C h a p t e r 4. M u l t i - o b j e c t i v e B o u n d e d C o n t r o l o f U n c e r t a i n N o n l i n e a r S y s t e m s : a n I n v e r t e d P e n d u l u m E x a m p l e

St~phane Dussy and Laurent E1 Ghaoui

Laboratoire de Math~matiques Appliqu~es, Ecole Nationale Sup~rieure de Tech- niques Avang~es, 32, Blvd. Victor, 75739 Paris. France.

1. I n t r o d u c t i o n

1.1 P r o b l e m s t a t e m e n t

We consider a parameter-dependent nonlinear system of the form

k = A(x,~( t ) )x + B . (x ,~ ( t ) )u + Bw(x,~(t))w, y = Cy(x,~(t))x + Dy~v(x,~(t))w, z = c z ( ~ , ~ ( t ) ) x + Dz,(~,~(0)~, ¢i = Eix, i = l , . . . , N .

(1.1)

where x is the state, u the command input, y the measured output, z the controlled output, ¢i, i = 1 , . . . N , some outputs, and w is the disturbance. The vector ~ contains time-varying parameters, which are known to belong to some given bounded set 79. Also, we assume that A(., .), Bu(', .), Cy(., .), Bw(., .), Dyw(', ") and Dzu(', ") are rational functions of their arguments. We seek a dynamic, possibly nonlinear, output-feedback controller, with input y and output u, such that a number of specifications are satisfied. To define our specifications, we consider a given polytope ~ of initial conditions, and a set of admissible disturbances, chosen to be of the form

{ I/0 } : (1.2) W ( W m a x ) - - W e J~2([0 OO[) w(t)Tw(Tt) dt < Wma x ,

where Wm~x is a given scalar. To ~ , W and the uncertainty set 79, we asso- ciate the family X(xo, Wm~×) of trajectories of the above uncertain system, in response to a given z0 E P. Our design constraints are as follows. S.1 The system is well-posed along every trajectory in X(x0, Wmax), that is, for every t >_ 0, the system matrices A(x,(( t)) , B~(x,~(t)), etc, are well- defined. S.2 Every trajectory in A'(x0, 0) decays to zero at rate ~, that is lim e~tx(t) =

t--+c~ 0. S.3 For every trajectory in X(xo, Wmax), the command input u satisfies vt > 0, Ilu(t)ll2 < ~m~x.

56 StSphane Dussy and Laurent E1 Ghaoui

S.4 For every trajectory in X(Xo, Wmax), the outputs ¢i, i = 1 , . . . N , are bounded, that is, [l~i(t)[l~ < %max for every t > 0. S.5 For every trajectory in X(0, Wm~x), the closed-loop system exhibits good disturbance rejection, ie a £2-gain bound 7 from z to w must hold, that is for every w(t) E ~V(Wmax):

fo°° Z(t)T z(t)dt <_ 7~ fo°~ wT (t)w(t)dt.

We emphasize that the above specifications should hold robustly, that is, for every ~(t) e 7).

The following sections describe an LMI-based controller synthesis method for the above problem, that is based on the results of [12]. We illustrate this method with an inverted pendulum example. The method seeks a controller that is dependent on the measured parameters and/or nonlinearities, and which yields a closed-loop system robust with respect to the unmeasured parameters and/or nonlinearities. This kind of control law can be termed "robust measurement-schedtiled". It is based on a Linear Fractional Repre- sentation (LFR) of the system (see [17, 6]). The LFR model can be con- structed systematically, starting from the system's dynamic equations. The LFR results in a model where the system is described by a "nominal" lin- ear time-invariant system connected to an uncertainty diagonal matrix via (possibly fictitious) input p and output q. This matrix depends on the non- linearities and/or uncertainties (grouped here in the vector ~(t)).

The LMI approach to the gain-scheduling methodology was first intro- duced by Packard in [16, 3] for Linear Parameter-Varying (LPV) systems, and applied to H~-control in [18]. The extension to nonlinear systems can be found in [12]. Design examples (for LPV systems) can be found in [1, 4]. In [8], we have applied the method to a simplified inverted pendulum ex- ample. Here, we consider a more complicated system with more stringent conditions and improved performances.

1.2 M a in f e a t u r e s

There are several interesting features in the above approach.

- Efficient numerical solution. LMI optimization problems can be solved very efficiently using recent interior-point methods [15, 19, 10] (the global opti- mum is found in modest computing time). This brings a numerical solution to problems when no analytical or closed-form solution is known.

- A systematic method. The approach is very systematic and can be applied to a very wide variety of nonlinear control design problems. This includes (and is not reduced to) systems whose state-space representation are ra- tional functions of the state vector.

- Extension to uncertain nonlinear systems. It is possible to extend the method to cases when some parameters in the state-space representation of the system are uncertain.

Control of Uncertain Systems with Bounded Inputs 57

- Multiobjective problems. The approach is particularly well-suited to prob- lems where several (possibly conflicting) specifications are to be satisfied by the closed-loop system. This is made possible by use of a common Lya- punov function proving that every specification holds.

In view of the above features, the approach opens the way to CAD tools for the analysis and control of nonlinear systems, eg the public domain toolbox MRCT [9, 7] built on top of the software l m i t o o l [10].

The method is based on using the same Lyapunov function to enforce (possibly competing) design specifications. This may lead to conservative results. This disadvantage is traded off in several ways.

First, other methods, such as Hoo control, do not allow to impose directly several constraints. One must form a single criterion, in the form of an Hoo- norm bound, that is deemed to reflect all specifications. Forming this one criterion is sometimes difficult, since a (sometimes large) number of param- eters have to be chosen, and sometimes these parameters have no obvious relationship with the desired constraints.

In an LMI-based design, we form a set of LMI constraints, each one re- flecting one specification. In each constraint appears a parameter which de- termines how coercive the specification is--increasing the parameter amounts to relax the constraint. For example, if we impose a bound Um~× on the com- mand input norm, then Um~x is a measure of how stringent the norm bound is. The various parameters (command input Um~x, L~ gain bound V, etc) can be interpreted as design parameters.

The approach thus reduces the synthesis problem to choosing a few de- sign parameters, in order to meet the specifications. Note that all the design parameters have a direct physical interpretation, and that there are only a small number of them.

The possible conservatism of the LMI approach can be reduced if we choose the design parameters iteratively, as follows. At the first step of the design process, we may set the various parameters to higher values than those imposed by the actual specifications. In a second step, we may perform an (LMI-based) analysis, or a simulation, on the closed-loop system, in order to check if the latter obeys the specifications. (Since LMI-based design tools are more conservative than LMI-based analysis tools, it seems that this will be often the case. If not, we can identify (using our analysis or simulation) which specifications are violated, and decrease the corresponding design parameters.

1.3 P a p e r o u t l i n e

We first detail the inverted pendulum problem in §2. and write an LFR for the system in §3.. In §4., LMI-based conditions ensuring the feasibility of the bounded multi-objective robust control problem are derived. Finally, §5. contains numerical results and simulations.

58 St~phane Dussy and Laurent E1 Ghaoui

1.4 No ta t i ons

or a real matrix P, P > 0 (resp. P > 0) means P is symmetric and positive-definite (resp. positive semi-definite) ; then, p1/2 denotes its sym- metric square root. diag(A, B) denotes the block-diagonal matrix written with A, B as its diagonal blocks. For any matrix A E R pxq, the matrix .N'A denotes any matrix Z such that A T z = 0 and rank[A Z] is maximum. For r = [r l , . . . , rn], ri E N, we define the sets For r = [ r l , . . . , rn], ri E N, we define the sets

79(r) = {A = diag(51I~x,...,Snlr~), 15/E R } , B ( r ) = { B = d i a g ( B 1 , . . . , B n ) , [ B, E P J ' × r ' } , S ( r ) = { S E B ( r ) l S i > O , i = l , . . . , n } .

(1.3)

Finally, the symbol Co{v1,.. . ,VL} denotes the polytope with vertices vl, • .., VL, and for P > 0 and a positive scalar )~, Sp A denotes the ellipsoid ERA = {X [ xT p x <_ )~}.

2. T h e i n v e r t e d p e n d u l u m e x a m p l e

The inverted pendulum is built on a cart moving in translation without con- straint (see figure 2.1). Our goal is to stabilize the it in the vertical position, with an uncertain but constant weight m.

0

Fig. 2.1. Inverted pendulum on a cart

In fig.2.1, ~ is the position of the cart, 0 is the angle position of the pendulum with the vertical axis, u is the control input acting on the cart translation and w is the disturbance. We seek to control the angle and the

Control of Uncertain Systems with Bounded Inputs 59

translation position, 0 and ( . The nonlinear dynamic equations of the system are:

[tacos 2 0 - 4 (M + rn)]0

[mcos 2 0 - ] ( M + m)]~

c] = ] r

= mcosOsinOd 2 - ~ ( M + re )s in0 cos ou(t) +

= m g s i n O c o s O - 4 m l d 2 s i n O - ~u(t)

(2.1) We assume that the mass m is constant and unknown-but-bounded: m E

[m 0 - Am , m0 -t- Am], where too, A m are given. The parameter vector here is taken to be ~ - ( m - m o ) / A m , so that ~ E D = [ - 1 , 1]..Our specifications are described by S.1-5, with a, Um~x, ~/and Zmax = [O,~a~ Oread] T given. The polytope of admissible initial conditions is determined by two given scalars 00,~0:7 ) = C o { v + , - v + , v _ , - v _ } , with v+ = [4-80 "4-~0] T. Moreover, the system is also perturbed by a disturbance w(t) E ~:2([0,00[).

We seek a control law that stabilizes both the angle position and the translation motion of the cart around a given equilibrium position.

3. Construct ion of the L F R

3.1 L F R o f t h e o p e n - l o o p s y s t e m

The first step is to derive a Linear-Fractional Representation (LFR) for the system. In other words, we seek a set of equations

= A x + B p p + B u u + B ~ w , q = C q x + D q p p + D q u u + D q ~ w , y = C y x + D v p p + D y w w , z = C z z + D z p p + D z u u , p = A(x,~(t))q, A E :D(r).

(3.1)

that is formally equivalent to the nonlinear equations (2.1). We will refer to [8] where the framework for the construction and the normalization of an LFR for such a system is described.

Introducing (I1 = cos 0, ~2 sin 0 = = - g - , ~3 0sinS, (~4 "-- m e [ m o - , b m , m o + Am], we may rewrite (2.1) in

60 St6phane Dussy and Laurent E1 Ghaoui

0 3gtMT&~)6u

-- 315451-4(M+~4)1 0

3g1645152 --4t26. 31646~-4(M T54)t 3/~,$~-4(J

0 36 x

31545~-4(M+64)l U(t) + + 0

--4l 31646~-4( M +64)1

1 3t~4616~

3164~-4(M+6~)l 0 Lt63

-4(M+64)/

0 0

0 0

0 1

0 0

31~I4(~ 1-4(M+54)1 0 31

w(t).

(3.2)

The coefficients of the above state matrices are rational functions of the parameters 5i, i = 1 , . . .4 , so that we may derive an LFR for the system. This LFR has the form (3.1) with A, Bp, ..., given in Appendix A.1 and with A = diag(Am, Au) = diag(5116, 52, 53, 5414) where A m = diag(51/6,52) is measured in real time, while Au = diag(53,5414) is not.

3.2 L F R o f t h e c o n t r o l l e r

We take our measurement-scheduled controller to be in the LFR format

z = _A£ + By__y + Bpf,_ "Z(O) = O, "q : _Cqx -b Dquy -{- Dqpp, (3.3) u : CuT,

where p, q are fictitious measured inputs and outputs of the controller. Note the crucial assumption here: the controller is scheduled with respect to the measured element in the nonlinear block A, namely Am (y). In our benchmark example, this means that the controller matrices will be scheduled on the value of 61 = cos 0 and (f2 = sin e 0

3.3 L F R of t h e c losed- loop s y s t e m

With the chosen controller structure, the closed-loop system assumes the form

= .A(,~)~ + Bw(fi)w, (3.4)

where ~ = Ix T .~T]T Here, A, /3w are given rational functions of the nonlin- earity element A, where the part A,~(y) appears twice as it is present in the open-loop system and in the controller (see Fig. 3.1).

z~ = d iag(A u (x), Am(y), Am (y)). (3.5)

Introducing fiT = [ pT ~T ], ~ = [ qT .qT ] , we obtain the following LFR for the closed-loop system:

Control of Uncertain Systems with Bounded Inputs 61

= 2 ~ + ~ + b~w,

z = G~+9~p~, = ,~q, ,~ = d i a g ( A u ( x ) , A m ( y ) , A m ( y ) ) ,

(3.6)

where the matrices A,/~p, etc, depend affinely in the design matrix variables Cu and K, defined by

[ ] A Bp B u K = Cq Dqp D--qu "

The closed-loop system is shown in Fig. 3.1.

W "-t- ~'X i Z I y r

u _t LTI

/ L T I I_

Fig. 3.1. Closed-loop system.

4. Synthesis Conditions

4.1 Analysis via quadra t i c Lyapunov f lmct ions

We seek quadratic Lyapunov functions ensuring specifications S.1-5 for the closed-loop system. We thus consider a matrix /3 ¢ R n×n, with /3 > 0, and associated Lyapunov function V(£ ~) = ~T/3~. Assume that, for a given nonzero trajectory in the set X(z0, Wmax), V satisfies

9 + 2av + v(z~z - v%Tw) < 0. (4.1) When u = O, the above condition ensures the a-stability of the system. When u # 0 and a -- O, it guarantees a maximum £~-gain from w to z, and when

62 StSphane Dussy and Laurent E1 Ghaoui

u # 0 and a # 0, it both ensures the a-stability of the unperturbed system and guarantees maximum L2-gain from w to z when the initial conditions are set to zero.

We can readily infer from (4.1) that, for a given nonzero trajectory in the set X(xo, Wmax), with x0 E £p,~, the following condition holds

(4.2) V(~(T)) < A + v 7 Wm~ x.

The above bound implies that when initiating in an ellipsoid Ep, x, the aug- mented state of the system $(t) will be confined in the bigger ellipsoid Ep,(x+,7:~i.x ). In other words, it suffices to ensure that inequality (4.1) holds to prove that every given nonzero trajectory in the set X(xo, Wmax) will never escape this bigger ellipsoid. This "pseudo-invariance condition" will be Used in the following sections.

4.2 S t a b i l i t y a n d d i s t u r b a n c e r e j e c t i o n c o n d i t i o n s

As seen in [12, 8], it is possible to formulate the synthesis conditions to ensure the specifications 8.2 and 8.5 in the form of a set of LMI conditions asso- ciated with a non-convex constraint. These inequalities depend in particular on P and Q, the upper-left n × n blocks of P and/5-1 , respectively.

T h e o r e m 4.1. Let A/" be a matrix whose columns span the null-space of [Cy Dyp Dyw] T. The synthesis conditions are

0 Su , T = 0 Tu 6 S(r) and Y such that

I Q >-0' I T >-0' (4.3)

A T p + PA+CTqSCq PBp +CTSDqp PBw +CTSDqw +C TCa + 2aP +C T D~e

Af T , D~'p SDqp - S T Af < O, + DTzp S Dzp Dqp S Dqw

T * * DqwSDqw -- 72I (4.4)

AQ+QAT+B~y+yTB. T T T T T QC4 + Y Dqu QCT~ + Y m.. --2 T +BpTB T + 7 B~B~ + 2aQ T --2 T +BpTDqp + 7 B~Dq~ BpTDT~ * DqpTDq T - T DqpTD~p

--2 T .... +7 DqwDqw * * DzpTDz T - I

< O, (4.5)

Su Tu = Ivu. (4.6)

Control of Uncertain Systems with Bounded Inputs 63

4.3 C o m m a n d inpu t b o u n d

T h e o r e m 4.2. A sufficient condition to have llu(t)tt2 < Urea× for everyt > 0 is that the matrices P, Q, Y as defined in § 4.2 and variables )% p > 0 also satisfy

x0 6 £p,~, (4.7) 2 2 Ft()~ -}-/2"/ Wmax) "- 1, (4.8)

y T Q I > O. (4.9) 0 I P

Proof: We defined in § 4.2 that P and Q are the upper-left n × n blocks of/5 and/5-1. Precisely,/5 and Q are parameterized with arbitrary matrices M and L as follows (see [12] for more details)

P= Mr ~ O= , L T -~ ,

where P and Q are such that C~-I and C~-, are respectively the projection of the ellipsoids Cp and C~ on the subspace of the controller state. Then, the matrix Y defined in § 4.2 also depends on the the parameterization matrices via Y = -CuLT. Let assume that the above conditions (4.7), (4.8) and (4.9) hold. Then, using Schur complement [5], (4.9) is equivalent to

P > 0 and y T Q -1 >_ O.

Then, with Y =-CuL T, we readily obtain

:; ] ~?~max P > 0 and ---=T _ t )L_ T > O.

C u L - I (Q

With -~ = L T (Q - P - i)- i L (which stems from the matrix equality/5(~ = I), we may infer the following equivalent conditions

[01 P > 0 a n d Cu #U2max >0 .

Using Schur complements, we obtain

--~T-- T - - - 1

p > o andV~ER,~ ~ Q ~_gT ~ > 0 . ]ZUmax

The assumption x0 6 Sp,~ implies that ~ 6 £p,~. Therefore, the state of the

controller will never escape the ellipsoid obtained by projecting the state- space of the augmented system on the subspaze of the controller space. In other words, that means that ~ 6 E~-,,~. Then, with u = C~Z, we can write

64

[]

St~phane Dussy and Laurent E1 Ghaoui

Vx0 ~ gp,~, uTu < u~n~×.

Remark : As seen in [5], the condition x0 E £p,x for every x0 E Co{v1, . . . , VL} can be written as the following LMIs

v TPvj<_)~, j = I , . . . L .

4.4 O u t p u t b o u n d

T h e o r e m 4.3. A sufficient condition to have, for every t > O, IIQ(t)I]2 = tlEix(t)]12 <_ ~;,m~×, i = 1, . . . N, is that the matrix Q as defined in § 4.2 and variables A, ~ > 0 also satisfies (d.7), (4.8) and

#¢~max -- ETQEi >- O, i = 1 . . . . g . (4.10)

Proof." Let assume that the above conditions hold. Then using Schur complements with the additional condition Q > 0, (4.10) is equivalent to

Ei 2 > 0 , i : 1 . . .N . P q , m a x - -

Another Schur complement transformation leads to

V x E R n, x T Q - l x > x T ETE~ - 2 x , i = l , . . . N .

#q,m~×

Now, if x0 E £p,x, then x E £p,~. Furthermore, we know from (4.3) that p >_ Q- t , which implies that x E £Q-1,~. This achieves the proof since then

Vx E R n s.t. x0 E gP, X, Q,max ~_ xTETEix = gTgi, i = 1 , . . .N .

[]

4.5 Solving for the synthes is condi t ions

Every condition above is an LMI, except for the non-convex equations (4.6) and (4.8). When (4.3) holds, enforcing these conditions can be done by im- posing TrSuTu + #(A + W~max72) = uu + 1 with the additional constraint

[ P / ] I ,~+72W2ma x _> 0. (4.11)

In fact, the problem belongs to the class of "cone-complementarity problems", which are based on LMI constraints of the form

F ( V , W , Z ) > O , [ V I ] - I W >- 0, (4 .12)

Control of Uncertain Systems with Bounded Inputs 65

where V, W, Z are matrix variables (V, W being symmetric and of same size), and F(V, iV, Z) is a matrix-valued, affine function, with F symmetric. The corresponding cone-complementarity problem is

minimize TrVW subject to (4.12). (4.13)

The heuristic proposed in [11], which is based on solving a sequence of LMI problems, can be used to solve the above non-convex problem. This heuristic is guaranteed to converge to a stationary point.

A l g o r i t h m 7i:

1. Find V, W, Z that satisfy the LMI constraints (4.12). If the problem is infeasible, stop. Otherwise, set k = 1.

2. Find Vk, Wk that solve the LMI problem

minimize Tr(Vk-IW + Wk-IV) subject to (4.12).

3. If the objective Tr(Vk-lWk + Wk-lVk) has reached a stationary point, stop. Otherwise, set k = k + 1 and go to (2).

4.6 R e c o n s t r u c t i o n of t h e c o n t r o l l e r

When the algorithm exits successfully, ie when Su T~ "-" I and #(A+W2m~x72) _~ 1 (note that a stopping criterion is given in [11]), then we can reconstruct an appropriate controller by solving another LMI problem [12, 8] in the controller variables K only, Cu being directly inferred from the variables Y, P, Q. Note that analytic controller formulae are given in [13].

5. N u m e r i c a l r e s u l t s

Our design results are based on two sets of LMIs: the first set focuses on the s-stabili ty of the system for given initial conditions. It corresponds to the case v -- 0 in § 4.1. The second one guarantees a maximum/:2-gain for the system, as defined in §4.2, with (~ -- 0. (It may be too conservative to try and enforce both a decay-rate and an/:2-gain condition via the same set of LMIs.) The bounds (respectively in rad.s -1 and in tad) on the states that appear in the nonlinearities are Zmax -= [0max ~max] T : [0.5 1] T, and the system parameters are M -- 1, l -- 0.6, g = 10, rn - 0.2(1 4- 0.4(frn) with ]5,n] _< 1. The units used for the system parameters and for the plots are kilogram for weights, meter for lengths, second for time, radian for angles and Newton for forces. The design parameters are c~ = 0.05, X0 -- [00 00 (0 ~0] T = [0 0.2 0 0.2] T. We seek to minimize the bound Um~x on the command input and to achieve good disturbance rejection. The numerical results were obtained using the public domain toolbox MRCT [9, 7] built on top of the software lm±tool [10] and the Semidefinite Programming package SP [19]. The output-feedback

66 St~phane Dussy and Laurent E1 Ghaoui

2:t t k l k

1 .5 a .,

0 •

~ m e , $ e ~ d s

0 . 2 5 .

o.2 t" 0 . 1 5 ~

~ 0.1

i o.o5 ', ~ 0

• .41.05 4 s ~

- O . t

- 0 . 1 5 i , = 1 2 3

30

i i * i i 4 5 6 7 8 10

T i m e , s e c o n d s

Fig. 5.1. Closed-loop system responses (cart position and angle position) with w - 0 and nonzero initial conditions. Extremal responses (OF) in plain line, nominal response (OF) in dotted line, nominal response (SF) in dashed line.

Control of Uncertain Systems with Bounded Inputs 67

0.2

>~-0.2 ,l

-0,6

-0.8

| '

,~. 2

- 3 ~

, ~ 6 ~ 8 ; 1'0 Time, seconds

L

Time. seconds 8 9

Fig. 5.2. Closed-loop system responses (angle velocity and command input) with w ----- 0 and nonzero initial conditions. Extremal responses (OF) in plain line, nom- inal response (OF) in dotted line, nominal response (SF) in dashed line.

68 Stdphane Dussy and Laurent El Ghaoui

4 ~ .........

3

2

-2

i i % ;o 2o 3o 20 go 60

T~ne, a e c o n d s

0.25

0,2

0,15

~. 0.1

. 0 , 0 5

o ; , I f ~ I l l ; . . . .

-o.,~[, V v v v V V V V V -0,2 0 5 10 15 20 25 30 35 40

Time, seconds

E

~o

Fig. 5.3. Closed-loop system responses (cart position and angle position) with disturbance w(t) and zero initial conditions. Extremal responses (OF) in plain line, nominal response (OF) in dotted line, nominal response (SF) in dashed line.

Control of Uncertain Systems with Bounded Inputs 69

4r

~ O t l l l l ~ l I t l _----

M 0 5 10 15 ~ 25 ~ ~

Time, e . ~ o ~

1

ti 0.8 0.6 0.4 0.2

J:H/ ~'4f/ ~.6 ~ . 8

T ~ , seconds

Fig. 5.4. Closed-loop system responses (command input and perturbation) with disturbance w(t) and zero initial conditions. Extremal responses (OF) in plain line, nominal response (OF) in dotted line, nominal response (SF) in dashed line.

7"0 Stdphane Dussy and Laurent E1 Ghaoui

controller is given by the LFR (3.3), where the gain matrices are described in Appendix A.2.

We have shown the closed-loop system responses with w -= 0 and a nonzero initial condition 00 = 0.2tad, z0 = 0.2m in Fig. 5.1-??, and with zero initial conditions and w(t) = 0.1cos(~rt), 0 < t < 20 in Fig. ??. We plotted in plain line the envelopes of the responses for different values of the weight m varying from -40% to +40% of the nominal value. We also plotted the response of the nominal system (m = 0.2kg) with the output- feedback law (dotted line), and with a state-feedback control law (dashed line) u = [17.89 4.65 0.06 0.40]x computed as described in [8]. Both robust stabilization and disturbance rejection are achieved for a 40% variation of the weight around its nominal value. We can note that the bounds on 6(t) and on ~(~), respectively 0max = 0.5tad and ~m~x = lrad/s , are enforced. Also note that the state-feedback controller requires less energy SF (urea × = 3.SNewton)

OF than the output-feedback one (urea x = 6.2Newton).

6. Concluding remarks

In this paper, an LMI-based methodology was described to ensure various specifications of stability, disturbance rejection, input and output bounds, for a large class of uncertain, perturbed, nonlinear systems. The proposed control law is allowed to depend on parameters or states that are measured in real time, while it remains robust in respect to the unmeasured ones. This methodology has been successfully applied to the control of an uncertain, perturbed inverted pendulum. It is very systematic and handles multiobjective control. One important aspect of the method is that it allows computing trade-off curves for multiobjective design.

Recently, accurate robustness analysis tools based on multiplier theory have been devised, see e.g. Megretsky and Rantzers [14] and Balakrishnan [2]. A complete methodology for nonlinear design should include these tools for controller validation.

References

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2. V. Balakrishnan. Linear Matrix Inequalities in robustness analysis with multi- pliers. Syst. ~ Contr. Letters, 25(4):265-272, July 1995.

3. G. Becker and A. Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. Syst. (J Contr. Letters, 23(3):205-215, September 1994.

Control of Uncertain Systems with Bounded Inputs 71

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72 St6phane Dussy and Laurent E1 Ghaoui

A . L i n e a r F r a c t i o n a l R e p r e s e n t a t i o n

A.1 L F R of the open- loop sys tem

The LFR of the inverted pendulum is described by (3.1) with the following matrices.

A =

=

3 Bp--- 4 M

V q p ~-

0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 0 0 0 1

B u =

, D q u =

0 0 o

0 0 0 1 T 0 1

-g- o 0

0

0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0

- 1 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ~-~ 0 0 0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 o o o o 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~- 0 0 0 1 0 0 1 0 ~- 0 0

, B,~= ~ ,

--~ 0 Ml

0 0

- 3 4 M 0 0

- 3

~ V q w = 4 M l

0 0 0 0 1

M ,

0 0 0 0 0 0 o ,ool 0 0 0 0 0 0 0 0 0 -~ 1 0

0 0 g 0 0 0 0 0 0 0 0 0

3~ 0 0 4Ag 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 3 0

I M 4 M 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -4 -1 0

31M M -g 0 0 -~

Control of Uncertain Systems with Bounded Inputs 73

A .2 L F R o f t h e c o n t r o l l e r

The LFR of the controller is described by (3.3) with the following gain ma- trices.

A =

n

Bp = 0.01

- 8 . 8 - 3 . 7 - 1 . 2 -12 .7 -54 .2 -19 .7 -7 .6 -13 .0 39.3 19.0 -24 .1 -168.3

-1115 -443 -181 -1062

B u =

D-'-qp = 10 -3

, C . = 10 - 3 28.9 ' - 9 . 6

-22 .9 0.1 94.8 -1 .4

-395 44 -53 .7 1.5

1159 280 D-qu = 0.1 -82 .5 1.6 8868 -157 '

- 315 1.4 -732 5780

13.4 -0 .1 - 1 . 4 0.0

0.I 0.5 - 3 . 2 24.0 - 1 . 7 - 1 . 7 --0.0 - 0 . 4 - 2 . 5 12.5 -53 .5 3.3 3.9 0.I - 6 . 8 -22 .4 114 -267 22.7 21.4 - 4 . 0 6.3 20.4 - 8 9 111.3 -10 .4 -10 .6 2.4

-0 .6 - 0 . 3 0.6 0.9 4.7 1.2 - 2 . 7 - 8 . 4

- 3 . 9 0.4 - 0 . 4 9.2 Cq = 0.01 -8 ,1 -8 .2 13.2 -0 .8 ,

- 5 . 7 - 3 . 7 6.2 7.5 0.1 0.0 0.1 0.4

-0 .1 - 0 . 0 0.I 0,3

- 3 0 -4 .8 22.5 15.9 -8 .1 -0 .1 - 0 . 0 12.0 - 1 8 - 3 0 - 2 2 13.7 0,6 0.6 - 2 7 - 1 5 -10 .5 1,1 - 1 . 3 -0 .5 0.9 - 4 . 8 - 3 . 4 -2 .3 3.1 - 0 . 5 -0 .3 0.2 2,2 -7 .2 - 1 4 8.5 2.6 -1 .1 0.2

- 0 . 0 - 0 . 0 - 0 . 0 0.0 - 0 . 0 - 0 . 0 0.0 - 0 . I - 0 . i - 0 . 0 0.I - 0 . i - 0 . I - 0 . 0