A LINEAR MATRIX INEQUALITY APPROACH TO H ∞ CONTROLLER ORDER REDUCTION WITH STABILITY AND PERFORMANCE PRESERVATION R. Amirifar, N. Sadati Electrical Engineering Department Sharif University of Technology Azadi Ave., P.O. Box 11365-9363, Tehran, IRAN e-mail:
[email protected] Keywords: Order reduction, Robust control, H ∞ controller, LMI optimization.
to be truncated is performed which compensates for the zero frequency contribution of the truncated modes with an additional D-matrix term.
Abstract
More advanced methods use the Hankel singular values to perform a balanced realization of the system to be reduced, such that the controllability and observability gramians become equal [16]. Then the order of the system is reduced employing an optimal Hankel-norm approximation. This can be done by utilizing a minimized H ∞ –norm error bound for difference between the full order and reduced order system [12] or a frequency-weighted Hankel-norm approximation [15]. The problem with these more sophisticated methods is that obtaining a balanced realization of the system may be numerically ill-conditioned. Although by using a descriptor approach or an ordered Schur decomposition method, one can bypass this numerically delicate balanced reduction step [17,18]. A different approach is also pursued by Hyland and Bernstein [13] where the reduced order models are derived using a quadratic criterion which results in two modified Lyapunov equations which are coupled by an oblique projection. A variety of different methods based on these techniques is presented in MATLAB [2,5]. A minor issue is that not all of them posses ability to cope with an unstable controller. A major handicap, however, is the fact that when the order of the controller is reduced, the properties of the closed-loop full order system are not preserved. As a result, the robustness properties achieved with the full order controller are not guaranteed, and the robust performance is lost in most cases. This illustrates the need for alternative ways to reduce the order of controllers while thay guarantee the robust performance.
H ∞ controller order reduction with stability and performance preservation pose unique challenges to the designers. In this paper, a new approach for controller order reduction based on minimization of the rank of a matrix variable, subject to linear matrix inequality constraints, is presented. In this approach, the rank of a residue matrix of a high-order controller subject to the error between the loop gain of the closed-loop nominal system and the loop gain of the closedloop system with the reduced order controller is minimized. However, since solving this matrix rank minimization problem is very difficult, the rank objective function is replaced with the nuclear-norm that can be reduced to a semidefinite program, so that it can be solved efficiently. It is shown that the reduced order controller preserves the performances and stability of the nominal closed-loop system. The proposed approach is applied to an H ∞ highorder controller which is designed for an active suspension system. The performance and stability achieved by the reduced order controller is compared with those achieved by the high-order controller. The comparsion is based on both simulation and experimental results obtained by digital controller implementation.
1 Introduction A major difficulty associated with H ∞ synthesis method is the order of the resulting controllers. They are difficult in hardware and software implements and therefore, the reduced order controllers are easier to be implemented. One possibility to reduce the order of the controller is to apply a model order reduction technique. These techniques remove a specified number of states from a system based on various different approaches. Relatively simple methods transform the dynamics of a system to bidiagonal form in such a way that the elements of the dynamics matrix are ordered by increasing magnitudes. Then the system is either simply truncated to the desired order or a residualization of the states
The newer order reduction methods guarantee the preservation of closed-loop stability and performance. An approach for controller order reduction based on the closedloop identification, was presented by Karimi and Landau [14] where reduced order controller are derived by the minimization of the output error between the closed-loop nominal system and the closed-loop system with the reduced order controller which result, preserves the performances of the nominal closed-loop system. Some progress was also achieved by Enns [10], by introducing the weighting functions into the classical balanced reduction technique, in
order to preserve the closed-loop stability. Anderson and Liu [1] extended this approach to find the weighting functions for the controller order reduction problem that can maintain the closed-loop performance. By directly constraining the performance and stability of the closed-loop system, during the controller order reduction process, these properties are better preserved. Figure 1 shows the closed-loop system where K (s ) is the controller, G (s ) is
the plant model, r (t ) is the reference input, d (t ) is the output disturbance and y (t ) is the system’s output.
T ( s) − Tˆ ( s) will be minimized. The reduced order controller that is obtained from this minimization problem will preserve the performance in disturbance rejection and tracking. This paper consists of 6 sections. In section 2, it is shown how the controller order reduction problem can be formulated as a semidefinite program (SDP) so that it can be solved by using standard existing softwares [9,21]. The stability analysis of the closed-loop system with the reduced order controller is discussed in section 3. Simulation results are given in section 4. In section 5 the results of the real-time experiments are presented. Finally section 6 concludes the paper.
2 SDP formulation reduction problem
Now consider the following norm:
|| S −1 ( s )[S ( s ) − Sˆ ( s )]Sˆ −1 ( s ) || ∞
(2)
|| S −1 ( s )[T ( s ) − Tˆ ( s )]Sˆ −1 ( s ) || ∞
(3)
where S (s ) is the output sensitivity function of the nominal closed-loop system, and Sˆ ( s ) is the output sensitivity function of the closed-loop system with the reduced order controller. T (s) is the transfer function of the nominal closed-loop system, and Tˆ (s) is the transfer function of the closed-loop system with the reduced order controller defined by:
Sˆ ( s) =
1 1 + G ( s) Kˆ ( s)
order
(1)
where Kˆ ( s ) is the reduced order controller. This norm is equal to the following two norms:
1 S (s) = 1 + G (s) K (s)
controller
Let G (s ) be a nominal system and K (s ) be a high-order controller that satisfy the performance property and stability of the closed-loop system. Now consider the family of proper rational controllers given by:
Figure 1: Closed-loop system
|| G ( s )[ K ( s ) − Kˆ ( s )] || ∞
of
Kˆ ( s ) = R0 +
n
Ri
∑ s− p i =1
(8) i
where Ri , p i ∈ C are a set of complex numbers with conjugate symmetry. p i ’s are considered as fixed scalars and residues Ri ’s are the variables that are used for reducing the order of controller. Now, by defining the residue matrix as: R = Diag( Ri )
(9)
The order of a minimal state space realization of the rational function Kˆ ( s ) can be given by: deg[ Kˆ ( s )] = Rank ( R)
(10)
(4)
The goal is to determine the values of Ri that minimize the deg[ Kˆ ( s )] subject to some set of preserving performance and some constraints.
(5)
Now, in order to preserve the performance, the reduced order controller should satisfy the following condition: || G ( s )[ K ( s ) − Kˆ ( s )] || ∞ ≤ ε
G( s) K (s) T (s) = 1 + G (s) K (s )
(6)
G ( s ) Kˆ ( s ) Tˆ ( s ) = 1 + G ( s ) Kˆ ( s )
(7)
(11)
This condition can be approximated by the following LMIs:
Eqs. (2) and (3) show that by minimizing a weighted norm of K ( s ) − Kˆ ( s ) in (1), a weighted norm of S ( s ) − Sˆ ( s ) and
| G ( jω k )[ K ( jω k ) − Kˆ ( jω k )] | ≤ ε ,
k = 1,..., N
(12)
where N is a finite number of points in the frequency range of interest. The controller order reduction problem can then be given by:
min
min
Rank ( R ) | G ( jω k )[ K ( jω k ) − Kˆ ( jω k )] | ≤ ε ,
subject to
k = 1,..., N R j = Ri
Y R ≥0 subject to ∗ Z R ε E ( jω k ) ≥ 0, E ∗ ( jω ) ε k
(13)
p j = pi
for
where the optimization variables are the Ri ’s. Note that G ( jω ) Kˆ ( jω ) is a linear function of the R ’s. k
R ∈ Co R j = Ri
Definition 1 The nuclear-norm of a matrix R is defined as: N
|| R || ∗ =
∑σ
i ( R)
(14)
(18)
k = 1,..., N
i
k
Trace(Y ) + Trace(Z )
for
p j = pi
where R ∈ C n×n , Y = Y ∗ ∈ C n×n , Z = Z ∗ ∈ C n×n . Also:
i =1
E ( jω k ) = G ( jω k )[ K ( jω k ) − Kˆ ( jω k )]
where σ i (R) denote the singular values of R . In [11], it is shown that by solving the nuclear-norm minimization problem, a lower bound on the optimal value of the rank minimization problem can be obtained. Therefore, the problem in (13), can be expressed as a heuristic problem given by: min subject to
t || R ||∗ ≤ t | G ( jω k )[ K ( jω k ) − Kˆ ( jω k )] | ≤ ε ,
(15)
k = 1,..., N R j = Ri
for
(19)
Many other conditions on the reduced order controller can be considered as LMIs in the constraint set C o .
3 Stability analysis Theorem 1 Assume that K ( s ) is a high-order controller achieving stability of the closed-loop nominal system. If: i) K ( s ) and Kˆ ( s ) have the same number of poles in
Re( s ) > 0 and no poles on the imaginary axis; and ii)
p j = pi
|| G ( s )[ K ( s ) − Kˆ ( s )] || ∞ < || S ( s ) || ∞ −1
The heuristic problem (15) is a convex problem and can be handled using a variety of convex optimization algorithms. Now, the following lemma will be used to express this problem as an SDP, when the constraints are given by LMIs. The advantage of such a formulation is that available SDP solvers can be used to readily solve the problem.
(20)
then Kˆ ( s ) stabilizes G (s ) . Proof : Regarding the closed-loop system with Kˆ ( s)
replacing K ( s ) as being equivalent to that of Figure 2 .
Lemma 1 The following problems are equivalent: i) min t
subject to
where
X ∈C
m× n
|| X || ∗ ≤ t X ∈ Co
(16) Figure 2: Rearrangement of closed-loop system with reduced order controller.
and t ∈ R are the variables.
It can be concluded using this redrawing [1,6] (and it is now well known ) that if:
ii) min
Trace(Y ) + Trace(Z )
X Y subject to ∗ ≥0 Z X X ∈ Co
(17)
where X ∈ C m×n , Y = Y ∗ ∈ C m×m and Z = Z ∗ ∈ C n×n are the variables, and the constraint set C o is expressed as LMIs. See [11] for the proof of this lemma. Using lemma 1 and Schur complements [3], the controller order reduction problem can be expressed as following SDP:
||
1 G ( s )[ K ( s ) − Kˆ ( s )] || ∞ < 1 1 + G (s) K (s)
(21)
then Kˆ ( s ) stabilizes G (s ) . Now, by invoking the H ∞ -norm submultiplicative inequality: ||
Hence,
1 || ∞ . || G ( s )[ K ( s ) − Kˆ ( s )] || ∞ < 1 1 + G (s) K (s)
(22)
|| G ( s )[ K ( s ) − Kˆ ( s )] || ∞ < || S ( s ) || ∞ −1
(23)
Remark 1 Condition i) can be considered in the constraint set C o in (18) as:
Ri ≠ 0 for Re( p i ) > 0 , Kˆ ( p ) ≠ 0 for Re( p ) > 0 i
i
(24)
Remark 2 The approximation problem posed is not fully appropriate for controllers with unstable or poles on the imaginary axis. Consider a controller containing a pure integrator. The approximation problem posed demands that any approximation also contain a pure integrator with preciesly the same residue. This show that the approximation problem is in some way unnecessarily restrictive, since a good approximation in practice need not meet this requirement.
Figure 4: Input sensitivity constraint (solid), and inverse of the input sensitivity weighting function (dashed-dot).
4 Simulation results In this section, the proposed approach is illustrated by using a continuous-time model, namely G1 ( s ) , of EJC benchmark problem.
Figure 5: Output sensitivity constraint (solid), and inverse of the output sensitivity weighting function (dashed-dot).
4.1 Model reduction
G1 ( s ) consists of several vibration modes, whereas the first mode around 200 rad / sec , and the second mode around 1020 rad / sec , are the most important ones. Considering the effects of these modes when designing the controller, a loworder model ,namely G 2 ( s) , is obtained by truncating the high frequency modes. Figure 3 shows the magnitude Bode plot of G1 ( s ) and G 2 ( s ) .
Figure 3: High-order (nominal) model G1 ( s ) (solid), and low-order model G 2 ( s ) (dashed-dot).
4.2
H ∞ controller designing
The control objective is presented in terms of the constraints for the closed-loop sensitivity functions. The magnitude Bode plot of the inverse of the weighting functions and the constraints on the output and input sensitivity functions are shown in Figures 4 and 5. The resulting continuous-time high-order H ∞ controller K ( s ) is of the 12 th order and it approximately achieves the performances and stability of the nominal closed-loop system. Figures 6 and 7 show the magnitude Bode plot of the closed-loop sensitivity functions.
Figure 6: Output sensitivity constraint (solid), and output sensitivity function (dashed-dot), with 12 th order continuous-time controller .
Figure 7: Input sensitivity constraint (solid), and input sensitivity function (dashed-dot), with 12 th order continuous-time controller.
4. 3 Controller order reduction
In this section, the proposed reduction technique is demonstrated on the 12 th order controller K ( s ) , obtained from H ∞ controller design. The high-order continuous-time model G1 ( s ) is used, which is normalized so that || G1 ( s ) || ∞ = 1 . The poles of K ( s ) appear in five complex conjugate pairs and two real numbers. Also, original controller K ( s ) has a zero at z1 = −π Fs , where Fs = 800 Hz is the sampling frequency. By considering the constraint Kˆ ( z ) = 0 as an LMI in the constraint set C , this 1
o
zero can be preserved in the reduced order controller Kˆ ( s ) . The poles of the original controller are kept, and the residue matrix is modified. The goal is to reduce the order, while respecting a loop gain transfer function error and a constraint set C o . YALMIP is used to solve the resulting SDP. The SDP in (18) is solved with ε = 0.05 ( − 26dB ). The result is a 4 th order controller which has a zero at z1 . Figure 8 shows the magnitude Bode plot of | K ( jω ) − Kˆ ( jω ) | which k
performance is approximately preserved with the 4 th order discrete-time controller.
k
−1
satisfies its upper bound, i.e. , | G1 ( jω k ) | , with ε = 0.05 . Figures 9 and 10 show that the closed-loop performances are approximately preserved for the 4 th order controller. It should be noted that at some frequencies where | S ( jω k ) | is small enough, these perfomances are better preserved.
−1 Figure 8: | G1 ( jω ) | (solid), and | K ( jω ) − Kˆ ( jω ) | (dashed).
Figure 11: Output sensitivity constraint (solid), and output sensitivity function (dashed-dot), with 4 th order discrete-time controller ( ω c =950 rad/sec ).
Figure 12: Input sensitivity constraint (solid), and input sensitivity function (dashed-dot), with 4 th order discrete-time controller ( ω c =950 rad/sec).
5 Real-time experimental results The results of the real-time experiments are shown in Figures 13-16. The closed-loop performances are approximately preserved at the same frequencies as considered during the controller design and the controller order reduction process.
Figure 9: Output sensitivity constraint (solid), and output sensitivity function (dashed-dot), with 4 th order continuous-time controller.
Figure 13: Output sensitivity constraint (solid), and output sensitivity function (dotted), with 12 th order discrete-time controller, using real-time experiment ( ω c =1200 rad/sec). Figure 10: Input sensitivity constraint (solid), and input sensitivity function (dashed-dot), with 4 th order continuous-time controller.
4. 4
Controller discretization
The discrete-time controller is obtained by converting the continuous-time controller using the prewarp method. This method converts the continuous-time model to discrete-time by using the bilinear (Tustin) approximation method with frequency prewarping. The critical frequency ω c must be specified. Figures 11 and 12 show that the closed-loop
Figure 14: Input sensitivity constraint (solid), and input sensitivity function (dotted), with 12 th order discrete-time controller, using real-time experiment ( ω c =1200rad/sec).
Figure 15: Output sensitivity constraint (solid), and output sensitivity function (dotted), with 4 th order discrete-time controller, using real-time experiment ( ω c =950rad/sec).
Figure 16: Input sensitivity constraint (solid), and input sensitivity function (dotted), with 4 th order discrete-time controller, using real-time experiment ( ω c =950rad/sec).
6 Conclusion In this paper, a controller order reduction technique using linear matrix inequalities (LMIs) has been presented and implemented on an EJC benchmark problem. The controller order reduction problem is reduced to an SDP, so it can efficiently be solved. In the proposed approach, the reduced order controller preserves the performance and stability of nominal closed-loop system. The fixed parts of the high-order controller which should be preserved in the reduced order controller, and many other conditions on the reduced order controller which can be expressed as LMIs, can easily be treated. In the forthcoming paper presented by the authors, the proposed approach is extended to the multivariable case and the results are compared with those obtained using the other approaches available for reducing the order of a controller.
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