H2 optimal ripple-free deadbeat controller design - Semantic Scholar
H2 optimal ripple-free deadbeat controller design Mario E. Salgado∗, Diego A. Oyarz´ un† and Eduardo I. Silva‡ December 26, 2006
Abstract This paper deals with the design ripple-free deadbeat controllers for sampled-data systems. Given a prescribed settling time, the design problem is tackled by computing a ripple-free deadbeat controller that optimizes an H2 performance criterion. This criterion accounts for the quality of the tracking response and for the control energy necessary to achieve the deadbeat behavior. Our main result shows that the optimal performance improves as the settling time grows, revealing a fundamental tradeoff inherent to this control technique.
1
Introduction
Deadbeat control is a well established discrete-time control technique since some decades ago (Kuˇcera 1979, Kuˇcera 1993, Hartley, Veillette & Cook 1996). Its main feature is that it ensures that the tracking error settles to zero in a finite number of samples. Although deadbeat control allows perfect tracking in a finite time horizon and much design methods have been proposed, see for example (Zhao & Kimura 1986, Zhao & Kimura 1988, Funahashi & Katoh 1992), in the context of sampled-data control systems it may lead to poor loop performance. Indeed, deadbeat control only ensures that the error sequence vanishes at samples (beyond the settling time) and no considerations about the intersample behavior are done. Therefore, intersample ripple may appear in the continuous-time output, which is an unwanted feature for the control system’s performance. This issue is dealt with in (Sirisena 1985, Urikura & Nagata 1987, Casavola, Mosca & Zecca 1999), where parameterizations of ripple-free deadbeat controllers are given. The basic idea behind this approach is that to avoid any intersample ripple after the settling time, the control sequence must also reach its steady state in, at most, the same number of samples (Franklin & Emami-Naeini 1986). The results in (Sirisena 1985, Urikura & Nagata 1987, Casavola et al. 1999) apply to general plants with arbitrary reference signals, provided that the sampled-data model satisfies the internal model principle for the corresponding reference (Goodwin, Graebe & Salgado 2001). If we aim to design a ripple-free deadbeat controller with certain prescribed settling time, there are, in the general case, an infinite number of solutions. This issue has motivated researchers to seek for design methods that allow to adjust the controller in such a way that different design criteria can be ∗ [email protected], Department of Electronic Engineering, Universidad T´ ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile. † [email protected], Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland. ‡ [email protected], School of Electrical Engineering and Computer Science, University of Newcastle, NSW 2308, Australia.
1
accounted for. This has been tackled in a state-space framework in (Emami-Naeini & Franklin 1982, Lam, Tso & Tsing 1997), by posing the problem in the context of linear quadratic regulator theory. In (Paz & Elaydi 1998, Paz 2006) the problem is faced using a frequency domain approach, leading to optimization problems that deal with robustness and performance objectives which are solved using LMI and linear programming algorithms. An interesting result in this topic can be found in (Casavola et al. 1999), where the authors give a systematic procedure to design a ripple-free deadbeat controller that minimizes a combined measure formed by the l∞ norm of the tracking error and the l1 norm of the control input. An insightful conclusion of that work is that the optimal value of the mixed l∞ /l1 index is a decreasing function of the settling time, which reveals an essential tradeoff in the deadbeat design problem. A related result is reported in (Zhao & Kimura 1994), where a ripple-free deadbeat controller that ensures a minimal control energy is proposed and the authors conclude that the minimal energy can be decreased at the expense of increasing the settling time. This paper deals with optimal ripple-free deadbeat control for single-input/single-output, linear and time invariant plants in a sampled-data control setup. Firstly, we derive a simple characterization of ripple-free deadbeat controllers for constant reference signals. This parameterization is comparable to that of (Nobuyama 1993) and is useful to derive the analytical solution of an optimal control problem arising from the minimization of a two objective quadratic performance measure. This measure accounts for the energy of both the tracking error and control signal. This implies that the derived control law exhibits ripple-free deadbeat behavior while attaining an optimal performance in terms of both tracking quality and control effort. We stress that the latter property is important from an application point of view and that the excessive control magnitudes, typical from deadbeat control, have been one of the main criticisms to this control technique. The main result of this work is proving that the optimal performance is a non-increasing function of the settling time. This is an interesting feature that emphasizes the design tradeoffs derived in (Casavola et al. 1999) and (Zhao & Kimura 1994). In this sense, the contribution of this work follows the philosophy of that of (Casavola et al. 1999) and stand as a complement to them. It is revealed that the compromise between settling time/performance appears not only in a l∞ /l1 framework, but also in a quadratic sense, constituting an fundamental tradeoff inherent to deadbeat control. The paper is organized as follows: Section 2 defines the main notation and the assumptions made throughout this paper. In Section 3 we derive a characterization of ripple-free deadbeat controllers for step reference signals that is used in Section 4 to propose a design procedure. The main result of this article is given in Section 5. Finally, the concluding remarks of this work are given in Section 6.
2
Notation and assumptions In this paper we deal with proper continuous-time models of the form Gc (s) =
Bc (s) −sτ e , Ac (s)
τ ≥0
(1)
where Bc (s) and Ac (s) are coprime polynomials in s. Since we are interested in a sampled-data control setup, we consider a zero-order hold discrete-time model for GP c (s) given by G(z) = B(z)/A(z), where Pm n B(z) and A(z) are coprime, B(z) = i=0 bi z i and A(z) = i=0 bi z i . The following assumption is made throughout this paper
2
Assumption 1 (a) The sampling is nonpathological, i.e. if pi , i = 1, 2, . . . , n is a pole of Gc (s), then it holds ={pi } 6= κωs ; ∀κ ∈ Z, i = 1, 2, . . . , n, (2) where ωs is the sampling frequency. (b) B(1) 6= 0. Assumption 1(a) is made to ensure that no natural modes of the continuous-time model are lost through the sampling process, while Assumption 1(b) is a necessary and sufficient condition for being able to track constant reference signals. Furthermore, to simplify to subsequent derivations, without loss of generality we assume that B(z) is appropriately scaled such that B(1) = 1. We also introduce the decomposition A(z) = A− (z)A+ (z), where A− (z) has all its roots in |z| < 1, A+ (z) has all its roots in |z| ≥ 1 and deg {A+ (z)} = n+ .
3
Ripple-free deadbeat control
We are interested in the one degree of freedom sampled-data control architecture represented in Figure 1, where Gh0 (s) is the model of the zero order sample and hold device and C(z) is the transfer function of the controller. R(z) +
E(z)
C(z)
U (z)
Gh0 (s)
M (s)
Gc (s)
Yc (s)
− Y (z)
∆
Figure 1: Sampled data control loop. If the reference signal is a step function, i.e. r(k) = vµ(k), v ∈ R, then achieving a ripple-free deadbeat response means that yc (t) satisfies yc (t) = v , ∀t > N ∆, where N ∈ N0 is called the deadbeat horizon of the control system. Hence, a controller is of ripple-free deadbeat class if it provides perfect steady state tracking at D.C. and forces the output of the plant to settle in a finite number of samples, while avoiding any intersample ripple beyond the deadbeat horizon. Ripple in a deadbeat response arises when the controller cancels the minimum phase zeros of G(z). Those cancelled zeros appear as closed loop poles and generate natural modes in the control input that in turn, appear in the intersample response of the continuous-time output. In the context of sampled-data control systems, this issue is of major significance, since the discrete-time model G(z) usually contains sampling zeros located in the negative real axis and therefore, its cancellation leads to oscillatory modes in the deadbeat response. Since the sampling is assumed to be nonpathological, a necessary and sufficient condition to avoid the intersample ripple is that the control sequence also settles in, at most, N samples (see (Sirisena 1985, Franklin & Emami-Naeini 1986)). This condition is equivalent to u(k) = uss , ∀k > N , where uss is the steady state value of the control sequence. We thus conclude that a stabilizing controller C(z) with integral action provides a ripple-free deadbeat response if and only if it is such that the complementary sensitivity, T (z), and control sensitivity, Su (z), are FIR transfer functions of N th order.
3
From previous results (Nobuyama 1993) it is known that the minimum value for N is given by Nmin = n + n+ , so that the deadbeat horizon is, in general, N = Nmin + `, with ` ∈ N. We next give a parameterization of ripple-free deadbeat controllers for constant reference signals that will prove to be useful for the purpose of this work. Lemma 1 Consider the control system of Figure 1, then the controller C(z) achieves ripple-free deadbeat control in N = Nmin + ` samples if and only if C(z) =
Po (z) + X` (z)A(z) , Lo (z) − X` (z)B(z)
where X` (z) is any FIR transfer function of `th order and Co (z) = order achieving ripple free deadbeat control such that
(3) Po (z) Lo (z)
is a biproper controller of nth
A(z)Lo (z) + B(z)Po (z) = z Nmin A− (z),
(4)
Lo (1) = X` (1) = 0.
(5)
and
Proof • Sufficiency. Here we shall prove that with C(z) defined as in (3), and satisfying (4)-(5), the controller achieves ripple-free deadbeat behavior. We first note that (4) is the Diophantine equation that arises from solving a standard pole assignment problem (Goodwin et al. 2001). Here, Nmin closed loop poles are assigned to the origin and the rest are equal to those stable poles of G(z). This implies that the controller Co (z) is stabilizing and must cancel every stable pole of G(z). Hence we may let Po (z) = P˜o (z)A− (z), so that (4) can be written as A+ (z)Lo (z) + B(z)P˜o (z) = z Nmin .
(6)
Therefore, we have that the complementary and control sensitivities (Goodwin et al. 2001) are given by B(z)(P˜o (z) + A+ (z)X` (z)) , z Nmin A(z)(P˜o (z) + A+ (z)X` (z)) Su (z) = . z Nmin T (z) =
(7) (8)
C(z) is stabilizing since it does not cancel any unstable pole of G(z) and both T (z) and Su (z) are stable. In addition, we observe that if X` (z) is a FIR transfer function of `th order satisfying (5), then T (z) and Su (z) are FIR of order N = Nmin + `, with T (1) = 1, which are necessary and sufficient conditions for ripple-free deadbeat control. • Necessity. From the Youla-Kuˇcera parameterization (Goodwin et al. 2001) we have that any stabilizing controller for G(z) can be expressed as C(z) =
Po (z) + Q(z)A(z) , Lo (z) − Q(z)B(z) 4
(9)
where Q(z) is any stable and proper transfer function and Po (z)/Lo (z) is any stabilising controller. We thus choose that controller to satisfy (4) with Lo (1) = 0 and hence ³ ´ B(z) P˜o (z) + A+ (z)Q(z) T (z) = (10) z Nmin ³ ´ A(z) P˜o (z) + A+ (z)Q(z) Su (z) = (11) z Nmin However, ripple-free deadbeat requires that T (z) and Su (z) to be FIR of N th order, i.e. Q(z) = X` (z) must be FIR of `th order, and also T (1) = 1, which implies that X` (1) = 0. ¤¤¤ Expression (3) shows that the proposed deadbeat controller cancels every stable pole of the plant model (recall that Po (z) = A− (z)P˜o (z)) and that it does not cancel any zero, which is necessary to obtain a ripple-free deadbeat response. Also, the cancellation of A− (z) using C(z) implies that the set closed loop poles is the union of two subsets: one containing only N poles at the origin, and the other including all stable plant poles. This implies that the response to input disturbances does not exhibit deadbeat behavior. Nonetheless, if the focus is in the response to reference signals or output disturbances, then only the poles at the origin matter and the ripple-free deadbeat response is achieved. The problem of obtaining a deadbeat response even for input disturbances is more general than the one treated in this paper and the reader is referred to (Milonidis & Karcanias 2006) for recent results on the topic. The characterization of Lemma 1 provides a general form of a ripple-free deadbeat controller as a function of the free FIR transfer function X` (z). Since X` (z) must be a FIR transfer function of `th order, we may write it as X(z) =
D` (z) , z`
(12)
where D` (z) is a polynomial such that deg {D` } ≤ ` and D` (1) = 0, ∀ ` ∈ N, D` (0) 6= 0, ∀ ` 6= 0.
(13) (14)
The parameter X` (z) is completely determined by the polynomial D` (z), so that, instead of X` (z), in the sequel we will treat D` (z) as the free parameter. If we are interested in achieving a minimum horizon ripple-free deadbeat response, then the associated controller is unique and can be obtained by choosing ` = 0. Therefore D0 (z) = 0 is the only feasible choice for the free polynomial in (12). However, if we allow larger deadbeat horizons, additional degrees of freedom appear in the controller. These are given by the polynomial coefficients of D` (z), and may be chosen to satisfy additional design criteria. In this paper we are interested in adjusting the polynomial D` (z) in such a way that a two objective quadratic cost function is optimized. A parameterization of ripple-free deadbeat controllers such as that of Lemma 1 is not new in the literature and its structure follows that same guidelines as that of (Nobuyama 1993). It has been reformulated in this paper using the complex variable z (in (Nobuyama 1993) the authors work in z −1 ) 5
in a more compact way, and a simpler proof based on the Youla-Kuˇcera parameterization has been included for completeness. From Lemma 1 we have that the relative degree of the controller is equal to that of X` (z). Therefore, the relative degree of the controller can be modified by choosing an appropriate order for D` (z). In the sequel, we will always consider that deg {D` (z)} = `. This is advantageous since if we choose a biproper X` (z), then for a fixed ` the available number of free parameters in D` (z) is maximized, which is beneficial from an optimization point of view. In addition, biproper controllers are known to provide implementation benefits such as the inclusion of anti-windup mechanisms (Goodwin et al. 2001), which have made them popular in practical applications (e.g. PID controllers).
4
Optimal controller design
In this section, we provide a methodology to design a ripple-free deadbeat controller that achieves a good transient performance. The main issue to solve this problem is how to choose the polynomial D` (z) in (12), in such a way that an appropriate transient response is achieved. Following a similar idea to that in (Casavola et al. 1999), we tackle this problem by choosing a polynomial D` (z) that minimizes a performance index representative of some standard control objectives. The essential distinction of our approach with respect to (Casavola et al. 1999) is the usage of an H2 measure as the target function to be optimized. We can measure the quality of the transient performance through the cost function N X
Je =
e(k)2 ,
(15)
k=0
where e(k) = Z −1 {E(z)} is the tracking error sequence. The quantity Je is the energy of the tracking error sequence and has been widely used as a tracking performance measure (Morari & Zafiriou 1989, Toker, Chen & Qiu 2002) in the context of performance bounds computation. On the other hand, a common criticism made to deadbeat control is that it usually demands a large control energy. This observation suggests the introduction of an optimality criterion that weights the control effort necessary to achieve the deadbeat response. To that end, in an analogous fashion as in (Zhao & Kimura 1994), we define the cost functional Ju =
N X
(u(k) − uss )2 .
(16)
k=0
This measure accounts for the control effort by means of the energy of its deviation from the steady state value. It is simple to prove that both quantities, Je and Ju , can be conveniently expressed in the frequency domain as ¯¯ ¯¯ ¯¯ S(z) ¯¯2 ¯ ¯ Je (D` (z)) = ¯¯ v ¯¯ , (17) z − 1 ¯¯2 ¯¯ ¯¯ ¯¯ Su (z) − A(1) ¯¯2 v ¯¯¯¯ , Ju (D` (z)) = ¯¯¯¯ (18) z−1 2 where S(z) is the sensitivity function and the argument of both cost functions has been made explicit to emphasize their dependence on the free polynomial D` (z). In the sequel, without loss of generality 6
we assume that v = 1. We are interested in including both partial costs, tracking error and control effort, in our optimal control problem. To that end we define the two objective cost functional J (D` (z)) = λJe (D` (z)) + (1 − λ)Ju (D` (z)) ,
(19)
where 0 ≤ λ ≤ 1. We are now able to state the optimization problem of our interest. Problem 1 Given a discrete-time plant model G(z) and a prescribed deadbeat horizon N = Nmin + `, find a polynomial D`o (z) such that D`o (z) = arg
min
D` (z)∈P`
J (D` (z)) ,
(20)
where P` is the set of polynomials of `th order and D`o (z) satisfies (13) and (14). As stated in the previous section, if we choose ` = 0 there is a unique feasible polynomial that solves Problem 1. Hence, in such a case, the controller is uniquely determined and no optimization is possible; if we choose larger deadbeat horizons, then the coefficients of D` (z) may be computed as a solution of Problem 1. For that purpose we define m+n+ e
H (z) = B(z)A+ (z) =
X
hei z i ,
(21)
hui z i ,
(22)
i=0 n+n+
H u (z) = A(z)A+ (z) =
X i=0
z N − z ` B(z)P˜o (z) , z−1 z N A(1) − z ` A(z)P˜o (z) F u (z) = , z−1 F e (z) =
(23) (24)
where Po (z) = P˜o (z)A− (z) and Po (z) is defined in Lemma 1. From the proof of Lemma 1 we have that P˜o (z) satisfies (6), so that P˜o (1) = 1 and therefore both F e (z) and F u (z) are always polynomials. PN −1 PN −1 Thus, we may let F e (z) = i=0 fie z i and F u (z) = i=0 fiu z i . With these definitions, the analytic solution to Problem 1 can be found, as shown next. Theorem 1 Consider the parameterization of ripple-free deadbeat controllers given in Lemma 1. Then, the optimal polynomial that solves Problem 1 is given by h D`o (z) = (z − 1) 1
i z
···
z `−1
†
† √ √ λΓ` λγ` √ , √ 1 − λΨ` 1 − λψ`
(25)
where (·) denotes the Moore-Penrose pseudoinverse (Golub & Van Loan 1996), and Γ` ∈ RN ×` ,
7
Ψ` ∈ RN ×` , γ` ∈ RN ×1 and ψ` ∈ RN ×1 are defined as he0 0 ··· 0 he he0 ··· 0 1 . . . . . e .. . h1 . e .. . . hm+n . . 0 + .. e e , . h 0 h Γ` = m+n+ 0 .. e . 0 0 h1 . .. .. .. . . . . . 0 0 hem+n+ 0 0(n−m)×`
hu0 hu1
0 hu0
. .. hu1 .. hu . n+n+ Ψ` = 0 hun+n+ 0 0 . .. . . . 0 0
f0e
0 0
..
.
.. .
..
.
0
..
.
hu0
..
.
hu1
e f1 γ` = .. , . e fN −1
··· ···
.. . 0
.. .
,
(27)
hun+n+
f0u
u f1 ψ` = .. , . u fN −1
with 0(n−m)×` ∈ R(n−m)×` denoting the zero matrix of the specified dimensions.
8
(26)
(28)
Proof: From Lemma 1 and (12) we have that z N − B(z)P (z) , zN A(z)P (z) , Su (z) = zN S(z) =
(29) (30)
where P (z) = z ` P˜o (z) + A+ (z)D` (z). The constraint (13) can be satisfied if we write D` (z) as ˜ ` (z), D` (z) = (z − 1)D n o ˜ ` (z) is an arbitrary polynomial such that deg D ˜ ` (s) = ` − 1. Substituting (29)-(31) in where D and (18) yields ¯¯2 ¯¯ ¯¯ z N − z ` B(z)P˜ (z) ¯¯ ¯¯ o ˜ ` (z)¯¯¯¯ , Je (D` (z)) = ¯¯ − B(z)A+ (z)D ¯¯ ¯¯ z−1 2 ¯¯ ¯¯2 ¯¯ e ¯¯ e ˜ = ¯¯F (z) − H (z)D` (z)¯¯ , 2 ¯¯ N ¯¯2 ¯¯ z A(1) − z ` A(z)Qo (z) ¯¯ ˜ ` (z)¯¯ , Ju (D` (z)) = ¯¯¯¯ − A(z)A+ (z)D ¯¯ z−1 2 ¯¯ ¯¯2 ¯¯ u ¯¯ u ˜ = ¯¯F (z) − H (z)D` (z)¯¯ , 2
(31) (17)
(32) (33) (34) (35)
where H e (z), H u (z), F e (z) and F u (z) are defined in (21)-(24), and we have used the fact that z −N preserves the H2 norm. Expressions (33) and (35) can be vectorized as follows ¯¯£ ´¯¯2 ¤³ ¯¯ ¯¯ Je (D` (z)) = ¯¯ 1 z . . . z N −1 γ` − Γ` d˜` ¯¯ , (36) 2 ¯¯£ ´¯¯2 ¤³ ¯¯ ¯¯ Ju (D` (z)) = ¯¯ 1 z . . . z N −1 ψ` − Ψ` d˜` ¯¯ , (37) 2
where γ` , Γ` , ψ` and Ψ` are defined in (26)-(28), and d˜` ∈ R`×1 is such that £ ¤T d˜` = d˜0 d˜1 . . . d˜`−1 ,
(38)
˜ ` (z) = P`−1 d˜i z i . Substituting (36) and (37) in (19) and using elementary properties of the H2 and D i=0 norm yields ¯¯· √ ¸ · √ ¸ ¯¯2 ¯¯ ¯¯ λγ` λΓ` J (D` (z)) = ¯¯¯¯ √ − √ d˜` ¯¯¯¯ , (39) 1 − λψ` 1 − λΨ` e where ||·||e denotes the Euclidean norm. Thus, the original optimization problem can be posed in a standard least squares framework as d˜o` = arg min J (D` (z)) . d˜` ∈R`×1
9
(40)
The optimal vector d˜o` is then given by (Goodwin & Payne 1977) · √ ¸† · √ ¸ λΓ ` o ˜ √ λγ` d` = √ . 1 − λΨ` 1 − λψ`
(41)
The result (25) follows by substituting (41) in (31). ¤¤¤ Last theorem is comparable to those results of (Casavola et al. 1999, Paz & Elaydi 1998, Paz 2006), in which useful design methodologies based on optimization routines are proposed. Nonetheless, the framework used throughout this paper allows us to obtain an analytical solution to the two objective optimal control problem of our interest, which draws a difference with the previous existent work on the topic. This issue, together with the fact we use an H2 performance index, constitute the main distinctions of this work with respect to (Casavola et al. 1999, Paz & Elaydi 1998, Paz 2006). Theorem 1 provides a ripple-free deadbeat controller design procedure that can be summarized as follows: i. Choose a deadbeat horizon N = Nmin + ` and a scalar weight 0 ≤ λ ≤ 1. ii. Compute the unique solution (Po (z), Lo (z)) of the diophantine equation (4) such that Po (z) = P˜o (z)A− (z), Lo (1) = 0 and deg {Po (z)} = deg {Lo (z)} = n. iii. Compute the polynomials F e (z), H e (z), F u (z) and H u (z) using (21)-(24). iv. The optimal controller is obtained by computing D`∗ (z) from (25) and substituting it in (3) using (12).
5
Optimal performance
Having derived an explicit formula for the optimal ripple-free deadbeat controller in the previous section, we now turn our attention to the performance properties of the optimal design. An useful result would be to derive an analytical expression for the optimal performance achieved by the ripplefree deadbeat controller proposed of last section. This would allow to explore which dynamical features of the plant model are those that have a deleterious impact on the optimal performance achievable by a ripple-free deadbeat controller. Unfortunately, this is a difficult task so we restrict ourselves to show a qualitative feature of the optimal controller that sheds light on the design tradeoffs that appear in ripple-free deadbeat control. This constitutes the main result of this paper and is stated in next theorem. Theorem 2 Let G(z) be a discrete-time plant model and suppose that the ripple-free deadbeat controller defined by Lemma 1 is computed using the optimal polynomial D`o (z) that solves Problem 1. Then, the optimal cost J (D`o (z)) is a non-increasing function of `.
10
Proof: To keep a compact notation let us define · √ ¸ λΓ` M` = √ , 1 − λΨ` · √ ¸ λγ` √ m` = . 1 − λψ`
(42) (43)
It should be observed that M` ∈ R2N ×` and has full column rank (which can be verified from (26) and (27)). These facts imply that ³ ´−1 M` † = M` T M` M` T . (44) Therefore, from (39) and (41), it is simple to prove (Goodwin & Payne 1977) that the optimal cost for a horizon N is given by µ ¶ ³ ´−1 J (D`o (z)) = m` T I2N − M` M` T M` M` T m` . (45) Analogously, if we increase the deadbeat horizon in one sample, the optimal cost can be computed from µ ¶ ³ ´−1 ¢ ¡ o (z) = m`+1 T I2N+2 − M`+1 M`+1 T M`+1 M`+1 T m`+1 . J D`+1 (46) From the definition of Γ` and Ψ` in (26) and (27) we observe that "√ # λ heo 01×` M`+1 = , q T` M` where q ∈ R(2N +1)×1 is given by h√ √ e q= λ he1 · · · λ hm+n+
01×(n−m+`)
√
1 − λ hu0
···
(47)
√
1 − λ hun+n+
01×`
iT
,
(48)
and £ T` = e1
···
eN
eN+2
···
e2N+1
¤T
∈ R(2N +1)×2N ,
(49)
with ei denoting the ith element of the canonical basis of R2N +1 . Similarly, from (23), (24), (28) and (43), it follows that the vector m`+1 can be conveniently expressed as · ¸ 0 m`+1 = (50) . T` m ` Equation (47) implies that · M`+1 T M`+1 =
¸ λhe0 2 + qT q qT T` M` . M ` T TT q M` T M` 11
(51)
With some algebra after substituting (47), (50)-(51) in (46) and applying the matrix inversion lemma (Goodwin et al. 2001) it follows that ¡ o ¢ 1 T T J D`+1 (z) = (T` m` ) Λ (T` m` ) − (T` m` ) ΛqqT Λ (T` m` ) , κ
(52)
where ³ ´−1 T T Λ = I2N +1 − (T` M` ) (T` M` ) (T` M` ) (T` M` ) ,
(53)
κ = λhe0 2 + qT Λq.
(54)
Using the fact that T` T T` = I2N in (52) and ¡comparing ¢ with (45), we recognize the optimal cost for o horizon N in the first term of (52), so that J D`+1 (z) can be expressed as µ ¶ ¶2 ³ ´−1 1 T T T T √ m` I2N − M` M` M` M` T q . (55) κ ¡ o ¢ Therefore it can be concluded that J D`+1 (z) ≤ J (D`o (z)) , ∀` ≥ 0 and the result follows using an inductive procedure. ¡ o ¢ J D`+1 (z) = J (D`o (z)) −
µ
¤¤¤ The result of Theorem 2 reveals an essential tradeoff present in the design of ripple-free deadbeat control systems. If the focus of the designer is to have a deadbeat response with short settling time, last theorem implies that this can only be done at the expense of sacrificing performance. Similarly, one can only improve the transient performance by extending the settling time (and the complexity of the controller). This kind of design tradeoff had already been reported in (Casavola et al. 1999) for a mixed l∞ /l1 performance measure that accounts for the same elements as those in this work, i.e. tracking error and control input. Therefore, our result stands as a complement to these previous findings, adding an insightful conclusion to the topic, namely, that the performance vs. settling time tradeoff appears not only in a l∞ /l1 framework, but also in the context of H2 performance measures. It should be emphasized that our conclusion cannot be intuitively derived from the results in (Casavola et al. 1999), since essentially the l∞ /l1 and H2 approaches measure different aspects of the transient response.
6
Conclusions
In this paper a ripple-free deadbeat controller design methodology for step references has been proposed and analyzed. The ripple-free property of the controller is in the sense that, after the settling time, no ripples appear in the output of the plant. The design algorithm is applicable to stable and unstable plants under sampled data control and is such that, for a fixed settling time, the controller is adjusted to minimize a quadratic cost function. This cost function penalizes a combination of the energy of both the tracking error and control signal, and allows to pose a convex optimization problem that is solved with aid of standard properties of the H2 norm. This also allows to derive an explicit formula for the optimal controller. 12
Since the controller is obtained from an optimization procedure, the only design parameters in this strategy are the settling time and a weighting factor. The most significative result of this paper is showing that the optimal cost function is a non-increasing function of the settling time for any value of the weighting factor. This conclusion is a complement to those similar derived in previous literature (Casavola et al. 1999) using different norms and therefore, it reveals that the settling time/performance tradeoff is essential of deadbeat control.
Acknowledgements The authors gratefully acknowledge the support received from Universidad T´ecnica Federico Santa Mar´ıa (PAC program and grant UTFSM 230623) and from CONICYT through grant Fondecyt 1060437.
References Casavola, A., Mosca, E. & Zecca, P. (1999), ‘Robust ripple-free deadbeat control design’, International Journal of Control 72(6), 564–573. Emami-Naeini, A. & Franklin, G. (1982), ‘Deadbeat control and tracking of discrete-time systems’, IEEE Transactions on Automatic Control 27(1), 176–181. Franklin, G. F. & Emami-Naeini, A. (1986), ‘Design of ripple free multivariable robust servomechanisms’, IEEE Transactions on Automatic Control 31(7), 661–664. Funahashi, Y. & Katoh, H. (1992), ‘Robust-tracking deadbeat control’, International Journal of Control 56(1), 213–225. Golub, G. & Van Loan, C. (1996), Matrix computations, 3rd edn, The John Hopkins University Press, Baltimore, Maryland. Goodwin, G. C., Graebe, S. & Salgado, M. E. (2001), Control System Design, Prentice Hall, New Jersey. Goodwin, G. & Payne, R. (1977), Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York. Hartley, T., Veillette, R. & Cook, G. (1996), Techniques in deadbeat and one-step ahead control, in ‘Digital Control Systems Implementation and Computational Techniques: Control and Dynamic Systems & Advances in Theory and Applications’, Vol. 79, Academic Press NY, chapter 3. Kuˇcera, V. (1979), Discrete Linear Control: The Polynomial Equation Approach, John Wiley & Sons, Inc. New York, NY, USA. Kuˇcera, V. (1993), ‘Diophantine equations in control — a survey’, Automatica 29, 1361–1375. Lam, J., Tso, H. K. & Tsing, N. K. (1997), ‘Robust deadbeat regulation’, International Journal of Control 67(4), 587–602. Milonidis, E. & Karcanias, E. (2006), ‘Finite settling time stabilisation for multivariable discrete-time systems: a polynomial equation approach’, International Journal of Control 79(12), 1565–1580. 13
Morari, M. & Zafiriou, E. (1989), Robust process control, Prentice Hall Inc., Englewood Cliffs, New Jersey. Nobuyama, E. (1993), ‘Parametrization of all ripple-free deadbeat controllers’, Systems and Control Letters 21(3), 217–244. Paz, R. (2006), ‘Ripple-free tracking with robustness’, International Journal of Control 79(6), 543–568. Paz, R. & Elaydi, H. (1998), ‘Optimal ripple-free deadbeat controllers’, International Journal of Control 71(6), 1087–1104. Sirisena, H. R. (1985), ‘Ripple-free deadbeat control of SISO discrete systems’, IEEE Transactions on Automatic Control 30(2), 168–170. Toker, O., Chen, L. & Qiu, L. (2002), ‘Tracking performance limitations in LTI multivariable discretetime systems’, IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications 49(5), 657–670. Urikura, S. & Nagata, A. (1987), ‘Ripple-free deadbeat control for sampled-data systems’, IEEE Transactions on Automatic Control 32(6), 474–482. Zhao, Y. & Kimura, H. (1986), ‘Dead-beat control with robustness’, International Journal of Control 43(5), 1427–1440. Zhao, Y. & Kimura, H. (1988), ‘Two-degrees-of-freedom dead-beat control system with robustness’, International Journal of Control 48(1), 303–315. Zhao, Y. & Kimura, H. (1994), ‘Optimization of control input for deadbeat control systems’, International Journal of Control 59(4), 1107–1117.
14
Abstract This paper deals with the design ripple-free deadbeat controllers for sampled-data systems. Given a prescribed settling time, the design problem is tackled by computing a ripple-free deadbeat controller that optimizes an H2 performance criterion. This criterion accounts for the quality of the tracking response and for the control energy necessary to achieve the deadbeat behavior. Our main result shows that the optimal performance improves as the settling time grows, revealing a fundamental tradeoff inherent to this control technique.
1
Introduction
Deadbeat control is a well established discrete-time control technique since some decades ago (Kuˇcera 1979, Kuˇcera 1993, Hartley, Veillette & Cook 1996). Its main feature is that it ensures that the tracking error settles to zero in a finite number of samples. Although deadbeat control allows perfect tracking in a finite time horizon and much design methods have been proposed, see for example (Zhao & Kimura 1986, Zhao & Kimura 1988, Funahashi & Katoh 1992), in the context of sampled-data control systems it may lead to poor loop performance. Indeed, deadbeat control only ensures that the error sequence vanishes at samples (beyond the settling time) and no considerations about the intersample behavior are done. Therefore, intersample ripple may appear in the continuous-time output, which is an unwanted feature for the control system’s performance. This issue is dealt with in (Sirisena 1985, Urikura & Nagata 1987, Casavola, Mosca & Zecca 1999), where parameterizations of ripple-free deadbeat controllers are given. The basic idea behind this approach is that to avoid any intersample ripple after the settling time, the control sequence must also reach its steady state in, at most, the same number of samples (Franklin & Emami-Naeini 1986). The results in (Sirisena 1985, Urikura & Nagata 1987, Casavola et al. 1999) apply to general plants with arbitrary reference signals, provided that the sampled-data model satisfies the internal model principle for the corresponding reference (Goodwin, Graebe & Salgado 2001). If we aim to design a ripple-free deadbeat controller with certain prescribed settling time, there are, in the general case, an infinite number of solutions. This issue has motivated researchers to seek for design methods that allow to adjust the controller in such a way that different design criteria can be ∗ [email protected], Department of Electronic Engineering, Universidad T´ ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile. † [email protected], Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland. ‡ [email protected], School of Electrical Engineering and Computer Science, University of Newcastle, NSW 2308, Australia.
1
accounted for. This has been tackled in a state-space framework in (Emami-Naeini & Franklin 1982, Lam, Tso & Tsing 1997), by posing the problem in the context of linear quadratic regulator theory. In (Paz & Elaydi 1998, Paz 2006) the problem is faced using a frequency domain approach, leading to optimization problems that deal with robustness and performance objectives which are solved using LMI and linear programming algorithms. An interesting result in this topic can be found in (Casavola et al. 1999), where the authors give a systematic procedure to design a ripple-free deadbeat controller that minimizes a combined measure formed by the l∞ norm of the tracking error and the l1 norm of the control input. An insightful conclusion of that work is that the optimal value of the mixed l∞ /l1 index is a decreasing function of the settling time, which reveals an essential tradeoff in the deadbeat design problem. A related result is reported in (Zhao & Kimura 1994), where a ripple-free deadbeat controller that ensures a minimal control energy is proposed and the authors conclude that the minimal energy can be decreased at the expense of increasing the settling time. This paper deals with optimal ripple-free deadbeat control for single-input/single-output, linear and time invariant plants in a sampled-data control setup. Firstly, we derive a simple characterization of ripple-free deadbeat controllers for constant reference signals. This parameterization is comparable to that of (Nobuyama 1993) and is useful to derive the analytical solution of an optimal control problem arising from the minimization of a two objective quadratic performance measure. This measure accounts for the energy of both the tracking error and control signal. This implies that the derived control law exhibits ripple-free deadbeat behavior while attaining an optimal performance in terms of both tracking quality and control effort. We stress that the latter property is important from an application point of view and that the excessive control magnitudes, typical from deadbeat control, have been one of the main criticisms to this control technique. The main result of this work is proving that the optimal performance is a non-increasing function of the settling time. This is an interesting feature that emphasizes the design tradeoffs derived in (Casavola et al. 1999) and (Zhao & Kimura 1994). In this sense, the contribution of this work follows the philosophy of that of (Casavola et al. 1999) and stand as a complement to them. It is revealed that the compromise between settling time/performance appears not only in a l∞ /l1 framework, but also in a quadratic sense, constituting an fundamental tradeoff inherent to deadbeat control. The paper is organized as follows: Section 2 defines the main notation and the assumptions made throughout this paper. In Section 3 we derive a characterization of ripple-free deadbeat controllers for step reference signals that is used in Section 4 to propose a design procedure. The main result of this article is given in Section 5. Finally, the concluding remarks of this work are given in Section 6.
2
Notation and assumptions In this paper we deal with proper continuous-time models of the form Gc (s) =
Bc (s) −sτ e , Ac (s)
τ ≥0
(1)
where Bc (s) and Ac (s) are coprime polynomials in s. Since we are interested in a sampled-data control setup, we consider a zero-order hold discrete-time model for GP c (s) given by G(z) = B(z)/A(z), where Pm n B(z) and A(z) are coprime, B(z) = i=0 bi z i and A(z) = i=0 bi z i . The following assumption is made throughout this paper
2
Assumption 1 (a) The sampling is nonpathological, i.e. if pi , i = 1, 2, . . . , n is a pole of Gc (s), then it holds ={pi } 6= κωs ; ∀κ ∈ Z, i = 1, 2, . . . , n, (2) where ωs is the sampling frequency. (b) B(1) 6= 0. Assumption 1(a) is made to ensure that no natural modes of the continuous-time model are lost through the sampling process, while Assumption 1(b) is a necessary and sufficient condition for being able to track constant reference signals. Furthermore, to simplify to subsequent derivations, without loss of generality we assume that B(z) is appropriately scaled such that B(1) = 1. We also introduce the decomposition A(z) = A− (z)A+ (z), where A− (z) has all its roots in |z| < 1, A+ (z) has all its roots in |z| ≥ 1 and deg {A+ (z)} = n+ .
3
Ripple-free deadbeat control
We are interested in the one degree of freedom sampled-data control architecture represented in Figure 1, where Gh0 (s) is the model of the zero order sample and hold device and C(z) is the transfer function of the controller. R(z) +
E(z)
C(z)
U (z)
Gh0 (s)
M (s)
Gc (s)
Yc (s)
− Y (z)
∆
Figure 1: Sampled data control loop. If the reference signal is a step function, i.e. r(k) = vµ(k), v ∈ R, then achieving a ripple-free deadbeat response means that yc (t) satisfies yc (t) = v , ∀t > N ∆, where N ∈ N0 is called the deadbeat horizon of the control system. Hence, a controller is of ripple-free deadbeat class if it provides perfect steady state tracking at D.C. and forces the output of the plant to settle in a finite number of samples, while avoiding any intersample ripple beyond the deadbeat horizon. Ripple in a deadbeat response arises when the controller cancels the minimum phase zeros of G(z). Those cancelled zeros appear as closed loop poles and generate natural modes in the control input that in turn, appear in the intersample response of the continuous-time output. In the context of sampled-data control systems, this issue is of major significance, since the discrete-time model G(z) usually contains sampling zeros located in the negative real axis and therefore, its cancellation leads to oscillatory modes in the deadbeat response. Since the sampling is assumed to be nonpathological, a necessary and sufficient condition to avoid the intersample ripple is that the control sequence also settles in, at most, N samples (see (Sirisena 1985, Franklin & Emami-Naeini 1986)). This condition is equivalent to u(k) = uss , ∀k > N , where uss is the steady state value of the control sequence. We thus conclude that a stabilizing controller C(z) with integral action provides a ripple-free deadbeat response if and only if it is such that the complementary sensitivity, T (z), and control sensitivity, Su (z), are FIR transfer functions of N th order.
3
From previous results (Nobuyama 1993) it is known that the minimum value for N is given by Nmin = n + n+ , so that the deadbeat horizon is, in general, N = Nmin + `, with ` ∈ N. We next give a parameterization of ripple-free deadbeat controllers for constant reference signals that will prove to be useful for the purpose of this work. Lemma 1 Consider the control system of Figure 1, then the controller C(z) achieves ripple-free deadbeat control in N = Nmin + ` samples if and only if C(z) =
Po (z) + X` (z)A(z) , Lo (z) − X` (z)B(z)
where X` (z) is any FIR transfer function of `th order and Co (z) = order achieving ripple free deadbeat control such that
(3) Po (z) Lo (z)
is a biproper controller of nth
A(z)Lo (z) + B(z)Po (z) = z Nmin A− (z),
(4)
Lo (1) = X` (1) = 0.
(5)
and
Proof • Sufficiency. Here we shall prove that with C(z) defined as in (3), and satisfying (4)-(5), the controller achieves ripple-free deadbeat behavior. We first note that (4) is the Diophantine equation that arises from solving a standard pole assignment problem (Goodwin et al. 2001). Here, Nmin closed loop poles are assigned to the origin and the rest are equal to those stable poles of G(z). This implies that the controller Co (z) is stabilizing and must cancel every stable pole of G(z). Hence we may let Po (z) = P˜o (z)A− (z), so that (4) can be written as A+ (z)Lo (z) + B(z)P˜o (z) = z Nmin .
(6)
Therefore, we have that the complementary and control sensitivities (Goodwin et al. 2001) are given by B(z)(P˜o (z) + A+ (z)X` (z)) , z Nmin A(z)(P˜o (z) + A+ (z)X` (z)) Su (z) = . z Nmin T (z) =
(7) (8)
C(z) is stabilizing since it does not cancel any unstable pole of G(z) and both T (z) and Su (z) are stable. In addition, we observe that if X` (z) is a FIR transfer function of `th order satisfying (5), then T (z) and Su (z) are FIR of order N = Nmin + `, with T (1) = 1, which are necessary and sufficient conditions for ripple-free deadbeat control. • Necessity. From the Youla-Kuˇcera parameterization (Goodwin et al. 2001) we have that any stabilizing controller for G(z) can be expressed as C(z) =
Po (z) + Q(z)A(z) , Lo (z) − Q(z)B(z) 4
(9)
where Q(z) is any stable and proper transfer function and Po (z)/Lo (z) is any stabilising controller. We thus choose that controller to satisfy (4) with Lo (1) = 0 and hence ³ ´ B(z) P˜o (z) + A+ (z)Q(z) T (z) = (10) z Nmin ³ ´ A(z) P˜o (z) + A+ (z)Q(z) Su (z) = (11) z Nmin However, ripple-free deadbeat requires that T (z) and Su (z) to be FIR of N th order, i.e. Q(z) = X` (z) must be FIR of `th order, and also T (1) = 1, which implies that X` (1) = 0. ¤¤¤ Expression (3) shows that the proposed deadbeat controller cancels every stable pole of the plant model (recall that Po (z) = A− (z)P˜o (z)) and that it does not cancel any zero, which is necessary to obtain a ripple-free deadbeat response. Also, the cancellation of A− (z) using C(z) implies that the set closed loop poles is the union of two subsets: one containing only N poles at the origin, and the other including all stable plant poles. This implies that the response to input disturbances does not exhibit deadbeat behavior. Nonetheless, if the focus is in the response to reference signals or output disturbances, then only the poles at the origin matter and the ripple-free deadbeat response is achieved. The problem of obtaining a deadbeat response even for input disturbances is more general than the one treated in this paper and the reader is referred to (Milonidis & Karcanias 2006) for recent results on the topic. The characterization of Lemma 1 provides a general form of a ripple-free deadbeat controller as a function of the free FIR transfer function X` (z). Since X` (z) must be a FIR transfer function of `th order, we may write it as X(z) =
D` (z) , z`
(12)
where D` (z) is a polynomial such that deg {D` } ≤ ` and D` (1) = 0, ∀ ` ∈ N, D` (0) 6= 0, ∀ ` 6= 0.
(13) (14)
The parameter X` (z) is completely determined by the polynomial D` (z), so that, instead of X` (z), in the sequel we will treat D` (z) as the free parameter. If we are interested in achieving a minimum horizon ripple-free deadbeat response, then the associated controller is unique and can be obtained by choosing ` = 0. Therefore D0 (z) = 0 is the only feasible choice for the free polynomial in (12). However, if we allow larger deadbeat horizons, additional degrees of freedom appear in the controller. These are given by the polynomial coefficients of D` (z), and may be chosen to satisfy additional design criteria. In this paper we are interested in adjusting the polynomial D` (z) in such a way that a two objective quadratic cost function is optimized. A parameterization of ripple-free deadbeat controllers such as that of Lemma 1 is not new in the literature and its structure follows that same guidelines as that of (Nobuyama 1993). It has been reformulated in this paper using the complex variable z (in (Nobuyama 1993) the authors work in z −1 ) 5
in a more compact way, and a simpler proof based on the Youla-Kuˇcera parameterization has been included for completeness. From Lemma 1 we have that the relative degree of the controller is equal to that of X` (z). Therefore, the relative degree of the controller can be modified by choosing an appropriate order for D` (z). In the sequel, we will always consider that deg {D` (z)} = `. This is advantageous since if we choose a biproper X` (z), then for a fixed ` the available number of free parameters in D` (z) is maximized, which is beneficial from an optimization point of view. In addition, biproper controllers are known to provide implementation benefits such as the inclusion of anti-windup mechanisms (Goodwin et al. 2001), which have made them popular in practical applications (e.g. PID controllers).
4
Optimal controller design
In this section, we provide a methodology to design a ripple-free deadbeat controller that achieves a good transient performance. The main issue to solve this problem is how to choose the polynomial D` (z) in (12), in such a way that an appropriate transient response is achieved. Following a similar idea to that in (Casavola et al. 1999), we tackle this problem by choosing a polynomial D` (z) that minimizes a performance index representative of some standard control objectives. The essential distinction of our approach with respect to (Casavola et al. 1999) is the usage of an H2 measure as the target function to be optimized. We can measure the quality of the transient performance through the cost function N X
Je =
e(k)2 ,
(15)
k=0
where e(k) = Z −1 {E(z)} is the tracking error sequence. The quantity Je is the energy of the tracking error sequence and has been widely used as a tracking performance measure (Morari & Zafiriou 1989, Toker, Chen & Qiu 2002) in the context of performance bounds computation. On the other hand, a common criticism made to deadbeat control is that it usually demands a large control energy. This observation suggests the introduction of an optimality criterion that weights the control effort necessary to achieve the deadbeat response. To that end, in an analogous fashion as in (Zhao & Kimura 1994), we define the cost functional Ju =
N X
(u(k) − uss )2 .
(16)
k=0
This measure accounts for the control effort by means of the energy of its deviation from the steady state value. It is simple to prove that both quantities, Je and Ju , can be conveniently expressed in the frequency domain as ¯¯ ¯¯ ¯¯ S(z) ¯¯2 ¯ ¯ Je (D` (z)) = ¯¯ v ¯¯ , (17) z − 1 ¯¯2 ¯¯ ¯¯ ¯¯ Su (z) − A(1) ¯¯2 v ¯¯¯¯ , Ju (D` (z)) = ¯¯¯¯ (18) z−1 2 where S(z) is the sensitivity function and the argument of both cost functions has been made explicit to emphasize their dependence on the free polynomial D` (z). In the sequel, without loss of generality 6
we assume that v = 1. We are interested in including both partial costs, tracking error and control effort, in our optimal control problem. To that end we define the two objective cost functional J (D` (z)) = λJe (D` (z)) + (1 − λ)Ju (D` (z)) ,
(19)
where 0 ≤ λ ≤ 1. We are now able to state the optimization problem of our interest. Problem 1 Given a discrete-time plant model G(z) and a prescribed deadbeat horizon N = Nmin + `, find a polynomial D`o (z) such that D`o (z) = arg
min
D` (z)∈P`
J (D` (z)) ,
(20)
where P` is the set of polynomials of `th order and D`o (z) satisfies (13) and (14). As stated in the previous section, if we choose ` = 0 there is a unique feasible polynomial that solves Problem 1. Hence, in such a case, the controller is uniquely determined and no optimization is possible; if we choose larger deadbeat horizons, then the coefficients of D` (z) may be computed as a solution of Problem 1. For that purpose we define m+n+ e
H (z) = B(z)A+ (z) =
X
hei z i ,
(21)
hui z i ,
(22)
i=0 n+n+
H u (z) = A(z)A+ (z) =
X i=0
z N − z ` B(z)P˜o (z) , z−1 z N A(1) − z ` A(z)P˜o (z) F u (z) = , z−1 F e (z) =
(23) (24)
where Po (z) = P˜o (z)A− (z) and Po (z) is defined in Lemma 1. From the proof of Lemma 1 we have that P˜o (z) satisfies (6), so that P˜o (1) = 1 and therefore both F e (z) and F u (z) are always polynomials. PN −1 PN −1 Thus, we may let F e (z) = i=0 fie z i and F u (z) = i=0 fiu z i . With these definitions, the analytic solution to Problem 1 can be found, as shown next. Theorem 1 Consider the parameterization of ripple-free deadbeat controllers given in Lemma 1. Then, the optimal polynomial that solves Problem 1 is given by h D`o (z) = (z − 1) 1
i z
···
z `−1
†
† √ √ λΓ` λγ` √ , √ 1 − λΨ` 1 − λψ`
(25)
where (·) denotes the Moore-Penrose pseudoinverse (Golub & Van Loan 1996), and Γ` ∈ RN ×` ,
7
Ψ` ∈ RN ×` , γ` ∈ RN ×1 and ψ` ∈ RN ×1 are defined as he0 0 ··· 0 he he0 ··· 0 1 . . . . . e .. . h1 . e .. . . hm+n . . 0 + .. e e , . h 0 h Γ` = m+n+ 0 .. e . 0 0 h1 . .. .. .. . . . . . 0 0 hem+n+ 0 0(n−m)×`
hu0 hu1
0 hu0
. .. hu1 .. hu . n+n+ Ψ` = 0 hun+n+ 0 0 . .. . . . 0 0
f0e
0 0
..
.
.. .
..
.
0
..
.
hu0
..
.
hu1
e f1 γ` = .. , . e fN −1
··· ···
.. . 0
.. .
,
(27)
hun+n+
f0u
u f1 ψ` = .. , . u fN −1
with 0(n−m)×` ∈ R(n−m)×` denoting the zero matrix of the specified dimensions.
8
(26)
(28)
Proof: From Lemma 1 and (12) we have that z N − B(z)P (z) , zN A(z)P (z) , Su (z) = zN S(z) =
(29) (30)
where P (z) = z ` P˜o (z) + A+ (z)D` (z). The constraint (13) can be satisfied if we write D` (z) as ˜ ` (z), D` (z) = (z − 1)D n o ˜ ` (z) is an arbitrary polynomial such that deg D ˜ ` (s) = ` − 1. Substituting (29)-(31) in where D and (18) yields ¯¯2 ¯¯ ¯¯ z N − z ` B(z)P˜ (z) ¯¯ ¯¯ o ˜ ` (z)¯¯¯¯ , Je (D` (z)) = ¯¯ − B(z)A+ (z)D ¯¯ ¯¯ z−1 2 ¯¯ ¯¯2 ¯¯ e ¯¯ e ˜ = ¯¯F (z) − H (z)D` (z)¯¯ , 2 ¯¯ N ¯¯2 ¯¯ z A(1) − z ` A(z)Qo (z) ¯¯ ˜ ` (z)¯¯ , Ju (D` (z)) = ¯¯¯¯ − A(z)A+ (z)D ¯¯ z−1 2 ¯¯ ¯¯2 ¯¯ u ¯¯ u ˜ = ¯¯F (z) − H (z)D` (z)¯¯ , 2
(31) (17)
(32) (33) (34) (35)
where H e (z), H u (z), F e (z) and F u (z) are defined in (21)-(24), and we have used the fact that z −N preserves the H2 norm. Expressions (33) and (35) can be vectorized as follows ¯¯£ ´¯¯2 ¤³ ¯¯ ¯¯ Je (D` (z)) = ¯¯ 1 z . . . z N −1 γ` − Γ` d˜` ¯¯ , (36) 2 ¯¯£ ´¯¯2 ¤³ ¯¯ ¯¯ Ju (D` (z)) = ¯¯ 1 z . . . z N −1 ψ` − Ψ` d˜` ¯¯ , (37) 2
where γ` , Γ` , ψ` and Ψ` are defined in (26)-(28), and d˜` ∈ R`×1 is such that £ ¤T d˜` = d˜0 d˜1 . . . d˜`−1 ,
(38)
˜ ` (z) = P`−1 d˜i z i . Substituting (36) and (37) in (19) and using elementary properties of the H2 and D i=0 norm yields ¯¯· √ ¸ · √ ¸ ¯¯2 ¯¯ ¯¯ λγ` λΓ` J (D` (z)) = ¯¯¯¯ √ − √ d˜` ¯¯¯¯ , (39) 1 − λψ` 1 − λΨ` e where ||·||e denotes the Euclidean norm. Thus, the original optimization problem can be posed in a standard least squares framework as d˜o` = arg min J (D` (z)) . d˜` ∈R`×1
9
(40)
The optimal vector d˜o` is then given by (Goodwin & Payne 1977) · √ ¸† · √ ¸ λΓ ` o ˜ √ λγ` d` = √ . 1 − λΨ` 1 − λψ`
(41)
The result (25) follows by substituting (41) in (31). ¤¤¤ Last theorem is comparable to those results of (Casavola et al. 1999, Paz & Elaydi 1998, Paz 2006), in which useful design methodologies based on optimization routines are proposed. Nonetheless, the framework used throughout this paper allows us to obtain an analytical solution to the two objective optimal control problem of our interest, which draws a difference with the previous existent work on the topic. This issue, together with the fact we use an H2 performance index, constitute the main distinctions of this work with respect to (Casavola et al. 1999, Paz & Elaydi 1998, Paz 2006). Theorem 1 provides a ripple-free deadbeat controller design procedure that can be summarized as follows: i. Choose a deadbeat horizon N = Nmin + ` and a scalar weight 0 ≤ λ ≤ 1. ii. Compute the unique solution (Po (z), Lo (z)) of the diophantine equation (4) such that Po (z) = P˜o (z)A− (z), Lo (1) = 0 and deg {Po (z)} = deg {Lo (z)} = n. iii. Compute the polynomials F e (z), H e (z), F u (z) and H u (z) using (21)-(24). iv. The optimal controller is obtained by computing D`∗ (z) from (25) and substituting it in (3) using (12).
5
Optimal performance
Having derived an explicit formula for the optimal ripple-free deadbeat controller in the previous section, we now turn our attention to the performance properties of the optimal design. An useful result would be to derive an analytical expression for the optimal performance achieved by the ripplefree deadbeat controller proposed of last section. This would allow to explore which dynamical features of the plant model are those that have a deleterious impact on the optimal performance achievable by a ripple-free deadbeat controller. Unfortunately, this is a difficult task so we restrict ourselves to show a qualitative feature of the optimal controller that sheds light on the design tradeoffs that appear in ripple-free deadbeat control. This constitutes the main result of this paper and is stated in next theorem. Theorem 2 Let G(z) be a discrete-time plant model and suppose that the ripple-free deadbeat controller defined by Lemma 1 is computed using the optimal polynomial D`o (z) that solves Problem 1. Then, the optimal cost J (D`o (z)) is a non-increasing function of `.
10
Proof: To keep a compact notation let us define · √ ¸ λΓ` M` = √ , 1 − λΨ` · √ ¸ λγ` √ m` = . 1 − λψ`
(42) (43)
It should be observed that M` ∈ R2N ×` and has full column rank (which can be verified from (26) and (27)). These facts imply that ³ ´−1 M` † = M` T M` M` T . (44) Therefore, from (39) and (41), it is simple to prove (Goodwin & Payne 1977) that the optimal cost for a horizon N is given by µ ¶ ³ ´−1 J (D`o (z)) = m` T I2N − M` M` T M` M` T m` . (45) Analogously, if we increase the deadbeat horizon in one sample, the optimal cost can be computed from µ ¶ ³ ´−1 ¢ ¡ o (z) = m`+1 T I2N+2 − M`+1 M`+1 T M`+1 M`+1 T m`+1 . J D`+1 (46) From the definition of Γ` and Ψ` in (26) and (27) we observe that "√ # λ heo 01×` M`+1 = , q T` M` where q ∈ R(2N +1)×1 is given by h√ √ e q= λ he1 · · · λ hm+n+
01×(n−m+`)
√
1 − λ hu0
···
(47)
√
1 − λ hun+n+
01×`
iT
,
(48)
and £ T` = e1
···
eN
eN+2
···
e2N+1
¤T
∈ R(2N +1)×2N ,
(49)
with ei denoting the ith element of the canonical basis of R2N +1 . Similarly, from (23), (24), (28) and (43), it follows that the vector m`+1 can be conveniently expressed as · ¸ 0 m`+1 = (50) . T` m ` Equation (47) implies that · M`+1 T M`+1 =
¸ λhe0 2 + qT q qT T` M` . M ` T TT q M` T M` 11
(51)
With some algebra after substituting (47), (50)-(51) in (46) and applying the matrix inversion lemma (Goodwin et al. 2001) it follows that ¡ o ¢ 1 T T J D`+1 (z) = (T` m` ) Λ (T` m` ) − (T` m` ) ΛqqT Λ (T` m` ) , κ
(52)
where ³ ´−1 T T Λ = I2N +1 − (T` M` ) (T` M` ) (T` M` ) (T` M` ) ,
(53)
κ = λhe0 2 + qT Λq.
(54)
Using the fact that T` T T` = I2N in (52) and ¡comparing ¢ with (45), we recognize the optimal cost for o horizon N in the first term of (52), so that J D`+1 (z) can be expressed as µ ¶ ¶2 ³ ´−1 1 T T T T √ m` I2N − M` M` M` M` T q . (55) κ ¡ o ¢ Therefore it can be concluded that J D`+1 (z) ≤ J (D`o (z)) , ∀` ≥ 0 and the result follows using an inductive procedure. ¡ o ¢ J D`+1 (z) = J (D`o (z)) −
µ
¤¤¤ The result of Theorem 2 reveals an essential tradeoff present in the design of ripple-free deadbeat control systems. If the focus of the designer is to have a deadbeat response with short settling time, last theorem implies that this can only be done at the expense of sacrificing performance. Similarly, one can only improve the transient performance by extending the settling time (and the complexity of the controller). This kind of design tradeoff had already been reported in (Casavola et al. 1999) for a mixed l∞ /l1 performance measure that accounts for the same elements as those in this work, i.e. tracking error and control input. Therefore, our result stands as a complement to these previous findings, adding an insightful conclusion to the topic, namely, that the performance vs. settling time tradeoff appears not only in a l∞ /l1 framework, but also in the context of H2 performance measures. It should be emphasized that our conclusion cannot be intuitively derived from the results in (Casavola et al. 1999), since essentially the l∞ /l1 and H2 approaches measure different aspects of the transient response.
6
Conclusions
In this paper a ripple-free deadbeat controller design methodology for step references has been proposed and analyzed. The ripple-free property of the controller is in the sense that, after the settling time, no ripples appear in the output of the plant. The design algorithm is applicable to stable and unstable plants under sampled data control and is such that, for a fixed settling time, the controller is adjusted to minimize a quadratic cost function. This cost function penalizes a combination of the energy of both the tracking error and control signal, and allows to pose a convex optimization problem that is solved with aid of standard properties of the H2 norm. This also allows to derive an explicit formula for the optimal controller. 12
Since the controller is obtained from an optimization procedure, the only design parameters in this strategy are the settling time and a weighting factor. The most significative result of this paper is showing that the optimal cost function is a non-increasing function of the settling time for any value of the weighting factor. This conclusion is a complement to those similar derived in previous literature (Casavola et al. 1999) using different norms and therefore, it reveals that the settling time/performance tradeoff is essential of deadbeat control.
Acknowledgements The authors gratefully acknowledge the support received from Universidad T´ecnica Federico Santa Mar´ıa (PAC program and grant UTFSM 230623) and from CONICYT through grant Fondecyt 1060437.
References Casavola, A., Mosca, E. & Zecca, P. (1999), ‘Robust ripple-free deadbeat control design’, International Journal of Control 72(6), 564–573. Emami-Naeini, A. & Franklin, G. (1982), ‘Deadbeat control and tracking of discrete-time systems’, IEEE Transactions on Automatic Control 27(1), 176–181. Franklin, G. F. & Emami-Naeini, A. (1986), ‘Design of ripple free multivariable robust servomechanisms’, IEEE Transactions on Automatic Control 31(7), 661–664. Funahashi, Y. & Katoh, H. (1992), ‘Robust-tracking deadbeat control’, International Journal of Control 56(1), 213–225. Golub, G. & Van Loan, C. (1996), Matrix computations, 3rd edn, The John Hopkins University Press, Baltimore, Maryland. Goodwin, G. C., Graebe, S. & Salgado, M. E. (2001), Control System Design, Prentice Hall, New Jersey. Goodwin, G. & Payne, R. (1977), Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York. Hartley, T., Veillette, R. & Cook, G. (1996), Techniques in deadbeat and one-step ahead control, in ‘Digital Control Systems Implementation and Computational Techniques: Control and Dynamic Systems & Advances in Theory and Applications’, Vol. 79, Academic Press NY, chapter 3. Kuˇcera, V. (1979), Discrete Linear Control: The Polynomial Equation Approach, John Wiley & Sons, Inc. New York, NY, USA. Kuˇcera, V. (1993), ‘Diophantine equations in control — a survey’, Automatica 29, 1361–1375. Lam, J., Tso, H. K. & Tsing, N. K. (1997), ‘Robust deadbeat regulation’, International Journal of Control 67(4), 587–602. Milonidis, E. & Karcanias, E. (2006), ‘Finite settling time stabilisation for multivariable discrete-time systems: a polynomial equation approach’, International Journal of Control 79(12), 1565–1580. 13
Morari, M. & Zafiriou, E. (1989), Robust process control, Prentice Hall Inc., Englewood Cliffs, New Jersey. Nobuyama, E. (1993), ‘Parametrization of all ripple-free deadbeat controllers’, Systems and Control Letters 21(3), 217–244. Paz, R. (2006), ‘Ripple-free tracking with robustness’, International Journal of Control 79(6), 543–568. Paz, R. & Elaydi, H. (1998), ‘Optimal ripple-free deadbeat controllers’, International Journal of Control 71(6), 1087–1104. Sirisena, H. R. (1985), ‘Ripple-free deadbeat control of SISO discrete systems’, IEEE Transactions on Automatic Control 30(2), 168–170. Toker, O., Chen, L. & Qiu, L. (2002), ‘Tracking performance limitations in LTI multivariable discretetime systems’, IEEE Transactions on Circuits and Systems—Part I: Fundamental Theory and Applications 49(5), 657–670. Urikura, S. & Nagata, A. (1987), ‘Ripple-free deadbeat control for sampled-data systems’, IEEE Transactions on Automatic Control 32(6), 474–482. Zhao, Y. & Kimura, H. (1986), ‘Dead-beat control with robustness’, International Journal of Control 43(5), 1427–1440. Zhao, Y. & Kimura, H. (1988), ‘Two-degrees-of-freedom dead-beat control system with robustness’, International Journal of Control 48(1), 303–315. Zhao, Y. & Kimura, H. (1994), ‘Optimization of control input for deadbeat control systems’, International Journal of Control 59(4), 1107–1117.
14
