Pulse response based control for positive definite systems

Pulse Response Based Control for Positive Definite Systems

Raul Gonzalez-Lima

Departamento de Engenharia Mecânica, Escola Politécnica da Universidade de São Paulo,

Av. Prof. Mello Moares, 2231 - 05508-900 - São Paulo SP Brazil

Pulse Response Based Control (PRBC) is a recently developed minimum time control method for flexible structures. The flexible behavior of the structure is represented through a set of discrete time sequences, which are the responses of the structure due to rectangular force pulses. The rectangular force pulses are given by the actuators that control the structure. The set of pulse responses, desired outputs, and force bounds form a numerical optimization problem. The solution of the optimization problem is a minimum time piecewise constant control sequence for driving the system to a desired final state.

The method was developed for driving positive semi-definite systems. In case the system is positive definite, some final states of the system may not be reachable. Necessary conditions for reachability of the final states are derived for systems with a finite number of degrees of freedom. Numerical results are presented that confirm the derived analytical conditions. Numerical simulations of maneuvers of distributed parameter systems have shown a relationship between the error in the estimated minimum control time and sampling interval

Introduction

Methods for maneuvering flexible structures require, in general, an explicit state-space model, or a modal model. Very high sampling frequencies are required to maneuver flexible structures if no simplifications are made to the analytical approach. Several simplifications are normally imposed over the analytical approach (Jankovski and Brussel, 1992). Often, a discrete-time model is used and only a few rigid regions of the structure are attempted to be controlled (Cetikunt and Wu, 1992; Skaar and Tucker, 1992). Pulse Response Based Control (PRBC) is a minimum time control method that attempts to control a few rigid regions of the flexible structure through an input-output discrete-time model (Bennighof and Subramaniam, 1992; Bennighof and Chang, 1991).

Force pulses are applied by the actuators of the structure. The response at some points of the structure due to the applied force pulses is sampled, forming discrete-time pulse response sequences. The flexible behavior of the structure is represented through a set of measured pulse response sequences. Another discrete-time sequence, a desired output sequence, consistent with a desired final state, is specified. The pulse response data, desired output sequence, and force bounds define a numerical optimization problem. The solution of the optimization problem is a minimum time control sequence for driving a linear system to the desired final state.

Nomenclature

A - Coefficient Matrix

B - Force Bound

B - Input Matrix

C - Observation Matrix

D t – Time Interval and Pulse Width

h - Modal State Vector

f₁ - First Natural Frequency

f₂ - Second Natural Frequency

h - Pulse Response Sequence

H - Pulse Response Matrix

k - Generic Time Instant

n - Number of Control Steps

P - Controllability Matrix

Q - Observability Matrix

q+1- Number of Elements inthe Desired Output Sequence

T₁ - First Natural Period

T₂ - Second Natural Period

u - Control Sequence

Uss - Constant Force after Controlled Era

y - Desired or Observed Output Sequence

A description of the SISO version of the method illustrates the concept behind PRBC. The MIMO version of the method was proposed by Bennighof and Chang (1991). Consider a SISO linear system with input sequence u(k), output sequence y(k), and pulse response sequence h(k). The response of a linear system at instant n, y(n), is computed by a convolution sum, Eq.(1):

(1)

The optimization problem, which describes PRBC method for a SISO system, can be stated by Eqs. (2)-(4):

minimize n

subject to

(2)

(3)

(4)

with respect to u( i ) for i = 1, ..., n

where n denotes the number of control steps, y is the system output, h is the pulse response sequence, q+1 is the number of constraints imposed, B is the maximum force, and u_ss is a possible steady nonzero force required after the maneuver of positive definite systems.

Therefore, PRBC consists of determining the minimum number of control steps n, given the pulse response sequence h(k), the control bound B, and desired output sequence y(k), such that the constraints in Eqs.(2)-(4) are satisfied. The optimization problem is solved by linear programming (Bennighof and Subramaniam, 1992; Best and Ritter, 1985; Gonzalez-Lima, 1996).

It is well known that discrete-time representations of finite-dimensional time-continuous linear systems become uncontrollable or unobservable depending on the sampling interval (Chen, 1984). Due to possible singularities on the controllability matrix, the final state may not be reachable depending on the sampling interval of the pulse response matrix.

The method PRBC was originally proposed for maneuvering positive semi-definite flexible structures, common in aerospace applications. In robotics, flexible structures are frequently positive definite. Significant differences are observed when positive definite systems are contrasted with positive semi-definite systems.

Numerical simulations of maneuvers of flexible structures show that a large deviation from the minimum control time can result if the sampling interval is not carefully chosen.

Decomposition of H

The set of equations, Eq. (2), can be written in matrix form as

y = Hu (5)

where, matrix H is formed with the pulse response sequence h(k), y is the output sequence, and u is the control sequence.

The Hankel matrix H can be decomposed in terms of controllability matrix and the observability matrix of the system. It proves convenient to develop the analysis in terms of states in modal coordinates. The modal state is defined as

(6)

where 2p is the order of the system, h_i denotes the modal coordinate of the ith natural mode, and h_i denotes the first derivative in time of the ith natural mode.

From the linear system theory (Brogan, 1991), the controllability matrix of a constant coefficient system relates the control sequence to the state vector in modal coordinates in accordance to Eq. (7).

h (t_f ) = Pu (7)

The observability matrix Q relates the state vector to the observed output sequence y

y = Qh (t_f ) (8)

Inserting Eq. (7) into Eq. (8)} and comparing with Eq. (5) leads to a possible decomposition of the pulse response matrix H

H = QP (9)

The decomposition of the pulse response matrix into the product of the controllability matrix P times the observability matrix Q suggests an analysis of PRBC through controllability and observability concepts.

Controllability

Consider the undamped oscillator in Figure (1). The discrete-time system equations, in modal coordinates, become

h( k +1) - Ah( k ) + Bu( k ) (10)

and, the discrete-time controllability matrix, defined through Eq.(7)is

(11)

Matrix A is diagonal

(12)

and matrix B can be computed by

(13)

where matrix A_c is a diagonal state transition matrix of the time-continuous positive definite system and B_c is the input vector of the time-continuous system.

The discrete-time controllability matrix P and (A-I) have the first two rows filled with zeros when the time interval is equal to the period of the first natural frequency Dt = 1 / f₁

(14)

where x_i, for i=1, ,8, denote complex values not important for the analysis. The last two rows of P will be filled with zeros if the time interval is equal to the second natural frequency Dt = 1 / f₂.

Consequently, states with non-zero modal displacement will not be reachable in such conditions. Rest-to-rest maneuvers of positive definite systems with non-zero final displacement have non-zero final modal displacement. No minimum control time can be computed through PRBC in such conditions. Numerical simulations confirm that the minimum control time increases when the time interval approaches a period of a natural mode (Gonzalez-Lima, 1996), see Figure 2. Therefore, the time interval is a critical parameter for maneuvering positive definite systems using PRBC (Gonzalez-Lima, 1997).

The rest final state on positive semi-definite systems is represented through zero modal coordinates of flexible modes. Zero modal coordinates are reachable even when the mode is not controllable. The sampling interval is not critical for maneuvering positive semi-definite systems in the absence of measurement noise.

Observability

The necessary conditions for observability of finite dimensional systems are now explored through the two degrees of freedom undamped system of Figure 1. The observed output is related to the modal state by

y( k ) = CM _h ( k ) (15)

where y denotes a vector of observed outputs, M is the matrix of eigenvectors of A, and h is the modal state vector defined by Eq. 6. The matrix of eigenvectors is denoted by

(16)

Consider that the observed outputs of the system are the displacement x₁ and the velocity x₁. Consequently, the observation matrix C becomes

(17)

The observability matrix of the two degrees of freedom system is

(18)

which is written in explicit form

(19)

The observability matrix can be transformed into a real matrix with the same rank by post-multiplying it by a non-singular matrix K

(20)

An analysis of the resulting matrix, QK shown in Eq. 21, leads to some conclusions on whether to observe one signal, say x₁, or both x₁ and x₁. For instance, if the sampling interval is Dt = 1 / 2 f₁, half of the first natural period, the first, third and seventh rows have rank three. The second, fourth, sixth, and eighth rows have rank three. Observing one signal results in a rank deficient observability matrix. The larger the number of independent scalar observed outputs smaller is the possibility of rank-deficiency in the observability matrix.

(21)

If the quotient of the natural frequencies f₂ / f₁ is an odd integer, the rank of the observability matrix becomes 2. There are only two linearly independent rows in QK, Eq.22.

(22)

The sampling interval Dt shall be smaller than half of the smaller natural period of the finite dimensional system to keep the controllability matrix and the observability matrix full rank.

Example: Distributed Parameter System

It may be argued that a two degree of freedom system does not reflect the reality of a flexible structure with an infinite number of flexible modes. Numerical simulations of the translation of a flexible beam are performed for values of the time interval ranging from .5 to 26 ms. A diagram of the physical model is shown in Figure 3 and the parameters of the flexible structure are shown in Table 1. The mathematical model is a partial differential equation solved by eigenfunction superposition (Gonzalez-Lima, 1996).

Figure 4 is a graph of minimum control time vs. time interval. The natural periods of the system are shown in Table 2. It suggests that the smaller the sampling interval, smaller is the error in the minimum control time computed through PRBC. A sampling interval must be smaller than 2.6 milliseconds to keep the relative error in the minimum control time within 20.

Conclusions

The time interval of the discrete-time pulse response sequence is a critical parameter for maneuvering positive definite systems through PRBC. The sampling interval must be smaller than the natural periods of finite dimensional systems. Numerical simulations of maneuvers of distributed parameter systems suggest that the error on the estimated minimum control time diminishes as the sampling interval diminishes. It is shown also that some singularities in the observability matrix may be avoided observing more scalar outputs.

Acknowledgement

The financial support from Conselho Nacional para o Desenvolvimento da Pesquisa (CNPq) Brazil made possible the present research.

Manuscript received: May 1997, Technical Editor: Agenor de Toledo Fleury.

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• Gonzalez-Lima, Raul, Experimental Investigation of Minimum Time Translation of a Flexible Beam using Pulse Response Based Control, Ph.D. Dissertation, The University of Texas at Austin, May 1996.
• Jankovski, K. P. and H. Van Brussel, Discrete-Time Inverse Dynamics Control of Flexible Joint Robots, Journal of Dynamic Systems, Measurements, and Control, June 1992, v. 114, pp. 229-233.
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Publication Dates

• Publication in this collection
  18 Dec 2000
• Date of issue
  2000

History

• Received
  May 1997