a unity feedback digital controller of fixed order or structure whose parameters need to be determined. Such a problem is common in applications. For example ...
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThCIn4.3
Fixed Order Multivariable Discrete-Time Control L.H. Keel and S.P. Bhattacharyya
Abstract— In this paper we develop a new algorithm to compute stabilizing sets for a fixed order digital controller for a multivariable plant. This computation is crucial in applications and few results are available. Our algorithm is based on a) the Tchebyshev representation of the unit circle image of a polynomial, b) recent results on sign-definite decomposition and c) bounded phase results from Robust Stability. These are combined to capture the unit circle image set of a polynomial over an interval set of controllers. The algorithm can be used recursively to obtain inner approximations of stabilizing sets. Examples are included to illustrate the algorithm.
I. INTRODUCTION We consider a discrete time multivariable plant attached to a unity feedback digital controller of fixed order or structure whose parameters need to be determined. Such a problem is common in applications. For example consider a two-input two-output plant with a PID controller in each error channel. Here the six parameters representing the two controllers need to be determined. A first step in this process is to find the set of stabilizing values or the stabilizing set since every design must lie in this set. Although the above problem is of longstanding interest there are few effective results available for fixed order multivariable controllers. Recent results on this problem includes [1], where a generalization of the KYP lemma designed to be valid over prescribed frequency ranges was developed to deal with fixed order controller synthesis. The H∞ design [2] and the quantifier elimination (QE) approaches [3] are also used to study fixed order controller design problem. In [4], Neimark’s D-Decomposition technique [5], [6] was revisited and applied to design fixed order controllers. The fixed order controller design problem is also attempted by using LMI techniques [7]. In [8], we obtained some interesting results for continuous time systems using some concepts of sign definite decomposition. In the present paper we also use these concepts along with Tchebyshev representations of the unit circle images of polynomials, and vertex results from robust stability to develop an algorithm for determining stabilizing sets. II. PRELIMINARIES We begin with a brief account of some basic results on sign-definite decomposition; more details may be found in [9]. Given a set of polynomial functions of a box of This work was supported in part by DOD Grant W911NF-08-0514 and NSF Grant CMMI-0927664 Center of Excellence in Information Systems, Tennessee State University, Nashville, TN 37209 Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
parameters, the problem of determining whether the set of polynomial functions is positive over the box is known as the robust positivity problem [10], [11]. This is an important problem in designing and analyzing control systems. For example, the Routh criterion determines stability of a control system by examining the signs of the entries of the first column of the Routh table. These entries are typically polynomial functions of parameters of both plant and controller and must be sign invariant over the parameter space box. Let x := [x1 , x2 , · · · , xl ] (1) be a real vector and f (x) be a real polynomial function of x. We consider the problem of determining whether f (x) is positive for all x ∈ X where + X := x : x− (2) k ≤ xk ≤ xk , k = 1, 2, · · · , l .
We can assume without loss of generality that the xi are positive since a coordinate translation can reposition the box X in the first orthant. Introduce the sign-definite decomposition f (x) = f + (x) − f − (x) +
(3)
where f (x) and f (x) are uniquely determined polynomial functions of the parameter x with positive coefficients. Define − − x− := x− 1 , x2 , · · · , xl , + + (4) x+ := x+ 1 , x2 , · · · , xl . −
Based on the above sign definite decomposition, we have the following. More details including proofs are in [8]. Lemma 1 For all x ∈ B, the following inequalities hold: f + (x− ) ≤ f + (x) ≤ f + (x+ )
(5)
f − (x− ) ≤ f − (x) ≤ f − (x+ ).
(6)
The function f can be represented in the (f − , f + ) plane by associating f (x) with the point f − (x), f + (x) as shown below (Figure 1). Consider the rectangle formed by the four points in the (f − , f + ) plane A = f − (x− ), f + (x− ) , B = f − (x− ), f + (x+ ) , C = f − (x+ ), f + (x+ ) , D = f − (x+ ), f + (x− ) .
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ThCIn4.3 f + (·)
f (·) > 0
functions fi (x) are to be robustly positive, one can determine the corresponding Xi+ and find ∩i Xi+ .
L f (·) = 0
III. MAIN RESULTS The controller design problem for discrete-time systems can be effectively solved by using the Tchebyshev representation [12]. Consider a polynomial
f + (x) f (x)
f (·) < 0
p(z) = an z n + zn−1 z n−1 + · · · + a1 z + a0 . f − (x) Fig. 1.
(9)
The Tchebyshev representation is obtained by evaluating the polynomial along the unit circle
f − (·)
p(z)|z=ejθ ,
f + (·) and f − (·) representation
for θ ∈ [0, 2π].
(10)
In the standard literature [12], [13], the Tchebyshev polynomials are obtained by letting u = − cos θ for u ∈ [−1, 1]. In this paper, we use instead From Lemma 1, we have the following results.
u := − cos θ + 1,
+
Lemma 2 For every x ∈ B, (f (x), f (x)) lies inside ABCD (Figure 2): −
f (B)
f + (·) B
u ∈ [0, 2],
(11)
to maintain u ≥ 0. This will be useful when we apply the sign-definite decomposition. Thus, we have p(z)|z=ejθ = p(ejθ )|cos θ=−u+1 =: pc (u)
(12)
pc (u) := Rp (u) + jvTp (u)
(13)
C
and where v := and with A
ck (u) := cos kθ, sin kθ sk (u) := sin θ
D
f − (·) Fig. 2.
p u(2 − u),
(14)
(15)
and
A rectangle ABCD
Rp (u) := an cn (u) + an−1 cn−1 (u) + · · · + a1 c1 (u) + a0 , Tp (u) := an sn (u) + an−1 sn−1 (u) + · · · + a1 s1 (u). Lemma 3 For all x ∈ B, if f + (x− ) − f − (x+ ) > 0, > 0, f (x) < 0, if f + (x+ ) − f − (x− ) < 0.
It can be shown that 1 dck (u) sk (u) := − · , k du ck (u) := −(u − 1)ck (u) − u(2 − u)sk (u).
We can now introduce the following. Recursive Algorithm When the the vertices B and D (Figure 2) lie on opposite side of L (Figure 1), that is, f + (x+ ) − f − (x− ) > 0 f + (x− ) − f − (x+ ) < 0
(7)
and it is not possible to conclude robust positivity or negativity. In this case, the box X can be decomposed into smaller boxes Xk , k = 1, 2, · · · , m so that X = ∪m k=1 Xk
(8)
and the above test applied to each Xk . This can be repeated recursively to generate subsets X + and X − of X such that f (x) > 0 for all x ∈ X + and f (x) < 0 for all x ∈ X − . In general, X + or X − are unions of boxes but are not necessarily box like or even connected. If a number of
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k 1 2 3 4 5 .. .
ck (u) −u + 1 2u2 − 4u + 1 −4u3 + 12u2 − 9u + 1 8u4 − 32u3 + 40u2 − 16u + 1 −16u5 + 80u4 − 140u3 + 100u2 − 25u + 1 .. .
k 1 2 3 4 5 .. .
sk (u) 1 −2u + 2 4u2 − 8u + 3 3 −8u + 24u2 − 20u + 4 16u4 − 64u3 + 84u2 − 40u + 5 .. .
(16)
(17) (18)
ThCIn4.3 Consider the polynomial family
Define
P := {P (z, x) : x ∈ X }
R+ (u) := r¯0 + r¯1 u + r¯2 u2 + r¯3 u3 + · · · R− (u) := r 0 + r1 u + r2 u2 + r 3 u3 + · · · T + (u) := t¯0 + t¯1 u + t¯2 u2 + t¯3 u3 + · · ·
(19)
where X is a box in the first orthant. A typical element of the family is
2
(30)
3
T (u) := t0 + t1 u + t2 u + t3 u + · · · . −
P (z, x) = a0 (x) + a1 (x)z + a2 (x)z 2 + · · ·+ an (x)z n . (20)
Then we have for u ∈ [0, 2], and x ∈ X
Then we have P (z, x)|z=ejθ = P (ejθ , x)|cos θ=−u+1 =: Pc (u, x)
R− (u) ≤ R(u, x) ≤ R+ (u)
(22)
We also take sign-decomposition for ck (u) and sk (u), and we let − ck (u) = c+ k (u) − ck (u), − sk (u) = s+ k (u) − sk (u).
Theorem 1 For every u ∈ [0, 2], the complex plane image set {Pc (u, x) : x ∈ X } is bounded by the rectangle enclosed by {v1 (u), v2 (u), v3 (u), v4 (u)}: v1 (u) := (R− (u), vT − (u)), v2 (u) := (R− (u), vT + (u)), v3 (u) := (R+ (u), vT + (u)),
(23)
v4 (u) := (R+ (u), vT − (u)).
Using the above Tchebyshev representation, we now have Pc (u, x) = R(u, x) + jvT (u, x)
(24)
where R(u, x) := r0 (x) + r1 (x)u + r2 (x)u2 + · · · T (u, x) := t0 (x) + t1 (x)u + t2 (x)u2 + · · · .
(25)
Proof: The proof follows from the construction of the polynomials R− (u), R+ (u), T − (u), and T + (u) along with the fact that v ≥ 0 for all u ∈ [0, 2]. It is also depicted in Figure 3. T (u, x)
Using the sign definite decomposition + − rk (x) := a+ ck (u) − c− k (x) − ak (x) k (u) + − − = a+ k (x)ck (u) − ak (x)ck (u) {z } |
vT + (u) P (u∗ , x)
+ rk (x)
+ + − − a− k (x)ck (u) + ak (x)ck (u) {z } |
(31)
This leads to the following result.
and using sign-definition decomposition, we have − ak (x) = a+ k (x) − ak (x), for all k.
T − (u) ≤ T (u, x) ≤ T + (u).
(21)
vT − (u)
− rk (x)
= rk+ (x) − rk− (x) + − tk (x) := a+ sk (u) − s− k (x) − ak (x) k (u) − − = a+ (x)s+ k (u) − ak (x)sk (u) {z } | k
(26) R− (u) Fig. 3.
R+ (u)
R(u, x)
Image set of Pc (u, x) at u = u∗ .
t+ (x) k
+ + − − a− k (x)sk (u) + ak (x)sk (u) | {z }
Example 1 Consider the polynomial family
t− (x) k
− = t+ k (x) − tk (x).
P (z, x) = z 4 + (−2x1 + 0.8 − x2 ) z 3
(27)
+ x21 + 0.25 − 1.6x1 + 2x1 x2 v − 0.8x2 z 2 + 0.8x21 + 0.2 − x21 x2 − 0.25x2 + 1.6x1 x2 z + −0.8x21 x2 − 0.2x2
Define r¯k := rk+ (x+ ) − rk− (x− ) r k := rk+ (x− ) − rk− (x+ ) − − + t¯k := t+ k (x ) − tk (x ) + − + tk := tk (x ) − t− k (x ).
(28)
(xo1 , xo2 ) = (0.3, 0.5)
Then rk ≤ rk (x) ≤ r¯k ,
with the parameters x1 and x2 varying ±20% around the nominal values
so that tk ≤ tk (x) ≤ t¯k , for all k.
(29)
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x1 ∈ [0.24, 0.36], x2 ∈ [0.4, 0.6].
ThCIn4.3
1
1
0.9
0.95
0.8
0.9
0.7
0.85 ∗
0.6
0.8
Pc(u , x)
0.5
v2
vT(u, x)
vT(u, x)
Pc(u, x)
v3
0.75
0.4
0.7
0.3
0.65
0.2
0.6 v1
0.1 0 −0.5
0
Fig. 4.
0.55
v4 0.5 R(u, x)
1
0.5 2.5
1.5
Image set of Pc (u, x) and the overbounding set.
2.6
Fig. 5.
Example 2 Consider P (z, x) = z 5 + x4 z 3 + x3 z 2 + x2 z + x1 where the parameter x varies by ±20% around the nominal value:
2.8
2.9
3 R(u, x)
3.1
3.2
3.3
3.4
3.5
Image set of Pc (u, x) and the overbounding set.
u ∈ [0, 2]:
The figure below illustrates the image capturing property of Theorem 1. It may be interesting to specialize the above result to the case of interval polynomials. It is known that neither the stability of the Kharitonov polynomials nor even the stability of all the vertex polynomials guarantees the Schur stability of the entire interval polynominal family [14]. The following example shows that the image of an interval polynomial family is contained in the box (v1 (u), v2 (u), v3 (u), v4 (u)) for a fixed value of u and therefore, the condition developed here is a good sufficient condition for Schur stability of a interval polynomial family.
2.7
v1 (u) := (R− (u), vT − (u)), v2 (u) := (R− (u), vT + (u)), v3 (u) := (R+ (u), vT + (u)), v4 (u) := (R+ (u), vT − (u)).
Proof: From the Mikhailov criterion, the image set must go through n quadrants and exclude the origin for all u ∈ [0, 2]. The phase condition given above guarantees that the image set excludes the origin. This together with the Schur stability of a nominal polynomial guarantees the stability of the entire family. The following example illustrates that the recursive algorithm given in (8) can be used to partition the parameter space to tighten the sufficiency of the condition. Example 3 Consider the family
{xo1 , xo2 , xo3 , xo4 } = {0.3, 0.5, 0.5, 0.8}.
p(z, x) = z 4 + (0.5 + x1 )z 3 + x2 z 2 + x2 z + x1
In Figure 5, it is shown that the image set of Pc (u, x) for a fixed u is contained in the box (v1 (u), v2 (u), v3 (u), v4 (u)).
where
The above considerations naturally lead the following theorem on robust stability of the family P (z, x), x ∈ X .
and the polynomial with the nominal values is Schur stable. The following Figures 6, 7, and 8 shows that the condition gets tighter as the number of partitions increases. From Figure 8, we conclude that the family is Schur stable.
Theorem 2 A family of polynomials ( ) n X k P = p(z, x) = ak (x)z : x ∈ X k=0
where the box X lies in the 1st orthant of the parameter space, is Schur stable if 1. at least one polynomial in the family is Schur stable and 2. the phase difference between the vertices, (v1 (u), v2 (u), v3 (u), v4 (u)), is less than π for each
x = {[x1 , x2 ] : [0.3, 0.5] ± 20%}
Example 4 Consider the following 2 input - 2 output discrete-time plant 0.1 0.2 0.1 z − 0.1 P (z) := z − 0.3 0.4 z − 0.2 z − 0.2 −1 z − 0.1 0 0.1 0.2 = 0 z − 0.2 0.3 0.4
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=: Dp−1 (z)Np (z)
ThCIn4.3 and the controller
No partitions 3
k2 k2 k1 z − k1 C(z) := z − k2 k2 z − k1 z − k1 z − k1 0 k2 k2 = 0 z − k1 k2 k2
2.5 2 1.5
Imag
1 0.5
=: Nc (z)Dc−1 (z).
0
The characteristic polynomial of the closed-loop system is −0.5 −1 −1.5 −2 −3
−2
−1
0
1
2
3
4
Real
Fig. 6. Image set of p(u, x) and the rectangles (no partition in parameter space).
δ(z) = det [Dp (z)Dc (z) + Np (z)Nc (z)] 3 1 3 4 2 3 = z + − − 2k1 z + + k2 + k1 + k1 z 2 10 50 5 3 2 13 1 k2 z + − k1 − k2 k1 − k1 − 25 10 100 13 1 2 k + k2 k1 . + 50 1 100 The closed-loop system is verified to be stable for k1 = 0.1 and k2 = 0.4. The question we ask is whether the closedloop system is stable for all values of
4 partitions 3 2.5 2
k1 ∈ [0, 0.15] and k2 ∈ [0.3, 0.5].
1.5
After bisecting each parameter range 8 times, we have the following plot depicting the maximum phase difference between all vertices of union of boxes at each u (see Figure 9). Since the plot does not reach π radian, we conclude that the entire image excludes the origin and therefore, the characteristic polynomial is Schur stable for all k1 ∈ [0, 0.15] and k2 ∈ [0.3, 0.5].
Imag
1 0.5 0 −0.5 −1 −1.5 −2 −3
Maximun phase differences between vertices −2
−1
0
1
2
3
2.5
4
Real
Fig. 7. space).
2
Image set of p(u, x) and the rectangles (4 partition in parameter
1.5
16 partitions Radian
3 2.5
1 2 1.5 0.5 Imag
1 0.5 0 0
0
0.2
0.4
0.6
0.8
1 u
1.2
1.4
1.6
1.8
2
−0.5
Fig. 9.
−1
Maximum phase difference between vertices.
−1.5 −2 −3
−2
−1
0
1
2
3
IV. CONCLUDING REMARKS
4
Real
Fig. 8. Image set of p(u, x) and the rectangles (16 partition in parameter space).
The discrete-time fixed order multivariable control problem has been addressed here and some new methods to determine the stabilizing sets in the controller parameter space have been developed using sign definite decomposition. Our
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ThCIn4.3 treatment is conceptually similar to the case of the robust Hurwitz stability given in [8] and this is due to the use of the Tchebyshev representation. The algorithm given here could be enhanced by adding some necessary conditions that give outer approximations of the stabilizing sets. Another issue of importance for further research is the determination of performance attaining subsets. R EFERENCES [1] S. Hara, T. Iwasaki, and D. Shiokata, “Robust PID control using generalized KYP synthesis: Direct open-loop shaping in multiple frequency ranges,” IEEE Control Systems Magazine, vol. 26, pp. 80 – 91, February, 2006. [2] T. Iwasaki and R.E. Skelton, “All fixed order H∞ controllers: observer based structure and covariance bounds,” IEEE Transactions on Automatic Control, Vol. 40, pp. 512 - 516, March 1995. [3] P. Dorato, “Quantified multivariable polynomial inequalities: The mathematics of (almost) all practical design problems,” Proceedings of the Sixth IEEE Mediterranean Conference on Control and Systems, Alghero, Italy, June 9 - 11, 1998. [4] E.N. Gryazina and B.T. Polyak, “Stability regions in the parameter space: D-decomposition revisited,” Automatica, vol. 42, no. 1, pp. 13 – 26, January, 2006. [5] Yu. I. Neimark, Stability of Linearized Systems (in Russian), LKVVIA, Leningrad, 1949. ˘ [6] D.D. Siljak, Nonlinear Systems: The Parameter Analysis and Design, Wiley, New York, 1969. [7] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SAIM, Philadelphia, PA, 1994. [8] L.H. Keel and S.P. Bhattacharyya, “Fixed order multivariable controller synthesis: A new algorithm,” Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, December 911, 2008. [9] C. Elizondo-Gonzalez, “Necessary and sufficient conditions for robust positivity of polynomial functions via sign decomposition,” Proceedings of the 3rd IFAC Symposium on Robust Control Design (ROCOND 2000), Prague, Czech Republic, 2000. ˘ [10] D.D. Siljak and D.M. Stipanovic, “SPR criteria for uncertain rational matrices via polynomial positivity and Bernstein’s expansions,” IEEE Transactions of Circuits and Systems, vol.48, no. 11, pp. 1366 - 1369, 2001 ˘ [11] D.D. Siljak and D.M. Stipanovic, “Robust D-stability via positivity,” Automatica, vol. 35, no. 8, pp. 1477 - 1484, 1999. [12] L.H. Keel, J.I. Rego, and S.P. Bhattacharyya, “A new approach to digital PID controller design,” IEEE Transactions on Automatic Control, vol. 48, no. 4, pp. 687 - 692, 2003. [13] G. P´olya and G. Szeg¨o, Problems and Theorems in Analysis II: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry, Springer-Verlag, New York, NY, 1976. [14] S.P. Bhattacharyya, H. Chapellat, and L.H. Keel, Robust Control: The Parametric Approach, Prentice Hall PTR, Upper Saddle River, NJ, 1995.
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