Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 199f

Singular 7-1, suboptimal control for a class of two-block interconnected nonlinear systems

Morten Dalsmo Department of Engineering Cybernetics

Norwegian University of Science and Technology N-7034 Trondheim, Norway

E-mail: Morten.Dalsmo@itk.ntnu.no

FM04 2 : l O

Willem C.A. Maas Department of Applied Mathematics

University of Twente 7500 AE Enschede, The Netherlands

E-mail: Maas@math.utwente.nl

Abstract In this paper singular H, suboptimal control for a

class of two-block interconnected nonlinear systems is addressed. Under the assumption that the regular H , suboptimal control problem is solvable for one of the blocks, an auxiliary nonlinear system is defined. Then it is shown that a feedback solution to the singular H , suboptimal control problem for the auxiliary sys- tem also applies to the original problem. An explicit feedback solution is given for a special case when the auxiliary system can be strongly input-output decou- pled. Application to the special case when one of the system blocks is passive is considered.

1. Introduction In the last few years, nonlinear 3-1, control methods

have been investigated by many authors (see eg [6,15]). The problem is difficult because it involves solving non- linear partial differential inequalities. In [13, 141 the idea of solving the singular H, control problem using a state decomposition was applied to linear systems. The singular nonlinear H , control problem received atten- tion just recently. In [7, 81 a cheap control approach was considered for nonlinear systems. In [lo] the de- composition ideas from linear theory were used to solve the nonlinear problem. Also in [l] a decomposition idea was used.

In the present paper singular 'H, suboptimal con- trol for a class of two-block interconnected nonlinear systems is investigated. The 71, control problem we investigate is totally singular, ie there is no direct feedthrough term from the control inputs to the to-be- controlled variables. We consider the situation where the regular H , suboptimal control problem is solvable for one of the blocks. Motivated by the results in [lo], we define an auxiliary nonlinear system from our orig- inal two-block system. We show that a feedback solu- tion to the singular 'Idm suboptimal control problem for this auxiliary system, also applies to the singular 71, suboptimal control problem for the original two-block system. The advantage of the auxiliary problem to the original problem is that the auxiliary penalty variable has lower dimension than the original penalty variable. In many cases this fact will simplify the problem con- siderably. Inspired by the result for nonlinear SISO system given in [ll, 121, we give an explicit state feed- back solution to the problem in a special case in which the auxiliary system can be strongly input-output de- coupled.

0-7803-3590-2/96 $5.00 0 1996 IEEE 3782

It should be mentioned that the class of two-block in- terconnected nonlinear systems that is investigated in the present paper is quite general, and are often en- countered in applications. As an example of this, we study the application of our theory to the interesting case when one of the system blocks is passive.

The present paper is organized as follows. In the next section we define the problem. In the third section we present the main results. In the forth section we con- sider the case when one of the system blocks is passive.

2. Problem statement The system under consideration is an affine nonlinear

system, which we will denote by C, given by state space equations of the form

5 = f (z) + g(z)u + k ( z ) d , f (20) = 0 z = h(z) , h(z0) = 0 (2.1)

E : { where z = z l , . . . , PIT are local coordinates for a smooth (C" \ state space manifold M , U E Iw" are the control inputs, d E Rr the disturbance inputs and z E Rs the penalty variables. f, g, k and h are all Ck with k 2 2 . We will assume that there exist local coor- dinates 5 = [d, . . . , zq, zq+l . . . ,PIT, 1 5 q < n, such that C takes the form

for x1 = [z , . . . , xqIT and 2 2 = [xq+l,. . . ,znIT. More- over, z1 E Rsl and 22 E Rsz, where s1 2 1 and s = s1 + s2. Hence, in these coordinates it follows that

and finally

Thus, C can be viewed as a two-block interconnected system where the system blocks are two general non- linear systems: a system C1 where x1 is the state, z1 is the output and ( z z , ~ ) are the inputs and a system CZ where x2 is the state, 22 is the output and ( U , X I , d ) are the inputs

As mentioned in the introduction, the general inter- connected system model (2.2) is often encountered in applications. C1 may eg represent the kinematic part of a nonlinear mechanical control system wheras Cz rep- resents the dynamics. Another example is a nonlinear control system where CZ is a model of the actuator dy- namics.

We will now state the problem that we want to con- sider in the present paper.

Problem 2.1 Let y be a fixed positive constant. Solve the state feedback singular H , suboptimal control prob- lem associated with the nonlinear system C, ie find a nonlinear static state feedback

U = a($), a ( x 0 ) = 0 (2.6)

such that the closed loop system (2.2), (2.6) has La- gain < y from d to z in the sense that for every x E M there exists a constant K ( x ) , 0 5 K ( x ) < 00, with K(x0) = 0, such that

for all d E L2[0,T] and all T 2 0 with z ( t ) denoting the output of the closed loop system (2.2), (2.6) for initial condition x(0) .

Remark 2.1 Our definition of the state feedback sin- gular 'H, suboptimal control problem is consistent with (151.

Now, consider the system C and define the pre- Hamiltonian K-, : T* M x Iwm x R' --t Iw as

K&, P , U , d ) = P T ( f ( X I + g(x)u + k ( z ) d ) 1 1 2 2

- -y2dTd + - z T z (2.8)

where x = [ x l , . . . , P I T , p = [p ' , . . . ,p"IT are natu- ral coordinates for the cotangent bundle T * M . For the sake of comprehension, we will write down the follow- ing well-known result, which follows immediately from Theorem 3.3 and Proposition 3.4 in [15].

Proposition 2.1 Consider the nonlinear closed loop system (2.2), (2.6). Suppose that there exists a non- negative C 1 solution V : M + R+ to the differential dissipation inequality

K r ( x , V z ( x ) , a (x ) , d ) 5 0, V ( x 0 ) = 0, for all d E R'

Then the state feedback (2.6) solves Problem 2.1. More- over, i f the closed loop system (2.2), (2.6) is zero- state observable with xo as the zero-state ie z ( t ) = 0, d(t) f 0 implies that z ( t ) E XO), then V ( x I > 0 for all x # X O , and xg is a locally asymptotically stable equi- librium when d ( t ) 0. If V ( x ) is proper, then xo is a globally asymptotically stable equilibrium when d ( t ) 0.

(2.9)

3. Main results For the derivation of our main results in this section

the following assumption will be used.

Assumption 3.1 Consider the two-block intercon- nected system C given by (2.2). Let M I be the subman- ifold of M with local coordinates x1 = [x', . . . , xq]IT. Then there exists a non-negative C" (k 2 v 2 2) solu- tion P : Ml 4 Iw+ to the Hamilton-Jacobi inequality

P Z , ( ~ l ) f l ( X l )

+ zPzl ( 2 1 ) +l(Xl)kr(z l ) - 91(51)g~(x1) P Z ( X 1 ) K I 1 + p r ( x l ) h l ( " l ) L 0, P(Zl,O) = 0 (3.1)

where 20 = [xro, z,',,]".

In consequence of Assumption 3.1, the regular 'Ft, sub- optimal control problem for the subsystem

is solvable when zz is regarded as the control input. The C"-' state feedback solution is then given by (cf. 1151)

z; = -gT(xl)P; ( 2 1 ) (3.3)

Motivated by the results in [lo], we will now use this fact to define an auxiliary nonlinear system from the original two-block interconnected system C. The aux- iliary nonlinear system, which we will denote by E, is again an affine nonlinear system given by state space equations of the form

where

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and

(3.7) 1 d = d - -kT(2i)P,'I (21)

Y2

and f(x) as

(3.9)

system (3.4) can be written more compactly as

Lemma 3.1 Let z E R" -and E denote the penalty variable of C and C respectively. T h e n the fol- lowing inequality i s satisfied

1 - T - 2-z z 2. Pz,(xl)(fl(xl) +91(21)22) 1

Proof: We have

which inserted in (3.12) gives the inequality (3.11).

Lemma 3.2 Consider equation (2.8), and let K-, and K7 denote the pre-Hamiltonian of C and 2 respectively. T h e n it follows that

Proof: First note that f(z) + k ( x ) d = f (x) + k(z)d. Thus,

- -

since Pz7(x1) = 0. An immediate consequence of the last lemma is given

in the following theorem.

Theorem 3.1 If the static state feedback

U = a(%), a(x0) = 0 (3.18)

solves the singular E , suboptimal control problem for t he auxiliary sy s t em c in the sense that there exists a non-negative C1 solution W : M -+ R+ t o the differen- tial dissipation inequality

f o r all d E RT, t hen the state feedback (3.18) also solves Problem 2.1 and the differential dissipation inequality

K,(x, VT(x), a ( x ) , d ) 5 0, V(zo) = 0 , f o r all d E R' (3.20)

holds for the non-negative C1 f unc t ion V M + R+ given by V = W + P .

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Proof: in equation (3.14), we get

By substituting W,'(x) for p and a(.) for u

for all d E Rr , which means that

Kr(", V % , , ~ ( X ) , d ) i 0 (3.22)

for all d E RT. Moreover, V ( x 0 ) = 0, so the result a

We will now consider the singular 3-1, suboptimal control problem given by the auxiliary system 2. The advantage of this problem to the original problem is that the auxiliary penalty variable 2 has lower dimen- sion than the original penalty variable z. This can eg be utilized as described in the following. Consider the system E and let L,h(x) denote the s2 x m matrix given by

L,h(x) = [ Lgh:(x) ] where h(z) = hl z), . . . , hs,(z)IT and s2 = s - SI. From equation U 2.4 and equation (2.5) it is easily seen that the rank of the s x m matrix L,h(x) is always less than s because its first s1 rows-are zero. This is no t the case for the s2 x m matrix L,h(x). If s2 5 m, L,h(x) may have rank equal to s2, which is a very nice prop- erty with respect to the design of the feedback law. For simplicity, we will just consider the specid- case when L,h(x) is invertible in this paper. Then C can be strongly input-output decoupled (see eg [5]) by the feedback

U = L, lh(z ) ( -L fh(x) + v ) (3.24)

follows directly from Propsition 2.1.

(3.23)

L&sz ( x )

where

(3.25)

We will use this fact to solve the singular 3-1, sub- optimal control problem for the auxiliary system E in this special case. Our solution is inspired by the results given in [ll, 121 for nonlinear SISO systems. Regarding the solution to this problem under more general condi- tions than invertibility of L,h(x), we refer to [ll, 121 for the SISO case and [lo] for an extension to the MIMO case.

Proposition 3.1 Consider the auxiliary system c. Assume that sa = m and that the s2 x s2 matrix L,h(x) is invertible for all x E M . Define the feedback gain matrix C(z) by

s 2 (3.26) C(X) = I + -D(x) 4Y2

where the diagonal matrix D ( x ) is given by

Then, by using the state feedback

U = a(.) = -L,lh(x) [L fh (X) + C ( x ) h ( x ) ] (3.28)

the differential dissipation inequality

~ ? ~ ( x , ~ ~ ( x ) , a ( x ) , c I ) 5 0, ~ ( x o ) = 0, for all Z E IW' (3.29)

holds for

W ( x ) = 5 i T ( x ) h ( x ) 4 (3.30)

Hence, from Theorem 3.1 it follows that the state feed- back (3.28) also solves Problem 2.1 and that

V ( x ) = p(x)h(x) + P(x) (3.31)

is a solution to the associated diflerential dissipation inequality (2.9).

Proof: The closed loop system equations (3.10), 3.28 together with a completion of squares argument I 1 see 111) gives

&(x , W 3 4 , a ( x ) , 4 = W d x ) [m + d x ) a ( x ) + k ( 4 4

= i h T ( x ) [Lfh(") + L g h ( x ) a ( x ) + Lkh(X)4

= $ T ( X ) [-C(x)h(x) + L & ) 4

- i r 2 d T d + i z T f

- i r 2 d T J + $ETf

-ir-y2dTd+

= - L + h T ( ~ ) D ( ~ ) h ( ~ ) 2 47 + $hT(x)Lkh(x)d - iy2dTJ

5 0

Moreover, W ( x 0 ) = 0 since h(x0) = 0, and the result follows. a Example 3.1 Assume now that there exist local coor- dinates x = [x', . . . x " - ~ , xn-m+l , . . . , xnIT such that C takes the following form

i i = f i ( x 1 ) + gi(xi)xz + k i ( x i )d X 2 = - T - l ~ 2 + T-'U

(3.32)

for 2 1 = [ x l , . . . , xn--m]T and x2 = [xn-"+l,. . . , xnIT. Then, the second equation is a simple MIMO linear ac- tuator model, where U E Rm is the control input (to the

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actuator), 2 2 E Rm is the output from the actuator and T = TT > 0 is an m x m matrix. In this case, again under Assumption 3.1, Z is found to be

- Z = h ( x ) = ~2 - ~ 1 ( ~ 1 ) (3.33)

where a 1 ( q ) = [a11(x1), . . . , a1,(xl)lT E Rm is de- fined by

a1 ( 2 1 ) = -g%1 P; ( 2 1 ) (3.34)

This means that L,h(x) = T-l > 0. Hence, it follows from Proposition 3.1 and a straightforward calculation that the following state feedback law

U = - T C ( X ) [ X ~ - a l ( ~ l ) ]

+ xz + TLf la i ( x ) (3.35)

where C ( x ) is given b y equation (3.26) and

~ ( x ) = diag ( I IGlm(x1) t12 , . . . , IIGlalm(x1)I12}

& ( x ) = f l ( 2 1 ) +91(x1)22 + +1(21)kT(Zl )P~(Xl ) 1 Y

solves Problem 2.1 in this special case.

4. Application to nonlinear systems with passive subsystems

Before concluding the present paper, we also want to make a note about an important special case that is reg- ularly encountered in applications. A (physical) non- linear system is often composed of smaller and simpler subsystems which have nice system theoretic proper- ties such as passivity. We will now consider the general two-block interconnected system C when the subsystem C1 is passive. Using the same ideas as in [15] for loss- less systems and in [3] for passive systems, we get the following result.

Proposition 4.1 Consider the two-block intercon- nected system C as given in (2.2). Suppose that the subsystem C1 is passive in the sense that it has the K Y P property (cf. [ 2 ) ie there exists a non-negative storage function v E c’d , with Vl(z1,o) = 0, such that

~ 1 Z l ~ ~ l ~ f l ~ ~ l ~ I 0 (4.1) K s l ( ~ l ) g l ( x I ) = hY(s1) ( 4 4

and that kl(x1) is given by

kl(Z1) = Cgl(x1) 07- kl(X1) = [ Cgl ( z1 ) 0 ] (4.3)

for c E R. If y > then the finction

solves the Hamilton- Jacobi inequality (3.1) and

(4.4)

(4.5)

Proof: The result follows directly from a simple cal- m

In order to illustrate how this result can be applied to a physical nonlinear system, we will now give a sim- ple example for a rigid robot in the presence of torque disturbances.

Example 4.1 Consider the rigid robot dynamics equa- tions in state space form, including the presence of torque disturbances d E R3

culation by using (4.1), (4.2) and (4.3).

( X I = 2 2

x1 and 2 2 represent joint position and velocities, M(x1) is the positive definite inertia matrix, C(x1, xz)x2 the Coriolis and centrifugal forces and e(x1) the gravity load. z is the penalty variable.

It is simple to observe that C, can be viewed as an interconnected two-block system where one block repre- sents the robot dynamics and the other block represents the robot kinematics. Let 2 1 be the state of CTl , the kinematics, and x2 be the state of Cr2, the dynamics. I t is straightforward to see that the subsystem C1, is passive as defined in Propsition 4.1, with storage func- tion V ( x 1 ) = ~llx1112. Also, kl(x1) = 0 in this special case, which means that c = 0. Thus, by using Proposi- tion 4.1, the auxiliary system corresponding to the rigid robot system (4.8) is found to be

where

g(2) = k(s) = (4.11)

and

h ( x ) = 2 1 + 2 2 (4.12)

In this case s2 = m = 3 and L,h(x) = M-l(x1) > 0, Thus, we can apply the state feedback law proposed in Proposition 3.1. A straightforward calculation gives

L&x) = 2 2 - M-l(x1) (G(x1, x2)x2 + e(x1)) (4.13)

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Also L&(x) = M-’ (x l ) , and we see from (3.27) that the matrix D(x1) in this case is given by

=

1 [ l l M 2 y 0 0 0 II MZ1 ( 2 1 ) 1 1 2

0 l l M 2 (x1 ) 1 1 2 where M*;’(xl) denotes the i’th column of the ma- trix M-l(x1) (remember that M- l (x1) is symmetric). Therefore, from Proposition 3.1 it follows that the state feedback law

U = -Kxl (x1)xl - Kx, (x l , x2)22 + e(x1) (4.14)

where the feedback gain matrix Kxl ( X I ) is defined b y n

(4.15)

and the feedback gain matrix K x z ( 2 1 , x2) is defined by

Kx2(x1,x2) = M(X1) - C(Xl,X2) 3 + I + - D ( z ~ ) (4.16)

4Y2 solves the state feedback singular ‘H, suboptimal control problem for the rigid robot system (4.8). Moreover, the closed loop system (4.8), (4.14) is obviously zero-state observable and the function

1 1

2 4 (4.17) V ( x ) = - X T X 1 + - ( 2 1 + X 2 ) y . l + 2 2 )

is proper. This means that the closed loop system is globally asymptotically stable at the equilibrium point 0 when d ( t ) 0 .

Remark 4.1 In Example 4.1 the passive block of the two-block interconnected system was linear and ex- tremely simple. However, it is not dificult to find physi- cal nonlinear systems in which eg a block that represents the kinematics is nonlinear and complicated. This is the case for the rigid body tracking problem in [4]. In that problem, it is possible to show that the block that represents the error kinematics is passive. After real- izing this, it is straightforward to see how the method proposed in the present paper can be used to give an al- ternative solution to the rigid body tracking problem in [4]. A solution to a slightly different problem is given an [9].

5. Conclusions In the present paper we have given a method to solve

a (totally) singular ‘H, suboptimal control for a class of two-block interconnected nonlinear systems. Under the assumption that the regular 3-1, suboptimal con- trol problem is solvable for one of the blocks, an auxil- iary nonlinear system has been defined and it has been shown that by solving the singular ‘H, suboptimal con- trol problem for this system then also the problem for the original system is solved. An explicit solution has been given for a special case in which the auxiliary sys- tem can be strongly input-output decoupled. Also the potential importance of passive blocks in connection with the proposed method has been pointed out and illustrated with a simple rigid robot example.

6. Acknowlegdements The work of the first author was supported by the Re-

search Council of Norway. The authors would also like to thank Arjan van der Schaft (Univiversity of Twente) for giving a simple example that served as one of the starting points for the work in the present paper.

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