Selection of LQG/LTR Weighting Matrices through Constrained Optimization David Haessig GEC-Marconi Systems, Inc. Wayne, NJ 07474-0932 (201)305-2766
[email protected].
Abstract A method for numerically determining the weighting matrices used in the design of an LQG/LTR compensator is illustrated. A constrained optimization problem is formulated and solved to generate a controlled system that satisfies performance and robustness constraints, and minimizes a frequency-domain performance measure representing a desirable system characteristic. Also shown is that this method can be used to balance the principal gains over a wide range of frequencies.
balancing of the principal gains. In [3] Athens shows how to compute the process noise sensitivity matrix E to balance the principal gains at low and high frequencies, and in [4] Maciejowski uses singular value decomposition to compute the E that balances the principal gains at any specific frequency. The technique being presented here, the constrained optimization of a performance measure involving the principal gains, can in some cases balance the principal gains over a range of frequencies. It does so in the example.
Problem Formulation Introduction During the 80's important contributions were made to the body of control theory dealing with the design of fixed dimensional linear time-invariant compensators for multi-input multi-output systems. Many of these advances were made by integrating time-domain optimization-based approaches (LQG) with frequency-domain approaches. The Linear Quadratic Gaussian with Loop Transfer Recovery (LQG/LTR) methodology pioneered by Stein and others [1] is such an integrated frequency- and time-domain approach. When applying the LQG/LTR procedure, one begins by shaping the principal gains of a Target Feedback Loop (TFL) so that they meet (performance and stability robustness) design requirements that must ultimately be met by the actual closed-loop system. Then a step is taken to cause the principal gains of the actual system to approach or recover those of the TFL. This recovery step involves no design judgment at all. One simply follows the procedure, and recovery occurs. The designers skill comes to bear in the shaping of the principal gains of the Target Feedback Loop. This involves (1) augmentation of the plant model with additional dynamics to effect large changes in the loop shapes and changes in their slope, and (2) the selection of design matrices which also affect the loop shapes more subtly. This last step involves selecting one point from within the region in the design parameter space producing designs that satisfy the design constraints. But which point should the designer choose? He may, for example, want to minimize overshoot in all channels and will hunt around in the design parameter space to find the point that best achieves that goal while not violating the design constraints. In doing this he has, in essence, added an additional optimality criterion that he drives toward an extremum manually. This paper contains an example in which this type of optimization is done computationally, using a constrained optimization algorithm. A similar technique is described in [2]. There is an additional reason why one might want to use a technique like the one illustrated here it naturally results in a
Also a doctoral student at New Jersey Institute of Technology
Performance and Robustness Constraints: The problem is to design an LQG/LTR based compensator for the Rosenbrock system that:
achieves a disturbance attenuation factor of at least 40 dB from DC to 0.1 rad/sec, and
is robust against multiplicative perturbations at the plant output:
G ( I + o ) G where G is the open-loop transfer function and o is a stable perturbation matrix acting at the plant output and bounded by: I O M( ) where M() is one (or zero dB) out to 10 rad/sec and then increases by 20 dB/decade beyond 10 rad/sec. These performance requirements are illustrated in Figure 1. Performance Constraint
40 dB
20
Robustness Constraint
0 -20
.1
1.0 (rad/sec)
10
-20 dB/decade
Figure 1 Performance and Stability Robustness Boundaries Frequency Domain Performance Criteria: In our example, we would like the closed-loop response of the system to approach that of an ideal low-pass filter as closely as possible in all channels. A measure of how closely the closed-loop system comes to approximating this ideal filter is the integral of the difference between the two. Defining the scalar I() as the frequency response of the ideal low-pass filter, and using the standard notation GK to represent the open-loop frequency response, T = GK(sI-GK)-1 to represent the closed-loop frequency response, and
( T ) and ( T ) to represent the maximum and minimum singular values of T, we define the following performance measure:
0 | I( ) - s (T) | d(log )
(1)
where s is the principal gain straying the farthest from the ideal filter at each frequency. The independent variable of integration is set equal to log to equally emphasize low and high frequency deviations from I() as they appear on a semi-log plot. The areas summed by integration are crosshatched in Figure 2.
1 Gain 0
c
Figure 2 -- The frequency-domain performance measure Rosenbrock System: The open-loop plant is linear, timeinvariant, and defined by the following system matrices: 1 1 0 A 0 1 0 0 0 1
1 6 0 B 2 3 1 0 1 2
3 3 4 1 2 C 0 2 1
0 0 D 0 0
where x Ax Bu Ew, y Cx Du v , where and w are additive gaussian white noises, and where E is the process noise distrubition matrix.
Constrained Optimization Definition of optimization problem: Recovery of the loop transfer function principal gains at the plant output involves the shaping of the principal gains of the Kalman filter return ratio, and recovery of these gains through the design of the full-state regulator. Design of the Kalman filter involves the selection of three constant matrices the plant and measurement covariance matrices, W and V, and the plant noise distribution matrix Ea. These matrices contain the parameters that the optimization routine will manipulate in finding the optimum solution. Not all of the elements in Ea , V, and W should be selected as parameters that the optimization routine will vary. The smallest subset of parameters that provides the necessary design freedom should be used, since this simplifies the task that the optimizer must accomplish. We chose to fix the process noise covariance matrix W at I2, and to allow the independent elements of V and the upper 3x2 submatrix of Ea to vary. The lower 2x2 submatrix of Ea was fixed at I2 . This allowed the optimization routine to vary the relative level of measurement to process noise and to vary the relative amount of process noise driving the plant to that driving the augmented integrators. i.e.:
e11 e 21 E a e31 1 0
e12 e 22 v11 e32 V v12 0 1
v12 1 0 W v 22 0 1
The vector of 9 design parameters to be determined through optimization is thus [e11 e12 e21 e22 e31 e32 11 12 22 ]. Initial Feasible Solution. An initial feasible solution was found by setting Ea = [02x3 I2]’, V = I2 , and W = I2. The closest one can come to meeting the design specifications is with = 100. The filter principal gains for this case are shown in Figure 3.
Loop Shaping by Plant Augmentation: A constant gain matrix Go-1 = (CA-1B)-1 is appended to the input of the plant to draw the principal gains together at low frequencies. Also, the plant is augmented with integrator states , x a = [ x ] , to increase the principal gains at low frequencies. This leads to the augmented plant model:
x a = Aa xa + Ea w + Ba u y = Ca xa where:
A BG -1 o Aa = 0 A w
Ca C 0
Ba
BG -1 o 0
E1 Ea = E 2
with Aw = -1x10-4I2 rather than zero to avoid numerical difficulties. It is this augmented system for which the Target Feedback Loop (Kalman filter) is designed.
Figure 3 - Principal gains of the initial feasible solution, the filter return ratio before optimization
The filter gain for this initial feasible solution, with and given by Kf = lqe(Aa , Ea , Ca , 100I2 , I2 ), is:
2.564 3.094 1.825 4.365 K f 2.677 1.800 4.615 8.871 8.871 4.615 The filter return ratio plotted in Figure 3 was computed in accordance with GK C a (sI A a ) 1 K . f
f
Constrained Optimization Solution: The optimal solution was obtained using Matlab’s constrained optimization function constr(), starting with the initial feasible solution given above. The cutoff frequency of the ideal low-pass filter in (1) was set to 5 rad/sec. The solution to which it converged is the following:
.262 .658 .551 1.30 E a .825 1.95 0 1 0 1
.946 .0487 V .0487 .905
The corresponding Kalman gain is:
5.325 - 3.505 10.656 10.740 - 4.908 K f = 8.515 - 10.290 0.2728 0.2778 - 10.524
Figure 4 Principal gains of the filter return ratio after optimization. Note that optimization has balanced the principal gains everywhere. This completes the 1st step in the design of the compensator following the LQG/LTR procedure. The recovery step can be accomplished following the procedure defined in [1,3,4].
Conclusions In this paper LQG/LTR weighting matrices are computed using constrained optimization, minimizing a user defined performance criterion while satisfying performance and stability robustness constraints. This approach is illustrated on the 2 x 2 Rosenbrock System in an example in which the closed-loop frequency response is driven as closely as possible to that of an ideal lowpass filter. A secondary beneficial effect that this technique has is that of balancing the principal gains.
Note what has happened to the principal gains (see Figure 4). They are balanced everywhere! The optimization of a frequency-domain performance measure which penalized the worst-case principal gain quite naturally pushed all of the principal gains together. And the performance and robustness constraints are all met.
Acknowledgments
It is qualitatively obvious that these principal gains minimize Eq. (1), the performance measure. A system having principal gains that roll-off at -20 dB/decade everywhere could satisfy the design constraints. However, to minimize (1) and still achieve the 40 dB performance bound the principal gains must drop more rapidly between 0.5 and 3 rad/sec and then return to nearly -20 dB/decade at the crossover to achieve the guaranteed stability robustness properties of the Kalman filter. Both principal gains have crossover frequencies of 4 rad/sec and bandwidths of 5, which evidently minimizes (1).
References
The idea of using a frequency-domain performance criteria and constrained optimization to define an optimal LQG/LTR design was suggested by Dr. Tim Chang of NJIT.
[1] Doyle, J.C. and Stein, G., "Multivariable Feedback Design: Concepts for a Classical Modern Synthesis,", IEEE Trans. on Auto. Control, Vol. AC-26, Feb. 1981, pp.4-16. [2] Frangos, C. and Yavin, Y, “Design Methodology for Linear Optimal Control Systems,” J. Guidance, Control, & Dynamic Systems, Vol. 15, No. 5, Sept.-Oct., 1992. [3] Athens, M., "A Tutorial on the LQG/LTR Method", Proc. American Control Conference, Seattle, WA, June 1986. [4] Maciejowski, J.M., Multivariable Feedback Design, AddisonWesley, 1990.
