Abstract. The vibration control of structures may be approached in many ways. Often a robust controller is difficult to design as spillover from modes not ...
Smart Mater. Struct. 8 (1999) 285–291. Printed in the UK
PII: S0964-1726(99)02637-3
The relationship between positive position feedback and output feedback controllers Michael I Friswell† and Daniel J Inman‡ † Department of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, UK ‡ Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0261, USA Received 3 August 1998 Abstract. The vibration control of structures may be approached in many ways. Often a
robust controller is difficult to design as spillover from modes not considered in the controller design can cause the closed loop system to be unstable. One great advantage of positive position feedback (PPF) control is that the frequency response of the controller rolls off quickly, making the closed loop system robust to spillover. Unfortunately methods to design a PPF controller are not as advanced as other approaches in the control literature. This paper shows that a PPF controller may be formulated as an output feedback controller, and both centralized and decentralized controllers are considered. Optimal control methods are used to demonstrate how control design algorithms for output feedback systems may be used to design PPF controllers. The method is demonstrated using a single-degree-of-freedom system.
1. Introduction
The major problems associated with the control of flexible structures arise from two sources: the structure must be considered as a distributed parameter system with many modes in the frequency range of interest, and there are likely to be many actuators and sensors on the structure. The high modal densities give rise to the well known phenomenon of spillover (Balas 1978), where contributions from unmodelled modes affect the control of the modes of interest. The effects of spillover are worse where the structure, sensors or actuators are poorly modelled and the numbers of sensors and actuators are low. Positive position feedback (PPF) control was introduced by Goh and Caughey (1985) as a robust control solution as it is not sensitive to spillover, is not destabilized by finite actuator dynamics and stability may be assured by consideration of the structure’s stiffness properties only. In the single-degree-of-freedom form PPF introduces a second order auxiliary system, which is forced by the displacement response of the structure. The displacement response of the auxiliary system is then fed back to give the force input to the structure. The controller is essentially a second order compensator, and conditions to ensure the closed loop system is stable depend only the natural frequency of the structure, which is easily and accurately measured. The transfer function of the controller rolls off 0964-1726/99/030285+07$19.50
© 1999 IOP Publishing Ltd
quickly at high frequencies making the approach well suited to controlling the first mode of a structure with well separated modes, as the controller is insensitive to the un-modelled high frequency dynamics. An underdamped controller will mainly respond near to the controller natural frequency, making it possible to control higher modes of the structure, possibly using independent modal space control (Meirovitch and Baruh 1982). One disadvantage of applying a PPF controller is that the control makes the structure more flexible, which can lead to larger steady state errors (Friswell et al 1997). Baz et al (1992), Baz and Poh (1996) and Hwang et al (1996) used first order auxiliary systems to control the structure in modal space. Ensuring stability in these cases requires consideration of the closed loop poles via the Routh–Hurwitz criterion. The great advantage in using a second order auxiliary system is a symmetry with the physical system, that allows a physical interpretation of the controller and more easy implemented stability checks. Although the first order auxiliary systems work fine for a single-degree-of-freedom structure, they are more difficult to apply independently to each of the modal co-ordinates in modal space control. The response of a first order compensator will roll off at high frequency, but the effect on any un-modelled low frequency modes will be higher than the mode of interest. Thus these controllers lose the advantage of inherent robustness to spillover from the low frequency 285
M I Friswell and D J Inman
modes, and it is difficult to design controllers for each mode independently.
2. The single degree of freedom case
Suppose a structure may be modelled as the single-degreeof-freedom, single-input–single-output (SISO) system, x¨ + 2ζ ωn x˙ + ωn2 x = bu
(1)
where x is the state of the system and u is the control input. It is assumed that the response x is measured. Equation (1) is written in modal form where ζ and ωn are the damping ratio and natural frequency of the structure, and b determines the level of force into the mode of interest. The positive position feedback (PPF) control is implemented using an auxiliary dynamic system as √ 2 2 q = gωnf x q¨ + 2ζf ωnf q˙ + ωnf √ g 2 u= ω q b nf
Consider the multi-degree-of-freedom equations of motion of a structure, where the sensor outputs and force inputs are grouped. This grouping allows the implementation of decentralized control, where the local actuator force is derived solely from the local sensor outputs. The equations of motion for this structure are derived and then the conditions for stability are discussed. The equations of motion for centralized control may be generated as a special case, by taking just a single group.
3.1. Equations of motion The motion of the structure is described by the following equations, Mx ¨ + Dx˙ + Kx =
nc X
Bi u i
i=1
(2)
where ζf and ωnf are the damping ratio and natural frequency of the controller and g is a constant. The particular form of equation (2) is convenient for the subsequent analysis. The appearance of the controller natural frequency on the right-hand side of equation (2) is not necessary, and may be eliminated by scaling of the controller degree of freedom, q. All possible single-degree-of-freedom auxiliary systems are covered by the general form of equation (2). Bang and Agrawal (1994) and Friswell et al (1997) also added a term in the measured structure response to the input force in equation (2). Combining equations (1) and (2) gives the equations of motion in their usual form, which are, assuming no external force, 0 x˙ x¨ 2ζ ωn + 0 2ζf ωnf q˙ q¨ √ 2 − gωnf ω2 x 0 = . (3) + √n 2 2 q 0 − gωnf ωnf The ‘stiffness’ matrix in equation (3) is symmetric, which is a consequence of the scaling of the auxiliary degree of freedom. This is a stable closed loop system if the ‘stiffness’ matrix is positive definite, that is if the eigenvalues of this matrix are all positive. This condition is satisfied if and only if 2 < ωn2 . gωnf
3. A multi-degree-of-freedom structure
(4)
Notice that the stability condition depends only on the natural frequency of the structure, and not on the damping. This condition may also be derived from the determinant of the symmetric stiffness matrix (Inman 1995). Alternatively stability may be formally proved using the Routh–Hurwitz criterion (Caughey 1995).
y i = Ci x
for i = 1, . . . , nc .
Here x consists of the degrees of freedom of the structure, and M, D and K are the usual mass, damping and stiffness matrices. The actuators and sensors are split into nc groups, so that separate, decentralized controllers may be designed. The inputs to the structure from the ith controller, ui , and outputs from the sensors for the ith controller, yi , are shown as vectors, although the case of a series of SISO controllers is not excluded. The structure input matrix Bi distributes the ith force to the required degrees of freedom, and Ci is the structure’s output matrix. If the actuators and sensors are collocated then Bi = CTi . The PPF approach to the controller design requires nc auxiliary systems given by Mi q¨i + Di q˙i + Ki qi = Bci yi
ui = Cci qi
(6)
where all the matrices in equation (6) are chosen during the controller design. The controller for each sensor/actuator group may have different orders, that is the number of degrees of freedom in each auxiliary system may be different. If the damping in these auxiliary systems (controllers) is assumed to be proportional then, without loss of generality since a modal transformation may be applied to equation (6), the auxiliary mass, damping and stiffness matrices may be assumed diagonal, with the mass matrix as the identity (if the modes are mass normalized). Also the input and output matrices, Bci and Cci , for the controller are not uniquely determined as the force input to each controller mode may be arbitrarily scaled. In this way the number of parameters that may be chosen is reduced to the minimum required, as the full problem will have non-unique parameters. Combining equations (5) and (6) gives, Mx ¨ + Dx˙ + Kx =
nc X
Bi Cci qi
i=1
Mi q¨i + Di q˙i + Ki qi = Bci Ci x for i = 1, . . . , nc . 286
(5)
(7)
Relationship between PPF and output feedback controllers
In matrix form equations (7) become, assuming no external force, x¨ x˙ M D q¨ q˙ M1 D1 1 1 M2 D2 q¨2 + q˙2 . . .. .. .. . . . . Mnc Dn c q¨nc q˙nc K −B1 Cc1 −B2 Cc2 . . . −Bnc Ccnc x q K1 0 ... 0 −Bc1 C1 1 −Bc1 C2 0 K1 ... 0 q2 = 0. + . .. .. .. .. .. . . . . . . . −Bcnc Cnc 0 0 ... Knc qnc To enable the analogy with a completely mechanical system the ‘stiffness’ matrix in equation (8) should be symmetric. This occurs if the controller ‘input’ and ‘output’ matrices satisfy Bci Ci = CTci BTi . (9) For structures with collocated sensors and actuators, this occurs if the controller ‘inputs’ and ‘outputs’ are also collocated, that is Bci = CTci . 3.2. Stability The form of equation (8) is significant. The mass and damping matrices are block diagonal and consist of the mass and damping matrices of the structure and controllers. Since these will be chosen to be positive definite (or possibly semi-definite damping) the whole system will be stable if the stiffness matrix of the whole system is positive definite (Fanson and Caughey 1990). Thus the closed loop system is stable if K −B1 Cc1 −B2 Cc2 . . . −Bnc Ccnc K1 0 ... 0 −Bc1 C1 0 K2 ... 0 > 0. −Bc2 C2 . . . . .. .. .. .. . . . 0 0 ... Kn c −Bcnc Cnc (10) Equation (10) provides a test for a given PPF controller to ensure stability. The influence of the controller is governed by the magnitude of the ‘input’ and ‘output’ matrices, Bci and Cci . If these are zero, then the system stiffness matrix is diagonal, and assuming the controller stiffness matrices, Ki , are positive definite the system is stable. As the magnitude of the controller ‘input’ and ‘output’ matrices is increased, the system becomes more flexible and a point will be reached where the closed loop system is unstable. This occurred in the single-degree-of-freedom case when the controller gain, g, is increased and violates the condition given in equation (4). 4. PPF as an output feedback controller
PPF is a compensator with a pre-determined structure that has some advantages as a controller that were outlined earlier. Although the design of PPF controllers in the single-degreeof-freedom case is quite straightforward, the design methodologies for multi-degree-of-freedom structures are not well developed. This section aims to cast a PPF controller in state space form, where the auxiliary degrees of freedom
(8)
are incorporated into the state vector. In this way PPF control becomes equivalent to output feedback, and standard control methods may then be applied to obtain the parameters of the PPF controllers. 4.1. The single-degree-of-freedom case Equations (1) and (2) may be written is the standard state space form, in terms of the ‘state’ z , the ‘output’ y and the ‘control’ uˆ , as, z˙ = Az + Buˆ (11) y = Cz where x x˙ √ z= 2 √gωnf q 2 q˙ gωnf 0 1 −ωn2 −2ζ ωn A= 0 0 0 0 " # 1 0 0 0 C= 0 0 1 0 0 0 0 1
y
0 1 0 0
√ x 2 gωnf q = √ 2 gωnf q˙ 0 0 0 0 B= 1 0 0 1
and the control law (full output feedback) is 4 uˆ = [ gωnf
2 −ωnf
−2ζf ωnf ] y .
(12)
Notice that the auxiliary degree of freedom has been scaled so that all the unknown parameters are contained in the feedback gain matrix, and thus the matrices for the state space model are all known. The unknown parameters, g, ωnf and ζf , only appear in the feedback gain matrix, given in equation (12). Furthermore there is a unique correspondence between this gain matrix and the PPF controller parameters, providing the signs of the elements in the gain matrix are correct. 4.2. The multi-degree-of-freedom case Assume that the controller ‘output’ matrices Cci are fixed. This is not as restrictive as it may appear. If Cci is square then we may transform the auxiliary states so that the output matrix is the identity. For the case where the controller input and output are collocated, the transformed mass matrix will 287
M I Friswell and D J Inman
be symmetric. The equations of motion (8) may be written in the standard state space form for decentralized control as z˙ = Az +
nc X
ˆ i uˆ i B (13)
i=1
ˆ iz yˆ = C
for i = 1, . . . , nc
x x˙ q1 z = q˙1 . .. q˙nc
0 −1 −M0 K 0 0 0 .. .
0 0 0
yˆ i =
I
ˆ1 = C "
C1
0 0 C2
0 0
0 0 0 0 0 0
0 0
yi qi q˙i
)
0
M−1 B2 Cc2
I
0 0 0 .. .
0 0 0 0 .. .
0 I
0 0
0 0 0 0
0 0 0
I
0 0 0 0 I
0 0 0 0 0
0
I
0 0 0 0 I
0 .. .
0 0 0 ˆB2 = 0 0 I .. .
0 0 .. .
"
ˆ2 = C
(
I 0 −M−1 D M−1 B1 Cc1 0 0 0 0 0 0 0 0 .. .. . .
ˆ1 = B
... ... ... ... ... ... .. .
etc
# ... ... ... # ... ... ,.... ...
where the unknown gain matrices Gi contain the unknown parameters of the auxiliary system, and are given by
−1 −1 Gi = M−1 i Bci − Mi Ki − Mi Di .
Baz and Poh (1996) used optimal control to obtain the optimum controller parameters in an independent modal space approach, based on first order auxiliary systems. They wrote the system and controller equations in state space form, and applied standard optimal control. They formulated the problem as full state feedback where one of the gains was constrained to be zero and optimized the remaining parameters. There are three major problems with the formulation of Baz and Poh. First, having one of the gains as zero is not a standard procedure in optimal control, although this could easily be reformulated as output feedback. More seriously, the weighting matrices for the optimal control were not related to weights applied to the original physical system. Also the approach is difficult to generalize to multidegree-of-freedom, decentralized systems. The equations of motion for the single-degree-offreedom system, with a second order compensator, have already been placed in state space form (equation (11)). Only the second order compensator will be considered here; the equivalent procedure for the first order compensator may be easily generated in a similar way. The standard optimal control method produces the feedback law in equation (12) (that is, gives our parameters g, ωnf and ζf ) so that the following penalty function is minimized: Z ∞ (16) y T Qy + uˆ R uˆ dt J = 0
Notice that all the quantities required to generate these matrices are known. The controllers are assumed to be of the form uˆ i = q¨i = Gi yˆ i (14)
(15)
Thus the standard output feedback control methods will generate the gain matrices Gi , which may then be used to calculate the matrices for the auxiliary systems. In general, there will be insufficient parameters in Mi , Di and Ki to reproduce the gain matrix Gi . Thus the gain matrix must be approximated in some sense, or the form of the gain matrix in equation (15) must be used as a constraint in the controller design algorithm. For the important case of a single sensor and single actuator per group, an exact solution is possible. 288
The idea of optimal controllers in state space formulations is common, for example the linear quadratic regulator (LQR). These methods can be employed to determine the parameters of the auxiliary systems in PPF control. 5.1. The SISO case
where
A=
5. PPF as an optimal controller
where R must be positive in the single-input case. In terms of the original PPF control problem this is equivalent to minimizing Z
∞
J =
αx 2 + βu2 + χ u˙ 2 + δ u¨ 2 dt
(17)
0
where the weights in the standard optimal control problem are α 0 0 1 1 0 Q = 0 b2 β (18) R = 2 δ. b 1 0 0 χ 2 b Dosch et al (1995) and Inman (1995) used the absolute value of system response and control effort rather than the square, and optimized the parameters numerically. Using the cost function given in equation (17) does allow standard optimal control methods to be used. Note that since R 6= 0, some weight must be given to the second derivative of the control effort. This weight has the effect of smoothing the control signal, and may be set to a value based on the speed of the actuator dynamics.
Relationship between PPF and output feedback controllers
The optimal parameters for the general case are obtained by minimizing the following cost function: J =
nc Z X i=1
ζ
5.2. The general case
∞
yiT αi yi +uTi βi ui +u˙ Ti χi u˙ i +u¨ Ti δi u¨ i dt
0
(19) for given positive semi-definite weighting matrices. The terms in the derivatives of the control forces are included based on the experiences above. We have the constraints as the equations of motion (5) and the dynamics of the auxiliary systems, equation (6). If we fix the controller ‘output’ matrices Cci then we can write the equations in state space form with the unknown parameters in the controller gain matrix. Suppose that each group has only a single force input and a single sensor measurement. Then the structure input matrices Bi are column vectors and the structure output matrices Ci are row vectors. We can then assume, without loss of generality (well apart from assuming proportional damping), that the auxiliary systems are written in mass normalized modal co-ordinates, and that the controller output matrices Cci are row vectors where every element is unity. The optimal decentralized control problem minimizes J =
nc Z X i=1
∞
yˆ iT Qi yˆ i + uˆ Ti Ri uˆ i dt.
(20)
0
The weighting matrices for this optimal control problem are, from equation (13), " # 0 0 [αi ] T Qi = Cci [βi ]Cci 0 0 (21) CTci [χi ]Cci 0 0 Ri = CTci δi Cci . 5.3. The solution to the optimal output feedback control problem
δ
Figure 1. The effect of control weight on the optimum PPF compensator damping ratio.
Output feedback control in decentralized systems is more difficult than in centralized systems. Often systems are considered to be interconnected, where the interactions may be considered as disturbances. While this is suitable for many systems considered in the control engineering community where the interactions are weak, structures usually exhibit global modes and strong interactions between the sensor and actuator groups (Friswell et al 1996). In optimal control the decentralized nature of the controller may be formulated as extra constraints on the controller gain matrix. These constraints generally take the form of requiring the gain matrix to be block diagonal, which is equivalent to some of the gain terms being zero. These constraints may be easily implemented using general optimization codes. Sandell (1976) introduced an algorithm for the optimal decentralized control using state feedback. The algorithm of Moerder and Calise (1985) may be extended to the decentralized case relatively easily (Lewis and Syrmos 1995). Jiang and Moore (1996) considered a gradient flow approach to optimizing the gain matrix. 6. A simple example
The solution to the LQR optimal control problem with state feedback is well known. Two problems occur in the design of an optimal controller for the structure considered here: output feedback is required, and a decentralized controller is required. Lewis and Syrmos (1995) consider the output feedback aspects in detail. With full state feedback, the gain matrix is obtained as a closed form solution based on the solution to a Lyapunov equation. With state feedback, the gain does not depend on the initial conditions of the state vector, whereas output feedback requires optimization based on the average performance, or on the worst case performance. With output feedback the gain, and two other matrices, are given by the solution of three coupled matrices, two of which are Lyapunov equations. These equations must be solved iteratively, either by standard optimization methods, or the algorithm of Moerder and Calise (1985). These iterations must be initialized using an initial gain that produces a stable closed loop system, which could be obtained from pole placement, or by zeroing elements in an optimal state feedback controller. For structural dynamics, where some structural damping is present, it may be sufficient to start with a zero initial gain matrix.
The methods outlined in this paper will be tested on a single-degree-of-freedom system, where a PPF controller based on a single-degree-of-freedom auxiliary system is used. Although very simple this system demonstrates the approach proposed in this paper in a very visible way. The system corresponds to equations (1) and (2) in the second order form, or equivalently equation (11) in state space form. The parameters of the system are shown in table 1. Optimal control will be applied with weighting parameters shown in table 1, which correspond to the parameters in equation (17). The weighting of the second derivative of the control force, δ, will be varied to show the effect of requiring less control effort. The gain matrix is computed in state space form using the approach outlined by Lewis and Syrmos (1995). As indicated earlier, optimal control using output feedback may only be solved in an average sense, and requires an expected value of the initial state vector. In the following simulations the system is supposed to be at rest initially, with a disturbance to the displacement only. Other initial conditions may be tried, which would give slightly different gain matrices. For example, if the disturbances to the system were thought to 289
M I Friswell and D J Inman Table 1. Parameters and results for the single-degree-of-freedom example.
System parameters Optimal control weights
PPF controller parameters
δ δ δ δ δ
=1 = 10 = 100 = 1000 = 10 000
ωn = 1 rad s−1 α=1
ζ = 0.01 β=0
b=1 χ =0
ωnf
ζf
g
δ variable R 2 x dt
R
1.0015 1.0014 1.0014 1.0014 1.0014
0.8664 0.8670 1.6351 4.3383 11.511
0.9971 0.9972 0.9972 0.9972 0.9972
1.121 1.122 1.677 3.511 5.739
0.285 0.285 0.145 0.046 0.011
u2 dt
R
u¨ 2 dt
0.562 0.561 0.295 0.103 0.033
Figure 2. The time response for a compensator designed based on
Figure 3. The time response for a compensator designed based on
correspond to impulsive forces, the initial velocity could be assumed non-zero, and the initial displacement zero. Equation (12) is then used to compute the natural frequency, damping ratio and gain for the auxiliary system. The range of the control weighting parameter δ is 1 to 10 000. Throughout this range the compensator natural frequency, ωnf , and gain, g, change very little, and indeed the results shown in table 1 give the complete range of values. The compensator damping ratio changes considerably, as shown in figure 1. For a large range of the lower values of δ the compensator damping ratio is approximately constant, but for larger values the damping ratio increases and the auxiliary system becomes overdamped. Figure 2 shows the time response for a low value of control weight and figure 3 shows the equivalent results for a high value of control weight. The weighting of the control effort is readily apparent.
Indeed, it has been shown that a PPF controller may be formulated as an output feedback controller. The cases of both centralized and decentralized controllers have been considered. If an auxiliary system in the PPF controller has two or more degrees of freedom, then the elements of the corresponding output feedback controller gain are not independent. The application of the optimal output feedback controller to the system has been considered with respect to the implications for the corresponding PPF controller. An example, applying optimal control to a single-degreeof-freedom system, has shown the power of the proposed approach.
control weight δ = 1.
7. Conclusions
This paper has considered the relationship between the positive position feedback (PPF) and output feedback control. 290
control weight δ = 100.
Acknowledgment
Dr Friswell gratefully acknowledges the support of the EPSRC through the award of an Advanced Fellowship. References Balas M J 1978 Active control of flexible systems J. Optimisation Theory Appl. 25 415–36
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