Multirate Output Functional Observer in Linear ... - Semantic Scholar
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Multirate Output Functional Observer in Linear Feedback for Systems with Unknown Inputs Neeli Satyanarayana1 and S Janardhanan2 observing states and unknown disturbances by differentiating the output measurement was investigated by Park and Stein [10] for linear systems and by Callafona et.al in [7]. In [11], a combined state and disturbance estimator is proposed which does not require differentiation of the measured outputs. Xiong and Saif [12], extended the method proposed in [11] to reduced-order ones. However, observing a linear combination of the state and disturbance vector is amenable for the underlying task and the observer problem can be solved with less computational effort. The objective of this paper is to provide a solution for this problem. The proposed method is combination of the functional observer design and multirate output sampling (MROS) technique. This paper is organized as follows. The background knowledge about the design of both dynamical and nondynamical functional observer for systems with unknown disturbance inputs is presented in Section II. The motivation and problem statement appears, in section III. In section IV, a preliminary results is introduced, in particular the formation of augmented system for the case of slowly varying disturbance. This form is used to derive necessary and sufficient conditions and present the procedure for the design of functional observer to estimate the given functional using multirate sampling of the plant output in section V. A numerical example is included in section VI to illustrate the design procedure and to demonstrate its simplicity. Finally, a conclusions are drawn in section VII followed by references.
Abstract— This paper proposes a new method to design a scalar dynamical functional observer based feedback control, for a class of linear sampled-data systems with additive disturbances. The control action consists of the function of system state and disturbance components. The design of the function of state is based on the pole-placement technique and the function of disturbance is for the complete disturbance-counteraction on system performance. This work is based on the combination of functional observer design and multirate output sampling technique. Necessary and sufficient conditions for existence of the proposed controller are derived and the design procedure is given. It is shown that the proposed method relaxes some of the conditions of disturbance decoupled functional observer design, provided that the variation of disturbance in the two consecutive sampling instances is not significant. Numerical example is considered to illustrate the design procedure and its simplicity. Index Terms— Multirate output sampling, dynamical functional observer, linear feedback, unknown inputs.
I. INTRODUCTION In system modeling, there is always some degree of uncertainties such as nonlinearities, parameter changes, actuator faults, interconnecting terms in large-scale systems, and unknown external excitation which can be conveniently represented as unknown inputs which also termed as additive disturbances in the plant dynamics [1], [2]. The implementation of a feedback control law to make the closed-loop system to be asymptotically stable due to the specified poleplacement with the system performance is completely free from the effects of disturbance, is based on the assumption that all of the system states are available (measurable), y(t) = x(t) and the information about the disturbance is known a priori [3]. However, in most physical systems, this premise is not always the case. An arbitrary output, y(t) = Cx(t) was considered, for the case of a constant disturbance, by Davison and Smith [4], for the case of a polynomial-type disturbances, by Davison [5]. Shreyas Sundaram in his thesis [6], proposed the design of delayed observers to estimate the state vector and unknown inputs. In [8], an observer based technique is proposed, to estimate the state variables and also the unknown disturbance inputs with measurement of control inputs and outputs. This technique has been effectively applied in design of the controller, to reduce structural fatigue in large wind turbines by Parker and Johnson [9]. The problem of simultaneously
II. BACKGROUND A. Disturbance decoupled functional observer 1) Dynamical functional observer: Consider the sampleddata system representation for a sampling rate of τ sec x(k + 1) = Φτ x(k) + Γτ u(k) + Dτ v(k) y(k) = Cx(k)
(1b) n
r
rd
where x(k) ∈ R , u(k) ∈ R , v(k) ∈ R , and y(k) ∈ Rm , with matrices being of appropriate dimensions. The existing disturbance decoupled functional observer which will estimate the linear function of the state vector, follows directly from the continuous-time equivalent of [14] as
1 Neeli
Satyanarayana is Assistant Professor with the Department of Electrical and Electronics Engineering, Amity University, Noida, India
z(k + 1) = F z(k) + Gy(k) + Hu(k)
(2a)
w(k) = P y(k) + Qz(k)
(2b)
and is required to have w(k) → Lx(k), as k → ∞, where L is an (r × n) matrix; r < n. Obviously, such a functional observer have the same structure as that of deterministic
[email protected] 2S
Janardhanan is with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India [email protected]
978-1-4799-3271-9/$31.00 ©2014 AACC
(1a)
451
system [13]. From (1b) and (2b), the necessary condition for w(k) → Lx(k) is z(k) → T x(k) for a constant T . The necessary and sufficient conditions for z(k) → T x(k) is [14] T Φτ − F T = GC,
(3a)
H = T Γτ ,
(3b)
T Dτ = 0,
(3c)
F is stable.
(3d)
given in [30] cannot always be satisfied with small values of N . Apart from the vast results on scalar functional observer designs for deterministic systems, there is little contribution for the estimation of scalar linear functional observers for systems with disturbances. Therefore, we consider the problem of estimating (functional observation) any given scalar function of the system state and disturbance to implement linear feedback for single-input systems with additive disturbance and taking the advantage of using MROS technique. Consider a SIMO continuous-time linear time-invariant system described by
Substitute z(k) → T x(k) into (2b), then L = P C + QT
(4)
for w(k) → Lx(k). The observer with T Dτ = 0 has good robustness against control disturbance, known as disturbance decoupled functional observers [15]–[17] having dynamical in nature. The problem of designing the disturbance decoupled functional observers has found applications in decentralized estimation [19] and indirectly in the context of fault diagnosis [20], [21]. Ha et al. [18], presented a sliding-mode control of systems with disturbance using pole placement [22] and linear dynamical functional observer design algorithm [23]. However, dynamic observers may increase the complexity of the system. 2) Non-dynamical functional observer: In general, the conventional sampled-data systems assume that the sampling periods of hold and sampler are equal and synchronous at time. However, the sampling periods of the sampled-data system need not necessarily be the same. Systems having different sampling periods at the input and output of the plant are called multirate systems [25], [26]. In multirate output sampling (MROS) technique [27], the plant output is sampled at a faster rate as compared to the update of control input. Using MROS, wherein the update of control signal at sampling time τ sec and the output is measured at sampling time ∆/N sec, Janardhanan and Kariwala [28] derived the system states in terms of past output and control input without incurring the added complexity of the dynamic disturbance decoupled observers, where N is greater than the observability index [29] of the system. However, this technique was equivalent to a full order observer rather than a functional observer. Recently, Janardhanan and Satyanarayana [30] gave necessary and sufficient conditions for existence of non-dynamical functional observer in terms of past multirate output and control input as, wm (k) = Pm yk + Qm u(k − 1)
x(t) ˙ = Ac x(t) + bc u(t) + dc v¯(t)
(6a)
y(t) = Cx(t)
(6b)
where x(t) ∈ Rn is the system state, u(t) is the scalar control input, v¯(t) ∈ Rrd is the unknown disturbance input, and y(t) ∈ Rm is the measurable output of the system. For system (6), the system matrices have appropriate dimensions, and matrix C is of full row rank. Suppose that the sampling interval is τ sec, and a zero-order-hold is adopted for the aforementioned continuous-time model. Denoting x(k) = x(kτ ), y(k) = y(kτ ), u(k) = u(kτ ), and v(k) = v(kτ ), where k ≥ 0 is an integer, the discrete-time model can then be given by x(k + 1) = Φτ x(k) + γτ u(k) + dτ v(k)
(7a)
y(k) = Cx(k)
(7b)
Rτ Rτ where Φτ = eAc τ , γτ = 0 eAc t dt bc , dτ = 0 eAc t dt dc . Remark 1: To R τget closed form of expression, we define a v(k) such that 0 eAc t dc v¯((k + 1)τ − t) dt = dτ v(k). Assumption 1: System (7) is stabilisable and detectable. Assumption 2: It is assumed that the disturbance affecting the system is a matched. This implies dτ = γτ ρ with ρ being a scalar value. In physical terms, this assumption implies that the disturbance affecting the system enters through the input channels. For single input system with assumption 2, the disturbance is scalar. Hence, for further discussions, we consider (7) with scalar disturbance signal, i.e., rd = 1. The control objective is to asymptotically stabilize the system and counteract the effects of disturbance. The control u(k) consists of two components, up (k), ud (k), where up (k), is the control responsible for asymptotic stability of closed loop system, assumed to be a linear state function l10 x(k), and ud (k), is the disturbance-counteraction control in the form l2 v(k). Due to non-availability of x(k) and v(k), we propose a new design of dynamical functional observer to estimate,
(5)
where wm (k) is an estimate of Lx(k), Pm ∈ Rr×N m , and Qm ∈ Rr×r . Here, the condition on N being the greater than the observability index is need not be satisfied.
u(k) = l10 x(k) + l2 v(k)
III. M OTIVATION AND P ROBLEM F ORMULATION
(8)
where l10 is row vector of length n and l2 is a scalar.
For system with unknown disturbance inputs, the functional observer (2) have the same structure as that of deterministic case [13]. However, it necessitates an extra condition T Dτ = 0. On the other hand, a non-dynamical functional observers are also proposed based on MROS, as an alternative to dynamical observers. However, the conditions
IV. P RELIMINARIES Let the system (6) be sampled at ∆ = τ /N sec
452
x(k + 1) = Φx(k) + γu(k) + d∆ v(k)
(9a)
y(k) = Cx(k)
(9b)
where N is a positive integer. The matrices of the τ –system and ∆–system have the relation −1 −1 NX NX Φτ = ΦN , γτ = Φi γ, dτ = Φi d∆ . i=1
V. M AIN RESULT For all sampling periods of τ , the inverse of matrix Φτ of the sampled-data system (7) exists. Hence, (14a) can be re-arranged as
i=1
¯ −1 ¯ −1 ¯τ u(k) xv (k) = Φ τ xv (k + 1) − Φτ γ
Then, the multirate system representation with the input sampling time of τ sec and the output of sampling time ∆sec can be shown to be x(k + 1) = Φτ x(k) + γτ u(k) + dτ v(k)
(10a)
yk+1 = C0 x(k) + d0 u(k) + cd v(k)
(10b)
Substitute (16) in (14b) gives ¯ −1 ¯ −1 ¯τ )u(k) yk+1 = Cov Φ τ xv (k + 1) + (d0 − Cov Φτ γ
¯ v (k) y¯k = Cx
0 Cd∆ .. .
cd =
C
NP −2
Φi d∆
, yk+1 =
y(kτ ) y(kτ + ∆) .. .
¯ −1 y¯k = yk − (d0 − Cov Φ ¯τ )u(k − 1) τ γ u(k) = l0 xv (k) where, l0 = l10 multirate model
y((k + 1)τ − ∆)
Based on the results presented in previous sections, we derive necessary and sufficient conditions for the existence and give a constructive procedure for estimation of the functional (20) to implement the feedback control for system (7) by proposing the dynamical functional observer of the following form:
(12)
zp (k + 1) = Fp zp (k) + Gp y¯k + hp u(k) wp (k) =
¯τ = Γ
Cov =
Φτ 0 γτ 0 C0
dτ 1
∈ R(n+1)×(n+1)
∈ R(n+1) cd ∈ RN m×(n+1)
p0p y¯k
+
qp0 zp (k)
(22a) (22b)
where wp (k) is the estimate of l0 xv (k) and zp (k) will estimate Tp xv (k). Fp , Gp , hp , p0p , and qp0 are constant matrices of appropriate dimensions to be designed. The next two lemmas establish conditions for this is to be true. Lemma 1: The observer state vector, zp (k) of (22a) is an estimate of Tp xv (k) if and only if the following conditions hold. (a) Fp is stable matrix, ¯ ¯ τ − Fp Tp = Gp C, (b) Tp Φ (c) hp = Tp γ¯τ . Proof: Define the error in observer state vector
(13)
(14b)
where
(21b)
A. Structure of controller
From (11), (12), and the preceding definition of xv (k), we have ¯ τ xv (k) + γ¯τ u(k) xv (k + 1) = Φ (14a)
¯τ = Φ
(21a)
Now the problem of observing the functional, l10 x(k)+l2 v(k) from the input and output signals of sampled-data system (7) turns out to be estimation of (20) from (21).
Define the state vector of the augmented system (11) as x(k) xv (k) = v(k)
yk+1 = Cov xv (k) + d0 u(k)
(20) l2 , and xv (k) is the state vector of a
¯ τ xv (k) + γ¯τ u(k) xv (k + 1) = Φ ¯ v (k) y¯k = Cx
Assumption 3 ( [28]): It is assumed that the disturbance is slowly varying, i.e., |v(k + 1) − v(k)| is not significant. Hence, v(k) would be a good estimate of v(k + 1). Using the lifted technique and assumption 3, the augmented model is then given by x(k + 1) Φτ dτ x(k) γτ = + u(k) v(k + 1) 0 1 v(k) 0 (11)
and functional (8), x(k) u(k) = l10 l2 v(k)
(19)
The functional (13) to be estimated is rewritten as
i=0
We re-arrange the multirate output (10b) x(k) + d0 u(k) yk+1 = C0 cd v(k)
(18)
¯ −1 and where, C¯ = Cov Φ τ
i=0
(17)
Rearranging (17) and shifting one sampling period backwards results into
where matrices, C0 ∈ RN m×n , d0 ∈ RN m , cd ∈ RN m and the multirate output vector yk+1 ∈ RN m are given as 0 C Cγ CΦ .. C0 = , d = .. 0 . . NP −2 N −1 i CΦ C Φγ
(16)
ed (k) = zp (k) − Tp xv (k)
(15a)
(23)
The error (23) is governed by (15b)
¯ τ )xv (k) ed (k + 1) = Fp ed (k) + (Fp Tp + Gp C¯ − Tp Φ ¯ τ )wp (k) (24) + (hp − Tp Γ
(15c) 453
Algorithm 1 Design algorithm for the proposed functional observer based feedback controller. 1: Consider the continuous–time system (6). 2: Select an appropriate sampling (input update) period τ , choose output multiplicity, N . 3: Given, τ and N , obtain the multirate output representation of the system (10). 4: Now formulate the augmented system model (21). Let ¯τ, Γ ¯ τ , C). ¯ this model represented by (Φ 5: Assign the order of the functional observer based controller to be { n+1 N m } − 1. 6: Choose the matrix Fp with stable eigenvalues (for simplicity, choose Fp to be diagonal with distinct eigenvalues). n+1 7: Take qp0 = [1 1 1...1] ∈ R1×({ N m }−1) . Solve for p0p ∈ n+1 n+1 R1×N m , Gp ∈ R({ N m }−1)×N m , Tp ∈ R({ N m }−1)×n ¯ τ − Fp Tp = GC¯ and l0 = p0p C¯ + qp0 Tp , using from Tp Φ scheme of algorithm proposed in [24]. 8: Finally, obtain hp = Tp γ ¯τ .
If Condition (b) and Condition (c) hold, then ed (k + 1) = Fp ed (k)
(25)
In addition, if condition (a) is also satisfied, lim ed (k) = 0
(26)
k→∞
Conversely, suppose ed (k) → 0 for all x(0), u(k), and zp (0). Letting x(0) and u(k) vanish establishes Condition (a). Condition (b) and Condition (c) must hold, otherwise stabilizability of (21) and existence of u(k) would make ed (k) arbitrarily large. Lemma 2: The observer output wp (k) estimates l0 xv (k) if zp (k) → Tp xv (k) with error ed (k) and l0 = p0p C¯ + qp0 Tp
(27)
Proof: Observing wp (k) − l0 xv (k)
=p0p y¯k + qp0 zp (k) − l0 xv (k) =q 0 ed (k) + (p0 C¯ + q 0 Tp − l0 )xv (k) p
p
p
γτ ∈ R5 , dτ ∈ R5 , and C ∈ R1×5 are 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 Φτ = ; γτ = dτ = −1 0 0 0.5 0 0 0 0 1 2 2 −1 1.5 −0.25 −1 C = 1 0 1 −1 0
The conditions (25), (27), ensure wp (k) is an asymptotic estimate of l0 xv (k). B. Order of the proposed controller and its design procedure Luenberger has shown in [13] that any specified scalar linear functional l0 xv (k), where xv (k) is state vector of (21), can be estimated by the functional observer of order ν − 1, with arbitrary dynamics, where ν is the observability index ¯ τ , C). ¯ Namely, ν is the smallest integer for of the pair (Φ which the matrix C¯ ¯τ C¯ Φ O(Φ¯ τ ,C),ν = ¯ .. . ν−1 ¯ ¯ C Φτ
,
The sampling period is 0.3sec. The open-loop poles of this system are at, −1.2082 ± 1.0189i, 0.1930 ± 0.8601i and 1.304. We propose to stabilize the system by pole-placement technique which is a linear combination of the state vector, where the functional gain is, l10 = −2.515
2.156
−2.043
0.099
2.086
with nullify the effect of matched input disturbance by ρ = l2 = −1. It has been proposed to implement the feedback based control law of the form
has rank(n + 1). For sampled-data system (7) with m outputs, multirate output sampling with output multiplicity N , produces N m successive outputs over the sampling interval [iτ, (i + 1)τ ), i = 1, 2, · · · . The order of the proposed multirate output functional observer based controller (22a) is the dimension of the vector, zp (k). For almost all sampling periods τ sec, the x } denotes dimension of zp (k) is { n+1 N m } − 1, where { the smallest integer greater than or equal to x. Based on the preceding lemmas and predetermined order, the design procedure for the proposed functional observer is given by algorithm 1.
u(k) = l10 x(k) + l2 v(k) The output is samples at an interval of ∆ = 0.3 N sec, where output multiplicity N = 3 and input updated once in every τ sec, the output of multirate model, given as 1 0 1 −1 0 −0.8 −0.673 x(k) yk+1 = 1.26 −0.213 1.48 0.98 0.118 1.12 −0.017 −1.141 0 0 + 0.462 u(k) + 0.462 v(k) 0.281 0.281
VI. N UMERICAL E XAMPLE Consider the sampled-data representation of continuoustime plant (7) with single input and matched disturbance. The system matrices of sampled-data form will be Φτ ∈ R5×5 ,
The system matrices of the augmented model (21) is formu454
lated as
30
0 0 0 0 2 0
1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 −1 , γ¯τ = −1 0 0 0 1 0.5 0.5 2 −1 1.5 −0.25 −1 2 0 0 0 0 1 0 0.5 0.25 −0.875 0.5 0.5 −2.625 C¯ = 0.41 0.53 −0.641 −0.043 0.63 −1.83 0.60 0.38 0.105 −0.65 0.49 −0.87
¯ Φτ =
25 20 15 Output
10 5 0 −5 0
Given, n = 5, m = 1 and N = 3, the order of functional observer is { n+1 N m }−1 = 1. Choose the stable matrix Fp with eigenvalue at 0.1 and qp0 = 1. Hence the proposed multirate output sampling functional observer-based controller will take the form
wp (k) =
130.951
−55.058
Gp =
−6.321
79.795 −149.422
Tp =
−50.26
−4.4
8.492
76.3
−62.23 −32.9
249.3
also hp = −223.488. an initial For the simulation purpose, 0 state of xv (0) = 0 1 0.3 −1 0 0.2 and slowly varying disturbance signal v(k) = 5 sin(k/20) exp(−k/500) is used. Figs. 1, and 2, show the simulation results of the
estimation error
5
0
2
Fig. 1.
4 6 Sampling period, τ sec
8
4
Trajectory of output
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10
0
3.5
R EFERENCES
15
−5
3
In this paper, we have proposed a new method to design a scalar linear functional observer based controller for systems with unknown disturbance inputs. The proposed method is simple and is based on multirate sampling of the plant output. Under the assumption of slowly varying disturbance, an augmented system has been constructed. With respect to other procedure [14], the design procedure, proposed in this work does not require the solution of Tp dτ = 0, yet preserves the behavior of disturbance decoupled observers. The proposed method guarantees the existence of functional controller provided if conditions of Luenberger observer [13] is satisfied. Extension of proposed method to multivariable system is under investigation.
¯ τ −Fp Tp = where, p0p , Gp and Tp are obtained by solving Tp Φ 0 0 ¯ 0 ¯ Gp C and l = pp C + qp Tp , according to the algorithm 1, we obtain
1.5 2 2.5 Sampling period, ∆ sec
VII. C ONCLUSIONS
+ zp (k)
p0p =
1
Fig. 2.
zp (k + 1) = 0.1zp (k) + Gp y¯k + hp u(k) p0p y¯k
0.5
10
Trajectory of estimation error
error in estimation of functional and system output. These simulation results demonstrate that our proposed design of functional observer-based controller will asymptotically estimate the given functional and closed–loop system will be stable. 455
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Multirate Output Functional Observer in Linear Feedback for Systems with Unknown Inputs Neeli Satyanarayana1 and S Janardhanan2 observing states and unknown disturbances by differentiating the output measurement was investigated by Park and Stein [10] for linear systems and by Callafona et.al in [7]. In [11], a combined state and disturbance estimator is proposed which does not require differentiation of the measured outputs. Xiong and Saif [12], extended the method proposed in [11] to reduced-order ones. However, observing a linear combination of the state and disturbance vector is amenable for the underlying task and the observer problem can be solved with less computational effort. The objective of this paper is to provide a solution for this problem. The proposed method is combination of the functional observer design and multirate output sampling (MROS) technique. This paper is organized as follows. The background knowledge about the design of both dynamical and nondynamical functional observer for systems with unknown disturbance inputs is presented in Section II. The motivation and problem statement appears, in section III. In section IV, a preliminary results is introduced, in particular the formation of augmented system for the case of slowly varying disturbance. This form is used to derive necessary and sufficient conditions and present the procedure for the design of functional observer to estimate the given functional using multirate sampling of the plant output in section V. A numerical example is included in section VI to illustrate the design procedure and to demonstrate its simplicity. Finally, a conclusions are drawn in section VII followed by references.
Abstract— This paper proposes a new method to design a scalar dynamical functional observer based feedback control, for a class of linear sampled-data systems with additive disturbances. The control action consists of the function of system state and disturbance components. The design of the function of state is based on the pole-placement technique and the function of disturbance is for the complete disturbance-counteraction on system performance. This work is based on the combination of functional observer design and multirate output sampling technique. Necessary and sufficient conditions for existence of the proposed controller are derived and the design procedure is given. It is shown that the proposed method relaxes some of the conditions of disturbance decoupled functional observer design, provided that the variation of disturbance in the two consecutive sampling instances is not significant. Numerical example is considered to illustrate the design procedure and its simplicity. Index Terms— Multirate output sampling, dynamical functional observer, linear feedback, unknown inputs.
I. INTRODUCTION In system modeling, there is always some degree of uncertainties such as nonlinearities, parameter changes, actuator faults, interconnecting terms in large-scale systems, and unknown external excitation which can be conveniently represented as unknown inputs which also termed as additive disturbances in the plant dynamics [1], [2]. The implementation of a feedback control law to make the closed-loop system to be asymptotically stable due to the specified poleplacement with the system performance is completely free from the effects of disturbance, is based on the assumption that all of the system states are available (measurable), y(t) = x(t) and the information about the disturbance is known a priori [3]. However, in most physical systems, this premise is not always the case. An arbitrary output, y(t) = Cx(t) was considered, for the case of a constant disturbance, by Davison and Smith [4], for the case of a polynomial-type disturbances, by Davison [5]. Shreyas Sundaram in his thesis [6], proposed the design of delayed observers to estimate the state vector and unknown inputs. In [8], an observer based technique is proposed, to estimate the state variables and also the unknown disturbance inputs with measurement of control inputs and outputs. This technique has been effectively applied in design of the controller, to reduce structural fatigue in large wind turbines by Parker and Johnson [9]. The problem of simultaneously
II. BACKGROUND A. Disturbance decoupled functional observer 1) Dynamical functional observer: Consider the sampleddata system representation for a sampling rate of τ sec x(k + 1) = Φτ x(k) + Γτ u(k) + Dτ v(k) y(k) = Cx(k)
(1b) n
r
rd
where x(k) ∈ R , u(k) ∈ R , v(k) ∈ R , and y(k) ∈ Rm , with matrices being of appropriate dimensions. The existing disturbance decoupled functional observer which will estimate the linear function of the state vector, follows directly from the continuous-time equivalent of [14] as
1 Neeli
Satyanarayana is Assistant Professor with the Department of Electrical and Electronics Engineering, Amity University, Noida, India
z(k + 1) = F z(k) + Gy(k) + Hu(k)
(2a)
w(k) = P y(k) + Qz(k)
(2b)
and is required to have w(k) → Lx(k), as k → ∞, where L is an (r × n) matrix; r < n. Obviously, such a functional observer have the same structure as that of deterministic
[email protected] 2S
Janardhanan is with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India [email protected]
978-1-4799-3271-9/$31.00 ©2014 AACC
(1a)
451
system [13]. From (1b) and (2b), the necessary condition for w(k) → Lx(k) is z(k) → T x(k) for a constant T . The necessary and sufficient conditions for z(k) → T x(k) is [14] T Φτ − F T = GC,
(3a)
H = T Γτ ,
(3b)
T Dτ = 0,
(3c)
F is stable.
(3d)
given in [30] cannot always be satisfied with small values of N . Apart from the vast results on scalar functional observer designs for deterministic systems, there is little contribution for the estimation of scalar linear functional observers for systems with disturbances. Therefore, we consider the problem of estimating (functional observation) any given scalar function of the system state and disturbance to implement linear feedback for single-input systems with additive disturbance and taking the advantage of using MROS technique. Consider a SIMO continuous-time linear time-invariant system described by
Substitute z(k) → T x(k) into (2b), then L = P C + QT
(4)
for w(k) → Lx(k). The observer with T Dτ = 0 has good robustness against control disturbance, known as disturbance decoupled functional observers [15]–[17] having dynamical in nature. The problem of designing the disturbance decoupled functional observers has found applications in decentralized estimation [19] and indirectly in the context of fault diagnosis [20], [21]. Ha et al. [18], presented a sliding-mode control of systems with disturbance using pole placement [22] and linear dynamical functional observer design algorithm [23]. However, dynamic observers may increase the complexity of the system. 2) Non-dynamical functional observer: In general, the conventional sampled-data systems assume that the sampling periods of hold and sampler are equal and synchronous at time. However, the sampling periods of the sampled-data system need not necessarily be the same. Systems having different sampling periods at the input and output of the plant are called multirate systems [25], [26]. In multirate output sampling (MROS) technique [27], the plant output is sampled at a faster rate as compared to the update of control input. Using MROS, wherein the update of control signal at sampling time τ sec and the output is measured at sampling time ∆/N sec, Janardhanan and Kariwala [28] derived the system states in terms of past output and control input without incurring the added complexity of the dynamic disturbance decoupled observers, where N is greater than the observability index [29] of the system. However, this technique was equivalent to a full order observer rather than a functional observer. Recently, Janardhanan and Satyanarayana [30] gave necessary and sufficient conditions for existence of non-dynamical functional observer in terms of past multirate output and control input as, wm (k) = Pm yk + Qm u(k − 1)
x(t) ˙ = Ac x(t) + bc u(t) + dc v¯(t)
(6a)
y(t) = Cx(t)
(6b)
where x(t) ∈ Rn is the system state, u(t) is the scalar control input, v¯(t) ∈ Rrd is the unknown disturbance input, and y(t) ∈ Rm is the measurable output of the system. For system (6), the system matrices have appropriate dimensions, and matrix C is of full row rank. Suppose that the sampling interval is τ sec, and a zero-order-hold is adopted for the aforementioned continuous-time model. Denoting x(k) = x(kτ ), y(k) = y(kτ ), u(k) = u(kτ ), and v(k) = v(kτ ), where k ≥ 0 is an integer, the discrete-time model can then be given by x(k + 1) = Φτ x(k) + γτ u(k) + dτ v(k)
(7a)
y(k) = Cx(k)
(7b)
Rτ Rτ where Φτ = eAc τ , γτ = 0 eAc t dt bc , dτ = 0 eAc t dt dc . Remark 1: To R τget closed form of expression, we define a v(k) such that 0 eAc t dc v¯((k + 1)τ − t) dt = dτ v(k). Assumption 1: System (7) is stabilisable and detectable. Assumption 2: It is assumed that the disturbance affecting the system is a matched. This implies dτ = γτ ρ with ρ being a scalar value. In physical terms, this assumption implies that the disturbance affecting the system enters through the input channels. For single input system with assumption 2, the disturbance is scalar. Hence, for further discussions, we consider (7) with scalar disturbance signal, i.e., rd = 1. The control objective is to asymptotically stabilize the system and counteract the effects of disturbance. The control u(k) consists of two components, up (k), ud (k), where up (k), is the control responsible for asymptotic stability of closed loop system, assumed to be a linear state function l10 x(k), and ud (k), is the disturbance-counteraction control in the form l2 v(k). Due to non-availability of x(k) and v(k), we propose a new design of dynamical functional observer to estimate,
(5)
where wm (k) is an estimate of Lx(k), Pm ∈ Rr×N m , and Qm ∈ Rr×r . Here, the condition on N being the greater than the observability index is need not be satisfied.
u(k) = l10 x(k) + l2 v(k)
III. M OTIVATION AND P ROBLEM F ORMULATION
(8)
where l10 is row vector of length n and l2 is a scalar.
For system with unknown disturbance inputs, the functional observer (2) have the same structure as that of deterministic case [13]. However, it necessitates an extra condition T Dτ = 0. On the other hand, a non-dynamical functional observers are also proposed based on MROS, as an alternative to dynamical observers. However, the conditions
IV. P RELIMINARIES Let the system (6) be sampled at ∆ = τ /N sec
452
x(k + 1) = Φx(k) + γu(k) + d∆ v(k)
(9a)
y(k) = Cx(k)
(9b)
where N is a positive integer. The matrices of the τ –system and ∆–system have the relation −1 −1 NX NX Φτ = ΦN , γτ = Φi γ, dτ = Φi d∆ . i=1
V. M AIN RESULT For all sampling periods of τ , the inverse of matrix Φτ of the sampled-data system (7) exists. Hence, (14a) can be re-arranged as
i=1
¯ −1 ¯ −1 ¯τ u(k) xv (k) = Φ τ xv (k + 1) − Φτ γ
Then, the multirate system representation with the input sampling time of τ sec and the output of sampling time ∆sec can be shown to be x(k + 1) = Φτ x(k) + γτ u(k) + dτ v(k)
(10a)
yk+1 = C0 x(k) + d0 u(k) + cd v(k)
(10b)
Substitute (16) in (14b) gives ¯ −1 ¯ −1 ¯τ )u(k) yk+1 = Cov Φ τ xv (k + 1) + (d0 − Cov Φτ γ
¯ v (k) y¯k = Cx
0 Cd∆ .. .
cd =
C
NP −2
Φi d∆
, yk+1 =
y(kτ ) y(kτ + ∆) .. .
¯ −1 y¯k = yk − (d0 − Cov Φ ¯τ )u(k − 1) τ γ u(k) = l0 xv (k) where, l0 = l10 multirate model
y((k + 1)τ − ∆)
Based on the results presented in previous sections, we derive necessary and sufficient conditions for the existence and give a constructive procedure for estimation of the functional (20) to implement the feedback control for system (7) by proposing the dynamical functional observer of the following form:
(12)
zp (k + 1) = Fp zp (k) + Gp y¯k + hp u(k) wp (k) =
¯τ = Γ
Cov =
Φτ 0 γτ 0 C0
dτ 1
∈ R(n+1)×(n+1)
∈ R(n+1) cd ∈ RN m×(n+1)
p0p y¯k
+
qp0 zp (k)
(22a) (22b)
where wp (k) is the estimate of l0 xv (k) and zp (k) will estimate Tp xv (k). Fp , Gp , hp , p0p , and qp0 are constant matrices of appropriate dimensions to be designed. The next two lemmas establish conditions for this is to be true. Lemma 1: The observer state vector, zp (k) of (22a) is an estimate of Tp xv (k) if and only if the following conditions hold. (a) Fp is stable matrix, ¯ ¯ τ − Fp Tp = Gp C, (b) Tp Φ (c) hp = Tp γ¯τ . Proof: Define the error in observer state vector
(13)
(14b)
where
(21b)
A. Structure of controller
From (11), (12), and the preceding definition of xv (k), we have ¯ τ xv (k) + γ¯τ u(k) xv (k + 1) = Φ (14a)
¯τ = Φ
(21a)
Now the problem of observing the functional, l10 x(k)+l2 v(k) from the input and output signals of sampled-data system (7) turns out to be estimation of (20) from (21).
Define the state vector of the augmented system (11) as x(k) xv (k) = v(k)
yk+1 = Cov xv (k) + d0 u(k)
(20) l2 , and xv (k) is the state vector of a
¯ τ xv (k) + γ¯τ u(k) xv (k + 1) = Φ ¯ v (k) y¯k = Cx
Assumption 3 ( [28]): It is assumed that the disturbance is slowly varying, i.e., |v(k + 1) − v(k)| is not significant. Hence, v(k) would be a good estimate of v(k + 1). Using the lifted technique and assumption 3, the augmented model is then given by x(k + 1) Φτ dτ x(k) γτ = + u(k) v(k + 1) 0 1 v(k) 0 (11)
and functional (8), x(k) u(k) = l10 l2 v(k)
(19)
The functional (13) to be estimated is rewritten as
i=0
We re-arrange the multirate output (10b) x(k) + d0 u(k) yk+1 = C0 cd v(k)
(18)
¯ −1 and where, C¯ = Cov Φ τ
i=0
(17)
Rearranging (17) and shifting one sampling period backwards results into
where matrices, C0 ∈ RN m×n , d0 ∈ RN m , cd ∈ RN m and the multirate output vector yk+1 ∈ RN m are given as 0 C Cγ CΦ .. C0 = , d = .. 0 . . NP −2 N −1 i CΦ C Φγ
(16)
ed (k) = zp (k) − Tp xv (k)
(15a)
(23)
The error (23) is governed by (15b)
¯ τ )xv (k) ed (k + 1) = Fp ed (k) + (Fp Tp + Gp C¯ − Tp Φ ¯ τ )wp (k) (24) + (hp − Tp Γ
(15c) 453
Algorithm 1 Design algorithm for the proposed functional observer based feedback controller. 1: Consider the continuous–time system (6). 2: Select an appropriate sampling (input update) period τ , choose output multiplicity, N . 3: Given, τ and N , obtain the multirate output representation of the system (10). 4: Now formulate the augmented system model (21). Let ¯τ, Γ ¯ τ , C). ¯ this model represented by (Φ 5: Assign the order of the functional observer based controller to be { n+1 N m } − 1. 6: Choose the matrix Fp with stable eigenvalues (for simplicity, choose Fp to be diagonal with distinct eigenvalues). n+1 7: Take qp0 = [1 1 1...1] ∈ R1×({ N m }−1) . Solve for p0p ∈ n+1 n+1 R1×N m , Gp ∈ R({ N m }−1)×N m , Tp ∈ R({ N m }−1)×n ¯ τ − Fp Tp = GC¯ and l0 = p0p C¯ + qp0 Tp , using from Tp Φ scheme of algorithm proposed in [24]. 8: Finally, obtain hp = Tp γ ¯τ .
If Condition (b) and Condition (c) hold, then ed (k + 1) = Fp ed (k)
(25)
In addition, if condition (a) is also satisfied, lim ed (k) = 0
(26)
k→∞
Conversely, suppose ed (k) → 0 for all x(0), u(k), and zp (0). Letting x(0) and u(k) vanish establishes Condition (a). Condition (b) and Condition (c) must hold, otherwise stabilizability of (21) and existence of u(k) would make ed (k) arbitrarily large. Lemma 2: The observer output wp (k) estimates l0 xv (k) if zp (k) → Tp xv (k) with error ed (k) and l0 = p0p C¯ + qp0 Tp
(27)
Proof: Observing wp (k) − l0 xv (k)
=p0p y¯k + qp0 zp (k) − l0 xv (k) =q 0 ed (k) + (p0 C¯ + q 0 Tp − l0 )xv (k) p
p
p
γτ ∈ R5 , dτ ∈ R5 , and C ∈ R1×5 are 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 Φτ = ; γτ = dτ = −1 0 0 0.5 0 0 0 0 1 2 2 −1 1.5 −0.25 −1 C = 1 0 1 −1 0
The conditions (25), (27), ensure wp (k) is an asymptotic estimate of l0 xv (k). B. Order of the proposed controller and its design procedure Luenberger has shown in [13] that any specified scalar linear functional l0 xv (k), where xv (k) is state vector of (21), can be estimated by the functional observer of order ν − 1, with arbitrary dynamics, where ν is the observability index ¯ τ , C). ¯ Namely, ν is the smallest integer for of the pair (Φ which the matrix C¯ ¯τ C¯ Φ O(Φ¯ τ ,C),ν = ¯ .. . ν−1 ¯ ¯ C Φτ
,
The sampling period is 0.3sec. The open-loop poles of this system are at, −1.2082 ± 1.0189i, 0.1930 ± 0.8601i and 1.304. We propose to stabilize the system by pole-placement technique which is a linear combination of the state vector, where the functional gain is, l10 = −2.515
2.156
−2.043
0.099
2.086
with nullify the effect of matched input disturbance by ρ = l2 = −1. It has been proposed to implement the feedback based control law of the form
has rank(n + 1). For sampled-data system (7) with m outputs, multirate output sampling with output multiplicity N , produces N m successive outputs over the sampling interval [iτ, (i + 1)τ ), i = 1, 2, · · · . The order of the proposed multirate output functional observer based controller (22a) is the dimension of the vector, zp (k). For almost all sampling periods τ sec, the x } denotes dimension of zp (k) is { n+1 N m } − 1, where { the smallest integer greater than or equal to x. Based on the preceding lemmas and predetermined order, the design procedure for the proposed functional observer is given by algorithm 1.
u(k) = l10 x(k) + l2 v(k) The output is samples at an interval of ∆ = 0.3 N sec, where output multiplicity N = 3 and input updated once in every τ sec, the output of multirate model, given as 1 0 1 −1 0 −0.8 −0.673 x(k) yk+1 = 1.26 −0.213 1.48 0.98 0.118 1.12 −0.017 −1.141 0 0 + 0.462 u(k) + 0.462 v(k) 0.281 0.281
VI. N UMERICAL E XAMPLE Consider the sampled-data representation of continuoustime plant (7) with single input and matched disturbance. The system matrices of sampled-data form will be Φτ ∈ R5×5 ,
The system matrices of the augmented model (21) is formu454
lated as
30
0 0 0 0 2 0
1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 −1 , γ¯τ = −1 0 0 0 1 0.5 0.5 2 −1 1.5 −0.25 −1 2 0 0 0 0 1 0 0.5 0.25 −0.875 0.5 0.5 −2.625 C¯ = 0.41 0.53 −0.641 −0.043 0.63 −1.83 0.60 0.38 0.105 −0.65 0.49 −0.87
¯ Φτ =
25 20 15 Output
10 5 0 −5 0
Given, n = 5, m = 1 and N = 3, the order of functional observer is { n+1 N m }−1 = 1. Choose the stable matrix Fp with eigenvalue at 0.1 and qp0 = 1. Hence the proposed multirate output sampling functional observer-based controller will take the form
wp (k) =
130.951
−55.058
Gp =
−6.321
79.795 −149.422
Tp =
−50.26
−4.4
8.492
76.3
−62.23 −32.9
249.3
also hp = −223.488. an initial For the simulation purpose, 0 state of xv (0) = 0 1 0.3 −1 0 0.2 and slowly varying disturbance signal v(k) = 5 sin(k/20) exp(−k/500) is used. Figs. 1, and 2, show the simulation results of the
estimation error
5
0
2
Fig. 1.
4 6 Sampling period, τ sec
8
4
Trajectory of output
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10
0
3.5
R EFERENCES
15
−5
3
In this paper, we have proposed a new method to design a scalar linear functional observer based controller for systems with unknown disturbance inputs. The proposed method is simple and is based on multirate sampling of the plant output. Under the assumption of slowly varying disturbance, an augmented system has been constructed. With respect to other procedure [14], the design procedure, proposed in this work does not require the solution of Tp dτ = 0, yet preserves the behavior of disturbance decoupled observers. The proposed method guarantees the existence of functional controller provided if conditions of Luenberger observer [13] is satisfied. Extension of proposed method to multivariable system is under investigation.
¯ τ −Fp Tp = where, p0p , Gp and Tp are obtained by solving Tp Φ 0 0 ¯ 0 ¯ Gp C and l = pp C + qp Tp , according to the algorithm 1, we obtain
1.5 2 2.5 Sampling period, ∆ sec
VII. C ONCLUSIONS
+ zp (k)
p0p =
1
Fig. 2.
zp (k + 1) = 0.1zp (k) + Gp y¯k + hp u(k) p0p y¯k
0.5
10
Trajectory of estimation error
error in estimation of functional and system output. These simulation results demonstrate that our proposed design of functional observer-based controller will asymptotically estimate the given functional and closed–loop system will be stable. 455
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