A robust linear-quadratic moving averaging controller ...

A robust linear-quadratic moving averaging controller for strongly nonlinear systems

Andrzej Latocha AGH University of Science and Technology Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Krakow, Poland.

Abstract

A new method is proposed for designing a linear observer for a class of strongly nonlinear systems. The algorithm is based on a new approach to the linearization of the bounded-input, bounded-output, (b.i.b.o.) nonlinear systems. The use of the linearquadratic regulator (LQR) in real systems is dicult in terms of the non-linearity of the system, the uncertainty of the input/output signal, the astatic state matrix, the limited precision of the numerical calculation, and the limited power for the sytem control. This article proposes a new approach to the design of an linearquadratic regulator based on the innite impulse response moving average (IIR-MA) of the linearizable observer for linearization feedback. An innovation in this paper is the algorithm of design of robust linear observer for a class of strongly nonlinear systems. Keywords:

1

automatic, control, identication, modeling, estimation, ltration

Introduction

The standard assumptions made in the literature are applied to the strongly nonlinear system used in this paper. An linear-quadratic regulator (LQR) controller is an optimal controller in the operating point of linear systems. The use of an LQR to control nonlinear systems requires linearization feedback based on a linear observer. The classical solution is based on the piecewise linearization of the operating point. For strongly nonlinear systems, outside the linearization point, classical linearization algorithms do not clearly result in the stability of the system. The stability of a strongly nonlinear system outside the linearization point can be ensured by formulating a linearization task that includes the non-linearity of the system on an interval around the point of stabilization. A nonlinear system, in an input/output sense, is a map N from C[0, ∞], or the space of continuous functions from [0, ∞], to C[0, ∞]. Thus, given the input u ∈ C[0, ∞], we assume that the output of the nonlinear system N is also a continuous function, denoted by yN (.), dened by [0, ∞] [1], [2], [3]: yn = N (u) ∈ C[0, ∞]

1

(1.1)

u0 = C[0, ∞]

(1.2)

The goal is to obtain the optimal approximation of the non-linear system for the input reference signal to be outputted by the linear system. The class of linear systems marked by W , which we will solve for the optimal approximation, is represented by the following convolution (1.3): ˆ∞ ω(t)u0 (t − τ )dτ,

yL (t) = (W (u0 )(t) :=

(1.3)

−∞

where, for standard assumptions, u(t) ≡ 0, t < 0. The kernel of the convolution was chosen to minimize the mean squared error (1.4) [1]. 1 e(ω) = limT →∞ T

ˆT [W u0 (τ ) − N u0 (τ )]2 dτ

(1.4)

0

To solve the optimization problem (1.4) for assumptions [1] [p. 128]: 1. The stability of the system is provided with limited input and output (b.i.b.o.). For a given b, there is m0 (b) such that |u(t)| < b ⇒ |N u(t)| < m0 (b) for all t ∈ [0, ∞]. We assume a limi-tation of nonlinearity; therefore, the bounded input to the nonlinear system generates a limited signal as an output. 2. Causal stable approximators exist. The class of linear systems determines the possibility of generating a nonlinear system. The limited entry u(·) in the linear system W will generate a limited output. ˆ∞ |ω(τ )|dτ < ∞

(1.5)

0

An optimal causal, stable, linear system approximant exists that solves the optimization problem of (1.4), with convolution kernel ω ∗ (·). Then ω ∗ satises Theorem 1.

(1.6), [1].

φω∗ (τ ) ≡ φN (τ )

(1.6)

where φω∗ (τ ) is shorthand for the cross correlation between u0 (·) and N u0 (·), dened exactly as in the equation with W ∗ u0 replaced by N u0 (·). The terminology of a stationary deterministic signal used in this paper follows that of Wiener and his theory of generalized harmonic analysis [1]. The problem of approximating state x ∈ Rn of the linear system can be presented as follows: x˙ = Ax + Bu

(1.7)

y = Cx,

(1.8)

and 2

based on the knowledge that input u ∈ R and output y ∈ Rn have known solutions, provided only when C and A is an observable pair (1.9).    rank  

C CA

.. .

CAn−1

   =n 

(1.9)

This paper provides estimates of forms of (1.10) and (1.11) that fulll the principle of causality in a class of strongly nonlinear systems. ξ˙ = f (ξ, u)

(1.10)

ψ = h(ξ)

(1.11)

The eld of automation, beyond mathematical equations, takes the resources that are available for physical implementation into account. This is why the term "strongly nonlinear systems " is often encounterd in the literature. Strongly nonlinear systems, in the eld of automation, are understood to be nonlinear parabolic equations (1.10), (1.11) [10], as studied by Browder [11] and by Lions [12] using the theory of monotone operators dened on a reexive separable Banach space. In this paper, the problem of optimally approximate nonlinear system is extended to noncontinuous systemsthat are nondierentiable or weakly nondierentiable in the classical sense. For this reason, the assumptions are dened (1.4) (1.12) [1]: |u(t)| < bT ⇒ |N u(t)| < mT (bT ) ∀t ∈ [0, T ].

2

(1.12)

Problem formulation

The classic piecewise linearization for systems of strong nonlinearity calculates the control vector in posteriori, not taking into account whether the dynamic transfer to the system has met the controllability condition (2.1, 2.2) in priori. Ω = [B AB ... An−1 B]

(2.1)

det(Ω) 6= 0

(2.2)

Therefore, the new approach can perform the linearization around the operating point by using an innite impulse response moving average lter (IIR-MA) formed with an autoregressive moving-average model that includes exogenous inputs (ARMAX), as it satises the conditions (2.1, 2.2) to achieve linearization feedback on a moving average horizon.

3

2.1

Nonlinear system structure

This paper is based on the classic structure of the Hammerstein system (Fig. 2.1). The structure consists of a nonlinear steady-state block f (.) and a linear dynamic block G(z).

Fig. 2.1: The system structure to design of linearizable observer. 2.2

Linear block

For the linearization algorithm, we assume a linear block of the structure of the output-error model (Fig. 2.2). We examine the linear estimator in an (IIR-MA) ARMAX model.

Fig. 2.2: Output-error model. We consider a parametric model (2.3) y(k) = ϕT (k)θ k ∈ N, k > 0

(2.3)

where y(k) is a measurand value, ϕ(k) is the n-dimensional vector of the data, andθ is an n-dimensional vector of an unknown coecient. Thus: (2.4) To account for noise and the inaccuracy of the model, a larger number of data points (oversized data) improves the accuracy of the estimation. For N  n, the system is overdetermined, and there is no exact solution. For oversized data, the data matrix will not be square. In this case, the data matrix can be replaced by a pseudo square matrix. Therefore (2.5), [1], [2],[3] Y = ΦT θ

ε(k) = y(k) − ϕT (k)θ.

(2.5)

The least squares error (LSE) estimator θˆ is dened as a vector that minimizes the cost function (2.6). N

1 1 1X 2 ε (k) = εT ε = ||ε||2 V (θ) = 2 t=1 2 2

4

(2.6)

||  || is the Euclidean vector norm. For the positive denite matrix ΦT Φ, the cost

function (2.6) has a minimum as follows:

(2.7)

θ = (ΦT Φ)−1 ΦT Y, 1 T [Y Y − Y T Φ(ΦT Φ)−1 ΦT Y ], 2 E = Y − Φθ,

V (θ) =

0=

dV = −Y T Φ + θT (ΦT Φ). dθ

(2.8) (2.9) (2.10)

The matrix ΦT Φ is known as the Gramian matrix of Φ, is a positive semi-denite matrix. Matrix ΦT Y is known as the moment matrix of the regressand by regressors. Finally, θˆ is the coecient vector of the least-squares hyperplane, expressed as follows [1]  −1 T θˆ = ΦT Φ Φ Y (2.11) Thus, we consider an equation at the discrete time domain. The discrete system can be described by the ARMAX model. y(k) = z −n

C(z −1 ) B(z −1 ) u(k) + ε(k) A(z −1 ) A(z −1 )

(2.12)

We assume the following: ε=

C(z −1 ) ε(k). A(z −1 )

(2.13)

By describing the system using a dierence equation, the following equation is obtained: y(k) + a1 y(k − 1) + ... + an y(k − n) + ε = b1 u(k − 1) + ... + bm u(k − m), k  n, n ≥ m; m, n ∈ N; u ∈ R; y ∈ R; ε ∈ R (2.14)

where the linearization error ε = 0. By applying discrete Z-transform, the transfer functions of zero initial condition and ε = 0 obtained as follows: ˆb1 z m−1 + ˆb2 z m−2 + ... + ˆbm−1 z + ˆbm Y (z) ˆ G(z) = = , U (z) 1+a ˆ1 z n−1 + a ˆ2 z n−2 + ... + a ˆn−1 z + a ˆn

(2.15)

The discrete transfer function obtained from the denition of the discrete z operator requires the assumption that the signal does not grow faster than the exponential function (2.16). Z[f ∗ (t)] = Z[f (kT )] = F (z),

∞ X

F (z) =

f (kT )z −k

k=−∞ 2

T ∈ R; f (k) < k!; f (k) < eak ; a > 0, a ∈ R

5

(2.16)

2.3

Linear estimator of the nonlinear block

The nonlinear part f (.) of the structure (Fig. 2.1) used in this paper represent the nonlinearity of the u(k) signal, which is the noncontinuous function at the beginning of a linearized interval [5]. The nonlinearity f (.) is estimated using the proposed function (2.17) which optimally carries out the u˜(k) (2.18) signal from the zero initial condition to its actual state on the linearized interval and has an insignicant impact on the dynamics of the system (2.20). The proposed function (2.17) arbitrarily imposes zero initial conditions at the beginning of the linearized interval. 1 ; η ∈ N, η > 1 zη

h(z) =

(2.17)

u ˜(k) = Z −1 [h(z)U (z)]; kN, u ˜∈R

(2.18)

k = j, j + 1, ..., N ; N ∈ N; j ∈ N

(2.19)

N −j n

(2.20)

Function (2.17) is dened as the zero input initial reconstructor (ZIIR). 2.4

Linear system design

The nonlinear system does not fulll the linearity principle of causality. The goal is to have an arbitrary imposition of the rules of causality on the input signal (2.18) and an added error of estimation on the output (2.22) (2.23) (Fig. 2.1). y(k) = z −n

B(z −1 ) C(z −1 ) u ˜(k) + ε(k), −1 A(z ) A(z −1 )

ε=

C(z −1 ) ε(k), A(z −1 )

(2.21) (2.22)

y(k) + a ˆ1 y(k − 1) + ... + a ˆn y(k − n) + ε = ˆb1 u ˜(k − 1) + ... + ˆbm u ˜(k − m), (2.23) ε = 0, h i−1 ˜T Φ˜i ˜T Yi ; i ∈ N. θˆi = Φ Φ i i

(2.24)

We assume that the processes are related which, in the absence of noise, takes the following form: ˆ i (z) = G

ˆb1 z m−1 + ˆb2 z m−2 + ... + ˆbm−1 z + ˆbm , 1+a ˆ1 z n−1 + a ˆ2 z n−2 + ... + a ˆn−1 z + a ˆn ˆ i (z)U˜i (z)]. Yˆi = Z −1 [G

6

(2.25) (2.26)

The optimality function (2.17) can be calculated from condition (2.28). N −j 1 X (yj+k − yˆj+k )2 ei = N −j

(2.27)

η = arg inf (ei )

(2.28)

k=0

ηi

The optimal estimates of the linearizable system can be calculated from the following condition: (2.29)

θˆ = arg inf (ei ) θˆi

An optimally linearized system around the operating point is obtained by the following function: (2.30)

ˆ ˆ G(z) = f (θ,η). 2.5

Linearizable observer

The optimal LQ control of a strongly nonlinear system is based on an optimal linearizable observer (2.30), [1]. By converting the model (2.25), (ZOH), to the continuous time domain and use thr controllable form equation of a state space (2.31), we obtain the following equation: 

−an−1  1 ˆ  A= ··· ···

−an−2 ··· 1 ···

··· ··· ··· 1

  1 −a1  ···  ˆ= 0 ,B  ··· ···  0 0

    , Cˆ = cn 

.

···

c1



(2.31)

ˆx + Bu ˆ + L(y − Cˆ x x ˆ= Aˆ ˆ) + e˙ L ; x ˆ ∈ R n ; u ∈ Rm ; y ∈ Rr

(2.32)

eL = x ˆ−x

(2.33)

The optimal solution can be obtained by the following: ˆ L → 0. e˙ L = (Aˆ − LC)e

(2.34)

The observer equation is solved by algorithm pole placement gain selection using Ackermann's formula [14].

3 3.1

Numerical experiments LQ stabilization controller

In order to satisfy condition (3.1) in the system, we dene the cost function (3.2) on an innite horizon. 7

ˆ = [B ˆ AˆB ˆ Aˆ2 B ˆ . . . Aˆn−1 B]; ˆ det(Ω) ˆ 6= 0 Ω

(3.1)

ˆ∞ xT Qˆ x + uT Ru)dt, J = (ˆ

(3.2)

0

ˆ −1 B ˆ T P + Q = 0, AˆT P + P Aˆ − P BR

(3.3)

ˆ T P. K = R−1 B

(3.4)

Q ≥ 0, R > 0, P ≥ 0 are symmetric, positive, (semi-) denite matrices. Solving

Equation (3.3) gives the control law (3.5) for the linearization feedback (2.32). u = −K x ˆ 3.2

(3.5)

LQ tracking controller

For trajectory tracking, we can assume that eˆ is small and can linearize around eˆ = 0, giving the linearizable observer equation (3.6), .

ˆx + Bu ˆ + L(y − u − Cˆ x x ˆ= Aˆ ˆ) + eˆ˙ L

(3.6)

eˆL = x ˆ−x

(3.7)

and where the optimal solution for the trajectory tracking can be obtained as follows: y − u − Cˆ x ˆ = 0.

(3.8)

Thus, we assume structure of the nonlinear system model for control to be as is shown in (Fig. 3.1).

Fig. 3.1: Structure of a nonlinear system controlled by linearization feedback LQR. 8

The observer is calculated by pole placement gain selection using Ackermann's (acker) formula for closed loop poles at the values specied in vector P (3.9) (3.10) (3.11) [14].

3.3

P = eig(A − BK)

(3.9)

G = (acker(A, C, P ))

(3.10)

F = A − GC

(3.11)

Control of chaotic systems

This example is based on algebraically simple chaotic ows. And an attractor for the chaotic system can be found in [8], Fig. 20, and Equation 12. Consider a nonlinear dynamic system modeled by the dierential equation (3.12). ...

x (t) + 0.5¨ x(t) + x(t) ˙ − sgn(x(t)) + x(t) = u(t)

(3.12)

The system (3.12) is linearized on the interval u ∈ (−1...1); y ∈ [−34.5488... − 17.1261]; 4t = 0.01; k = 911...1100; testing function: u(t) = sin(220t); and linearized system obtained as follows:  −0, 0405 −0, 4675 , 0 0 1 0   1 ˆ =  0 , B 0



0, 0512 1 Aˆ =  0

(3.13) (3.14)

and Cˆ =



2, 8876e − 05

0, 0087



0, 1622

.

(3.15)

The linearization feedback can be obtained: K=



1, 6520

1, 2800

9

0, 06165



.

(3.16)

Fig. 3.2: Controls for system stabilization and trajectory tracking. 3.4

Bounded-input, bounded-output control with hysteresis

Given the nonlinear system described by Equation (3.17), and controlled by bounded output with hysteresis, x ¨(t) + 0.01x(t) ˙ + x(t) + x2 (t) = u(t); uR[−2.95...−hyst 0hyst ...2.95],

(3.17)

where the control signal with relay hysteresis (3.17) are modeled as shown in (Fig. 3.3).

Fig. 3.3: Model of the control signal with hysteresis (relay) (3.17). The system (3.17) is linearized on the interval u ∈ (−1...1); y ∈ [0...0.0410]; where 4t = 0.01; k = 87...1100; testing function u(t) = sin(220t); in linearized system is obtained as follows:  1.0547 −0.2086 , 0 0 1 0   1 ˆ =  0 , B 0



−7.8304 1 Aˆ =  0

(3.18) (3.19)

and Cˆ =



0.000011

0.0058



0.0536

.

(3.20)

The linearization feedback can be obtained by K=



0.4681

3.7752

10

0.3669



.

(3.21)

Fig. 3.4: Bounded-output control for system stabilization.

Fig. 3.5: Bounded-output control for trajectory tracking.

4

Conclusion

Piecewise linearization, at a minimum segment, gives optimal estimators at point of linearization. For nonlinear systems, the optimal local estimators can be used to calculate the control a priori. The calculated control gives the system locally dynamic at points of linearization. We obtain this from the previous piecewise linearization to the prior segment by taking the error of the initial conditions into account. For strongly nonlinear systems, the piecewise linearization, at a minimum segment calculated a posteriori, takes the problems that the calculation of the control a priori creates into account and provides the controllability condition of the system. The linearization, at the proposed algorithm, averages the dynamics of the observer at the interval. The nonlinear system can be linearized by the proposed algorithm in any state, which satises the controllability condition. The proposed linearization leads to the dynamic observer of a higher order that allows for the calculation of the controller. To do this, the proposed linearization algorithm provides a controllability condition for the linearization feedback for a wide neighborhood of the local state input-output injection. The presented algorithm generalizes linearization for nonlinear systems. However, a problem arises for short horizons in terms of the piecewise linearization, as wider linearization horizons provide stability for LQ control problems, as shown in Section 3. To solve the 11

issue of the designed linearizeble observer, we do not need to solve partial dierential equations (i.e., the standard solution in mathematics of dynamic systems) in the aforementioned assumptions (i.e., the real systems) of uncertainty signals, the estimates of a state space, and a driftless system. As shown in this work, sufciently accurate and stable solutions, in the eld of control systems engineering can be obtained through linear regression. This paper proves that the accuracy of strongly nonlinear systems, the structure of the system need not be considered. However, stability is a necessary consideration. In summary, this paper provided a new linearization algorithm for strongly nonlinear systems.

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