Discrete Linear Quadratic Control of Uncertain Switched System

2017 5th International Conference on Control, Instrumentation, and Automation (ICCIA)

Discrete Linear Quadratic Control of Uncertain Switched System Maryam Baluchzadeh Phd student Department of Electrical Engineering Ferdowsi University of Mashhad Mashhad, Iran Email: [email protected]

Ali Karimpour Associate Professor Department of Electrical Engineering Ferdowsi University of Mashhad Mashhad, Iran Email: [email protected]

Abstract— This paper presents optimal control of uncertain switched systems in presence of the external disturbance and parametric uncertainty and generally every uncertainty due to modeling error. In this paper, the switching signal is selected a priori and control input is designed to minimize a given cost function. Discrete Linear Quadratic (DLQ) control has been efficiently applied to certain systems as an optimal control. There would seem to be some difficulties to apply discrete linear quadratic control to uncertain switched system. To overcome the problems, this paper presents an appropriate model. Then uncertainties of the uncertain switched system are compensated by robust time-delay controller. Then control input is designed by discrete linear quadratic control. The stability analysis and simulation verify effectiveness of the proposed control approach. Keywords- Discrete Linear Quadratic; uncertain switched system; modeling error.

1 INTRODUCTION The optimal control problem of switched control system is one of the challenging research topics and has been attracting much attention in the control community. Switched linear systems belong to a special class of hybrid systems, which comprises a collection of subsystems described by linear dynamics (differential/ difference equations), together with a switching rule that species the switching between the subsystems. Such systems can be used to describe a wide range of physical and engineering systems in practice [1-2]. Switched linear systems not only provide a challenging forum for academic research, but also bridge the gap between the treatment of linear systems and that of highly complex and/or uncertain systems. Optimal control of switched system is studied in different papers [3-10]. For the simplicity, the uncertainties are not considered in the most papers. As a matter of practical viewpoint, most of practical control systems contain uncertainties. This paper introduces an optimal control of uncertain switched systems with external input by combining discrete linear quadratic control with robust time-delay controller. Since

978-1-5386-2134-9/17/$31.00 ©2017 IEEE

Naser Pariz Professor Department of Electrical Engineering Ferdowsi University of Mashhad Mashhad, Iran Email: [email protected]

the discrete linear quadratic control is model based, at first a linear discrete model is introduced. External disturbance and parametric uncertainty and generally every uncertainty due to modelling error are considered as a lumped uncertainty. The uncertainties are compensated by robust time-delay controller then discrete linear quadratic control is performed. The rest of this paper is organized as follows: modeling of uncertain switched system is presented in section I. Section II develops the optimal control of uncertain switched system includes time delay controller and discrete linear quadratic control. Section III presents stability analysis. Section IV illustrates simulation results. section V concludes the paper. 2 MODELING OF UNCERTAIN SWITCHED SYSTEM by

Consider a class of uncertain switched systems described

 x = ai x + hi x + bi u + d + fi ( x, x )

(1)

Where i, x , u , ai , hi , bi , d and fi ( x, x ) are active subsystem, state, control input, first order derivative coefficient of state in the ith subsystem, state coefficient in the ith subsystem, coefficient of input gain in the ith subsystem, random disturbance and nonlinear and uncertain function include nonlinear terms due to ith subsystem modeling error, respectively. Using nominal terms in (1) obtains

 x = aˆi x + hˆi x + bˆi u + ϕi

(2)

where aˆi , bˆi and hˆi are the nominal terms for the real terms ai , bi and hi , respectively. where the lumped uncertainty ϕi is expressed as

ϕi = (ai − aˆi ) x + (hi − hˆi ) x + (bi − bˆi )u + d + fi ( x, x )

(3)

Lumped uncertainty ϕi include parametric uncertainty, external disturbance and uncertain function include nonlinear terms due to ith subsystem modeling error.

Assumption 1: The desired trajectory must be smooth in the sense that state and its derivatives up to a necessary order are available and all uniformly bounded.

From (2), we derive the state-space model (4)

ˆ E + Bˆ U + gϕ E = A i i i

ˆ ,B ˆ and g are state, control input, the where E , U , A i i state matrix in the ith subsystem, the input gain matrix in the ith subsystem and uncertainty coefficient matrix, respectively.

0 x ˆ E =  ,A i = ˆ  hi  x 

1 0 0   , Bˆ i =  ˆ  , g =   aˆi  bi  1 

(5)

From (4), one can obtain a discrete switched system using a sampling period σ which is a small positive constant.  as Substituting kσ into t for approximating E  E = (E(t + σ ) − E(t )) / σ in (4), we obtain a discrete switched system ˆ E + Bˆ U + g ϕ Ek +1 = A i,k k i,k k k i,k Where ˆ = I +σ A ˆ (σ k ) , Ek = E(kσ ) , A i ,k i

Smoothness of the desired trajectory can be guaranteed by proper trajectory planning. As a necessary condition to design a robust controller, the matching condition must be satisfied: Matching condition: the uncertainty must enter the system the same channel as the control input. Then, the uncertainty is said to satisfy the matching condition or equivalently is said to be matched. We ensure the matching condition since in the system (6), the lumped uncertainty ϕi , k enters the switched system the same channel as the control input. As a necessary condition to design a robust control, the external disturbance d and nonlinear function f i ( x, x ) in (1) must be bounded. Assumption 2: The external disturbance d , nonlinear function f i ( x, x ) and control input are bounded as

(6)

Bˆ i,k = σ Bˆ i (σ k )

To make the dynamics of tracking error well-defined such that the switched system can track the desired trajectory, we make the following assumption.

,

U k = U(σ k ) and g k = σ g . ϕi , k is lumped uncertainty. 3 OPTIMAL CONTROL OF UNCERTAIN SWITCHED SYSTEM For applying the optimal control law ,full knowledge about the system considered is a necessary prerequisite. If the system is affected by uncertainties and disturbances, the optimal controller loses its effectiveness. An efficient way to tackle uncertainties is to integrate the optimal controller (DLQ) with robust time-delay controller to ensure robustness. In this section, an optimal Control of uncertain switched systems with external input is introduced. In order to, DLQ is combined with robust time-delay controller. The uncertainties include external disturbance and parametric uncertainty and generally every uncertainty due to modeling error. In order to employ the DLQ control for the uncertain switched system (6), we have to compensate the uncertainty ϕi , k .

3.1 robust time-delay controller The discrete linear quadratic control (DLQ) has been efficiently applied on certain linear systems as an optimal control. In order to employ the DLQ control for the uncertain switched systems (6), we have to compensate the uncertainty ϕi , k . For this purpose, a robust time-delay controller is used to compensate the uncertainty. This type of uncertainty estimation was successfully used to estimate the uncertainty in the robust impedance control of a hydraulic suspension system [11], the control of flexiblejoint robots [12], minimum-norm and time-optimal repetitive control [13] and optimal control of robot manipulators [14].

(7)

d ≤ d max fi ( x, x ) ≤ f max u (t ) ≤ umax where d max , f max and

u max are positive constants.

Assumption 3: All subsystems of switched system (1) are internally stable. A two-term control law is proposed. The first term is an optimal DLQ controller and the second term is a robust controller. Thus, system (6) is presented as

ˆ E + Bˆ u + E k +1 = A i ,k k i , k 1, k ˆ B u +g ϕ i , k 2,k

(8)

k i,k

where ui ,1 ( k ) and ui ,2 ( k ) are the first and second terms of the control input. Performance of the optimal control is improved if the lumped uncertainty ϕi , k is compensated. The uncertainty is perfectly compensated if

Bˆ i,k u2,k = −g k ϕi,k

(9)

Since ϕi , k is not known, control law (9) cannot be defined. To estimate the uncertainty, we obtain from (8) (10)

ˆ E − Bˆ u − Bˆ u g k ϕi , k = Ek +1 − A i,k k i , k 1, k i , k 2, k Since Ek +1 is not available in the kth step, g k ϕi, k cannot be calculated. Instead, the previous value of g k ϕi, k is used as

ˆ g k −1ϕi ,k −1 = Ek − A i , k −1E k −1 − Bˆ i , k −1u1, k −1 − Bˆ i ,k −1u2, k −1

As a result, we have

(11)

ˆ K k = [ R + Bˆ *i , k p k Bˆ i , k ]−1 Bˆ *k p k A i,k

The term g k −1ϕi , k −1 can be calculated since all terms in the RHS of (11) are known and available. Thus, we propose the robust control law

Bˆ i,k u2,k = −g k −1ϕi ,k −1

where p k is calculated as

We express the second term in the control law by substituting (11) into (12) to yield

The algorithm starts from k = 0 in (19) where p−1 = 0 . Then, K k is calculated as (18). Next, u1,k is computed from (16).

(13)

Bˆ i ,k −1u1, k −1 + Bˆ i , k −1u2, k −1

4

3.2 Discrete Linear Quadratic Control If the uncertainty can be well compensated, then the DLQ controller can be efficiently used as an optimal controller. Substituting (12) into (8) yields ˆ E + Bˆ u + E k +1 = A i ,k k i , k 1, k

STABILITY ANALYSIS

The block diagram of proposed control approach is shown in fig1. To examine the stability analysis, the state-space model is derived from (1) as:

(14)

x = Ai x + ρi

g k ϕi , k − g k −1ϕi , k −1 The time-delay robust controller fulfils its role since its effect is well detected in (14). The uncertainty from g k ϕi, k

 x x =  dx     dt 

(16)

)

1 * 1 N −1 * E N SE N +  [ E*k QEk + u1, k Ru1, k + 2 2 k =0 ˆ E + Bˆ u − E λ*k +1 A i,k k i , k 1, k k +1 +

(

*

Then, solution of equation (20) is: x(t ) = e Ai t x(0) + 0t e A i (t −τ ) ρi (τ ) dτ

(17)

for t ≥ 0

(22)

According to a proof given by [16], x(t ) is bounded as

)

( Aˆ i,k Ek + Bˆ i,k u1,k − Ek +1 )

( fi ( x, x ) + bi u + d )

bound of norm of ρi .

The gain matrix K k is calculated by minimizing a given cost function [15] of the form

(

T

(21)

1 ai 

Under Assumptions 1-2, ρi in equation (21) is bounded ρi is bounded such that ρi ≤ pmax where pmax is the upper

Then, the DLQ controller is given by

J=

0 Ai =   hi

ρi = [ 0 1]

(15)

u1,k = −K k E k

(20)

where

in (8) is reduced to g k ϕi ,k − g k −1ϕi ,k −1 in (14). In order to apply the DLQ, a nominal model in the form of linear switched system is suggested from (14) as ˆ E + Bˆ u E k +1 = A i,k k i , k 1, k

(19)

ˆ* p A ˆ − pk = Q + A i , k k −1 i ,k ˆ * p Bˆ [R + Bˆ * p Bˆ ]−1 Bˆ * p A ˆ A i , k k −1 i , k i , k k −1 i ,k i , k k −1 i ,k

(12)

ˆ Bˆ i ,k u2, k = −Ek + A i , k −1Ek −1 +

(18)

x(t ) ≤ e Ai t x (0) + pmax

λ k +1 ]

= e Ait x(0) + pmax A i −1 (I − e A i t )

with respect to Ek , u1,k and λ k where λ k is the Lagrange multiplier, Q and R are symmetric positive definite matrices.

umax

u

d x(k )

Actual system

x

−umax

u2 ( k )

Fig.1. Block diagram of proposed control approach

(23)

t A i ( t −τ ) dτ 0e



Robust time delay controller Discrete linear quadratic control

∀

Under Assumptions 3, Ai is Hurwitz. Since A i is Hurwitz

2 d

→ 0 as t → ∞ , thus, the upper bound of x(t)

asymptotically approaches pmax Ai −1 as t → ∞ , which is a limited value. Thus, system (21) is bounded-input, boundedoutput (BIBO) stable system. Therefore, the system (1) with following details provides a bounded state under BIBO stability.

1.5 disturbance

e

Ai t

-0.5 0

100

500

active subsystem

1.8 1.6 1.4 1.2 1 0

100

200

300

400

500

k

Consider the scalar switched system described by

Fig.3. switching signal

(24)

1

-3

-4

5

Simulation1. The final law includes (13), (16) for optimal controlling of uncertain switched system (24) is simulated. The initial states are considered as [0.5 0.5]. The results, including the histories of first state, second state and the first control input, second control input, cost function are shown in Fig.4, Fig.5, Fig.6, Fig.7 and Fig.8, respectively. The errors of first state and second state are vanished well after 30sec and come

2

-1

-6

8

under 1.542 × 10−2 , 8.174 × 10 −2 , respectively.

Table 1. parameters of the switched system Subsystem ai hi (i)

bi

The uncertainty may include the external disturbance and parametric uncertainty and generally every uncertainty due to modeling error. To consider the parametric uncertainty, all parameters of the nominal model used in the control law are given as %80 of the real ones. Nonlinear function due to modeling error in first and second subsystem are considered 0.1sin( x 2 ) and 0.1cos( x 2 ) , respectively. Sampling period is σ = 0.1 . The external disturbance is a random signal with the mean 0 and standard deviation 0.5 with a period of 10s as shown in Fig. 2. The matrices Q and R in cost function (17) Fig.1. Block diagram of proposed control approach

100 0  are Q =   , R = 10 . Switching signal is selected a  0 10  priori as shown in Fig. 3. The uncertainty is unknown; however, we have to use an example of a bounded uncertainty to check the performance of the proposed control system.

0.6 x

1

0.4 1

The parameters of the switched system are given in Table

x

1.

400

2

The performance of the proposed method is evaluated through the following two examples.

 x = ai x + hi x + bi u + d + fi ( x, x )

200 300 iteration

Fig.2. disturbance

active subsystem

5 SIMULATION RESULTS

0.5 0

The robust time-delay control law (13) has a main role in compensating the uncertainty. If there exists a much difference between the nominal model (15) and the actual system (6), the closed-loop system (6) is subject to a large uncertainty. The residual uncertainty in the closed-loop system (14) is reduced from a large value of g k ϕi, k to a small value of g k ϕi ,k − g k −1ϕi ,k −1 due to using the robust time-delay control law (13). As a result, the performance of control system is improved by reducing the residual uncertainty. The residual uncertainty g k ϕi ,k − g k −1ϕi ,k −1 will be very small when the uncertainty is smooth and sampling time is very short.

1

0.2 0 -0.2 0

100

200 300 iteration

400

Fig.4. Trajectory of system state of x

500

0.5 x

2

x

2

0

-0.5

-1 0

100

200 300 iteration

400

Fig.5. Trajectory of system state of

500

x

Simulation2. The effect of the robust time-delay controller in compensating the uncertainty is evaluated in this simulation. For this purpose, the time-delay controller is removed. The control law (16) for optimal controlling of uncertain switched system (24) is simulated. The initial states are considered as [0.5 0.5]. The results, including the histories of first state, second state and the control input, cost function are shown in Fig.9, Fig.10, Fig.11 and Fig.12, respectively. After 30sec , The errors of first state and second state come under the 1.001× 10−1 , 1.083 × 10−1 , respectively. Since cost functions are converges when switched systems are completely certain. In this paper, because of the presence of uncertainties, cost functions are increased as shown in fig. 8 and fig. 12. Compared to Simulation 1, the errors and cost function are increased than the ones in simulation1.

0.4 u

1

0.6

1

0.2

x

1

0.4

x

1

u

0 -0.2 -0.4 0

0.2 0

100

200 300 iteration

400

500

-0.2 0

Fig.6. Control effort of DLQ

100

200 300 iteration

400

500

Fig.9. Trajectory of system state of x

0.2 u

0.5

2

x

0.1

2

0 2

u

2

0 x

-0.1

-0.5

-0.2 -0.3 0

100

200 300 iteration

400

500

-1 0

100

200 300 iteration

400

500

Fig.7. Control effort of robust time delay control Fig.10. Trajectory of system state of

x

400 0.3 u

300 0.2

200

0.1 u

cost function

J

0

100 0 0

-0.1

100

200 300 iteration

400

Fig.8. Cost function in first simulation

500 -0.2 0

100

200 300 iteration

400

Fig.11. Control effort of DLQ

500

500

J

cost function

400 300 200 100 0 0

100

200 300 iteration

400

500

Fig.12. Cost function in second simulation

6 CONCLUSIONS This paper presented discrete linear quadratic control of uncertain switched systems that it includes external disturbance and parametric uncertainties and generally every uncertainty due to modeling error. The model uncertainty is efficiently compensated using a discrete robust time-delay controller. Then control input is designed by using discrete linear quadratic control. The robust controller has played an important role to improve the performance of the control system by reducing the residual uncertainty in the closed-loop system. The Stability analysis and simulations verified effectiveness of the proposed control approach. REFERENCES [1]

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