI-Baiyat and M. K. Sain

Volterra / Wiener Approach. The Johns Hopkins University Press, Baltimore. Sain, M.K., and J.L. Peczkowski (19 82). Nonlinear Multivariable Design by Total Synthesis. In Proceedings American Control Conference IEEE New York, pp. 252-260-.-----" Sain, M.K., and J.L. Peczkowski (1985). Nonlinear Cont r ol by Coord i nated Feedback Synthesis with Gas Turbine Applications . In Proceedings American ~ Conference, IEEE, New York, pp. 1121-1128. Sandberg, I.W. (1982). Expansions for Nonlinear Systems. Bell Systems Technical ~, ~, 159-199. Schetzen, M. (1965). Synthesis of a Class of Nonlinear Systems. International :!.:..~, .!.' 401-414. Shanmugam, K.S. and M. Lal (1976 ) . Analysis and Synthesis of a Cla s s of Nonlinear Systems. IEEE Transactions .£!!. Circuits ~ Systems, 23, 17-25. -Smith, W. W. and W.J. Rugh (1974). On the Structure of a Class of Nonlinear Systems. IEEE Transactions .£!!. Automati c Control, ~, 701706. APPE NDIX: BRIEF REVIEW OF REALIZATION During the last two decades, a great deal of interest has been shown in the realization problem of nonlinear systems. One of the earliest works on the subject i s that of Schetzen (1986). There,

the nonlinear realization problem was solved by using an interconnection of multipliers, adders, and time-invariant linear systems. Related investigations are due to Smith and Rugh (1974), Shanmugam and Lal (1976), snd Harper and Rugh (1976). Gilbert (1978) considered degree 2-homogeneous systems and gave necessary and sufficient conditions for finite dimensional realization of such systems. Gilbert (1983) also studied minimal realization and gave a procedure for constructing such a realization as well as necessary and sufficient conditions for existence. Mitzel with his co-workers (1979) has studied the question of realizability of a homogeneous degree n system. Necessary and sufficient conditions were given for the existence of a bilinear realization for such systems. In addition to that a procedure was also given for constructing the realization. The procedure was based on a special type of factorization of the nth homogeneous system. In Rugh (1981), an alternative approach was given for realizing the nth order homogeneous as well as the polynomic systems, which are sequences of homogeneous systems. The new approach is called the backward shift realization and it was introduced to the literature of bilinear realization by Frazho (1980). ACKNOWLEDGEMENT This work was supported in part by National Science Foundation Grant ECS 84-05714.

Nonlinear, Des i red

\ h ~ ~--------1 Linea r

10 Fig . Sa

20

10

Time .

Fi g . 6a

20 Time .

De s ir ed / L·

No nlinea r, Des ir ed

.5

Lin ea r

10

Fig . Sb.

20 Time .

10 Fi g . 6b.

20 Time .