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for a physical plant to be controlled, whether or not its internal dynamics and ... 2Department of Mathematics, North Central College, Naperville, IL. 60540, USA.
16th IEEE International Conference on Control Applications Part of IEEE Multi-conference on Systems and Control Singapore, 1-3 October 2007

WeB01.6

On Estimation of Plant Dynamics and Disturbance from Input-Output Data in Real Time Qing Zheng1, Student Member, IEEE, Linda Q. Gao 2, and Zhiqiang Gao 1,3, Member, IEEE Abstract— This paper is concerned with the question of, for a physical plant to be controlled, whether or not its internal dynamics and external disturbances can be realistically estimated in real time from its input-output data. A positive answer would have significant implications on control system design, because it means that an accurate model of the plant is perhaps no longer required. Based on the linear extended state observer (LESO), it is shown that, for a nth order plant, the answer to the above question is indeed yes. In particular, it is shown that the estimation error 1) converges to the origin asymptotically when the model of the plant is given; 2) is bounded and inversely proportional to the bandwidth of the observer when the plant model is mostly unknown. Note that this is not another parameter estimation algorithm in the framework of adaptive control. It applies to a large class of nonlinear, time-varying processes with unknown dynamics. The solution is deceivingly simple and easy to implement. The results of the mathematical analysis are verified in a simulation study and a motion control hardware test. Index Terms— Extended state observer, unknown dynamics estimation, disturbance observer, uncertain systems, stability analysis.

I. I NTRODUCTION State observers, also known as estimators, play a central role in modern control theory. Given the input-output data, the values of internal variables of a physical plant, which are often inaccessible instrumentation wise, are made available through state observers. Such information extracted by the state observers proved to be invaluable in control system design, as well as fault detections. See, for example, [1], [2] for recent surveys. A presumption in existing state observer design is that an accurate mathematical model of the plant has been obtained. While this is a common assumption made in academia, it could pose some rather considerable challenges time and cost wise in engineering practice. The sometimes prohibitive cost and limitation associated with obtaining a good mathematical model drove a significant number of researchers and engineers to seek alternatives, such as fuzzy logic control (FLC) and artificial neural networks (ANN), but there is more than an inconvenience at stake here. If the primary purpose of employing feedback control is to counter the uncertainties in physical devices so that a precise and consistent behavior can be obtained from a system that consists of devices with 1 The Center for Advanced Control Technologies, Department of Electrical and Computer Engineering, Cleveland State University, Cleveland, OH 44115, USA. 2 Department of Mathematics, North Central College, Naperville, IL 60540, USA. 3 The Corresponding author. E-mail: [email protected]. Tel:1-216-6873528, Fax:1-216-687-5405.

1-4244-0443-6/07/$20.00 ©2007 IEEE.

inconsistent and only partially known dynamics, why does most of the modern control theory insist on having a precise mathematical model prior to any analysis and design? As the well-known control theorist Roger Brockett puts it: “If there is no uncertainty in the system, the control, or the environment, feedback control is largely unnecessary” [3]. The assumption that a physical plant, without feedback, behaves rather closely as its mathematical model describes, as the point of departure in control system design, does not reflect either the intent of feedback control, or the physical reality. This issue is perhaps the central one in the on-going theory vs. practice debate [4]. The everyday users of feedback control, however, are concerned about if there is a viable alternative, an alternative that does not completely abandon most advances made in modern control theory, as in the case of FLC and ANN, but does away with the obsession with mathematical model. Recognizing the vulnerability of the reliance on accurate mathematical model, some researchers over the years have investigated the problem of real time estimation of the disturbance, i.e. the part of the plant that is not described by the mathematical model. The ingenious idea is that if such discrepancy between the plant and its model can be computed and compensated for in real time, the closed-loop system behavior will not be hinged upon the accuracy of the plant model. One class of such disturbance observers is the unknown input observer (UIO) [5]-[12], where the disturbance is treated as an augmented state of the plant. Given the model of the plant and the processes that generates the disturbance, the state observer is designed to estimate both the original states and the augmented one. The disturbance is then rejected by using its estimated value obtained by the observer. Another class of observers is known as the disturbance observer (DOB) [13]-[19], where the disturbance is estimated by using the inverse of the nominal transfer function of the plant. It is shown that, for some specific design choices, the DOB and the UIO are equivalent [18]. The UIO, being in the state space form, has the additional merit of estimating the internal state of the plant. There are also scattered reports of different variations of the ideas in UIO and DOB, such as the adaptive robust controller [20], the adaptive inverse controller [21], and model-based disturbance attenuation method [22]. For the most part, however, the disturbance is assumed to be independent of the states, or the dynamics, of the plant. That is, it is external to the plant. Some researchers did speculate that such observers can be used to estimate the discrepancy between the plant and its model [23]-[25]. But the attempt to

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WeB01.6 validate such claims quickly run into mathematical difficulty, in part because that these observers are all designed based on the nominal model of the plant and it is not clear to what extent the plant is allowed to deviate from its model without jeopardizing performance and stability. In this paper, it is established that not only the external disturbance but also the plant dynamics can be estimated in real time. In doing so, we cross the boundary between system identification and observer design. Instead of identifying the plant dynamics off-line, we propose to estimate the combined effect of plant dynamics and external disturbance from input-output data. If successful, the impact of this will be quite significant. It could mean that the uncertainty (robust control) problem, the adaptive control problem (time varying dynamics), and disturbance rejection problem etc. will all be handled in this single framework. The limitations of this approach will also need to be formulated. The basis of our approach, i.e. the extended state observer (ESO), was first proposed by Han in the context of the active disturbance rejection control (ADRC) [26]-[29]. The ADRC as a new design paradigm was first introduced to the English literature in [30] and further simplified and explicated in [31] and [32], respectively. Central to this novel design framework is the ability to estimate both unknown dynamics and disturbances in real time, which is the focus of this paper. Because ESO was originally formulated using nonlinear gains, the analysis of it was difficult. Employing the linear and parameterized linear extended state observer (LESO) in [31], the convergence or the bound of the estimation error is analyzed in this paper. In particular, at one extreme, the asymptotic tracking of LESO is shown where the mathematical model of the plant is given. At the other extreme, where the plant dynamics is largely unknown, the error bound of LESO is derived. Of course most practical scenarios fall in between. The paper is organized as follows. The analysis of LESO error dynamics is given in Section II, with or without a detailed mathematical model. Software simulation and hardware test results are shown in section III. The paper ends with a few concluding remarks in section IV. II. A NALYSIS OF LESO E RROR DYNAMICS Consider the following description of a physical process y (n) (t) = f (y (n−1) (t) , y (n−2) (t) , · · · , y (t) , w (t)) + bu (t) .

(1)

Without loss of generality, let us say this equation describes a motion system, where y is the position output, u is the input force, w is an extraneous unknown input force (known as external disturbance in its traditional sense), and b is  a constant. Let f y (n−1) (t) , y (n−2) (t) , ·· · , y (t) , w (t) be denoted as f y (n−1) , y (n−2) , · · · , y, w , or simply f . Similarly, assuming f is differentiable, define  d  (2) h = f y (n−1) , y (n−2) , · · · , y, w = f˙. dt

Then (1) can also be represented in state space form as x˙ 1 = x2 .. . x˙ n−1 = xn x˙ n = xn+1 + bu x˙ n+1 = h y = x1

(3)

T

where x (t) = [x1 (t) , x2 (t) , · · · , xn+1 (t)] ∈ Rn+1 , u ∈ R and y ∈ R are the state, input and output of the system, respectively. A. Convergence of the LESO with the Given Model of the Plant With u and y as inputs and the function h given, the LESO of (3) is given as ˆ2 + l1 (x1 − x ˆ1 ) x ˆ˙ 1 = x .. . ˆn + ln−1 (x1 − xˆ1 ) x ˆ˙ n−1 = x ˆn+1 + ln (x1 − x ˆ1 ) + bu x ˆ˙ n = x x ˆ˙ n+1 = ln+1 (x1 − x ˆ1 ) + h (ˆ x1 , x ˆ2 , · · · , x ˆn+1 , w)

(4)

T

ˆ2 (t) , · · · , x ˆn+1 (t)] ∈ Rn+1 , and where x ˆ (t) = [ˆ x1 (t) , x li , i = 1, 2, · · · , n + 1, are the observer gain parameters to be chosen. In particular, let us consider a special case where the gains are chosen as   (5) [l1 , l2 , · · · , ln+1 ] = ωo α1 , ωo2 α2 , · · · , ωon+1 αn+1

with ωo > 0. Here αi , i = 1, 2, · · · , n + 1, are selected such that the characteristic polynomial s n+1 + α1 sn + · · · + αn s + αn+1 is Hurwitz. For tuning simplicity, we just let n+1 sn+1 + α1 sn + · · · + αn s + αn+1 = (s + 1) where (n+1)! αi = i!(n+1−i)! , i = 1, 2, · · · , n + 1. It results in the characteristic polynomial of (4) to be λo (s) = sn+1 + ωo α1 sn + · · · + ωon αn s + ωon+1 αn+1 = (s + ωo )n . (6) This makes ωo , which is the observer bandwidth, the only tuning parameter to be adjusted and the implementation process much simplified, compared to other observers. Next, we wish to examine under what conditions, the states of the observer converge to those of the original system. For the sake of simplicity, the effect of the external disturbance, w, is ignored (set w = 0) in the context of stability proof. Let x ˜i (t) = xi (t) − x ˆi (t) , i = 1, 2, · · · , n + 1. From (3) and (4), the observer tracking error can be shown as

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˜1 x ˜˙ 1 = x˜2 − ωo α1 x .. . x ˜˙ n−1 = x˜n − ωon−1 αn−1 x ˜1 x ˜˙ n = x˜n+1 − ωon αn x ˜1 x ˜˙ n+1 = h (x) − h (ˆ x) − ωon+1 αn+1 x ˜1 .

(7)

WeB01.6 Now let us scale the tracking error x ˜ i (t) by ωoi−1 , i.e., let x ˜i (t) εi (t) = ωi−1 , i = 1, 2, · · · , n + 1, then the error equation o (7) can be rewritten as

where



   B=  

h (x) − h (ˆ x) ε˙ = ωo Aε + B n ωo  1 0 ··· −α1  −α2 0 1 ···   . .. . . .. .. A =  . . .   −αn 0 ··· 0 0 −αn+1 0 · · ·  0 0   .. . .   0  1

(8) 0 0 .. .



   ,  1  0

(10)

Since the function h is globally Lipschitz, that is, there exists a constant c′ such that |h (x) − h (ˆ x)| ≤ c′ x − x ˆ, it follows that |h (x) − h (ˆ x)| x − x ˆ 2εT P B ≤ 2εT P Bc′ . (11) n n ωo ωo x ˜ x When ωo ≥ 1, one has x−ˆ = = ωon ωon √2 2 2 2 4 2 2n  ε1 +ε2 ωo +ε3 ωo +···+εn+1 ωo  ≤ ε. Therefore, we obtain ωn o

|h (x) − h (ˆ x)| ωon

≤ 2εT P Bc′ ≤ c ε 2

ε˙ = ωo Aε + B 

   where A =   



0 0 .. .



= − (ωo − c) ε .

(16)

1 0 .. .

0 1 .. .

··· ··· .. .

−αn −αn+1

0 0

··· ···

0 0

0 0 .. .



   , and B =  1  0

       .    0  1 Theorem 2: Assuming that h (x) in (2) is a bounded function of x, then there exist a constant σ > 0 and a finite xi (t)| ≤ σ, i = 1,2, · ·· , n + 1, ∀t ≥ T1 > T1 > 0 such that |˜ 0 and ωo > 0. Furthermore, σ = O ω1k , for some positive o integer k. Proof. Solving (16), we can obtain [33] 

t

eωo A(t−τ ) B

0

h (x (τ )) dτ . ωon

(17)

Let

2

2

h (x) ωon

−α1 −α2 .. .

(12)

≤ −ωo ε + c ε

(15)

and Equation (8) is now

ε (t) = eωo At ε (0) +

where c = 1 + P Bc  . From (10) and (12), one has V˙ (ε)

(14)

˜2 − ωo α1 x ˜1 x ˜˙ 1 = x .. . x ˜˙ n−1 = x ˜n − ωon−1 αn−1 x ˜1 n ˙x ˜n = x ˜n+1 − ωo αn x ˜1 x ˜˙ n+1 = h (x) − ωon+1 αn+1 x ˜1

x − x ˆ ωon

2

′ 2

ˆ2 + l1 (x1 − x ˆ1 ) xˆ˙ 1 = x .. . ˆn + ln−1 (x1 − x ˆ1 ) xˆ˙ n−1 = x xˆ˙ n = x ˆn+1 + ln (x1 − x ˆ1 ) + bu xˆ˙ n+1 = ln+1 (x1 − x ˆ1 ) . Consequently, the observer tracking error (7) becomes

V˙ (ε) = 2εT P ε˙

2εT P B

In many real world scenarios, the information on the plant dynamics such as h is mostly unknown. In this case, can the LESO still track its target in some sense? Note that the observer in (4) now takes the form of

and

The characteristic polynomial of A is s n+1 + α1 sn + · · · + αn s + αn+1 and A is Hurwitz according to α i , i = 1, 2, · · · , n + 1, selection. Theorem 1: Assuming that the function h (x) in (2) is globally Lipschitz, then there exists a constant ω o > 0, such that lim x ˜i (t) = 0, i = 1, 2, · · · , n + 1. t→∞ Proof. Since A is Hurwitz, there exists a unique positive definite matrix P such that A T P + P A = −I. Choose the Lyapunov function as V (ε) = ε T P ε. Hence V˙ (ε) = ∂V (ε) ˙ where ∂ε ε,   ∂ εT P ε ∂V (ε) = = 2εT P, (9) ∂ε ∂ε and h (x) − h (ˆ x) 2 . = −ωo ε + 2εT P B n ωo

B. Convergence of the LESO with Plant Dynamics Largely Unknown

p (t) =

(13)

From (13), V˙ (ε) < 0 if ωo > c. Therefore, lim x˜i (t) = t→∞ 0, i = 1, 2, · · · , n + 1, for ωo > c. Q.E.D.



0

t

eωo A(t−τ ) B

h (x (τ )) dτ . ωon

(18)

Since h (x (τ )) is bounded, that is, |h (x (τ ))| ≤ δ, where δ is a positive constant, for i = 1, 2, · · · , n + 1, it follows that

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WeB01.6 [34]

obtain

 t  ω A(t−τ )  o B i |h (x (τ ))| dτ 0 e |pi (t)| ≤ ωn  t  ω A(t−τ ) o  δ 0 e o B i dτ ≤ n ωo    δ  = −A−1 B i + A−1 eωo At B i n+1 ωo δ  −1    −1 ωo At   A B i + A e ≤ B i . n+1 ωo (19)   −α1 1 0 ··· 0  −α2 0 1 ··· 0     .. .. , one can obtain .. .. . . Since A =  . . .  . .    −αn 0 ··· 0 1  0 0 −αn+1 0 · · ·  1 0 0 0 · · · − αn+1  1 0 0 · · · − α1   αn+1    2 0 1 0 · · · − ααn+1 , and A−1 =    .. ..   .. .. ..   . . . . . αn 0 0 ··· 1 − αn+1    α 1   −1   n+1 i=1  A B  = i  αi−1  αn+1 i=2,···, n+1

where ν =

max

i=2,··· ,n+1

≤ ν 

1 αn+1 ,

αi−1 αn+1

(20)  . Since A is Hurwitz,

there exists a finite time T1 > 0 such that   1  ωo At   ≤ n+1  e ij  ωo

for all t ≥ T1 , i, j = 1, 2, · · · , n + 1. Hence  ω At    e o B ≤ 1 i ωon+1

for all t ≥ T1 , i =1, 2, · · · , n + 1. Note that ... s1,n+1 s11  .. .. .. ωo A. Let A−1 =  . . . s · · · s n+1,1  n+1,n+1  ... d1,n+1 d11   .. .. ..  . One has . . . dn+1,1

···



ωon+1

(24)

 ω At   e o ε (0)  ≤ |ε1 (0)| + |ε2 (0)| + · · · + |εn+1 (0)| i ωon+1 εsum (0) = (25) ωon+1 for all t ≥ T1 , i = 1, 2, · · · , n + 1. From (17), one has   |εi (t)| ≤  eωo At ε (0) i  + |pi (t)| . (26) Let x ˜sum (0) = |˜ x1 (0)| + |˜ x2 (0)| + · · · + |˜ xn+1 (0)|. Ac(t) and Equations (24)-(26), we have cording to ε i (t) = ωx˜ii−1 o    x˜sum (0)  δν δµ  |˜ xi (t)| ≤  n+1  + n−i+2 + 2n−i+3 ωo ωo ωo = σ (27)

for all t ≥ T1 , i = 1, 2, · · · , n + 1. Q.E.D. III. S IMULATION AND H ARDWARE T ESTS Two examples are given below for illustration purposes. One is a simulation study applying LESO to a nonlinear plant to see how it performs with three different levels of knowledge on the plant dynamics. The other is a hardware test showing the effectiveness of the LESO in a real motion control environment. A. A Simulation Case Study y¨ = y˙ 3 + y + d + u.

(28)

y¨ = f + bu

(29)

Rewrite (28) as (22) T1 depends on 

  and eωo At =

where f represents the summation of the plant dynamics y˙ 3 + y and the external disturbance d. Note that for a second order plant, the LESO in (4) and (14) is of the third order, where x ˆ 3 is an estimate of f . With a well-tuned observer, the control law u=

u0 − x ˆ3 b0

(30)

where b0 is the approximate value of b, should approximately reduces the original plant (29) to a double integral one, i.e. y¨ ≈ u0 .

(31)

With the plant reduced to (31), a simple PD controller of the form ˆ1 ) − kd x ˆ2 u0 = kp (r − x

i=2,··· ,n+1

(23)

for all t ≥ T1 , i = 1,  2, · · · , n + 1, where µ = αi−1 1 max , 1 + αn+1 αn+1 . From (19), (20), and (23), we

i=2,··· ,n+1

δµ δν n+1 + 2n+2 ωo ωo

for all t ≥ T1 , i = 1, 2, · · · , n + 1. Let εsum (0) = |ε1 (0)| + |ε2 (0)| + · · · + |εn+1 (0)|. It follows that

(21)

 −1 ω At   |si,1 | + |si,2 | + · · · + |si,n+1 |  A e o B ≤ i ωon+1    n+11  ωo α n+1 i=1  = αi−1  1  n+1 1 +  α n+1 ω o



Consider the following nonlinear system

dn+1,n+1

µ

|pi (t)|

(32)

is usually sufficient to make the output track r, the desired trajectory. As shown in [31], [32], the controller gains can be simply selected as kp = ωc2 , kd = 2ωc , where ωc is the controller bandwidth. A simple tuning method based on ω o

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WeB01.6 LADRC performance Output

1.5 1

Tracking error

0

1

2

LESO1 LESO2 3LESO3 4

0

1

2

3

4

5

6

7

8

0

1

2

3

4 Time(s)

5

6

7

8

0.5 0

Control signal

and ωc is given in [31], [32]. Combining the LESO in (4) or (14) with the control law in (30) and (32), this control structure is denoted as the linear ADRC, or LADRC. The LESO tracking performance is demonstrated in Fig. 1 under three different scenarios: 1) f is completely unknown; 2) only partial internal dynamics information of the plant is given, i.e. f partial = y˙ 3 ; 3) the internal dynamics of the plant fin is completely known, i.e. f in = y˙ 3 + y is given. In this simulation, the tuning parameters are ω c = 4.5 rad/sec and ωo = 20 rad/sec. Fig. 1 shows the observer errors for three cases using a step input at t = 1 second as the excitation and a pulse disturbance with the amplitude of ±20, the period of 4 seconds, the pulse width 5% of the period, and the phase delay of 4 seconds. The LADRC performance with different LESOs is shown in Fig. 2. From Fig. 1 and Fig. 2, it can be observed that the observer error decreases as more model information is incorporated into the LESO, and so does the tracking error of the control loop.

5

6

7

8

1 0.5 0 −0.5 50 0 −50

Fig. 2. The LADRC performance with different LESOs for the nonlinear system. ( LESO1: without plant information; LESO2: with partial plant information; LESO3: with complete plant information.)

second-order plant of the form

0 −0.02

Error of f and fˆ

Error of y˙ and yˆ˙

Error of y and yˆ

The errors between actual and estimated information 0.02

y¨ (t) = f (y (t) , y˙ (t) , w (t)) + bu (t) 0

1

2

LESO1 LESO2 3 4 LESO3

0

1

2

3

4

5

6

7

8

0

1

2

3

4 Time(s)

5

6

7

8

5

6

7

8

2 0 −2 50 0

−50

Fig. 1. The errors between actual and estimated information. ( LESO1: without plant information; LESO2: with partial plant information; LESO3: with complete plant information.)

B. A Motion Control Hardware Test In this case study, an industrial motion control test bed [35] is used to verify if the plant dynamics can indeed be estimated in real time, as shown in the mathematical analysis. The experimental setup includes a PC-based control platform and a DC brushless servo system. The servo system includes two motors (one as an actuator and the other as the disturbance source), a power amplifier, and an encoder which provides the position measurement. The inertia, friction, and backlash are all adjustable, making it convenient to test the control algorithms. A Pentium 133 M Hz PC running in DOS is programmed as the controller. It contains a data acquisition board for digital to analog conversion and a counter board to read the position encoder output in the servo system. The sampling frequency is 1 KHz. The output of the controller is limited to ±3.5V . The drive system has a dead zone of ±0.5V . The system is approximated as a

(33)

where y (t) is the position output, b is a constant, u (t) is the control voltage sent to the power amplifier that drives the motor, w (t) is the external disturbance, and f (y (t) , y˙ (t) , w (t)) represents the combined effect of internal dynamics and external disturbances of the plant. How close (31) is to a double integral plant will be an indicator of how effective the observer is. Fig. 3 shows the output comparison of an ideal double integrator, simulation test and hardware test of the original system (33) compensated by (30), where xˆ 3 is obtained from (14), with b 0 = 25 and ωo = 300 rad/sec. The closeness of the three curves in Fig. 3 confirms, beyond doubt, that the LESO as shown in (14) is capable of extracting information from the input-output data on the unknown dynamics and disturbances. Moreover, this can be done in such a simple and effective manner that it makes the practical application straightforward. IV. C ONCLUDING R EMARKS It is demonstrated in this paper that, for a large class of physical processes, both the unknown plant dynamics and external disturbances can be estimated using a unique state observer. The convergence properties of this observer is analyzed in two cases: with and without a detailed mathematical model of the plant. It is shown that the asymptotic convergence is assured in the former and boundedness of the error in the later. The results in this paper solidify the foundation for an alternative control design paradigm [30], [32], one that is not bound by the prevailing notion that an accurate mathematical description of the physical process is required. The analytical study is furthered supported by both the simulation and experimental results, showing that the plant dynamics and disturbance can be realistically estimated in real time based on the plant input-output data and some limited knowledge of the plant structure and order.

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WeB01.6 The outputs of an ideal double integrator, simulation test, hardware test 9 ideal double integrator simulation test hardware test

8 7 6

Output y

5 4 3 2 1 0 −1

0

0.5

1

1.5

2 Time(s)

2.5

3

3.5

4

Fig. 3. The output comparison among an ideal double integrator, simulation test, and hardware test.

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