Open access peer-reviewed chapter

Simultaneous H ∞ Control for a Collection of Nonlinear Systems in Strict-Feedback Form

Written By

Jenq-Lang Wu, Chee-Fai Yung and Tsu-Tian Lee

Submitted: October 29th, 2015 Reviewed: May 5th, 2016 Published: October 19th, 2016

DOI: 10.5772/64105

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Abstract

Based on the control storage function approach, a constructive method for designing simultaneous H∞ controllers for a collection of nonlinear control systems in strict-feedback form is developed. It is shown that under mild assumptions, common control storage functions (CSFs) for nonlinear systems in strict-feedback form can be constructed systematically. Based on the obtained common CSFs, an explicit formula for constructing simultaneous H∞ controllers is presented. Finally, an illustrative example is provided to verify the obtained theoretical results.

Keywords

  • nonlinear control systems
  • simultaneous H∞ control
  • state feedback
  • storage functions
  • strict-feedback form

1. Introduction

The simultaneous H control problem concerns with designing a single controller which simultaneously renders a set of systems being internally stable and satisfying an L2-gain specification. In the last decades, there have been some researchers studying the simultaneous H control problem in linear case, see references [16]. In references [1] and [2], necessary and sufficient conditions for the simultaneous H control via nonlinear digital output feedback controllers were derived by using the dynamic programming approach. In reference [3], a numerical design method was proposed for designing simultaneous H controllers. In reference [4], it was shown that the simultaneous H control problem is equivalent to a strong H control problem. In reference [5], linear periodically time-varying controllers were employed for simultaneous H control. In reference [6], a simultaneous H control problem was solved via the chain scattering framework.

All the results mentioned earlier are derived for linear systems case. Till now, only few results have been reported about simultaneous H control of nonlinear systems, see references [7, 8]. In reference [7], a control storage function (CSF) method was developed for designing simultaneous H state feedback controllers for a collection of single-input nonlinear systems. Necessary and sufficient conditions for the existence of simultaneous H controllers were derived. Moreover, an explicit formula for constructing simultaneous H feedback controllers was proposed. The CSF approach was first introduced in reference [9]. It is motivated by the control Lyapunov function (CLF) method (please see references [1018]) for designing stabilizing controllers of nonlinear control systems. One difficulty in applying CSFs/CLFs for solving control problems is that how to derive CSFs/CLFs for nonlinear systems is an open problem unless they are in some particular forms. No systematic methods for constructing CSFs have been proposed in reference [7]. It is important to identify those nonlinear systems whose corresponding CSFs/CLFs exist and can be constructed systematically. In reference [8], the CSF approach was applied to design simultaneous H controllers for a collection of nonlinear control systems in canonical form. It was shown that under mild assumptions, CSFs can be constructed systematically for nonlinear systems in canonical form; and simultaneous H control for such systems can be easily achieved. In this chapter, we further study the simultaneous H control problem for nonlinear systems in strict-feedback form. It is known that the strict-feedback form is more general than the canonical form. Moreover, a restrictive assumption made in reference [8] is relaxed in this chapter. Based on the CSF approach and by using the backstepping technique, we develop a systematic method for constructing simultaneous H state feedback controllers. The proposed results in reference [8] are special cases of the results presented in this chapter.

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2. Problem formulation and preliminaries

In this section, the simultaneous H control problem to be solved will be formulated and some preliminaries will be presented. For simplifying the expressions, we use the same notations x, u, w, and zto denote the states, control inputs, exogenous inputs, and the controlled outputs of all the considered systems.

2.1. Problem formulation

Consider a collection of nonlinear control systems:

x˙=fi(x)+g1i(x)w+g2i(x)uz=h1i(x)+k11i(x)w, i=1,,q,E1

where x=[x1,x,,xn]TRnis the state, wRmis the disturbance input, uRis the control input, zRris the controlled output, fi:RnRn,g1i:RnRn×m,g2i:RnRn,h1i:RnRr, and k11i:RnRr×m,i=1,,q, are smooth functions. Here we denote the i-th system in Eq. (1) as system Si. For all i=1,…,q, suppose that fi (0) = 0 and h1i(0) = 0. For convenience, define x¯j=[x1,x2,,xj]TRj,j=1,,n. Suppose that fi(x), g1i(x), and g2i(x), i=1,…,q, have the following forms:

fi(x)=[x2+θi1(x1)            xj+1+θij(x¯j)xn+θi(n1)(x¯n1)θin(x)],g1i(x)=[00 0ρi(x)],g2i(x)=[00 0ηi(x)].E2

where θij:RjR,ρi:RnR1×m, and ηi:RnR,i=1,2,,q,j=1,,n, are smooth functions with θij(0) = 0 and ηi(x)0for each xRn. Assume that all functions ηi(x),i=1,,q,have the same sign. Without loss of generality, suppose that ηi(x)>0,i=1,,q. By Eq. (2), the qpossible models can be explicitly expressed as

x˙1=x2+θi1(x1)    x˙j=xj+1+θij(x¯j)    x˙n1=xn+θi(n1)(x¯n1)x˙n=θin(x)+ρi(x)w+ηi(x)u,z=h1i(x)+k11i(x)w,               i=1,2,..,q. E3

Suppose that the following assumption holds.

Assumption 1:γ2Ik11iT(x)k11i(x)>0.xRnand i{1,...,q}.

It is clear that we can always find a positive (semi)definite function U(x) such that, for all i∈{1,…q},

h1iT(x)h1i(x)+h1iT(x)k11i(x)(γ2Ik11iT(x)k11i(x))1k11iT(x)h1i(x)U(x), xRn.E5

The objective of this chapter is to find a continuous function p:RnRsuch that the state feedback controller

u=p(x)E4

internally stabilizes the systems in Eq. (3) simultaneously; and, for each T> 0 and for each wiL2[0, T], all closed-loop systems, starting from the initial state x(0) = 0, satisfy (for a given γ > 0)

 0 TzT(t)z(t)dtγ^2 0 TwT(t)w(t)dtforsomeγ^<γ.E5

2.2. Control storage functions

Here we review some important concepts about the CSF method introduced in references [7, 9].

Definition 1[7, 9]: Consider the system Siin Eq. (1). A smooth, proper, and positive definite function Vi:RnRis a CSF of Siif, for each xRn\{0}and each wRm,

infuR{Vi(x)x(fi(x)+g1i(x)w+g2i(x)u)+(h1i(x)+k11i(x)w)T(h1i(x)+k11i(x)w)γ2wTw}<0.E8

For ensuring the continuity of the obtained simultaneous H controllers, the L2-gain small control property(L2-gain SCP) has been introduced in reference [7].

Definition 2[7]: A CSF Vi:RnRof Sisatisfies the L2-gain SCP if for each ε > 0, there is a δ1 > 0 and a δ2 > 0 such that, if x≠ 0 satisfies x<δ1and wsatisfies w<δ2, there is some uwith |u| < ε satisfying

Vi(x)x(fi(x)+g1i(x)w+g2i(x)u)+(h1i(x)+k11i(x)w)T(h1i(x)+k11i(x)w)γ2wTw<0E9

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3. Main results

For a single system, it has been shown in reference [7] that the existence of CSFs is a necessary and sufficient condition for the existence of H controllers. Therefore, for the existence of simultaneous H controllers for the systems in Eq. (3), the existence of CSFs for these systems is necessary. In references [7] and [9], no systematic methods have been proposed for constructing CSFs. Here, based on the backstepping method, we first derive CSFs explicitly for the systems in Eq. (3).

Let

s1(x1)=x1E10
V^1(x1)=12s12(x1)E12

It is easy to show that we can find a function φ1:RRand a positive definite function μ1:RRsuch that

x1(ϕ1(x1)+θi1(x1))μ1(x1), for alli= 1,,q.E13

For j=2,…,n, let

sj(x¯j)=xjϕj1(x¯j1)E14
V^j(x¯j)=V^j1(x¯j1)+12sj2(x¯j)E15

Similarly, we can find functions φj:RjR,j=2,,n1, and positive definite function μj:RjR,j=2,,n1, such that

l=1j1V^j(x¯j)xl(xl+1+θil(x¯l))+V^j(x¯j)xj(ϕj(x¯j)+θij(x¯j))l=1jμl(x¯l),for alli=1,,p.E16

Then, it is clear that the function

V^(x)12j=1nsj2(x¯j)E17

is positive definite, and radially unbounded.

Now, we discuss the existence of common CSFs for the systems in Eq. (3). For convenience, we say that a continuous function ν(x¯j)is dominated by a continuous function υ(x¯j)if there exists a constant c> 0 such that ν(x¯j)<cυ(x¯j)for all x¯j0.

Theorem 1:Consider the systems in Eq. (3). Suppose that Assumption 1holds. If the functions φj:RjR, j=1,…,n-1, are such that U(x)|sn(x)=0is dominated by l=1n1μl(x¯l), then there exists a common CSFthat satisfies the L2-gain SCPfor all the systems in Eq. (3).

Proof:Let V(x)=KV^(x), where K> 0 will be specified later. For system Si, define the corresponding Hamiltonian function as

Hi(x,w,u)V˙(x)+(h1i(x)+k11i(x)w)T(h1i(x)+k11i(x)w)γ2wTw.E18

By the backstepping method, we can show that

Hi(x,w,u)    =Kj=1nsj(x¯j)s˙j(x¯j)+(h1i(x)+k11i(x)w)T(h1i(x)+k11i(x)w)γ2wTw    Ki=1n1μj(x¯j) +Ksn(x)(sn1(x¯n1)+θin(x)+ρi(x)w+ηi(x)ul=1n1ϕn1(x¯n1)xl(xl+1+θil(x¯l)))        +h1iT(x)h1i(x)+h1iT(x)k11i(x)w+wTk11iT(x)h1i(x)wT(γ2Ik11iT(x)k11i(x))w.E19

After some manipulations, we have

Hi(x,w,u)ai(x)+bi(x)u(wwi*(x))T(γ2Ik11iT(x)k11i(x))(wwi*(x))                  ai(x)+bi(x)u,E6

where

ai(x)=Kj=1n1μj(x¯j)+Ksn(x)(sn1(x¯n1)+θin(x)l=1n1ϕn1(x¯n1)xl(xl+1+θil(x¯l)))+h1iT(x)h1i(x)+(K2sn(x)ρiT(x)+k11iT(x)h1i(x))T(γ2Ik11iT(x)k11i(x))1(K2sn(x)ρiT(x)+k11iT(x)h1i(x))bi(x)=Kηi(x)sn(x)wi*(x)=(γ2Ik11iT(x)k11i(x))1(K2sn(x)ρiT(x)+k11iT(x)h1i(x))E21

Therefore, V(x)=KV^(x)is a CSF of Siif

x0such thatbi(x)=0ai(x)<0.E24

As U(x)|sn(x)=0is dominated by l=1n1μl(x¯l), we can choose a K> 0 such that

U(x)|sn(x)=0 and x0Kj=1n1μj(x¯j)<0E25

Notice that bi(x) = 0 if and only if sn(x) = 0. Therefore,

ai(x)|bi(x)=0 and x0Kj=1n1μj(x¯j)+U(x)|sn(x)=0 and x0<0.E7

This shows that V(x) is a CSF for the i-th system in Eq. (3). Since Eq. (7) holds for all i∈ {1,…,q}, V(x) is a common CSF for all the systems in Eq. (3).

Now we prove that V(x) satisfies the L2-gain SCP. Note that if we can find a continuous stabilizing feedback law di(x) with di(0) = 0 such that Hi(x,w,di(x))<0for each xRn\{0}and each wRm, then V(x) satisfies the L2-gain SCP. Let

di(x)=1ηi(x)(sn1(x¯n1)+θin(x)l=1n1ϕn1(x¯n1)xl(xl+1+θil(x¯l))+ρi(x)(γ2Ik11iT(x)k11i(x))1(K4sn(x)ρiT(x)+k11iT(x)h1i(x)))μ^n(x)E27

where the continuous function μ^n(x)with μ^n(0)=0is such that sn(x)μ^n(x)>0if sn(x) ≠ 0, and

Kj=1n1μj(x¯j)Kηi(x)sn(x)μ^n(x)+U(x)<0x0.E28

Note that such μ^n(x)always exists since U(x)|sn(x)=0is dominated by l=1n1μl(x¯l). Clearly, di(x) is continuous in Rnand di(0) = 0. By Eq. (6), we have

Hi(x,w,di(x))ai(x)+bi(x)di(x)=Kj=1n1μj(x¯j)Kηi(x)sn(x)μ^n(x)   +h1iT(x)h1i(x)+h1iT(x)k11i(x)(γi2Ik11iT(x)k11i(x))1k11iT(x)h1i(x)Kj=1n1μj(x¯j)Kηi(x)sn(x)μ^n(x)+U(x)<0,  x0, w.E29

This implies that V(x) satisfies the L2-gain SCP and completes the proof.

To derive simultaneous H controllers, define (for i=1,…,q)

pi(x){ai(x)+ai2(x)+βibi4(x)bi(x),if sn(x)00,if sn(x)=0E30

where βi> 0, i=1,…,q, are given constants. Since V(x) satisfies the L2-gain SCP, the functions pi(x), i=1,…,q, are continuous in Rn[16]. We have the following results.

Theorem 2:Consider the collection of systems in Eq. (3). Suppose that Assumption 1holds. If the functions φj:RjR,j=1,,n1, are such that U(x)|sn(x)=0is dominated by l=1n1μl(x¯l), then a continuous function p:RnRexists such that the feedback law defined in Eq. (4) internally stabilizes the collection of systems in Eq. (3) simultaneously; and moreover, all the closed-loop systems satisfy the L2-gain requirement specified in Eq. (5). In this case,

u=p(x){mini{1,2,...,q}{pi(x)}, if sn(x)>0   0,                   if sn(x)=0maxi{1,2,...,q}{pi(x)}, if sn(x)<0E8

is a simultaneous H controller for all the systems in Eq. (3).

Proof: Since the functions pi(x), i=1,2,…,q, are continuous in Rn, from the definition of p(x), its continuity is obvious. In the following, we first prove the achievement of L2-gain requirement [Eq. (5)], and then the internal stability of all the closed-loop systems.

A. L2-gain requirement

Since Hi(x,w,u)ai(x)+bi(x)u, if we can show that

ai(x)+bi(x)p(x)<0x0,i=1,,q,E9

Then, with the controller defined in Eq. (8), all the closed-loop systems satisfy the L2-gain requirement specified in Eq. (5).

1.sn(x) = 0 and x≠ 0.

In this case, u= p(x) = 0 and bi(x) = 0. Then, by Eq. (7),

ai(x)+bi(x)p(x)=ai(x)<0,i=1,,q.E33

2.sn(x)>0.

In this case, since bi(x) > 0, we have

ai(x)+bi(x)p(x)=ai(x)+bi(x)minj{1,2,...,q}{pj(x)}ai(x)+bi(x)pi(x)=ai2(x)+βibi4(x)<0,i=1,,q.E34

3.sn(x) < 0

Similarly, in this case we can show that

ai(x)+bi(x)p(x)<0,i=1,,q.E35

These discussions imply that Eq. (9) holds. That is, all the possible closed-loop systems satisfy the L2-gain requirement specified in Eq. (5).

B. Internal stability

To prove internal stability, notice that Eq. (6) implies that, along the trajectories of system Siunder w= 0,

Hi(x,0,p(x))=V(x)x(fi(x)+g2i(x)p(x))+h1i(x)Th1i(x)                       ai(x)+bi(x)p(x)<0  x0.E36

That is, for each i∈ {1,…,q}, along the trajectories of system Si, we have

V˙(x)=V(x)x(fi(x)+g2i(x)p(x))<0x0.E37

This shows that all the closed-loop systems are internally stable.

Remark 1:The systems considered in reference [8] are special cases of the systems considered in this chapter. If we let θij(x¯j)=0,i=1,2,,q, and j=1,2,…,n-1, the systems in Eq. (3) will reduce to the systems considered in reference [8]. On the other hand, in reference [17], it is assumed that U(s) is in quadratic form. In this chapter, we relax this restrictive assumption.

Remark 2:In this chapter, we consider the case that the controlled output zis independent of the control input u. In this situation, a much simpler formula (not a special case of the formula in reference [7]) is proposed for constructing simultaneous H controllers. In the case that the controlled output zdepends on u, necessary and sufficient conditions for the existence of simultaneous H controllers and a formula for constructing simultaneous H controllers can be derived by the results in reference [7].

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4. An illustrative example

Consider the following nonlinear systems:

Si:{x˙1=x2+θi1(x)x˙2=θi2(x)+ρi(x)w+ηi(x)uz=h1i(x)+k11i(x)w,                i=1,2,and 3E10

where

θ11(x)=x1,θ21(x)=x1sin(x1),θ31(x)=x1cos(x1)θ12(x)=x12+x23,θ22(x)=x1(12x2),θ32(x)=x1cos(x2)+2x2sin(5x1)ρ1(x)=1+x1,ρ2(x)=x1x2,ρ3(x)=x1x22η1(x)=1+(x1+x2)2,η2(x)=2cos(x1),η3(x)=2+x22h11(x)=x1cos(x22),h12(x)=x1sin(x1),h13(x)=x2k111(x)=1+cos(x1),k112(x)=1,k113(x)=1+sin(5x2)E39

It can be shown that

h1iT(x)h1i(x)+h1iT(x)k11i(x)(γ2k11iT(x)k11i(x))1k11iT(x)h1i(x)U(x),     i=1,2, and 3E45

with

U(x)=95x12+95x22.E46

Let γ = 3. It can be verified that Assumption 1holds. Let

s1(x¯1)=x1φ1(x1)=2x1μ1(x1)=x12s2(x¯2)=x2φ1(x¯1)=2x1+x2.E48

Then,

V^(x)=12(s12(x1)+s22(x¯2))E51

is positive, definite, and radially unbounded. By choosing K= 10, it can be shown that

Kμ1(x1)+U(x)|s2(x¯2)=0<0,x10.E52

Therefore,

V(x)=KV^(x)=5(s12(x1)+s22(x¯2))E53

is a common CSF for the three systems in Eq. (10). For i= 1, 2, and 3, define

ai(x)=Kμ1(x)+Ks2(x)(s1(x1)+θi2(x)ϕ1(x1)x1(x2+θi1(x)))+h1iT(x)h1i(x)+(K2sn(x)ρiT(x)+k11iT(x)h1i(x))T(γ2Ik11iT(x)k11i(x))1(K2sn(x)ρiT(x)+k11iT(x)h1i(x))bi(x)=Kηi(x)s2(x)E54

and (with β1 = β2 = β3 = 0.1)

pi(x){ai(x)+ai2(x)+βibi4(x)bi(x),  if s2(x)0   0,                                        if s2(x)=0.E56

From Theorem 2, the following controller

u=p(x){min{p1(x),p2(x),p3(x)},  if 2x1+x2>00,                                      if 2x1+x2=0max{p1(x),p2(x),p3(x)},  if 2x1+x2<0E11

is a simultaneous H controller for the three systems in Eq. (10). With arbitrarily chosen disturbance inputs, Figures 13 show the states, control inputs, disturbance inputs, and controlled outputs of these three systems starting at different initial states with the same controller defined in Eq. (11). It can be seen that all the three closed-loop systems are internally stable and satisfy the required L2-gain specification. That is, the controller defined in Eq. (11) is indeed a simultaneous H controller for the three systems in Eq. (10).

Figure 1.

Responses of the systemS1 controlled by the controller defined inEq. (11).

Figure 2.

Responses of the systemS2 controlled by the controller defined inEq. (11).

Figure 3.

Responses of the systemS3 controlled by the controller defined inEq. (11).

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5. Conclusions

In this chapter, a systematic way for constructing simultaneous H state feedback controllers of nonlinear control systems in strict-feedback form is proposed. It is shown that the existence of common CSFs guarantees the existence of simultaneous H controllers. An explicit formula for constructing simultaneous H controllers is derived. The simulation example is given for verifying the theoretical results. The simulation results show, as expected, that the designed controller can simultaneously stabilize the considered systems and such that all closed-loop systems satisfy the specified disturbance attenuation requirement. Possible further works include considering nonlinear control systems in more general forms, applying the approach to time-varying case, and considering the output feedback case.

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Written By

Jenq-Lang Wu, Chee-Fai Yung and Tsu-Tian Lee

Submitted: October 29th, 2015 Reviewed: May 5th, 2016 Published: October 19th, 2016