optimal control of mechanical systems subjected to periodic loading via chebyshev polynomials

OPTIMAL CONTROL APPLICATIONS & METHODS, VOL. 14, 75-90 (1993)

OPTIMAL CONTROL OF MECHANICAL SYSTEMS SUBJECTED TO PERIODIC LOADING VIA CHEBYSHEV

POLYNOMIALS

PAUL JOSEPH, R. PANDIYAN AND s. c. SINHA* Department of Mechanical Engineering, Auburn University, Auburn, AL 36849, U.S.A.

SUMMARY

Controllers using full-state feedback and observer-based feedback for mechanical systems subjected to periodic follower loads are designed in this paper. The designs are accomplished using an algebraic method incorporating Chebyshev polynomials coupled with principles of optimal control theory. The major advantage of this method is that it transforms differential equations with periodic coefficients to linear algebraic equations. Also, a dual-system approach to design observers for such periodic systems turns out to be a novel technique and appears to have been employed for the first time. As an example of a mechanical system a triple inverted pendulum subjected to a periodic follower load is chosen and it is shown that both types of controllers can be successfully designed. The computational aspects and the suitability of this method for higher-order mechanical systems are studied by applying it to one through five-mass inverted pendulum models. The efficiency of the method is checked against the Runge-Kutta, Adams-Moulton and Gear numerical algorithms available in the IMSL software package by comparing the CPU time taken to evaluate the characteristic exponents of the Floquet transition matrix (FTM) as well as the norm of the control vector at the end of one period. It is shown that the proposed technique becomes much more efficient for higher-order systems.

KEY WORDS Optimal control Chebyshev polynomials Linear periodic systems

1. INTRODUCTION

Often mechanical systems are subjected to follower or non-follower-type periodic loading. Even for linear systems, the dynamics and control problems associated with such systems are quite challenging. For example, it is well known that a follower force can give rise to divergence as well as flutter-type instabilities even if the force is time-independent. The control of large-dimensional systems subjected to periodic follower loads poses numerous problems of varying nature. Even though control methodologies for time-varying systems have been reported in the past, they are not suitable for practical applications, especially for higher- order systems, whereas optimal control theory3 has been found adaptable to the need, but obtaining the solution of the matrix Riccati differential equation is an exhaustive process. It has been shown that an algebraic method using orthogonal polynomials and the optimal control strategy blends together well and provides a method for controller design of general time-varying Using this method, the differential equations of motion with time- varying coefficients can be transformed to a set of linear algebraic equations. An important

* Author to whom correspondence should be addressed.

0 143 -2087/ 931 020075- 16$13 .OO 0 1993 by John Wiley & Sons, Ltd.

Received 27 March 1992 Revised 27 July 1992

P. JOSEPH, R . PANDIYAN AND S. C. SINHA 76

class of such problems is represented by systems where periodic coefficients appear in the differential equations. Satisfactory controller designs for higher-order periodic mechanical systems have not been attempted so far using Floquet theory. As an example of a higher-order periodic mechanical system a triple inverted pendulum subjected to a follower periodic load is considered. The method is also applied to four- and five-mass inverted pendulum models to show the computational efficiency of the method when applied to higher-order periodic systems.

Several authors have studied the control problem associated with time-invariant inverted pendulum models. Mori et d. and Schaefer and Cannon’ investigated the observer-based control of a single inverted pendulum supported on a cart and moving on a horizontal rail where a feedforward-feedback controller was used. Furuta et al. lo designed and constructed a digital controller to stabilize a double inverted pendulum in the upright position moving on an inclined rail. Sturgeon and Loscutoff” used an observer in the control algorithm that measured all the states of a double inverted pendulum. In yet another work, the attitude control of a triple inverted pendulum was considered by Furuta et al. l2 wherein the control was implemented by applying torques at the upper two hinges and making the lowest hinge free for rotation. Later, Meier Zu Farwig and Unbehauen” discussed the discrete computer control of a triple inverted pendulum hinged on a cart and moving on a horizontal rail. Maletinsky et af. l4 designed an observer-based controller where the observer measured the position of the cart and the angle between the cart and the first pendulum. However, the control of a multi-link inverted pendulum under periodic loads has not been reported.

Designs of full-state feedback controllers and observer based controllers via optimal control theory for mechanical systems such as a triple inverted pendulum subjected to a periodic follower load are presented in this paper. Owing to the appearance of periodically varying terms in the equations of motion, knowledge of Floquet theory15 is essential for the analysis. The method of analysis has been based on optimal control theory for time-varying systems l6

and the computational technique developed by Sinha and Wu17 and Wu18 for systems with periodic coefficients, where Chebyshev polynomials have been used to transform the system equations to a set of algebraic equations. The method developed by the above authors has been found to be very efficient 18*19 in the stability analysis of large-scale periodic systems. Combining their Chebyshev expansion procedure with principles of optimal control theory, efficient full-state feedback controllers can be designed for such periodic systems. Furthermore, a novel approach to designing observer-based controllers for linear periodic systems has been presented wherein Chebyshev polynomials and the dual-system approach have been used to compute the observer gain matrices. Finally, the computational aspects and the suitability of the method for higher-order mechanical systems have been shown via an application of the method to one- through five-mass inverted pendulum models.

2. METHOD OF ANALYSIS

2. I . Controller design for periodic systems

Consider a linear time-varying system in state space form

i ( t ) = A ( t ) x ( t ) + B(t )u( t )

and the corresponding output equation

Y(t ) = C ( t ) x ( t )

MECHANICAL SYSTEMS SUBJECTED TO PERIODIC LOADlNG 77

where the system is periodic, i.e. A ( t ) = A ( t + T), T being the period of the system. Considering a linear quadratic regulator (LQR) problem, the performance index to be minimized is selected as

where S(tf) is a symmetric positive semidefinite n x n matrix, Q ( t ) is a symmetric positive semidefinite n x n matrix, R ( t ) is a symmetric positive definite q x q matrix and tf is the time at which final state constraints are met.

Following the development given by Kwakernaak and Sivan, l6 the system Hamiltonian can be written as

(4) H* = i ( x T ( t ) Q ( t ) x ( t ) + u T ( t ) R ( t ) U ( t ) ) + pT(t)(A(t)x(t) + B ( t ) u ( t ) )

~ ( t ) = - R- ( t ) ~ ~ ( t ) p ( t )

and the control vector to minimize H* is given by

(5 1 where the adjoint variable p(t) and the state variable x ( t ) , both n x 1 vectors, satisfy the Hamiitonian equation

(6) A(t) -B(t)R-'(t)BT(t) x ( t ) [:{:;] = [ - Q ( t ) -A%) I[ P(t) 1

The above equation can also be expressed as

%(t)= (V + W ( t ) ) $ ( t ) (7)

where V is a 2n x 2n constant matrix and W ( t ) is a 2n x 2n periodic time-variant matrix defined in Appendix I by equations (37) and (38) respectively; W ) , a 2n x 1 vector, is defined as

= [x(l) P(t)l (8) Adapting the technique developed by Sinha and Wu," the elements of % ( t ) and W ( t ) can be expanded in terms of shifted Chebyshev polynomials of the first kind defined over the period [0, TJ. Thus

m- 1

r=O Xi(t)= C bfS,*(t)=s*'(t)b', i = 1 , 2 , 3 ,..., 2n (9)

m - 1

r=O Wij(t) = d?s,*(t) = S*'(t)d", i, j = 1,2,3, ..., 2n (10)

where . .

d i j = ( d t df ... dij 11 m - 1 b '= (bb b\ * - * bL-1)*, s**( t ) = (so*(t) sl*(t) ... ~ : - ~ ( t ) )

Here bf are unknown expansion coefficients of Ri ( t ) , d? are known expansion coefficients of Wi,(t) and $(t) are the shifted Chebyshev polynomials of the first kind." Now, for convenience in algebraic manipulation, a 2n x 2nm Chebyshev polynomial matrix is defined as

9 ( t ) = I o s**( t ) (1 1)

where 0 represents the Kronecker product2' and I is a 2n x 2n identity matrix. Using the

78 P. JOSEPH. R . PANDIYAN AND S. C. SINHA

definitions in equations (9)-(11), X ( t ) and W ( t ) can be rewritten as

X( t ) = 9(t)6

W(t)X(t)= S(t)Q6 W ( t ) = S(t)D

where 6 = (b' b2 b3 ..- b2"IT is a 2nm x 1 vector, D = [d" d i 2 d i 3 ..., 2n, is a 2nm x 2n matrix and 0 is a 2nm x 2nm product operation matrix.

a"], i , j = 1 , 2, 3 , 4,

The integral form of equation (7) is

where [ represents a dummy variable. Substituting equations (12)-(14) in (15) and then following the approach of Sinha and Wu, l7 one can obtain a set of linear algebraic equations of the form

(I - 2)6 = X(0) (16)

where z is a 2nm x 2nm constant matrix defined in Appendix I by equation (39) and 6 is the vector of unknown Chebyshev coefficients.

For the controller design of periodic systems, one needs to find the state transition matrix 9 ( t , 0) associated with the linear system given by equation (7). This requires a set of solutions with 2n initial conditions: Xi(0) = (l,O, 0, ..., 0), (0, l , O , ..., 0), ( O , O , l,O, ..., 0) , ..., (0, 0, ..., 1). It is to be noted that all bi corresponding to the above set of initial conditions can be determined simultaneously by defining the right-hand side of equation (19) in matrix form. Then the state transition matrix is given by

9(t, 0) = S(t)B (17) -

where B = [61 6 2 b 3 ... b2 "3. Therefore the 2n x 1 solution of equation (6) is given by

where 9ij(t,O) are blocks of the state transition matrix written in its partitioned form and 9(O,O)=I . It has to be noted that this state transition matrix (STM) is valid only for 0 ,< t < T, since the shifted Chebyshev polynomials of the first kind are defined over the interval [0, T I . When t > T, the STM can be evaluated using Floquet theory' as

9 ( t , 0 ) = *(+,O)(@(T,O))" (19)

w h e r e t = n T + $ , $ € [ O , T ] a n d n = l , 2 , 3 , ... . As given in Reference 16, the adjoint variable is computed as

p ( t ) = (922 - S912)-'(S911 - 9 2 1 ) ~ ( t ) = K(t)x(t)

K(t) = (*22 - S912)-'@911 - 9 2 1 )

(20)

Thus the gain matrix K(t) is given by

(21)

~ ( t ) = - R-'(t)BT(t)K(t)x(t) (22)

(23)

Hence the optimal control law as given in equation (5 ) can be computed as

Substituting equation (22) in equation (l), we obtain the closed-loop system equation as

X(t ) = (A(?) - B(t)R-'(t)BT(t)K(t))x(t)

MECHANICAL SYSTEMS SUBJECTED TO PERIODIC LOADING 79

By suitably adjusting the matrices S, Q and R, the performance characteristics required of the control system can be achieved.

2.2. Observer design for periodic systems

From a literature review it is found that observer designs for general linear time-varying continuous systems have been attempted mostly by using canonical transformations. 21322 In this approach an invertible linear transformation L ( t ) is used to convert the system matrices to a companion form, where L(t) is known as the invertible Lyapunov transformation matrix and is both continuous and bounded. Constructing such a Lyapunov transformation matrix even for a small-dimensional periodic system is a formidable task except for the special case of commutative systems. 23

A simpler method utilizing the dual-system approach which is widely used in the design of observers for time-invariant systems is proposed in this paper for periodically varying systems. The technique proposed here makes use of orthogonal polynomials coupled with optimal control theory applied to the dual system. In the past, orthogonal polynomial^^^*^^ have been used to design observers only for small-scale time-invariant systems.

It is shown in Reference 16 that the observer equation for the system defined by equation (1) is the form

i ( t ) = F ( t ) S ( t ) + C ( t ) y ( t ) + H(t )u ( t ) (24)

where %(t) is the n x 1 observer state vector. Matrices F(t) and H ( t ) are chosen such that F ( t ) = A ( t ) - E ( t ) C ( t ) and H ( t ) = B ( t ) . Defining the error state e ( t ) = x ( t ) - 9(t) , the error dynamics of the system is given by

e ( t ) = ( A ( t ) - c ( t ) C ( t ) ) e ( t ) (25) The gain matrix E ( t ) is selected such that the error vector asymptotically goes to zero as time tends to infinity. Noticing that the above equation is analogous to the dual system of equation ( I ) , the gain matrix G(t ) is obtained as shown below.

The dual system is defined as

i ( t ) = A T ( t ) z ( t ) + C T ( t ) u ( t ) y ( t ) = B T ( t ) z ( t )

where z ( t ) and y ( f ) are state and output vectors respectively. Applying optimal control theory to the above equation, a Hamiltonian equation similar to equation (6) is obtained as

(28) AT@) - CT( t )R- ' ( t )C( t )

where p ( t ) is the new adjoint variable of equation (26).

the solution is represented by Equation (28) is solved using the same algebraic method described earlier in Section 2. I and

where &jj(t, 0) are the elements of the state transition matrix written in its partitioned form and &(O, 0) = I. Since the solution is valid only for 0 < t < T, Floquet theory is applied to extend the solution beyond one period. Now the adjoint variable is computed as

(30) p(t) = ( 6 2 2 - S&12)-'(~&11 - & 2 1 ) x ( t ) = G ( t ) z ( t )

80 P. JOSEPH, R. PANDIYAN AND S. C. SINHA

and hence the dual-system gain matrix G ( t ) is given by

G ( t ) = (622 - S612)-1(s611 - 6 2 , ) (31)

Therefore the observer gain E ( t ) = G'(t) by Theorem 4.8 of Reference 16. Using the relations y ( t ) = C(t)x(t) and u(t ) = - P(t ) f i ( t ) , the observer-based closed-loop system can be written as

(32)

where K ( t ) = R-'(t)BT(t)K(t) and K ( t ) is obtained from equation (21) as described in Section 2.1.

A ( t ) - B ( t ) R ( t ) EI:I) = [ c T ( t ) C ( t ) A ( t ) - G'(t)C(t) - B ( t ) ~ ( t )

3. APPLICATIONS

As an example of a periodic mechanical system, consider the inverted triple pendulum subjected to a periodic follower load as shown in Figure 1 . Following Lagrange's formulation, the linearised equations of motion of the system are obtained as

(33) Mij + (K + K * ( t ) ) q = u(t)

where

-1 l - y ( l - a ) 1 -1 2 - Y - ( i+aY)

Also, y = Plfk, P = PI + P2 cos(wt) and u(r) = (ul( t ) , uz(t), u3(t)) is the control torque vector applied. Note that in the above equations the stiffness matrix has time-varying periodic terms with period T = 2 4 ~ . The set of second-order equations (33) can be rewritten in state space form as

(34) X(t) = A(t)x( t ) + Bu(t)

where

x ( t ) = [vi(t) 72(f) ~ 3 ( t ) fii(t) i/2(t) f i 3 ( t ) l T

0 1 0 0 0 0 0

0 0 0

B = [ -1 8 _8 2 -1 !] 0 - 1

with & = k/m12. Using the procedure described in Section 2.1, equation (34) can now be transformed into a

set of linear algebraic equations and the corresponding full-state optimal gain matrix can be


P*Cos(wt)

Figure 1 . A three-mass inverted pendulum subjected to a periodic load

easily computed. Under the assumption that only the state variables [TI ‘72 q3] are measured, the output equation is given by

y ( t )= [: 0 : 1 0 : 0 : 0 : : I [ ] 0 (35)

On the basis of the above equation, an observer-based controller is also designed as per the procedure explained in Section 2.2.

82 P. JOSEPH, R. PANDIYAN AND S. C. SlNHA

40.0

20.0

a a 0.0 i;

-20.0

-40.0

4 E

- X I - xz - xs x4 X6 XB

...............

...............

0.0 1.0 2.0 9.0 4.0 6.0

* ( P d d r J

Figure 2. Static instability of the triple inverted pendulum

10.0

6.0

0.0

-6.0

-10.0

PI-1.0. P2-0.7. KbW-1.0. AIpho-1.0. OnUgQ-1.0 I I I I

- Xf -x.? - xs

x4 xs

............... ----_

0.0 Z.0 4.0 3.0 8.0 fO.0

* (P-) Figure 3. Full-state feedback control (static case)

MECHANICAL SYSTEMS SUBJECTED TO PERIODIC LOADING

Pl-1.0, Pa-0.7, -1.0. A(p)m-l.O, m # a - l . O 10.0

- X I obsnvmd - XS o b s w d ............... X4 obsmud -~ X6 obmved --- xb obsmmd 6.0

f 0.0

I

83

1600.0

1000.0

600.0

0.0

-soo.o

-1 000.0

- 1600.0

mu (P.rl0d.J

Figure 4. Observer-based control (static case)

Pl-2.0. PI-0.7, -8.0. I-- 1.0, hru#wZ.O

- xz - XS X I X6 X6

............... ---. __--

0.0 0.s 1.0 1.6 0.0 1.6 3.0

nma fP&)

Figure 5. Kinetic instability of the triple inverted pendulum

84

8.0

4.0

2.0

5 a 0.0 P

-2.0

-4.0

-6.0

P. JOSEPH, R. PANDIYAN AND S. C. SINHA

PI-2.0. p2-0.7. mar-2.0. Alpha-1.0. Onnga-9.0

I I I I I

- Xl - x2

............... X I X6 ---

I I I I

0. 0 2.0 4.0 8.0 8.0 10.0 - f P d & )

Figure 6. Full-state feedback control (kinetic case)

8.0

4.0

2.0

5 p 0.0 h

-2.0

-4.0

-8.0

PI -2.0. PZ-0.7, mar-2.0. Alpha- 1.0. Onuga-2.0 I I I I

XI obsmmd - X2 obtmmd - XS o b t m 0 d X I obssmsd

---- X 6 obsmvsd ---- X 8 olumvsd

...............

u

0.0 2.0 4.0 8.0 8.0 10.0

mns f P d & )

Figure 7. Observer-based control (kinetic case)


The above-designed controllers should cater to both the failure modes, namely static and kinetic instability of the inverted pendulum, 26 common to this kind of structure. The static instability is a buckling failure and the kinetic instability is a flutter failure. The type of instability is usually determined from the characteristic exponents of A(t ) evaluated at the end of one period. If all the exponents are real or a combination of purely real and purely imaginary, then the system fails in a static sense. If at least one pair of the exponents becomes complex, then the system fails in flutter. It is to be noted that the type of failure of the structure depends on the parameters p l , p2, CY and w. For the given system, the values of m, 1 and k are assumed to be constants.

The controllers are tested by considering some typical parameter sets such as (i) p1 = 1 *O, p2 =0.7 , k = 1.0, a = 1.0 and w = 1.0, for which the system fails in buckling, and (ii) PI = 2.0, P2 = 0.7, k = 2.0, CY = 1.0 and w = 2.0, for which the system fails in flutter. The corresponding response characteristics of the system with and without the controllers are shown in Figures 2-7. It is found that the dual-system approach has been very successful in designing observers for higher-order periodic mechanical systems. Results have also been obtained for four- and five-mass inverted pendulum models, but the details are omitted for brevity.

4. DISCUSSION OF RESULTS AND CONCLUSIONS

Controller designs based on full-state feedback and observer-based feedback for mechanical systems subjected to periodic loading under failure conditions have been successfully achieved using an algebraic method incorporating Chebyshev polynomials. In particular, the observer design proposed in this study is simple and efficient. The proposed method exploits the advantages of evaluating the state transition matrix as an explicit function of time in terms of Chebyshev polynomials. Therefore the optimal gain matrices are also expressed as explicit functions of time in closed forms for a given set of parameters. Of course, such results cannot be obtained through a numerical integration approach.

Figures 2 and 5 show the open-loop behaviour of the triple inverted pendulum subjected to a periodic follower load for the buckling and flutter modes respectively. The closed-loop responses of the system with the designed controllers are depicted in Figures 3, 4, 6 and 7. The response characteristics of the system with full-state feedback control are shown in Figures 3 and 6 . From Figures 4 and 7 it is quite evident that the dual-system approach to designing observers for periodically time-varying systems is an accurate and efficient procedure for controller designs. These controllers have also been implemented via numerical integration and the results closely match those obtained by the algebraic method. It is well known that the characteristic exponents of linear periodic systems determine the stability behaviour and therefore the computation of these exponents constitutes suitable measure for comparison of different algorithms. Figure 8 shows the convergence of the percentage error in obtaining the largest characteristic exponent by the algebraic method as a function of the number of terms used in the Chebyshev expansion for a four-mass inverted pendulum problem. The percentage error is based on the values obtained by a sixth-order Runge-Kutta scheme with an optimum tolerance of lo-''. It is seen that 18 terms are sufficient to achieve this accuracy.

Furthermore, the computational characteristics of the algebraic scheme have been studied by considering one through five-mass inverted pendulum models. Table I shows the CPU time comparison between the algebraic method and standard numerical techniques implemented on a Cray X-MP/24 and Sun Sparc Station to evaluate the real exponents of the Floquet transition matrix (FTM) for the various inverted pendulum models. Another comparison evaluating the

0.0

-3.0

-6.0

P. JOSEPH, R. PANDIYAN AND S. C. SlNHA

E

-9.0

Number of terms used in Chebyshev Expansion

Figure 8. Convergence of characteristic exponent for the four-mass problem

Euclidean norm of the control vector at the end of one period is shown in Table 11. In both these comparisons a 15-term Chebyshev expansion has been used so that the real parts of the characteristic exponents of the FTM retained five-digit numerical accuracy. It is evident from the tables that as the size of the system becomes larger and larger, the efficiency of the algebraic method improves on both systems.

In conclusion, a very efficient, simple and straightforward approach to the design of controllers for periodically time-varying linear systems has been suggested. The approach is based on the idea that the periodic coefficients as well as the state vectors can be expanded in terms of Chebyshev polynomials. State feedback as well as observer-based controller designs are totally transformed into algebraic problems which yield a definite computational advantage, especially for higher-order systems. Another added feature of the present method is that it provides a straightforward one-level controller design, unlike the method reported in


Table I. CPU time (seconds) taken to determine the largest characteristic exponent of the inverted pendulum models

Inverted Algebraic Runge-Kutta Adams-Moulton pendulum method method method Gear method model (size of FTM) A B A B A B A B

One mass 0.0080 0.09 0.0134 0.09 0.0150 0.17 0.0170 0.18

Two-mass 0.0287 0.18 0.0383 0.27 0.0422 0.60 0.0530 0.63

Three-mass 0.0541 1.54 0.0964 2.27 0.1250 2.30 0.2154 2.84

Four-mass 0.1305 3.37 0.1936 5.69 0.2686 6.21 0.4642 6.98

Five-mass 0.2801 13.77 0.4875 17.72 0.4922 19.73 1.0251 21.70

(4 x 4)

(8 x 8)

(12 x 12)

(16 x 16)

(20 x 20)

A, Cray X-MP/24; B, Sun SparcStation 1.

Table 11. CPU time seconds taken to determine the norm of the control vector at the end of one period for the inverted pendulum models

Inverted Algebraic Runge-Kutta Adams-Moulton pendulum method method method Gear method model (size of FTM) A B A B A B A B

One-mass 0.0078 0.08 0.0129 0.09 0.0140 0.18 0.0160 0.17 (4 x 4) Two-mass 0-0256 0.17 0.0367 0.25 0.0407 0-58 0.0516 0.61 (8 x 8) Three-mass 0.0478 1.51 0.0920 2.24 0-1210 2.29 0.2111 2-83 (12 x 12) Four-mass 0.1195 3.31 0.1842 5.62 0.2589 6-15 0.4547 6.91 (16 x 16) Five-mass 0.1761 13.68 0,4653 17-63 0.4795 19.64 1.0224 21.61 (20 x 20)

~~ ~

A, Cray X-MP/24; B, Sun Sparstation 1.

Reference 27 where the authors themselves acknowledge that ‘the theory used is considerably more complicated than the time-invariant case and requires two levels of iterations in calculating the optimal feedback gains’. Yet another advantage of this approach is to be able to place the poles of the closed-loop linear periodic systems optimally without transforming the equations into a canonical form which is neither simple nor a unique process. It is also anticipated that since the control problem has been transformed into an algebraic one, the implementation of this procedure through parallel processing machines will provide some practical real-time applications in the future.


ACKNOWLEDGEMENTS

Financial support for this work was provided by the Army Research Office, monitored by Dr. Gary L. Anderson under contract number DAAL03-89-k-0172. The Cray computer time provided by Alabama Supercomputer Network and Auburn University is also acknowledged.

APPENDIX I

In equation (6) let A(t) = A0 + Ao(t)

- Q ( t ) = Qo + Qo( t ) -AT(t) = A0 + Ao(t)

- B ( ~ ) R - ' B ~ ( ~ ) = RO + R o w

where Ao, Qo, &, &, A&), &(t), oo(t) and Ao(t) are n x n matrices. Then

The matrix z appearing in equation (16) can be written as

Z = P + R (39)

where

Here CA, CR, CQ and CA are the coefficient matrices of Ao(t), &(t), 000) and Ao(t) respectively and from Reference 17, GT, the rn x rn integration operational matrix, and Q(dj), the rn x m product operational matrix, are given as

(42)

1/4(m - 1) 0

(43)

; 0 0 . . -- 8 1 0 8 0 . .

- a 0 i'r . . - 16 1 0 - 8 0 . .

1 2

1 1 6

--

. . . . GT= [ .

. . . . (-l)rn/2rn(rn - 2) . . . . . -1/4(m-2)

do d1/2 d2/2 . . do+d2/2 (dl+d3)/2 . .

d2 (di +d3)/2 do+d4/2 . . (dm-2 + dm)/2

. . dO+dZm-2/2

. .

. .

. . d m - 1 ( d m - 2 + dm)/2


APPENDIX 11: NOMENCLATURE

n x n periodic system matrix (b' bZ b3 ... - b2"IT, 2nm x 1 vector of Chebyshev coefficients [bl b2 b 3 ... bzn] , 2nm x 2n Chebyshev coefficient matrix n x q input gain matrix r x n output matrix 2nm x 2n expansion coefficient matrix x ( t ) - a( t ) , error between actual state and observed state A(t ) - GT( t )C( t ) , n x n matrix r x n gain matrix G'(t), n x r observer gain matrix Hamiltonian B(t) 2n x 2n identity matrix quadratic performance index kllm torsional stiffness of inverted pendulum n x n controller gain matrix constant stiffness matrix periodic stiffness matrix link length of inverted pendulum Lyapunov transformation matrix mass mass matrix n x 1 adjoint vector n x 1 adjoint vector P I + PZ cos(wt), periodic load n x n symmetric positive semidefinite matrix q x q symmetric positive definite matrix n x n symmetric positive definite matrix 2n x 2nm Chebyshev polynomial matrix shifted Chebyshev polynomials of first kind period final time input vector n x n constant matrix n x n periodic matrix n x 1 state vector from equation (1) 2n x 1 vector of initial conditions in equation (16) 2n x 1 augmented state vector from equation (7) n x 1 observer state vector from equation (24) r x 1 output state vector from equation (2) output state vector of dual system from equation (27) n x 1 dual-system state vector from equation (26) 2nm x 2nm constant matrix Kronecker product


Y PI1 k rl * ( t , o), &(t, 0) w frequency of periodic load IL O G I L G T t: dummy variable

state vector of triple inverted pendulum from equation (33) state transition matrices

REFERENCES

1. D’Angelo, H.. Linear Time-varying systems: Analysis and Synthesis, Allyn and Bacon, Boston, MA, 1970. 2. Brogan, W. L., Modern Control Theory, Prentice-Hall, Englewood Cliffs, NJ, 1985. 3. Kalman, R. E., ‘Contributions to the theory of optimal control’, SOC. Mat. Mex. Bol., 5 , 102-119 (1960). 4. Liu, C.-C. and Y. P. Shih, ‘Analysis and optimal control of time-varying systems via Chebyshev polynomials’,

5 . Hwang, C. and M.-Y. Chen, ‘Suboptimal control of linear time-varying multi-delay systems via shifted Legendre

6. Razzaghi, M. and M. Razzaghi, ‘Solution of linear two-point boundary value problems and optimal control of

7. Nagurka, M., S. Wang and V. Yen, ‘Solving linear quadratic optimal control problems by Chebyshev-based state

8. Mori, S., H. Nishihara and K. Furuta, ‘Control of a mechanical system, control of a pendulum’, Int. J. Control,

9. Schaefer, J. F. and R. H. Cannon, ‘On the control of unstable mechanical systems’, Proc. 3rd Congr. IFAC,

10. Furuta, K., H. Kajiwara and K. Kosuge, ‘Digital control of a double inverted pendulum on an inclined rail’, In( .

11. Stuergeon, W. R. and M. V. Loscutoff, ‘Application of modal control and dynamic observers to control of a

12. Furuta, K., T. Ochiai and N. Ono, ‘Attitude control of a triple inverted pendulum’, Int. J. Control, 39,

13. Meier Zu Farwig, H. and H. Unbehauen, ‘Discrete computer control of a triple inverted pendulum’, Optim.

14. Maletinsky, W., M. F. Senning and F. Wiederkehr, ‘Observer based control of a double pendulum’, Proc. IFAC

15. Coddington, E. A, and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. 16. Kwakernaak, H. and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York, 1972. 17. Sinha, S. C. and D.-H. Wu, ‘An efficient computational scheme for the analysis of periodic systems’, J. Sound

18. Wu, D.-H., ‘Development of an efficient computational technique for the analysis of linear dynamic systems with

19. Sinha, S. C., D.-H. Wu, V. Juneja and P. Joseph, ‘Analysis of dynamic systems with periodically varying

20. Bellman, R., Introduction to Matrix Analysis, McGraw-Hill, New York, 1970, p. 235. 21. Alfhaid, M. and G. K. F. Lee, ‘Pole assignment in linear time-varying systems using state feedback’, Proc. 22nd

Asilomar Con$ on Signals, Systems and Computers, Pacific Grove, CA, October-November 1988, pp. 241-245. 22. O’Reilly, J., Observers for Linear Systems, Academic, London, 1983. 23. Lukes, D. L., Differential Equations: Classical to Controlled, Academic Press, New York, 1982. 24. Horng, J. and 1. R. Horng, ‘Shifted Legendre series analysis of linear optimal control systems incorporating

25. Chang, Y.-F. and T.-T. Lee, ‘General orthogonal polynomials analysis of linear optimal control systems

26. Leipholz, H., Stability Theory, Academic, New York, 1972. 27. Calise, A. J., M. E. Wasikowski and D. P. Schrage, ‘Optimal output feedback for linear time-periodic systems’,

Int. J . Control, 38, 1003-1012 (1983).

polynomials’, Int. J . Syst. Sci., 16, 1517-1537 (1985).

time-varying systems by shifted Chebyshev approximations’, J. Franklin Inst., 327, 321-328 (1990).

parameterization’, Proc. 1991 Am. Control Conf., Boston, MA, June 1991. pp. 104-109.

23, 673-692 (1976).

1966, Paper 6C1.

J. Control, 32, 907-924 (1980).

double inverted pendulum’, Proc. JACC, Stanford, CA, 1972, pp. 857-865.

1351-1365 (1984).

control appl. methods, 11, 157-171 (1990).

World Congr., 1981, pp. 3383-3387.

Vibr., 151, 91-117 (1991).

periodically varying parameters’, Ph. D. Thesis, Auburn University, 1991.

parameters via Chebyshev polynomials’, J. Vibr. Acoust., 115, 96-102 (1993).

observers’, Int. J. Syst. Sci., 16, 863-867 (1985).

incorporating observers’, Int. J. Syst. Sci., 16, 1521-1535 (1985).

J. Guid.. Control, Dyn., 15, 416-423 (1992).

optimal control of mechanical systems subjected to periodic loading via chebyshev polynomials

Documents