Robust Stabilization for Discretized PID Control Systems ... - IEEE Xplore
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThBIn5.2
Robust Stabilization for Discretized PID Control Systems with Transmission Delay Yoshifumi Okuyama Abstract— This paper deals with a designing problem of nonlinear discrete-time and discrete-value (discretized) control systems on an integer grid. Especially, in this study, robust stabilization based on the discretized PID control accompanied with transmission delay is discussed. The transmission delay is assumed to be uncertain and varying within prescribed bounds. The robust stabilization is examined in a frequency domain with respect to the worst case. In this study, the concept of a modified Nyquist and Hall diagram (off-axis M-circles) for nonlinear control systems is applied to the design procedure. Numerical examples for continuous plants are provided to verify the design method.
I. I NTRODUCTION At present, almost all feedback control systems are realized using discretized (discrete-time and discretevalue) signals. However, the analysis and design of discretized/quantized control systems has not been entirely elucidated. An attempt to elucidate the quantized control systems was presented first in a paper [1]. Since then, the problem of mitigating the quantization effects in quantized control systems has been studied [2], [3], [4], [5]. However, few results have been obtained for the stability analysis of the nonlinear discrete-time feedback system [6], [7]. In our previous papers [8], [9], [10], a robust stability condition of nonlinear discretized control systems that accompany discretizing units (quantizers) at equal spaces was examined in a frequency domain. In the study, it was assumed that the discretization is executed on the input and output sides of a nonlinear element (sensor/actuator), and that the sampling period is chosen of such a size suitable for the discretization in the space. This paper describes a designing problem of such nonlinear discrete-time control systems on an integer grid [11], [12]. Especially, in this study, robust stabilization based on the discretized PID control accompanied with transmission delay is discussed. There are some results for stabilization of linear discrete systems with transmission delay in the state space, e.g, [13]. In this paper, the robust stabilization of a discretized control system is examined in a frequency domain because the transmission delay will be uncertain and varying within prescribed bounds. In the design procedure, the concept of a modified Nyquist and Hall diagram (off-axis M-circles) for nonlinear control systems in [14] is applied. II. D ISCRETIZED C ONTROL S YSTEM The discretized control system to be considered here is represented by a sampled-data (discrete-time) feedback Yoshifumi Okuyama is a director of Humanitech Laboratory Co., Ltd., 115-7, Nakatsuura, Hachiman-cho, Tokushima, 770-8072, Japan [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
u u† + her e† D1 - N (e† ) - D2 - C(z) − 6
y
H
h H H S
P (s) † vℓ
u†c
+ ? d† T (z) † h v +
P (z) Fig. 1.
Discretized control system with transmission delay.
−ε
Fig. 2.
ε
Discretized nonlinear characteristics on an integer grid.
system as shown in Fig. 1. In the figure, D1 D2 and H denote the discretizing units and the zero-order hold, respectively, which are usually performed in A/D (D/A) conversion. Here, N (·) is a nonlinear characteristic which is determined by sensor and actuator elements, and C(z) and T (z) are a digital controller (compensator) based on the PID control scheme and a transmission system with time-delay, respectively. Moreover, P (s) is a linear continuous plant (physical system to be controlled). The relationship between e and u† = N (e) is a stepwise nonlinear characteristic on a grid pattern as shown in Fig. 2. In this paper, a round-down discretization, which is usually executed on a computer, is applied. Therefore, the relationship between e† and u† is indicated by small circles on the stepwise nonlinear characteristic. In Fig. 1, each symbol e, u, y, · · · indicates the sequence e(k), u(k), y(k), · · · , (k = 0, 1, 2, · · · ) in discrete time, but for continuous value. On the other hand, each symbol e† , u† , v † , · · · indicates a discrete value that can be assigned to an
5120
ThBIn5.2 integer number, e.g.,
On the other hand, the transfer characteristic of transmission system T (z) is given by
†
e ∈ {· · · , −3γ, −2γ, −γ, 0, γ, 2γ, 3γ, · · · }, u† ∈ {· · · , −3γ, −2γ, −γ, 0, γ, 2γ, 3γ, · · · },
T (z) = z −L ,
(1)
†
v ∈ {· · · , −3γ, −2γ, −γ, 0, γ, 2γ, 3γ, · · · },
where γ is the resolution of each variable. In the above expression, it is assumed that the input and output signals of the nonlinear characteristic have the same resolution in the discretization. Here, e† and u† also represent the sequence e† (k) and u† (k). Without loss of generality, hereafter, we assume γ = 1.0. On the other hand, the time variable t is defined as t ∈ {0, h, 2h, 3h, · · · } for the sampling period h. Hereafter, we assume h = 1.0. In other words, the following integer time sequence is defined: k ∈ Z+ ,
Z+ := {0, 1, 2, 3, · · · }.
where L ∈ Z+ is a transmission delay(lag). In this paper, the transmission delay L is assumed to be uncertain and varying within prescribed bounds as follows: Lℓ ≤ L ≤ Lu . The following discussion will be given with respect to the upper bound Lu (i.e., usually the worst case). IV. ROBUST S TABILITY C ONDITION As shown in Fig. 2, the stepwise nonlinear characteristic is expressed as: N (e) = Ke + g(e),
PID control scheme has been widely used in practice and theory irrespective of whether it is continuous or discrete in time [15], [16], since it is a basic feedback control technique. The discrete-time PID controller C(z) is given by the following algorithm: u† (j) + Cd ∆u† (k),
(2)
j=0
where ∆u† (k) = u† (k) − u† (k − 1) is a backward difference in integer numbers, and each coefficient is defined as Kp , Ci , Cd ∈ Z+ ,
Z+ = {0, 1, 2, 3 · · · }.
Here, Kp , Ci , and Cd correspond to Kp , Kp h/TI , and Kp TD /h in the following (discrete-time z-transform expression) PID algorithm: TD h −1 + (1 − z ) . (3) C(z) = Kp 1 + TI (1 − z −1 ) h We use algorithm (2) without division because the variables u† , uc , and coefficients Kp , Ci , Cd are integers. Using the z-transform expression, equation (2) is written as:
(6)
|g(e)| ≤ β |e|,
0 ≤ β < ∞,
(8)
for |e| ≥ ε. When considering the robust stability in a global sense, it is sufficient to consider the nonlinear term (8) for |e| ≥ ε because the nonlinear term (7) can be treated as a disturbance signal [8]. (In this study, a fluctuation or an offset of error is assumed to be allowable in |e| < ε.) Equation (8) represents a sectorial nonlinearity for which the equivalent linear gain exists in a limited range. It can also be expressed as follows: 0 ≤ g(e)e ≤ βe2 . (9) With respect to the nonlinear sector β, the following theorem is given. [Theorem-1] If there exists a q ≥ 0 in which the sector parameter β with respect to nonlinear term g(·) satisfies the following inequality, the discrete-time control system with sector nonlinearity (8) is robust stable in an ℓ2 sense: β < β0 = Kη(q0 , ω0 ) = max min Kη(q, ω), q
ω
(10)
when the linearized system with nominal gain K is stable. (That is, the allowable sector can be given as [0, β0 ] from (10).) The η-function is written as follows: η(q, ω) := −qΩV + q 2 Ω2 V 2 + (U 2 + V 2 ){(1 + U )2 + V 2 } , U2 + V 2 ∀ω ∈ [0, ωc ]. (11)
In the closed form, controller C(z) can be given as
1 + Cd (1 − z −1 ) (4) 1 − z −1 for discrete-time systems. When comparing equations (3) and (4), Ci and Cd become equal to Kp h/TI and Kp TD /h, respectively.
(7)
for |e| < ε, and
uc (z) = C(z)u(z) = Kp + Ci (1 + z −1 + z −2 + · · · ) + Cd (1 − z −1 ) u(z). C(z) = Kp + Ci ·
0 < K < ∞,
|g(e)| ≤ g¯ < ∞,
III. PID C ONTROLLER AND T RANSMISSION S YSTEM
k
(5)
It can be partitioned into the following two sections:
That is, the variables e† (k), u† (k), and v † (k) are defined on a grid pattern that is composed of integers in the time and controller variables space.
u†c (k) = Kp u† (k) + Ci
and vℓ† (k) = v † (k − L),
Here, U and V are the real and the imaginary parts of the loop transfer function:
5121
KP (ejωh )T (ejωh )C(ejωh ) = KG(ejωh )e−jωLu (12) = KG(ejωh )(cos ωLu − sin ωLu ) = U (ω) + jV (ω).
ThBIn5.2 Moreover, Ω(ω) is the distorted frequency of angular frequency ω and is given by √ ωh 2 jωh δ(e ) = jΩ(ω) = j tan , j = −1 (13) h 2
between equations (16) and (17) with respect to the continuous values is shown by the block diagram in Fig. 3. Thus, the loop transfer function from v ∗ to e∗ can be given by W (β, q, z), as shown in Fig. 4, where
and ωc is a cut-off frequency. Here, δ corresponds to the bilinear transformation: 2 1 − z −1 · . (14) h 1 + z −1 in the z plane. (As will be shown in the proof, neutral points of the time sequences are considered here on account of a stepwise nonlinear characteristic.) With respect to the theorem, the following assumption must be provided on the base of the relatively fast sampling and the slow response of the controlled system. [Assumption] The absolute value of the backward difference of error e(k) does not exceed γ, i.e., δ(z) :=
|∆e(k)| = |e(k) − e(k − 1)| ≤ γ.
(15)
If condition (15) is satisfied, ∆e† (k) is exactly ±γ or 0 because of the discretization. That is, the absolute value of the backward difference can be given as |∆e† (k)| = |e† (k) − e† (k − 1)| = γ or 0.
The assumption stated above will be satisfied by the following examples. The phase trace of backward difference ∆e† is shown, for example, in Fig. 8. (Proof) The proof of this theorem is derived from the following new sequences ∆e† (k) , = +q· h ∆e† (k) ∗† † vm (k) = vm (k) − βq · . h
e∗† m (k)
e†m (k)
W (β, q, z) =
† where q is a non-negative number, e†m (k) and vm (k) are † † neutral points of sequences e (k) and v (k), and ∆e† (k) is the backward difference of sequence e† (k). The relationship
(18)
and r′ , d′ are transformed exogenous inputs. Here, the variables such as e∗ , v ∗ , u′ and y ′ written in Fig. 4 indicate the z-transformed ones. Based on the loop characteristic in Fig. 4, the following inequality can be given with respect to z = ejωh : ′ e∗m (z)2,p ≤ c1 rm (z)2,p + c2 d′m (z)2,p ∗† + sup |W (β, q, z)| · vm (z)2,p . (19) z=1
′ (z) rm
d′m (z)
denote the z-transformation for the and Here, neutral points of sequences r′ (k) and d′ (k), respectively. Moreover, c1 and c2 are positive constants. By applying inequality (e.g., Lemma-2 in [12]), ∗† vm (k)2,p ≤ βe∗m (k)2,p
(20)
the following expression is obtained: 1 − β · sup |W (β, q, z)| e∗m (z)2,p z=1
′ ≤ c1 rm (z)2,p + c2 d′m (z)2,p .
(21)
Therefore, if the following inequality (i.e., the small gain theorem with respect to ℓ2 gains) is valid, |W (β, q, ejωh )| = 1 (1 + jqΩ(ω))(U (ω) + jV (ω)) K + (K + jβqΩ(ω))(U (ω) + jV (ω)) < β .(22)
(16) (17)
(1 + qδ(z))P (z)T (z)C(z) , 1 + (K + βqδ(z))P (z)T (z)C(z)
the sequences e∗m (k), em (k), e(k) and y(k) in the feedback system are restricted in finite values when exogenous inputs r(k), d(k) are finite and p → ∞. From the square of both sides of inequality (22), the result of theorem is given. V. O FF -A XIS M-C IRCLES
e
` - 1 + qδ
e∗- g ∗ (·)
+ v ∗f 6+
v-
- βqδ g(e) Fig. 3.
+g r′ 6−
Nonlinear subsystem.
e∗ - ∗ g (·) y′
Fig. 4.
The design method adopted in this paper is based on the classical parameter tuning in the modified Hall diagram. This method can be conveniently designed, and it is significant in a physical sense (i.e., mechanical vibration and resonance). In the previous papers [10], [11], the inverse function was 1 used instead of the η-function, i.e., ξ(q, ω) = . Using η(q, ω) the notation, inequality (10) can be rewritten as follows: K . β
(23)
When q = 0, the ξ-function can be expressed as: √ U2 + V 2 ξ(0, ω) = = |Sc (ejωh )|, (1 + U )2 + V 2
(24)
M0 = ξ(q0 , ω0 ) = min max ξ(q, ω)
1, the following is obtained from (27). 2 M2 M2 + (V − λ)2 = + λ2 , U+ 2 2 M −1 (M − 1)2 qΩM λ= 2 ≥ 0. (28) M −1
Although the distorted frequency Ω is a function of ω, the term qΩ = cq ≥ 0 is assumed to be a constant parameter in this paper. Thus, it can been seen that (28) represents off2 2 axis circles with their center at (−M /(M −1), λ) and with radius of M 2 /(M 2 − 1) + λ2 . Figure 5 shows an example of the modified Hall diagram for 0 ≤ cq ≤ 4.0 and M = 1.4, and Nyquist curves of a control system with transmission delay (KG(ejωh ) · e−jωL , L = 0, 2, 4). Here, N1 is a vector locus that contacts with an M -circle at the peak value (Mp = ξ(0, ωp ) = 1.4). On the other hand, N2 is a vector locus that contacts with a circle C on the real axis, where all the M -circles cross the real axis. The latter case corresponds to the discrete-time system in which Aizerman’s conjecture is valid [14]. At the continuous saddle point where is also the phase-crossover
ξ(q, ω) ≤ ξ(q, ω0 ),
∀ω ∈ [0, ωc ].
(31)
The right-hand side of (31) is a peak value for angular frequency ω. Here, ω0 is not always determined as only one frequency, and may not be a smooth (differentiable) point of the frequency range depending on the q-value. Nonetheless, inequality (31) is also satisfied for q = q0 . Therefore, the following inequality is obtained: ξ(q0 , ω) ≤ ξ(q0 , ω0 ) = M0 , ∀ω ∈ [0, ωc ].
(32)
It can be shown that equation (30) in Theorem-2 is equivalent to (10) and (11). VI. N UMERICAL E XAMPLES [Example-1] Consider the following continuous plant and transmission system: P (s) =
K1 , (s + 0.1)(s + 1.0)
T (z) = z −L ,
(33)
where K1 = 0.002 = 2.0 × 10−3 and L = 5.0 ∼ 10.0. The discretized nonlinear characteristic (discretized sigmoid, i.e. arc tangent [17]) is as shown in Fig. 2. For C-language expression, it can be written as e† = γ ∗ (double)(int)(e/γ) u = 0.4 ∗ e† + 3.0 ∗ atan(0.6 ∗ e† ) u† = γ ∗ (double)(int)(u/γ),
TABLE I PID PARAMETERS FOR E XAMPLE -1 (gM [dB]: GAIN MARGINS , pM [deg]:
5123
PHASE MARGINS ,
(i) (ii) (iii) (iv) (v) (vi)
Kp 60 60 50 60 50 50
Mp : PEAK VALUES , β0 : ALLOWABLE SECTORS ).
Ci 0 3 3 3 3 2
Cd 0 0 30 0 30 40
L 10 10 10 15 15 15
β0 0.75 0.49 0.80 0.15 0.35 0.52
gM 4.87 3.41 5.15 1.21 2.55 3.64
pM 100.0 45.3 55.1 19.5 34.2 58.5
Mp 1.45 2.32 1.48 7.40 3.21 2.06
ThBIn5.2 ∆e
e
Pc
N2
N1 Fig. 8.
Phase traces for Example-1 (|∆e| ≤ 1, ∆e† (k) = ±1 or 0).
N3 Fig. 6. Off-axis circles and Nyquists plots for Example-1 (M = 1.45, cq = 0.0, · · · , 4.0).
(ii) (iii)
Pc
(i)
N4
N5 N6
Fig. 7.
Fig. 9. Off-axis circles and Nyquists plots for Example-1 (M = 2.06, cq = 0.0, · · · , 4.0).
Step responses for Example-1.
where (int) and (double) denote the conversion into an integral number (round-down discretization) and the reconversion into a double-precision real number, respectively. Here, the resolution is γ = 1.0. When choosing the nominal gain K = 1.0 and the threshold ε = 2.0, the sectorial area of the stepwise nonlinear characteristic for ε ≤ |e| can be determined as [0.5, 1.5] drawn by dotted lines in the figure. As is described in sections II and III the sampling period is chosen as h = 1.0, and the worst case of transmission delay Lu = 10.0 is considered. Figure 6 shows Nyquist plots of KG(ejωh ) · e−jωLu on the modified Hall diagram. Here, N1 , N2 , and N3 are cases (i), (ii), and (iii), respectively. The PID parameters are specified as shown in Table I. The gain margins gM [dB], the phase margin pM [deg] and the peak value Mp can be obtained from the phase crossover points Pc , the gain crossover points Gc , and the points of contact in regard to the M -circles, respectively. The max-min value β0 is calculated from (10) as follows: β0 = max Kη(q, ω0 ) = Kη(q0 , ω0 ) = 0.8.
for example (iii). Therefore, the allowable sector for nonlinear characteristic g(·) is given as [0, 1.8]. The stability of discretized control system (iii) will be guaranteed. In this example, the continuous saddle point (29) appears at point Pc (i.e., Aizerman’s conjecture is satisfied). Thus, the allowable interval of equivalent gain Kℓ can be given as 0 < Kℓ < 1.8. Figure 7 shows step responses for the three cases. In this figure, the time-scale line is drawn in 10h increments because of avoiding indistinctness. Sequences of the input u† (k) and the output u†c (k) of PID controller are also shown in the figure. Here, uc (k) is graduated in 1/100. Figure 8 shows phase traces, i.e., sequences of (e(k), ∆e) and (e† (k), ∆e† ). As is obvious from Fig. 8, assumtion (15) is satisfied. Since step response (i) remains an off-set, responses for cases (ii) and (iii) are stabilized and improved well by using the discretized PID action as specified in Table I. On the other hand, when the transmission delay increases from 10.0 to 15.0, (i.e., case (iv) in Table I), the robust stability is not guaranteed because the allowable sector for nonlinear characteristic is given as [0, 1.15]. (As for case (v),
5124
ThBIn5.2
(iv) (vi)
(v)
Pc
Fig. 10.
N1
N3
Step responses for Example-1.
N2
∆e Fig. 12. Off-axis circles and Nyquist plots for Example-2 (M = 1.6, cq = 0.0, · · · , 4.0).
e
(ii) (iii)
(i)
Fig. 11.
Phase traces for Example-1 (|∆e| ≤ 1, ∆e† (k) = ±1 or 0).
the allowable sector is given as [0, 1.35].) However, response for case (vi) is stabilized and improved well by using the discretized PID action. Figure 10 shows step responses for the three cases, and Figure 11 shows phase traces. Obviously, assumption (15) is satisfied in this example.
Fig. 13.
[Example-2] Consider the following continuous plant and transmission system: K2 (s + 0.5)(−s + 1.0) , P (s) = (s + 0.1)(s + 0.2)(s + 1.0)
T (z) = z
−L
Step responses for Example-2.
∆e
, (34)
where K2 = 0.001 = 1.0 × 10−3 and L = 5.0 ∼ 10.0. The same nonlinear characteristic and the nominal gain are chosen as shown in Example-1. Figure 12 shows Nyquist plots of KG(ejωh ) · e−jωLu on the modified Hall diagram. Here, N1 , N2 and N3 are cases (i), (ii), and (iii), and the PID parameters are specified as shown in Table II. In this example, the continuous saddle point (29) does not appear at Pc (i.e., Aizerman’s conjecture is not satisfied). Figure 13 shows step responses for the three cases, and Figure 14 shows phase traces. Also in this example, assumption (15) is satisfied. Although the allowable interval of equivalent linear gain is 0 < Kℓ < 1.75 (e.g., case (iii)), the allowable sector for nonlinear characteristic is [0, 1.64] as shown
5125
e
Fig. 14.
Phase traces for Example-2 (|∆e| ≤ 1, ∆e† (k) = ±1 or 0).
ThBIn5.2 TABLE II PID PARAMETERS FOR E XAMPLE -2 (gM [dB]: GAIN MARGINS , pM [deg]: PHASE MARGINS , Mp : PEAK VALUES , β0 : ALLOWABLE SECTORS ).
(i) (ii) (iii) (iv) (v) (vi)
Kp 50 50 40 50 40 30
Ci 0 2 2 2 2 2
Cd 0 0 20 0 20 40
L 10 10 10 15 15 15
β0 0.66 0.40 0.64 0.12 0.34 0.55
gM 4.3 2.8 4.5 1.0 2.6 4.4
pM 84.8 38.8 52.5 13.9 33.4 39.2
(iv) (vi) (v)
Mp 1.62 2.75 1.65 9.0 3.29 1.95
Fig. 16.
Step responses for Example-2.
accompany with transmission delay could be achieved. The design method will be applied to digital and discrete control systems. R EFERENCES
N4 N6 N5 Fig. 15. Off-axis circles and Nyquist curves for Example-2 (M = 1.9, cq = 0.0, · · · , 4.0).
in Table II. This is a counter example for Aizerman’s conjecture. However, since the sectorial area of the stepwise nonlinear characteristic is [0.5, 1.5], the robust stability of the nonlinear control system is guaranteed. The step responses are stabilized and improved well as shown in Fig. 13. On the other hand, when the transmission delay increases from 10.0 to 15.0, (i.e., cases (iv) in Table II), the robust stability is not guaranteed because the allowable sector for nonlinear characteristic is given as [0, 1.12]. (Note that the allowable interval of equivalent linear gain in this case is 0 < Kℓ < 1.42.) The step responses for (iv) and (v) actually fluctuate as shown in Fig. 16. However, the step response for (vi) is stabilized and improved using PID action. VII. C ONCLUSION In this paper, we have described robust stabilization for discretized PID control systems that are accompanied with transmission delay. The robust stabilization was examined in a frequency domain with respect to the worst-case delay. The design procedure is based on the modified Nyquist and Hall diagram and its parameter specifications in the frequency domain. The stability margins and the steady-state characteristic of the control system were specified directly in the diagram. In consequence, the robust stabilization for continuous plants using the discretized PID control
[1] R. E. Kalman, “Nonlinear Aspects of Sampled-Data Control Systems”, Proc. of the Symposium on Nonlinear Circuit Analysis, vol. VI, pp.273313, 1956. [2] R. E. Curry, Estimation and Control with Quantized Measurements, Cambridge, MIT Press, 1970. [3] D. F. Delchamps, “Stabilizing a Linear System with Quantized State Feedback”, IEEE Trans. on Automatic Control, vol. 35, pp. 916-924, 1990. [4] N. Elia and S. K. Mitter, “Stabilization of Linear System with Limited Information”, IEEE Trans. on Automatic Control, vol. 46, pp. 13841400, 2001. [5] M. Fu, “Robust Stabilization of Linear Uncertain Systems via Quantized Feedback”, IEEE Int. Conf. on Decision and Control, TuA06-5, 2003. [6] C. A. Desoer and M. Vidyasagar, Feedback System: Input-Output Properties, Academic Press, 1975. [7] C. J. Harris and M. E. Valenca, The Stability of Input-Output Dynamical Systema, Academic Press, 1983. [8] Y. Okuyama, “Robust Stability Analysis for Discretized Nonlinear Control Systems in a Global Sense”, Proc. of the 2006 American Control Conference, Minneapolis, USA, pp. 2321-2326, 2006. [9] Y. Okuyama, “Nonlinear Phenomena and Stability Analysis for Discretized Control Systems”, Proc. of IASTED Int. Workshop on Modern Nonlinear Theory, Montreal, Canada, pp. 350-355, 2007. [10] Y. Okuyama et al., “Robust Stability Evaluation for Sampled-Data Control Systems with a Sector Nonlinearity in a Gain-Phase Plane” Int. J. of Robust and Nonlinear Control, Vol. 9, No. 1, pp. 15-32, 1999. [11] Y. Okuyama, “Robust Stabilization and PID Control for Nonlinear Discretized Systems on a Grid Pattern”, Proc. of the 2008 American Control Conference, Seattle, USA, pp. 4746-4751, 2008. [12] Y. Okuyama, “Discretized PID Control and Robust Stabilization for Continuous Plant”, Proc. of the 17th IFAC World Congress, Seoul, Korea, pp. 1492-1498, 2008. [13] Z. Jiandong, “Stabilization of Linear Discrete Systems with Transmission Delay”, Proc. of the Chinese Control Conference, pp. 15-18, 2007. [14] Y. Okuyama et al., “Robust Stability Analysis for Non-Linear Sampled-Data Control Systems in a Frequency Domain”, European Journal of Control, Vol. 8, No. 2, pp. 99-108, 2002. [15] A. Datta, M.T. Ho and S.P. Bhattacharyya, Structure and Synthesis of PID Controllers, Springer-Verlag, 2000. [16] F. Takemori and Y. Okuyama, “Discrete-Time Model Reference Feedback and PID Control for Interval Plants” Digital Control 2000: Past, Present and Future of PID Control, Pergamon Press, pp. 260-265, 2000. [17] Y. Okuyama et al., “Amplitude Dependent Analysis and Stabilization for Nonlinear Sampled-Data Control Systems”, Proc. of the 15th IFAC World Congress, T-Tu-M08, 2002.
5126
ThBIn5.2
Robust Stabilization for Discretized PID Control Systems with Transmission Delay Yoshifumi Okuyama Abstract— This paper deals with a designing problem of nonlinear discrete-time and discrete-value (discretized) control systems on an integer grid. Especially, in this study, robust stabilization based on the discretized PID control accompanied with transmission delay is discussed. The transmission delay is assumed to be uncertain and varying within prescribed bounds. The robust stabilization is examined in a frequency domain with respect to the worst case. In this study, the concept of a modified Nyquist and Hall diagram (off-axis M-circles) for nonlinear control systems is applied to the design procedure. Numerical examples for continuous plants are provided to verify the design method.
I. I NTRODUCTION At present, almost all feedback control systems are realized using discretized (discrete-time and discretevalue) signals. However, the analysis and design of discretized/quantized control systems has not been entirely elucidated. An attempt to elucidate the quantized control systems was presented first in a paper [1]. Since then, the problem of mitigating the quantization effects in quantized control systems has been studied [2], [3], [4], [5]. However, few results have been obtained for the stability analysis of the nonlinear discrete-time feedback system [6], [7]. In our previous papers [8], [9], [10], a robust stability condition of nonlinear discretized control systems that accompany discretizing units (quantizers) at equal spaces was examined in a frequency domain. In the study, it was assumed that the discretization is executed on the input and output sides of a nonlinear element (sensor/actuator), and that the sampling period is chosen of such a size suitable for the discretization in the space. This paper describes a designing problem of such nonlinear discrete-time control systems on an integer grid [11], [12]. Especially, in this study, robust stabilization based on the discretized PID control accompanied with transmission delay is discussed. There are some results for stabilization of linear discrete systems with transmission delay in the state space, e.g, [13]. In this paper, the robust stabilization of a discretized control system is examined in a frequency domain because the transmission delay will be uncertain and varying within prescribed bounds. In the design procedure, the concept of a modified Nyquist and Hall diagram (off-axis M-circles) for nonlinear control systems in [14] is applied. II. D ISCRETIZED C ONTROL S YSTEM The discretized control system to be considered here is represented by a sampled-data (discrete-time) feedback Yoshifumi Okuyama is a director of Humanitech Laboratory Co., Ltd., 115-7, Nakatsuura, Hachiman-cho, Tokushima, 770-8072, Japan [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
u u† + her e† D1 - N (e† ) - D2 - C(z) − 6
y
H
h H H S
P (s) † vℓ
u†c
+ ? d† T (z) † h v +
P (z) Fig. 1.
Discretized control system with transmission delay.
−ε
Fig. 2.
ε
Discretized nonlinear characteristics on an integer grid.
system as shown in Fig. 1. In the figure, D1 D2 and H denote the discretizing units and the zero-order hold, respectively, which are usually performed in A/D (D/A) conversion. Here, N (·) is a nonlinear characteristic which is determined by sensor and actuator elements, and C(z) and T (z) are a digital controller (compensator) based on the PID control scheme and a transmission system with time-delay, respectively. Moreover, P (s) is a linear continuous plant (physical system to be controlled). The relationship between e and u† = N (e) is a stepwise nonlinear characteristic on a grid pattern as shown in Fig. 2. In this paper, a round-down discretization, which is usually executed on a computer, is applied. Therefore, the relationship between e† and u† is indicated by small circles on the stepwise nonlinear characteristic. In Fig. 1, each symbol e, u, y, · · · indicates the sequence e(k), u(k), y(k), · · · , (k = 0, 1, 2, · · · ) in discrete time, but for continuous value. On the other hand, each symbol e† , u† , v † , · · · indicates a discrete value that can be assigned to an
5120
ThBIn5.2 integer number, e.g.,
On the other hand, the transfer characteristic of transmission system T (z) is given by
†
e ∈ {· · · , −3γ, −2γ, −γ, 0, γ, 2γ, 3γ, · · · }, u† ∈ {· · · , −3γ, −2γ, −γ, 0, γ, 2γ, 3γ, · · · },
T (z) = z −L ,
(1)
†
v ∈ {· · · , −3γ, −2γ, −γ, 0, γ, 2γ, 3γ, · · · },
where γ is the resolution of each variable. In the above expression, it is assumed that the input and output signals of the nonlinear characteristic have the same resolution in the discretization. Here, e† and u† also represent the sequence e† (k) and u† (k). Without loss of generality, hereafter, we assume γ = 1.0. On the other hand, the time variable t is defined as t ∈ {0, h, 2h, 3h, · · · } for the sampling period h. Hereafter, we assume h = 1.0. In other words, the following integer time sequence is defined: k ∈ Z+ ,
Z+ := {0, 1, 2, 3, · · · }.
where L ∈ Z+ is a transmission delay(lag). In this paper, the transmission delay L is assumed to be uncertain and varying within prescribed bounds as follows: Lℓ ≤ L ≤ Lu . The following discussion will be given with respect to the upper bound Lu (i.e., usually the worst case). IV. ROBUST S TABILITY C ONDITION As shown in Fig. 2, the stepwise nonlinear characteristic is expressed as: N (e) = Ke + g(e),
PID control scheme has been widely used in practice and theory irrespective of whether it is continuous or discrete in time [15], [16], since it is a basic feedback control technique. The discrete-time PID controller C(z) is given by the following algorithm: u† (j) + Cd ∆u† (k),
(2)
j=0
where ∆u† (k) = u† (k) − u† (k − 1) is a backward difference in integer numbers, and each coefficient is defined as Kp , Ci , Cd ∈ Z+ ,
Z+ = {0, 1, 2, 3 · · · }.
Here, Kp , Ci , and Cd correspond to Kp , Kp h/TI , and Kp TD /h in the following (discrete-time z-transform expression) PID algorithm: TD h −1 + (1 − z ) . (3) C(z) = Kp 1 + TI (1 − z −1 ) h We use algorithm (2) without division because the variables u† , uc , and coefficients Kp , Ci , Cd are integers. Using the z-transform expression, equation (2) is written as:
(6)
|g(e)| ≤ β |e|,
0 ≤ β < ∞,
(8)
for |e| ≥ ε. When considering the robust stability in a global sense, it is sufficient to consider the nonlinear term (8) for |e| ≥ ε because the nonlinear term (7) can be treated as a disturbance signal [8]. (In this study, a fluctuation or an offset of error is assumed to be allowable in |e| < ε.) Equation (8) represents a sectorial nonlinearity for which the equivalent linear gain exists in a limited range. It can also be expressed as follows: 0 ≤ g(e)e ≤ βe2 . (9) With respect to the nonlinear sector β, the following theorem is given. [Theorem-1] If there exists a q ≥ 0 in which the sector parameter β with respect to nonlinear term g(·) satisfies the following inequality, the discrete-time control system with sector nonlinearity (8) is robust stable in an ℓ2 sense: β < β0 = Kη(q0 , ω0 ) = max min Kη(q, ω), q
ω
(10)
when the linearized system with nominal gain K is stable. (That is, the allowable sector can be given as [0, β0 ] from (10).) The η-function is written as follows: η(q, ω) := −qΩV + q 2 Ω2 V 2 + (U 2 + V 2 ){(1 + U )2 + V 2 } , U2 + V 2 ∀ω ∈ [0, ωc ]. (11)
In the closed form, controller C(z) can be given as
1 + Cd (1 − z −1 ) (4) 1 − z −1 for discrete-time systems. When comparing equations (3) and (4), Ci and Cd become equal to Kp h/TI and Kp TD /h, respectively.
(7)
for |e| < ε, and
uc (z) = C(z)u(z) = Kp + Ci (1 + z −1 + z −2 + · · · ) + Cd (1 − z −1 ) u(z). C(z) = Kp + Ci ·
0 < K < ∞,
|g(e)| ≤ g¯ < ∞,
III. PID C ONTROLLER AND T RANSMISSION S YSTEM
k
(5)
It can be partitioned into the following two sections:
That is, the variables e† (k), u† (k), and v † (k) are defined on a grid pattern that is composed of integers in the time and controller variables space.
u†c (k) = Kp u† (k) + Ci
and vℓ† (k) = v † (k − L),
Here, U and V are the real and the imaginary parts of the loop transfer function:
5121
KP (ejωh )T (ejωh )C(ejωh ) = KG(ejωh )e−jωLu (12) = KG(ejωh )(cos ωLu − sin ωLu ) = U (ω) + jV (ω).
ThBIn5.2 Moreover, Ω(ω) is the distorted frequency of angular frequency ω and is given by √ ωh 2 jωh δ(e ) = jΩ(ω) = j tan , j = −1 (13) h 2
between equations (16) and (17) with respect to the continuous values is shown by the block diagram in Fig. 3. Thus, the loop transfer function from v ∗ to e∗ can be given by W (β, q, z), as shown in Fig. 4, where
and ωc is a cut-off frequency. Here, δ corresponds to the bilinear transformation: 2 1 − z −1 · . (14) h 1 + z −1 in the z plane. (As will be shown in the proof, neutral points of the time sequences are considered here on account of a stepwise nonlinear characteristic.) With respect to the theorem, the following assumption must be provided on the base of the relatively fast sampling and the slow response of the controlled system. [Assumption] The absolute value of the backward difference of error e(k) does not exceed γ, i.e., δ(z) :=
|∆e(k)| = |e(k) − e(k − 1)| ≤ γ.
(15)
If condition (15) is satisfied, ∆e† (k) is exactly ±γ or 0 because of the discretization. That is, the absolute value of the backward difference can be given as |∆e† (k)| = |e† (k) − e† (k − 1)| = γ or 0.
The assumption stated above will be satisfied by the following examples. The phase trace of backward difference ∆e† is shown, for example, in Fig. 8. (Proof) The proof of this theorem is derived from the following new sequences ∆e† (k) , = +q· h ∆e† (k) ∗† † vm (k) = vm (k) − βq · . h
e∗† m (k)
e†m (k)
W (β, q, z) =
† where q is a non-negative number, e†m (k) and vm (k) are † † neutral points of sequences e (k) and v (k), and ∆e† (k) is the backward difference of sequence e† (k). The relationship
(18)
and r′ , d′ are transformed exogenous inputs. Here, the variables such as e∗ , v ∗ , u′ and y ′ written in Fig. 4 indicate the z-transformed ones. Based on the loop characteristic in Fig. 4, the following inequality can be given with respect to z = ejωh : ′ e∗m (z)2,p ≤ c1 rm (z)2,p + c2 d′m (z)2,p ∗† + sup |W (β, q, z)| · vm (z)2,p . (19) z=1
′ (z) rm
d′m (z)
denote the z-transformation for the and Here, neutral points of sequences r′ (k) and d′ (k), respectively. Moreover, c1 and c2 are positive constants. By applying inequality (e.g., Lemma-2 in [12]), ∗† vm (k)2,p ≤ βe∗m (k)2,p
(20)
the following expression is obtained: 1 − β · sup |W (β, q, z)| e∗m (z)2,p z=1
′ ≤ c1 rm (z)2,p + c2 d′m (z)2,p .
(21)
Therefore, if the following inequality (i.e., the small gain theorem with respect to ℓ2 gains) is valid, |W (β, q, ejωh )| = 1 (1 + jqΩ(ω))(U (ω) + jV (ω)) K + (K + jβqΩ(ω))(U (ω) + jV (ω)) < β .(22)
(16) (17)
(1 + qδ(z))P (z)T (z)C(z) , 1 + (K + βqδ(z))P (z)T (z)C(z)
the sequences e∗m (k), em (k), e(k) and y(k) in the feedback system are restricted in finite values when exogenous inputs r(k), d(k) are finite and p → ∞. From the square of both sides of inequality (22), the result of theorem is given. V. O FF -A XIS M-C IRCLES
e
` - 1 + qδ
e∗- g ∗ (·)
+ v ∗f 6+
v-
- βqδ g(e) Fig. 3.
+g r′ 6−
Nonlinear subsystem.
e∗ - ∗ g (·) y′
Fig. 4.
The design method adopted in this paper is based on the classical parameter tuning in the modified Hall diagram. This method can be conveniently designed, and it is significant in a physical sense (i.e., mechanical vibration and resonance). In the previous papers [10], [11], the inverse function was 1 used instead of the η-function, i.e., ξ(q, ω) = . Using η(q, ω) the notation, inequality (10) can be rewritten as follows: K . β
(23)
When q = 0, the ξ-function can be expressed as: √ U2 + V 2 ξ(0, ω) = = |Sc (ejωh )|, (1 + U )2 + V 2
(24)
M0 = ξ(q0 , ω0 ) = min max ξ(q, ω)
1, the following is obtained from (27). 2 M2 M2 + (V − λ)2 = + λ2 , U+ 2 2 M −1 (M − 1)2 qΩM λ= 2 ≥ 0. (28) M −1
Although the distorted frequency Ω is a function of ω, the term qΩ = cq ≥ 0 is assumed to be a constant parameter in this paper. Thus, it can been seen that (28) represents off2 2 axis circles with their center at (−M /(M −1), λ) and with radius of M 2 /(M 2 − 1) + λ2 . Figure 5 shows an example of the modified Hall diagram for 0 ≤ cq ≤ 4.0 and M = 1.4, and Nyquist curves of a control system with transmission delay (KG(ejωh ) · e−jωL , L = 0, 2, 4). Here, N1 is a vector locus that contacts with an M -circle at the peak value (Mp = ξ(0, ωp ) = 1.4). On the other hand, N2 is a vector locus that contacts with a circle C on the real axis, where all the M -circles cross the real axis. The latter case corresponds to the discrete-time system in which Aizerman’s conjecture is valid [14]. At the continuous saddle point where is also the phase-crossover
ξ(q, ω) ≤ ξ(q, ω0 ),
∀ω ∈ [0, ωc ].
(31)
The right-hand side of (31) is a peak value for angular frequency ω. Here, ω0 is not always determined as only one frequency, and may not be a smooth (differentiable) point of the frequency range depending on the q-value. Nonetheless, inequality (31) is also satisfied for q = q0 . Therefore, the following inequality is obtained: ξ(q0 , ω) ≤ ξ(q0 , ω0 ) = M0 , ∀ω ∈ [0, ωc ].
(32)
It can be shown that equation (30) in Theorem-2 is equivalent to (10) and (11). VI. N UMERICAL E XAMPLES [Example-1] Consider the following continuous plant and transmission system: P (s) =
K1 , (s + 0.1)(s + 1.0)
T (z) = z −L ,
(33)
where K1 = 0.002 = 2.0 × 10−3 and L = 5.0 ∼ 10.0. The discretized nonlinear characteristic (discretized sigmoid, i.e. arc tangent [17]) is as shown in Fig. 2. For C-language expression, it can be written as e† = γ ∗ (double)(int)(e/γ) u = 0.4 ∗ e† + 3.0 ∗ atan(0.6 ∗ e† ) u† = γ ∗ (double)(int)(u/γ),
TABLE I PID PARAMETERS FOR E XAMPLE -1 (gM [dB]: GAIN MARGINS , pM [deg]:
5123
PHASE MARGINS ,
(i) (ii) (iii) (iv) (v) (vi)
Kp 60 60 50 60 50 50
Mp : PEAK VALUES , β0 : ALLOWABLE SECTORS ).
Ci 0 3 3 3 3 2
Cd 0 0 30 0 30 40
L 10 10 10 15 15 15
β0 0.75 0.49 0.80 0.15 0.35 0.52
gM 4.87 3.41 5.15 1.21 2.55 3.64
pM 100.0 45.3 55.1 19.5 34.2 58.5
Mp 1.45 2.32 1.48 7.40 3.21 2.06
ThBIn5.2 ∆e
e
Pc
N2
N1 Fig. 8.
Phase traces for Example-1 (|∆e| ≤ 1, ∆e† (k) = ±1 or 0).
N3 Fig. 6. Off-axis circles and Nyquists plots for Example-1 (M = 1.45, cq = 0.0, · · · , 4.0).
(ii) (iii)
Pc
(i)
N4
N5 N6
Fig. 7.
Fig. 9. Off-axis circles and Nyquists plots for Example-1 (M = 2.06, cq = 0.0, · · · , 4.0).
Step responses for Example-1.
where (int) and (double) denote the conversion into an integral number (round-down discretization) and the reconversion into a double-precision real number, respectively. Here, the resolution is γ = 1.0. When choosing the nominal gain K = 1.0 and the threshold ε = 2.0, the sectorial area of the stepwise nonlinear characteristic for ε ≤ |e| can be determined as [0.5, 1.5] drawn by dotted lines in the figure. As is described in sections II and III the sampling period is chosen as h = 1.0, and the worst case of transmission delay Lu = 10.0 is considered. Figure 6 shows Nyquist plots of KG(ejωh ) · e−jωLu on the modified Hall diagram. Here, N1 , N2 , and N3 are cases (i), (ii), and (iii), respectively. The PID parameters are specified as shown in Table I. The gain margins gM [dB], the phase margin pM [deg] and the peak value Mp can be obtained from the phase crossover points Pc , the gain crossover points Gc , and the points of contact in regard to the M -circles, respectively. The max-min value β0 is calculated from (10) as follows: β0 = max Kη(q, ω0 ) = Kη(q0 , ω0 ) = 0.8.
for example (iii). Therefore, the allowable sector for nonlinear characteristic g(·) is given as [0, 1.8]. The stability of discretized control system (iii) will be guaranteed. In this example, the continuous saddle point (29) appears at point Pc (i.e., Aizerman’s conjecture is satisfied). Thus, the allowable interval of equivalent gain Kℓ can be given as 0 < Kℓ < 1.8. Figure 7 shows step responses for the three cases. In this figure, the time-scale line is drawn in 10h increments because of avoiding indistinctness. Sequences of the input u† (k) and the output u†c (k) of PID controller are also shown in the figure. Here, uc (k) is graduated in 1/100. Figure 8 shows phase traces, i.e., sequences of (e(k), ∆e) and (e† (k), ∆e† ). As is obvious from Fig. 8, assumtion (15) is satisfied. Since step response (i) remains an off-set, responses for cases (ii) and (iii) are stabilized and improved well by using the discretized PID action as specified in Table I. On the other hand, when the transmission delay increases from 10.0 to 15.0, (i.e., case (iv) in Table I), the robust stability is not guaranteed because the allowable sector for nonlinear characteristic is given as [0, 1.15]. (As for case (v),
5124
ThBIn5.2
(iv) (vi)
(v)
Pc
Fig. 10.
N1
N3
Step responses for Example-1.
N2
∆e Fig. 12. Off-axis circles and Nyquist plots for Example-2 (M = 1.6, cq = 0.0, · · · , 4.0).
e
(ii) (iii)
(i)
Fig. 11.
Phase traces for Example-1 (|∆e| ≤ 1, ∆e† (k) = ±1 or 0).
the allowable sector is given as [0, 1.35].) However, response for case (vi) is stabilized and improved well by using the discretized PID action. Figure 10 shows step responses for the three cases, and Figure 11 shows phase traces. Obviously, assumption (15) is satisfied in this example.
Fig. 13.
[Example-2] Consider the following continuous plant and transmission system: K2 (s + 0.5)(−s + 1.0) , P (s) = (s + 0.1)(s + 0.2)(s + 1.0)
T (z) = z
−L
Step responses for Example-2.
∆e
, (34)
where K2 = 0.001 = 1.0 × 10−3 and L = 5.0 ∼ 10.0. The same nonlinear characteristic and the nominal gain are chosen as shown in Example-1. Figure 12 shows Nyquist plots of KG(ejωh ) · e−jωLu on the modified Hall diagram. Here, N1 , N2 and N3 are cases (i), (ii), and (iii), and the PID parameters are specified as shown in Table II. In this example, the continuous saddle point (29) does not appear at Pc (i.e., Aizerman’s conjecture is not satisfied). Figure 13 shows step responses for the three cases, and Figure 14 shows phase traces. Also in this example, assumption (15) is satisfied. Although the allowable interval of equivalent linear gain is 0 < Kℓ < 1.75 (e.g., case (iii)), the allowable sector for nonlinear characteristic is [0, 1.64] as shown
5125
e
Fig. 14.
Phase traces for Example-2 (|∆e| ≤ 1, ∆e† (k) = ±1 or 0).
ThBIn5.2 TABLE II PID PARAMETERS FOR E XAMPLE -2 (gM [dB]: GAIN MARGINS , pM [deg]: PHASE MARGINS , Mp : PEAK VALUES , β0 : ALLOWABLE SECTORS ).
(i) (ii) (iii) (iv) (v) (vi)
Kp 50 50 40 50 40 30
Ci 0 2 2 2 2 2
Cd 0 0 20 0 20 40
L 10 10 10 15 15 15
β0 0.66 0.40 0.64 0.12 0.34 0.55
gM 4.3 2.8 4.5 1.0 2.6 4.4
pM 84.8 38.8 52.5 13.9 33.4 39.2
(iv) (vi) (v)
Mp 1.62 2.75 1.65 9.0 3.29 1.95
Fig. 16.
Step responses for Example-2.
accompany with transmission delay could be achieved. The design method will be applied to digital and discrete control systems. R EFERENCES
N4 N6 N5 Fig. 15. Off-axis circles and Nyquist curves for Example-2 (M = 1.9, cq = 0.0, · · · , 4.0).
in Table II. This is a counter example for Aizerman’s conjecture. However, since the sectorial area of the stepwise nonlinear characteristic is [0.5, 1.5], the robust stability of the nonlinear control system is guaranteed. The step responses are stabilized and improved well as shown in Fig. 13. On the other hand, when the transmission delay increases from 10.0 to 15.0, (i.e., cases (iv) in Table II), the robust stability is not guaranteed because the allowable sector for nonlinear characteristic is given as [0, 1.12]. (Note that the allowable interval of equivalent linear gain in this case is 0 < Kℓ < 1.42.) The step responses for (iv) and (v) actually fluctuate as shown in Fig. 16. However, the step response for (vi) is stabilized and improved using PID action. VII. C ONCLUSION In this paper, we have described robust stabilization for discretized PID control systems that are accompanied with transmission delay. The robust stabilization was examined in a frequency domain with respect to the worst-case delay. The design procedure is based on the modified Nyquist and Hall diagram and its parameter specifications in the frequency domain. The stability margins and the steady-state characteristic of the control system were specified directly in the diagram. In consequence, the robust stabilization for continuous plants using the discretized PID control
[1] R. E. Kalman, “Nonlinear Aspects of Sampled-Data Control Systems”, Proc. of the Symposium on Nonlinear Circuit Analysis, vol. VI, pp.273313, 1956. [2] R. E. Curry, Estimation and Control with Quantized Measurements, Cambridge, MIT Press, 1970. [3] D. F. Delchamps, “Stabilizing a Linear System with Quantized State Feedback”, IEEE Trans. on Automatic Control, vol. 35, pp. 916-924, 1990. [4] N. Elia and S. K. Mitter, “Stabilization of Linear System with Limited Information”, IEEE Trans. on Automatic Control, vol. 46, pp. 13841400, 2001. [5] M. Fu, “Robust Stabilization of Linear Uncertain Systems via Quantized Feedback”, IEEE Int. Conf. on Decision and Control, TuA06-5, 2003. [6] C. A. Desoer and M. Vidyasagar, Feedback System: Input-Output Properties, Academic Press, 1975. [7] C. J. Harris and M. E. Valenca, The Stability of Input-Output Dynamical Systema, Academic Press, 1983. [8] Y. Okuyama, “Robust Stability Analysis for Discretized Nonlinear Control Systems in a Global Sense”, Proc. of the 2006 American Control Conference, Minneapolis, USA, pp. 2321-2326, 2006. [9] Y. Okuyama, “Nonlinear Phenomena and Stability Analysis for Discretized Control Systems”, Proc. of IASTED Int. Workshop on Modern Nonlinear Theory, Montreal, Canada, pp. 350-355, 2007. [10] Y. Okuyama et al., “Robust Stability Evaluation for Sampled-Data Control Systems with a Sector Nonlinearity in a Gain-Phase Plane” Int. J. of Robust and Nonlinear Control, Vol. 9, No. 1, pp. 15-32, 1999. [11] Y. Okuyama, “Robust Stabilization and PID Control for Nonlinear Discretized Systems on a Grid Pattern”, Proc. of the 2008 American Control Conference, Seattle, USA, pp. 4746-4751, 2008. [12] Y. Okuyama, “Discretized PID Control and Robust Stabilization for Continuous Plant”, Proc. of the 17th IFAC World Congress, Seoul, Korea, pp. 1492-1498, 2008. [13] Z. Jiandong, “Stabilization of Linear Discrete Systems with Transmission Delay”, Proc. of the Chinese Control Conference, pp. 15-18, 2007. [14] Y. Okuyama et al., “Robust Stability Analysis for Non-Linear Sampled-Data Control Systems in a Frequency Domain”, European Journal of Control, Vol. 8, No. 2, pp. 99-108, 2002. [15] A. Datta, M.T. Ho and S.P. Bhattacharyya, Structure and Synthesis of PID Controllers, Springer-Verlag, 2000. [16] F. Takemori and Y. Okuyama, “Discrete-Time Model Reference Feedback and PID Control for Interval Plants” Digital Control 2000: Past, Present and Future of PID Control, Pergamon Press, pp. 260-265, 2000. [17] Y. Okuyama et al., “Amplitude Dependent Analysis and Stabilization for Nonlinear Sampled-Data Control Systems”, Proc. of the 15th IFAC World Congress, T-Tu-M08, 2002.
5126
