- Email: [email protected]
Design of Dynamical Controllers for Electromechanical Hamiltonian Systems
Copyright ® IFAC Robust Control De:sign, Prague, Czech Republic, 2000
DESIGN OF DYNAMICAL CONTROLLERS FOR ELECTROMECHANICAL HAMILTONIAN SYSTEMS
Werner Haas and Kurt Schlacher '
• Department of Automatic Control Johannes Kepler University of Linz Altenbergerstr. 69, A-4040 Linz Email: [email protected] Phone: ++43 732 2468 9730
Abstract: Linear Hamiltonian control systems with collocation of sensor and actuator are considered. Based on a frequency domain approach a controller design algorithm is stated. The design leads to a controller with internal dynamics which uses the output of the system and its first time derivative. The presence of internal dynamics in the controller is an extension of the usual PD-control law and a main result of the work. The design is based on the special properties of the proposed class of systems. In particular, this class of Hamiltonian systems is passive. It is shown that the design leads to strict ly passive controllers for a certain choice of the design parameters. This is another significant result and offers a way for robust L 2- stabilization even in the case of infinite dimensional systems. Some feat ures of the controller design are discussed with respect to an a pplication, the control of an infinite dimensional system. Copyright @2000 IFA C
Keywords: Hamiltonian control systems, dynamical controllers, passivity, infinite dimensional systems, frequency domain
insensitive to uncertainties in the system. This approach leads to PD- controller - see e.g. (Hill and Moylan, 1995). (Kelkar and Joshi, 1996), (Morris and Juang, 1995) , (Nijmeijer and Van der Schaft, 1991) . Such a control law can be interpreted as adding damping and extra potential energy to t he system.
1. INTRODUCTION
This work is concerned with the control of linear Hamiltonian control systems with a single input and a single output. The actuator and sensor are collocated and the passivity of the system is assumed. Basically such a class covers internally conservative electromechanical systems. This conservative idealization is usually a starting point for analysis, as well as for control design . In the control design of such systems there exists two common ways. One way is the use of a mathematical model for the controller design. Here, special care has to be taken for the robust stability problem due to less accurate models e.g. the eigenfrequencies are uncertain quantities. This is the approach behind many H oo-designs see e.g. (Green and Limebeer, 1995). The second way is based on physical considerations, which is
The intention of the work is to present a t hird way which combines both methods. More precise, the proposed design is based on a mathematical model of the Hamiltonian control system and takes into account physical considerations . The feedback law u = -kyY - Cvv with ky E R , Cv E R(s) uses the so called natural output y and the corresponding first time derivative v = y. Moreover, a proper design gives t he controller quantities ky and Cv where the dynamical part Cv is strictly passive. This fac t offers a robust £2-
763
stabilization of Hamiltonian control systems even for infinite dimensional systems. The paper is structured as follows. In section 2 an application of an electromechanical Hamiltonian control system of infinite dimension is stated. Hamiltonian control systems have special properties caused by the conservation of the internal energy. They are discussed in section 3. Next, the passivity approach is discussed and a well known stability theorem is stated: Every passive system can be stabilized using a strictly passive controller. In the proposed case the feedback of the time derivative of the natural output with any strictly passive controller stabilizes the control loop. Then, this result is extended using the control law u -kyY - evv. In section 4 a design method is presented based on a 1i2-optimization technique. In general, the result of this design leads to improper and therefore non realizable controllers. But in the case of Hamiltonian control systems the controller can be realized using the natural output and its first time derivative. Moreover, it will be shown that the design leads to a strictly passive controller for a certain choice of the input parameters. Section 5 presents the control design with respect to the application.
Fig. 2. The cantilever- beam. space-wise distribution along the x- axIs IS not uniform. The shaping function of the actuator is chosen such that the steady-state deflection due to the lateral load F is annihilated by the steadystate deflection due to the control voltage u. The mathematical modeling - see (Haas et al., 1998) for details - leads to an Euler-Bernoulli partial differential equation
=
2
8 w (x, t) /J. &2
8x (O , t) = 0,
N - - .- -
--1---1
-i il - -iij - -tii - -. .-,,- -" '---i:P.-Fi--.--P; - - i- i- ;-i-l-Pi - -!iF -
---- -- -
outpur-1't:;--;; --;;---j-;; --;;---i; --;; --;; ---i; --;'; - ;;;;.;;;~i#:~
&W(l t)=_F(t) _ ku(t) 8x 3 ' B B
(2) and material parameters /J., B, k . For the following theoretical investigations the values of these parameters are chosen with /J. = 1, B = 1, and k = 1.
Remark 1. The effect of the actuator voltage u can be summarized as a fictive actuator force at the free end of the beam. So, the disturbance F and the control input u act at the same point. The resUlting problem thus consists in controlling a distributed parameter system via boundary control. Hamiltonian control systems require a collocation of sensor and actuator which leads to the concept of the natural output, see e.g. (Nijmeijer and Van der Schaft, 1991 ). For the proposed application the natural output y is given as the deflection of the free end y(t) = w(l,t), see (Haas et al., 1996), which can be measured by means of a shaped piezoelectric sensor. A discussion of the shaping functions is out of the scope of this work.
actuat9r layer .-_ _~ Y+
--- -- _:j'--------- -- ----- --
-2
aw
The application under consideration consists of a cantilever beam-type structure (also called piezoelectric bender) with piezoelectric layers and sensors for the active damping of vibrations in lateral direction. The outer pair of piezoelectric layers is used as actuator and the inner one as sensor see Fig. 1. The ± signs indicate the polarization
rt-..---,,--,-,:'i,,-,,--.--
(1)
82 w (l ,t) = 0, 8x
w(O ,t)=o ,
PIEZOELECTRIC ELEMENTS
sensor
+
with the boundary conditions
2. CANTILEVER BEAM WITH
sensor layer
°
4
B8 w (x,t) _ 8x4 -,
ru
1u
se~n~-~~~-:t-71~-~~;e:--r~---m--m-~--~~;~t~~~:~t~~~-;-~~a~-y:e:r~~-~-Y+
Using the Laplace transform the system of equations (1), (2) can be solved which gives the transfer function P y/u = Y/ P with
Fig. 1. The piezoelectric bender.
=
of the layers. This piezoelectric bender of length l is clamped at one end and free at the other one. The relative displacement in the lateral zdirection with respect to the undeformed beam (the symmetry axis of which corresponds to the x-axis) is denoted as w - see Fig. 2. Applying the control voltage - V+ ~ u ~ V+ leads to a bending of the structure in the (x, z)-plane. All piezoelectric layers are shaped, in other words the
P(s) = cos
FJS sinh FJS -
cosh
FJS sin FJS -
(_js)3/2 (cosJ-jscoshJ-js+
1) (3 )
=
and j A. A hat refers to the corresponding Laplace transform. In section 3 will be shown that all poles and zeros of a Hamiltonian system lie
764
Fact 1. The poles and zeros of P are purely imaginary. The number of poles is given with 2nH and the number of zeros is 2 (nH - 1) . Moreover, there are no poles at the origin s = o.
on the imaginary axis. Hence, any reasonable low order model PI of P has the form
In (1 +
R (s)
1 k=1
="3
I
IT k=1
(S/W Z ,k)2)
(1 + (S/W P,k)2)
'
1 EN.
(4) To show this, the energy conservation of Ho is taken into account. Thus, all poles of P are located on the imaginary axis. Moreover, V > 0 and C > 0 implies that there are no poles at the origin. Next, the output is differentiated twice which gives Y. = eT·q = eTC p, ii = -e[CVq + e[Celu.
with the purely imaginary zeros jWz,k and poles jWp,k. The numerical values for the two poles and the first zeros are given as Wp,l = 3.52, W p ,2 = 22.03, Wz,l = 15.42. The expression (4) indicates that transfer functions of Hamiltonian systems satisfy some special properties. This fact will be pointed out in detail in the next section. The investigations are restricted to finite dimensional systems which are required for the controller design.
l
(8) The second relation in Eq. (8) offers a way to calculate u as a function of ii. Hence, the relative degree of P is two and the number of zeros is 2 (nH - 1). The location of the zeros follows from the fact that the free inverse system of a Hamiltonian system is a Hamiltonian system, too - for the general nonlinear case see (Nijmeijer and Van der Schaft, 1991 ). To show this the input y of the inverse system is chosen with y = 0 which leads to the restrictions ql = 0, tll = 0 and Cp = O. The latter restriction states a relation for the coordinate PI as a function of the coordinates P2 , . .. PnH· Next, the coordinates ql and PI are eliminated from the Hamiltonian (5). Let q and 'j5 be the new vector of generalized coordinates and generalized momenta, respectively. ij consists of the coordinates qi, i E {2, .. . ,nH} of q and 'j5 of the coordinates Pi, i E {2, ... ,nH } of p. By a suitable partition of C and V the Hamiltonian of the free inverse system can be obtained
3. PROPERTIES OF HAMILTONIAN SYSTEMS The attention is focused to a special class of single input, single output, and linear Hamiltonian control systems with a Hamiltonian of the form
H(p,q,u)
er
= Ho -yu 1 1 = _pT Cp + _qT'y' q _ yu
2
(5)
2
with positive definite matrices C and V. Ho is the Hamiltonian of the free system and given by the sum of kinetic and potential energy, q is the vector of the generalized coordinates, and p denotes the vector of generalized momenta. The system has nH degrees of freedom. A duality is defined between the input u (a generalized force) and the output y in such a way that y is equal to the corresponding generalized coordinate q;, i E {I, . .. ,nH}. A motivation for this choice can be deduced from the fact that the rate of Ho due to the motion of the system is given as
dHo
.
ill =uq;
l
fl,o = -:? I-T(C 22
-
CTC - lC) -+ 2 I q-TV22q· 12 11 12 P
Remark 2. The Nyquist plot of P is on the real axis of the complex plane. This is a consequence of fact 1. Further, let v = y then the Nyquist plot of the transfer function 1; (s) /u (s) = sP is on the imaginary axis.
(6)
see e.g. (Nijmeijer and Van der Schaft, 1991). Hence, the increase of the internal energy is equal to the total external work performed on the system. This is usually called the case of collocated sensors and actuators and y is called the natural output. The corresponding Hamiltonian equations of motions are given in state-space form as
Remark 3. The positive definiteness of C and V implies that the internal energy Ho is a positive definite function. Definition 1. A dynamic system is said to internally passive if there exists a non negative storage function E such that
Jy(t)u(t)dt~E(x(T))-E(x(O)) T
ei denotes the i-unit vector and the number i specifies the collocated output y = qi. For simplicity, a reordering is made such that y = ql. The following properties are of importance for the proposed controller design. Let P = fj/u be the transfer function of the Hamiltonian system (7).
(9 )
o for Vu E £2, "IT
~
o.
Fact 2. Assuming that Ho is pC~lltlve definite, then the Hamiltonian system will be passive.
765
To show this, the equation dHo/ dt integrated which gives
J
uq
is
are simple and their residues are non negative and c) Re(G(jw)) ~ 0 for all w E (-00,00).
(10)
Strict passivity of a linear time invariant system is equivalent to strict positive realness.
T
y (t) u (t) dt = Ho (T) - Ho (0) .
o Thus, the internal energy Ho is a suitable storage function - see remark 3 - and the comparison of the equations (9) and (10) proves the fact 2. An interesting feature is that the inequality is replaced by an equality. The fact 2 has immediate consequences on the £z--stability of the control loop which will be discussed in the next section.
Definition 3. A BIBO-stable transfer function G is said to be strictly positive real if and only if Re (G (jw)) > 0 for all w E (-00,00). The following stability result is sufficient.
Theorem 1. (Vidyasagar, 1993) The feedbacl< system in Fig. 4 is £z--stable if the following conditions are satisfied: a) G is positive real and b) H is strictly positive real.
3.1 Stabilization of Hamiltonian Systems The rate of the internal energy of a Hamiltonian control system is given by equation (dHo/dt = uq. Hence, the D-control law u = -k"v leads to a decrease of the internal energy by dHo/dt = -k v v 2 , with kv > O. Now, a control law is assumed of the form it,
= -kyf) -
CJ)
with
Let G be given as G = sP with P as the transfer function of the Hamiltonian control system (7). In spite of fact 2 - see also fact 1 and remark 2 - the positive realness of G is deduced. Consequently any strictly positive real Cv and ky = 0 stabilizes the feedback system in Fig. 3. But what happens in the case ky =F 0 ?
ky En, Cv En (8). (11 )
R (s) denotes the set of all transfer functions over the real numbers R. In the following sufficient conditions are stated such that the control law (ll) stabilizes the feedback system given in Fig. 3. Here, d denotes the disturbance signal. First,
Fact 3. The feedback system given in Fig. 3 is £z--stable if Cv is strictly positive real and ky is non negative. To show this, a control law of the form is assumed u = -kyY - z, with ky ~ 0 using the natural output Y = ql ' Then, the system - see equation
(7) -
[~] = [-(V+~yelen ~] [~] [e~]
Z
can be obtained with the new Hamiltonian of the free system
Fig. 3. The proposed controller scheme.
IT IT 1 2 Ho='2 P GP+'2 q Vq+'2 kyq1 '
consider the simple feedback system in Fig. 4, with G, HEn (8). A well known result is that
Hence, the feedback of ql leads to a modified potential energy. Due to V > 0, ky ~ 0, the new system is another Hamiltonian system and the feedback of ql does not destroy the £z-stability.
Remark 4. Passivity is a physical property of a system and independent of changes of parameters. Moreover, any reasonable low order model of an infinite dimensional Hamiltonian system is itself a Hamiltonian system - see equation (4).
Fig. 4. A simple feedback system. the feedback interconnection of a passive system and a strictly passive finite-gain system is £2stable. Passivity of a linear time invariant system is equivalent to positive realness.
Remark 4 offers a way for the £z-stabilization of infinite dimensional systems based on the control law (11) and the control scheme as given in Fig. 3. For this matter a low order Hamiltonian model is required, for example the transfer function P2 as
Definition 2. A transfer function G is said to be positive real if and only if a) G is analytic in Re (s) > 0, b) the poles on the imaginary axis
766
given in equation (4 ). Then any strictly positive real Cv and ky 2: 0 leads to a £2- stable control system. Moreover, the original system P of infinite dimension is stabilized, too.
roots of the Hurwitz polynomial n" ( -s) satisfy the property Re (Si ) = - 0' . Due to the coprimeness of d and n there exists a solution of the Diophantine equation (14). Moreover, the minimal degree solution is unique.
4. CONTROLLER DESIGN
Remark 6. The proposed design can be applied to any transfer function P. In general, this would give improper and therefore non realizable Cv. If P corresponds to a Hamiltonian system, then we get a proper Cv of order 2nH - 2 , where nH is the number of freedoms .
In this section the controller design procedure is stated. The algorithm is presented and some important properties of the controller will be discussed. The proof of the design is beyond the scope of this paper and is presented in (Haas et al., 1999). Note, [s] denotes the set of polynomials in s.
n
As mentioned in section 3.1 there is particular interest in strictly passive Cv . Of course, the passivity property has not been taken into account for the derivation of the controller design algorithm . Nevertheless, the following fact holds.
The aim of the design is to find values for Cv E (s), ky E such that the 1i2--cost function
n
n
J(u + q~y2 + q~v2) e 00
J =
inf
C v ,k.
2
2" tdt
(12 )
Fact 4. For a certain choice of the input parame0', qy and qv there exist strictly passive Cv
o
ters
is minimized with respect to the impulse disturbance response. Here, 0' 2: 0 denotes the shift factor and qy 2: 0, qv > 0 are weights. The corresponding control configuration is shown in Fig. 3. The controller design starts with the transfer function P of a Hamiltonian control system. In the case of infinite dimensional systems P is replaced by a suitable low order approximation A. The result of the optimization task (12) leads to the following algorithm.
To show this, the special choice qv > 0, qy = 0' = 0 is the starting point. This gives n (-s) = n (s), d ( -8) = d (s) and the spectral factor a follows from
a (-s) a (s) -
a(8)
Algorithm for the controller design:
+ q; (s -
d" (-8) d" (s)
+ n" (-s) n" (s) q (s) .
f (s) = n (8)
and
e (8) = qv8n (8) .
Finally, the controller follows with ky = 0 , Cv = qv which is a D- controllaw and strictly passive. The controller depends on the parameters 0', qy and qv in a continuous way. Hence, for small 0' > 0, qy > 0 strictly passive controllers can be expected, too.
(13)
Remark 7. For small 0' > 0, qy > 0 the polynomials e, f are almost equal to the numerator of the shifted system P". Thus, the controller introduces additional dynamics which corresponds to the zeros of the Hamiltonian system.
• Diophantine equation: Find the minimal degree solution e, fEn [s] of
a(s)n"(-8)
+ e (8) n (8)
with solution
• Spectral Factorization: Calculate the Hurwitz polynomial a from
=
= d(s) + qvsn (8).
(d (8) + qvsn (8)) n (8) = f (8) d (8)
0')( -s - 0') .
a(-8)a(8)
with
Inserting this relation in Eq. (14) leads to
• Input: Specify P = nld E n (s) and the quantities 0', qy and qv' • Shift operation: Substitute 8 by 8-0'. This gives a shifted P" = n" Id" E n (8) and calculate the polynomial
q (8) = q~
see Eq. (13) -
= d2 (s) - q;s2n2 (s)
= f(8)d" (8) +e(8)n"(s). (14 )
• Calculate the controller with . e (8) g (8) (8 )' v - f (8) ky=hm.-0
c _
f
5. APPLICATION (CONTINUED)
where g is given as e (8) - kyf (s)
= 8g (8) .
In section 2 an application of a infinite dimensional Hamiltonian control system was stated. The following considerations are devoted to the controller design. The proposed design requires a low
Remark 5. All poles and zeros of P" are located in the closed right half plane - see fact 1. The
767
order model of the transfer function P and it turns out that
P2=~
x 10" 1.5 , . - - - - , - - - - . - - - - . - - - - - ,
l+(~f
3(1 + (3.~2) 2) (1 + (22"03) 2)
0.5
is suitable - see Eq. (4) . Two designs are included to demonstrate some properties of the controller design.
o -0.5
• Controller 1: The choice qv = 0.5, qy = 10, a = 1 of the design parameters gives the controller ky = 5.92, 5
Cv = 2.27
( 18 57) 5
C6 67)
2 2
-I
-1.5 0
5
1 + 3.89 . 10- 5 15 42
2
function of the Hamiltonian system . Moreover, a proper choice of the design parameters leads to strictly passive controllers. This allows a robust stabilization of infinite dimensional systems, too. Finally, an application shows some features of the design .
+ (15 42) 2 2' 5 1 + 2.59 . 10- 15~42 + ( 15 42) 5
=1
1.5
Fig. 6. Impulse response of the different controllers.
+ 0.20 18 57 + 1 . + 0.2516567 + 1
• Controller 2: The choice qv = 1, qy = 0, a = 0.0001 of the design parameters gives the controller ky ~O, Cv
0.5
5
5
7. REFERENCES
which is almost a D- control law .
Green, M. and D. Limebeer (1995). Linear Robust Control. Prentice Hall. Haas, W., A. Kugi, K. Schlacher and M. Paster (1998). Experimental results of the control of structures with piezoelectric actuators and sensors. In: Proceedings of the MOVIC 98, Zurich. Haas, W., K. Schlacher and Irschik H . (1996). Control of multi-layered structures via special distributed piezoelectric actuators. In: Proceedings of the First European Conference, Barcelona. Haas, W., M. Krommer and Irschik H. (1999). Design of dynamical controllers for linear hamiltonian systems. In: Proceedings of DETC'99/VIB-8009, 1999 ASME Design Engineering Technical Conferences, Las Vegas. Hill, D . and P. Moylan (1995). Stability of nonlinear dissipative systems. IEEE Transactions on Automatic Control 21(5), 708-711. Kelkar, A. and S. Joshi (1996). Control of Nonlinear Multibody Flexible Space Structures. Vol. 221 of Lecture Notes in Control and Information Sciences. Springer . Morris, K. and J. Juang (1995). Dissipative controller designs for second order systems. IEEE Transactions on .4utomatic Control 39(5),105&-1063. Nijrneijer, H. and A. Van der Schaft (1991) . Nonlinear Dynamical Control Systems. Springer. Vidyasagar, M. (1993). Nonlinear Systems Analysis. Prentice Hall.
To show the strict passivity (strict positive realness) the Fig. 5 presents the Nyquist plots of the Cv. The performance of both controllers is shown imaginary part
real part ·1
.(!,
fJ
(J ;;
Fig. 5. Nyquist plots of Cv' using the time responses - see Fig. 6. Here, the transfer function P6 is taken into account for the simulation.
6. CONCLUSION A controller design was introduced which uses the special properties of a Hamiltonian control system. The design is based on a 1i2 -optimization problem and the related algorithm uses the frequency domain approach. The result is a controller with "PIM3tructure" but additional internal dynamics. The internal dynamic of the controller corresponds to the zeros of the transfer
768
DESIGN OF DYNAMICAL CONTROLLERS FOR ELECTROMECHANICAL HAMILTONIAN SYSTEMS
Werner Haas and Kurt Schlacher '
• Department of Automatic Control Johannes Kepler University of Linz Altenbergerstr. 69, A-4040 Linz Email: [email protected] Phone: ++43 732 2468 9730
Abstract: Linear Hamiltonian control systems with collocation of sensor and actuator are considered. Based on a frequency domain approach a controller design algorithm is stated. The design leads to a controller with internal dynamics which uses the output of the system and its first time derivative. The presence of internal dynamics in the controller is an extension of the usual PD-control law and a main result of the work. The design is based on the special properties of the proposed class of systems. In particular, this class of Hamiltonian systems is passive. It is shown that the design leads to strict ly passive controllers for a certain choice of the design parameters. This is another significant result and offers a way for robust L 2- stabilization even in the case of infinite dimensional systems. Some feat ures of the controller design are discussed with respect to an a pplication, the control of an infinite dimensional system. Copyright @2000 IFA C
Keywords: Hamiltonian control systems, dynamical controllers, passivity, infinite dimensional systems, frequency domain
insensitive to uncertainties in the system. This approach leads to PD- controller - see e.g. (Hill and Moylan, 1995). (Kelkar and Joshi, 1996), (Morris and Juang, 1995) , (Nijmeijer and Van der Schaft, 1991) . Such a control law can be interpreted as adding damping and extra potential energy to t he system.
1. INTRODUCTION
This work is concerned with the control of linear Hamiltonian control systems with a single input and a single output. The actuator and sensor are collocated and the passivity of the system is assumed. Basically such a class covers internally conservative electromechanical systems. This conservative idealization is usually a starting point for analysis, as well as for control design . In the control design of such systems there exists two common ways. One way is the use of a mathematical model for the controller design. Here, special care has to be taken for the robust stability problem due to less accurate models e.g. the eigenfrequencies are uncertain quantities. This is the approach behind many H oo-designs see e.g. (Green and Limebeer, 1995). The second way is based on physical considerations, which is
The intention of the work is to present a t hird way which combines both methods. More precise, the proposed design is based on a mathematical model of the Hamiltonian control system and takes into account physical considerations . The feedback law u = -kyY - Cvv with ky E R , Cv E R(s) uses the so called natural output y and the corresponding first time derivative v = y. Moreover, a proper design gives t he controller quantities ky and Cv where the dynamical part Cv is strictly passive. This fac t offers a robust £2-
763
stabilization of Hamiltonian control systems even for infinite dimensional systems. The paper is structured as follows. In section 2 an application of an electromechanical Hamiltonian control system of infinite dimension is stated. Hamiltonian control systems have special properties caused by the conservation of the internal energy. They are discussed in section 3. Next, the passivity approach is discussed and a well known stability theorem is stated: Every passive system can be stabilized using a strictly passive controller. In the proposed case the feedback of the time derivative of the natural output with any strictly passive controller stabilizes the control loop. Then, this result is extended using the control law u -kyY - evv. In section 4 a design method is presented based on a 1i2-optimization technique. In general, the result of this design leads to improper and therefore non realizable controllers. But in the case of Hamiltonian control systems the controller can be realized using the natural output and its first time derivative. Moreover, it will be shown that the design leads to a strictly passive controller for a certain choice of the input parameters. Section 5 presents the control design with respect to the application.
Fig. 2. The cantilever- beam. space-wise distribution along the x- axIs IS not uniform. The shaping function of the actuator is chosen such that the steady-state deflection due to the lateral load F is annihilated by the steadystate deflection due to the control voltage u. The mathematical modeling - see (Haas et al., 1998) for details - leads to an Euler-Bernoulli partial differential equation
=
2
8 w (x, t) /J. &2
8x (O , t) = 0,
N - - .- -
--1---1
-i il - -iij - -tii - -. .-,,- -" '---i:P.-Fi--.--P; - - i- i- ;-i-l-Pi - -!iF -
---- -- -
outpur-1't:;--;; --;;---j-;; --;;---i; --;; --;; ---i; --;'; - ;;;;.;;;~i#:~
&W(l t)=_F(t) _ ku(t) 8x 3 ' B B
(2) and material parameters /J., B, k . For the following theoretical investigations the values of these parameters are chosen with /J. = 1, B = 1, and k = 1.
Remark 1. The effect of the actuator voltage u can be summarized as a fictive actuator force at the free end of the beam. So, the disturbance F and the control input u act at the same point. The resUlting problem thus consists in controlling a distributed parameter system via boundary control. Hamiltonian control systems require a collocation of sensor and actuator which leads to the concept of the natural output, see e.g. (Nijmeijer and Van der Schaft, 1991 ). For the proposed application the natural output y is given as the deflection of the free end y(t) = w(l,t), see (Haas et al., 1996), which can be measured by means of a shaped piezoelectric sensor. A discussion of the shaping functions is out of the scope of this work.
actuat9r layer .-_ _~ Y+
--- -- _:j'--------- -- ----- --
-2
aw
The application under consideration consists of a cantilever beam-type structure (also called piezoelectric bender) with piezoelectric layers and sensors for the active damping of vibrations in lateral direction. The outer pair of piezoelectric layers is used as actuator and the inner one as sensor see Fig. 1. The ± signs indicate the polarization
rt-..---,,--,-,:'i,,-,,--.--
(1)
82 w (l ,t) = 0, 8x
w(O ,t)=o ,
PIEZOELECTRIC ELEMENTS
sensor
+
with the boundary conditions
2. CANTILEVER BEAM WITH
sensor layer
°
4
B8 w (x,t) _ 8x4 -,
ru
1u
se~n~-~~~-:t-71~-~~;e:--r~---m--m-~--~~;~t~~~:~t~~~-;-~~a~-y:e:r~~-~-Y+
Using the Laplace transform the system of equations (1), (2) can be solved which gives the transfer function P y/u = Y/ P with
Fig. 1. The piezoelectric bender.
=
of the layers. This piezoelectric bender of length l is clamped at one end and free at the other one. The relative displacement in the lateral zdirection with respect to the undeformed beam (the symmetry axis of which corresponds to the x-axis) is denoted as w - see Fig. 2. Applying the control voltage - V+ ~ u ~ V+ leads to a bending of the structure in the (x, z)-plane. All piezoelectric layers are shaped, in other words the
P(s) = cos
FJS sinh FJS -
cosh
FJS sin FJS -
(_js)3/2 (cosJ-jscoshJ-js+
1) (3 )
=
and j A. A hat refers to the corresponding Laplace transform. In section 3 will be shown that all poles and zeros of a Hamiltonian system lie
764
Fact 1. The poles and zeros of P are purely imaginary. The number of poles is given with 2nH and the number of zeros is 2 (nH - 1) . Moreover, there are no poles at the origin s = o.
on the imaginary axis. Hence, any reasonable low order model PI of P has the form
In (1 +
R (s)
1 k=1
="3
I
IT k=1
(S/W Z ,k)2)
(1 + (S/W P,k)2)
'
1 EN.
(4) To show this, the energy conservation of Ho is taken into account. Thus, all poles of P are located on the imaginary axis. Moreover, V > 0 and C > 0 implies that there are no poles at the origin. Next, the output is differentiated twice which gives Y. = eT·q = eTC p, ii = -e[CVq + e[Celu.
with the purely imaginary zeros jWz,k and poles jWp,k. The numerical values for the two poles and the first zeros are given as Wp,l = 3.52, W p ,2 = 22.03, Wz,l = 15.42. The expression (4) indicates that transfer functions of Hamiltonian systems satisfy some special properties. This fact will be pointed out in detail in the next section. The investigations are restricted to finite dimensional systems which are required for the controller design.
l
(8) The second relation in Eq. (8) offers a way to calculate u as a function of ii. Hence, the relative degree of P is two and the number of zeros is 2 (nH - 1). The location of the zeros follows from the fact that the free inverse system of a Hamiltonian system is a Hamiltonian system, too - for the general nonlinear case see (Nijmeijer and Van der Schaft, 1991 ). To show this the input y of the inverse system is chosen with y = 0 which leads to the restrictions ql = 0, tll = 0 and Cp = O. The latter restriction states a relation for the coordinate PI as a function of the coordinates P2 , . .. PnH· Next, the coordinates ql and PI are eliminated from the Hamiltonian (5). Let q and 'j5 be the new vector of generalized coordinates and generalized momenta, respectively. ij consists of the coordinates qi, i E {2, .. . ,nH} of q and 'j5 of the coordinates Pi, i E {2, ... ,nH } of p. By a suitable partition of C and V the Hamiltonian of the free inverse system can be obtained
3. PROPERTIES OF HAMILTONIAN SYSTEMS The attention is focused to a special class of single input, single output, and linear Hamiltonian control systems with a Hamiltonian of the form
H(p,q,u)
er
= Ho -yu 1 1 = _pT Cp + _qT'y' q _ yu
2
(5)
2
with positive definite matrices C and V. Ho is the Hamiltonian of the free system and given by the sum of kinetic and potential energy, q is the vector of the generalized coordinates, and p denotes the vector of generalized momenta. The system has nH degrees of freedom. A duality is defined between the input u (a generalized force) and the output y in such a way that y is equal to the corresponding generalized coordinate q;, i E {I, . .. ,nH}. A motivation for this choice can be deduced from the fact that the rate of Ho due to the motion of the system is given as
dHo
.
ill =uq;
l
fl,o = -:? I-T(C 22
-
CTC - lC) -+ 2 I q-TV22q· 12 11 12 P
Remark 2. The Nyquist plot of P is on the real axis of the complex plane. This is a consequence of fact 1. Further, let v = y then the Nyquist plot of the transfer function 1; (s) /u (s) = sP is on the imaginary axis.
(6)
see e.g. (Nijmeijer and Van der Schaft, 1991). Hence, the increase of the internal energy is equal to the total external work performed on the system. This is usually called the case of collocated sensors and actuators and y is called the natural output. The corresponding Hamiltonian equations of motions are given in state-space form as
Remark 3. The positive definiteness of C and V implies that the internal energy Ho is a positive definite function. Definition 1. A dynamic system is said to internally passive if there exists a non negative storage function E such that
Jy(t)u(t)dt~E(x(T))-E(x(O)) T
ei denotes the i-unit vector and the number i specifies the collocated output y = qi. For simplicity, a reordering is made such that y = ql. The following properties are of importance for the proposed controller design. Let P = fj/u be the transfer function of the Hamiltonian system (7).
(9 )
o for Vu E £2, "IT
~
o.
Fact 2. Assuming that Ho is pC~lltlve definite, then the Hamiltonian system will be passive.
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To show this, the equation dHo/ dt integrated which gives
J
uq
is
are simple and their residues are non negative and c) Re(G(jw)) ~ 0 for all w E (-00,00).
(10)
Strict passivity of a linear time invariant system is equivalent to strict positive realness.
T
y (t) u (t) dt = Ho (T) - Ho (0) .
o Thus, the internal energy Ho is a suitable storage function - see remark 3 - and the comparison of the equations (9) and (10) proves the fact 2. An interesting feature is that the inequality is replaced by an equality. The fact 2 has immediate consequences on the £z--stability of the control loop which will be discussed in the next section.
Definition 3. A BIBO-stable transfer function G is said to be strictly positive real if and only if Re (G (jw)) > 0 for all w E (-00,00). The following stability result is sufficient.
Theorem 1. (Vidyasagar, 1993) The feedbacl< system in Fig. 4 is £z--stable if the following conditions are satisfied: a) G is positive real and b) H is strictly positive real.
3.1 Stabilization of Hamiltonian Systems The rate of the internal energy of a Hamiltonian control system is given by equation (dHo/dt = uq. Hence, the D-control law u = -k"v leads to a decrease of the internal energy by dHo/dt = -k v v 2 , with kv > O. Now, a control law is assumed of the form it,
= -kyf) -
CJ)
with
Let G be given as G = sP with P as the transfer function of the Hamiltonian control system (7). In spite of fact 2 - see also fact 1 and remark 2 - the positive realness of G is deduced. Consequently any strictly positive real Cv and ky = 0 stabilizes the feedback system in Fig. 3. But what happens in the case ky =F 0 ?
ky En, Cv En (8). (11 )
R (s) denotes the set of all transfer functions over the real numbers R. In the following sufficient conditions are stated such that the control law (ll) stabilizes the feedback system given in Fig. 3. Here, d denotes the disturbance signal. First,
Fact 3. The feedback system given in Fig. 3 is £z--stable if Cv is strictly positive real and ky is non negative. To show this, a control law of the form is assumed u = -kyY - z, with ky ~ 0 using the natural output Y = ql ' Then, the system - see equation
(7) -
[~] = [-(V+~yelen ~] [~] [e~]
Z
can be obtained with the new Hamiltonian of the free system
Fig. 3. The proposed controller scheme.
IT IT 1 2 Ho='2 P GP+'2 q Vq+'2 kyq1 '
consider the simple feedback system in Fig. 4, with G, HEn (8). A well known result is that
Hence, the feedback of ql leads to a modified potential energy. Due to V > 0, ky ~ 0, the new system is another Hamiltonian system and the feedback of ql does not destroy the £z-stability.
Remark 4. Passivity is a physical property of a system and independent of changes of parameters. Moreover, any reasonable low order model of an infinite dimensional Hamiltonian system is itself a Hamiltonian system - see equation (4).
Fig. 4. A simple feedback system. the feedback interconnection of a passive system and a strictly passive finite-gain system is £2stable. Passivity of a linear time invariant system is equivalent to positive realness.
Remark 4 offers a way for the £z-stabilization of infinite dimensional systems based on the control law (11) and the control scheme as given in Fig. 3. For this matter a low order Hamiltonian model is required, for example the transfer function P2 as
Definition 2. A transfer function G is said to be positive real if and only if a) G is analytic in Re (s) > 0, b) the poles on the imaginary axis
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given in equation (4 ). Then any strictly positive real Cv and ky 2: 0 leads to a £2- stable control system. Moreover, the original system P of infinite dimension is stabilized, too.
roots of the Hurwitz polynomial n" ( -s) satisfy the property Re (Si ) = - 0' . Due to the coprimeness of d and n there exists a solution of the Diophantine equation (14). Moreover, the minimal degree solution is unique.
4. CONTROLLER DESIGN
Remark 6. The proposed design can be applied to any transfer function P. In general, this would give improper and therefore non realizable Cv. If P corresponds to a Hamiltonian system, then we get a proper Cv of order 2nH - 2 , where nH is the number of freedoms .
In this section the controller design procedure is stated. The algorithm is presented and some important properties of the controller will be discussed. The proof of the design is beyond the scope of this paper and is presented in (Haas et al., 1999). Note, [s] denotes the set of polynomials in s.
n
As mentioned in section 3.1 there is particular interest in strictly passive Cv . Of course, the passivity property has not been taken into account for the derivation of the controller design algorithm . Nevertheless, the following fact holds.
The aim of the design is to find values for Cv E (s), ky E such that the 1i2--cost function
n
n
J(u + q~y2 + q~v2) e 00
J =
inf
C v ,k.
2
2" tdt
(12 )
Fact 4. For a certain choice of the input parame0', qy and qv there exist strictly passive Cv
o
ters
is minimized with respect to the impulse disturbance response. Here, 0' 2: 0 denotes the shift factor and qy 2: 0, qv > 0 are weights. The corresponding control configuration is shown in Fig. 3. The controller design starts with the transfer function P of a Hamiltonian control system. In the case of infinite dimensional systems P is replaced by a suitable low order approximation A. The result of the optimization task (12) leads to the following algorithm.
To show this, the special choice qv > 0, qy = 0' = 0 is the starting point. This gives n (-s) = n (s), d ( -8) = d (s) and the spectral factor a follows from
a (-s) a (s) -
a(8)
Algorithm for the controller design:
+ q; (s -
d" (-8) d" (s)
+ n" (-s) n" (s) q (s) .
f (s) = n (8)
and
e (8) = qv8n (8) .
Finally, the controller follows with ky = 0 , Cv = qv which is a D- controllaw and strictly passive. The controller depends on the parameters 0', qy and qv in a continuous way. Hence, for small 0' > 0, qy > 0 strictly passive controllers can be expected, too.
(13)
Remark 7. For small 0' > 0, qy > 0 the polynomials e, f are almost equal to the numerator of the shifted system P". Thus, the controller introduces additional dynamics which corresponds to the zeros of the Hamiltonian system.
• Diophantine equation: Find the minimal degree solution e, fEn [s] of
a(s)n"(-8)
+ e (8) n (8)
with solution
• Spectral Factorization: Calculate the Hurwitz polynomial a from
=
= d(s) + qvsn (8).
(d (8) + qvsn (8)) n (8) = f (8) d (8)
0')( -s - 0') .
a(-8)a(8)
with
Inserting this relation in Eq. (14) leads to
• Input: Specify P = nld E n (s) and the quantities 0', qy and qv' • Shift operation: Substitute 8 by 8-0'. This gives a shifted P" = n" Id" E n (8) and calculate the polynomial
q (8) = q~
see Eq. (13) -
= d2 (s) - q;s2n2 (s)
= f(8)d" (8) +e(8)n"(s). (14 )
• Calculate the controller with . e (8) g (8) (8 )' v - f (8) ky=hm.-0
c _
f
5. APPLICATION (CONTINUED)
where g is given as e (8) - kyf (s)
= 8g (8) .
In section 2 an application of a infinite dimensional Hamiltonian control system was stated. The following considerations are devoted to the controller design. The proposed design requires a low
Remark 5. All poles and zeros of P" are located in the closed right half plane - see fact 1. The
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order model of the transfer function P and it turns out that
P2=~
x 10" 1.5 , . - - - - , - - - - . - - - - . - - - - - ,
l+(~f
3(1 + (3.~2) 2) (1 + (22"03) 2)
0.5
is suitable - see Eq. (4) . Two designs are included to demonstrate some properties of the controller design.
o -0.5
• Controller 1: The choice qv = 0.5, qy = 10, a = 1 of the design parameters gives the controller ky = 5.92, 5
Cv = 2.27
( 18 57) 5
C6 67)
2 2
-I
-1.5 0
5
1 + 3.89 . 10- 5 15 42
2
function of the Hamiltonian system . Moreover, a proper choice of the design parameters leads to strictly passive controllers. This allows a robust stabilization of infinite dimensional systems, too. Finally, an application shows some features of the design .
+ (15 42) 2 2' 5 1 + 2.59 . 10- 15~42 + ( 15 42) 5
=1
1.5
Fig. 6. Impulse response of the different controllers.
+ 0.20 18 57 + 1 . + 0.2516567 + 1
• Controller 2: The choice qv = 1, qy = 0, a = 0.0001 of the design parameters gives the controller ky ~O, Cv
0.5
5
5
7. REFERENCES
which is almost a D- control law .
Green, M. and D. Limebeer (1995). Linear Robust Control. Prentice Hall. Haas, W., A. Kugi, K. Schlacher and M. Paster (1998). Experimental results of the control of structures with piezoelectric actuators and sensors. In: Proceedings of the MOVIC 98, Zurich. Haas, W., K. Schlacher and Irschik H . (1996). Control of multi-layered structures via special distributed piezoelectric actuators. In: Proceedings of the First European Conference, Barcelona. Haas, W., M. Krommer and Irschik H. (1999). Design of dynamical controllers for linear hamiltonian systems. In: Proceedings of DETC'99/VIB-8009, 1999 ASME Design Engineering Technical Conferences, Las Vegas. Hill, D . and P. Moylan (1995). Stability of nonlinear dissipative systems. IEEE Transactions on Automatic Control 21(5), 708-711. Kelkar, A. and S. Joshi (1996). Control of Nonlinear Multibody Flexible Space Structures. Vol. 221 of Lecture Notes in Control and Information Sciences. Springer . Morris, K. and J. Juang (1995). Dissipative controller designs for second order systems. IEEE Transactions on .4utomatic Control 39(5),105&-1063. Nijrneijer, H. and A. Van der Schaft (1991) . Nonlinear Dynamical Control Systems. Springer. Vidyasagar, M. (1993). Nonlinear Systems Analysis. Prentice Hall.
To show the strict passivity (strict positive realness) the Fig. 5 presents the Nyquist plots of the Cv. The performance of both controllers is shown imaginary part
real part ·1
.(!,
fJ
(J ;;
Fig. 5. Nyquist plots of Cv' using the time responses - see Fig. 6. Here, the transfer function P6 is taken into account for the simulation.
6. CONCLUSION A controller design was introduced which uses the special properties of a Hamiltonian control system. The design is based on a 1i2 -optimization problem and the related algorithm uses the frequency domain approach. The result is a controller with "PIM3tructure" but additional internal dynamics. The internal dynamic of the controller corresponds to the zeros of the transfer
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